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Full text of "Handbook vol.1 2008"

THfc INTERNATIONAL HANDBOOK 
MATHEMATICS TEACHER EDUCATION 



VOLUME 1 

Knowledge and Beliefs in Mathematics 

Teaching and Teaching Development 

Peter Sullivan and Terry Wood (Eds.) 




BIBLIOTECA CICATA LEGARIA - IPN 






Publishers 




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KNOWLEDGE AND BELIEFS IN MATHEMATICS TEACHING AND TEACHING 
DEVELOPMENT 



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BIBLIOTECA - LEGARIA 



The International Handbook of Mathematics 
Teacher Education 



Series Editor: 

Terry Wood 
Purdue University 
West Lafayette 
USA 



This Handbook of Mathematics Teacher Education, the first of its kind, addresses the learning of 
mathematics teachers at all levels of schooling to teach mathematics, and the provision of activity and 
programmes in which this learning can take place. It consists of four volumes. 

VOLUME I: 

Knowledge and Beliefs in Mathematics Teaching and Teaching Development 

Peter Sullivan, Monash University, Clayton, Australia and Terry Wood, Purdue University, West 

Lafayette, USA (eds.) 

This volume addresses the "what" of mathematics teacher education, meaning knowledge for 

mathematics teaching and teaching development and consideration of associated beliefs. As well as 

synthesizing research and practice over various dimensions of these issues, it offers advice on best 

practice for teacher educators, university decision makers, and those involved in systemic policy 

development on teacher education. 

paperback: 978-9O-8790-54I-5, hardback: 978-90-8790-542-2, ebook: 978-90-8790-543-9 

VOLUME 2: 

Tools and Processes in Mathematics Teacher Education 

Dina Tirosh, Tel Aviv University, Israel and Terry Wood, Purdue University, West Lafayette, USA 

(eds.) 

This volume focuses on the "how" of mathematics teacher education. Authors share with the readers 

their invaluable experience in employing different tools in mathematics teacher education. This 

accumulated experience will assist teacher educators, researchers in mathematics education and those 

involved in policy decisions on teacher education in making decisions about both the tools and the 

processes to be used for various purposes in mathematics teacher education. 

paperback: 978-90-8790-544-6, hardback: 978-90-8790-545-3, ebook: 978-90-8790-546-0 

VOLUME 3: 

Participants in Mathematics Teacher Education: Individuals, Teams, Communities and Networks 

Konrad Krainer, University of Klagenfurt, Austria and Terry Wood, Purdue University, West Lafayette, 

USA (eds.) 

This volume addresses the "who" question of mathematics teacher education. The authors focus on the 

various kinds of participants in mathematics teacher education, professional development and reform 

initiatives. The chapters deal with prospective and practising teachers as well as with teacher educators 

as learners, and with schools, districts and nations as learning systems. 

paperback: 978-90-8790-547-7, hardback: 978-90-8790-548-4, ebook: 978-90-8790-549-1 

VOLUME 4: 

The Mathematics Teacher Educator as a Developing Professional 

Barbara Jaworski, Loughborough University, UK and Terry Wood, Purdue University, West Lafayette, 

USA (eds.) 

This volume focuses on knowledge and roles of teacher educators working with teachers in teacher 

education processes and practices. In this respect it is unique. Chapter authors represent a community 

of teacher educators world wide who can speak from practical, professional and theoretical viewpoints 

about what it means to promote teacher education practice. 

paperback: 978-90-8790-550-7, hardback: 978-90-8790-551-4, ebook: 978-90-8790-552-1 



Knowledge and Beliefs in Mathematics 
Teaching and Teaching Development 



Edited by 

Peter Sullivan 

Monash University, Clayton, Australia 

and 

Terry Wood 

Purdue University, West Lafayette, USA 



ftp. CICAtA - iw 

BIBUOTECA - LEGARIA 




SENSE PUBLISHERS 
ROTTERDAM / TAIPEI 



A C.I. P. record for this book is available from the Library of Congress. 



ISBN 978-90-8790-541-5 (paperbaqjj^ 
ISBN 978-90-8790-542-2 (hardback) 
ISBN 978-90-8790-543-9 (e-book) 



Published by: Sense Publishers, ^g» ( V°5 

P.O. Box 21858, 3001 AW B^igrdarn, The. Netherlands , i ,_ 
' nttp://www.sensepubhshers.com , vn s^C'- iV) , -T L 

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The photo on the front cover, can be described as: 

Human being as tree: transforming energies flowing both ways between air and earth Teacher as tree, 

enabling the flow of energies between mathematical thinking and learner thinking In the distance, a 

human 'tree' reminding us of the need for 'transformers' to 'step down the voltage' 

what Dewey called 'psychologising the subject matter'; what is currently exercising so many people 

under the banner of 'teacher knowledge in and for teaching' 

Tree as image or metaphor for transition, from the world of this volume to the world of professional 

activity 

John Mason 2008 



Printed on acid-free paper 



All rights reserved © 2008 Sense Publishers 

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by 
any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written 
permission from the Publisher, with the exception of any material supplied specifically for the purpose 
of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 



TABLE OF CONTENTS 



Preface ix 

Knowledge for Teaching Mathematics: An Introduction 1 

Peter Sullivan 

Section 1: Mathematics Discipline Knowledge for Teaching 

Chapter 1: Mathematical Discipline Knowledge Requirements for Prospective 
Primary Teachers, and the Structure and Teaching Approaches of 
Programs Designed to Develop That Knowledge 1 3 

Mike Askew 

Chapter 2: Mathematical Preparation of Prospective Elementary Teachers: 
Practices in Selected Education Systems in East Asia 37 

Yeping Li, Yunpeng Ma, and Jeongsuk Pang 

Chapter 3: Discipline Knowledge Preparation for Prospective Secondary 
Mathematics Teachers: An East Asian Perspective 63 

Li Shiqi, Huang Rongjin, and Shin Hyunyong 

Chapter 4: Mathematics for Secondary Teaching: Four Components of 
Discipline Knowledge for a Changing Teacher Workforce 87 

Kaye Stacey 

Section 2: Mathematics for and in Teaching 

Chapter 5: Pedagogical Content Knowledge: Useful Concept or Elusive 

Notion 117 

Anna Graeber and Dina Tirosh 

Chapter 6: The Mathematics Teacher as Curriculum Maker: 

Developing Knowledge for Enacting Curriculum 1 33 

Doug Clarke 

Chapter 7: Learning to Design for Learning: The Potential of Learning Study 
to Enhance Teachers' and Students' Learning 153 

Ulla Runesson 



TABLE OF CONTENTS 

Chapter 8: Beliefs about Mathematics and Mathematics Teaching 1 73 

Helen J. Forgasz and Gilah C. Leder 

Section 3: Knowledge of Equity, Diversity 
and Culture in Teaching Mathematics 

Chapter 9: The Social Production of Mathematics for Teaching 195 

Jill Adler and Danielle Huillet 

Chapter 10: Development of Mathematical Knowledge and Beliefs of 

Teachers: The Role of Cultural Analysis of the Content to Be Taught 223 

Paolo Boero and Elda Guala 

Section 4: Assessment of, and Research on, Teacher Knowledge 

Chapter 1 1 : Assessment of Mathematical Knowledge of Prospective 

Teachers 247 

Anne D. Cockburn 

Chapter 12: Researching Teachers' Mathematics Disciplinary Knowledge 273 
Tim Rowland 

Critical Response to Volume Chapters 

Chapter 1 3: PCK and Beyond 30 1 

John Mason 



PREFACE 



It is my honor to introduce the first International Handbook of Mathematics 
Teacher Education to the mathematics education community and to the field of 
teacher education in general. For those of us who over the years have worked to 
establish mathematics teacher education as an important and legitimate area of 
research and scholarship, the publication of this handbook provides a sense of 
success and a source of pride. Historically, this process began in 1987 when 
Barbara Jaworski initiated and maintained the first Working Group on mathematics 
teacher education at PME. After the Working Group meeting in 1994, Barbara, 
Sandy Dawson and I initiated the book, Mathematics Teacher Education: Critical 
International Perspectives, which was a compilation of the work accomplished by 
this Working Group. Following this, Peter de Liefde who, while at Kluwer 
Academic Publishers, proposed and advocated for the Journal of Mathematics 
Teacher Education and in 1998 the first issue of the journal was printed with 
Thomas Cooney as editor of the journal who set the tone for quality of manuscripts 
published. From these events, mathematics teacher education flourished and 
evolved as an important area for investigation as evidenced by the extension of 
JMTE from four to six issues per year in 2005 and the recent 1 5 th ICMI Study, The 
professional education and development of teachers of mathematics. In preparing 
this handbook it was a great pleasure to work with the four volume editors, Peter 
Sullivan, Dina Tirosh, Konrad Krainer and Barbara Jaworski and all of the authors 
of the various chapters found throughout the handbook. 

Volume 1, Knowledge and Beliefs in Mathematics Teaching and Teaching 
Development, edited by Peter Sullivan, examines the role of teacher knowledge and 
beliefs. This important aspect of mathematics teacher education is at present the 
focus of extensive research and policy debate globally. This is the first volume in 
the series and is an excellent beginning to the handbook. 

REFERENCES 

Jaworski, B., Wood, T., & Dawson, S. (Eds.) (1999). Mathematics teacher education: Critical 

international perspectives. London: Falmer Press. 
Sullivan, P. & Wood, T. (Eds). (2008). International handbook of mathematics teacher education: Vol. 

I. Knowledge and beliefs in mathematics teaching and teaching development. Rotterdam, the 

Netherlands: Sense Publishers. 
Wood, T. (Series Ed), Jaworski, B., Krainer, K., Sullivan, P., & Tirosh, D. (Vol Eds.) (2008). 

International handbook of mathematics teacher education. Rotterdam, the Netherlands: Sense 

Publishers. 

Terry Wood 
West Lafayette, IN 
USA 



vn 



PETER SULLIVAN 

KNOWLEDGE FOR TEACHING MATHEMATICS 

An Introduction 



This introduction to Volume 1 of the Handbook draws on some teachers' answers 
to prompts about a particular mathematics question to highlight the challenge and 
complexity of describing the knowledge that mathematics teachers need in order to 
be able to teach. It points the way to the various chapters in the volume that 
provide theoretical and practical perspectives on the many dimensions of this 
knowledge for teaching. 

VOLUME 1 OVERVIEW 

This volume presents research and theoretically informed perspectives on 
Knowledge and Beliefs in Mathematics Teaching and Teaching Development. The 
chapters together address the "what" of mathematics teacher education, meaning 
knowledge for mathematics teaching and teaching development and consideration 
of associated beliefs. As well as synthesising research and practice over various 
dimensions of these issues, the volume offers advice on 'best practice' for teacher 
educators, university decision makers, and those involved in systemic policy 
decisions on teacher education. 

There are four sections. The first, about mathematics discipline knowledge for 
teaching, contains chapters on mathematics discipline knowledge from both East 
Asian and Western perspectives, with separate chapters addressing 
primary/elementary teacher education and secondary teacher education, along with 
a chapter on approaches for assessing this mathematics knowledge of prospective 
teachers. The second section describes ways of thinking about how this 
mathematical knowledge is used in teaching. It includes chapters on pedagogical 
content knowledge, on knowledge for and about mathematics curriculum 
structures, the way that such knowledge can be fostered with practising teachers, 
on a cultural analysis of mathematical content knowledge, and on beliefs about 
mathematics and mathematics teaching. The third section outlines frameworks for 
researching issues of equity, diversity and culture in teaching mathematics. The 
fourth section contains a description of an approach to methods of researching 
mathematics discipline knowledge of teachers. 

This introduction is not an attempt to summarise the chapters. (I encourage you 
to read a perceptive description of the various chapters and their emphases, along 
with insights into ways of progressing thinking about knowledge for teaching, in 
the review chapter written by John Mason). Nor is this introduction an overview of 

P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 1-9. 

© 2008 Sense Publishers. All rights reserved. 



PETER SULLIVAN 

the various dimensions of knowledge that teachers might be expected to have. The 
different sections on the volume illustrate those. Nor is it a summary of the 
respective research perspectives or ways of viewing knowledge for teaching. The 
abstracts and the respective chapters do that. 

Rather, this introduction argues that the issues of teacher knowledge and belief 
are important and complex. It presents a rationale for anyone with an interest in 
mathematics teacher education to read the volume. To begin, I describe why 
teacher knowledge matters, then through a particular mathematics question present 
three perspectives on the knowledge needed for teaching mathematics, and finally 
consider what this means for teacher education. 

TEACHER KNOWLEDGE MATTERS 

The challenge of describing succinctly the knowledge required for teaching is 
reflected in the debate within the mathematics education community on key issues 
and characteristics of effective mathematics teaching. On one side of the debate, 
there is substantial support for a need to intertwine conventional discipline-based 
learning with physical, personal and social dimensions, and the imperative to 
develop expertise relevant for demands of global economy and the nature of 
thinking required. As an example, there are explicitly stated demands in Australian 
curriculum documents, such as: students should demonstrate useful mathematical 
and numeracy skills for successful general employment and functioning in society, 
and develop understanding of the role of mathematics in life, society and work, as 
well as developing flexible and creative approaches to solving problems. Those on 
this side of the debate recommend that students work on questions illustrating the 
usefulness of mathematics and those that foster creativity and engagement. The 
other side of the debate takes a more explicitly mathematical perspective with 
attention to the principles, patterns, processes, and generalisations that have 
conventionally formed the basis of the mathematics curriculum. It can be assumed 
that proponents of this side of the issue would anticipate teachers using classroom 
experiences that focus students' attention onto the mathematics. 

It is stressed that this debate is far from academic. Schools in Western contexts, 
at least, are confronting serious challenges from disengaged students (e.g., Russell, 
Mackay, & Jane, 2003), with the implication that more interesting, functionally 
relevant tasks can enhance engagement (see Klein, Beishuizen, & Treffers, 1998). 
At the same time, there is a serious decline in the number of students entering 
university level mathematics courses (at least in the countries where the debate 
rages), threatening international competitiveness and innovation, fostering calls for 
more mathematical rigour at secondary level. As society, commerce, technology 
and more or less everything else is becoming more complex, to understand the 
complexity and contribute to developments requires an understanding of 
mathematics, and not only the formal processes, but also the power of 
generalisation, the nature of problem solving, and the demands for creativity, 
adaptability and the on-going nature of learning. 



KNOWLEDGE FOR TEACHING MATHEMATICS 

Essentially this debate is about the nature of discipline knowledge and the nature 
of learning, and is evidenced in various countries such as through the "Math Wars" 
in the United States (Becker & Jacobs, 1998) and in other disciplines, such as in 
the concerns about the teaching of reading in Australia (Reid & Green, 2004). 

In any case, advances in technology make everyday living, the work 
environment, and mathematics itself more, not less, important and complex; future 
citizens need to be better educated than previous generations. For a better educated 
population we need teachers who can cope with this increased complexity, and so 
the knowledge that teachers bring to their classrooms matters. Again this 
knowledge is not just about the formal processes that have traditionally formed the 
basis of mathematics curriculums in schools and universities but the capacity to 
adapt to new ways of thinking, the curiosity to explore new tools, the orientation to 
identify and describe patterns and commonalities, the desire to examine global and 
local issues from a mathematical perspective, and the passion to communicate a 
mathematical analysis and world view. 

LOOKING AT A MATHEMATICS QUESTION FROM THREE PERSPECTIVES 

To illustrate the complexity and challenge of identifying the expected knowledge 
for teaching mathematics, three perspectives on the knowledge needed for teaching 
are described in the context of a particular mathematics question that I, with Doug 
Clarke (see Chapter 6, this Handbook, this volume) and Barbara Clarke (see 
Chapter 10, this Handbook, volume 3), asked of several teachers. As part of a 
survey we gave to teachers, one question invited teachers to respond to a prompt 
that sought insights into the extent to which they could describe the content of a 
particular mathematics question or idea, and the ways that they might convert the 
question to a lesson. 

Knowing the Mathematics 

The first part of the prompt to the teachers in the survey, including the mathematics 
question was as follows: 

The following is a description of an idea that might be used as the basis of a 
lesson 

2 201 
Which is bigger — or ? 

3 301 



Suppose hypothetical ly I, as a mathematics teacher, might consider using this 
question as the basis of a lesson. The first step is to work out the answer. If I am 
not sure that I can work out the answer, then I probably will not use the question. 
There are, of course a number of ways of finding the answer, and these are 
categorised as one of two types: 



PETER SULLIVAN 

Type A: Algorithmic 

This type of response includes converting both denominators to 903, and 
either by cross multiplying or by converting the respective fractions to 
equivalent fractions with the denominator of 903, the fractions can then be 
directly compared. 

A different approach, that could also be termed algorithmic, would be to 
convert the fractions to decimals, possibly using a calculator, and then 
comparing the decimal representations. Note that this is actually harder than it 
looks in that the decimals to be compared are 0.6677774086 and 
0.666666667, which is possibly more complex for the target students than the 
original task. 

Type B: Intuitive 

This would include methods such as realising that the real comparison is 

200 201 

and . Since this is complex, a simpler comparison can be made, 

300 301 

2 3 

such as — and — where I has been added to the numerator and 

3 4 

denominator, which clearly makes the second fraction larger. This can then be 

tried with different examples, and an inference made about the original 

comparison 

Another method was described by Doug Clarke who had posed the question 

to some teachers in the U.S. One of them answered: 

We could think of 200/300 as a basketball player's free throw success 
rate, as 200 successful throws out of 300. In moving to 201/301, the 
basketball player has had one more throw, which was successful. His 
average must therefore have improved, and so 201/301 must be larger. 

A third approach is directly intuitive. I have posed the question to both 
students and teachers, and it is interesting that many give the correct answer, 
but cannot explain their correct answer. For them it is actually intuitive or 
obvious that the second is larger. It is not for me. 

Mathematics teacher educators hope that all mathematics teachers would be able 
to solve the question by at least one or more of the algorithmic methods. Clearly 
we would also hope that mathematics teachers would at least consider that intuitive 
type methods might be possible in this case, perhaps prompted to consider this 
possibility by the unusual numbers in the fractions. For both algorithmic and 
intuitive approaches, the knowledge required is mathematical and is not specific to 
teachers or teaching. We would expect that mathematicians, for example, would 
give more accurate and more diverse intuitive solutions than the general 
population, and probably more than would teachers. 



KNOWLEDGE FOR TEACHING MATHEMATICS 

Mathematical Knowledge for Teaching 

Many of the authors in this volume refer to the work of Deborah Ball exemplified 
by the research reported by Hill, Rowan, and Ball (2005) exploring what they call 
mathematical knowledge for teaching which includes what they term specialised 
content knowledge. In the case of the question, given above, this mathematical 
knowledge for teaching includes the process of teachers working out how to use 
such a question in their teaching. For this, among other things, teachers need ways 
to describe what is needed to solve the question. For this, specialised knowledge is 
needed. In our project, we asked teachers (referring to the above question): 

If you developed a lesson based on this idea, what mathematics would you 
hope that the students would learn? 

The fundamental concept can be described as comparing fractions, with the 
question offering opportunities for students to seek alternate or intuitive strategies, 
as well as considering formal approaches to comparisons, such as finding a 
common denominator or converting to decimals. 

We asked 107 primary and junior secondary teachers to articulate what they saw 
as the content by responding to the above prompt. Many responses were 
disappointing, ranging from vague to inaccurate. Table 1 summarises their replies. 
The first category, where the respondents were able to describe what the question 
was about, was acceptable. The other categories, whether teachers had a limited 
and inaccurate view (for example, a number of teachers wrote "place value"), or a 
vague view (for example, one teacher wrote "What a fraction is. Fractions are just 
another way to write a number where they are used in everyday life"), did not 
convey that they understood the focus or potential of the question. 

Table 1. Categorisation of responses describing the content of the question (n = 107) 

No. of responses 

Students learn various ways of comparing 38 
A single specific concept (not comparing) 39 
General and only vaguely related multiple concepts 30 

Only a little over one third of the teachers were able to specify the content of the 
question in a meaningful way, and there was little difference in the proportion of 
primary and junior secondary teachers in this category, perhaps confirming that 
responding to this prompt requires more than the ability to answer the question. 

To be effective, teachers would need to be able to identify specific concepts 
associated with a question so that they can match it to curriculum documents, can 
talk to other teachers about it, and maybe even explain the concepts to the students. 
We would not, for example, expect teachers to have the specialised knowledge of 
researchers on ways that the learning of fraction concepts develops, for example. 
But we would expect teachers to be able to match a question they had seen, such as 
the one above, with curriculum documents or other literature. 



PETER SULLIVAN 

The knowledge to do this is not just the mathematics, which is clearly a 
prerequisite, but is about the curriculum, the language used, connections within the 
curriculum, what make this question unexpectedly difficult, and even more general 
knowledge about strategies such as "make the problem simpler". 

Mathematics Knowledge, Mathematics Knowledge for Teaching, and Pedagogy 

Once teachers are confident that they can answer the question, and they have the 
words to describe what the question is about, there is the challenge of converting 
the question to a learning experience for their students. Most chapters in this 
volume draw on the notion, from Shulman (1986), of pedagogical content 
knowledge. Thankfully, most of the chapters explore the subtlety of the construct, 
and elaborate its meaning and implication for teacher learning. It is unfortunate that 
the term has often be used to include everything other than specialised 
mathematical knowledge, but the following chapters seek to elaborate the nature of 
this aspect of teachers' knowledge. In the survey mentioned above, the notion of 
teachers' pedagogical content knowledge was explored through the following 
prompt (also relating to the above question): 

Describe, briefly, a lesson you might teach based on this idea. 

We had hoped to see descriptions of lessons that would allow students to see that 
there were various ways to solve the task, and ideally teachers would plan 
specifically to allow students to explore the task in their own way at some stage. 
The categories used for their responses are presented in Table 2, along with the 
numbers of teachers responding in each category: 

Table 2. Categorisation of the lessons based on the question (N=I07) 

Responses 

A teacher-centred lesson, incorporating a specific strategy for teaching 20 

the task 

Student-centred, perhaps using a meaningful example, emphasising 14 

student generation of strategies and discussion (or the process) 

Real life or concrete examples but only vaguely related to the concepts 38 

Teacher-centred but with a general strategy not specific to the task 9 

Nothing or don't know 23 

Around 19% of the hypothetical lessons could be described as good traditional 
teaching, with a higher proportion of the secondary teachers (25%) than primary 
(16%) in this category. Given that the teachers identified an appropriate focus for 
the task, such lessons have a good chance of producing useful learning for those 
students. Nevertheless, such teachers may well miss key opportunities that the 
above question offers. For example, the fact that the question can be solved in 



KNOWLEDGE FOR TEACHING MATHEMATICS 

different and non-algorithmic ways is itself powerful learning for students, and yet 
this traditional approach can create the impression that there is just one way. 

About 13% of the lessons were of the "reform" style (the second category), with 
similar percentages of both primary and secondary teachers. Assuming that the 
teachers implement their hypothetical lesson effectively, this would be likely to 
foster learning, utilising the potential of the question. 

The other hypothetical lessons do not create confidence that these teachers can 
transform a mathematics question into an effective mathematics lesson. For 
example, 35% of the hypothetical lessons could be described as meaningless use of 
relevant or real life examples. This is not what an emphasis on relevance is 
intended to achieve. Taken at face value, it could be suspected that most students 
would not learn effectively during such lessons in that the connection between the 
representations and the concepts may be difficult for students to ascertain for 
themselves. 

It should be noted that the responses perhaps underestimate the potential of 
teachers to respond to the prompts. The prompts were part of a larger survey, 
although it is noted that the teachers were not rushed, and there was no advantage 
in finishing quickly. It is also possible that the teachers did not know how much 
detail we sought; the responses were therefore categorised leniently. 

WHAT THIS MEANS FOR TEACHER KNOWLEDGE 

It seems that the key focus of the debate should not be so much on traditional 
versus reform. Indeed, a well structured traditional lesson explicitly teaching a 
process for answering the above question, followed by structured practice, may 
well result in productive learning, so long as the teacher is aware that there are 
multiple ways of approaching the question. Nor should the debate focus on utility 
versus high level mathematics. Even though many teachers commented on the 
difficulty of identifying a practical context, the question is focused on the 
development of a mathematical idea, and perhaps a way of working. So a focus on 
mathematics does not detract from learning that can be adaptable and transferable. 
The debate should focus on identifying what teachers need to know to teach 
mathematics well. This is clearly complex and multidimensional. 

To illustrate this complexity, the steps in a reform lesson based on the above 
question require sophisticated actions by teachers. For example, in the "reform" 
category of lesson in the above table, there are a number of key stages. Anne 
Watson and I (Chapter 5, this Handbook, Volume 2) describe these stages through 
a generic description of a lesson that might be suitable for such a question: 

Teacher poses and clarifies the purpose and goals of the question. If 
necessary, the possibility of student intuitive responses can be discussed. 
Students work individually, initially, with the possibility of some group work. 
Based on students' responses to the task, the teacher poses variations. The 
variations may have been anticipated and planned, or they might be created 
during the lesson in response to a particular identified need. The variations 



PETER SULLIVAN 

might be a further challenge for some, with some additional scaffolding for 

students finding the initial task difficult. 

The teacher leads a discussion of the responses to the initial question. 

Students, chosen because of their potential to elaborate key mathematical 

issues, can be invited to report the outcomes of their own additional 

explorations. 

The teacher finally summarises, with the students' input perhaps, the main 

mathematical ideas. 

The elements of this lesson all require specific actions by the teachers and therefore 
specific knowledge. 

CONCLUSION 

Overall these three perspectives - knowledge of mathematics, knowledge for 
teaching mathematics, and knowledge of pedagogy - on the one question given 
above highlight the challenge of specifying the knowledge and beliefs required for 
effective mathematics teaching. For example, among other things, we would expect 
teachers to be able to: 

- answer the above question correctly; 

- anticipate that intuitive methods are possible; 

- use relevant language to describe the content represented by the question; 

- match the content to the curriculum. 
We anticipate that teachers would also: 

- appreciate the difficulties that students generally, and their students in particular 
might experience; 

- be aware that lessons can be structured so that the learning is the product of 
students' exploration, as distinct from listening to explanations; 

- know how to pose tasks; 

- know to stand back, and to wait before offering guidance; 

- assess student learning; and 

- conduct effective discussion and reviews. 

And these qualities have not even started to explore the complexity of knowledge 
about pedagogy, student management, interpersonal relationships, historical 
perspectives, cultural influences and differences, social disadvantage, linguistic 
challenges, and so on. 

While it is necessary that teachers know the relevant mathematics, this is clearly 
not sufficient. There is much more, and the challenge for mathematics teacher 
educators is to find ways to describe the scope and depth of knowledge. In the 
following chapters you will find information about ways of researching such 
knowledge and beliefs, ways of describing the knowledge, and ways of working 
with prospective and practising teachers to increase their knowledge for 
mathematics teaching. 



KNOWLEDGE FOR TEACHING MATHEMATICS 

REFERENCES 

Becker, J. P., & Jacobs, B. (1998). 'Math War' developments in the United States (distributed by e- 

mail). 
Hill, H. C, Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching 

on student achievement. American Educational Research Journal, 42, 371-406. 
Klein, A. S., Beishuizen, M., & Treffers, A. (1998). The empty number line in Dutch second grades: 

Realistic versus gradual program design, Journal for Research in Mathematics Education, 29, 443- 

464. 
Reid, J., & Green, B. (2004). Displacing method(s)? Historical perspectives in the teaching of reading. 

Australian Journal of Language and Literacy, 27(1), 12-26. 
Russell, V. J., Mackay, T., & Jane, G. (2003). Messages from MYRAD (Middle Years Research and 

Development) - Improving the middle years of Schooling, 1ARTV. 
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 

75,4-14. 



Peter Sullivan 
Faculty of Education 
Monash University 
Australia 



SECTION 1 

MATHEMATICS DISCIPLINE KNOWLEDGE 
FOR TEACHING 



MIKE ASKEW 



1. MATHEMATICAL DISCIPLINE KNOWLEDGE 

REQUIREMENTS FOR PROSPECTIVE PRIMARY 

TEACHERS, AND THE STRUCTURE AND TEACHING 

APPROACHES OF PROGRAMS DESIGNED TO 

DEVELOP THAT KNOWLEDGE 



This chapter examines the research evidence for the sort of mathematics discipline 
knowledge that primary teachers might need in order to teach effectively and 
improve student learning. One conclusion is that, despite the wealth of research 
into this domain, the exact nature of such mathematical knowledge is still not 
clear, and the evidence for the impact on learning outcomes is equivocal. I argue 
that instead of trying to set out the mathematical content for prospective teachers, 
teacher education might be better directed at helping teachers develop a certain 
mathematical sensibility. I also go on to argue that trying to draw distinctions 
between content knowledge and pedagogical content knowledge may no longer be 
helpful. 

INTRODUCTION 

Concern about the mathematical knowledge of primary (elementary) teachers is not 
recent. Over a century ago Dewey (1904, 1964) argued that teachers need to be 
familiar with the nature of inquiry in particular domains. (I am taking 'primary' as 
schooling for children aged five to 11, and shall use the terms primary and 
elementary as interchangeable.). But it was not always thus. 

In medieval universities no distinction was made between knowledge of a 
discipline and knowing how to teach it - if you knew the former, it was assumed 
that you would be able to teach it (McNamara, Jaworski, Rowland, Hodgen, & 
Prestage, 2002). Today, apprenticeship models of learning still draw no distinction 
between being an expert crafts-person and being able to induct an apprentice into 
the craft. With the introduction of schooling for greater numbers of children, 
however, a separation of knowing and teaching evolved. Discipline knowledge, 
rather than taken as a given, begins to be problematised. Teaching moved from 
being part of ongoing practices within disciplines; teaching became a practice in its 
own right. This meant the need to impart knowledge about disciplines that one was 
not necessarily part of. Knowing (doing) and teaching were severed. 



P. Sullivan and T. Wood (eds.). Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 13-35. 

© 2008 Sense Publishers. All rights reserved. 



MIKE ASKEW 

Once this separation of knowing and teaching, of content and pedagogy 
occured, then questions about the extent and form of the discipline knowledge for 
teaching emerge. 

Common sense suggests that an effective teacher of mathematics would need to 
have understanding of the discipline, so it is hardly surprising that researchers are 
interested in the mathematics that primary teachers may need to know. But is 
common sense correct? The evidence for exactly what primary teachers (both 
prospective and practising) do need to know about mathematics is still debated. In 
this chapter I explore something of the current state of this debate and suggest 
some directions that the argument may go. 

Although the overall focus of the chapter is on prospective teachers much of the 
research in this area has explored the mathematics of practising teachers. So I also 
draw on this research. Partly for pragmatic reasons: the research on prospective 
primary school teachers is less extensive than that on practising teachers. Partly for 
theoretical reasons: looking at what experienced primary teachers need to know 
may help clarify what might need to be addressed with prospective teachers. My 
gaze thus encompasses studies both of prospective and practising teachers. 

Two thousand and eight seems an opportune time to be taking stock of the 
research into the mathematical subject knowledge of primary teachers, as the field 
might be considered to have recently 'come of age', if one takes 1986 as a birth- 
date. Marking 1986 is not arbitrary. First, that year marks the publication of what is 
possibly the seminal paper on knowledge for teaching, Lee Shulman's 'Those who 
understand: knowledge growth in teaching' (Shulman, 1986). Also published that 
year was the 3rd Edition of the Handbook of Research on Teaching (Wittrock, 
1986). Connolly, Clandinin, and He (1997, p. 666) note that in that edition of the 
Handbook there are only two, relatively minor, references to research into teacher 
knowledge and suggest that since then the field has 'exploded'. Since 1997 this 
explosion has not diminished: this chapter is necessarily selective of the literature. 

As well has the huge expansion of studies over the last 20 years or so, another 
significant change continues to foreground the importance of teacher subject 
knowledge. In many parts of the world, there has been increasing political and 
policy involvement in the setting out and defining of the subject knowledge for 
teaching generally and mathematics specifically (Poulson, 2001). Defining and 
codifying of subject knowledge for teaching is taken by many policy makers as 
central to improving standards of pupil learning. In England, for example, much 
policy effort and public funding has gone into attempts to improve teachers' 
subject knowledge: centrally determined curricula have been set out for teacher 
educators to follow, prospective teachers have had their subject knowledge 
'audited'; and online teacher tests of 'numeracy' have been developed. It is worth 
asking whether research findings warrant such efforts. 

PRIMARY TEACHERS: THE CASE FOR DISCIPLINE KNOWLEDGE? 

Elsewhere in this volume Kaye Stacey, and Shiqi Li and colleagues present the 
case for mathematics subject knowledge and secondary school teachers; and 



14 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

Yeping Li and colleagues do the same for primary level teachers. As 1 argue below 
the evidence for the importance of subject knowledge in primary schools is more 
equivocal. 

Across the studies into teacher knowledge, two particular questions recur: 

What is the relationship between teachers' mathematics subject 

knowledge and the teaching and learning of mathematics in primary 

schools? 

What sort of mathematics subject knowledge do primary school teachers 

need to know? 
In a fully rational world the second question would only be addressed once the 
first had been answered. But, if as indicated, common-sense suggests that there 
must be a relationship between discipline knowledge and the ability to teach, then 
work can proceed on the second question prior to the first being answered. And 
indeed, despite mixed answers to the first question, work continues unabated on the 
second. For example, in a major study into young children learning mathematics. 
Kilpatrick and colleagues (National Research Council, 2001) address these 
questions in reverse order, looking first at 'knowledge of mathematics' and then 
'teachers' mathematical knowledge and student achievement' (pp. 372-377). With 
regard to knowledge of mathematics, the authors claim that content knowledge is 
'the cornerstone of teaching for proficiency' and 

improving teachers' mathematical knowledge and their capacity to use it to 
do the work of teaching is crucial in developing students' mathematical 
proficiency, (p. 372) 

Just over a page later, however, in examining the links between teacher content 
knowledge and student achievement, the same writers report: 

For the most part, the results have been disappointing: Most studies have 
failed to find a strong relationship between the two." (p. 373) 

A North American review of research asked what kind of subject matter 
preparation is needed for prospective teachers (Wilson, Floden, & Ferrini-Mundy, 
2001). Drawing on rigorous selection for inclusion of studies, the authors could 
only find seven studies, four of which included mathematics and of these four only 
one addressed the mathematics subject knowledge of elementary teachers (Darling- 
Hammond, 2000). Wilson and her colleagues report that the conclusions from these 
studies in establishing the impact of discipline knowledge are contradictory and 
"undermine the certainty often expressed about the strong link between college 
study of a subject matter area and teacher quality" (p. 6). 

It is unusual to find a paper on teachers' subject knowledge that does not pay 
homage to Shulman's work on forms of knowledge for teaching. Less often 
acknowledged is the cautiousness that Shulman and his colleagues' expressed over 
generalising the range of their work to primary school teachers (Grossman, Wilson, 
& Shulman, 1989; Shulman, 1987; Wilson, Shulman, & Richert, 1987). In his 1987 
paper, Shulman expresses his belief that findings from the research carried out with 
secondary school teachers may well apply to primary teachers, but was "reluctant 

15 



MIKE ASKEW 

to make that claim too boldly" (p. 4). A little later, Grossman and colleagues 
(1989) point out that there are considerable differences between teaching only one 
subject in secondary schools and the demands of teaching several subjects in 
primary schools and that imputing implications from their work in secondary 
schools 'for elementary school teaching should be drawn cautiously' (p. 28). 

Despite such caveats, one thing is clear from the research evidence: many 
prospective and practising primary teachers have, or express, a lack of confidence 
in their mathematical knowledge. Wragg, Bennett, and Carre (1989) surveyed 
teachers from 400 primary schools in Great Britain and found a self-reported lack 
of confidence in their knowledge of the mathematics required to teach the national 
curriculum (although these same teachers expressed an even greater lack of 
confidence about science). Other research by the same team assessed prospective 
teachers' knowledge of mathematics (and other subjects) at the beginning and end 
of their teacher education. For many their knowledge was limited both times. 

In a study of teaching 4- to 7-year-olds, Aubrey (1997) found that teachers 
claimed not to have extensive knowledge of mathematics. Many other studies have 
found similar results (Bennett & Turner-Bisset, 1993; Rowland, Barber, Heal, & 
Martyn, 2003; Simon & Brown, 1996). 

What such studies, particularly, but not exclusively from the USA and UK, 
established is a deficit model of teachers' knowledge, leading to at least two 
impacts. Firstly, focusing on what primary teachers did not know fanned the flames 
of a 'crisis' in the knowledge base of the teaching forces. Even now, findings are 
couched in terms of the negative: Stacey, Helme, Stienle, Baturo, Irwin, and Bana 
(2003) report that 'only 80% of the sample tested as experts' (p. 205). While it 
might be reasonable to expect 100% to be proficient, at 80% the pot is certainly 
much more full than empty. Secondly, it was imputed that improving teachers' 
subject knowledge would necessarily improve teaching and learning (Askew, 
Brown, Rhodes, Wiliam, & Johnson, 1997a; Brown, Askew, Baker, Denvir, & 
Millett, 1998). However, such a conclusion needs to be treated cautiously. 

With colleagues (Askew & Brown, 1997; Brown et al., 1998) I have argued that 
while there may be evidence that teachers lack subject knowledge when this is 
assessed outside the context of the classroom: whether or not this is a hindrance in 
practice is more difficult to establish. For example, in our study of effective 
teachers of numeracy (number sense) we observed only two occasions where 
teachers appeared to be hampered by a lack of mathematical knowledge, out of 86 
mathematics lessons observed in total. We concluded that while some teachers of 
younger children may have real problems with their understanding of subject 
knowledge, it was not clear how much this actually impacted on their effectiveness 
(Askew et al. (1997a, p. 59). 

Bennett and Carre (1993) found that prospective primary teachers who were 
mathematics specialists did display greater subject knowledge than those 
specialising in other areas (music and early years). However when they observed 
all these prospective teachers actually teaching they found little difference in their 
practices that distinguished the mathematics specialists from the non-specialists 
teaching mathematics. These observations took place whilst the teachers were still 



16 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

in training so we do not know if there was a longer term impact of the mathematics 
specialists' greater subject knowledge. It may be that novice teachers share a 
common set of concerns, irrespective of their subject specialism. Concerns centred 
around becoming familiar with the curriculum, planning lessons, managing 30 or 
more children, and building productive classroom relationships. Such concerns 
initially may over-ride attention to the particularities of subject content; in moving 
from being novice teachers to experienced ones, specialised content knowledge 
may begin to have an impact. 

Nevertheless, it would seem that a certain threshold of discipline knowledge is 
necessary for effective teaching. Whilst a major longitudinal study carried out in 
the 1960s could find no association between teachers' study of higher level 
mathematics and their students' achievement, the director later argued that teachers 
need to attain a certain level of mathematical understanding, but that beyond a 
certain level further study of mathematics did not lead to increased student gains 
(Begle, 1979). A later study found associations between student attainment and the 
number of mathematics courses that their teachers had studied, but only up to a 
certain number of courses (although this was with secondary teachers, there is no 
reason to suppose the results would be different for primary) (Monk, 1994). 

A major difficulty in establishing the impact of discipline knowledge is the 
means by which teachers' mathematical knowledge has been identified and 
quantified. Until recently studies have often relied on proxies such as highest level 
of formal qualification in mathematics or courses taken rather than looking 'inside 
the black box' at the mathematics that teachers actually draw on when teaching. 

The fact that such proxies have not demonstrated a link between subject matter 
knowledge and teaching expertise or pupil outcomes is hardly surprising as there 
are at least two problems. 

Firstly, the assumption that examination results are an accurate reflection of 
someone's level of understanding. Success on formal mathematics examinations 
may be attained through a base of procedural rather than conceptual understanding, 
and it may be that is the view of mathematics that such 'successful' teachers 
develop. This conjecture is supported by research we carried out in England. 
Observations of lessons conducted by teachers with higher formal mathematical 
qualifications did tend to be more procedural in their content. These same teachers, 
in interview, expressed difficulty in understanding why some pupils had problems 
with mathematics. Further, there was a slight negative association between the 
gains over the course of a year that the pupils made on a specially designed number 
assessment and the highest level of mathematical qualification of their teachers: the 
higher the level of qualification, the lower the gains the pupils made (Askew, 
Brown, Rhodes, Wiliam, & Johnson, 1997b). 

Secondly, even if qualifications are an accurate measure of understanding, are 
they the ones that teachers will need to draw on in the classroom? As discussed 
below, it is only recently that researchers have begun to try and 'unpack' the 
knowledge specifically needed from primary teaching. 

To summarise, there is agreement in the literature on prospective and practising 
primary teachers' mathematical knowledge in the conclusion that a certain lack of 



17 



MIKE ASKEW 

knowledge of mathematics is associated with less successful teaching and lower 
student attainment. 

The flip-side of this argument - that more subject knowledge is linked to better 
teaching and learning - is less well established. In part this is a result of the proxies 
used as measures to examine subject knowledge. More recent studies have begun 
to examine more closely the nature and content of discipline knowledge needed by 
primary teachers, and it is to these that I now turn. 

DISCIPLINE KNOWLEDGE: FROM WHETHER TO WHAT? 

In this section I look at ways that various writers have classified mathematics 
discipline knowledge for primary teachers. The work of Ball and her colleagues 
has been particularly influential in the study of the mathematics for teaching and 
most classifications bear strong resemblances to frameworks that they have set out. 
For example, in an early work, Ball (1990) argues for attention to the distinction 
between knowledge of mathematics - the various meanings attached to different 
representations and associated procedures. Alongside this, teachers also need to 
have knowledge about mathematics - the means by which 'truth' is established 
within the discipline. This distinction echoes Lampert's (1986) separation of 
procedural and principled knowledge, the former involving 'knowing that', and 
includes the rules and procedures of mathematics ('knowledge of mathematics' in 
Ball's terms). Principled knowledge is more conceptual - the knowing why of 
mathematics. Other similar distinctions are between the substantive (facts and 
concepts) and the syntactic (nature of knowledge growth in the field through 
inquiry and, in the case of mathematics, proof) (Shulman, 1986). 

Even earlier was Skemp's (1976) setting out of instrumental or relational 
understanding. As is often the case, a dominant metaphor here is that of a map. For 
Skemp, instrumental understanding was analogous to being given step-by-step 
instructions for getting from A to B; many people's understanding of traditional 
algorithms would be described as instrumental. Relational understanding is more 
akin to having a map; so if one gets lost then one has the wherewithal to figure out 
the way back to the right path. 

Thompson, Philipp, Thompson, and Boyd (1994) discuss the difference between 
calculational and computational, although this distinction is more akin to beliefs 
about the curriculum than about mathematical knowledge per se. It could be 
perfectly possible for a teacher to have a rich and varied understanding of the 
mathematics curriculum and yet still hold that learning basic computational skills is 
the goal of primary mathematics teaching. 

The work of Grossman, Wilson, and Shulman (1989) is based around three 
dimensions (Thompson et al., 1994) rather than two. Like later writers they stress 
the importance of understanding the organising principles of a discipline, and also 
factual knowledge and central concepts. 

Distinctions between conceptual/procedural, instrumental/relational and so forth 
seem reasonable, but closer examination reveals some difficulties. Suppose a 
teacher knows how to carry out a multiplication algorithm accurately but cannot 



18 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

articulate how it works. Is the ability to correctly carry out the calculation any less 
'conceptual' than being able to explain how it works? Much of mathematics relies 
on procedural fluency, so there is a danger in perceptions of conceptual 'good', 
procedural 'bad'. 

Also, looking at the mathematics that teachers draw on in mathematics lessons 
could be as much a measure of their beliefs about the role of mathematics in the 
curriculum as about their knowledge of mathematics per se. For example, 
researchers have identified teachers with good understanding of mathematics but 
who still adopted 'transmission' style teaching approaches rather than work on 
crafting student explanations (D. Ball, 1991; D. L. Ball, 1991). 

Thus a number of distinctions have been drawn up which are now largely taken 
as descriptive most notably between discipline knowledge and pedagogical 
knowledge. Within discipline knowledge the distinction is made between 'facts, 
concepts and procedures' - the established cannon of mathematical knowledge - 
and the way that the community of mathematicians has come to establish this 
knowledge. Within pedagogic knowledge there is a similar separation of 
knowledge of how to teach particular mathematical topics (didactics) and 
knowledge of how the individual learner might develop understanding 
(psychology). In Davis and Simmt's (2006) terms these are assumed distinctions 
between established/dynamic and collective/individual. They argue that the 
distinctions between 'formal disciplinary knowledge and instructional knowledge 
... between established collective knowledge and dynamic individual 
understandings ... are inherently problematic' (p. 293). I return to this point later. 

Despite the prevalence of the claims for there to be (at least) two sides to the 
coin of mathematics subject knowledge, surprisingly little attention has been paid 
to setting out the range of procedural/factual knowledge that it might be reasonable 
for primary teachers to know. This aspect appears, by and large, to be taken for 
granted: either this is the bulk of subject knowledge that teachers have learned 
through their own mathematics education or they gain it through teaching and 
becoming familiar with such knowledge as embodied in curriculum materials. 

Our own research at King's College, London, supports the conjecture that 
procedural knowledge of mathematics is not the issue. As part of a five-year 
longitudinal study of teaching and learning numeracy in primary schools (the 
Leverhulme Numeracy Research Programme, Millett, Brown, & Askew, 2004), we 
interviewed a group of teachers about a variety of mathematics problems. Their 
responses indicated that they could, on the whole, arrive at correct answers. But 
when it came to probing the rationales behind their answers, it became clear that 
their solutions were based on instrumental/procedural approaches. The difficulty 
here is not that teachers were unable to reach correct answers, but that they were 
not able to generalise from their answers. 

Where attention has turned to looking in detail at the content of teachers' subject 
knowledge the focus is often on the quality of understandings rather than the actual 
content of them. As Freudenthal (1975) expressed it, it may be that more nuanced 
understandings are required. Ball and Bass (2003) similarly argue that 



19 



MIKE ASKEW 

mathematics-for-teaching knowledge is not a case of knowing more than or to a 
greater 'depth' than that expected of students; it needs to be qualitatively different. 

Ma (1999) expresses qualitative difference through her construct of 'profound 
understanding of fundamental mathematics' (PUFM) such understanding being 
exhibited by the Chinese teachers in her study, whilst the USA teachers presented 
knowledge that was lacking in conceptual underpinning and 'unconnected'. 
Powerful though Ma's evidence is, it still only partially sets out what such PUFM 
might look like. Her detailed qualitative work addresses a small number of the 
topics in the primary mathematics curriculum. And the teachers involved in Ma's 
study were specialist mathematics teachers and so not representative of the 
majority of primary school teachers. 

Nevertheless, the metaphor of teachers needing to have 'connected' 
mathematical knowledge resonates through other research. Our study of effective 
teachers of numeracy in primary schools engaged teachers in constructing 'concept 
maps' of understanding number (Novak & Gowin, 1984). In interviews, we asked 
teachers to list as many topics in learning about number in the primary years as 
they could. Each topic was noted on a separate 'stickie'. The teachers then 
'mapped' these out onto a larger sheet of paper. Once the locations of these topic 
'landmarks' had been chosen, the teachers drew arrows connecting topics: arrows 
could be one or two headed to indicate directions of links, and topics could have 
multiple connections. Finally, and importantly, they were asked to label the arrows 
to provide concise descriptions of the nature of the connections. 

We analysed the completed concept maps in terms of the numbers of 
connections identified, the range of the connections and the quality of the 
descriptions of the links. We found that there was a strong association between the 
complexity of the maps that teachers produced and the average gains in scores that 
their classes attained on a numeracy assessment over the course of a year. Where 
teachers had more 'connected' maps, the gains for classes were higher (Askew et 
al., 1997a). Writing recently, Hough, O'Rode, Terman, and Weisglass (2007) look 
at how concept maps can also be used with prospective teachers to help them think 
about connections in mathematics. 

Like our study, Ma's conclusions about teachers' knowledge are based on 
interviews rather than observation of actual classroom practice. Ball and her 
colleagues have turned to taking an approach more grounded in practice by looking 
at the demands made in classrooms (Ball, Hill, & Bass, 2005). For example, by 
looking in detail at the standard algorithm for long multiplication they draw out the 
knowledge of language and representations that teachers may need in teaching the 
algorithm. 

Will working on areas such as those identified by Ball et al. improve teaching? 
It may be that the mathematical behaviours demonstrated by effective teachers are 
still proxies for something else - a mathematical sensibility - that cannot be 
reduced to a list of mathematical topics. And transfer across the curriculum is an 
issue. For example, an item from the Ball research asks teachers to judge which of 
several 'non-standard' vertical algorithms would work for any multiplication. If 



20 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

teachers are able to answer this correctly, does that indicate a similar ability to 
unpack non-standard methods for, say, division? 

One of the few reports to set out the actual range of subject knowledge that 
prospective elementary school teachers might be expected to be confident in is the 
report of the Conference Board of the Mathematical Sciences (2001). In terms of 
number and operations, the authors' argue that 

Although almost all teachers remember traditional computation algorithms, 
their mathematical knowledge in this domain generally does not extend much 
further. ... In fact, in order to interpret and assess the reasoning of children 
learning to perform arithmetic operations, teachers must be able to call upon 
a richly integrated understanding of operations, place value, and computation 
in the domains of whole numbers, integers, and rationals. (p. 58) 

When it comes to adumbrating the learning needed, there is little that looks 
different to what one might find in a typical primary mathematics syllabus. 
Teachers, for example, are expected to develop 'a strong sense of place value in the 
base- 10 number system' which includes 'recognizing the relative magnitude of 
numbers' (p. 58). Readers are left to decide for themselves exactly what constitutes 
a 'strong sense'. 

Part of the problem here is the codification of networks of knowledge into 
discrete lists of points (death by a thousand bullet points). Working with a group of 
teachers to elicit the range of representations that might be grouped together as 
multiplication, Davis and Simmt (2006) conclude that 'multiplication was not the 
sum of these interpretations. ... we conjecture that access to the web of 
interconnections that constitute a concept is essential for teaching' (p 301). They 
challenge attempts to delineate and list the elements of mathematical knowledge: 

What is multiplication? has no 'best' or 'right' answer. Responses, rather, are 
matters of appropriateness or fitness to the immediate situation. The 
underlying notion of adequacy that is at work here stands in contrast to the 
pervasive assumption that mathematical constructs are unambiguous and 
clearly defined, (p. 302) 

Formulating content lists is prey to another difficulty: the assumption that the 
traditional curriculum will continue to be the bedrock of primary mathematics. 
That central to the curriculum is 'children learning to perform arithmetic 
operations' and, as Ball, Hill, and Bass. (2005) assert, that they need to be 'able to 
use a reliable algorithm to calculate an answer' (p. 17). Some curricula, for example 
in Australia, have already tossed the 'reliable algorithm' for long division onto the 
curriculum scrap-heap, with the view that new technologies have produced more 
efficient and reliable methods for carrying out such calculations. It may be the case 
that other algorithms are regarded similarly in the future. Assuming that the 
primary curriculum will not (need not?) change very much in the near future may 
become a self-fulfilling prophecy if too much effort goes into specifying the 
knowledge that teachers need to know. 



21 



MIKE ASKEW 

Rather than acquire a 'body' of mathematical knowledge, perhaps primary 
teachers need something else - a mathematical sensibility - that would enable them 
to deal with existing curricula but also be open to change. 

A FRAMEWORK FOR WORKING WITH PROSPECTIVE TEACHERS 

Rather than expecting primary teachers to enter the profession knowing all the 
discipline knowledge that they might ever need to draw on, is it more reasonable to 
expect them to learn new aspects of the discipline, as and when they need to? As 
Sullivan (2003) argues 'so long as teachers have the orientation to learn any 
necessary mathematics, and the appropriate foundations to do this, then prior 
knowledge of particular aspects of content may not be critical' (p. 293). 

How then might we encourage this sort of orientation? What sort of foundations 
might it need? At the risk of adding to an already burgeoning number of 
frameworks for considering mathematics subject knowledge for teaching, I set out 
what I consider to be key elements for prospective primary teachers. My starting 
point is adapted from Whitehead (1929, reprinted 1967). He argues that the 
curriculum should be a process, a cycle of exploration and inquiry, a blend of 
'romance, precision and generalization' (p. 17). Whitehead suggested that these 
three elements should, to an extent, follow each other in that order; I look at them 
in the order of precision, generalization and romance. To illustrate my arguments 1 
draw on examples from the domain of multiplicative reasoning. 

Precision 

The Oxford English Dictionary definition of precision involves being 'definite, 
exact, accurate and free from vagueness'. Accuracy is thus part of precision but not 
the whole of it. Accuracy does, however, provide a starting point for considering of 
precision, and accuracy is arguably the part of the mathematics curriculum with 
which most primary teachers are most familiar. Indeed for many it is the main 
purpose of the mathematics curriculum: getting right answers (and, unfortunately 
for some learners, getting them right quickly). It is also part of the curriculum dear 
to the hearts of policy makers. Any perceived attempts to diminish this aspect of 
the curriculum are met with cries of lowering of standards and 'fuzzy' teaching. 
Hence the continuing debate in some parts of the world on whether or not 'standard 
algorithms' should be taught. 

But accuracy without insight is limiting. As discussed earlier, as part of a large, 
longtitudinal study (Millett, Askew, & Simon, 2004) my colleagues and I 
interviewed 12 teachers about a number of mathematical tasks. In the main, they 
were able to find accurate answers to the tasks. But their approaches to finding 
answers revealed a lack of insight into how and why correct answers can come 
about. A question we asked about factors and divisibility illustrates the difficulty. 

Inscribed in the English national curriculum for mathematics is the expectation 
that children are taught rules of divisibility. We presented the teachers a selection 
of numbers and asked them to figure out which digits were divisors of each of the 

22 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

numbers (calculators were available). For example, they had to decide which digits 
were factors of 165. The teachers all were confident that 165 would not be divisible 
by 2, or 4 ("it's odd") and that ending in 5 meant it was divisible by 5. They knew 
that there was a rule for checking divisibility by 3, although some needed help to 
recall it. 

However, deciding whether or not 6 was a factor of 165 was not immediately 
apparent to them. Ten of the twelve teachers needed to carry out dividing by 6 to 
check whether or not there was a remainder. Asked whether knowing that 165 was 
not divisible by 2 could help in deciding if 6 was a factor, they reasoned along the 
lines of "no, because the fact that it was divisible by 3 might have made 6 a factor". 

The final number we presented was 3 2 x 5 2 x 7. This time, all 12 teachers had to 
multiply the product out to 1575 and then check each digit in turn. When they had 
established that 3, 5, 7 and 9 were factors we asked whether, looking back at the 
original number, they might have been able to predict any of these results. No, was 
the general response. The sense of why they could not have predicted this outcome 
was summed up by the teacher who said: "No, because you never know that the 
way the 3s, 5s and 7 were multiplied together may have led to it being divisible by, 
say, 2." 

None of the teachers appeared particularly surprised at the result that having 
started with a combination of 3s, 5s and 7 the only factors to emerge were 3, 5, 7 
and 9. These teachers displayed a marked lack of curiosity about the connection 
between the initial product and their answers. This issue of being mathematically 
curious is one that I return to below. 

But precision is more than accuracy. For instance teachers need to be aware of 
the importance of precise language in describing mathematical action. In a lesson 
observed as part of this same research programme, a teacher was modelling 
division with a class of 8- and 9-year-olds. She had a number of children standing 
holding empty boxes and was modelling how many boxes would be filled if 7 
cubes from 42 were put into each box; the model was of division as quotition 
(measurement). However, throughout the modelling the teacher kept up a running 
commentary about how the cubes were being 'shared out' amongst the boxes: the 
language of division as partition. While problems like 'How many bags are needed 
to put 42 apples into bags of 7?' and "How many apples will be in each bag if 42 
are shared between 7 bags?' can both ultimately be represented by 42 6, children 
are likely initially to solve these in different ways. Teachers need to appreciate the 
need for precision and not muddling the two models through inappropriate 
marrying up of words and actions. 

Generalization 

Without awareness of the move into the general, teaching and learning primary 
mathematics is unlikely to move beyond an emphasis on getting correct answers. 
For example, consider exploring with children whether or not the equation 
45 x 24 = 90 x 12 
is true or not. 

23 



MIKE ASKEW 

Working from a base of precision, one way to answer this is to test it out; 
calculating the product on each side of the equation will reveal the equation to be 
correct. Inspection of the numbers involved may suggest a connection between 
them. This could be tested out with other similar examples, arriving at a conjecture 
that doubling one number and halving the other preserves the answer. This 
empirical approach to generalising does not establish why the conjecture holds 
true, or whether or not it will continue to hold. And it does not provoke curiousity 
into whether or not this can be generalised further - would trebling one number 
and 'thirding' the other also work? A generalisation is reached, but it is a rule- 
without-reason. 

Consider, instead, modelling this with an array. I appreciate that some readers 
may think that this strays into the realm of pedagogic-content knowledge here, 
rather than content knowledge. Arrays provide a powerful model for examining the 
structure of multiplication and so, I suggest, are as much part of content knowledge 
as pedagogic. A few simple diagrams, as shown in Figure 1, with an open array 
quickly establishes the veracity of the equation. 



24 



45 



45 



12 


12 







90 



Figure I: Arrays for examining the structure of multiplication. 

This analytic approach to generalising (Schmittau, 2003) goes beyond the 
'specialise (through lots of examples) generalise model' to seeing the general-in- 
the-specific. It is a precise argument, but not based on the precision of answers. 
The move to the general is a short one — the actual dimensions of the rectangles are 
immaterial and the argument can, literally, be seen to hold whatever dimensions 
are used to label the sides. Generalising is an adjunct to precision. 



24 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

But the use of the array goes beyond establishing the specific result. It opens up 
the possibility of other constructions: do we only have to slice our rectangle into 
two pieces? What about three slices? Or four? Curiosity is opened up. 

Romance 

Mathematics has beauty and romance. It's not a boring place to be, the 
mathematical world. It's an extraordinary place; it's worth spending time 
there. (Marcus du Sautoy, Professor of Mathematics, University of Oxford) 

Beauty maybe considered a quality pertaining to the other - the quality of 
mathematics that gives pleasure. Or it may be in the eye of the beholder - how the 
subject finds beauty in mathematics. Whichever, there is a 'distance' between 
subject and object. Beauty can be admired from afar. Romance, however, implies 
intimacy, a certain reciprocity, an entering in to a relationship with the other. Care 
and curiosity are elements of entering into a romantic relationship with another 
person that I suggest can be applied to romance with mathematics. 

Romance and Care 

Noddings (1992) questions whether it is too anthropomorphic to talk of caring for 
an abstract discipline like mathematics. Her basic position is that of establishing 
caring relationships, so a caring relationship with mathematics may indeed be 
problematic, and that 'strictly speaking, one cannot form a relation with 
mathematics' (p. 20). After all, how can mathematics care 'back'? Yet, she argues, 
we can talk meaningful about caring for mathematics, that 'oddly, people do report 
a form of responsiveness from ideas and objects. The mathematician Gauss was 
"seized" by mathematics' (p. 20). And Bertrand Russell is said to have described 
mathematics as his chief source of happiness. 
Noddings goes on to argue that 

teachers should talk with students about the receptivity required in caring 
about mathematics. People can become engrossed in mathematics, hear it 
"speak to them," be seized by its puzzles and challenges. It is tragic to 
deprive students of this possibility. (Noddings, 1992, p. 152) 

Initial education for prospective teachers often engages them in mathematical 
inquiry. But to what extent does such work engross, seize or challenge teachers 
with mathematics? Davis and Simmt (2003) suggest that a common purpose of 
engaging teachers in mathematical inquiry is predominantly pedagogical, so that 
teachers are better placed to model to students 'what it means to engage with 
mathematical problems and processes'. This, they argue is problematic because 
treating teaching as 

a modelling activity seems to be rooted in the assumption of radical 
separations among persons in the classroom. The teacher models, the learner 
mimics, but their respective actions are seen to be separable and to spring 



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MIKE ASKEW 

from different histories, interests and so on. (Simmt, Davis, Gordon, & 
Towers, 2003) 

In my experience of engaging teachers in mathematical inquiry, they talk not so 
much in terms of how the experience will help them model for their students but 
more of re-entering the experience of being a learner - the joys and frustrations of 
engaging in such work. The emphasis is still individualistic with a focus on the 
learner experience rather than the mathematics. Either response - modelling or 
empathy - not only separates teachers and students, it enable teachers to maintain a 
distance from the mathematics. The focus of mathematical inquiry with primary 
school teachers needs to be on the mathematics. 

Building on Noddings' work I add the distinction between caring /or and caring 
about. It probably is unreasonable to expect all primary school teachers to care for 
mathematics - people develop different appetites for different subjects. But they 
do, I think, have a duty to care about mathematics: to recognise and acknowledge 
the role that mathematics has played and continues to play in shaping the world we 
live in. One step in promoting caring about is the development of curiosity. 

Romance and Curiosity 

When a National Curriculum was first established in England, funding was 
provided for many primary teachers to engage in 20 days of mathematical 
professional development. Local funding for this was conditional on teachers 
working on their discipline knowledge. As the 20 days released from school were 
spread over two terms there was sufficient time to establish relationships with 
teachers that allowed for working on aspects of mathematics that they might 
usually shy away from. 

On one such course that I conducted, the following incident occurred towards 
the end when the teachers and I had got to know each other quite well. Working on 
a particular inquiry I had given out some calculators. One of the teachers, Ursula, 
called me over. 

"My cheap calculator that I bought at a garage has got a 1 /x button on, so why 
doesn't this expensive one?" 

"Well, it does," I replied " it's that x"' button." 

This seemed an opportunity to explore powers and so I stopped everyone to go 
through an argument as to why x' 1 is equated with 1/x. 

While the teachers' nods during my explanation suggested that they were 
following my argument, afterwards there was a lot of muttering at Ursula's table. 

"Is there anything you are not clear about?' I enquired. 

"No, we follow your argument," Ursula replied. "But we were just saying to 
each other, 'why would anyone ever want to do that in the first place?'" 

This incident has stayed with me over the years, acting as a touchstone for 
several issues. First it highlights that teachers' emotional relationship with 
mathematics cannot be separated from their intellectual, cognitive, knowledge of 
the subject. This was a group of teachers who had begun to work with the 
mathematics intellectually - they were willing to 'play the game' of developing 



26 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

mathematical ideas with me. But they were not engaged with the game in the sense 
of deriving satisfaction from the pleasure of playing it, as evidence by the asking of 
why would anyone want to do that. Hodgen (2004) highlights the importance of 
desire and imagination in developing and transforming teachers' relationships with 
mathematics. 

We talk about engagement with mathematics as if the playing around with ideas 
in and of itself is sufficient to make teachers and pupils want to carry on with the 
play. As Simmt, Davis, Gordon, and Towers (2003) put it, teachers have an 
'obligation' to be curious about mathematics. Prospective teachers are encouraged 
to be curious about students' responses to mathematics but this needs to be 
counterbalanced with a curiosity about mathematics itself. Visiting a class recently, 
the teacher, in slightly exasperated terms, asked me 'why, just when you think the 
children have got it, they can just as easily lose it?' When I asked to elaborate on 
what the 'it' was, she talked of a boy who could correctly say whether or not a 
number up to 8 was odd or even, but had answered that nine was also even. 1 
suspected that he may have over-generalised and, sure enough when I asked him 
why he thought nine was even, he demonstrated how nine tallies could be set out 
into three 'even' (that is, same sized) sets. Part of being curious here is about being 
interested in the pupil's thinking, but there is also the curiosity of wondering if it is 
possible to construct a mathematics where nine could be considered even. Such 
mathematical curiosity, when present 'compels teacher attendance to student 
articulations, it opens up closed questions, and ... can trigger similar contributions 
from learners' (Simmt et al., 2003, p. 181). 

Simmt and her colleagues go on to argue that 'curiosity is not an innate 
proclivity, but can be learned, to some extent at least' and that this learning of 
curiosity comes about through collective activity - a theme I return to in the final 
section. 

One of the delicious aspects of entering into a new romantic relationship is 
finding out about the history of the other. Being curious about the history of 
mathematics is another aspect of subject knowledge that I would argue is 
necessary. 

Knowledge of the history of mathematics can help teachers appreciate that there 
is no one single story, no 'truth' of the way that mathematics has developed. For 
example the story of 'Pascal's' triangle challenges the popularly held belief that 
mathematics is the result of individual activity and inspiration. Although key 
theories are attributed to individuals - Newton Pascal, Pythagoras - these 
individuals were part of ongoing communities, collectives. Not only did they stand 
on the shoulders of giants, they rubbed shoulders with their peers. The 
development of mathematics is a collective endeavour. 

The story of the development of mathematics as one of the emergence and 
invention of ideas, either to solve problems or simply through 'playing' with 
mathematical objects, may challenge dominant views of the linearity of learning 
mathematics. This perspective may help teachers to appreciate that there is no 
'truth' about the way that students learn mathematics. As Davis and Sumara argue 
the distinction between established knowledge (the curriculum) and knowledge that 



27 



MIKE ASKEW 

learners are developing or product versus process may not be the most appropriate 
distinction. That the actual distinction may be more a matter of scale than quality: 
it's the time span of development that is different rather than the substance of the 
knowledge per se (Davis & Sumara, 2006). Appreciating the history of 
mathematics problematises the distinction between the established canonical body 
of knowledge and the tentative knowledge of learners 

A commonly held misconception amongst students of history is that the 
discipline is simply about the chronological listing of events. In contrast, historians 
are more concerned with states of being than lists of events. This view of history 
also has implications for the mathematics classroom. Teaching mathematics is 
commonly perceived as a series of lessons covering a collection of mathematical 
topics. Perhaps we should be attending more (or at least equally) to what 'states' 
we want our classrooms to be in to allow for the emergence of mathematics. 

This points to the need to shift the discussion away from the individualistic view 
of teachers, to a recognition that knowledge and learning is collective rather than 
individual. We have to shift the attention away from the what of mathematics 
subject knowledge that teachers 'need' to the why of what are they learning about 
the way that they do mathematics. What does it mean to part of a community of 
mathematicians rather than an isolated acquirer of mathematical knowledge? 
Before examining this question, I look at some of the theoretical and 
epistemological stances that might underpin much of the work in this area to date. 
Through making these explicit, directions for future work begin to emerge. 

DISCIPLINE KNOWLEDGE: DISCOVERED OR CREATED? 

An issue arising from early work into subject matter knowledge was a lack of 
theorising of how discipline knowledge might inform and come to be played out in 
practice. As Leinhardt and Smith point out, there was a lack of problematising the 
issue of transfer: "No one asked how subject matter was transformed from the 
knowledge of the teacher into the content of instruction" (Leinhardt & Smith, 
1985, p. 8). But this assumes that there are different knowledge 'packages' to be 
transported and transformed. It is built upon an objectivist epistemology. 

Shulman's models of knowledge has elements of an objectivist epistemology: 
knowledge comprises objects located in the minds of individual teachers. While 
philosophers of mathematics question an objectivist epistemology of mathematics 
(Ernest, 1998) research that continues to treat subject knowledge as object may fix 
us in a teacher-centred pedagogy, with the assumption that the main source of 
learning comes about through the pre-exisiting knowledge of the teacher. 

Davis and Simmt (2006) argue that the practices of research mathematicians 
involve creating concise expressions of mathematics, through 'compressing' 
information. Teachers, in contrast, have the opposite task and need to be "adept at 
prying apart concepts, making sense of the analogies, metaphors, images, and 
logical constructs that give shape to a mathematical construct" (p. 301). 

Ball and Bass (2000) regard this ability to 'unpack' mathematics as an aspect of 
pedagogical content knowledge. This raises the question of how easy and/or 



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MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

necessary it is to separate discipline knowledge from pedagogic knowledge. If 
mathematical constructs are shaped through "analogies, metaphors, image, and 
logical constructs" then are these part of pedagogical content knowledge or part of 
subject knowledge? And does it matter? 

The answer comes down to a philosophical position on the epistemology of 
mathematics. If one believes that there are idealised mathematical forms that exist 
independently of representations, illustrations, examples and so forth, that there is a 
signifier/signified distinction (Walkerdine, 1988), then such 'unpacking' (or re- 
packaging) is going to be seen as a pedagogic skill. If, on the other hand, one views 
mathematics as a 'language game' (Wittgenstein, 1953) only brought into being 
through representations, illustrations, examples and not existing outside these, then 
this is an aspect of subject knowledge as much as pedagogic. 

As McNamara (1991) argues, all mathematics is a form of representation. 
Similarly, I have questioned whether attempts to classify different types of teacher 
knowledge and to separate these from beliefs is possible, or desirable, as all such 
propositional statements are brought into being through discourse and any 
classification into discrete categories can only be established within social relations 
(Askew, 1999). 

Thus distinctions between subject knowledge, pedagogic knowledge, 
semantic/syntactic, product/process can be regarded as the being constructed 
within the discourse of research literature, rather than being discovered as 
independently existing 'objects. In line with Vygotsky's observation, psychology 
creates the very objects that it investigates. 

The search for method becomes one of the most important paradoxes of the 
entire enterprise of understanding the uniquely human forms of psychological 
activity. In this case, the method is simultaneously prerequisite and product, 
the tool and the result of the study. (Vygotsky, 1978, p. 65) 

Vygotsky thus challenges the view that the method of inquiry in psychology is 
separate from the results of that inquiry, the traditional 'tool for result' position 
(Newman & Holzman, 1993). Instead 

As "simultaneously tool-and-result", method is practiced, not applied. 
Knowledge is not separate from the activity of practicing method; it is not 
"out there" waiting to be discovered through the use of an already made tool. 
... Practicing method creates the object of knowledge simultaneously with 
creating the tool by which that knowledge might be known. Tool-and-result 
come into existence together; their relationship is one of dialectical unity, 
rather than instrumental duality. (Holzman, 1997, p. 52) 

Tool-and-result means that no part of the practice can be removed and looked at 
separately. Like the classic vase and faces optical illusion neither the faces nor the 
vase can be removed and leave the other. There is no mathematical discipline 
knowledge that can be removed from the way that it has been studied and looked at 
separately. There is no content knowledge separate from pedagogic knowledge. 



29 



MIKE ASKEW 

There is simply the practice of working together on the problem. This 
'practice of method' is entire in itself and '(t)he practice of method is, among 
other things, the radical acceptance of there being nothing social (-cultural- 
historical) independent of our creating it' (Newman & Holzman, 1997, p. 
107). 

The most important corollary of the 'practice of method' is, for Newman and 
Holzman the priority of creating and performing over cognition: "We are 
convinced that it is the creating of unnatural objects-performances-which is 
required for ongoing human development (developing)" (Newman & Holzman, 
1997, p. 109). 

In a similar vein, Adler (1998) refers to knowing being a dynamic and 
contextual ised process, in contrast to 'knowledge' as more static and abstract. The 
key distinction here is that knowing is only manifested in practice, and unfolds 
within a social practice. The dynamics of this unfolding are different from those of 
'testing' teachers' subject knowledge in contexts that differ greatly from being in 
the classroom: one-to-one interviews, multiple choice questions and so forth. We 
might question the assumption that knowledge can be 'retrieved' outside the 
context in which it is used: there is no retrieval, there is only practice. 

Tool-and-result presents a double challenge to the research on teacher subject 
knowledge: firstly that the research itself constructs objects of knowledge, and, 
secondly, that knowledge in classrooms emerges within ongoing discourse. 
Vygotsky's theory dissolves the notion that teachers 'carry' a store of mathematical 
knowledge that they 'apply' in classrooms, and which mediates between the 
established cannon of mathematical knowledge and the emergent mathematics of 
the classroom. 

Davis and Simmt (2006) argue that constructed distinctions along the lines of 
product and process are not helpful and that, from a complexity perspective, the 
accepted cannon of codified mathematical knowledge and the emergent 
mathematical understandings of learners are 'self-similar'. This does not mean that 
either is reducible to the other, but any difference between the two is more a matter 
of scale than quality. Arguing that 'learning' needs to be extended beyond being 
regarded as only happening in the heads of individuals, Davis and Simmt consider 
mathematics as a learning system. Any distinction between discipline knowledge 
and knowledge of how learners develop mathematical understanding is a matter of 
timescale: mathematical discipline is only relatively stable in comparison to 
mathematics in classrooms. It becomes impossible to separate subject knowledge 
from pedagogic knowledge: "we must consider both teachers' knowledge of 
established mathematics and their knowledge of how mathematics is established as 
inextricably intertwined" (Davis & Simmt, 2006, p. 300, original emphasis). 

This is not to argue for a shift to a view of 'learner-centred' classrooms. As 
Davis points out the question is not one of whether or not classrooms are learner- 
centred or teacher-centred but whether or not they are mathematics centred. The 
argument for primary school teachers to engage in mathematical activity may be 
more to do with helping them better understand how mathematical knowledge is 



30 



MATHEMATICAL DISCIPLINE KNOWLEDGE FOR PRIMARY TEACHERS 

established than with the substance of discipline knowledge that they actually 
acquire. 

FROM THE INDIVIDUAL TO THE COLLECTIVE 

Finally I examine whether the enterprise of looking at /for prospective primary 
teachers' subject knowledge needs to move in different directions. To move away 
from the individualistic/cognitive focus that dominates so much of the literature 
towards a collective/situated perspective. That classrooms are constituted as they 
are with a single teacher to a large group of students is an historically contingent 
norm rather than a necessary condition for the organisation of teaching and 
learning. As such, the location of the 'problem' of discipline knowledge within the 
heads of individual teachers is a result of this contingency. Rather than continuing 
to seek to solve the problem by working on what is 'in' teachers heads perhaps a 
more productive approach is to look to how schooling is constituted, to accept that 
the knowledge of mathematics for teaching, particularly in primary schools, is 
likely to be distributed across a group of teachers and to work with this. 

Although studies of classroom communities have begun to take social, situated 
and collective perspectives (Lerman, 2000), studies of teachers' subject knowledge 
remain, by and large, within the paradigm of individual cognition. Even those 
studies that have begun to look at the mathematics knowledge that teachers 
actually use in classrooms are still driven by a desire to 'extract' such knowledge 
from its history and context, and codify and canonise it. 

Adler (1998) argues that learning to become a mathematics teacher means 
becoming able to talk within the discourses of mathematics teaching and to talk 
about such discourses. Thus there is more than simply learning new knowledge and 
such talk needs to be developed through collective engagement. 

Much of the research into subject knowledge is predicated (tacitly) on the 
assumption that it is necessary for one individual (the teacher) to be able to make 
sense of another individual's (the pupil) cognitions. With a shift to recognising the 
importance of the social in developing individual understandings, this 'head-to- 
head' model is only part of the picture. Perhaps even the focus on individual pupils 
needs to be questioned: "The 'learning system' that the teacher can most directly 
influence is not the individual student, but the classroom collective" (Davis & 
Simmt, 2003, p. 164) 

If the classroom collective rather than the individual student should be the focus 
of the teacher, then should the teaching collective rather than the individual teacher 
be the focus of the researcher? This is as much a political question as a theoretical 
one. It is notable that teacher development in places like the USA and UK focuses 
on the development of the individual, in line with the dominant political culture of 
'self-actualisation'. Other traditions of development, for example lesson study in 
Japan (Lewis, 2002) already attend more to the collective than the individual. 

Attending to the distributed nature of discipline knowledge means paying 
greater attention to the communities within which teachers are located. Millet, 
Brown, and Askew (2004), examine the importance of the professional community 



31 



MIKE ASKEW 

of teachers within schools. Some of the schools tracked over five years in the 
course of the Leverhulme Numeracy Research Programme were able to able 
successfully to 'share' mathematical expertise through setting up teams of teachers 
with collective responsibility for mathematics across the school. 

Other research studying the effectiveness of 90 primary school teachers in 
teaching mathematics, I and colleagues (Askew et al., 1997b) identified one school 
where the pupil gains over the course of a year on an assessment of numeracy were 
consistently high across all classes. This was despite the fact that not all teachers in 
the school demonstrated particularly strong discipline knowledge. One factor that 
we conjecture contributed to these consistent gains was the strength of support 
provided by two teachers who shared the responsibility for mathematics across the 
school and who had complementary strengths. One had a strong mathematical 
background gained through studying a science degree. The other had studied the 
psychology and pedagogy of primary mathematics over several years of ongoing 
involvement in professional development. 

One claim arising from the research into teacher knowledge is that lack of 
subject knowledge leads to over-reliance on textbooks in the classroom. It may be 
that rather than lack of subject knowledge, use of textbooks is the result of teachers 
seeking 'surrogate' classroom partners. In the absence of another adult being 
physically present in the room, the voice of the textbook may provide the next best 
thing. The issue may not be one of helping individual teachers become better 
'equiped' to scale the peaks of mathematics lessons, but acknowledging their need 
for fellow climbers. 

This is not to deny the place of the individual, but to recognise that individual 
cognition is part of a wider network. As Davis and Simmt express it: 

mathematical knowing is rooted in our biological structure, framed by bodily 
experiences, elaborated within social interactions, enabled by cultural tools, 
and part of an ever-unfolding conversation of humans and the biosphere, (p. 
315) 

It seems there is plenty to keep researchers in this field going for many more 
years yet. 

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Representations of knowledge in teaching. In J. Calderhead (Ed.), Exploring teachers' 

thinking (pp. 84-103). London: Cassell. 
Wittgenstein, L. (1953). Philosophical investigations (translated by G. E. M. Anscombe). 

Oxford: Blackwell. 
Wittrock, M. (Ed.). (1986). Handbook of research on teaching. New York: Macmillan. 
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Mike Askew 

Department of Education and Professional Studies 

King's College 

London, UK 



35 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 



2. MATHEMATICAL PREPARATION OF 
PROSPECTIVE ELEMENTARY TEACHERS 

Practices in Selected Education Systems in East Asia 



Providing prospective elementary school teachers with adequate preparation in 
mathematics is undoubtedly important, but the respective emphases remain 
contested. This chapter summarises extensive information from our study 
undertaken to examine and review the practices in mathematical preparation of 
prospective elementary teachers in six selected education systems in East Asia, 
with detailed information collected from Mainland China and South Korea. To 
further our understanding of the possible effectiveness of teacher preparation 
programmes in East Asia, we also investigated prospective elementary school 
teachers ' beliefs and knowledge in elementary school mathematics together with 
teacher educators' opinions about mathematical preparation of elementary school 
teachers in the case of Mainland China and South Korea. It was found that many 
variations are evidenced in programme set-up and curriculum structure for 
preparing elementary school teachers both across and within the six education 
systems. Nevertheless, mathematics is generally seen as an important subject in 
elementary teacher preparation programmes in all six education systems. The 
performance of prospective teachers sampled in Mainland China and South Korea 
also supports the perception that prospective elementary teachers in East Asia 
have a strong preparation in mathematics content knowledge. Furthermore, it 
seems that teacher educators in Mainland China and South Korea still seek to 
provide even more training in mathematics, especially in school mathematics, for 
prospective elementary teachers. The culture of valuing mathematical preparation 
in elementary teacher education programmes presents a salient feature of these 
education systems in East Asia. 

INTRODUCTION 

It is well recognised that teachers and their teaching are key to the improvement of 
students' mathematics learning (e.g., National Commission on Teaching and 
America's Future [NCTAF], 1996; National Council of Teachers of Mathematics 
[NCTM], 1991; Sowder, 2007). Educational research over the past decades, 
especially in the United States, has seen a dramatically increased emphasis on 
preparing and developing teachers who will take the main role in providing high 
quality classroom instruction (e.g., Sikula, 1996; Townsend & Bates, 2007). Yet, a 



P. Sullivan and T. Wood (eds.). Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 37-62. 

© 2008 Sense Publishers. All rights reserved. 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

recent review of research literature on mathematics teacher preparation indicates 
that we still do not know much about the effects of prospective teacher preparation 
in content and methods on their classroom instruction (Wilson, Floden, & Ferrini- 
Mundy, 2001). Nor is it clear to teacher educators which courses prepare 
prospective teachers to help them take up the challenge of developing effective 
mathematics instruction (e.g., Ball, Hill, & Bass, 2005). As issues related to 
mathematics teacher preparation are not restricted to specific regions; worldwide 
efforts to improve teacher education programmes have led to the increased interest 
in learning about teacher education practices in other education systems, especially 
those that have consistently produced students with high achievement in school 
mathematics (e.g., Silver & Li, 2000; Stewart, 1991). 

Over the past decades, students in East Asia have attracted mathematics 
educators' attention worldwide for their superior performance in mathematics as 
documented in several large-scale international comparative studies (e.g., Beaton, 
et al, 1996; Lapointe, Mead, & Askew, 1992; Mullis, Martin, Gonzalez, & 
Chrostowski, 2004). Efforts to search for possible contributing factors led to a 
great deal of interest in examining mathematics teachers and their teaching 
practices in East Asia (e.g., Ma, 1999; Stigler & Hiebert, 1999). Existing cross- 
national studies have revealed remarkable differences between Japan, Hong Kong 
and the United States in mathematics classroom instruction (Hiebert, et al., 2003; 
Stigler & Hiebert, 1999), as well as differences between Mainland China and the 
United States in teachers' knowledge of mathematics for teaching (e.g., An, Kulm, 
& Wu, 2004; Ma, 1999). In particular, Ma's study (1999) revealed that Chinese 
elementary teachers had a profound understanding of the fundamental mathematics 
they teach, whereas U.S. counterparts lacked a strong knowledge base in 
mathematics. As argued by Ma (1999), the results provide a partial explanation for 
what contributed to Chinese students' achievement in school mathematics, and are 
in alignment with current understanding that teachers and their teaching are key to 
the improvement of students' learning (e.g., NCTAF, 1996; NCTM, 1991). 
Paradoxically, according to Ma (1999), Chinese elementary teachers actually 
received less formal education, in terms of the number of years spent before 
entering the teaching profession, than their U.S. counterparts. The apparent cross- 
national discrepancy between teachers' knowledge and the number of years 
training they received in China and the United States remains open to multiple 
interpretations. One possible explanation is that Chinese teachers had better 
opportunities to continue to improve their mathematics knowledge for teaching 
over the years. At the same time, it is also unclear whether Chinese elementary 
teachers might enjoy a better start than their American counterparts upon the 
completion of their teacher education programmes. Much remains to be understood 
about teacher education practices in China, as well as other education systems in 
East Asia, that prepare teachers who eventually take a primary role in developing 
classroom instruction that helps generate students' high achievement in school 
mathematics. 

The value of exploring and understanding teacher education practices in an 
international context was recognised many years ago. As early as 1953, the 



38 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

International Council on Education for Teaching (ICET) was founded with a 
fundamental purpose of assisting the improvement of teacher education around the 
world. Through its world assemblies, ICET published proceedings on teacher 
education in different education systems (e.g., Klassen & Collier, 1972; Klassen & 
Leavitt, 1 976). Similar efforts can also be seen in recent years (e.g., Britton, Paine, 
Pimm, & Raizen, 2003; Cheng, Chow, & Mok, 2004; Eraut, 2000; Tisher & 
Wideen, 1990). Studies on teacher education policy and practices focused on 
various aspects of teacher education programmes including features and changes 
over different periods of time. However, not until a few years ago, there were few 
efforts to explore mathematics teacher education practices in an international 
context (e.g., Jaworski, Wood, & Dawson, 1999; Li & Lappan, 2002; Silver & Li, 
2000). Across education systems, different perspectives and approaches used in the 
preparation of elementary mathematics teachers are especially interesting. 
Different from the preparation at the secondary school level where mathematics 
teachers are normally content specialists, elementary school teachers are typically 
responsible for teaching all different content subjects in one education system but 
not in another. Cross-system variations in the practices of assigning elementary 
teachers instructional responsibility can be taken as a profound factor in 
influencing teachers' instructional practices, their professional development, and 
subsequent students' learning in different systems. At the same time, cross-system 
variations in assigning elementary teachers instructional responsibility can 
presumably suggest possible differences existed in preparation requirements 
provided to prospective elementary teachers across systems. Thus, an 
understanding of preparation requirements for elementary school teachers in other 
systems should provide a unique opportunity for teacher educators to learn possible 
alternative practices for improving teacher preparation. 

This chapter reviews current practices of prospective elementary school teacher 
preparation in several selected education systems in East Asia, including Hong 
Kong, Japan, Mainland China, Singapore, South Korea, and Taiwan. Although it is 
still not clear what courses prospective elementary school teachers should take, it is 
now generally recognised that teachers' mathematical knowledge for teaching is 
essential to effective classroom instruction (e.g., Ball, Hill, & Bass, 2005; RAND 
Mathematics Study Panel, 2003). It thus becomes the focus of this chapter to 
examine and discuss the practices in mathematical preparation of elementary 
(mathematics) teachers in these selected education systems in East Asia. 

In particular, we examine the programme structure in mathematics course 
offerings for prospective elementary school teachers with detailed information 
collected from Mainland China and South Korea. To further our understanding of 
the possible effectiveness of teacher preparation programmes in East Asia, we also 
investigate prospective elementary school teachers' beliefs and competence in 
elementary school mathematics together with teacher educators' opinions about 
mathematical preparation of elementary school teachers in the case of Mainland 
China and South Korea. Correspondingly, we took various approaches in our data 
collection. In addition to the collection and analysis of documents that relate to 
teacher preparation programmes in East Asia from literature and online resources, 



39 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

we also carried out surveys with a sample of prospective elementary school 
teachers and some mathematics teacher educators in Mainland China and South 
Korea. The results thus allow us to develop a better understanding of relevant 
practices in elementary teacher preparation in East Asia. 

This chapter is organised in four parts. In the first part, we provide contextual 
information about these six selected education systems in East Asia. We intend to 
establish whether elementary school teachers are content specialists or generalists 
and what requirements may be in place for becoming elementary school teachers 
across these education systems. Secondly, programme features for elementary 
teacher preparation are examined and discussed, with special attention to 
curriculum requirements in mathematics content training across these selected 
education systems. Detailed information related to programme requirements is 
exemplified through the case study of Mainland China and South Korea. Thirdly, 
we examine and discuss prospective elementary teachers' beliefs and knowledge in 
school mathematics based on the data collected from Mainland China and South 
Korea. In the last part, teacher educators' perceptions of the effectiveness of 
elementary teachers' mathematics preparation in Mainland China and South Korea 
are taken as insiders' views to discuss possible advantages and disadvantages of 
elementary teacher preparation practices in East Asia. 

ELEMENTARY TEACHERS IN SELECTED EDUCATION SYSTEMS IN EAST ASIA 

General Characteristics of Six Selected Asian Education Systems and Their 
Cultural Contexts 

The selected six education systems in East Asia, Hong Kong, Japan, Mainland 
China, Singapore, South Korea, and Taiwan, share some broad similarities. Apart 
from the well-known fact that students in all these education systems achieve well 
on large-scale international examinations in mathematics, they are all centralised 
education systems. In particular, the school mathematics curriculum framework is 
set up at the system level, and it serves as a guideline for school mathematics 
curriculum and instructional activities at all grade levels. Additionally, school 
textbooks used in these Asian education systems are required to bear an approval 
from a system- level authority. Moreover, these educational systems share the same 
cultural roots. Although there are a variety of ethnic groups living in some 
education systems such as Hong Kong, Mainland China, Singapore, and Taiwan, 
some education researchers have argued that there is a "Confucian Heritage 
Culture" (CHC) in this region (e.g., Bond, 1996; Wong, 2004). Confucius is a 
legendary figure in the Chinese history, whose thoughts in morals, government, 
and education have been influential in forming Chinese culture. There are some 
salient values, as summarised by Bond (1996) from relevant studies, common to 
the CHC regions in East Asia. Such common values include hierarchy, discipline 
and a strong achievement orientation. 

Although the six selected education systems share some broad similarities in 
student schooling and mathematics education (e.g., Schmidt, McKnight, Valverde, 
Houang, & Wiley, 1997), they differ in many other aspects, such as geographic 

40 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

locations, economy, language, and political system. For example, Hong Kong and 
Singapore were both British colonial territories (Brimer, 1988; Hayhoe, 2002; 
Thomas, 1988). Therefore, English is an official language in both Hong Kong and 
Singapore, especially for government and trade. Although Hong Kong began to 
adopt Mandarin as its official language with the return of sovereignty to China in 
1997, this change may take a long time (Adamson & Lai, 1997). Because of the 
role played by the English language and the influence of non-Confucian (i.e., 
Western) culture, many educational practices in Hong Kong and Singapore may be 
more like those in the West than is the case with Mainland China's practices. 
Likewise, educational researchers documented similarities and differences in 
schooling between Japan and Mainland China (e.g., Tobin, Wu, & Davidson, 
1989); mathematics classroom instruction between Japan and Taiwan (e.g., Stigler, 
Lee, & Stevenson, 1987); and teacher preparation between Hong Kong and Japan 
(e.g., Grossman, 2004). Therefore, the selection of these six Asian education 
systems provides an adequate basis to examine and understand possible variations 
in elementary teacher preparation in East Asia. 

Who Teaches Elementary School Mathematics and How to Become a Teacher? 

Elementary school teachers take different instructional responsibilities across these 
six education systems in East Asia. While elementary school teachers in Mainland 
China are often assigned to focus their teaching on a main subject (e.g., Chinese, 
mathematics), elementary school teachers in the other five education systems teach 
many different school subjects. Table 1 summarises elementary teachers' 
instructional responsibilities in these six education systems. 

Table 1. Some general characteristics related to elementary teachers in East Asia 



System 



H. K. 



Japan 



M. China Singapore S. Korea Taiwan 



Subject 
taught 


3 or more 
subjects 


Ail subjects 


lor 2 
subjects 


2-4 
subjects 


All subjects 


3 or more 
subjects 


Position 
qualification 


Certificate 


Certificate 
&exam 


Certificate 


Programme 
studies 


Certificate 
&exam 


Certificate 
&exam 


Years of 
education 


Varied 


4 years 


Varied 


Varied 


4 years 


4 years 



In Table 1 one can see that the assignment of elementary teachers' instructional 
responsibilities has a wide spectrum of variation across these six systems. In the 
case of Mainland China, elementary school teachers are prepared to be capable of 
teaching all content subjects at the elementary school levels as needed. This 
preparation is reflected in the programme requirements for prospective elementary 
school teachers, except special training programmes provided in music, art, 
physical education and English. In reality, the exact teaching assignment varies 
depending on the size of the schools. Typically, elementary school teachers are 
assigned to teach one main subject as a focus area, especially in large schools. In 
small and/or rural schools, elementary school teachers can be expected to teach one 



41 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

main subject like mathematics, plus one or two other subject areas such as art, 
physical education, or music. In contrast, teachers in Japan and South Korea are 
required to teach all subjects at the elementary school level (Park, 2005). ' It seems 
to be in-between for elementary teachers in Hong Kong, Singapore and Taiwan, 
where teachers often need to teach three or more subjects. The teachers' 
instructional assignments in some education systems, such as Singapore and 
Taiwan, are also different due to different perspectives of various school subjects 
in that system. For example, specialist teachers are required in Singapore for art, 
music, physical education or mother tongue languages (i.e., Chinese). All other 
elementary school teachers are expected to teach English, mathematics and either 
science or social studies or perhaps all four (Lim-Teo, 2002). Likewise, in Taiwan, 
about one third of elementary teachers are specialist teachers for courses like art, 
music, or science. The remaining elementary teachers are responsible for teaching 
all subject areas to their classes (Lin, 2000; Lo, Hung, & Liu, 2002). In contrast, 
mathematics is taken as a main school subject with a similar status as the mother- 
tongue language in Mainland China. Thus, elementary schools in Mainland China 
emphasise the quality of mathematics instruction and assign teachers to teach 
mathematics as a focus subject area. 

In order to become an elementary school teacher in these six education systems, 
it is consistent that a teacher certificate is required. Certainly, other variations exist 
for some specific requirements. In particular, Singapore is a city-nation and has a 
unique situation, where the Ministry of Education is the only entity that recruits 
prospective teachers for education and is also the only employer that provides all 
graduates jobs at different schools (Lim-Teo, 2002). As the education is provided 
by the only National Institute of Education (NIE) that works closely with 
Singapore Ministry of Education, the quality of prospective elementary teacher 
preparation is closely monitored through programme studies at NIE. Hong Kong, 
in a sense similar to Singapore, imposes no additional certification requirements 
beyond successful completion of preparation programme study (Wang, Coleman, 
Coley, & Phelps, 2003). 

In Japan, South Korea, and Taiwan, successful completion of teacher 
preparation programme studies at specific institutions is only one necessary step to 
possibly become employed as a teacher. In addition to the teacher certificate, 
prospective teachers are also required to take and pass specifically designed exams 
for regular employment. Such requirements are especially important in these 
systems because teaching is a highly sought after position. For example, being a 
teacher is traditionally valued and respected in South Korea. In comparison with 
other jobs, teaching is popular partly because of its security and stability. Once a 
person becomes a teacher, he/she can teach for a long period of time without 
dismissal. As a result, top students from high schools often intend to become 
prospective elementary school teachers. Up until the early 1990s, a graduate from 



Note that there are also some exceptions in South Korea where some subjects such as physical 
education, music, art, and English are assigned to specific teachers. The number of specific teachers is 
determined by the size of the schools, and the subjects are dependent upon the conditions of the schools 
and the preference of the specific teachers Such specific teachers are not necessarily content specialists. 



42 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

one of the 13 specific universities in South Korea could have become an 
elementary school teacher without taking any examination, and served for the 
specific province in which that university is located. Since then, however, the 
policy of no examinations has been changed so that every graduate has to pass a 
national examination to be an elementary school teacher. Correspondingly, there is 
no limitation for a prospective teacher to apply for a teaching position in other 
provinces regardless of the location of his/her graduating university. As a result, 
the competition is becoming intense, especially for those who want to be a teacher 
in popular cities such as Seoul. 

In Mainland China, a teacher certificate system was initiated in 1 996 and began 
its implementation in 2000. According to the Teacher's Act released in 1993, the 
minimum requirement for being qualified as an elementary school teacher is the 
completion of a 3-year programme of study at a normal school that admits junior 
secondary school graduates. As reported in the 2006 National Statistics Bulletin 
Board for Educational Development (Ministry of Education, Mainland China, 
2007), 98.87% of all 5,587,600 elementary teachers met this minimum 
requirement. 

Although the requirement of a teacher certificate is quite consistent across these 
education systems, more variations can be observed in terms of the years of 
preparation needed for obtaining a certificate both across and within these 
education systems. Comparatively, it is now common practice in Japan, South 
Korea, and Taiwan that prospective teachers take a four year B.A. or B.Sc. 
programme study. In the other three education systems, there are still mono- 
technical institutions like normal schools or junior normal colleges that offer 2-4 
year preparation programmes. In Singapore, it is still a common practice to offer 2- 
year Diploma of Education programmes for high school or polytechnic graduates 
together with the one-year Postgraduate Diploma in Education (Primary) 
programme and the regular 4-year B.A. or B.Sc. programmes. In Hong Kong, 
Japan, Mainland China, and Taiwan, there has been a move to upgrade elementary 
teacher preparation to 4-year B.A. or B.Sc. programmes and to become an open 
system where teacher preparation can be offered not only in normal colleges but 
also in comprehensive universities. Japan took such a step much earlier than Hong 
Kong (Grossman, 2004; Leung & Park, 2002), and then Taiwan in 1994 (Lin, 
2000) and Mainland China in 1999 (Department of State, Mainland China, 1999). 
For example, as reported in the newspaper of China Education (2006), from 1999 
to 2005, the number of normal schools for junior secondary school graduates 
decreased from 815 to 228, the number of junior normal colleges that offer 5 (or 
'3+2') or 3 years teacher preparation programmes decreased from 140 to 58, and 
the number of normal universities and comprehensive universities that offer 4-year 
B.A. or B.Sc. teacher preparation programmes increased from 87 to 303. Thus, 
while 4-year B.A. or B.Sc. preparation programmes have become more and more 
popular across these education systems in East Asia, other programmes that offer 
fewer years preparation are still in existence. To reflect different preparation that 
prospective elementary teachers have received, teacher certificates are often 
provided with some differentiations in different education systems. 



43 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

ELEMENTARY TEACHER PREPARATION IN EAST ASIA 

General Features of Elementary Teacher Preparation Programmes in East Asia 

In contrast to the centralised administration of school mathematics curriculum, 
teacher preparation programmes show many more variations both across and 
within these education systems in East Asia. While the variations are 
understandably large between teacher preparation programmes for elementary 
school level and ones for secondary school level, programmes for elementary 
teacher preparation alone present many different pictures both across and within 
these education systems. According to a classification system used by Park (2005), 
teacher education programmes can roughly be classified along two dimensions: (1) 
the type of university (i.e., mono-purpose institutes that focus on teacher 
preparation versus comprehensive universities) where elementary teacher 
education programmes are typically housed in an education system, and (2) the 
nature of programmes where prospective elementary teachers are required to 
complete: an integrated programme of study or post-baccalaureate training in 
education for those who have received their Bachelor degree in a subject content 
field. This two-dimensional classification system led Park (2005) to provide a 
general sketch of elementary and secondary school teacher preparation 
programmes in East Asia. In particular, the majority of elementary teacher 
preparation programmes are integrated programmes in nature and housed in mono- 
purpose institutions in Hong Kong, Japan, Mainland China, South Korea and 
Taiwan. Whereas in Singapore, it has both integrated programmes and post- 
baccalaureate diploma in education programmes for elementary teacher preparation 
that are housed in the only teacher education institution, National Institute of 
Education (NIE), as part of a comprehensive university (Lim-Teo, 2002). 

The two-dimensional classification framework used by Park (2005) is helpful to 
a certain degree for understanding the structure of elementary teacher preparation 
programmes in different education systems, but it does not reflect some rich 
variations in existing elementary teacher preparation programmes that have 
evolved over time in East Asia. For example, some of the elementary school 
teachers are educated through teacher education programmes offered in 
comprehensive universities in Hong Kong, Japan, and South Korea, but especially 
more in Mainland China and Taiwan. Depending on how well elementary teacher 
education programmes can serve the needs in different systems and cultural 
contexts, elementary teacher education programmes have been undertaking 
different development trajectories over the years. In the case of South Korea, the 
structure of elementary teacher preparation programmes has been fairly stable over 
the past decade. There are 1 1 unique national universities specialising in preparing 
only elementary school teachers who teach from grade 1 to grade 6. These 1 1 
universities are spread out across the nation, located in a main city per province. In 
addition, there are two comprehensive universities that also offer an elementary 
teacher preparation programme. One is the national university specialising in 
teacher education ranging from preschool through elementary to secondary 
education. The other is a private comprehensive university that offers not only 

44 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

teacher education programmes but also other programmes such as business and 
engineering. Compared to various programmes for preparing secondary school 
mathematics teachers offered by a lot of national and private universities (see 
Chapter 3 by Shiqi Li and colleagues in this volume), the system of preparing 
elementary school teachers is rather uniform and stable in South Korea. 

Quite different from the case of South Korea, elementary teacher preparation 
programmes in Mainland China have experienced some dramatic structural 
changes over the past decades at the levels of both institution and programme. 
Historically, all elementary school teachers were prepared exclusively through 
normal schools. These are typically 3-year mono-technical institutions that admit 
junior secondary school graduates who are interested in becoming elementary 
school teachers and pass an entrance examination. This type of programme 
dominated elementary teacher preparation in Mainland China from the late 1970s 
to the late 1990s. Over the years, there has been gradual upgrading of teacher 
preparation programmes with the start of junior normal colleges offering 5 (or 3+2) 
or 3-year preparation programmes in the mid of 1980s and 4-year B.A. or B.Sc. 
preparation programmes in universities in 1998. Currently, all three types of 
preparation programmes are in co-existence, but with rapid changes to upgrade 
elementary teacher preparation into 4-year B.A. or B.Sc. programmes. As of the 
end of 2002, it took less than four years for Mainland China to have a total of 130 
normal and comprehensive universities that offer such a B.A. or B.Sc. programme 
in elementary teacher preparation (Huang, 2005). 

Programme Requirements in Mathematics for Elementary Teacher Preparation 

The variations evidenced in preparation programmes both across and within these 
six education systems inevitably lead to variations in programme requirements for 
elementary teacher preparation. Apparently, different types of teacher education 
programmes existing in an education system would reflect different curriculum 
requirements. For example, there are three different types of elementary teacher 
preparation programmes available in Mainland China: 

(1) 3-year programmes offered by normal schools (N. S.); 

(2) 5 (i.e., '3+2') or 3-year programmes offered by junior normal colleges (J. N. 
C); and 

(3) 4-year B.A. or B.Sc. programmes offered by normal or comprehensive 
universities (N./C. U.). 

In Table 2, the structure and requirements are summarised for preparing 
elementary teachers in these three types of programmes. 



45 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

Table 2. Curriculum requirements of different types of programmes in Mainland China 





N. S. 


J.N. 


C. 


N./C. U. 


Programmes 


3 years 


3+2 years 


3 years 


4 years 


Gen. education 


16.1% 


14.5% 


22.3% 


varied 


Edu. major 


11.5% 


9.5% 


17.1% 


varied 


Content subjects 


50% 


44.7% 


14.9% 


varied 


Selectives 


8.6% 


15% 


32.8% 


varied 


Edu. activities 


13.8% 


16.3% 


12.9% 


varied 



In Table 2, we adapted a course structure that can roughly accommodate 
different types of elementary teacher preparation programmes. Normal schools 
provide 3-year preparation programmes for junior secondary school graduates. The 
programmes offer a common curriculum to all prospective teachers. Chinese and 
mathematics are two main subjects with 17.5% and 16.4% course hours of the 
whole curriculum, plus a methods course each in Chinese and mathematics 
respectively. Mathematics content courses basically cover fundamental theories of 
elementary mathematics and what students typically need to learn in high school 
such as elementary functions, analytical geometry, and introduction to calculus. 
Junior normal colleges are the type of mono-purpose institutes that provide either a 
5-year preparation programme for junior secondary school graduates or a 3-year 
program for secondary school graduates. For those junior secondary school 
graduates who enrol in the 5-year preparation programme, the first three years are 
used in a way similar to completing the typical three year secondary school study 
and followed by two more years' study in the programme. Thus, the 5-year 
preparation programme is also termed '3+2'. Both the 5-year and 3-year 
preparation programmes offer subject specifications beyond the common 
curriculum for all prospective teachers. In the 5-year programme, prospective 
teachers typically need to take five mathematics content courses (e.g., algebra, 
geometry, mathematical analysis) and one methods course as part of their common 
curriculum. 

For those majoring in mathematics education, they need to take three more 
courses in mathematics: probability and statistics, mathematical thinking and 
method, and analytical geometry. Adding these mathematics courses brings the 
percentage of course hours in mathematics total to about 22%. In the 3-year 
programme, only one mathematics course and one methods course are required in 
the common curriculum. For those majoring in mathematics and science, 
prospective teachers typically need to take five more mathematics courses. Adding 
the mathematics courses required in common curriculum, prospective teachers 
majoring in mathematics education are required to take about 26.5% course hours 
in mathematics. The Ministry of Education in Mainland China provided curriculum 
guidelines for the programmes offered in both normal schools and junior normal 
colleges, but not yet for the 4-year B.A. or B.Sc. programmes. Thus, more 
variations exist in such newly developed 4-year programmes housed in normal or 
comprehensive universities. 



46 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

Even within the same type of preparation programmes in an education system, it 
can be the case that there are large variations across institutions. Again in the case 
of Mainland China, the curriculum structure and requirements of 4-year B.A. or 
B.Sc. programmes can be classified into three sub-categories. The classifications 
are mainly based on the orientation in preparing elementary teachers as generalists 
or content specialists. See Table 3 for the curriculum structure and average course 
requirements obtained from several normal universities representing each of the 
three sub-categories: integrated, middle ground, having a focus area. (Note: the 
results presented in the following table may not add to 100% due to rounding 
errors.) 

Table 3. Sample curriculum of three different 4-year preparation models in Mainland China 







N./C. U. 




Programme 


Integrated 


Mid ground 


Focus area 


Gen. education 
Edu. major 
Content subjects 
Edu. activities 


32.2% 
31.4% 
23.9% 
12.4% 


30% 
17.6% 
41.1% 
11.2% 


33.8% 
21.7% 
33.9% 
10.5% 



As shown in Table 3 programmes in these three sub-categories have a similar 
curriculum structure, consisting of four components: general education courses; 
education major courses; content subject courses; and educational activities 
including student teaching and thesis. But there are many more variations in terms 
of the type and number of specific courses required. For the 'integrated' sub- 
category, the programme provides elementary teacher preparation as a whole and 
prospective teachers need to take courses in different subject areas. In contrast, the 
programme in the 'focus area' sub-category provides prospective elementary 
teachers choices of subject focus areas such as mathematics. The subject 
specification automatically leads to more content course requirements in a specific 
subject. The programme in the 'mid ground' sub-category basically provides 
grouped subject orientation. It typically includes two general subject orientations: 
science oriented and humanity oriented. The specific course structure and the 
number of courses provided vary from institution to institution. Across these 
programmes, the total credit hours required can range from 150 to 197. 

Understandably, the variations within a specific education system are not 
consistent across different education systems. Mainland China may be taken as a 
case with the most variations in terms of many different aspects, especially at the 
time when the system is undertaking some fundamental changes to upgrade its 
teacher education. Comparatively, other education systems have a smaller scale of 
variation. For example, in Taiwan, it is generally required that 4-year B.A. or B.Sc. 
preparation programmes need to have at least 148 credit hours. Among the 148 
credit hours, 97 credit hours are required for courses in the categories of general 
knowledge, basic subject, and professional education. These are the common 
course requirements for all prospective teachers who may plan to teach in any 



47 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

subject areas. The remaining 51 credit hours are for courses in the focus area of 
prospective teachers' own choice. The structure is similar to the sub-category of 
'focus area' programme in Mainland China (see Table 3). For the mathematics 
education major, prospective teachers are expected to take additional mathematics 
courses in Calculus, Algebra, Analysis, as well as mathematics education courses 
such as Study of Elementary School Mathematics Curricula (Lo, Hung, & Liu, 
2002). Although there are some variations in terms of specific curriculum content 
and courses required across institutions, these are all minor variations under the 
same curriculum guidelines. 

The case of South Korea is in between in terms of variations. Prospective 
elementary school teachers in South Korea have the option of choosing one school 
subject area, such as mathematics or music, as a focus area during their four year 
programme study. They can also choose general elementary education or computer 
education as their focus area, even though they are not specific school subjects. In 
fact, the universities offering elementary teacher education programmes are 
structured with departments according to different focus areas. Most of these 
universities have departments of moral education, Korean language education, 
social studies education, mathematics education, science education, physical 
education, music education, fine arts education, practical arts education, 
elementary education, English education, and computer education. 

Because elementary school teachers in South Korea need to teach every school 
subject, prospective elementary school teachers are required to take the same 
courses with only the exception of their focus areas. In general, the total credit 
hours required for a bachelor degree is about 145 and the credit hours for the focus 
area is 21. In other words, about 85% of the course requirements are the same for 
any prospective elementary school teacher. With regard to requirements common 
in mathematics, elementary teacher education programmes consist of the liberal 
arts courses and the major courses. Among the liberal art courses, prospective 
teachers take a compulsory course (2-3 credit hours) dealing with the foundations 
or basics of mathematics. Among the major courses, prospective teachers take 2 to 
4 compulsory courses (4-7 credit hours) in mathematics. 

The most common courses are (Elementary) Mathematics Education 1 with 2 
credit hours and (Elementary) Mathematics Education II with 3 credit hours. The 
former mainly deals with overall theories (including the national mathematics 
curriculum) related directly to teaching elementary mathematics, whereas the latter 
covers how to teach elementary mathematics tailored to multiple content areas such 
as number and operations. The course of Mathematics Education II often requires 
prospective teachers to analyse elementary mathematics textbooks and to practise 
teaching through mock instruction. Most institutions also offer elective courses in 
mathematics for the category of the liberal arts courses and the major courses. 
Taken together, a minimum of three courses or 6 credit hours in mathematics are 
required for all prospective elementary teachers. All of these mathematics-bound 
courses are offered by the department of mathematics education in each institution, 
except in the two comprehensive universities where both elementary and secondary 
teacher education programmes are provided. Across institutions, there are some 



48 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

variations in terms of the number of credit hours required in mathematics and 
names used to call different mathematics courses. For example, Mathematics 
Education I may be called Theory of Teaching Elementary School Mathematics or 
Understanding of Elementary Mathematics Education. Likewise, Mathematics 
Education II may be called Practice of Teaching Elementary School Mathematics 
or Practice of Elementary Mathematics Education or even structured as three sub- 
courses in one of the eight institutions surveyed. But in essence, these course 
requirements are remarkably similar across institutions. 

As seen in Table 1, elementary school teachers may teach many different school 
subjects in all these education systems, except in Mainland China where teachers' 
instructional responsibility is often focused on a main school subject like 
mathematics. In general, one may expect that in-service teachers' instructional 
responsibility in different education systems would be consistent with the training 
that prospective teachers receive. As illustrated in the above discussion, this is 
often not the case. For example, in Mainland China where practising teachers often 
need to have a focus teaching subject, teacher preparation programmes were 
traditionally set up to offer virtually the same curriculum of studies to all 
prospective elementary teachers in normal schools with 3-4 year programmes for 
junior secondary school graduates. This situation began to change when junior 
normal colleges were introduced with more years of preparation and selection of a 
focus subject area (National Education Committee, Mainland China, 1995; 
Ministry of Education, Mainland China, 2003). The new 4-year B.A. or B.Sc. 
preparation programmes started in 1999 in Mainland China also have many more 
variations as the system does not yet have a uniform curriculum. In contrast, 
elementary teachers in other education systems need to teach many different school 
subjects. However, teacher preparation programmes in these education systems 
often require prospective teachers to have a focus area through their studies. 

Instructional and Assessment Approaches Used in Mathematics Content and 
Methods Courses for Prospective Elementary Teachers 

Prospective teachers' learning is influenced not only by what they learn (i.e., 
course content) but also by the way they are taught (i.e., instructional approaches 
used in programme courses). As there is limited information available about 
instructional approaches used in courses for prospective teachers in East Asia, we 
conducted a survey with mathematics teacher educators in some selected 
institutions in Mainland China and South Korea. Several questions in the survey 
were specifically designed to ask mathematics teacher educators about the 
instructional and assessment formats used in mathematics content and methods 
courses for prospective elementary teachers. They were also asked about their 
opinions on any changes needed for instructional and assessment methods. 

In Mainland China, although instructional and assessment methods used can 
vary from one instructor to another, some common features are prevalent. Answers 
from 20 mathematics teacher educators in 10 institutions indicated that lectures are 
the dominant method used in teaching mathematics content courses, with 19 out of 



49 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

20 respondents indicating that lectures are used all the time or most of the time. 
The result is consistent with other observations in Mainland China (e.g., Chen, 
2003; Liu, 2005). Moreover, our survey also shows that most respondents indicated 
that they sometimes used the methods of cooperative learning and self-exploratory 
in mathematics content course instruction. In comparison, there are more different 
methods used in teaching mathematics methods courses. While 13 out of the 20 
respondents indicated that lectures are used all the time or most of the time, many 
others reported the use of case analysis, cooperative learning, and mock teaching. 
When asked whether any changes in instructional methods would be needed, more 
than 50% respondents indicated that changes are needed for methods course 
instruction but not for content courses. Many teacher educators preferred to include 
more case analyses to make connections with classroom instruction in methods 
course instruction. The results suggest that Chinese mathematics teacher educators 
would like to see more changes in methods course instruction but feel comfortable 
with the dominant use of lecture in teaching mathematics content courses. With 
regard to assessment formats, our survey results indicate that written tests and 
homework assignments were the most commonly used methods in both 
mathematic content and methods courses in Mainland China. Some teachers also 
reported the use of term papers and topic study as assessment approaches in both 
types of mathematics courses. Similar to the results reported for instructional 
methods, mathematics teacher educators saw the need for a change in assessment 
approaches more in the methods course but not in the content courses. In particular, 
suggestions were made to adopt multiple assessment approaches, emphasise the 
knowledge application in instruction, use mock teaching, and analyse textbooks. 
When asked how best to prepare prospective elementary teachers in mathematics 
methods courses in terms of both content covered and instructional approaches 
used, many teacher educators indicated the need to connect theory learning and 
instructional practices, follow curriculum changes in elementary mathematics, and 
use the instructional case analysis method. 

In South Korea, our survey results were obtained from eight mathematics 
teacher educators representing eight national universities out of the 13 institutions 
that have elementary teacher preparation programmes. The survey results present a 
picture that shares some similarities with the case in Mainland China. Although the 
formats for teaching both mathematics content and methods courses also varied 
from one respondent to another, the overall tendency of instructional formats of 
mathematics content and methods courses was a lecture-oriented style with an 
estimated use of more than 60% of the time for the content courses and 40% to 
75% of the time for the methods courses. More instructional methods were 
reported in use for the methods courses, which includes the use of presentation, 
cooperative learning, individual and group activities, seminar, and mock teaching. 
Likewise, the overall tendency of assessment formats was reported to use written 
tests in more than 70% of the time in conjunction with a few homework 
assignments for the content course, and in more than 50% of the time in 
conjunction with homework assignments and presentations for the methods course. 
When asked about any changes needed in instructional and assessment approaches 



50 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

used in the mathematics content courses, the answers included the use of more 
cooperative learning or projects, the design of interesting teaching materials, 
engaging students in classroom instruction activities, and the use of various 
assessment methods beyond written test and homework assignments. In 
comparison, there were more and different suggestions made for possible changes 
in the methods course. Some teacher educators insisted on the connection between 
the methods course instruction and elementary school mathematics classrooms. 
Beyond teaching general theories mainly by lecture, suggestions were made to 
analyse in detail the national mathematics curriculum and a series of elementary 
mathematics textbooks and to let prospective teachers experience know how to 
teach elementary mathematics as realistically as possible. Other suggestions 
include encouraging peer discussions among prospective teachers themselves, 
providing constructive feedback for each mock teaching, and using multiple 
teaching models applicable to elementary mathematics classrooms. With regard to 
the opinion about how to best prepare prospective elementary teachers in 
mathematics methods courses in terms of both content covered and instructional 
approaches used, the common answer was a combination of profound 
understanding of elementary mathematics contents (or big ideas) and experience of 
using multiple teaching models. 

PROSPECTIVE ELEMENTARY TEACHERS' BELIEFS AND KNOWLEDGE IN 
MATHEMATICS: SAMPLE RESPONSES FROM CHINA AND SOUTH KOREA 

Given that students in East Asia often performed well in international examinations 
of school mathematics, it may be reasonable to assume that prospective teachers in 
these high-achieving education systems should also be well prepared in 
mathematics and ready to teach. However, there have been some uncertainties over 
the years about the quality of prospective teacher preparation in Mainland China 
(e.g., Fang & Paine, 2000), Singapore (see Lim-Teo, 2002), and Taiwan (e.g., Lo, 
Hung, & Liu, 2002). To gain a better understanding of prospective elementary 
teachers' beliefs and knowledge in mathematics, we thus developed and conducted 
a survey in Mainland China and South Korea. 

The survey contains two main parts with five items for each part. Part 1 contains 
items on prospective teachers' knowledge of elementary mathematics curriculum 
and their beliefs in their preparation and mathematics instruction. Part 2 has five 
items that assess prospective elementary school teachers' mathematics knowledge 
and pedagogical content knowledge on the topic of division of fractions. Most 
items were taken from a previous study (Li & Smith, 2007), with some items 
adapted from school mathematics textbooks and others' studies (e.g., Hill, 
Schilling, & Ball, 2004; TIMSS 2003; Tirosh, 2000). Given the limited page space 
we have here, only a few items from each part and prospective teachers' responses 
to these questions are included to provide a glimpse of prospective teachers' beliefs 
and knowledge in mathematics. 

In Mainland China, the survey was given to both junior and senior prospective 
elementary teachers in 10 institutions. Five institutions are normal universities or 



51 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

colleges that offer 4-year B.A. or B.Sc. preparation programmes, and the other five 
are junior normal colleges that offer 5 (i.e., '3+2') or 3 years programmes for 
junior secondary school graduates and secondary school graduates respectively. 
361 responses were collected and used for data reporting, with 50.4% of responses 
from junior normal colleges and 49.6% from normal universities/colleges. In South 
Korea, the survey data was collected from 291 seniors in three universities that all 
offer 4-year B.A. or B.Sc. preparation programmes. These 291 seniors included 
those with mathematics education as their focus area, and ones with their focus 
areas in other subjects. Due to the sampling difference across these two education 
systems, the results reported here are not for comparison purposes and should not 
be taken as reflecting the overall situation in these two education systems. A 
thorough data analysis and detailed reporting is needed before a cross-system 
comparison can possibly be made. 

Prospective Teachers ' Knowledge in Elementary Mathematics Curriculum and 
Beliefs about Their Own Knowledge Preparation 

The following three items (note: only part of items 2 and 3 are included here) were 
included in Part 1. In general, responses from sampled prospective teachers in both 
Mainland China (361 respondents) and South Korea (291 respondents) show that 
prospective teachers often do not feel over-confident with their knowledge and 
readiness to teach elementary school mathematics. The results present a picture of 
sampled prospective teachers' self-confidence in two Asian education systems, 
which is quite different from the high confidence displayed by a sample of 
prospective teachers in the United States (Li & Smith, 2007). (Note: the results 
presented in the following tables may not add to 100% due to rounding errors.) 

I. How would you rate yourself in terms of the degree of your understanding of the 
National Mathematics Syllabus? 
(a) High (b) Proficient 

(c) Limited (d) Low 

Table 4. Percent of Sampled Prospective Teachers ' Self-rating of the Degree of their 
Curriculum Understanding 

High Proficient Limited Low 

Mainland China 3% 13% 60% 24% 
South Korea 2% 18% 66% 14% 

It can be seen in Table 4 that many sampled prospective teachers in both 
Mainland China and South Korea did not think that they know their national 
mathematics syllabus well. In particular, there are only 16% of sampled 
prospective teachers in Mainland China and 20% of those in South Korea self-rated 
with either "high" or "proficient" understanding of their national mathematics 
syllabus. Consequently, 84% of sampled prospective teachers in Mainland China 



52 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

and 80% in South Korea self-rated with either "limited" or "low" understanding of 
their national mathematics syllabus. 

2. The following list includes some topics that are often included in school mathematics. 
Choose the response that best describes whether primary school students have been 
taught each topic. 

(a) Mostly taught before grade 5 

(b) Mostly taught during grades 5-6 

(c) Not yet taught or just introduced during grades 5-6 

(d) Not included in the National Mathematics Syllabus 

(e) Not sure 

( ) Addition and subtraction of fractions 

( ) Multiplication and division of fractions 

( ) Representing decimals and fractions using words, numbers, or models 



Table 5. Percent of Sampled Prospective Teachers ' Choices about the Curriculum 
Placement of Three Content Topics 



Topic: Addition and Subtraction of Fractions 




Choice (a) Choice (b) Choice (c) Choice (d) 


Choice (e) 


M. China 
S. Korea 


71%* 26% 2% 0% 
79% 18% 1% 0% 


1% 
2% 


Topic: Multiplication and Division of Fractions 




Choice (a) Choice (b) Choice (c) Choice (d) 


Choice (e) 


M. China 
S. Korea 


20% 67% 8% 2% 
34% 61% 3% 1% 


3% 
2% 


Topic: Representing Decimals and Fractions with Words, Numbers, 


or Models 




Choice (a) Choice (b) Choice (c) Choice (d) 


Choice (e) 


M. China 
S. Korea 


42% 24% 11% 7% 
64% 18% 4% 3% 


17% 
11% 



Note (*): Bold means the correct choice for the topic in that education system. 

Table 5 presents a diverse picture for both Mainland China and South Korea. 
The results show that the majority of sampled prospective teachers in both 
Mainland China (71%) and South Korea (79%) know that the topic of "addition 
and subtraction of fractions" is taught before grade 5 in elementary mathematics 
curriculum. It becomes less certain to these sampled prospective teachers that the 
topic of "multiplication and division of fractions" is taught between grades 5-6 in 
Mainland China (67%) and South Korea (61%). For the topic of "representing 
decimals and fractions using words, numbers, or models", a high percentage of 
sampled prospective teachers in South Korea (64%) knew the correct placement of 
this topic in elementary school mathematics, but a smaller percentage of those 
sampled prospective teachers in Mainland China (42%) knew its correct placement 
and some of them (17%) were not sure. The result suggests that using non- 



53 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

numerical representations such as visual representations may not be a familiar area 
to many prospective teachers in the Chinese sample. 

3. Considering your training and experience in both mathematics and instruction, how 
ready do you feel you are to teach the following topics? 

(a) Very ready 

(b) Ready 

(c) Not ready 

( ) Primary school mathematics in general 

( ) Number - Representing decimals and fractions using words, numbers, or models 
( ) Number - Representing and explaining computations with fractions using words> 
numbers, or models 



Table 6. Percent of sampled prospective teachers ' choices of their readiness to teach 
elementary school mathematics 



Primary school mathematics in general 




Choice (a) Choice (b) Choice (c) 


M. China 
S. Korea 


13% 72% 15% 
6% 80% 14% 


Number - 


- representing decimals and fractions using words, numbers, or 
models 




Choice (a) Choice (b) Choice (c) 


M. China 
S. Korea 


13% 52% 36% 
7% 71% 22% 


Number - 


- representing and explaining computations with fractions using 
words, numbers, or models 




Choice (a) Choice (b) Choice (c) 


M. China 
S. Korea 


12% 44% 44% 
6% 70% 23% 



Table 6 shows that the majority of sampled prospective teachers in South Korea 
felt that they are ready to teach the three topics specified (80%, 71%, and 70% 
respectively). Much smaller percentages of those in South Korea felt that they are 
very ready to teach these three topics (6%, 7%, and 6%, respectively), and some 
would not feel confident (14%, 22%, and 23% for each of these three topics, 
respectively). The results from sampled prospective teachers in Mainland China 
present a different picture. In particular, a higher percentage of prospective 
teachers felt that they are very ready to teach these three topics (13%, 13%, and 
1 2% respectively). However, only about 1 5% felt that they are not ready to teach 



54 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

primary school mathematics in general, relatively high percentages of sampled 
prospective teachers felt that they are not ready to teach the other two topics with 
the use of different representations (36% and 44%, respectively). The results may 
well relate to what has been reported in Table 5, where more than 50% of 
respondents were unclear about the curriculum placement of the topic of 
'representing decimaJs and fractions using words, numbers, or models'. The results 
may also relate to the fact that both juniors and seniors from two different types of 
institutions (i.e., junior normal colleges and normal universities) were included in 
this survey result from Mainland China. Consequently, the results from Mainland 
China contain relatively smaller percentages of sampled prospective teachers who 
would think that they are ready to teach these three topics as specified. 

Prospective Teachers ' Knowledge in Elementary Mathematics: Division of 
Fractions 

Division of fractions is a difficult topic in elementary school mathematics (e.g., 
Ma, 1999), not only for school students but also for teachers (e.g., Li, 2008; Li & 
Smith, 2007). By focusing on this content topic, we developed a survey with test 
items that aim to assess prospective teachers' mathematics content knowledge and 
pedagogical content knowledge on this topic. In particular, the following two items 
were included in part 2 of our survey (note: only part of item 4 is included here, 
and the items are numbered here in terms of the sequence of items included in this 
chapter). In general, sampled prospective elementary teachers in both Mainland 
China (361 respondents) and South Korea (291 respondents) were quite successful 
in solving division of fraction items that assess teachers' mathematics knowledge. 
But the results from answering items targeted on teachers' pedagogical content 
knowledge present two quite different pictures in Mainland China and South 
Korea. Such differences are likely beyond sampling differences and can possibly 
relate to the different training provided for prospective elementary teachers in these 
two education systems. 

4. Solve the following problems (no calculator). Be sure to show your solution process. 

9 2 9 3 

(1). Say whether -. — is greater than or less than -■ — without solving. 

11 3 11 4 

Explain your reasoning. 

In the case of Mainland China, about 94% of the sampled prospective teachers 
provided the correct answer (i.e., the first numerical expression is greater than the 
second one). And the remaining 6% did not get the correct answer. Out of those 
getting the correct answer, about 64% did not carry out computations. The 
common explanations are (a) "If the dividend is the same, the smaller the divisor, 
the larger the quotient; the larger the divisor, the smaller the quotient." and (b) "2/3 
is smaller than 3/4". And many respondents provided both reasons. The remaining 
36% did the computation to reach the correct answer. 



55 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

In South Korea, about 95% of the sampled prospective elementary teachers 
answered correctly. They mentioned, "If the divisor is the smaller (and the 
dividend is the same), the result of the division is bigger". Some respondents 
demonstrated the fact that 2/3 is smaller than 3/4 in words (68%) or by drawing 
(5%). Others demonstrated using the common denominator 1 2, in words (9%) or 
by drawing (3%). About 5% of these teachers answered correctly used an analogy 
with natural numbers. There were about 5% of the sampled prospective teachers 
who answered incorrectly or did not answer at all. They either made a mistake on 
which fraction is bigger between 2/3 and 3/4, or did not infer correctly on what 
would be the result of division if the divisor was the smaller. 

2 o 1 2 1. 

5. How would you explain to your students why H 2. = — ? Why — ! — = 4 ? 

(item adapted from Tirosh, 2000.) 

For the sample prospective teachers in Mainland China, about 88% provided 
valid explanations for dividing a fraction by a natural number. However, the 
dominant explanation was based on the algorithm, "dividing a number equals to 
multiplying its reciprocal" (64%). The other 24% were dominated by explanations 
such as "dividing a whole into three equal parts, each part should be 1/3, so two 
parts should be 2/3. 2/3 * 2 means to equally divide 2/3 into 2 pieces, thus one 
piece should be 1/3", or "The half of 2/3 is 1/3". Similar patterns of success rates 
and explanation were observed with the second fraction division computation. In 
particular, about 82% of respondents provided valid explanations but the majority 
explained directly in terms of the same algorithm, that is, flip and multiply (73%). 
The other 9% provided their explanations mainly as "dividing a whole into six 
equal parts, each part is 1/6, four parts should be 4/6. In other words, 4/6 contains 
four 1/6. Thus, 4/6 * 1/6 = 4. Because 4/6 equals 2/3, so 2/3 * 1/6 = 4." A few 
students also tried to draw a picture such as a circle to help with their explanations. 
Some others either provided incomplete solutions or misunderstood the problem as 
to explain why 2/3 * 2 is smaller than 2/3 + 1/6. 

In the case of South Korea, about 98% of prospective teachers provided valid 
explanations with regards to dividing a fraction by a natural number. They came up 
with various methods such as drawing (45%), using manipulative materials (10%), 
using the meaning of division and/or fractions (8%), finding the common 
denominator (6%), number lines (6%), and using word problems (5%). In addition, 
about 15% of the respondents provided explanations in more than two ways. Only 
about 3% of the respondents predominately employed the algorithm of "flip and 
multiply". In contrast, about 88% of the sampled prospective teachers provided 
valid explanations with regards to dividing a fraction by a fraction. They again 
came up with various methods such as drawing (33%), finding the common 
denominator (23%), number lines (5%), using manipulative materials (5%), using 
the meaning of division (3%), using the relationship between fractions (1%), and 
using word problems (1%). Note that the percentage of respondents finding the 
common denominator was dramatically increased from 6% to 23%, whereas the 

56 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

percentage of respondents using manipulative materials was decreased from 10% 
to 5%. Although drawing was still the preferred approach, about 4% experienced 
difficulty and failed. For instance, an area model was drawn to represent 2/3 x 1/6 
instead of 2/3 + 1/6. Finally, about 10% of these prospective teachers used the "flip 
and multiply" algorithm as providing an explanation for the second division 
computation. 

MATHEMATICAL PREPARATION OF ELEMENTARY TEACHERS: SAMPLE 
CHINESE AND KOREAN TEACHER EDUCATORS' VIEWS 

Teacher preparation programmes are a moving target with constant changes in the 
structure and curriculum requirements. Such changes or needs for changes in an 
education system can be perceived quickly by its insiders but not outsiders of that 
system. We thus included some questions in our survey for mathematics teacher 
educators from Mainland China and South Korea to learn their views about 
mathematical preparation of elementary teachers in these two systems respectively. 

As discussed previously, Mainland China is in the process of restructuring and 
upgrading its teacher preparation programmes. Three different types of elementary 
teacher preparation are currently in existence and there are many variations across 
institutions even for the same type of 4-year B.A. or B.Sc. preparation 
programmes. The existing variations led to different perceptions held by different 
mathematics teacher educators in Mainland China. Nevertheless, many teacher 
educators thought that prospective teachers were serious about teaching and were 
willing to learn. Some thought that prospective teachers had adequate training in 
collegiate mathematics and developed their abilities in analysing and solving 
problems. 

At the same time, however, many teacher educators thought that some 
prospective teachers did not have solid training in basic but important school 
mathematics. The differences between these teacher educators' views are in line 
with sampled prospective teachers' self-rating where about 13% of prospective 
teachers felt that they are very ready to teach primary mathematics in general and 
about 15% felt not ready (see Table 6). Finally, many teacher educators thought 
that it is necessary to help prospective teachers connect their mathematics learning 
with current elementary mathematics curriculum reform and instruction. In fact, 
about 84% of 361 prospective teachers surveyed in Mainland China self-rated with 
either "limited" or "low" understanding of the national mathematics syllabus. 

In South Korea, all prospective elementary school teachers need to take one 
course (2-3 credit hours) where they learn the basics of mathematics, and then take 
two more courses of about 5 credit hours where they learn not only general theories 
and curriculum related to teaching elementary mathematics, but also teaching 
methods related to different mathematics content topics. With these course studies 
in mathematics, most of the teacher educators in the survey thought that 
prospective teachers still do not have enough mathematical training for their future 
teaching career. At the same time, they admitted that it is difficult for prospective 
teachers to have enough background knowledge in each school subject, mainly 



57 



YEPING LI, YUNPENG MA, AND JEONGSUK PANG 

because elementary school teachers need to teach every subject. Nevertheless, 
teacher educators expected that prospective teachers should have appropriate 
knowledge preparation in their focus area. Some teacher educators provided a 
cautionary note that the courses should be tailored to prospective elementary 
school teachers so that the mathematics contents should be different from those for 
secondary mathematics teachers. It was emphasised by sampled teacher educators 
that mathematics content courses should be designed in a way to help prospective 
teachers deepen their understanding of mathematics content taught in elementary 
school, not sophisticated university level mathematics such as algebra or analysis 
as a discipline, and to foster their pedagogical content knowledge. 

Most of the teacher educators surveyed in South Korea also thought that all 
prospective teachers should have more opportunities to learn mathematics content 
and method, partly because mathematics is one of the most important subjects in 
elementary school and it is taught more hours per week than other subjects such as 
social studies or music. With the perceived lack of sufficient mathematics content 
and method courses, most teacher educators expected prospective teachers' 
competence in mathematics to be average or below average. Some teacher 
educators worried about prospective teachers' lack of profound understanding of 
basic but important mathematical concepts. Others thought that prospective 
teachers would have confidence in elementary mathematics content, but not in 
collegiate mathematics (i.e., the compulsory course in the category of liberal arts 
courses). In fact, according to the mathematics knowledge survey, about 86% of 
the 291 sampled prospective teachers indicated that they are either ready or very 
ready to teach elementary school mathematics in general (see Table 6), given their 
education received in both mathematics content and method. At the same time, 
many teacher educators underlined the strengths of current teacher preparation in 
that prospective elementary teachers learn how to teach every subject. In particular, 
the comprehensive programmes help prospective teachers make good connections 
among multiple subjects for developing an inter-disciplinary approach, and have 
good understanding of children and general pedagogical knowledge. 

CONCLUSION 

Discussions presented previously show many variations in programme set-up and 
curriculum structure for preparing elementary school teachers both across and 
within these six education systems. Although there is a trend for all of these 
education systems to improve and upgrade their elementary teacher preparation, 
different development trajectories have been undertaken in different system 
contexts. Specific curriculum structure of teacher preparation is also influenced by 
different factors, including the type of degrees awarded and the orientation placed 
in preparing generalists or subject specialists. Nevertheless, mathematics is 
generally seen as an important subject in elementary teacher education in all these 
six education systems. Our preliminary survey results obtained from sample 
prospective teachers in Mainland China and South Korea also support the 
perception that prospective elementary teachers in East Asia have a strong 



58 



MATHEMATICS PREPARATION OF ELEMENTARY TEACHERS 

preparation in mathematics content knowledge. However, it seems that teacher 
educators still seek to provide even more training in mathematics, especially in 
school mathematics, to prospective elementary teachers. 

At the beginning of the chapter, we mentioned some cross-national studies (e.g., 
An, Kulm, & Wu, 2004; Ma, 1999) that has led to the ever- increasing interest in 
learning more about teacher education practices in East Asia. In particular, Ma's 
study suggested that the Chinese elementary school teachers in her sample 
demonstrated their profound understanding of fundamental elementary 
mathematics they teach. However, our preliminary survey results do not seem to 
support the hypothesis that prospective elementary teachers in Mainland China 
may have strong preparation in pedagogical content knowledge. Although the 
sample of practising teachers in Ma's study is different from the sample of 
prospective teachers in our survey, the results obtained from our survey can safely 
suggest that the training provided to prospective teachers will not automatically 
lead to the profound understanding of school mathematics demonstrated by 
practising teachers in Ma's study. It is thus possible, as Ma (1999) pointed out, that 
the Shanghai teachers' profound understanding of fundamental mathematics in her 
sample resulted from being subject specialists in teaching and having opportunities 
to study the teaching materials intensively together with other teachers that occurs 
after leaving their teacher preparation programme. 

ACKNOWLEDGEMENTS 

We would like to thank Xi Chen, Dongchen Zhao, and Shu Xie for their 
assistances in the process of preparing this chapter. We are also grateful to all the 
survey participants in Mainland China and South Korea for sharing their time and 
efforts. 

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Yeping Li 

Department of Teaching, Learning, and Culture 

Texas A&M University, U S. A. 

Yunpeng Ma 

School of Education 

Northeast Normal University, P. R. China 

JeongSuk Pang 

Department of Elementary Education 

Korea National University of Education, South Korea 



62 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 



3. DISCIPLINE KNOWLEDGE PREPARATION FOR 

PROSPECTIVE SECONDARY MATHEMATICS 

TEACHERS 



An East Asian Perspective 



This chapter presents and compares the structure and content of programmes for the 
preparation of prospective secondary mathematics teachers in China and Korea. It 
describes the structure of the programmes, presents samples of the actual programmes, 
and outlines some characteristics of innovative courses. It provides readers with a sense of 
the depth expected in the respective programme elements, and allows consideration of the 
respective perspectives on the discipline preparation expected of teachers in China and 
Korea, and the balance of this discipline preparation with other programme elements. 

INTRODUCTION 

The phenomenon that East Asian students have consistently outperformed their 
counterparts in mathematics in many international comparative studies (Mullis, 
Martin, Gonzalez, & Chrostowski, 2003; OECD, 2003) has aroused policy makers 
and researchers' interests in exploring the underlying reasons (Leung, 2005; Leung 
& Park, 2002; Park & Leung, 2003). In view of the fact that students learn most of 
their mathematics through their teachers, it is reasonable to explore mathematics 
teacher education in these countries. In examining the teacher education system in 
East Asia, it was found that there are diversities in terms of the mechanisms for 
preparing teachers. Some systems provide an integrated approach such as acquiring 
a teacher certificate through a four-year bachelor degree programme through a 
comprehensive university such as in Korea or Chinese Taipei (Taiwan), or through 
a Normal Institute as in Mainland China. While some systems adopt an end-on 
approach such as after completing a bachelor degree and then taking a one- or two- 
year PGCE (Post Graduate Certificate in Education) programme as in Hong Kong 
or Japan. Each model has its own strengths and weaknesses with regards to 
acquiring the subject knowledge and pedagogical knowledge (Leung, 2003; Park, 
2005). Furthermore, based on a close examination of mathematics teacher 
preparation in East Asia from a pedagogical perspective, Park (2005) called for the 
need for a proper balance between discipline knowledge and the pedagogy needed 
to transform it. Some studies indicated there are remarkable differences in 
pedagogical knowledge between Chinese and U.S. mathematics teachers (An, 
Klum, & Wu, 2004; Ma, 1999), and that these differences come from practising 
teachers' learning communities (Ma, 1999; Wang & Paine, 2003). Undoubtedly, 

P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 63-86. 

© 2008 Sense Publishers. All rights reserved. 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 

prospective teacher programmes and mechanisms play a crucial role in teacher 
professional development which is explored in the East Asia teacher education 
system. In this chapter, we provide a detailed examination of prospective 
mathematics teacher education systems and the core curricula of these 
programmes, underlining principles of designing and teaching these courses in 
Mainland China and Korea. Through this lens, we hope to offer an East Asian 
points of view on preparing prospective secondary mathematics teachers. 

THE STRUCTURE OF MATHEMATICS TEACHER EDUCATION PROGRAMMES 

This section outlines the structure of programmes for prospective mathematics 
teachers in China, and then in Korea. 

Mathematics Teacher Education in China 

Since adopting the "nine-year compulsory education system" in 1978, teacher 
education has become a daunting task. Through approximately 20 years of efforts, 
a three-staged process of "normal" education has been established and has made 
significant contribution to educating teachers from elementary to secondary level 
in China. It means: (I) primary school teachers are trained in secondary normal 
schools; (2) junior high school teachers are trained in three-year teacher colleges; 
and, (3) senior high school teachers are trained in four-year teacher colleges and 
normal universities. However, with the rapid development of the economy and 
technology in China, it has been an urgent agenda to upgrade and foster teachers' 
quality. In order to meet this challenge, the Ministry of Education (1998) 
documented an action plan for revitalising education in the 21 st century. Two 
projects were launched; one is referred to as the "Gardener Project" and aims to 
establish continuing teacher education systems for practising teachers. Meanwhile, 
the Ministry of Education (1999) enacted a decision on deepening education 
reform and whole advancing quality education in which comprehensive 
universities and non-normal universities were encouraged to engage in educating 
elementary and secondary teachers. This meant that the privilege of normal 
universities for teacher education changed. The Ministry of Education (2001) put 
forward a process to perfect an open teacher education system based on the 
existing normal universities and supported by other universities, and to integrate 
prospective teacher preparation and practising teachers' professional development. 
Through five years of research, teacher education has shown some changes: 

(1) integration of education of prospective and practising teachers; 

(2) opening of teacher education in all qualified universities rather than just normal 
universities; 

(3) forming a new three-staged teacher education: where primary school teachers 
are trained in the three-year teacher colleges or four-year teacher colleges; the 
junior and senior high school teachers are trained in four-year teacher colleges and 
normal universities, and some of the senior high school teachers are required to 



64 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

attain postgraduate level studies. In order to achieve this feature, the Academic 
Degree Committee of the Ministry of Education set a special degree, called a 
Professional Degree of Master of Education (PDME), for school teachers. 

The candidates are only required to sit for the Joint Examination given by the 
related universities, and do not need to take the National Postgraduate 
Entrance Examination. Their study is 2-4 years duration part-time, however, 
it is necessary to study at university full time for one year accumulatively (23 
credit points) and present a teaching-related dissertation at the end. 
Furthermore, in order to allow some non teaching experienced candidates 
who could be school teachers to obtain PDME, a new model, called 4+2 
(which means four-years undergraduate course and two-years postgraduate 
course) has been successfully utilised by Beijing Normal University since 
2003. (Gu, 2006) 

In 2004, there were more than 400 institutes conducting teacher education 
programmes and around 280 of them were teacher education universities or 
colleges. It was also found that one third of graduates who became teachers were 
from non-teacher education institutes (Yuan, 2004). This proportion has steadily 
increased in recent years. In addition, there is now a flexible and encouraging 
accreditation system for teacher recruitment in China. University degree holders 
who wish to become school teachers and can pass some related examinations, 
usually pedagogy, psychology and subject didactics, in order to be a secondary 
school teacher. 

Mathematics Teacher Education in Korea 

Education of Teachers 

In Korea, there are three kinds of bodies that exist for educating prospective 

teachers. In this section, 'teacher' means 'prospective teacher'. 

Teacher education universities are the typical bodies for educating teachers. 
Every graduate of these is offered the teaching certificate. There are three types of 
universities for teacher education. 

- Universities only for educating elementary school teachers. There are 1 1 such 
national universities. Ewha Woman's University, which is not only for educating 
teachers, but is the unique private university that is for educating elementary 
school teachers. 

- Colleges only for educating secondary teachers. Many universities (national and 
private) have a college for educating secondary teachers. 1 



In fact, there are 31 departments of mathematics education in Korea. Ewha Woman's University 
educates elementary teachers as well as secondary teachers. 



65 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 

- Korea National University of Education (KNUE) is the only university which 
educates teachers for all levels of schools: kindergarten, primary and secondary 
schools. 
There are also some programmes in general universities. The department of 
mathematics of general universities is not for teacher preparation. Usually, 
however, these departments do have some students who want to become teachers. 
A small portion of them are selected. They take the required pedagogical courses 
including field experience. At graduation, they are given the teaching certificate. 

There are also many graduate schools of education. Many are mainly for 
professional development for practising teachers, but some educate prospective 
teachers. The students in these programmes are mathematics majors. They take 
required pedagogical courses including field experience. At graduation, they are 
given the teaching certificate and a Master Degree of education. 

Employment of Teachers 

Any candidate achieving a teaching certificate needs to pass a national 
examination, called Teachers Employment Test (TET), to be a classroom teacher 
of national or public schools. The TET consists of two phases. The first phase is a 
paper examination. The questions on the test can be categorised into three areas: 
mathematics (50%), mathematics education (20%), and general education (30%). If 
an examinee is successful in the first phase, she/he can join the second phase. The 
second phase varies depending on the provincial education bureau. The 
performance and teaching ability of the teacher is the main concern of this phase. If 
successful in the TET, it is possible to enter the classroom as a teacher after a short 
period of orientation organised by each provincial education bureau. The TET has 
serious impact on the curriculum of teacher educating institutes. Park (in 
preparation) gives more detailed information of the TET. 

It has recently been announced that the TET will consist of three phases from 
2009. Through the first phase which will be mainly a paper examination, double 
the necessary number of teachers will be selected. At the second phase, which will 
be mainly comprehensive writing on mathematics and pedagogy, 130% will be 
selected. The final phase will vary depending on the provincial education bureau. 
However, the aptitude and teaching ability will be checked through an in-depth 
interview and field test. 

MATHEMATICS TEACHER EDUCATION PROGRAMMES 

This section outlines the content of programmes for prospective secondary 
mathematics teachers in China and Korea. 

Programmes for Preparing Secondary Mathematics Teachers in China 

Outline of Curriculum Development 

Due to historical reasons, from 1949 Mainland China was modelled after the 



66 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

former Soviet Union (Soviet Russia), including the education system. Thus, 
mathematics teacher education was heavily influenced by Russian mathematics 
philosophy, mathematics curriculum, and pedagogy of mathematics (Zhang & 
Wang, 2000). These influences on the evolution of mathematics education in 
Mainland China are mainly reflected in the following aspects: (1) regarding 
mathematics as an abstract, rigorous and wide application subject as suggested by 
Aleksandrov et al. (1964). Thus, logical deduction and formal mathematical 
operations have been emphasised in mathematics education until now. (2) The aim 
of education is "to transmit the most stable knowledge accumulated over thousands 
of years" to young generations, advocated by Russian educationalist Kailofu 
(1951). This is in line with the Chinese notion of teaching which is "to transmit, 

instruct, and disabuse" (Wt& Hf# , drM-ff jl§ikfi?38fc1fe). Thus, teacher-centred 
and whole classroom lecture has dominated classroom teaching at primary and 
secondary schools in Mainland China. 

(3) Providing a strong advanced mathematics content knowledge and study of 
primary mathematics is necessary for prospective teachers. 

In the 1950s, there were three compulsory advanced mathematics courses 
(called the "Old Three Advances"), advanced algebra, analytical geometry and 
mathematical analysis which are quite difficult for freshmen who have not learnt 
primary calculus at secondary school, particularly in learning the e-S definition of 
limit. However, through learning these three advanced mathematics courses, the 
students were trained in rigorous logical reasoning and mathematical literacy 
which lays a sound foundation of mathematics for their future career as secondary 
mathematics teachers. In the 1960s, the mathematics courses were extended to 
include some pure mathematics courses such as functional analysis, abstract 
algebra, and applied mathematics such as probability and statistics, partial 
differential equations, operation research, and computation methods. Thus, it was a 
key development of mathematics education in normal universities and has shaped 
the direction of further development. However, Mainland China has suffered from 
two political movements "The Great Leap Forward" (1957-1961) and "Cultural 
Revolution" (1966-1977) in which the education system was seriously damaged. 
Through some policies around 1980, the mathematics curriculum returned to that 
in the 1960s. Advanced algebra, analytical geometry and mathematical analysis as 
"old three advanced ones" were recognised as dominant courses, and function 
analysis, abstract algebra and topology were selected as "new three advanced 
mathematics courses". Thus, the mathematics content knowledge was further 
emphasised in the mathematics education in normal universities. In the late 1980s, 
related computer science courses became compulsory, and, in the 1990s, 
mathematical modelling, mathematics experiments and mathematics education 
technology became optional courses in departments of mathematics. Since then, 
modern technology and modern mathematics and mathematics education ideas 
have become important influences on the mathematics education in Mainland 
China. 



67 



LI SHJQI, HUANG RONGJIN, AND SHIN HYUNYONG 

Reformed Curriculum for Mathematics Education 

After the 1990s, with the advancement of mathematical sciences and information 
technology, mathematics education has been challenged in many ways in China. 
Theoretically, with the rapid development of information technology, 'doing' 
mathematics is no longer a business of using "one paper, one pencil and a brain" 
and teaching mathematics is not a matter of using chalk and board. It is 
mathematical modelling, mathematics experiments, and mathematical education 
technology which have come to the forefront of mathematics education. 
Mathematics is not only a tool for training the mind and expressing other subjects, 
but also comes to the fore by producing direct economic effect. With the 
integration of mathematical theory and mathematical technology, the functions of 
mathematics have been extended widely. Thus the ideas of mathematics education 
have to be changed from a traditional and static notion to a constructive and 
dynamic one. In addition, with the expansion of recruitment of students at 
universities during the 1990s, and the diversity of employment opportunities for 
mathematics graduates, the quality of the students in mathematics departments has 
unavoidably fallen. Meanwhile, the curricula at secondary schools were condensed 
and adjusted, some advanced mathematics such as calculus, probability and 
statistics, vectors and matrices, and mathematical modelling were included in the 
secondary mathematics curricula. Both internal and external factors have forced 
normal universities to undertake certain reforms to meet the challenges. Through a 
four-year study, a report was written on the curriculum reform in mathematics 
departments at normal universities (Zhang & Wang, 2000). The report claimed that 
the aims of mathematics education at normal universities were basically to train 
prospective mathematics teachers and to prepare students for further research or 
quantitative fields. As a consequence, it is necessary to focus on mathematical 
content knowledge in the department of mathematics. In addition, when 
emphasising mathematics subject matter knowledge, it is also important to keep 
one eye on studying primary mathematics with updating ideas. Moreover, it is 
more important to update the ideas on mathematics and mathematics education. 
Zhang and Wang (2000) suggest that mathematics curriculum at the department of 
mathematics should follow the following principles: (a) less and more refined 
foundation courses, (b) broader and concise specialisation courses, (c) multiple 
optional courses, and (d) high quality mathematics education courses. The 
curriculum schedules in the department of mathematics at different normal 
universities are diverse. The following Tables 1 , 2, 3 and 4 outline programmes for 
mathematics major undergraduate students at East China Normal University: 

Table I. Required courses 
Required Courses. Sub-total: 83 credit points (cps) 



Analytic geometry and higher algebra 


10 


C- Language 


3 


Calculus and mathematical analysis 


14.5 


General physical 


6 


Ordinary differential equation 


3 


General psychology 


2 


Partial differential equation 


3 


Pedagogy 


3 



68 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 



Probability and statistics 
Abstract algebra 
Differential geometry 
Complex function theory 
Mathematical experiment 
Mathematical modelling 



4 


Mathematics didactics 


3 


3 


Educational technology 


2 


3 


Teacher's spoken language 


1 


3 


Teaching practice 


6 


2.5 
3 


Thesis 


8 



There is a trend to provide students with more flexible choices in courses and 
specialisations. For example, at East China Normal University, according to 
students' interests and requirements, they are streamed into three optional majors: 
mathematics education, foundation mathematics, and applied mathematics. The 
following outlines some electives: 

Table 2. Elective courses with some limitations 



One series from two. Sub-total needed: 1 1 cps 
Pure mathematics series 



Applied mathematics series 



Higher algebra in depth 2 
Real function theory 3 

Functional analysis 3 

Topology 3_ 



Linear programming 
Computational approach 
Statistical approach 
Control theory 



2.5 
3.5 
2 
3 



Table 3. Elective courses with some limitations 



Three from six. Sub-total needed: 6 cps 



Ordinary differential equation in depth 

Classical geometry 

Galois theory (Abstract algebra in depth) 



2 Introduction to mathematics education 2 
2 Assessment of mathematics education 2.5 
2.5 Elementary mathematics in depth 2.5 



Table 4. Free elective courses 



Elective courses. Sub-total needed: 
Mathematics education series 



: 20 cps 

Pure mathematics series 



Computational & applied series 



Modern mathematics and 
school mathematics 
Elementary number theory 
Mathematics curriculum 
theory 

Philosophy and history of 
mathematics 
Problem solving and 
mathematical competition 
GSP and course ware 
Methodology of 
mathematics 



Overview of modern 
mathematics 
Calculus on manifold 

Algebraic curves 

Complex function theory 
in depth 

Modular theory and 
representation of group 
Lie algebra 



Numerical solution of 
differential equation 
Finite element method 

Matrix theory 

Applied partial differential 
equation 
Environmental 
mathematics modelling 
Biomathematics 



There are other series in mathematics and other courses in general 



The preparation of prospective mathematics teachers (i.e., in the department of 
mathematics at normal universities) exhibits the following characteristics: 



69 



LI SHIQ1, HUANG RONGJIN, AND SHIN HYUNYONG 

(a) providing prospective teachers with a foundation in profound mathematics 
knowledge and high advanced mathematics literacy; 

(b) emphasising review and study of primary mathematics. It was believed that a 
profound understanding of primary mathematics and strong ability of solving 
problems in primary mathematics were crucial to being a qualified 
mathematics teacher at secondary schools. Due to the tradition of examination- 
oriented teaching, a high level of problem solving ability is necessary for a 
qualified teacher; 

(c) teaching practicum is limited. A six-week teaching practicum can only provide 
prospective teachers with a preliminary experience of teaching in secondary 
schools. 

This reflects a belief that a solid mathematics base is vital for teacher 
preparation. Furthermore, higher mathematics courses are taken as a priority and 
privilege since prospective teachers will have less chance to learn them in their 
career lives. It is a main aim to foster prospective teachers with a bird's eye view of 
understanding elementary mathematics deeply rather than immediately connecting 
to what they will teach in schools, though there are special courses such as Modern 
Mathematics and School Mathematics, and Elementary Mathematics in Depth 
which connect higher mathematics to elementary mathematics. In contrast with the 
rigid requirement of mathematics, it is hoped that graduates learn teaching skills 
from their practical teaching when they become teachers. In fact, their professional 
development is strongly supported by a well-organised teacher continuing 
education (Huang, 2006). It is worth mentioning that every secondary school year 
in China has two semesters and each has 18 weeks for teaching and two for 
examinations (a 2 credit-point course means 18 periods of 90 minute lessons), 
much longer than most countries. Longer semesters provide prospective teachers 
sufficient time to acquire enough mathematical and pedagogical knowledge and to 
understand them in depth. 

Programmes for Preparing Secondary Mathematics Teachers in Korea 

Number theory, linear algebra, abstract algebra, analysis, complex analysis, 
differential geometry, topology, probability and statistics, discrete mathematics, 
and mathematics education are the compulsory courses for prospective teachers in 
Korea. Usually, the teacher education departments also offer more advanced 
courses for the ones mentioned above as well as some independent courses such as 
set theory and the history of mathematics. Even though the contents of the courses 
are partially influenced by the TET, they depend on the tradition and the faculty 
members of each institute. Each institute for teacher education has its own 
curriculum and content. 

In Korea, there are two types of professors at teacher education departments. 
Some are specialists of mathematics, and others are specialists of mathematics 
education. Usually each group has not much interest in the others' work. In other 



70 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

words, mathematicians only teach mathematics, and education specialists only 
teach education related topics without much interest in mathematics at university 
level. It is the students' duty to digest these two areas of knowledge to come up 
with the useful knowledge which will be needed for their future careers as 
mathematics teachers. However, it has been recently pointed out that this paradigm 
of teacher education is not effective. In fact, in 2005, the Korea Research 
Foundation launched a project to develop curriculum and teaching/learning 
materials for the departments of mathematics education at the various teacher 
education universities. Some products include: 

(1) suggestions for developing curriculum for mathematics departments of teacher 
education universities; 

(2) extended syllabi of 26 key courses for mathematics department of teacher 
education universities. Each syllabus consists of approximately 100 pages of 
description; 

(3) a list of approximately 300 non-professional books which are recommended 
for prospective or practising teachers of mathematics; 

(4) a list of approximately 200 web-sites which are recommended for prospective 
or practising teachers of mathematics. 

In particular, the syllabi have been developed according to the following 
suggestions which are proposed in the final report of the project: 

(1) To help students to understand the foundation of mathematics. 

(2) To help students know and utilise the effectiveness and usefulness of 
mathematics and eventually use and enjoy mathematics in real life. 

(3) To help students integrate all branches of mathematics and to connect each 
course with school mathematics. 

(4) To stimulate the intellectual interest of students to induce self investigation. 

(5) To incorporate technology in some courses. 

(6) To show a teaching programme (or model). 

The following is an example of the proposed course schedules for the 
department for educating secondary school mathematics teachers. The number of 
credits of all courses is three. 

Table 5. Proposed course schedules 

Year Courses for the First Semester Courses for the Second Semester 

. Calculus I Calculus II 

Set Theory Discrete Mathematics 

Linear Algebra and Its Applications Number Theory and Its Applications 

2 Geometry I Geometry II 
Analysis I Analysis II 

Abstract Algebra I £!£.**££ 

Differential Equations 

3 General Topology 



Complex Analysis 
Differential Geometry 
Applied Mathematics 



Probability and Statistics 
Foundations of Mathema 

Algebra for School Mathematics Analysis for School Mathematics 



c j.- c vi »l. •■ rJ .- Teaching and Learning of Mathematics 

Foundations of Mathematics Education t--ut- i 

Field Experience I 



71 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 



Geometry for School Mathematics Probability and Statistics for School 

History of Mathematics for School Teachers Mathematics 

Problem Solving in Mathematics Philosophy of Mathematics for School 

Field Experience II Teachers 



School Mathematics and Modem Mathematics 
Assessment in Mathematics 



The courses for seniors are designed to actively incorporate pedagogy into 
mathematics. However, this proposal has not yet been efFectively implemented in 
many departments. For this proposal to be successfully implemented it would need 
to be recommended that it be connected with the TET. 

CHARACTERISTICS OF INNOVATIVE COURSES 

To illustrate the dynamic nature of course development, the following sections 
presents examples of some innovative course in both China and Korea. 

Selected Innovative Courses in China 

In order to meet the reform of mathematics curriculum for preparing secondary 
mathematics teachers, some organisations are editing relevant textbooks to meet 
the special needs of mathematics school teachers. For example, the Higher 
Education Press has published a series of textbooks for mathematics teacher 
education, such as Mathematical Analysis, Higher Algebra and Analytical 
Geometry, Theory of Real Variable Function, Theory of Complex Variable 
Function, Introduction to Mathematics Education, Research on Secondary School 
Algebra, Research on Secondary School Geometry, Research on Secondary School 
Probability and Statistics, and Mathematics Education Technology etc. The 
following will describe and discuss some specific courses in terms of their design 
principles, organisation and features to demonstrate the characteristics of these 
reform-oriented courses at East China Normal University. 

Course 1 . Higher A Igebra and A nalytical Geometry 

Analytical geometry and higher algebra are two courses for freshmen at the 
department of mathematics at teacher-training universities and colleges in China. 
The analytical geometry course taught at universities mainly consists of three- 
dimensional topics while the higher algebra course usually includes two parts: 
linear algebra and polynomial theories. 

Since the late 1990s, some professors at the East China Normal University have 
developed a new course titled Higher Algebra and Analytical Geometry, which 
integrated the two traditional courses of Higher Algebra and Analytical Geometry 
(Chen, 2000). The rationale of designing the new course is that linear algebra, as 
the main content of higher algebra, has a profound geometric underpinning, while 
analytical geometry aims to study geometry problems with algebraic methods. As a 
result, the two courses are internally connected in terms of the mathematical ideas 
and methods. In an interview, Professor Chen Zhijie, an algebraist, the chief 

72 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

designer of the courses, presented the following information about the background 
and some unique features of the newly designed course. 

Historically, the development of algebra and geometry are tightly related and 
mutually supported side by side. The relationship between these two areas can be 
described as algebra providing the research methods for geometry while geometry 
providing visual representations of algebra. The latter one is more important than 
the former for future teachers. It was suggested that looking for a visual 
background from the perspective of geometry is helpful for students to understand 
abstract algebraic concepts. Many examples demonstrate that if an algebraic 
abstract concept can be visually presented by geometrical representations, students 
can get a deeper understanding of the concept and be more motivated of the study 
in general. Such a new course is intended to foster students' abilities of analysing 
and solving problems in integrated ways with algebra and geometry. 

One of the considerations of the integrated course is that, in practical teaching, 
there are some difficulties in teaching higher algebra and analytic geometry 
separately. Usually, the two courses are taught in parallel in the first semester at the 
department of mathematics. There are many overlapping related concepts between 
these two courses. Sometimes, the instructors of analytic geometry had to mention 
some new concepts, which would be formally taught later in a higher algebra 
course. Such a paradox may be due to the traditional artificial split of these 
courses. 

Based on this understanding, the attempt to integrate these two courses has relied 
on the geometric meaning of algebraic concepts. For example, in order to help 
students get a visual perception of a vector in concrete vector space, the 3- 
dimensional vector algebra is introduced first, and then, the concept of n-dimension 
vector space is introduced subsequently. Thus, students could be successful in 
learning n-dimensions of vector spaces based on the learning of 3 dimensions of 
vector space. 

Another example is the concept of the function of the determinant is introduced 
through calculating the directed volume of a parallel polyhedron which is formed 
by n vectors. This not only allows students to have an impressive perception of the 
geometric meaning of determinant, but also makes the definition of determinant as 
the alternating sum of the products of terms more obvious. This treatment provides 
the determinant of the product of matrices with a geometric explanation and further 
makes the determinant of matrix product equal to the product of determinants of 
matrices more visible 

In the same way, the concept of linear mapping is introduced before the 
definition of the matrix operations. Based on the one to one correspondent 
relationship between linear mapping and the matrix operations, operations of linear 
mapping can be used for defining relevant operations of matrices. So the properties 
of matrix operations and the properties of linear mapping can be justified at the 
same time, only one of them needs to be proved carefully. Thus the geometric 
meaning of matrices is manifest implicitly. 

In the process of curriculum reform, the problems of reselecting and 
reorganising the topics need to be considered. After some necessary adaptation, 



73 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 

most of content of linear algebra was maintained in the new course. However, the 
content of analytical geometry was tailored, and reshaped to a certain extent. Since 
spatial visualisation is vital for students' mathematical development in general, and 
for multivariate calculus in particular, the methods of drawing spatial graphs, the 
intersection of curved surfaces and its projection on the coordinate plane etc. are 
kept, though they may not be tightly related to some content in linear algebra. The 
mechanical proving of geometry theorems which is built on the theories of 
polynomials of several indeterminates should be introduced. 

It is important for prospective mathematics teachers to learn that the truth of 
plane geometry proposition could be proved by programming methods which can 
be realised with the help of some specific software. In order to discuss mechanical 
proof of the geometry theorem, some traditional topics in both analytic geometry 
and higher algebra were condensed, or deleted. 
The main content of this new course is as follows: 

Chapter 1 Vector algebra 

Chapter 2 Determinant 

Chapter 3 Linear equation system and linear subspace 

Chapter 4 Rank of matrix and the operations of matrices 

Chapter 5 Linear space and Euclidean space 

Chapter 6 Common curved surfaces in geometry space 

Chapter 7 Linear transformation 

Chapter 8 Functions of linear space 

Chapter 9 Coordinate transformation and point transformation 

Chapter 10 Polynomial of one indeterminate and integer factoring 

Chapter 1 1 Polynomial of several indeterminates 

Chapter 1 2 Polynomial matrix and Jordan canonical form 

Chapter 13 Discussion and application of Jordan canonical form 

Course 2. Mathematical Analysis 

Mathematical analysis is always regarded as one of the most important courses for 
mathematics majors at university. This course lasts three semesters (one and a half 
years), six periods in each week including four periods of formal classes and two 
periods of problem-solving classes. The course is thought to be of more importance 
because it includes many important mathematical concepts which form the basis 
for learning other courses of the mathematics major. Moreover, it provides students 
with a valuable opportunity to develop their conceptual thinking and rigorous 
deductive reasoning ability. As a result, the course maintains a higher demand for 
students' learning. For example, most of the important concepts are introduced 
systematically, and nearly all of the theorems are proved formally and rigorously. 
Higher cognitive level problems are assigned to students as homework. These 
requirements may imply the basic belief of mathematicians that mathematical 
analysis has a long history so that it is a classical and well-developed course. It is 
unnecessary to reform it substantially. So the course still keeps most of the content, 
as well as the structure and features, over the last 25 years. Only a few parts are 



74 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

modified. 

However, based on many years of teaching experience, nearly all instructors of 
the course have realised that it is not easy for freshmen students to master so much 
new knowledge, especially in their first semester in which calculus with one 
variable is taught. In order to help students learn this content easily and 
meaningfully, some examination of the content has been done, but the rigorous 
demands of the relevant content are still maintained. The topic of real number 
continuity in calculus is provided as a typical example. 

There is a group of six theorems in the topic of real number, including the 
Heine-Borel theorem, the rest of interval theorem, Cauchy criterion for 
convergence, monotone convergence theorem etc., as well as the proofs of 
equivalence. From the perspective of mathematicians, the topic about completeness 
theory of real number is a core in calculus, especially the related proofs. Although 
there are difficulties for students to understand these theorems and proofs, it is 
certain that they are crucial for mathematics majors. In fact, this topic is waived at 
some universities while it is taught in a traditional way at others. How to cope with 
such a topic was an issue at ECNU more than 20 years ago, and the reform has 
been conducted carefully. Now the teaching of some topics is treated in a special 
way to reduce the difficulties but the course still keeps the essential content 
consistent. 

Traditionally, this topic was introduced just after the topics of the limit of series 
and function. Now, it is delayed and taught after all the chapters of differential 
theories. In the new textbook, the six theorems and the proofs of equivalence are 
still formally introduced. Instructors may teach them in several classes and one or 
two of the proofs remain as students' home work. Therefore, even though some 
students who are not able to fully understand and master all of the proofs, at least 
they will know a set of theorems and the rigorous equivalent proofs between them. 
It is important for prospective teachers to know and believe that any mathematics 
propositions have to be proved mathematically, which may influence their teaching 
in schools. 

Course 3. History of mathematics and its implications for mathematics education 
Since the late 1980s, the history of mathematics is an elective course for 
mathematics education programmes in many normal universities. The teaching 
content was selected by the teachers who teach the course because there was no 
unified syllabus and textbook available. The majority of the universities used the 
teaching materials they developed themselves, while there were several institutes 
adopting the Chinese version of An Introduction to the History of Mathematics by 
Eves (1986). From the 1990s, more and more normal universities have adopted a 
textbook, entitled A History of Mathematics by Wenlin Li (1999). Traditionally, 
the teaching of the history of mathematics aims to introduce the history without 
paying attention to its connection with pedagogy of mathematics at secondary 
school level. Thus, the usefulness of this course is not so satisfactory for teacher 
education. In order to improve the appropriateness of mathematics history at 
normal universities, by adopting research findings of history and pedagogy of 



75 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 

mathematics (e.g., Fauvel & Mannen, 2000), some major adjustments of 
mathematics history courses have been made recently, and following aspects are 
emphasised: 

1. Through many historical cases, the importance of the history of mathematics in 
cultivating students' attitudes and values is discussed. 

2. The history of mathematics as instructional resources, and relevant historical 
topics in high school mathematics are described, such as the concept of function, 
trigonometric formulas, arithmetical and geometrical sequences, the binomial 
theorem, complex numbers, mathematical induction, the volume of the sphere, 
conic sections, classic probability, etc. 

3. Teaching designs based upon the history of mathematics, directly or indirectly, 
are introduced, such as teaching of linear equations, quadratic equations, system 
of linear equations, application of similar triangles, complex numbers, 
trigonometric formulae (e.g. Wang, 2006, 2007; Zhang, 2007) and so on. 

4. More attention is paid to historical parallelism (or the historical-genetic 
principle), which implies that an individual's mathematical understanding 
processes can be illuminated by the historical developments of mathematical 
ideas. Many famous mathematicians have paid keen attention to the importance 
of this issue since the end of the last century and have conducted empirical 
studies on the historical parallelism, i.e., symbolic algebra (Harper, 1987), actual 
infinity (Moreno & Waldegg, 1991), the concept of plane (Zormbala & 
Tzanakis, 2001), the concept of angle (Keiser, 2004), the limit of functions 
(Juter, 2006). Further empirical studies on relevant concepts such as imaginary 
numbers, infinite set, functions, tangent of curve (e.g., Wang et al., 2005, 2006) 
are described in the course. 

5. There are II historical topics in Optional Course 3 in the new curriculum at 
senior high school. Some of them are: Ancient Greek mathematics; Ancient 
Chinese mathematical treasures; Development of Calculus - epoch-marking 
mathematical achievements; puzzle problems for thousands of years - solution 
of Galois; Cantor's set theory; Development of random and probability thinking; 
Development of modern mathematics in China, etc. Thus, this course values both 
the history of mathematics and its implications for mathematics teaching at 
secondary schools through historical case studies for the new school curriculum 
reform. 
Below are two cases as examples. 

Case J: Linear equations with one unknown. In the section "Teaching designs 
based upon the history of mathematics", the concept of linear equations with one 
unknown was introduced through solving relevant historical problems classified 
under the following types: arithmetic operation problems; cooperation problems; 
journey problems; fixed sum problems; and remainder problems. Through solving 
these problems in turn, the development of the concept of linear equations with one 
unknown was reconstructed. Two examples of these problems are as follows: 

Example 1.(1) Alice, Bonnie, and Jessica can make 300, 250 and 200 bricks per day 
respectively. How long will it take for them working together to make 1500 bricks? 



76 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

(2) If Alice works one day, and Bonnie and Jessica work together with her, how long 
will it take to finish this task? Let the number of days be x, establish the equation 
with x respectively (Cooperation problem, adapted from The Greek Anthology 
(Paton, 1979). 

Example 2. ( 1) It takes 5 days for the first ship to sail from place A to B and 7 days for the 
second ship to sail from place B to A. If the two ships start the voyage from A and B 
respectively at the same time, in how many days will they meet each other? (2) If the 
second ship has sailed for two days before the first ship starts its voyage, in how 
many days will they meet each other? Let the number of days be x, create the 
equation with respect to x. (Journey problem, adapted from the Nine Chapters on the 
Mathematical Art [Guo, 2004] and Liber Abaci Siegler, 2002). 

Based on the previous problems, and historical origins, the concept of linear 
equations with one unknown was introduced. The main aim of this design is to help 
students experience the process of creating linear equation with one unknown to 
solve problems with situations from daily life, and to introduce the concept of 
linear equations. On the one hand, through solving relevant historical problems, 
students realise the necessity of learning the concept and stimulate the motivation 
of learning; on the other hand, through re-constructing the historical sequences, the 
learning is based on the students' knowledge preparation, and cognitive 
appropriateness, from easy to more difficult. 

Case 2. Introduction of imaginary numbers. In the same section of "Teaching 
designs based upon the history of mathematics", the history of the development of 
complex numbers is introduced. Usually, the concept of the imaginary number is 
introduced through solving the equation x 2 + 1 = , but it does not motivate 
students and convince them of the necessity of learning the topic. Therefore, the 
Leibniz problem (McClenon, 1923) is used to introduce this topic as follows: 
Thus far, students have explored the real number system. Then, are there any 
numbers different from real numbers? Please explore the following problem: 

Given x 2 + y 2 = 2 ■ xy = 2 , find: ( 1 ) x+ y = ? ( 2 ) the values of x and y. 

We can find the values of x-\ y (±-j6), but we cannot find the real value of x and 
y. The existence of x-\ y assures us that x and y must exist, but they are not real 

numbers. What are they? Let us take a closer look at the nature of x and y. Based 
on the equation x + y = j6 and xy = 2 , we can regard x and y as two roots of a 
quadratic equation with one unknown. But there is no real root with regard to this 
equation. It is necessary to introduce a new type of number in order to solve this 
problem. The key to solving this problem is to express the roots when the 
discriminate is negative. Therefore, the necessity of introducing the imaginary 
numbers has been dealt with properly. 

Course 4. Modern Mathematics and School Mathematics 

From the late 1980s, many teacher education universities in China have developed 

a series of special courses for prospective mathematics teachers. The aim of these 



77 



LI SH1QI, HUANG RONGJIN, AND SHIN HYUNYONG 

courses is to strengthen the connection between higher mathematics students learnt 
at university and the mathematics they will teach at school. Generally, learning 
higher mathematics will give an overall view of elementary mathematics. 
However, mathematics teacher educators believe that it is foundational and more 
helpful for prospective teachers to make some implicit connections between higher 
mathematics and elementary mathematics. For example, the underlying differences 
and conceptual development relationship between finite additivity of the concepts 
of length/area in elementary geometry and denumerably additivity of the concept 
of measurement in real function theory. Making sense of accurate and detailed 
connections will promote future teachers to understand allowing them to teach 
elementary mathematics with understanding and flexibly. 

At the Department of Mathematics at ECNU, a course titled Modern 
Mathematics and School Mathematics has been delivered for more than 15 years. 
Though it is an optional course, the majority of students who want to become 
mathematics teachers have taken this course. 
The main content of the course is: 

- Set theory as mathematics language: A brief history of set theory, its language, 
open sentence and quantifiers, power set concepts, calculation in sets, and the Z- 
F Axioms in set theory 

- Relations and functions: The Cartesian product, relations abstracted from real 
life, the representations of relations, equivalent relations and functions as a 
special relation 

- Mapping and its applications: The mapping and drawer principle, permutations, 
combinations and mapping, the computation formula of the number of 
surjections, the perspectives of mapping as scientific method 

- Quotient set and congruence: Models of quotient sets, algebraic calculations, 
problem solving with remainders and language 

- Mathematical induction: The Peano Axiom and mathematical induction 
principle, ordered set and well-ordered set, the equivalence between induction 
axiom and well-ordered axiom 

- Number systems: The principle of the extension of number systems, the system 
of N, Z, Q, R, C, 4-tuple and 8-tuple numbers etc. 

- Polynomial ring and factoring: The ideal subring, highest common factor, 
Euclidean algorithm, and prime factoring 

- The three great problems in geometric construction: The criterion of 
construction with a ruler and compass, extension field, and the three great 
problems in geometric construction 

- Set algebra and prepositional calculus: The shortcomings of traditional logic, 
algebra and set algebra, proposition and compound of proposition, qualifier and 
predicate 

- Vector geometry and mechanical proving in elementary geometry: Vector space, 
creating a geometric model, the algebraic definition of an angle, and brief 
introduction of mechanical proving, 

- Geometry foundations and geometric models: Euclidean elements, Hilbert 
axiomatic theory and geometry, hyperbolic geometry and elliptic geometry, 



78 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

Erlangen Programme, transformation groups and geometry, projective geometry 
and its applications in school mathematics 

- Length and area: Length axiom and area axiom, the area of a triangle and 
convex polygons, and measurement 

- Topology in brief: Local and global properties, topology space, Brouwer fixed 
point theorem, definition of dimension and fractal dimensions. 

This course introduces many different kinds of relationships between higher and 
elementary mathematics. Most of the content discusses the relationships between 
relevant concepts, such as the quotient set and congruence, ideal and common 
factors, mapping, relations and functions, extension field and three great problems 
in geometric construction, Peano Axiom and the mathematical induction principle, 
set algebra and propositional calculus, etc. When teaching related topics or 
concepts in secondary schools, teachers may call on their backgrounds from 
advanced mathematics perspectives and use such knowledge to teach students with 
profound understanding. Other kinds of relationships may not be so obvious and 
direct, but they may imply certain strategies or thinking skills, for example, 
relationships between vector geometry and mechanical proving, hyperbolic/elliptic 
geometry and Euclidean geometry, fractal dimension and plane/solid geometry etc. 
Realising these kinds of relationships may broaden prospective teachers' visions 
and also foster their mathematics thinking capabilities and problem solving skills. 

Selected Innovations in Korea 

The following outline some innovations in the Korean system. 

/. Directions of textbook development 

The general direction of developing textbooks is to present mathematics as a story 
combined with pedagogy. Based on the proposed curriculum, the Korea Society of 
Mathematical Education has decided to publish a series of textbooks for 
mathematics teachers. 

(I) Textbooks for mathematical courses 
The following four (Vol.1 through Vol. 4) books were published. 



Abstract Algebra for Secondary School teachers 

Mathematics Education at Kindergarten. 

Algebra for Elementary School Teachers. The main portion of elementary 

school mathematics is algebra. An independent course of algebra is required 

for all elementary prospective teachers. This is a proposed textbook for the 

course. 

Set Theory for School teachers. This book can be used at universities for 

educating secondary school mathematics teachers as well as at universities for 

educating elementary school mathematics teachers. 



79 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 

The following four books (Vol. 5 through Vol. 8) are scheduled to appear in March 

of 2008. 

5: Real Analysis for Secondary School Teachers 

6: Number Theory for Secondary School Teachers. Cryptographic content is 

included. Secure key distribution, coin flipping by telephone, and digital 

signature are some of the examples which are covered during the course. 
7: Mathematics for Elementary School Teachers. This book is being written by 

two authors, a mathematician and an education specialist. 
8: Teaching Materials for Secondary School Teachers. The scheduled content of 

this book is: the value of mathematics, the beauty of mathematics, the power 

of mathematics, probability, symmetry, dimension, and modesty of 

mathematics. 
The following two books are scheduled to be developed during 2008. 
9: Linear Algebra for Secondary School Teachers. Coding Theory is included. 

Syndrome decoding schemes and cyclic codes are some of the examples which 

are covered during the course. 
10: Physics for Mathematics Teachers: The role of mathematics in 

electromagnetic theory, relativity theory and quantum mechanics is briefly 

explained. The principal equations (Maxwell's equations, for example) in each 

theory are the main topics. 
The following books and more will be developed in the future.. 
1 1 : Geometry for Secondary School Teachers 
12: Probability and Statistics for Secondary School Teachers 
1 3: Topology for Secondary School Teachers 
14: History and Philosophy of Mathematics - For School Mathematics Teachers 

(2) Textbooks for pedagogical courses 
Books on mathematics curriculum, psychology, and teaching methods respectively, 
will be developed. 

2. A detailed description 
The text Abstract Algebra for Secondary School Teachers is used as an example. 
This book is used for two semesters (Abstract Algebra I and Abstract Algebra II). 
It consists of two parts, and each part is for a semester course. Each course consists 
of 45 class hours, and each class hour is for 50 minutes duration. The first part is 
on algebraic structures. Three structures of group, ring, and vector space are mainly 
discussed. The second part is on the applications of algebraic structures. The field 
of constructible numbers, the Galois Theory, and the construction of regular 
polygons are the main themes. The main content is: 

(1) The course is focused on 3 algebraic structures: group, ring, and vector space. 
The basic examples are Z, Q, R 2 , R } , Mat 2 (R) , Q[x], R[x]. The definitions, 

examples, and basic properties of these algebraic structures are introduced in 
parallel. The parallel discussion which is one of the characteristics of the book 
is to help prospective teachers to have an overall understanding of algebraic 
structures. 

(2) This course is neither for group theory nor for ring theory. Understanding 



80 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

important algebraic structures through basic examples and school mathematics 
is the main purpose. 

(3) Substructures (subgroup, subring, and subspace), quotient structures (with 
normal subgroup and ideal), and direct products (sums) are introduced in 
parallel. Some concepts (e.g., subset, partition by an equivalence relation, 
Cartesian product, nZ, Zn, and matrix and determinant) in set theory, number 
theory, and linear algebra are mentioned. 

(4) The homomorphism and linear mapping are introduced in parallel. Various 
functions (e.g., function in set theory and continuous functions in calculus) are 
mentioned. 

(5) Lagrange Theorem is proved. Students compare the theorem with the 
following facts; 

In set theory, |A'UK|=|A'|+ |K|- \Xp\Y\. 

In linear algebra, dim(A r + Y) = dimA r + dimT- d\m(Xr\Y). 

In ring (field) theory, [L:F]=[L: K][K : F] for F<K<L . 

(6) The basic properties of maximal ideals are proved and some finite fields (of 

order 4, 8, 9) are constructed. The prime fields Zp, p: prime, are mentioned. 

(7) Some topics which need number theory, group theory, and ring theory are 
discussed. 

(8) Field extension and constructive numbers are studied briefly. The co-work of 

algebra and geometry is explained. 

(9) Some students will take the advanced course (Abstract Algebra II). The course 

mainly deals with Galois Theory. To be ready for the course the students will 
be given some homework. 

Some features of the book are the following. It is pointed out that commutativity, 
associativity, and distributive property are deeply connected with finiteness and 
convergence. The following examples are presented. 

(1) Check the associativity and commutativity in the series: 

1. i+i. 1+...+ 1- i+... 

(2) Discuss the following argument: 

.. 1+2 + --+H 

hm 5 

"->- n 

= lim(— + — + ■•■ + — ) 
"-*- n n n 

,. I ,. 2 ,. n 

= lim— -+ hm — + •••+ hm— - 

n-+°° n n ~* aa n n ~* oa n 

= 0. 

(3) Discuss the following argument: 

(i) If we let x= 0.999..., then 10jc= 9.999..., so 9x= 9, thus x= 1. 

This book also discusses the co-work of algebra and geometry through the 

81 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 

following topics. 

(1) Some examples in analytic geometry. 

(2) Erlangen Programme by Felix Klein. 

(3) A proper picture explaining the equality: (a+ b) 2 = a 1 + 2ab + b 1 

(4) Construction of numbers. 

(5) Proper pictures explaining the following equalities: 

,111 

1 + - + — + — + --- = 2> 
2 2 2 2' 

,111 4 

1 + - + -^ + — - + ••• = -• 
4 4 2 4' 3 

This book proposes various types of problems. Essays, performance assessment, 
and group projects are some of the examples. Some students may want to study 
abstract algebra more deeply. This book is not deep and broad enough. Some 
additional mathematical books are recommended. 

3. Course management 

Each course has been designed to provide the necessary pedagogical content 
knowledge to prospective teachers. It seems to be inevitable for each course to be 
managed tightly. It seems to be helpful for some supplementary books to be 
recommended during the course. The following books (translated into Korean) are 
helpful for the course. 

• Fermat's Last Theorem (S. Singh 1 997). 

• The Man Who Loved Only Numbers (P. Hoffman 1998). 

• My Brain Is Open (B. Schechter 1998). 

• Knowing and Teaching Elementary Mathematics (L. Ma 1999). 

• Mystery of Aleph (A. D. Aczel 2000). 

• The Elegant Universe (B. Greene 1999). 

• Beautiful Mind (S. Nasar 1 998). 

• In Code (S. Flannery & D. Flannery 2000). 

• Flatland (E. Abbott, Annotated by I. Stewart 2002). 

Some students might read some of the above books as they take the previous 
courses. This course is mainly offered through mathematical lectures. It is 
recommended to give three or four lectures through PowerPoint which are 
specially designed for providing pedagogical knowledge. A special web site is 
devoted to this course. Various materials for further readings or in-depth study 
are provided here. 

CONCLUSION 

From the aforementioned descriptions and analysis, we can summarise that there 
are similarities and differences between the cases in China and Korea with respect 
to the programmes offered for prospective teachers, as well as the mechanism of 



82 



MATHEMATICS DISCIPLINARY KNOWLEDGE FOR SECONDARY TEACHING 

educating and recruiting mathematics teachers. On the basis of the curriculum 
framework in China and Korea, both traditional and innovative programmes 
emphasise the foundation of mathematics subject knowledge in terms of its 
systematic structure, and demand for logical reasoning. These features are echoed 
in other studies. For example, the Glenn commission (U.S. Department of 
Education, 2000) put forwarded that "High quality teaching requires that teachers 
have a deep knowledge of subject matter". But, the ways to reflect such a belief in 
practice depend on the specific context found in different countries. 

In the reform-oriented mathematics courses in the case of China, it seems that 
much of the endeavour taken was to make difficult advanced mathematics 
knowledge easier for prospective teachers to understand (e.g., the case of Higher 
Algebra and Analytic Geometry, and Mathematical Analysis), but the original aim 
is to maintain the difficulty level of the mathematics content. These mathematics 
courses pay less attention to exploring the implications for teaching secondary 
school mathematics; however, there are other courses offered students as a 
compensation or remedy, such as Modern Mathematics and School Mathematics, 
Research on Secondary School Algebra, Research on Secondary School Geometry, 
Research on Secondary School Probability and Statistics, which specifically deal 
with the connection between higher and elementary mathematics. The course on 
the history of mathematics plays an additional role to help future teachers deal with 
elementary mathematics topics from a mathematical historical perspective. 

The innovative mathematics courses in the case of Korea are integrated and 
devote much attention to both: internal connections between higher mathematics 
and relevant elementary mathematics, and connections between mathematics and 
pedagogy of teaching relevant content. Given that teacher preparation programmes 
tend to work in a zero-sum game environment where additional preparation in 
mathematics results in decreased preparation in some other areas such as pedagogy 
(Jermey et al., 2003), Korean innovative courses attempt to adopt an integrated 
approach to connect subject knowledge and pedagogy pertaining to the learning 
and teaching of specific mathematics content. This effort is well supported by 
Kant's words, "pedagogy without mathematics is empty, mathematics without 
pedagogy is blind" (cited from Park, 2005). Thus, the mathematics content and 
pedagogy should be emphasised properly. 

The practice in China may imply that mathematicians there still value the 
mathematical structure and nature of mathematics subject matter, and hope to 
provide students a refined and profound mathematics foundation, with a broad and 
concise mathematics background, and further try to help students to master 
mathematics more easily and properly. At the same time, they leave the 
responsibility of connecting higher mathematics to elementary mathematics and the 
responsibility of providing high quality mathematics pedagogical knowledge for 
mathematics educators. However, mathematicians in Korea seem to realise it is 
their responsibility to make the connection between higher and elementary 
mathematics explicit and teach subject content knowledge for teaching school 
mathematics. 

These reformed curricula are new and only implemented during the past years, 



83 



LI SHIQI, HUANG RONGJIN, AND SHIN HYUNYONG 

therefore there is no empirical data demonstrating intended results. But some initial 
responses from prospective teachers and instructors of those courses are positive. It 
is certain that these reformed programmes for prospective mathematics teachers 
will have promising influences on mathematics teacher education in the future. 

ACKNOWLEDGEMENT 

The chapter was supported by Eastern China Normal University (ECNU) 985 
project funded by China Ministry of Education. The authors would like to thank 
Professors Chen Zhijie, Pang Xuecheng and Wang Xiaoqin at ECNU for their help 
and references. 

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Li Shiqi 

Department of Mathematics 

East China Normal University 

Shanghai 

China 



Huang Rongjin 

Department of Teaching, Learning and Culture 

Texas A & M University 

US.A. 

Shin Hyunyong 

Department of Mathematics Education 

Korea National University of Education 

Korea 



86 



KAYE STACEY 



4. MATHEMATICS FOR SECONDARY TEACHING 

Four Components of Discipline Knowledge for a Changing 
Teacher Workforce 



This chapter addresses the mathematics required for teaching in secondary 
schools, from early adolescence to preparation for university. The chapter works 
from a vision of good mathematics learning which values working from reasons 
not just rules, and being able to use whatever mathematics has been learned for 
solving problems within and beyond mathematics. Four components of 
mathematical knowledge are needed for teaching: (i) knowing mathematics in a 
way that has special qualities for teaching; (ii) having experienced mathematics in 
action solving problems, conducting investigations and modelling the real world; 
(Hi) knowing about mathematics including its history and current developments; 
and (iv) knowing how to learn mathematics. The chapter includes a short survey of 
teacher certification requirements in some western countries, and also reviews 
some reports that highlight shortages of well-qualified mathematics teachers. The 
policy responses to this situation relate to certification requirements, as well to the 
adequate provision for practising teachers of experiences that address all four 
components of discipline knowledge for teaching mathematics. 

INTRODUCTION 

This chapter addresses the mathematics required for teaching in secondary schools. 
In the chapter, I consider secondary school as being for students aged 
approximately II to 18 and primary school for students aged approximately 5 to 
1 1. This chapter is concerned with preparation of teachers who will teach a broad 
spectrum of secondary mathematics through the middle and upper years of 
secondary school. Secondary school encompasses a very wide range of 
mathematics learning, from the early adolescent years when some students are still 
struggling with ideas of whole number place value up to the highest levels of 
school achievement when some students are prepared to enter the most demanding 
university mathematics courses. This wide range of mathematical content and 
student achievement necessitates a breadth of teaching tasks which in turn requires 
a broad range of mathematical knowledge for teaching. Given that other chapters in 
this volume deal with mathematics knowledge for teaching from an Asian 
perspective, this chapter is concerned with 'western' countries although a 
comprehensive review cannot be attempted because of the large variety of 



P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 87-1 13. 

© 2008 Sense Publishers. All rights reserved. 



88 

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A3DV1S 3A\OI 



MATHEMATICS FOR SECONDARY TEACHING 

special skills, at the interface between mathematics and teaching, which teachers 
uniquely need. In this chapter I address the content knowledge needed to support 
these skills. 

Anyone who reflects on what mathematics teachers should know does so in 
relation to two things - the mathematics that students should learn at school, and 
the most important features of mathematics as an activity. I believe that a good 
mathematics education engages all students at every level in age-appropriate 
activities that develop: 

- knowledge of facts 

- fluency and accuracy in routine procedural skills; 

- deep conceptual understanding; 

- understanding of the major applications of mathematics; 

- ability to communicate using clear and precise mathematical language; 

- ability to tackle non-routine problems systematically; 

- ability to apply what has been learned to solve problems in real world contexts; 

- ability to conduct investigations using mathematics; 

- logical reasoning and a conception of the nature of proof; 

- practical ability for measuring, estimating, drawing and constructing; 

- sensible use of calculators and computers; 

- appreciation of the dynamic role of mathematics in society and the processes by 
which mathematics grows; 

- confidence and a productive disposition, which inclines one to see mathematical 
activity as useful and worthwhile. 

This list does not specify the content of mathematics learning, but it express an 
orientation to the processes and outcomes of learning any mathematics, which 
values working from reasons not just rules and being able to use whatever 
mathematics has been learned for solving problems within and beyond 
mathematics. The list is in broad agreement the U.S. report Adding It Up 
(Kilpatrick, Swafford & Findell, 2001) which lists five strands of mathematical 
proficiency (conceptual understanding, procedural fluency, strategic competence, 
adaptive reasoning, productive disposition) although with a stronger emphasis on 
applications and problem solving outside mathematics. I take these orientations and 
values to apply to all mathematics education - in school and in the preparation of 
teachers. 

This chapter begins with a short survey of the mathematical knowledge required 
to be a fully qualified mathematics teacher in several western countries. I then 
discuss in turn several different aspects of the content knowledge needed for 
teaching, which are widely recognised in the literature as important. These are: 

- Knowing mathematics 

- Experiencing mathematics in action 

- Knowing about mathematics 

- Knowing how to learn mathematics 

In the final section, 1 will discuss how the recommendations given in earlier 
sections, which outline an ideal answer to the question 'what mathematics should 
mathematics teachers know?' are affected by the reality in many countries of a 



89 



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A3DV1S HAV^I 



MATHEMATICS FOR SECONDARY TEACHING 

required to have at least 50% mathematics or content strongly related to 
mathematics within their undergraduate degree. This is spelt out in broad terms in 
the UK "Qualifications to teach" (HREF3) as "Teachers should have a secure 
knowledge and understanding of the subject(s) they are trained to teach. For those 
qualifying to teach secondary pupils, this knowledge and understanding should be 
at a standard equivalent to degree level". Spain provides something of a contrast 
because while their teachers are required to undertake general then professional 
studies the particular discipline which may be the focus of their undergraduate 
general degree may not be linked to the subject they intend to teach. To become a 
secondary mathematics teacher, it is not necessary for the candidate's general 
degree qualification to have been in mathematics (HREF1). The requirements for 
initial teacher training in the UK are further described by McNamara, Jaworski, 
Rowland, Hodgen, and Prestage (2002), including the introduction of the 
"Numeracy Skills Test" for all primary and secondary teachers (not just 
mathematics teachers) by the central Teacher Training Agency as part of a wide- 
reaching standards agenda. They also conduct audits of the mathematics content of 
teacher training courses. 

France is an example of a system that has stronger national agreement on the 
content of mathematics that prospective teachers should study. In writing about the 
French requirements for mathematics teachers, Robert and Hache (2000) draw 
attention to the strong emphasis placed there on the teachers' personal knowledge 
of mathematics. To qualify as a mathematics teacher in France students undertake 
five years of training beginning at University and with final two years at IUFM 
(Institut Universitaire de Formation des Maitres). In the first three years, 
prospective mathematics teachers are expected to study a standard undergraduate 
mathematics syllabus including classical linear algebra, real analysis, topology, 
differential and integral calculus and options such as probability, complex analysis 
or numerical analysis. The first year at IUFM is again devoted to mathematics, 
preparing for a highly competitive theoretical examination with written and oral 
components. The written paper covers problems from the mathematics topics that 
students are expected to have covered during their university studies (analysis, 
topology and functional analysis, integration and differential calculus, groups, 
rings, and linear algebra, geometry) while the oral examination focuses on topics 
from the later part of the high school curriculum. Henry (2000) notes that this year 
helps students consolidate their knowledge and reorganise it for teaching. This 
theme of knowledge being reorganised for teaching is also evident in initiatives 
that are described in later sections. Clearly France expects a high level of 
uniformity in the students' undergraduate mathematics programmes. Only in the 
fifth year is there any consideration of what the profession will involve. Robert and 
Hache (2000) observe that in the French system it is accepted that a thorough 
knowledge of mathematics is the most important ingredient for teaching. They 
observe that there is a need for teachers to be able to do mathematics themselves 
but also to take account of the students. Despite their years of study in 
mathematics, it is not uncommon for teachers to have difficulty in presenting 



91 



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A3DV1S HAV^I 



MATHEMATICS FOR SECONDARY TEACHING 

KNOWING MATHEMATICS 

In this section and the three following, I discuss the mathematical discipline studies 
for teaching. This first section discusses the content; the next section discusses how 
it is essential for mathematics teachers to have experienced doing mathematics 
through open investigations and modelling the real world. In this regards it is often 
the case that good mathematics learning for prospective teachers is not different 
from good mathematics learning for other undergraduates. In the next two sections, 
I discuss what teachers should know about mathematics as a discipline and how 
they need to be able to continue to learn mathematics independently. 

There is no question that teachers need a sound understanding of mathematics 
for teaching in secondary schools, but there are important questions of how much, 
what topics, and what should be the qualities of their mathematical knowledge. As 
noted elsewhere, most jurisdictions specify only how much mathematics (or any 
other discipline area) a secondary teacher requires for certification. In this section, 
we discuss what topics and what qualities this knowledge should have. Before 
leaving the question of quantity of mathematics teaching however, I note that the 
specification of some study of university level mathematics indicates several 
values - that teachers should have strengthened their mathematical knowledge and 
skills beyond what is needed at the school level, that they should know more than 
their students, and that they should have some perspective on where school 
mathematics leads. It is also worthwhile noting that there is an assumption that 
those who have completed school mathematics know its content well. This 
assumption is reassessed below: students learning mathematics as mathematically 
immature adolescents are unlikely to have experienced it with the richness and 
perspective required for teaching. 

Commentaries and opinions on the mathematics that teachers should know 
reflect ideals of what constitutes good school mathematics and what constitutes 
good mathematical practice in a wider sense. Regardless of one's philosophical 
stance on the nature of mathematics, it is abundantly clear that the existing and 
ideal school and university curricula and the values about mathematics itself that 
people hold are socially constructed, with the result that recommendations for 
teacher education vary from time to time and place to place, always expressing 
what is seen as the ideal. In the era of the 'new mathematics' in school curricula 
and Bourbaki formalism amongst mathematicians, for example, people valued a 
highly logical approach to mathematics where even school children drew on set 
theory and definitions of fundamental objects to deduce their properties. As 
another example, from Cyprus, Stylianides, Stylianides, and Philippou (2007) 
document prospective teachers' knowledge of proof by mathematical induction. 
The article assumes that all mathematics teachers should have mastered various 
forms of proof and should be able to give their students rich experiences related to 
proof. Whilst I would like this to be true in my own country, many fully qualified 
mathematics teachers in Australia have little understanding of proof. It is also the 
case that proof and verification currently play a very small part in Australian 
school mathematics. The 1999 TIMSS video study (Hiebert et al., 2003), for 



93 



KAYE STACEY 

example, found practically no instances of proof or deduction in large random 
samples of Grade 8 lessons from Australia, USA and the Netherlands, although 
instances of proof were prominent in Japan. 

The most prominent recent recommendations of mathematical knowledge for 
teaching have been given by the Conference Board for the Mathematical Sciences 
(CBMS, 2001) in the USA. Here I discuss only their recommendations for high 
school teachers (grade levels 9-12). The CBMS report begins by reviewing 
research that shows that beyond a threshold, having taken additional subject matter 
courses has only a small effect on teachers' effectiveness. They take this as a 
challenge to rethink teacher education so that what prospective teachers learn will 
indeed increase their effectiveness in the classroom in the future. As a consequence 
they recommend emphasising the underlying nature of the subject matter, a deep 
understanding of the subject in a way that is organised for teaching, and awareness 
of historical, cultural and scientific roots to mathematical ideas and techniques. 
They consider that recent changes in areas of application of mathematics caused a 
broadening of school mathematics to include more statistics and discrete 
mathematics and the use of new technologies opened up new opportunities for both 
teaching strategies and content to be studied. Consequently, they conclude that 
future teachers need to know more mathematics than before, somewhat different 
mathematics than before, and to experience learning it in a new way for 
themselves. 

The CBMS report outlines the content of a mathematics major for prospective 
teachers. In doing this, it is mindful that mathematics departments may not have 
the student numbers to warrant special programmes for teachers. As a consequence 
they have to consider whether the mathematical knowledge required by prospective 
teachers is quite different to that required by students pursuing other mathematics- 
related professions. Although they initially state that it is quite different, they later 
propose that the recommendations for prospective teachers, which outline a major 
study broader than before and with stronger connections to school mathematics and 
with use of modern technology, may well serve today's US undergraduates better 
than traditional majors. They note that mathematics courses which traditionally 
aimed at preparing students for graduate school work in mathematics must now 
serve a much wider constituency including mathematics majors not planning 
graduate work, and undergraduates with other major studies, such as engineering or 
science. 

The broad content of the CBMS recommended university major is organised 
around five themes of the high school curriculum: algebra and number theory, 
geometry and trigonometry, functions and analysis, statistics and probability, and 
discrete mathematics and computer science. The report gives an explanation of the 
special role and emphasis of each of these areas in the education of teachers. They 
recommend use of new technology, and note that prospective teachers need to 
understand the differences between electronic calculation to advance learning, 
human computation to advance learning, and electronic computation as a practical 
expedient, as well as learning to use a wide range of mathematical software (e.g., 
CAS, school-level technology such as graphics calculators and dynamic geometry) 



94 



MATHEMATICS FOR SECONDARY TEACHING 

and learning some computer-related mathematics and the basics of computer 
science. In setting out proposed content for each of the courses, the principle that 
university study of mathematics should illuminate high school mathematics is 
strong. This is in agreement with Cooney and Wiegel's (2003) Principle 2 that 
prospective teachers should explicitly study and reflect on school mathematics. For 
example, in algebra and number theory suggested exercises are to explain how 
operations of school algebra link with formal axiomatic principles and to justify 
each step of a common procedure (such as solving a quadratic equation) with field 
or ring properties. The geometry and trigonometry course is used to demonstrate 
the nature of axiomatic reasoning, and uses computer graphics and robotics as 
applications to strengthen students' understanding of modern areas of application 
and use of technology. Looked at from the British mathematics tradition, the 
proposed major seems to be lacking an emphasis on the major applications of 
mathematics, especially relating to partial differential equations. There are also 
some personal favourite topics of mine that are omitted. For example, I would like 
to include the study of complex analysis because it gives a strong sense of how the 
real number line is embedded in the complex plane and because it explains the 
difficulties with the definitions of log and power functions for negative and 
fractional numbers; difficulties that impinge on high school mathematics. These 
examples serve to show the increasing pressure on curriculum time in the 
mathematics major. The enormous growth in mathematical sciences over the last 
century is necessarily changing the face of both undergraduate and school 
mathematics and the place of every component needs to be regularly reassessed. 

Although the CBMS report claims that their mathematics major designed for 
teachers may serve other undergraduates better than a traditional major in 
mathematics, they also recognise that teachers need some knowledge of 
mathematics that is unique to teaching. The main need is to assist prospective 
teachers to make insightful connections between the advanced mathematics they 
are learning and school mathematics. They recommend that this can be done within 
each course, but it can also be done by offering a capstone course, taught jointly 
with mathematics educators. Extensive suggestions for the contribution of each of 
the five themes to the capstone course are given. For example, linking to the 
functions and analysis theme, the report suggests that the capstone course looks at 
the concept of function as a unifying theme in mathematics, examines the role of 
computers as tools for graphing and computation, examines relations between 
exploration and proof, and offers some experience of mathematical modelling. 
Other suggestions include historical perspectives on the development of the idea of 
function from a 'formula' to a 'mapping' and the cognitive difficulties that modern 
students experience in making this same transition, examining the use of graphing 
technology in teaching calculus, and drawing connections between the functions 
used in different branches of mathematics including probability distributions and 
log-log plots. How all of this material, with more from the other four themes, could 
fit within the one capstone course is not addressed. The squeeze on time in the 
mathematics major comes not only from the expansion of mathematics, but also 
from the growing appreciation of the need for connections to school mathematics. 



95 



K.AYE STACEY 

Other moves to reform undergraduate education have also found good alignment 
between the needs of prospective teachers and the needs of the new population of 
undergraduates now in universities. Pierce, Turville, and Giri (2003) report on the 
process of review of mathematics courses in an Australian mathematics department 
where a significant proportion of mathematics majors are prospective teachers. 
They report that the review was a challenging and reinvigorating process, requiring 
analysis of what mathematicians valued, but also coming to see their teaching from 
the students' viewpoints. They argued "mathematical thinking, key skills and 
conceptual understandings were valued, so too was the exposure of students to 
various branches of mathematics and their applications. What we needed was to 
approach this learning from the students' perspective" (p. 155). Their review set 
goals for knowledge of mathematics, experience of the process of doing 
mathematics ("reason mathematically, communicate and solve problems"), and for 
knowledge about mathematics ("understand and appreciate the role of mathematics 
and its applications in the real world"). They also specified goals related to career 
development ("Education students should form a positive view of their potential 
careers as mathematics teachers") and goals related to improving their experience 
of learning mathematics ("incorporate up-to-date teaching technology and utilise 
methods that enhance student learning"). Pierce et al. (2003) adopted a thematic 
approach to their new curriculum, and planned for the use of realistic problems to 
introduce the need for theory. They gave courses non-traditional and enticing 
names such as "Logic and Imagination". Practical activities and current 
technologies were used to enhance the process, and the assessments were chosen to 
cater for different learning styles and to encourage a range of different skills of 
communication and analysis. In accordance with the results of many other 
investigations, surveys revealed that their earlier students often had a narrow 
perception of mathematics, focussing on routine processing, and so their 
reinvigorated curriculum for mathematics teachers (and other students) also aimed 
to heighten enthusiasm for mathematics by using engaging topics in both learning 
and assessment. The use of new technologies was one tool they used to change 
undergraduate students' perceptions. In particular they wanted to use new 
technologies to emphasise a view of mathematics as description and explanation, 
rather than mathematics as rules for symbol manipulation. Pierce et al. (2003) 
found that these changes resulted in an increase in both initial enrolments and 
retention rates, increasing awareness of the relevance of mathematics for other 
disciplines and every day life, reduction in mathematics anxiety, increasing interest 
in mathematical thinking and an improved understanding of mathematics. 
Initiatives such as this may produce teachers with better understanding— although 
of a possibly more limited curriculum—and with more enthusiasm for mathematics. 
In turn, by increasing mathematics enrolments, it may produce more mathematics 
teachers. 



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MATHEMATICS FOR SECONDARY TEACHING 

Attitudes to and Beliefs about Mathematics 

Initiatives such as that of Pierce et al. (2003) and the capstone course of CBMS 
intend to create more a more productive disposition towards mathematics in future 
teachers. There is a large literature (reviewed, for example, by Cooney and Wiegel, 
2003) which shows that teachers' attitudes to, and beliefs about, mathematics 
influence their teaching. Some of these attitudes and beliefs are inconsistent with 
the reform vision of mathematics set out above, especially as they are often 
dominated by the view of teaching and learning mathematics by 'rules without 
reasons'. As Lachance and Confrey (2003) observe, teacher reform efforts are 
often "attempting to get teachers to think about and to teach mathematics in ways 
which they have never experienced as learners" (p. 110). It is clear that all 
mathematics education, at school and within teacher education, contributes to the 
attitudes and beliefs that teachers bring to with them to their work and hence to 
those that they impart to students. Teacher education that supports the 
implementation of a better form of mathematics teaching will itself need to 
demonstrate the characteristics of that mathematics teaching. 

It is important, though, to remember that there is not only one good way to teach 
mathematics and that there are different valid views of what is most important 
about mathematics. Kendal and Stacey (2001) studied two experienced teachers in 
one school who were incorporating computer algebra systems into their 
introductory calculus teaching. They incorporated this new technology into their 
pedagogy in different ways, consistent with their different beliefs and 
understandings about mathematics. The different conceptions of mathematics 
influenced their particular choices while using technology, their emphasis, and how 
to incorporate the graphical and symbolic algebra capabilities of the calculator into 
their lessons. In turn, these choices affected what their students learned. Although 
both classes achieved almost identical overall achievement, they showed quite 
different strengths. One teacher enjoyed the exactness of mathematics and 
especially liked the capability of the computer algebra system to give exact 
answers (e.g., fractions, square roots). He privileged the teaching of mathematical 
procedures, to which he added new technology procedures. His class was better at 
recognising how differentiation could be applied to solve a problem. The other 
teacher privileged conceptual understanding of mathematical ideas supported by 
extensive use of technology graphing and consequently his students were superior 
at interpretation of mathematics. 

It is also important to recall that teachers will move beyond their own 
preferences in the interest of students. Teachers' own beliefs about mathematics 
can be mediated by beliefs about students and their needs, especially as their 
experience of students and teaching grows. Good teachers aim to give the best 
education for every individual student, regardless of their mathematical talent and 
approach to school work, and this can override the teacher's personal preferences. 
An excellent example of this was provided in a recent project (Stacey, Stillman & 
Pierce, no date) where we worked with teachers in six schools to enhance students' 
achievement and engagement through using real world problems and new 



97 



KAYE STACEY 

technology. At the end of an interview Meryl, one teacher in the project, explained 
that she used the project activities to teach in a way different to her own 
preferences because she judged doing so was in the interests of her students: 
Personally I like algebra. If I could choose to teach children where I don't 
need to have activities, where I don't need do a lot of modelling or real life 
problems, I would. I would just do the 'boring' algebra. I enjoy that most. 
But the percentage of students in a normal cohort who would gain from that 
is just a minor part. The majority gain more from this approach. 

The Quality of Knowledge 

The above discussion has focussed mostly on what teachers should know. 
However, it is well established that for this content knowledge to be effective for 
teaching, it requires certain characteristics. Shulman (1986) listed aspects such as 
the amount and organisation of knowledge, understanding the structures of the 
subject matter and how truth and falsehood are established, and being able to 
explain why a proposition is regarded as true, why it is worth knowing and how it 
relates to other propositions within and beyond the discipline. Many subsequent 
studies have shown that these characteristics are often lacking even in prospective 
teachers who have strong undergraduate mathematics backgrounds (see, for 
example, Ball, 1990; Goulding, Hatch, & Rodd, 2003). Concerns to ensure content 
knowledge has such characteristics lie behind proposals such as the CBMS 
capstone course. 

There is a growing body of research which aims to measure some of these 
qualities of teachers' subject matter knowledge. Connectedness is one that has 
received considerable prominence in recent literature. Chinnappan and Lawson 
(2005) present a framework which enabled them to characterise teachers' content 
knowledge and content knowledge for teaching. They demonstrated how to map 
teachers' content knowledge and their knowledge for teaching one topic (in this 
case in geometry) in a way which revealed and began to quantify the 
connectedness of that knowledge. This draws on research which shows that 
knowledge structures which are more comprehensive and more internally and 
externally connected are more likely to be useful in problem solving. In this case 
the 'problem solving' is the act of teaching itself. Their maps revealed qualitative 
and quantitative differences in the connectedness of knowledge of two well- 
qualified and experienced secondary mathematics teachers. Further developments 
in research on measuring the qualities of teachers' knowledge bases will help us 
understand which differences and which magnitudes of differences impact upon 
teaching effectiveness. 

EXPERIENCING MATHEMATICS IN ACTION 

The most influential argument for teachers to experience the process of doing 
mathematics was put forward by Polya (1962) in his books on how to solve 
mathematical problems and his exposition of problem solving heuristics. Polya's 

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MATHEMATICS FOR SECONDARY TEACHING 

work was inspired by his "concrete, urgent, practical aim: to improve the 
preparation of high school mathematics teachers" (p. vii) and so along with case 
studies of interesting problems and their solutions, he provided a section on "hints 
for teachers and teachers of teachers" (p. 209) drawn from his own teaching 
experiences. Writing in the era of the 'new math', Polya found the issue of what 
content should be offered to prospective high school teachers and to their students 
too controversial for agreement, but he proposed that knowledge of the process of 
doing mathematics was something upon which experts would agree. 

Our knowledge about any subject consists of information and of know how. If 
you have genuine bona fide experience of mathematical work on any level, 
elementary or advanced, there will be no doubt in your mind that, in 
mathematics, know-how is more important than mere possession of 
information. Therefore, in the high school, as on any other level, we should 
impart, along with a certain amount of information, a certain degree of know- 
how to the student. What is know-how in mathematics? The ability to solve 
problems - not merely routine problems but problems requiring some degree 
of independence, judgment, originality, creativity. [...] The teacher should 
know what he is supposed to teach. He should show his students how to solve 
problems - but if he does not know, how can he show them? The teacher 
should develop his students' know-how, their ability to reason; he should 
recognise and encourage creative thinking - but the curriculum he went 
through paid insufficient attention to his mastery of the subject matter and no 
attention at all to his know-how, to his ability to reason, to his ability to solve 
problems, to his creative thinking. Here is, in my opinion, the worst gap in 
the present preparation of high school mathematics teachers. To fill this gap, 
the teachers' curriculum should make room for creative work at an 
appropriate level (p. vii). 

Polya's recommendations for teaching teachers about mathematical problem 
solving and investigation have been the basis of much subsequent work. He 
recommended problems that did not require much knowledge beyond high school 
mathematics, but did require concentration and judgement. His book was organised 
around strategies for problem solving, illustrated by problems chosen to highlight 
the strategies. He also advised that prospective teachers should reflect on the 
classroom use of such problems. He therefore supplemented the 'look back' phase 
of doing mathematics (where problem solvers reflect on the mathematical solution) 
by an additional didactically-oriented phase. Polya's basic ingredients for teaching 
problem solving of experience, strategies and reflection have formed the basis of 
many subsequent endeavours, including those of Schoenfeld (1985), and Mason, 
Burton, and Stacey (1982). 

Since the time of Polya, explicit attention to the process aspects of mathematics 
has been evident as a goal in teacher preparation courses. This is evident, for 
example, in two of the 3 principles that Cooney and Wiegel (2003, p. 806) 
recommend for mathematics for prospective teachers: 



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- Principle 1: Preservice teachers should experience mathematics as a pluralistic 
subject. 

- Principle 2: Preservice teachers should explicitly study and reflect on school 
mathematics. 

- Principle 3: Preservice teachers should experience mathematics in ways that 
support the development of process-oriented teaching styles. 

Experiencing mathematics as a pluralistic subject (Principle 1) includes (among 
other things) experiencing mathematical investigation and problem solving, and 
Principle 3 calls for prospective teachers to be supported in making such 
experiences a reality for their own students. Recently, Ryve (2007) found that 6 of 
28 teacher education institutions in Sweden had a specific course directed to 
problem solving, although he noted different foci. Some courses emphasised the 
mathematical aspects of the tasks with which the prospective teachers engage. 
Others dealt with problem solving tasks in relation to the secondary school 
students' learning and behaviour. Others emphasised aspects of the problem 
solving process, such as interpretations of answers, multiple solutions etc. Other 
important differences are whether a course sets out to examine only problems 
challenging to school students or aims to extend prospective teachers' own 
problem solving, giving them a taste of the genuine problem solving experience 
which they aim to provide for school students. In my opinion, some attention to the 
latter (teachers' own experience of doing mathematics) is essential in order to 
equip them to provide rich experiences for school students. Stacey (2005), using 
examples from Singapore and three western countries, documents how the goal of 
teaching children to be better problem solvers (teaching for problem solving) has 
now generally been supplanted by the intention of 'teaching through problem 
solving' where students encounter new material about standard topics through 
investigations. They are expected to acquire problem solving and investigative 
approaches implicitly. It is my opinion that this approach is unlikely to give student 
teachers the explicit knowledge of the process of doing mathematics that they 
require for teaching it well, and that there is a need to provide prospective teachers 
with supported experience of doing mathematics at their own level, with discussion 
of strategies and reflection on successful elements. 

Ryve (2006) also notes the importance of studies which examine the different 
character of courses that are provided in teacher education. It is hoped that such 
studies might extend the primarily quantitative discussions of how much 
mathematics prospective teachers should learn, to examining qualitatively the 
nature of their participation in those courses and consequently how their views of 
mathematics could be extended. The literature provides many examples of how 
prospective teachers' views and approaches to solving mathematical problems are 
limited. Van Dooren, Verschaffel, and Onghena (2003) examined secondary 
prospective teachers solving algebraic and arithmetic word problems. The 
secondary teachers had good content knowledge, but their habits and attitudes 
whilst problem solving were stereotyped and some were not open to look for the 
alternative methods which their future pupils may use. They characterised the 



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prospective teachers, at both the beginning and end of their teacher preparation, as 
tending to have routine expertise rather than adaptive expertise. They concluded 
that prospective teacher education needed to promote experiences of doing 
mathematics that encouraged flexible thinking and created a well-organised body 
of professional knowledge. 

Despite the agreement that all mathematics students, and especially teachers, 
should have direct experience of the processes of mathematical discovery, 
investigation and application, the history of its implementation has been anything 
but smooth, in teacher education as well as in schools and universities. Burkhardt 
and Bell (2007), for example, describe the development of problem solving in UK 
school mathematics over more than a century. They highlight the many different 
interpretations of problem solving, from mathematical research by professionals; to 
the solution of 'riders' where knowledge of theorems and proofs is adapted to a 
novel problem situation; to conducting open investigations such as finding which 
numbers are not sums of consecutive numbers; to modelling the real world; to 
teaching for functional numeracy and mathematical literacy (as defined by OECD, 
2003). They highlight the gap between the goals for having students' experience 
any of the above forms of problem solving and the understanding by school 
systems of the nature of the change required and the conditions under which it 
could really be achieved. It is likely that the situation is similar in teacher 
education, with a mismatch of goals and reality. 

Burkhardt and Bell's article also draws attention to the fact that there is a lot 
more to the 'experience of doing mathematics' than engaging in the pure 
mathematics investigations in the spirit of Polya and successors (e.g., which 
numbers are not sums of consecutive numbers), or the investigations that reform- 
oriented teachers offer to develop students' understandings of particular concepts 
(e.g., Chick's (2007) finding the dimensions of a rectangle of given perimeter with 
maximum area or Teacher B in Kendal et al. (2001) setting his class the problem of 
finding a general rule for the derivative of x 2 , x 3 , x A and then x" by guessing rules 
from numerical values of the slopes of tangents). Mathematics is studied for its 
interest and its beauty and for its place in our cultural heritage, but its central role 
in the school curriculum is due to its usefulness. Teachers therefore need to 
understand deeply the way in which mathematics is applied. The presence of 
teachers coming into teaching as a second or subsequent career from a diversity of 
backgrounds is a strength here. 

Mathematical modelling, using mathematics to answer questions about the real 
world, has a distinctly different flavour to investigation within the real world. It can 
be understood as consisting of four steps: formulating a mathematical problem 
from the real world problem, solving the mathematical problem (using all the 
techniques developed within mathematics itself), interpreting the mathematical 
solution in real world terms, and evaluating the solution to see if this solution is 
adequate for the task. The intention of mathematical modelling is similar to that of 
mathematical literacy as defined by the OECD (2003) for its PISA study of 15 year 
olds, although the possibilities of actually assessing modelling are severely 
curtailed by the written timed test format in PISA. Teachers need experience of 



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mathematical modelling, in addition to pure mathematics investigations, because 
the process is distinctly different. Only in the simplest of word problems does a 
mathematical model capture all that is relevant to the real situation (e.g., 6 identical 
apples shared equally amongst 3 boys). The skill of the mathematical modeller is to 
identify what are likely to be the most critical variables, but the test for whether 
they are adequate choices can only come after a full modelling cycle when results 
are compared with reality. For example, to answer a question about how the times 
should be set. for traffic lights to clear traffic at a busy intersection is it enough to 
assume that cars pass through the lights every 2 seconds in each lane (this is the 
time difference that is recommended to learner drivers), or should it be assumed 
that the first car takes longer and then others pass through at a constant rate, or is it 
necessary to go for a probabilistic model where cars pass through according to a 
selected probability distribution? Even the choice to include the rate at which cars 
pass through the lights as a variable in the model is part of the formulation stage, 
subject to later verification as to its usefulness. As with pure mathematics 
investigations, providing prospective teachers with experience of mathematical 
modelling is a high priority that it often not achieved. When it is achieved, a study 
by Nicol (2002) shows there may be further work to do to translate this into lively 
teaching. Prospective teachers were able to see how mathematics was being used in 
workplaces, but when they created lessons from the experience, the mathematics 
remained decontextualised. I noticed the same phenomena with prospective 
teachers from an engineering background. They were expert in applying 
mathematics in their previous work, but do not see how they could use these 
experiences to motivate students and illustrate applications. 

KNOWING ABOUT MATHEMATICS 

Beyond knowledge of mathematics and experience of doing mathematics, teachers 
need to know about mathematics. This is an area where the preparation of teachers 
is readily seen to require something different to preparation for other professions. 
Teachers who know about mathematics - its history in both the East and the West, 
its ways of working, its major events, and so on - can enliven their teaching and 
assist students to understand how mathematics works, where it comes from and its 
role in society. Some knowledge about mathematics comes incidentally as we learn 
mathematics (assisted by teaching), but some such as the history, epistemology or 
philosophy of mathematics can be studied separately from mathematics. The 
CBMS (2001) report recognises this need in both its recommended capstone course 
as well as in the attention it pays to 'habits-of-mind goals' (p. 141) and 
mathematical thinking. 

There is long standing interest in courses on the history of mathematics. Since 
1976 there has been, for example, an International Study Group on the relations 
between the History and Pedagogy of Mathematics (HREF4) affiliated to the 
International Commission for Mathematical Instruction. One of their aims is to 
assist mathematics teachers to gain insights on how the history of mathematics may 
be integrated into teaching and may help students to learn mathematics. Materials 



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to support courses for prospective teachers are now becoming readily available on 
the Internet. For example, Mills (2007) describes an elective course intended for 
prospective teachers (although not exclusively), that is based on historical 
documents rather than a modern textbook. Mills' course aims to show how 
"mathematics is created by human beings and hence is connected with the culture, 
the times and the place where this creative activity takes place" (p. 195). Students 
study ancient Egypt (Rhind Papyrus), ancient Greece (Euclid) and medieval 
Europe (Fibonacci). In justifying his selection of elementary mathematics topics 
for the course, Mills (2007) notes that many more students can enjoy studying the 
history of elementary, rather than advanced mathematics. Whilst this is certainly 
the case, it is often not the case that the history of simple mathematics is itself 
simple. Matrices, for example, can be introduced in a straightforward way to junior 
secondary students as storage arrays for data or for coefficients of equations, but 
their important place in mathematics derives from advanced work by Cauchy, 
Lagrange and others (HREF5) on a variety of topics such as the use of differential 
equations to solve problems of celestial mechanics. It is not feasible to motivate the 
early study of matrices with this. A serious treatment of history can demand both 
difficult mathematical content and difficult historical material, both from the point 
of view of the prospective teacher and the instructor. As a consequence, the use of 
historical anecdotes to enrich the standard teaching of customary mathematical 
topics is a more widespread approach to increasing teachers' understanding of the 
history of mathematics than offering complete courses. 

Whereas it is common to use history as a source of enrichment when teaching 
many topics, the philosophy of mathematics seems more difficult to encompass in 
teacher education. It seems reasonable that prospective teachers should be able to 
supply good answers to questions such as: what is a mathematical object?; what is 
the nature of mathematical truth?; is mathematics created or discovered?; and why 
does mathematics model the real world so well? However, there are no simple 
answers here; these are difficult questions at the interface of philosophy, logic and 
mathematics, which require serious study. In addition, they are questions which 
rarely trouble the working mathematicians who generally provide education in 
mathematics for prospective teachers. Davis and Hersh (1981) observed that most 
mathematicians act as though they are Platonists, acting on a naive view that 
mathematical objects have an uncomplicated status, although when pressed they 
often retreat to a formalist view, where mathematics is viewed as a game played 
according to certain rules and where it is not required to specify any further 
meaning. In contrast to the concern with mathematical foundations that is attacked 
with the tools of logic and mathematics, many mathematics educators are keen to 
stress mathematics as a human endeavour and view it from a social perspective. 
For example, in their extensive article on mathematics for teacher education, 
Cooney and Wiegel (2003) promote the view that a fallibilist view of mathematics 
is the most productive to guide courses for prospective teachers - a view where 
truth in mathematics is seen as grounded not in pure reason, or on correspondence 
with data from the senses, but is a result of a social process. In their view, prevalent 
teachers' beliefs that mathematics is abstract, rigid, unchanging and not based in 



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human experience have arisen from their mathematical training. They propose that 
these beliefs present a major obstacle to reform of mathematics in schools, and so 
should be countered by the fallibilist (social) view. My belief is that the view of 
mathematics as abstract (only), unchanging, rigid, and unrelated to human 
experience is a consequence of an inadequate mathematics education, not attending 
to the four components outlined in this chapter, rather than a consequence of a non- 
fallibiiist philosophy. In a recommendation that I strongly endorse, Cooney and 
Wiegel also recommend that courses for prospective teachers should present 
mathematics as a pluralistic subject that includes elements of discovery and 
investigation alongside appropriate formalism, rather than being dominated by one 
view. 

Beyond history and philosophy of mathematics, what else should teachers know 
about mathematics? For teachers of science, it is important that they 'keep up to 
date' with their subject, and there is a great deal of information in the press to help 
them and other concerned citizens, but this is harder for mathematics teachers since 
advances in mathematics are generally highly technical, and only slowly, if ever, 
become part of the school mathematics curriculum. Moreover, teachers tend to 
view mathematics as uncontested, often only as what their own educational 
jurisdiction sets down for them to teach (Wilson, Cooney, & Stinson, 2005). This 
view is too limited. At a minimum, teachers need some personal experience that 
new mathematics and new applications of mathematics continue to be invented 
and/or discovered. Mathematics 'general knowledge' does not have an easy place 
in formal mathematical training. The Black-Scholes equation is used for pricing 
options in financial markets by hedging against losing bets. It is now said to be the 
world's most used formula and so one might expect that most mathematics 
teachers would know about it. (Bulmer (2002) provides a suitable introduction for 
teachers.) However, since pricing options belongs to the mathematics of finance, 
the Black-Scholes formula is not a central part of a mathematics major today. 
Moreover it has become important only in recent years, after many teachers have 
graduated. As a consequence, it is probably the case that most of the world's 
mathematics teachers have not heard of the world's most used formula. 

Infinity is a concept that fascinates people of all ages, including young students, 
and so one might expect that most mathematics teachers would have a good grasp 
of the infinite cardinals and how to do arithmetic with them. A teacher who 
understands why a mathematician says there is the same number of fractions as of 
all integers but more decimals than either, for example, can link into these 
widespread fascinations. Similarly, the fourth dimension is a fascinating idea and 
again one might expect that mathematics teachers could explain how the idea of a 
2-dimensional net of a 3-dimensional cube can be generalised to the 3-dimensional 
net of a 4-dimensional cube. Mathematicians have also shown in recent years that 
it can be useful to calculate non-integer dimensions, for example the fractal 
dimension of a coast line. These, and many other ideas which are similarly 
prominent in the popular imagination, rarely have a firm place in a mainstream 
mathematics major study, because they are not central to the development of 
knowledge to participate in advanced mathematics. None of these illustrations 



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above are of themselves essential knowledge for teaching (I expect that there will 
be individual readers disagreeing with my choice of each of these examples) but 
there is a cumulative effect of having teachers who can tap into the natural human 
interest in things mathematical and appreciate the ever increasing role of 
mathematics in society. This comes back to my point: mathematics teachers need 
to 'keep up to date' with mathematics and, over time, build up a wide general 
knowledge of mathematics. Consequently, there is a need for materials on new 
mathematical developments and on perennial favorites to be prepared for the 
teacher audience, and presented through teachers' journals and other media. 

KNOWING HOW TO LEARN MATHEMATICS 

Equally importantly, the need to keep up-to-date throughout a career points to the 
need for mathematics training at every level to develop skills in learning to learn 
mathematics. The CBMS report (2001) makes the same point: "Thus, college 
mathematics courses should be designed to prepare prospective teachers for the 
lifelong learning of mathematics, rather than to teach them all they will need to 
know in order to teach mathematics well" (p. 6). An initial preparation in any 
subject is not sufficient for an on-going career. Being an independent learner of 
mathematics, both to master new technical skills as required to deal with new 
topics in the curriculum, and to keep abreast of the concepts behind new 
developments, is as important for teachers as for any other professional. New 
demands from the workforce and national economies, new mathematical ideas and 
applications and new technologies for doing mathematics all contribute to making 
it a dynamic subject, with consequent needs for continual independent learning. 

What do we know about courses that develop good skills for independent 
learning of mathematics? Seaman and Szydlik (2007) developed a concept of 
'mathematical sophistication' to encompass a set of values and avenues for doing 
mathematics that is required to create fundamental understandings. They observed 
prospective primary teachers attempting to learn from an on-line resource and 
noted that many of them experienced difficulties that were due to characteristics 
such as not attempting to make sense of the relevant definitions provided, not 
focussing on giving meaning to the problem and attending precisely to language, 
and not using the explanations supplied. Although this was only a small study and 
it studied primary rather than secondary teachers, this study is relevant to this 
discussion because it points to the way in which independent learning of 
mathematics depends on an enculturation into mathematical practices and values. 
A detailed example of how understanding of the key role of mathematical 
definitions affects the mathematical work of teaching is given by Chick (2007). 
She reported episodes of teaching where teachers' knowledge of mathematics 
strongly influenced the outcome. In a classroom investigation, students who were 
finding rectangles with a certain property discarded a square as a solution, but the 
teacher skillfully drew their attention to how the square met the requirements 
(definition) of being a rectangle even though it had other properties as well. The 
point being made was that a definition provides minimal criteria, not a frill 



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description. Mathematical practice has many other instances, large and small, in 
which teachers who are not enculturated into its language and practices may find 
an obstacle in extending their own knowledge. For example, even simple words 
such as 'some' and 'either' have mathematical meanings that contrast to their 
everyday meanings. It is true in mathematics that some women are called Kylie 
Minogue (even if there is only one person with this name, mathematicians can still 
say some) and it is true that a red square fulfils the criterion of being "either red or 
square" (being both is not excluded by a mathematical either-or). 

The need for teachers to be able to learn mathematics independently is most 
critical for those teachers teaching 'out-of-field'. Unfortunately, these are the 
teachers who are likely to have the more difficulty with learning mathematics and 
also had less practice. With colleagues, I recently conducted professional 
development sessions for such teachers of junior secondary mathematics. In one 
instance, we wanted to focus on pedagogical content knowledge, to demonstrate 
the different degrees of difficulty of the three basic types of percentage problems. 
Given similar numbers, type 1 problems (given whole, given part and missing 
percentage) and type 2 problems (given whole, given percentage and missing part) 
are easier for students than type 3 problems (given percentage, given part and 
missing whole). In the professional development session, the demonstration was 
much more effective than we had intended. In every group, the type 3 problems 
could not be solved by a small but significant proportion of teachers currently 
teaching those grade levels that percentage is taught. It is likely that these teachers 
will avoid setting problems of this type for students, believing they are too 
difficult, and hence in effect setting expectations of students that are too low. 
Wilson and Berne (1999) comment on the dilemmas posed when voluntary 
professional development exposes teachers' lack of knowledge, and they also 
comment on the difficulties of research into teacher knowledge when it too can 
spotlight gaps in individual teachers' knowledge base. Even in professional 
development programmes with a strong focus on content knowledge, Wilson and 
Berne reported that opportunities to discuss mathematical content were repeatedly 
missed because instructors and fellow participants did not want to embarrass 
individuals. These sensitivities make it difficult for teachers to improve their 
content knowledge. Lachance and Confrey (2003) report similar observations, but 
note that teaching content in the context of new technology, new to all participants, 
eased the situation and fostered stronger helping relationships among teachers. 

Another instance from the same professional development series highlights how 
teachers' 'big picture' of mathematics affects their flexibility in adapting teaching 
ideas. We introduced these 'out-of-field' teachers to the dual number line to solve 
percentage and ratio and proportion problems. The dual number line is marked on 
one side with the absolute amount and on the other with the percentage. It is used 
widely in Singapore but not in Australia where the professional development was 
held. By marking the data on a dual number line, a student can organise the 
information and hence see how to solve proportional reasoning problems, including 
problems related to speed, density, prices given cost per gram, and so on. From the 
view point of more advanced mathematics, all of these problems are problems of 



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direct linear proportion, using the same mathematical techniques. After 1 
commented quickly that the dual number line was also useful for these problems, 1 
was surprised when teachers asked me how this could work. These teachers, 
learning mathematics independently and building only on their own school 
experience, have learned it topic by topic, without seeing the big picture of direct 
proportion and the multiplicative structure that links them all. After my short 
explanation of the similarity between the problems, it is likely that they over- 
generalised. They may have not seen that the dual number line needs serious 
modification to be used with other than direct proportion (e.g., conversion between 
Celsius and Fahrenheit). 

GOOD POLICY IN AN ERA OF TEACHER SHORTAGE 

The above sections focussed on the mathematics education that is desirable for 
mathematics teachers. In this section, I confront this discussion with reality. Le 
Metais (2002) notes that the trend of mathematics to decline in popularity in 
secondary schools across a number of western countries, has reduced the pool of 
specialist mathematics teachers. As a result, there are many non-specialists 
teaching mathematics, who may not have the expertise or confidence necessary to 
prepare and motivate students to pursue higher-level studies in mathematics. 
Stephens (2003) reiterates these concerns, and concludes that traditional pathways 
will not be able to provide sufficient mathematics teachers in the future, especially 
since the recent growth in occupations requiring a strong grounding in mathematics 
sciences exceeds supply. 

The qualifications of mathematics teachers in Australia is reported in detail by 
Harris and Jensz (2006) in a study conducted on behalf of the Deans of Science of 
Australian universities, as a result of their concern about teacher supply and 
quality. Harris and Jensz report that 75% of teachers of senior (grade levels 11-12) 
mathematics had studied some mathematics to third year at university. This may 
not necessarily be sufficient for a major, as they may have taken mathematics as a 
small component alongside other major studies. Harris and Jensz express deep 
concern that about 8% of all secondary mathematics teachers (grade levels 7-12) 
studied no mathematics at university and 20% of all mathematics teachers studied 
no mathematics beyond first year (p. iv). The report also notes that many teachers, 
including a third of those teaching only junior secondary mathematics (grade levels 
7-8) had studied no mathematics-specific education. Stephens (2003) cites data that 
over a quarter of high school students in the USA are taught by 'out-of-field' 
teachers. 

It is also the case that the decline in the number of students studying 
mathematics, the increasing availability of well-paid careers for those qualified in 
the mathematical sciences, and a decline in the popularity of teaching as a career, 
leads to prospective teachers with a wide range of undergraduate qualifications (not 
a traditional mathematics major) entering consecutive courses of teacher education. 
For example, the University of Melbourne submission to the Australian Review of 
Teachers and Teacher Education (University of Melbourne, 2002) reported on the 



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discipline qualifications of entrants to the post-graduate diploma in education from 
1998 to 2002. The submission noted that "demographics of current teacher 
education students do not conform to traditional expectations. They are older, 
better qualified and have substantial prior work experience" (p. 14). The report 
noted an increasing diversity of academic backgrounds, reflecting increasingly 
specialised undergraduate science training. Prospective mathematics and physics 
specialist teachers had an average age of 31 and about three quarters had 
previously worked in another profession, most commonly engineering. Only about 
a quarter of students had done their mathematics study within a Bachelor of 
Science degree, which indicates that they have mostly been trained in the 
applications of mathematics (e.g., engineering, applied science, information 
technology, commerce) rather than studying mathematics for its own sake. The 
decline in teachers with 'normal' qualifications is also indicated in a survey carried 
out in the Australian state of Victoria by the Mathematical Association of Victoria. 
The results of this survey indicated that a mathematics major in the undergraduate 
degree was held by 83% of teachers who had been teaching for more than 20 years 
but by only 61% of teachers who had been teaching less than 10 years (University 
of Melbourne, 2002). They have all met the requirements for amount of 
mathematics in the undergraduate degree as specified by state legislation (a sub- 
major), but the nature of this training is very different to what it might have been 
20 years ago. 

These trends in teacher supply raise important issues for answering the question 
of what mathematics teachers 'should' know. On the one hand, it is likely that most 
countries will have significant numbers of teachers who have studied little or no 
mathematics at university, and educational systems should plan to meet their needs 
for professional development in mathematics, as well as in other educational 
issues. These people will be an important part of the teacher workforce. On the 
other hand, a growing proportion of 'fully qualified' mathematics teachers will 
have studied mathematics in the service of another profession or discipline, rather 
than for its own sake. In answering the question of what mathematics teachers 
'should' know, their needs must also be considered. Their expertise in another 
discipline is an advantage. The presence of many teachers who have used 
mathematical sciences in the workplace, in many different occupations, provides a 
considerable resource for enriching school students' experience of the applications 
of mathematics and for understanding the careers in which mathematical expertise 
is useful. Teacher education must empower these teachers to use these prior 
experiences in the classroom. On the other hand, having their expertise outside of 
mathematics means that there may be large gaps in their mathematical knowledge 
as exemplified in the article on prospective teachers' knowledge of proof by 
Stylianides et al. (2007) mentioned previously. It is extremely unlikely that 
prospective teachers who come to mathematics teaching from engineering or 
commerce will have experience of proof. 

Reports such as that of the Deans of Science (Harris & Jensz, 2006) arise from 
concern about an inadequate supply of teachers well qualified in mathematics. 
They advocate that there should be stronger minimum standards of discipline study 



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and of pedagogy for those teaching mathematics. This is one of the common policy 
responses to the widely recognised need to improve the quality of the teacher 
workforce. Boyd, Goldhaber, Lankford, and Wyckoff (2007) have compared the 
effects of this policy response (to tighten teacher certification requirements) with 
the opposite response of easing requirements and introducing alternative ways of 
being certified as a teacher, as implemented in the states of the USA. They 
conclude that tightening requirements is effective only if the tighter requirements 
focus on characteristics that lead to better outcomes for students and if they do not 
deter potential applicants who may become excellent teachers. In other 
circumstances, the opposite approach of easing requirements becomes the more 
attractive policy. Whilst calling for further research related to such policies, they 
note that teachers' content knowledge in mathematics is known to improve 
students' mathematics learning (hence can be seen as a characteristic that leads to 
better outcomes for students), but they also claim that there is some evidence that 
teacher certification requirements shrink the pool of people pursuing teaching 
careers. As a consequence, whilst it is appealing for teacher educators to identify 
substantial lists of the knowledge that teachers ought to have, it may be that better 
outcomes for students overall may arise from looser initial requirements and good 
opportunities for growth within the profession. Stephens (2003) makes the point 
that even with full teacher education, the powerful blend of content knowledge and 
pedagogical content knowledge that is envisioned by the Conference Board CBMS 
recommendations is unlikely to be able to be attained in initial teacher education, 
due to time constraints, lack of clarity of responsibility between discipline and 
education components, and lack of experience of the prospective teachers 
themselves with school student's mathematical thinking. As a consequence, the 
responsibility for deep understanding of content knowledge in a pedagogical 
context has to be taken up by education for practising teachers. These issues need 
careful consideration at the local level for the formation of good policy. 

CONCLUSION 

In the first part of this chapter, I outlined what a good education in mathematics for 
prospective teachers would be like. Working from an ideal of mathematics as a 
subject that is taught for both its interest and its applications, and from a basis of 
reasons rather than rules, a vision of what teachers should know and how they 
might know it emerges. The knowledge of teachers was classified into four: 
knowledge of the content of mathematics; experience of doing mathematics; 
knowledge about mathematics as a discipline; and knowing how to learn 
mathematics. In planning the content of mathematics courses for teachers in the 
past, it seems it was assumed that teachers had an adequate knowledge of school 
mathematics by having been successful students. However, led by research on the 
minute-by-minute mathematical requirements of teaching and by research which 
shows only weak links between university studies of mathematics and 
effectiveness as a teacher, there has been a movement (encapsulated in the CBMS 
report) to make school mathematics a central concern of university studies for 



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teachers. The need to give prospective teachers an understanding of newly 
important mathematics (notably but certainly not restricted to statistics and 
computer science) has also put pressure on the traditional mathematics major, 
leading to a reassessment of the priorities for study. An important research question 
for the future will be to investigate whether courses along the lines envisaged by 
CBMS (2001) or initiatives as exemplified by Pierce et al. (2003) really do provide 
a major in mathematics that well-equips graduates for their chosen careers. In 
particular, it is important to investigate whether these initiatives do succeed in 
providing teachers with knowledge that really does enhance their teaching. Boyd et 
al. (2007) note that some knowledge of mathematics does improve student 
outcomes. Can we show that a theoretically excellent education for mathematics 
teaching actually makes a further measurable difference to student outcomes, and 
hence is worth requiring? 

The later section examined the reality of teacher qualifications for mathematics 
and considers projections that in the future, shortages of well qualified 
mathematics teachers are likely to persist. Whereas the first section addressed an 
ideal with a teacher undertaking a mathematical major, the reality is that many 
teachers who are fully qualified to teach mathematics in the eyes of regulating 
bodies will in fact have been trained as users of mathematics in the service of other 
professions. More seriously, many will not be educated as mathematics teachers at 
all, but will do their best, as they understand it, from their own experience of being 
a high school student or undertaking a small component of university mathematics. 
This situation raises a substantial series of research questions related to how gaps 
in teacher knowledge can be overcome. There are some concerning reports in the 
literature (e.g., Wilson & Berne, 1999) that professional development that is 
intended to focus on content knowledge often avoids facing up to the challenge. In 
the literature there are many examples of teachers or prospective teachers not 
knowing the mathematics that will teach. How serious a problem is this in the long- 
term? What sort of knowledge is likely to be gathered during a teaching career, by 
teaching topics and engaging with students' thinking and what support does this 
require? To what extent do teachers (with differing backgrounds) learn 
mathematics by teaching it, and what support materials would encourage this? The 
CBMS report identifies that there is a special knowledge required for teaching, 
principally an overview of school mathematics as in the capstone course and the 
type of content that I have outlined above in the 'knowing about mathematics' 
section. How can this best be gained by teachers without the expected content 
background? I have also noted that many mathematics teachers now have 
substantial experience of using mathematics in a previous career. How can the 
educational system harness this? 

In recent years, research has demonstrated how mathematical knowledge affects 
the minute-by-minute decisions that a teacher makes. Most of this research has 
been conducted with primary teachers, and most of the initiatives to address the 
identified deficiencies have also been aimed at primary teachers. Possibly this is 
because usually more of their education is within schools of education, and hence 
in the hands of those who conduct education research. Secondary teaching has been 



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MATHEMATICS FOR SECONDARY TEACHING 

neglected on both these accounts. There is less research and the avenues for change 
are administratively more complex within institutions for secondary majors. 
Secondary mathematics is extremely important for individual life chances and for 
national prosperity yet it is often taught as rules without reasons, turning students 
away. Attending to the mathematics knowledge of the secondary teacher 
workforce, consisting, as it does, of people with mixed mathematical backgrounds, 
should be a high priority for researchers, teacher educators, mathematicians and 
educational systems. 

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Burkhardt, H., & Bell, A. (2007). Problem solving in the United Kingdom, ZDM Mathematics 

Education, 59(5-6), 395-403. 
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the 30th annual conference of the Mathematics Education Research Group of Australasia (pp. 3- 

21). Hobart: MERGA. 
Chinnappan, M., & Lawson, M. (2005). A framework for analysis of teachers' geometric content 

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197-221. 
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Washington, DC: American Mathematical Society and Mathematical Association of America. 
Cooney, T., & Wiegel, H. (2003). Examining the mathematics in mathematics teacher education. In A. 

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mathematics education (pp. 795-828). Dordrecht, the Netherlands: Kluwer Academic Publishers. 
Davis, P., & Hersh, R. ( 1982). The mathematical experience. Boston, MA: Houghton Mifflin Company. 
Goulding, M , Hatch, G , & Rodd, M. (2003). Undergraduate mathematics experience: Its significance 

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393. 
Graham, K.., Li, Y., & Curran Buck, J. (2000). Characteristics of mathematics teacher preparation 

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Harris, K-L., & Jensz, F. (2006). The preparation of mathematics teachers in Australia: Meeting the 

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University of Melbourne. 
Henry, M. (2000). Evolution and prospects of preservice secondary mathematics teacher education in 

France. Journal of Mathematics Teacher Education, J. 271 -279. 
Hiebert, J. et al. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video 

study. Washington, DC: National Center for Education Statistics. 
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Ill 



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HREF3 Training and Development Agency for Schools. Qualifying lo leach. Professional standards for 

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Accessed Jan 6, 2008. 
HREF4 International Study Group on the relations between the history and pedagogy of mathematics 

http://www.clab.edc.uoc.gr/HPM/INDEX.HTM Accessed Jan 6, 2008. 
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Scotland. http://www-groups.dcs.st-andac.uk/~history/HistTopics/Matrices_and_determinants.html. 

Accessed Jan 6, 2008. 
Kendal, M., & Stacey, K. (2001). The impact of teacher privileging on learning differentiation with 

technology. International Journal of Computers for Mathematics Learning, 6, 143-165. 
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. 

Washington, DC: National Academy Press 
Lachance, A., & Confrey, J. (2003). Interconnecting content and community: A qualitative study of 

secondary mathematics teachers. Journal of Mathematics Teacher Education, 6, 1 07- 1 37. 
Le Metais, J. (2002). International Developments in Upper Secondary Education: Context, provision 

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Mason, J., Burton, L., & Stacey, K. ( 1 982). Thinking mathematically. London: Pearson. 
McNamara, O., Jaworski, B., Rowland, T., Hodgen, J., & Prestage, S. (2002). Developing mathematics 

teachers and teaching, http://www.maths-ed.org.uk/mathsteachdev/ 
Mills, T. M. (2007). A course on the history of mathematics The Australian Mathematics Society 

Gazette. 34(4), 195-201. 
Nicol, C. (2002). Where's the math? Prospective teachers visit the workplace. Educational Studies in 

Mathematics, SO, 289-309. 
Organisation for Economic Co-operation and Development. (2003). The PISA 2003 assessment 

framework: Mathematics, reading, science and problem solving knowledge and skills. Paris: Author. 
Pierce, R., Turville, C, & Giri, J. (2003). Undergraduate mathematics curricula - A new angle. New 

Zealand Journal of Mathematics, 32 (Supp. Issue), 155-162. 
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New York: John Wiley. 
Robert, A., & Hache, C. (2000). Connecting research to French mathematics teacher education. Journal 

of Mathematics Teacher Education, 3. 28 1 -290. 
Ryve, A. (2007). What is actually discussed in problem-solving courses for prospective teachers? 

Journal of Mathematics Teacher Education, 10, 43-6 1 . 
Schoenfeld, A. (1985). Mathematical problem solving. Orlando, FL: Academic Press. 
Seaman, C, & Szydlik, J. (2007). Mathematical sophistication among pre-service elementary teachers. 

Journal of Mathematics Teacher Education, 10, 167-182. 
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Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum documents. 

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Stephens, M. (2003). Regulating the entry of teachers of mathematics into the profession: Challenges, 

new models and glimpses into the future. In A. Bishop, M. Clements, C. Keitel, & F-K. Leung 

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Van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service teachers' preferred strategies for 

solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6, 27- 

52. 
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Education, 24, 173-209. 



Kaye Stacey 

Melbourne Graduate School of Education, 

University of Melbourne 

Australia 



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SECTION 2 
MATHEMATICS FOR AND IN TEACHING 



ANNA GRAEBER AND DINA TIROSH 

5. PEDAGOGICAL CONTENT KNOWLEDGE 

Useful Concept or Elusive Notion 



Shulman's 1985 presidential address at AERA is frequently credited with 
launching increased attention to knowledge unique to teaching. In that address, 
Shulman coined the term Pedagogical Content Knowledge (PCK) and described it 
as "the particular form of content knowledge that embodies the aspect of content 
most germane to its teachability" (1986a, p. 9). Since then, the notion of PCK has 
been widely used in framing and describing research and practice in teacher 
education in many fields of education, including mathematics teacher education. 
In this chapter we first describe Shulman and others ' attempts to define those 
aspects of teachers ' knowledge that are unique to teachers and are related to their 
students ' learning. We then consider the role of pedagogical content knowledge in 
mathematics teacher education. We end the chapter with reflections and future 
directions. 

We begin by posing a task for readers and ask that they take some time to construct 
such a task before continuing. 

Write a multiple-choice item concerning decimal numbers that teachers will 
be able to answer but which will be a challenge to those in most other 
professions including mathematicians. 

Hill, Schilling, and Ball (2004. p. 28) offer one example of such an item, as shown 
in Figure 1. 



Mr. Fitzgerald has been helping his students learn how to compare decimals. 
He is trying to devise an assignment that shows him whether his students 
know how to correctly put a list of decimals in order of size. Which of the 
following sets of numbers will best suit this purpose? 

a) .5 7 .01 11.4 

b) .60 2.53 3.14 .45 

c) .60 4.25 .565 2.5 

d) Any of these would work well for this purpose. They all 
require the students to read and interpret decimals. 

Figure 1. A task designed to tap an aspect of teacher knowledge. 



P. Sullivan and T. Wood (eds.). Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 117-132. 

© 2008 Sense Publishers. All rights reserved. 



ANNA GRAEBER AND DINA TIROSH 

Teachers' knowledgeable about teaching decimals are most apt to know that 
students can order the decimals in two of the options, a and b, by ignoring the 
decimal points and ordering the resultant whole numbers (e.g., for option a they 
consider 5, 7, I, 1 14 which they order correctly as 1,5, 7,1 14 - a correct order for 
.01, .5, 7, 1 1 .4). However such an approach to option c yields 2.5, .60, 4.25, .565, 
an incorrect ordering. Hill, Schilling, and Ball (2004) report that most non-teachers 
including mathematicians select option d, seeing no differences in the options 
offered. However teachers' familiarity with students' tendency to over generalise 
knowledge of whole numbers to decimal numbers example enables them to make 
the required distinction. This reliance on whole number ideas has been well 
documented (e.g., Grossman, A. S., 1983; Hiebert & Wearne, 1986; Sackur- 
Grisvard & Leonard, 1985) and has made its way into teacher education and 
teacher professional development materials (e.g., Barnett, Goldstein, & Jackson, 
1994; Sconiers, Isaacs, Konold & McFadden, 1999; Van de Walle, 2007). The 
knowledge required to answer the item above is therefore somewhat unique to 
teachers. It requires more than knowledge of mathematics, namely knowledge of 
how students approach such tasks. This is one aspect of a construct that is 
frequently labelled pedagogical content knowledge, a term most closely identified 
with Shulman. 

The driving question that led Shulman to formulate the term "pedagogical 
content knowledge" was - Are there specific types of knowledge needed for 
teaching? In this chapter we first describe Shulman and others' attempts to define 
those aspects of teachers' knowledge that are unique to teachers and are related to 
their students' learning. We also consider the role of pedagogical content 
knowledge in teacher education, and we end the chapter with reflections and future 
directions. 

WHAT IS PEDAGOGICAL CONTENT KNOWLEDGE? 

We begin by describing pedagogical content knowledge (PCK) as used by 
Shulman. We then outline how he and his colleagues' initial ideas were 
subsequently modified, critiqued, and expanded. Next we present some of the work 
of Ball and her colleagues that both expands on Shulman's notion of PCK and 
provides empirical support that such knowledge for teaching is indeed important 
for student learning. Finally we discuss the status of PCK in 2007. 

Early Formulations of PCK 

Shulman's 1985 presidential address at AERA is frequently credited with 
launching increased attention to knowledge unique to teaching. In that address 
(published in 1986), he proposed three categories of content knowledge: subject 
matter knowledge, pedagogical content knowledge and curricular knowledge. 
Shulman described PCK as "the particular form of content knowledge that 



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PEDAGOGICAL CONTENT KNOWLEDGE 

embodies the aspect of content most germane to its teachability" (p. 9). In the 1985 
address, Shulman considered PCK to include 

for the most regularly taught topics in one's subject area, the most useful 
forms of representation of those ideas, the most powerful analogies, 
illustrations, examples, explanations, and demonstrations ... an under- 
standing of what makes the learning of specific topics easy or difficult; the 
conceptions and preconceptions that students ... bring with them ... 
knowledge of strategies most likely to be fruitful in reorganising the 
understanding of learners. (1986a, pp. 9-10) 

Shulman also characterised PCK as "that special amalgam of content and pedagogy 
that is uniquely (emphasis added) the province of teachers, their own special form 
of professional understanding" (p. 8). 

In Shulman and his colleagues later writings (e.g., Grossman, P., 1990; Marks, 
1990; Wilson & Wineburg, 1988) the characterisation of PCK was expanded and 
elaborated, especially through the work of case studies of teachers working in 
different subject areas. P. Grossman (1990), who studied secondary English 
teachers, suggested four major components of PCK: knowledge of students' 
understanding, knowledge of curriculum, knowledge of instructional strategies, and 
purposes for teaching. Thus, P. Grossman expanded PCK to include aspects of 
Shulman 's category of curricular knowledge. She also raised issues about the 
relationship of beliefs and values to PCK. 

Adapting PCK to mathematics education, Marks (1990) interviewed fifth-grade 
teachers concerning their teaching of equivalence of fractions. He then suggested 
four components of PCK: students' understanding, subject matter for instructional 
purposes, media for instruction, and instructional processes. Marks elaborated on 
each of these, and upon close examination one finds that he has included aspects of 
Shulman 's original category of curriculum knowledge in three of these 
components. For Marks, "subject matter for instructional purposes" included: 
"purposes of math instruction, justifications for learning a given topic, important 
ideas to teach a given topic, prerequisite ideas for a given topic, and typical 'school 
math' problems" (p. 5). Another contribution of Marks was his elaboration of the 
notion of students' understanding to include student learning processes, typical 
understandings, common errors, and those things that are hard or easy for students. 
Here, Marks also included knowledge of particular students' understanding. These 
modifications are consistent with Shulman's 1987 reflection that "the knowledge 
base for teaching is not fixed and final" but that much "remains to be discovered, 
invented, and refined" (p. 12). 

Subsequent Extensions and Challenges 

Indeed, both extensions and criticisms of Shulman's notion of PCK ensued. 
Shulman's claim that PCK "goes beyond subject matter per se" (1986a, p. 9) was 
the target of some criticism. Others interpreted Shulman's early work as reflecting 



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ANNA GRAEBER AND DINA TIROSH 

only a direct instruction view of teaching or ignoring the role of teacher beliefs and 
values. Other researchers commented that the boundaries among PCK, content 
knowledge and pedagogical knowledge were blurred, or, that aspects of content 
knowledge or pedagogical knowledge should be included in PCK. 

McEwan and Bull (1991) objected to the notion of pedagogical content 
knowledge, arguing against the division of subject matter knowledge into scholarly 
and pedagogic forms. These authors claim that "all subject matter knowledge is 
pedagogic" (p. 318). In fact, McEwan and Bull argued that "the distinction 
between content knowledge and pedagogical content knowledge introduces an 
unnecessary and untenable complication into the conceptual framework on which 
the research is based" (p. 318). Their objection is philosophically based, describing 
Shulman's work as harbouring an objectivist theory of knowledge. McNamara 
(1990) also questioned whether the distinction between pedagogical content 
knowledge could or should be made, but he acknowledged Shulman's contribution 
of helping to focus the profession on the "essential purpose of teaching which is 
the passing on of knowledge (however broadly or narrowly defined)" (McNamara, 
1990, p. 157). 

Other critics such as Meredith (1993, 1995) and Cochran, DeRuiter, and King 
(1993) argued that Shulman reflected "a teacher-directed didactical model of 
teaching" (Meredith, 1995, p. 176), a view of teaching that we note is consistent 
with an objectivist theory of knowledge. However, unlike McEwan and Bull who 
would dismiss the very notion of PCK, Meredith found the framework useful but 
called for a broadening of the framework to permit alternative forms of teaching. 
Cochran et al. (1993) also found Shulman's framework to focus so heavily on the 
"transformation of subject matter" (p. 266) that they were driven to modify the 
notion of PCK to that of Pedagogical Content Knowing (PCKg). For these authors 
the use of the form "knowing" signified an active process that is more consistent 
with a constructivist view of learning. They defined PCKg as "a teacher's 
integrated understanding of four components of pedagogy, subject matter content, 
student characteristics and the environmental context of learning" (p. 266). 

Several researchers have proposed elaborations of aspects of PCK, especially 
attention to knowledge of students. In 1992, Fennema and Franke's chapter in the 
Handbook of Mathematics Teaching and Learning identified not only knowledge 
of content, but two other types of knowledge that are generally categorised within 
pedagogical content knowledge: knowledge of students; and knowledge of 
representations. Fennema and Franke's chapter raised teachers' knowledge of 
students to an important position. In considering teachers' knowledge of students, 
these authors presented evidence to show that knowledge of student's processes 
and thinking impacts teacher decision-making, allows teachers to attend to 
individual students, and influences educational outcomes. They also discussed 
teacher knowledge of mathematical representations in a manner that parallels 
Shulman's ideas about the importance of knowing useful representations, analogies 
and examples. In a similar way, Even and Tirosh (1995) emphasised the 
importance of studying the sources of knowledge that teachers use in responding to 



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PEDAGOGICAL CONTENT KNOWLEDGE 

students' questions, ideas or hypotheses. One major source that teachers relied 
upon in order to formulate responses was knowledge about students. In an effort to 
unpack specific components of "knowledge of students", Even and Tirosh 
identified two, knowing that and knowing why. Knowing that refers to research- 
based and experienced-based knowledge about students' common conceptions and 
ways of thinking about the subject matter. Knowing why refers to general 
knowledge about possible sources of these conceptions, and also to the 
understanding of the sources of the specific students' reaction in a specific case. 

A relatively recent framework derived from Shulman's work is offered by An, 
Kulm, and Wu (2004). These authors view PCK as including aspects of content, 
teaching, and curriculum; identifying teaching as the most important. An and 
colleagues characterise "profound pedagogical content knowledge" as broad and 
deep knowledge of teaching and curriculum. This concept parallels Ma's (1999) 
notion of "profound understanding (emphasis added) of fundamental mathematics" 
Ma characterised differences in understanding among teachers using the notion of 
"profound understanding of fundamental mathematics" as involving 
connectedness, multiple perspectives, basic ideas, and longitudinal coherence (p. 
122). While generally more closely associated with content knowledge, Ma's 
descriptions also include knowledge that might be classified as pedagogical content 
knowledge or curriculum knowledge. For example, she argued that teachers with 
"deep, vast, and thorough understanding do not invent connections between and 
among mathematical ideas but reveal and represent them in terms of mathematics 
teaching and learning" (p. 122). Because such teachers understand which ideas are 
basic, they "revisit and reinforce these ideas" (p. 122). Thus Ma's concern was not 
simply with teachers' knowledge of mathematics but also with knowing when and 
how to use such knowledge in teaching. The aspects of content that An and 
colleagues include in their PCK, are consistent with the way Ma views knowledge 
of content in use in teaching. 

An and colleagues acknowledged that knowledge of teaching includes preparing 
instruction and strategies for delivery of instruction. But, given their perspective of 
teaching for understanding, they identify "knowing students' thinking" as most 
critical. An and colleagues developed the following set of categories for knowing 
students' thinking: building on students' ideas, addressing students' 
misconceptions, engaging students in mathematics learning, and promoting 
students' thinking about mathematics (p. 155). These categories were defined by 
listing components of each. For example, the misconceptions category included 
"address students' misconceptions, use questions or tasks to correct 
misconceptions, use rule and procedure, draw picture or table, and connect to 
concrete model" (p. 1 55). 

PCK AND MATHEMATICAL KNOWLEDGE FOR TEACHING 

The notion of PCK dominated the literature for almost twenty years. In the last few 
years another construct built on Shulman's notion of PCK but specific to 
mathematics education has begun to gain much attention. The recent work of Ball 



121 



ANNA GRAEBER AND DINA TIROSH 

and Bass (2003) introduced the global term mathematical knowledge for teaching. 
This term was derived from their approach to uncovering the knowledge needed 
for teaching by looking at the work of teachers. Ball and Bass (2003) documented a 
year of teaching in a third-grade mathematics class in an attempt to tease out "core 
task domains of teachers work" (p. 6). As examples they cite "representing and 
making mathematical ideas available to students; attending to, interpreting, and 
handling students' oral and written productions; giving and evaluating 
mathematical explanations and justifications; and establishing and managing the 
discourse and collectivity of the class" (p. 6). The analyses of teacher work led to a 
framework of teacher knowledge that included two major aspects, subject matter 
knowledge and pedagogical content knowledge. Subject matter or content 
knowledge, comprises common content knowledge and specialised content 
knowledge. Common knowledge of content included what any well informed 
citizen would know. For example, what decimal is halfway between 1.1 and 1.11. 
An example of specialised content knowledge is shown below in Figure 2. Hill and 
colleagues (2007) describe such knowledge as "mathematical knowledge that is 
used in teaching, but not directly taught to students" (p. 132). Analysis of the 
student work and knowledge of whole number properties are needed to conclude 
that all three of the students have employed a method valid for any two whole 
numbers. The knowledge of mathematics is what is critical, but the item represents 
the type of mathematical task that is done by few adults other than teachers. 

Imagine that you are working with your class on multiplying large numbers. 
Among your students' papers, you notice that some have displayed their 
work in the following ways: 



Student A 


Student B 


Student C 


35 


35 


35 


\25 


x25 


\25 


125 


175 


25 


75 


700 


150 


875 


875 


100 
600 

875 



Which of these students is using a method that could be used to multiply any 
two whole numbers? (Consortium for Policy Research in Education, 2004, p. 
19) 

Figure 2. An item that was designed to tap specialised content knowledge. 

Pedagogical content knowledge comprises knowledge of content and students, 
knowledge of content and teaching, and knowledge of curriculum. The decimal- 
ordering item presented near the beginning of this paper is an item that was 
designed to measure knowledge of content and students. It taps into teachers' 



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PEDAGOGICAL CONTENT KNOWLEDGE 

knowledge of how "students learn content," the characterisation Hill, Sleep, Lewis 
and Ball (2007) gave for knowledge of content and students (p. 133). In the case of 
Figure 1, this is the knowledge that students will likely apply whole number ideas 
to decimals. The item shown in Figure 3 is an example of an item designed to test 
knowledge of content and teaching. As the numbers given in the four choices are 
all correctly categorised as prime or composite, this item depends less on teachers 
knowing which numbers are prime and which are composite (content knowledge) 
but more on how to choose examples and representations (an aspect of the work 
of teaching) given knowledge of students' thinking. Hill and colleagues argued 
that success with the item in Figure 3 is dependent on teachers knowing that 
students have difficulty categorising 2 as a prime and are "likely to think that odd 
numbers cannot be composite" (p. 133). Hence choice d), with 2 listed as a prime 
and with its odd composite, 9, is the only set that challenges both of these faulty 
ideas. 

While planning an introductory lesson on primes and composites, Mr. 
Rubenstein is considering what numbers to use as initial examples. He is 
concerned because he knows that choosing poor examples can mislead a 
student about these important ideas. Of the choices below, which set of 
numbers would be best for introducing primes and composites? (Mark one 
answer.) (Hill et al., 2007, p. 133) 





Primes 


Composites 


a) 


3,5,11 


6, 30, 44 


b) 


2,5 17 


8, 14, 32 


c) 


3,7,11 


4, 16, 25 


d) 


2, 7, 13 


9, 24, 40 


e) 


All of these would work equal! 



prime and composite numbers. 

Figure 3. An item that was designed to tap knowledge of content and teaching. 

Hill, Schilling, and Ball (2004) administered tests constructed around such 
categories and found that knowledge of content in a topical areas (e.g., number and 
operations or patterns, functions and algebra) differed from knowledge of students 
in that particular area. 

This and other analyses (Ball & Rowan, 2004; Hill, Rowan, & Ball, 2005) 
support Shulman's claim that knowledge for teaching includes a specialised 
knowledge of content. This seems at least a start in addressing earlier concerns 
about whether the distinction between content knowledge and pedagogical content 
knowledge could or should be made. 



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ANNA GRAEBER AND DINA TIROSH 

PCK IN 2007 

At the beginning of this section we raised the question: What is PCK? Here, we 
attempt to provide a response from the existing literature. Then we examine the 
extent to which the benefits Shulman anticipated from a framework of knowledge 
for teachers have accrued. 

In reviewing the ways in which researchers have extended and modified 
Shulman's framework, we note that the direction of this evolution was to widen the 
definition. Researchers included within PCK aspects of Shulman's domains of 
subject matter content knowledge and curricular knowledge. The more recent 
frameworks (e.g., An, Kulm & Wu, 2004; Ball & Bass, 2003) identify aspects of 
content, teaching and curriculum within PCK. Within each of these domains, 
different researchers have explicated and stressed different concepts. Thus, Ma 
(1999) gives the field one way to think about content knowledge for teaching and 
Even and Tirosh (1995) another. Much interest has focused on characterising the 
"knowledge of students" sometimes included within the broader domain of 
teaching. The five components Marks (1990) offered (see above) overlap but differ 
from An et al.'s (2004) four components: students' prior knowledge; common 
misconceptions and strategies for amending; methods of engaging students; and 
methods of promoting students' thinking. These differences and the lack, at least 
to-date, of a widely agreed upon characterisation of PCK, suggest that while 
progress has been made, much remains to be done. The fact that many researchers 
do not offer a definition of PCK but rather attempt to characterise it with lists or 
examples is another indication that the concept is still somewhat ill defined. 

In pursuing work on PCK, a number of questions, not all unrelated, have arisen. 
These include: 1) the role of beliefs and values in the development of a teacher's 
PCK, 2) whether different teaching/learning paradigms require different 
components of PCK, and 3) what are improved methods for assessing PCK. 

A crucial trait of a valuable framework of teacher knowledge is the extent to 
which it identifies that knowledge needed for student learning and understanding. 
In fact, some limited progress has been made in identifying knowledge that is 
related to student achievement. Hill, Rowan, Ball (2005) found that teachers' 
mathematical content knowledge for teaching was positively related to first and 
third graders' achievement on a standardised achievement test. It is also the case 
that there is some research showing that individual components of PCK are 
positively related to student achievement. While such studies were not directed at 
establishing a foundation of an entire PCK framework, they may nevertheless 
testify to the importance of some individual components of PCK (e.g., knowledge 
about how students generally learn a topic as well as knowledge of specific 
students' thinking [Fennema, Carpenter, & Franke, Levi, Jacobs, & Empson, 
1996]; and knowledge of and attention to misconceptions [Tsamir, 2005]). 

Shulman began his quest for a framework for teacher knowledge at a time that 
he and others lamented the lack of attention given to the role of subject matter in 
research used to study teaching. He wished to bring to the fore questions such as 



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PEDAGOGICAL CONTENT KNOWLEDGE 

"Where do teacher explanations come from?, How do teachers decide what to 
teach and how to represent it?, etc." Such questions certainly have been addressed 
in research since 1985. Research by Lampert (2001) and by Sowder, Philipp, 
Armstrong, and Schappelle (1998) are but two examples of different genres of 
research on teaching that attend to the role of subject matter. Further, Shulman's 
categories have been used to frame research on teachers' knowledge of various 
topics. For example, Watson (2001) collected information about teachers' 
knowledge of statistics using Shulman's knowledge types to structure her 
questionnaire. Kieran (2007) noted that Shulman's PCK construct is the most 
widely used framework in studies of algebra teachers' knowledge (p. 739). Jones, 
Langrall and Mooney (2007) structure their review of teacher knowledge related to 
probability around Shulman's notions of content knowledge and pedagogical 
content knowledge by giving special attention to the knowledge of student 
understanding aspect of PCK (p. 932). Clearly Shulman's work has influenced 
research on teaching and teachers that does focus on subject matter, and it has also 
given researchers a way to organise the findings of such research. 

A hope that Shulman had for a framework of teacher knowledge was the 
development of instruments that would test aspects of knowledge unique to 
teachers. He hoped for tests that could only be successfully completed by those 
specifically prepared to teach. If one believes that such knowledge is important to 
student learning and understanding, they could reasonably be used in a number of 
ways. For example, they could provide valuable information for policy makers, 
assist in the evaluation and modification of teacher education programmes, could 
conceivably be used in decisions about hiring, and provide information about 
individual teacher's strengths and needs. Given the paucity of empirical data on the 
various frameworks, it is clear that much research is needed in this area. The 
difficulty of obtaining such data, which involves testing of teachers and educated 
non-teachers, as well as teachers and their students, makes the task both practically 
and conceptually challenging. As noted above, Ball and colleagues have provided 
some data that positively relates aspects of teachers' PCK to student learning. 
Other projects attending to PCK, such as Knowing for Algebra Teaching (Ferrini- 
Mundy, Floden, McCory, Burrill, & Sandow, 2005,), Diagnostic Teacher 
Assessments in Mathematics and Science (Bush, Ronau, Brown, & Myers), have 
constructed measures for the ultimate purpose of relating teacher knowledge to 
student achievement. 

Shulman's overarching goal for a framework of teacher knowledge was that it 
be used to inform and legitimise the content of teacher education programmes 
(1986a, pp. 13-14) thereby establishing teaching as a profession with a well 
defined knowledge base, along with medicine and law. That is that the framework 
would both aid in the design of teacher education programmes and drive studies in 
order to establish a firm research base for teacher education. In the next section we 
consider the use of PCK components in mathematics teacher education. 



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PCK AND MATHEMATICS TEACHER EDUCATION 

Designing a teacher education programme around elements of PCK would be an 
enormous task. First there are numerous aspects of PCK to consider. Then there are 
questions about when and how such knowledge is included and conveyed. In this 
section we first consider challenges involved in attempting to use components of 
the PCK framework as a basis of designing a mathematics teacher education 
programme. In this example we draw on knowledge of students' thinking. We then 
describe how one researcher addressed each of those challenges. In that case the 
attempt was to draw on the PCK component, knowledge of representations/ 
explanations. 

Design Challenge 

Shulman proposed that teacher education should "draw upon the growing research 
on the pedagogical structure of students' conceptions and misconceptions, on those 
features that make particular topics easy or difficult to learn" (1986, p. 14). While 
this may sound like a rather straight-forward task, a closer examination reveals the 
many challenges that are involved in such an endeavour. 

For example, suppose one were to decide to include knowledge of student 
conceptions in a mathematics teacher education programme. In the last decades 
many researchers have investigated students' mathematical ideas and conceptions 
in many topic areas, and there are numerous theories about the origin and impact of 
students' naive conceptions on learning. Thus, among the first question one would 
face is, what (mis)conceptions' should one include? For example, a quick answer 
might be - choose the most salient conceptions. However, students' conceptions 
may differ according to the curricula they study, the classroom practices they 
experience, and other factors. The extent to which students' mathematical ways of 
thinking and difficulties are embedded in a particular approach to learning and 
teaching still needs to be studied. For instance, consider a curriculum in which 
percent is taught prior to decimal numbers. What types of (mis)conceptions might 
arise? Will they differ from those prompted by the more usual curriculum that 
places decimals before percents? Would this have any impact on the ordering-of- 
decimal difficulty highlighted in the task presented at the outset of this chapter? In 
addition to examples of (mis)conceptions in a specific domain, what, if any 
theories, concerning the origin or impact of (mis)conceptions should be included? 

Suppose one has made the decision about what to teach, another question is - 
how is this knowledge addressed with teachers? Some options are: reading about 
conceptions, watching videos that capture students expressing (mis)conceptions, 
observing a class lesson, tutoring student. Which of these options, in what 
combination, and in what order should they be employed? Practical issues such as 
time and curriculum space will limit these options. 

A third question that needs to be addressed is - when should this be addressed? 
Is a topic primarily for prospective or practising teachers' education? Which is 



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optimal? Fennema et al. have suggested that there is some knowledge that can 
"only be acquired in the context of teaching mathematics" (1996, p. 432). There 
are conflicting perspectives as to when in the teacher's development is it best to 
incorporate attention to specific aspects of PCK? 

Design Example 

Kinach (2002), in describing a process she employed to transform prospective 
teachers instructional explanations from merely instrumental to relational 
(following Skemp, 1978), provided her answers to the what, when, and how 
questions posed above. Kinach selected as her aim, the what, the transformation of 
explanations from instrumental to relational. She observed that many prospective 
teachers know a great deal of mathematics but only instrumentally, and if we wish 
them to teach with relational understanding, their education must include 
opportunities for making such a transformation. For this particular article (2002), 
Kinach chose to illustrate the process using the topic, subtraction of integers. She 
selected this topic because she found that it is accessible but one in which 
prospective teachers' knowledge is generally limited to instrumental 
understanding. Kinach noted that she has used the same process (described below) 
with other computational as well as non-computational topics. We draw on 
Kinach's work not to hold up her resulting intervention as exemplary, although it 
may be so, but rather to illustrate one way in which a researcher used information 
about pedagogical content knowledge to design experiences in a teacher education 
programme. 

For Kinach 'a methods course' answered the question of when to include this 
work. Her rationale was that in the teacher education curriculum the methods 
course is where "content and pedagogy are most likely to interact" (p. 65). Kinach 
noted that some teacher educators have argued that the methods course is not an 
appropriate site for changes in prospective teachers' views, because they are unable 
to reflect on such intangible educational goals (p. 68). However, Kinach argued 
that if a sufficiently powerful method is skilfully used; prospective teachers could 
make such a transformation. 

The process, the how; which Kinach developed, employed, and evaluated a 
number of times; involved the following five-steps: 

(1) Evoke an explanation of integer subtraction from each of the prospective 
teachers. 

(2) Using the Perkins and Simmons' (1988) framework of four levels of 
understanding (concept level, problem solving level, epistemic level, inquiry 
level), have the class determine the level of understanding reflected in the 
initial explanations. 

(3) Through discussion generate tension between the level of understanding 
reflected in the original explanation and that needed for effective teaching. 

(4) Ask the prospective teachers to construct a second explanation for integer 
subtraction in another context known to present difficulties (e.g., number- 
line). Evaluate these explanations, using Perkins and Simmons' framework, in 

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ANNA GRAEBER AND DINA TIROSH 

class discussion and reshape explanations to match the problem solving and 

epistemic levels of the framework. 
(5) Ask the prospective teachers to explain integer subtraction for a second 

context (e.g., algebra tiles). This second context is selected so that it will 

scaffold students' problems encountered in the number-line context. 
Based on an evaluation of the levels of explanation reflected in steps 1, 4, and 5 
above along with qualitative date gathered from video recordings of class 
discussions, student journal entries, and various homework assignments, Kinach 
concluded that this process was effective for helping the prospective teachers make 
the desired transformation to explanations that reflected relational understanding. 
This an example of an educator who drew on the PCK framework in designing 
experiences in teacher education. There is a need for empirical data for more 
ventures of this type. 

REFLECTIONS AND FUTURE DIRECTIONS 

We have traced, in this chapter, the evolution of PCK over the last twenty years. 
We began our discussion of PCK without noting that this notion has roots in earlier 
research and theory, Shulman (1987) cited Dewey, Scheffler, Green, Schwab, 
Fenstermacher, and others (p. 4) as researchers on whose work he drew. He also 
acknowledged that prominent others were working on similar questions (e.g., 
Leinhardt, Anderson & Smith) (Shulman, 1986a, pp. 8-9). This work continues by 
researchers within mathematics education (e.g., Wilson, Floden, & Ferrini-Mundy, 
2001; An, Kulm, & Wu, 2004) and in other subject matter areas (e.g., Grossman, 
P., 2005). 

The term, "pedagogical content knowledge" appears frequently in both scholarly 
papers and publications intended for less specialised audiences, a phenomenon that 
could be interpreted as an indication of the term's usefulness. It has helped frame 
the way researchers look at teachers' work and it has stimulated and guided 
considerable research on teacher knowledge. It has influenced mathematics teacher 
education and is beginning to influence teacher assessment (e.g., the recent work 
undertaken by Ball and colleagues in the United States and in the COACTIV 
project, the Cognitive Activation in the Classroom, undertaken in Germany by 
Baumert, Krauss, & Kunter, 2003) 2 . This term is now part of the mathematics 
educator's lexicon and has even found its way into the title of mathematics 
methods textbooks, (e.g., Elementary mathematics: Pedagogical content 
knowledge; Schwartz, 2008). However, we have shown that even within 
mathematics education those who have built upon Shulman's work in PCK have 
tended to use slightly different, broader definitions than did Shulman, and not all of 
these later definitions are equivalent to one another. This is one reason why the 
notion of PCK is still considered a bit elusive. Evidence that such knowledge is 
unique to teachers and more evidence that it is positively related to student 
achievement in mathematics will aid in the field's acceptance of PCK as a less 
elusive and a more useful construct. 



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We feel quite confident in conjecturing that in the coming years the mathematics 
education community will devote growing efforts to defining the specific types of 
knowledge needed for teaching mathematics and to provide evidence that these 
types of knowledge positively affect students' achievement. Our conjecture is 
based, among other factors, on the intense political pressure in many countries 
including Britain, Germany Israel, and the US, to raise "student achievement" in 
mathematics. The widespread acknowledgment of the critical role teachers have on 
student achievement leads to an increased interest in teacher knowledge and 
teacher "quality" that drives the search for ways of assessing teachers. Which 
teacher candidate should be hired? Which teachers are in "need" of professional 
development and on what should that professional development concentrate? Thus 
we believe that the current interest in defining the pedagogical, as well as subject 
matter, knowledge for teaching mathematics will remain. This effort appeals to 
researchers in mathematics education. For some it is a means of establishing the 
existence of important knowledge for teaching that is not simply more 
mathematics. As such it is also a part of the larger quest to establish a unique base 
of professional knowledge. 

We would like to end this chapter by stating that we view the attempts to define 
the construct PCK within mathematics education as part of that larger quest to 
establish a unique base of professional knowledge, a hallmark of a true profession, 
for teachers of mathematics. Progress has been made, but much remains to be 
done. Further research and theory development are needed around profound 
questions concerning PCK, questions that are closely related to those originally 
posed by Shulman (1986b): 
Are different components of PCK associated with different definitions of "quality 

teaching" in mathematics? 
How specific must the teaching for PCK be? Are general instances illustrated 

with examples sufficient, or must specific element of PCK be identified and 

taught for each topic in mathematics education? 
What is the standard for assessing the validity of proposed components of PCK 

for teaching mathematics? 
What methods of assessing teachers' PCK are both valid and practical in large- 
scale studies? 
Which components of PCK are best considered in the education of prospective 

teachers, and which are best learned in later professional development or, on 

the job? 

GENERAL NOTES 

1. We recognise that the use of the term "misconception" is particularly controversial in 
mathematics (as well as science) education. Some prefer naive conceptions, child theories, or 
alternative conceptions. 

2. See also Krauss, S., Kunter, M., Brunner, M, Baumert, J., Blum, W., Neubrand, M., Jordan, 
A., & Lowen, K. (2004). COACTIV: Professionswissen von Lehrkraften, kognitiv 
aktivierender Mathematikunterricht und die Entwicklung von mathematischer Kompetenz. In 
J. Dool & M. Prenzel (Eds.), Bildungsqualitat von Schule: Lehrerprofessionalisierung, 



129 



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Unterrichlsentwicklung und Schiilerforderung als Strategien der Qualitatsverbesserung (pp. 
3 1-53). Munster: Waxmann. 

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Anna O. Graeber 
College of Education 
University of Maryland, USA 

Dina Tirosh 

School of Education 

Tel Aviv University, Israel 



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6. THE MATHEMATICS TEACHER AS 
CURRICULUM MAKER 

Developing Knowledge for Enacting Curriculum 



This chapter outlines a process by which a teacher begins with the intended 
curriculum as outlined in curriculum frameworks, guidelines or standards and 
enacts it. There is a discussion, placed within a narrative involving the teaching of 
fraction content, of how a teacher might identify the big ideas within a topic, 
sequence concepts within that topic, recognise and enhance connections between 
concepts, and match the curriculum to the developing understanding of students. 
The discussion also includes a consideration of the kinds of knowledge which a 
teacher might draw upon when being a curriculum maker, and some of the 
constraints which may prevent a teacher from fully enacting this role. The chapter 
concludes with a discussion of effective approaches to preparing prospective and 
practising teachers to be active curriculum makers, through appropriate 
professional development. 

INTRODUCTION 

Few people would disagree that the teacher is the key to worthwhile mathematical 
experiences for students in our schools. Although a whole range of circumstances 
need to be in place for teachers to carry out their work effectively, there is 
overwhelming evidence that two teachers working in similar circumstances with 
similar supports and constraints can provide quite different learning experiences 
for students (e.g., McDonough & Clarke, 2003). 

In this chapter, I take the mathematics curriculum to be the teacher's 
experiences, intended or not, which occur within the mathematics classroom, and 
the processes on the part of the teacher leading up to the enactment of curriculum, 
including preparation and planning. This is a broader definition than that of some 
scholars. For example, Clements (2007) defined curriculum as "a specific set of 
instructional materials that order content used to support pre-K to grade 12 
classrooms" (p. 36). Of course, my definition does not include all mathematical 
learning opportunities for teachers, but it clarifies my emphasis in this chapter, and 
allows me to focus on the teacher's role in the enactment of the mathematics 
curriculum. 

A number of writers have distinguished between the intended curriculum, the 
implemented curriculum, and the attained curriculum (e.g., Robitaille et al., 1993). 



P. Sullivan and T. Wood (eds.). Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 133-151. 

© 2008 Sense Publishers. All rights reserved. 



DOUG CLARKE 

Gehrke, Knapp, and Sirotnik (1992) contributed the planned, enacted and 
experienced curriculum to this list, while Burkhardt, Fraser, and Ridgway (1990) 
referred to the ideal, adopted, implemented, achieved, and tested curriculum. Given 
my particular focus in this chapter on what the teacher knows and does my interest 
here is mainly in the implemented curriculum — the ways in which a teacher takes a 
syllabus or curriculum guidelines or standards and enacts them in the classroom. In 
the following sections, I provide a brief discussion of some of the influences and 
constraints on the work of teachers in relation to curriculum, before turning to the 
teacher's role in implementing curriculum. 

INFLUENCES AND CONSTRAINTS ON MATHEMATICS TEACHERS 

There are many factors which impact on mathematics teachers (primary or 
secondary) being able to teach in the way they would wish and achieving outcomes 
of the kind desired by educational institutions and systems. I discuss several of 
these factors below. 

The Canonical Curriculum 

In most countries, syllabus documents that define the content and approach to 
instruction are prepared centrally for school systems, and yet the mathematics 
curriculum in countries across the world do not generally reflect the diversity of 
cultures and contexts; in other words they are quite similar (Clarke, D. M., Clarke, 
B. A., & Sullivan, 1996). Howson and Wilson (1986) described the almost 
universal adoption of what they called the school mathematics "canonical 
curriculum". For example, they noted that in Japan and Mexico, where 
approximately 95% and 3% respectively complete secondary school, there is 
remarkable similarity in the year-by-year syllabuses. Of course, there are many 
factors leading to this situation. As Wilson (1992) noted, in discussing the situation 
in Africa: 

What did it matter that the chapters on stocks and shares, and rates and taxes 
were meaningless to an African pupil? Such relevance was simply not a 
criterion [...]. Any deviation would have been regarded by well-educated 
Africans as having their children fobbed off with a second-best or watered- 
down education, (p. 127) 

It appears that the curriculum is so well established in many countries that societies 
do not wish to change it for fear of not operating a 'proper' education. 

The Influence of State and National Assessment Schemes 

Assessment at state and national levels appears to have a major influence in most 
countries on both content and teaching methods, particularly if the testing is 'high 
stakes'. As Wilson (2007) noted, the grade levels at which high-stakes testing 
occurred were more likely to experience more profound changes in curriculum and 

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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

instruction than those grades which were not included in the evaluation. Phelps 
(2000) who examined data from 31 countries and provinces, concluded that there 
was an overall net increase internationally in all testing from 1974-1999, therefore 
the influence of such testing is likely to have increased over time. The influence of 
testing regimes on content and instructional practices is often channelled through 
the school system via the principal to the teacher in a top down approach; few 
teachers can claim to be unaffected by its influence. 

Provision of Time and Resources 

There are some basic requirements for teachers to be able to carry out their role. 
They need to be supplied with appropriate resources, including appropriate texts 
(student and teacher), relevant student materials (such as manipulative materials), 
relevant technology, and an appropriate physical environment. Just as importantly, 
teachers need time: to plan, alone and with colleagues, to assimilate new content 
and pedagogy into their teaching repertoire. They also need sufficient time on the 
school programme for students to engage with mathematics. In my own country for 
example, the issue of the 'crowded curriculum' is a common topic of conversation. 

Large Scale Curriculum Development Projects 

Around the world, there are a large number of curriculum development projects, 
increasingly linked to some kind of curriculum standards. In the United States for 
example, there were two large-scale, high-profile curriculum development efforts, 
the 'new math' and the recent curriculum reform of the 1990s. The "new math" 
emerged in the 1960s and had considerable impact on not only the U.S. but many 
countries around the world. Then again in the 1990s there was a strong move 
towards reforming mathematics education through changes presented in the 
Curriculum and Evaluation Standards for School Mathematics (National Council 
of Teachers of Mathematics, 1989), backed by substantial funding from the 
National Science Foundation (NSF) for developing, implementing and evaluating 
reform-based instructional materials to support the curriculum transformation 
(Hirsch, 2007). A major feature of these more recent curriculum projects was the 
strong commitment to funding to support teachers' professional development, in 
the belief that the absence of such financial support would lead to little real change 
in teaching and therefore little impact in the classroom (Bradley, 2007). However, 
many countries lack these extensive funds and/or political will to provide support 
in either instructional materials or professional development programmes. 

The Ways in Which the Role of Teachers Are Perceived by Curriculum Developers 

The form, size and style of curriculum documents developed for classroom 
teachers often provides insights into the ways in which the role of the teacher is 
perceived by the authors of such documents. Where teachers are seen as key 
players in curriculum implementation, such documents often take the form of 

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DOUG CLARKE 

general guidelines, upon which a teacher can place her/his stamp, as they work 
together with colleagues to adapt materials to the perceived needs of their students. 

On the other hand, developers of highly prescriptive materials possibly think of 
teachers as incompetent or lacking experience, which are seen to be able to be used 
by any teacher. These materials are derived from an era in which curriculum 
developers attempted to provide 'teacher-proof materials in order to bypass the 
influence of the teacher on student learning. This approach however does not allow 
for the impact of context and culture on the way in which materials might be 
implemented in classrooms. The distinctions between the types of curriculum 
discussed earlier (intended, enacted, experienced, etc.) is a recognition that the 
notion of teacher-proof materials is nonsense — not possible and, I would argue, not 
at all desirable. 

In the remainder of this chapter, 1 focus on the teacher using the intended 
curriculum and the process by which she implements it, with a related discussion 
about teacher knowledge. In this scenario, I attempt to paint a picture of what could 
be described as an 'ideal' situation of a well-equipped and skilful teacher, who I 
will call Ms X, engaged in the implementation of relevant, meaningful and 
worthwhile learning experiences for her students. It is not claimed that this 
description is necessarily the norm; in fact it is clearly not. But the hope is that the 
reader will consider ways in which more teachers could be equipped and supported 
to prepare and teach in this way. I then discuss the knowledge which I argue 
necessarily underpins such implementation. Finally, I discuss principles of 
professional development which might enhance such knowledge. 

A SCENARIO OF A TEACHER ENACTING THE CURRICULUM 

1 propose the following detailed scenario as an example of quality enactment of 
curriculum. The narrative is a fictitious account of a teacher's experience in 
teaching fractions, based on my own research, my reading, and the opportunity to 
observe some exemplary teachers in action. In order to make this discussion more 
concrete, 1 situate Ms X as about to teach the topic of comparing fractions (i.e., 
determining the relative size of two fractions) to a group of 7th grade students, 
around 13 years old, and describe the steps she might take in preparing for and 
teaching this topic. 

/. Reflect on what has happened in previous related lessons. 
When Ms X started to teach the topic of fractions to this 7 th grade class, she knew 
that her students had a reasonable understanding of the part-whole notion of 
fractions. By this term, 1 refer to part-whole comparison, which Lamon (2006) 
states "designates a number of equal parts of a unit out of the total number of equal 
parts into which the unit is divided" (p. 125). Ms X determined from her reading, 
that part-whole ideas may have been overemphasised in many classrooms in 
relation to other constructs of fractions, such as measure, quotient, and operator 
(Kieren, 1976; Lamon, 2006). Ms X participated in a two-part professional 
development programme several months earlier when these ideas had been 

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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

discussed. Two lessons ago, the teacher engaged the students in a problem solving 
activity focused on partitioning strategies. This activity consisted of the following: 
10 students come into the room one at a time, and choose to stand at one of three 
chairs, each having different amounts of chocolate (one block, two blocks, and 
three blocks, respectively), when all students are in the room, they are to share the 
chocolate at their chair with whoever else is standing at that chair (see Clarke, D. 
M., 2006, for more details of this activity). There is an assumption underpinning 
this activity, readily accepted on the part of students, that more chocolate is 
desirable. 

One major purpose of this problem solving task was for students to become 
aware (if they were not already) that, for example, three blocks of chocolate shared 
between five people automatically means that each person would receive 3/5 of a 
block of chocolate, without the need to partition, or imagine partitioning the 
chocolate to determine each share. This activity emphasised the construct of 
fractions as division or quotient, that is, that alb is the same as a ■*■ b. After doing 
the activity, most students admitted that prior to this task they had no idea of this 
notion, even though many had learned in the previous year to convert fractions to 
decimals with a calculator or by hand, thereby enacting this principle. 

At the end of the activity, in order to maximise the chances that the 
mathematical intent became clear to most students, Ms X formed a visual image 
which the students might retain to help them remember the notion. The teacher 
asked three students to hold a chair above their heads with the two chocolate 
blocks on it. She asked the class to observe the two blocks above the line of the 
chair (the line representing the vinculum), which was in turn above the three 
students, and pointed out "two blocks shared between three people is 2 (gesturing) 
over 3, so two-thirds". 

In reflecting upon this lesson, Ms X felt a need to consolidate these ideas the 
following day. It was her general practice to attempt to build connections between 
a lesson and those which came before and after. A key finding of the effective 
teachers in the numeracy study in the UK (Askew, Brown, Rhodes, Johnson, & 
Wiliam, 1997) was that the most effective teachers could be described as 
"connectionist". Therefore, in the next lesson, Ms X attempted to build on this 
understanding of fraction as division (or quotient) by the provision of another 
problem, this time involving pizzas. The problem had been posed as follows: 
"Three boys share one pizza evenly, and seven girls share three pizzas of the same 
size evenly. Who gets more pizza, a boy or a girl?" Ms X had been interested to see 
how many students would build upon what they had learned in the previous lesson, 
thereby comparing 1/3 and 3/7. Only a small number did so. In fact, common 
solutions were of the following two kinds: 

For the girls' situation to be equivalent to the boys, six girls would have to 
share three pizzas. So, we effectively have one extra pizza and one extra 
girl, so clearly the girls must receive more on average. 
'Scaling up' the boys' situation would mean nine boys and three pizzas, 
giving two more boys for the same number of pizzas as the girls. So the 
girls receive more. 

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DOUG CLARKE 

Of course, these two solutions were innovative and more straightforward than 
comparing 1/3 and 3/7, but it was interesting that few saw the link with the 
previous lesson. In debriefing upon the different strategies, some students had 
come to compare 1/3 and 3/7 but had 'hit a brick wall', unsure as to which was 
larger, 1/3 or 3/7? This led quite appropriately for Ms X to a focus on strategies for 
comparing fractions in the following lesson. 

2. Read through the written curriculum statements or guidelines (the intended 
curriculum). 

As discussed previously, written curricula come in a variety of levels in terms of 
details and prescription. This is one aspect which varies considerably around the 
world. In some countries (e.g., Japan), a relatively thin book outlines all that should 
be 'covered' in one year, while in other countries (e.g., the United States), 
curriculum guides, particularly textbooks, are often very detailed, large and thick. 
The curriculum guidelines for this teacher were relatively brief, on the assumption 
that the teacher would work with colleagues, drawing upon her reading and 
insights which had emerged from professional development. In relation to this 
topic, the documentation indicated that students would require an understanding of 
equivalent fractions, and a process for finding the common denominator of two 
fractions, and converting both to this common denominator. The guidelines also 
indicated that such comparisons should be made successfully with both proper and 
improper fractions. 

3. Consider what the 'big ideas ' might be for the topic of fractions overall. 

For any teacher, developing an understanding of what constitutes the 'big ideas' 
within a topic is an ongoing process, as curriculum statements, other readings, 
observations of students, conversations with colleagues, and teacher education and 
other professional development experiences come together. Sowder (2007) argued 
that effective mathematics teachers think about mathematics curricula "in terms of 
big ideas, such as proportional reasoning or the mathematics of change, around 
which to structure instruction" (p. 165). The kinds of big ideas which might be 
relevant here might include the following examples: 

Uses appropriate symbols to represent fractions, understanding the 

meaning attached to each part (e.g., denominator shows what 

'denomination' is being counted, the numerator 'enumerates' how many 

of these parts). 

Understands that fractions (including whole numbers, mixed numbers and 

improper fractions) are entities that can be counted (e.g., 4/5 represents 

four things called 'fifths') and can recognise and use counting patterns 

and equivalences. 

Recognises a/b as a divided by b. 

Readily compares and orders fractions. 

Relates a given fraction to key benchmarks (e.g., 0, 54, I). 



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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

4. Talk to colleagues about their experiences in teaching this topic. 

Other 7 th grade teachers indicated to Ms X that this is a difficult topic to teach, that 
they have had little success with it in the past, and that fractions generally is the 
least popular topic among students in the 7* grade curriculum. On a regular basis, 
the teacher used the IMPACT procedure (Clarke, D. J., 1989), which involves 
students responding every two weeks (for about ten minutes) to four or five 
questions of the kind, "What are you having most trouble with in mathematics at 
the moment?" "What is one new thing you can do in mathematics this week which 
you couldn't do last week?" and "How could we improve mathematics classes?" 
Ms X had noticed that even when students were not currently studying fractions, 
quite a few nevertheless nominated fractions as their greatest topic of concern. Her 
Grade 6 colleagues indicated that they taught the topic the previous year, but when 
assessed few students had a strong understanding, as revealed on a topic test after 
the completion of the topic. They indicated that their teaching approach had largely 
been instrumental (Skemp, 1976), involving converting fractions to common 
denominators, and that students seemed to do so all the time, even when the 
comparison should have been obvious. They indicated that they would be glad to 
hear of a better way of teaching this topic if Ms X finds one. 

5. Consider articles which show research findings in this area with related 
teaching implications. 

As part of a part-time study programme which Ms X was undertaking in her spare 
time, she had accumulated a number of 'research into practice' type articles. 
Articles which summarise research reinforced her perception that this is a difficult 
topic for teachers and students. They indicated that a major difficulty is that most 
students do not think of fractions as having a 'size', but are rather just one whole 
number over another whole number. One particular example was frequently cited 
in the literature. Students were asked to select "the nearest whole number to 7/8 + 
12/13" from the four choices of 1, 2, 19 and 21. The majority of 13-year olds in the 
USA chose 19 or 21, almost certainly found by adding the numerators and 
denominators respectively (Carpenter, Kepner, Corbitt, Lindquist, & Reys, 1980). 
The research literature did however point to sense-making strategies which a 
number of students use, and which can be shared with others. They also 
recommended greater use of number lines, which help students to see things like 
the relationships between whole numbers, fractions and percentages. Number lines 
also support the idea of the density of rational numbers, namely that between any 
two distinct rational numbers, there is always an infinite number of rational 
numbers. 

6. Gather some information from a sample of students on how they would choose to 
compare fractions. 

During recess, Ms X chose carefully a selection of fraction pairs, and asked several 
volunteers (some of the more typically capable ones, and some who tend to 
approach problems in innovative ways) individually how they would compare 
them. The pairs were 3/7 and 5/8, 2/4 and 4/2, and 5/6 and 7/8. It is important to 

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DOUG CLARKE 

note here the power of carefully chosen examples in eliciting and enhancing 
student thinking and bringing common misconceptions to the fore (Watson & 
Mason, 2005). As she worked with these students, she noticed that they found them 
very difficult, but several were using some number sense-based strategies and few 
were using common denominators. Some were using what research calls residual 
strategies (Clarke, Sukenik, Roche, & Mitchell, 2006), that is, deciding for each 
fraction how much would be needed to make the fraction up to I (the 'residual'). 
For 5/6, only 1/6 would be required, while for 7/8, 1/8 would be required — a 
smaller residual, so therefore 7/8ths is a larger fraction. There were also a couple of 
examples of benchmarking, where students compare each fraction to a well known 
benchmark, such as 0, 1/2 or I. One student compared 3/7 and 5/8 by indicating 
that 3/7 is a bit less than 1/2, while 5/8 is clearly more than 1/2, which is most 
encouraging. 

7. Use what has been learned from research and brief student interviews to 
structure a lesson. 

Ms X decided that there was sufficient potential in the different strategies which 
some students had chosen to use when she spoke to them individually at recess, to 
make the next lesson a small group working session, with a focus on developing 
efficient strategies for comparing fractions. She started the lesson by reminding the 
class that when they left off yesterday, there was some confusion about how to 
compare 1/3 and 3/7. In 'friendship groups', students were given six pairs to work 
on, agreeing on which fraction was larger in each case, with a written reason on 
why. While methods involving common denominators were acceptable, groups 
were challenged to produce an additional method for each pair, which made sense 
to them. Ms X reminded the class that her emphasis all year was on mathematics 
making sense to them, and also helping us to make sense of the world. The class 
was accustomed to discussions about any proposed methods in terms of three 
things: Is the method mathematically valid? Is it efficient? Is it generalisable to 
many similar problems? (Campbell, Rowan, & Suarez, 1998). Once most groups 
had completed their set of six problems, she asked one person from each group to 
share one of their results, which they believe to have yielded a particularly 
impressive strategy, which could be generalised to many other pairs. Ms X listed 
each strategy on the board, and discussed it with the students, using the three 
criteria above. 

8. Pose an assessment task for the first half of next lesson. 

As an advance organiser (Ausubel, I960), the teacher let the class know that in the 
next lesson they would be challenged to write five pieces of advice or hints for 
other students who are just learning how to compare fractions for the first time. For 
each piece of advice, they are to show what they mean with specific examples. 
This was to be their major assessment task for the week. 



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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

9. Encourage the students to share their knowledge with another group of students. 
In this school, there was a regular expectation that a given class will share what 
they have learned with another class, when some particularly helpful insights 
emerge. The teacher therefore organises for one of the groups who have 
particularly impressed her to share what they have found with another 7 th grade 
class who are just starting this topic, during a presentation to this class. She was 
aware of FreudenthaPs (1975) notion about students needing to create knowledge 
for themselves (in the same way that particular procedures have developed over 
hundreds of years) — what he termed reinvention as compared to the learning of 
ready-made material. She emphasises that "the telling teacher is not the telling 
teacher" and encouraged the group to create a way of scaffolding the students' 
understanding and not rush to "tell," so that in time they will develop similar 
understanding to that of her class. 

In the following section, I will start to unpack the kinds of knowledge which 
might lead to the hypothetical story I outlined above. 

WHAT KIND OF KNOWLEDGE MIGHT A MATHEMATICS TEACHER HOLD TO 
UNDERPIN THIS ENACTMENT OF CURRICULUM? 

In this section, I discuss the kinds of knowledge which might enable a teacher to 
follow the path described in the narrative of Ms X. 

Teachers ' Knowledge 

Various scholars have categorised the knowledge possessed by mathematics 
teachers in a variety of ways. The best known is Shulman (1987) who identified the 
components of teachers' knowledge (general and not specific to mathematics 
teaching) as: 

content knowledge; 

general pedagogical knowledge, with special reference to those broad 

principles and strategies of classroom management and organisation that 

appear to transcend subject matter; 

curriculum knowledge, with particular grasp of the materials and 

programmes that serve as the 'tools of the trade' for teachers; 

pedagogical content knowledge, that special amalgam of content and 

pedagogy that is uniquely the province of teachers, their own special form 

of professional understanding; 

knowledge of learners and their characteristics; 

knowledge of educational contexts, ranging from the workings of the 

group or classroom, the governance or financing of school districts, to the 

character of communities and cultures; and 

knowledge of educational ends, purposes, and values, and their 

philosophical and historical grounds (p. 8). 



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DOUG CLARKE 

Of course, it is not claimed that these are mutually exclusive, but I believe the 
distinctions are helpful. I focus on all of Shulman's categories now, relating them 
to the knowledge as evidenced by the decisions of the teacher in the above 
scenario. 

Content knowledge. There is little doubt that effective teachers of mathematics 
need a clear understanding of the mathematics they wish to teach students, at least 
at the level at which they propose to teach. By understanding, I take the definition 
of understanding from Hiebert et al. (1997), namely that we understand something 
if we see how it is related or connected to other things we know. Clearly, the way 
teacher knowledge is organised and accessed as well as the nature of that 
knowledge is important. It must also be acknowledged that in many countries 
(including Australia) there has been a shift in focus from a transmission model of 
teaching to an emphasis on teaching for understanding (Fennema & Romberg, 
1999) which changed the nature of student/teacher interaction. Moving to a more 
learner-centred approach from a teacher-centred stance places greater demands on 
teacher knowledge, as the lesson can take many possible directions, given the more 
responsive nature of the teaching process, and students' strategies and reasoning 
can provide additional challenges (Clarke, D. M, 1997). 

Brophy (1991) argued in relation to content knowledge that 

where (teachers') knowledge is more explicit, better connected, and more 
integrated, they will tend to teach the subject more dynamically, represent it 
in more varied ways and encourage and respond fully to students' comments 
and questions. Where their knowledge is limited, they will tend to depend on 
the text for content, de-emphasise interactive discourse in favour of seatwork 
assignments, and in general, portray the subject as a collection of static, 
factual knowledge, (p. 352) 

In the fraction narrative, Ms X needed a clear, connected knowledge of rational 
numbers, in order to deal with the ways in which students might choose to compare 
fractions. This is where the borders of content knowledge and pedagogical content 
knowledge become fuzzy (see Chapter 6, this volume). It is possibly helpful in 
determining the borders to consider what it is that teachers need to know in relation 
to this topic which is the special province of teachers, and unlikely to be possessed 
by an engineer, say. This will be discussed in more detail later. 

Deborah Ball and her colleagues at the University of Michigan have explored 
further what they call "mathematical knowledge for teaching" (Hill, Rowan, & 
Ball, 2005). In their work they focus on subject matter knowledge, currently 
breaking it into common content knowledge (what people with reasonable 
mathematical knowledge know), the specialised content knowledge of teachers, 
and another category, which they call horizon knowledge, which can be thought of 
as knowledge of the mathematics which the student is likely to meet in coming 
years, and how this might connect to their current topics (D. Ball, personal 
communication, 2/10/07). The bulk of Ball and colleagues' current work is on 
describing the specialised content knowledge of teachers through the use of 
"records of practice". 

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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

General pedagogical knowledge. One example of this kind of knowledge in 
mathematics teaching might involve the capacity to form, manage and maximise 
the outputs of small group work and other organisational approaches in 
mathematics. Being clear on why groups might be formed in a certain way on a 
certain day is part of this. In the Early Numeracy Research Project (Clarke, D. M. 
et a!., 2002), an extensive research, assessment and professional development 
project involving 35 'trial' schools and 353 teachers, data were collected at the end 
of the project from 220 teachers, on the ways in which trial school teachers 
organised their classes during the mathematics lesson. 

When asked in the final weeks of the project to indicate the most common way 
in which they organised their students to work on tasks in the mathematics 
classroom during the main part of the lesson, there was an almost even split 
between: individual work, with discussion being allowed or encouraged (34%); 
children working in pairs (33%); and children working in larger groups (two or 
more) (34%). Slighty less than three-quarters of the teachers indicated that they 
used all three working group sizes at different times. Teachers were asked to 
identify the most common way in which groups were formed when children 
worked in pairs or other small groups. The distribution of the various forms of 
group assignment were heterogeneous (31.5%); homogeneous (28.5%); 
heterogeneous with one special group (26.0%); and student choice (14.0%). 

Only 7.6% of teachers indicated that they used only one of these options. There 
was considerable variety in the ways in which groups were used and in how 
children were assigned to groups. Teachers indicated that the content and activities 
for the day were a major factor in these decisions. For example, one case study 
teacher paired "strong literacy children" with less strong literacy children for a 
mathematical activity with a high literacy demand. 

Although these pedagogical decisions are important, Brown (1999) noted that 
quality teaching is more than this: "Quality teaching is more important than 
classroom organisation [...] it's not whether it's whole class, small group or 
individual teaching, but rather what you teach and how you interact mathematically 
with children that seem to count" (p. 7). 

In the fraction narrative, the teacher made at least two conscious decisions that 
were influenced by her pedagogical knowledge. First, she made several efforts to 
find out where her students were in terms of their understanding and use of 
strategies, by talking to individual students and to other teachers. She also made the 
decision to use small group work (friendship groups), thus encouraging students to 
pool their knowledge in working on the challenging fraction pairs. If she had 
chosen to use small groups based on ability, it is likely that some groups would 
have struggled, because they were merely "pooling their weaknesses". 

Curriculum knowledge. This component of teacher knowledge also has clear 
links to pedagogical content knowledge. In knowing the materials and programmes 
that serve as the 'tools of the trade' for teachers as Shulman (1986) describes them, 
in the story above, this would include awareness of the curriculum guidelines, the 
kinds of manipulative material which are likely to be helpful in supporting 
learning, and the kinds of experiences which students are likely to have had in 



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DOUG CLARKE 

previous grades in relation to this topic, and where this topic might lead in 
subsequent years. We saw this, or the teacher's awareness for the need for this, in 
the fraction scenario, as Ms X referred to curriculum documents and to the 
experiences of other colleagues. 

Pedagogical content knowledge. This is the area of knowledge which has 
received the most attention in the mathematics education research and professional 
development literature in the last five years (Chick et al., 2006; Hill, Rowan, & 
Ball, 2005). Shulman's contribution was to identify and elaborate the notion of 
pedagogical content knowledge which refers to how specific knowledge can be 
interpreted in teaching situations (Cooney, 1994). 

Shulman (1986) summarised this knowledge as 

the most useful forms of representations of those ideas, the most powerful 
analogies, illustrations, examples, explanations, and demonstrations — in a 
word, the ways of representing and formulating the subject that make it 
comprehensible to others. ... Pedagogical content knowledge also includes an 
understanding of what makes the learning of specific topics easy or difficult: 
the conceptions and preconceptions that students of different ages and 
backgrounds bring with them to the learning of the most frequently taught 
topics and lessons ... if the preconceptions are misconceptions, which they so 
often are, teachers need knowledge of the strategies most likely to be fruitful 
in reorganising the understanding of learners, (p. 9) 

To Shulman's list of the components of pedagogical content knowledge, Hill, 
Rowan, and Ball (2005) add knowing how to use pictures or diagrams to represent 
mathematical concepts and procedures, providing students with explanations for 
common rules and mathematical procedures, and analysing students' solutions and 
explanations. 

In considering the fraction narrative, Ms X demonstrated her considerable 
pedagogical content knowledge. In Shulman's terms, she chose as an example the 
chocolate problem to encourage the students to explore, with obvious motivation, 
an important fraction construct. She collected information prior to the lesson (in 
class and out of it) on the kinds of difficulties which students face in studying this 
topic, and on some of the more important big ideas and strategies which might 
assist the students. Her thorough preparation increased the chances of helping 
students to reorganise their understanding where this was necessary. 

Knowledge of learners and their characteristics. A teacher can be aware of 
curriculum documents and general patterns of student developing understanding in 
mathematics. In order however to meet the needs of individuals in classrooms, the 
teacher needs to know the students as individuals — their interests, their level of 
understanding, and their disposition for mathematics. There is a variety of ways of 
gaining this information and updating it on a regular basis, and many of these 
methods are in common use. Of particular personal interest is the potential of the 
one-to-one, task-based assessment interview in providing a detailed picture of 
individual learners' level of understanding and preferred problem solving 
strategies. As Bobis et. al. (2005) noted, an important feature of some of the largest 



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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

research and professional development projects conducted in Australia and Mew 
Zealand in the past 10 years was the use of such interviews by classroom teachers. 
Although there was an assessment purpose behind the use of such interviews, the 
major emphasis was on assessment to inform teaching and thereby enhance 
learning 

The teacher in the fraction scenario demonstrated a number of strategies to gain, 
and subsequently use, a detailed understanding of her students and their levels of 
understanding and preferred strategies. I would argue that one important 
component of teacher knowledge which is not always acknowledged in the 
literature is realistic expectations of students, for both individuals and the group as 
a whole. This component seems to overlap with pedagogical content knowledge, 
curriculum knowledge, and knowledge of learners and their characteristics. In the 
Early Numeracy Research Project (Clarke, D. M. et al., 2002), teachers were 
offered 22 statements about possible mathematical knowledge of their students 
(e.g., "knows that 78 is 7 tens and 8 ones"). Completing these items in both the 
entry and exit questionnaires (approximately three years apart) enabled 
quantification of whether their expectations of what students could do had changed 
over the course of the project (see also Volume 2, Chapter 10). One result of the 
regular use of the Early Numeracy Research Project task-based, one-on-one 
assessment interview by teachers in the project appeared to be changes in 
expectations, in two respects. 

One change was in the "spread of expectations", where teachers seemed to be 
acknowledging the existence of considerable within-class variation. For example, 
in response to the questionnaires at the beginning and end of the project, all of the 
Grade 1 teachers at the end of the project (approximately 70 teachers) expected at 
least some of their children to "recall and use addition and subtraction facts to 20 
(including 0)" compared to 29% at the beginning who thought that none of their 
children could do so. (Grade 1 children are typically six-year olds.) 

There were also areas where the teachers had higher expectations of their 
children's understanding at the end of the project than at the beginning. For 
example, in relation to the item "knows that four hundred and two is written 402 
and knows why neither 42 or 4002 is correct"), at the beginning of the project 61% 
of teachers of Preps (the first year of school) expected no child at the end of the 
first year of school (generally five-year olds) to have this knowledge. But, after 
three years of the project, the figure was only 30%, indicating considerably 
increased expectations over time. There were similar increased expectations among 
Grade 1 and 2 teachers. 

A second factor which appeared crucial to teachers obtaining appropriate 
expectations for students was data provided by the research team about their own 
students' achievement of "growth points". The notion of growth points is discussed 
extensively in Volume 2 of this handbook series (see Barbara Clarke's chapter in 
that volume), but the idea was that the growing points were research-based 'big 
ideas' or strategies in students' developing mathematical understanding which had 
the following characteristics: 



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DOUG CLARKE 

reflect the findings of relevant research in mathematics education from 

around the world; 

emphasise important ideas in early mathematics understanding in a form 

and language readily understood and, in time, retained by teachers; 

reflect, where possible, the structure of mathematics; 

allow the description of the mathematical knowledge and understanding of 

individuals and groups; 

form the basis of planning and teaching; 

provide a basis for task construction for interviews, and the recording and 

coding process that would follow; 

allow the identification and description of improvement where it exists; 

enable a consideration of those students who may benefit from additional 

assistance; and 
• have sufficiently high 'ceiling' to describe the knowledge and 

understanding of all children in the first three years of school. 
By relating their own students' interview data to movement through these 
growth points, teachers were able to gain a sense of 'typical' performance and also 
variation across students, thereby enhancing the accuracy of their expectations. 
There is clear evidence in the literature that higher expectations have an impact on 
student outcomes. According to Brophy (1983), "higher expectations for student 
achievement are part of a pattern of differential attitudes, beliefs and behaviours 
that characterise teachers and schools who are successful in maximising their 
student learning gains" (p. 642). 

Knowledge of educational contexts is the least relevant of the Shulman 
categories to the current discussion, and therefore will not be elaborated here. In 
the same way, knowledge of educational ends, purposes, and values is a more 
difficult area of knowledge to articulate, and it provides considerable overlap with 
beliefs, which are discussed in great detail in Forgasz and Leder's chapter in this 
volume. Thompson (1992), in her extensive review of the literature on beliefs and 
conceptions, took beliefs as a subset of conceptions, and on occasions, used them 
interchangeably. The picture painted earlier of Ms X gives a sense of a teacher 
who acts in harmony with the dynamic, problem solving view of mathematics 
(Thompson, 1992). Having said that, we need to remember that inconsistencies 
between stated beliefs and practices can occur for a variety of reasons, including 
the constraints mentioned early in this chapter. The statement to Cockburn by a 
teacher that she already knew how to teach twice as well as she was able to do 
(Desforges & Cockburn, 1987) is a sobering statement in relation to this 
inconsistency. 

In the following section, I briefly report on several pieces of important research 
which link teacher knowledge of the kinds discussed above and other 
characteristics with student achievement. 



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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

ENHANCING TEACHER KNOWLEDGE OF CURRICULUM THROUGH PRINCIPLES 
OF PROFESSIONAL DEVELOPMENT 

There are no easy ways for a teacher to build up the kinds of knowledge for 
teaching articulated in the previous sections, and which are illustrated in the 
fraction scenario. However, there is growing research evidence on the principles 
which might underpin effective professional development. By 'effective 
professional development', I am referring to those approaches which lead to 
substantial professional growth for teachers and also improvement in student 
learning. 

Goals and Principles of Professional Development 

Sowder (2007) proposed six somewhat overlapping goals for professional 
development in mathematics education that address both beliefs and knowledge: 1 ) 
developing a shared vision; 2) developing mathematical content knowledge, 3) 
developing an understanding of how students think about and learn mathematics; 
4) developing pedagogical content knowledge; 5) developing an understanding of 
the role of equity in school mathematics; and 6) developing a sense of self as a 
teacher of mathematics (pp. 1 65-1 68). 

Clarke, D. M. (1994) summarised research on the professional development of 
mathematics teachers and from this synthesis offered 10 principles which appeared 
to increase the effectiveness of professional development programmes. They were 
as follows: 

1 . Address issues of concern and interest, largely (but not exclusively) identified 
by the teachers themselves, and involve a degree of choice for participants. 

2. Involve groups of teachers rather than individuals from a number of schools, 
and enlist the support of the school and district administration, students, parents 
and the broader school community. 

3. Recognise and address the many impediments to teachers' growth at the 
individual, school and district level. 

4. Using teachers as participants in classroom activities or students in real 
situations, model desired classroom approaches during sessions with practising 
teachers to project a clearer vision of the proposed changes. 

5. Solicit teachers' conscious commitment to participate actively in the 
professional development sessions and to undertake required readings and 
classroom tasks, appropriately adapted for their own classroom. 

6. Recognise that changes in teachers' beliefs about teaching and learning are 
derived largely from classroom practice; as a result, such changes will follow 
the opportunity to validate, through observing positive student learning, 
information supplied by professional development programmes. 

7. Allow time and opportunities for planning, reflection, and feedback in order to 
report successes and failures to the group, to share 'the wisdom of practice', 
and to discuss problems and solutions regarding individual students and new 
teaching approaches. 



147 



DOUG CLARKE 

8. Enable participating teachers to gain a substantial degree of ownership by their 
involvement in decision making and by being regarded as true partners in the 
change process. 

9. Recognise that change is a gradual, difficult and often painful process, and 
afford opportunities for ongoing support from peers and critical friends. 

10. Encourage participants to set further goals for their professional growth. 

It is interesting to reflect on this list in relation to the narrative of Ms X, and to 
consider the extent to which, 14 years later, I might wish to revise this list. The 
experience of the Early Numeracy Research Project and the more recent work on 
rational number (Clarke, D. M., Sukenik, Roche, & Mitchell, 2006) have led me to 
see the powerful effect on teachers' professional learning of the two related 
strategies of 1 ) providing a framework of research-based growth points and, 2) a 
related, task-based, one-to-one student interview. These two components enable 
teachers to internalise both a sense of the 'big ideas' in a given content area and the 
reasonable expectations of what students know and can do, while increasing 
teachers' understanding of the incredible range of understanding within a given 
group of students. 

In relation to the ten principles above, other writers outside the discipline of 
mathematics education have proposed similar principles (see, e.g., Elmore, 2002; 
Hawley & Valli, 1999), and there is no certainty that a programme with all of these 
principles in place would automatically achieve its aims, but it is clearly an 
important start. 

Models of Professional Development 

Sowder (2007) offered a helpful discussion of different approaches to professional 
development which have the potential to enhance "knowledge-in-practice" for 
teachers. The models included: a focus on student thinking (e.g., Cognitively 
Guided Instruction [CGI] Carpenter, Fennema, Franke, Levi, & Empson, 1999); a 
focus on curriculum (e.g., the various projects funded by the National Science 
Foundation in the 1990s in the USA); studying cases (e.g., Barnett, 1998); and 
formal course work. Of course, there are many more such models, one of the best 
known is that offered by Loucks-Horsley, Love, Stiles, Mundry, & Hewson (2003) 
which provides principles for the design of approaches to teacher professional 
development 

DRAWING UPON APPROPRIATE KNOWLEDGE IN ENACTING THE CURRICULUM 
-AN IMPORTANT, COMPLEX BUT REWARDING TASK WHEN DONE WELL 

In 1986, Shulman rejected the claim of George Bernard Shaw that "he who can 
does. He who cannot teaches," recognising that it gave no recognition to the highly 
specialised knowledge of good teachers. In this groundbreaking article, Shulman 
concluded with the statement, "those who can do. Those who understand, teach" 
(p. 14). In this chapter, I have attempted to describe the thoughtful artistry of a 
good teacher, to unpack the components of knowledge proposed by scholars, and 



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THE MATHEMATICS TEACHER AS CURRICULUM MAKER 

have offered some thoughts on how such knowledge might be enhanced. I close 
with a quote from Andy Hargreaves, who like Shulman, acknowledged the key role 
played by the teacher in mathematics learning — the teacher as curriculum maker: 

Teachers don't merely deliver the curriculum. They develop, define it and 
reinterpret it too. It is what teachers think, what teachers believe and what 
teachers do at the level of the classroom that ultimately shapes the kind of 
learning that young people get. (Hargreaves, 1994, p. ix) 



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Doug Clarke 

Faculty of Education 

Australian Catholic University, Australia 



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7. LEARNING TO DESIGN FOR LEARNING 

The Potential of Learning Study to Enhance Teachers ' and 
Students ' Learning 



The aim is to present an approach - learning study-for teacher cooperation in 
teaching and learning that has been shown to be a promising way to enhance 
teachers ' learning about students ' learning of a particular subject matter. A 
learning study is a hybrid of the Japanese lesson study model and design 
experiment. The teachers learn an inquiry approach to their teaching in which 
they systematically plan, enact and evaluate teaching and learning and are guided 
by some theoretical principles. A learning study centers on the object of learning 
and emphasises the learning problems the students have. In this paper I describe 
how a group of elementary teachers and their students learn about pre-algebra 
from a learning study. It is demonstrated that the teachers learned what was 
critical for students' learning the particular object of learning; how to write an 
algebraic expression for a given example and vice versa. These insights enabled 
the teachers to revise the lesson design in a way that enhanced students ' learning. 
In that respect, the students ' contribution, in terms of opening variation in the 
lesson, played a significant role. It is suggested that a parallel learning process of 
teachers and students ' learning occurred and that the richness of 'meaning ' of an 
algebraic expression was constituted and developed. 

INTRODUCTION 

How can teachers improve their knowledge about their students' learning 
obstacles? How can they learn about how their teaching affects students' learning 
and whether there was possibilities to learn that which was intended in the 
mathematics lesson? In this chapter I suggest an approach for teacher cooperation 
in teaching and learning that has been shown to be a promising way to enhance 
teachers' learning about students' learning of a particular subject matter. This 
approach - the learning study - centres on the object of learning, is grounded in a 
theoretical framework, and emphasises the learning problems of the students. It is a 
systematic planning process of reflection and revision, and entails follow-up and 
feedback. I describe the rationale behind the approach, the cyclic process the 
approach encompasses and provide an example from a learning study about the 
teaching of pre-algebra at elementary level. 



P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 153-172. 

© 2008 Sense Publishers. All rights reserved. 



ULLA RUNESSON 

LEARNING STUDY - AN APPROACH FOR ENHANCING STUDENT LEARNING 

In a learning study student learning is central. It has been reported that when 
teachers learn about students' understanding, their difficulties and strategies used, 
they can change their teaching in a way that facilitates students' learning 
(Carpenter, Fennema, Peterson, & Carey, 1988). In a learning study teachers try to 
learn from what takes place in the classroom. Hiebert, Morris, Berk, and Jansen 
(2007) have pointed out the potential of teachers learning from their teaching, and 
have proposed a framework for a teacher learning. However it should be noted, a 
learning study is more than reflective practice or cooperative effort among 
teachers. It is more than learning about students' strategies in mathematics. It is a 
systematic iterative process based on a theoretical framework for learning. It is not 
the lesson per se that is in focus, but rather the students' possibilities for learning 
during the lesson. 

The Guiding Theoretical Framework 

In a learning study the teachers try to develop their understanding of what it means 
to know something in mathematics, for instance, being able to do arithmetic 
calculations, operate with negative numbers, and so on. It involves asking 
questions about the object of learning. If you know what knowledge, understanding 
or skill you want the learners to develop and what that ability really implies, you 
are more likely to be able to help your students to learn. This includes taking the 
learners' perspective as well, by asking: What does it take to learn something from 
the point of view of the learners? 

In a learning study a particular theoretical framework-variation theory (Bowden 
& Marton, 1998; Marton & Booth, 1997; Marton & Tsui, 2004) is used as a 
guiding principle. Nuthall (2004) is one of several (e.g., Floden, 2001; Levin & 
O'Donnell, 1999) who have pointed out the necessity for teachers to have a theory 
as an explanatory framework for understanding how their actions in the classroom 
affect students' learning. 

Teachers need an explanatory theory of the teaching-learning relationship 
that meets these requirements in order to plan, carry out, and evaluate their 
daily activities effectively and efficiently, (p. 277) 

What kind of framework could help us to better understand the relation between 
what is taught and learned? It should not explain learning as an effect of teaching 
but should be powerful enough to help the teacher to know what to look for and 
interpret what they see when monitoring student learning. Nuthall (2004) advocates 
a theory which can help teachers to design learning. 

The theoretical framework on which a learning study is based seems to be 
powerful and appropriate in that respect. Variation theory has been shown to have 
potential to explain how learning is afforded in the classroom and how this is 
reflected in students' learning (Marton & Morris, 2002). Variation theory is mainly 
a framework for learning, but has been used as an analytical tool when studying 



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LEARNING TO DESIGN FOR LEARNING 

teaching and designing for learning (Lo, Chik, & Pang, 2006). A series of studies 
within this framework has addressed issues such as what matters for learning, what 
is possible to learn, what is learned in a specific learning situation, and how 
differences in learning outcomes can be understood in the light of the lesson. Since 
what is learned is the primary focus in a learning study, variation theory is 
particularly useful. From a variation theory perspective, learning always has an 
object: there is always something learned. 

But why mind about the object of learning? Brentano (1838-1917), a prominent 
figure within phenomenology, stated that there is "no hearing without something 
heard, no believing without something believed, no striving without something 
striven" (Spiegelberg, 1982 p. 37). In the same way, you can say that there is no 
learning without something learned. When teachers and learners interact in the 
learning situation, they interact about something. Embedded in the activity and 
interaction in a classroom there is always an object that may be learned. So, apart 
from describing the teaching-learning process and the interaction between the 
teacher and the learners, it is also possible to suggest what may be learned about 
the specific topic taught (Runesson, 2005). Hence, classroom interaction affords an 
object of learning, in terms of a potential for learning. Therefore, it is significant to 
keep in mind what that is. 

The object of learning is not equivalent to learning objectives. The object of 
learning refers to a certain capability that the students are expected to develop 
during a lesson, or during a limited sequence of lessons (Marton & Pang, 2006). 
Capabilities are not confined to theory or concepts only. They can also be 
associated with skills, values or attitudes. Every capability has two aspects, one 
general and one specific. The general aspect refers to the nature of the capability, 
for instance, 'understanding', 'calculating', 'explaining' or 'describing'. The 
specific aspect refers to what is acted upon, hence the subject matter; for instance, 
multiplication with two digits, linear equations or Pythagoras' theorem. To be able 
to afford vital learning possibilities for the learner, it is probably important for a 
teacher to focus on both aspects. Hence, teachers should be aware not only of what 
the students are trying to learn, but also of the way the students master that which 
is learned. 

This does not mean that the interaction or the activity is not important itself. The 
character of the interaction, for instance, students' participation in a classroom 
discussion, can pave the way for or delimit learning (Emanuelsson & SahlstrSm, 
accepted). However, descriptions of activities, particular methods and so on are 
neutral to the topic taught and learned. Notions such as "student-centred" or 
"interactivity" do not say anything about what is possible to leam. Learning can be 
seen as two-fold, that is, it has a how-aspect and a what-aspect. These are two sides 
of the same coin and can hardly be separated. Next I provide some ideas of 
variation theory that are relevant for guiding and can help teachers to plan and 
evaluate their teaching in a more systematic and deliberate way. 



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ULLA RUNESSON 

Discernment, Variation and Critical Aspect - Central Elements of the Framework 

From a variation theory perspective, learning is seen as a process of differentiation, 
rather than enrichment. This is in line with psychologists like Gibson and Gibson 
(1955) who pointed to the importance of differentiation when arguing that learning 
implies that "perception gets richer in differential responses" (p. 35). Thus, seeing 
learning as differentiation means seeing learning as a matter of being able to 
discern similarities and differences. From a variation theory perspective, to 
understand something in a certain way, it is necessary to discern certain necessary 
aspects simultaneously. For instance, there are probably several features that are 
necessary to discern to calculate the value of -5-(-3) of which the sign "-" is one. 
The "-" has different meanings in this expression. So, in line with Gibson and 
Gibson's line of reasoning, it is necessary to differentiate between "-" as a sign for 
the subtraction operation and "-" as a sign for the number. From a variation theory 
perspective the differentiation of the meaning is a critical aspect for learning. 
However, what is critical for learning to calculate -5- (-3) can probably not be 
prescribed on a general level, or be derived from mathematical theory alone. 
Knowledge about the object of learning always includes the learner; what s/he 
brings into the learning situation in terms of previous experiences and how s/he 
understands the object to be learned. The way something is perceived, understood 
or experienced is due to what extent the critical aspects are discerned by the 
learner. A student's failure or lack of understanding can be understood in the light 
of un-discerned aspects; for instance the learner does not differentiate the double 
meaning of the - sign in the operation above. So, the discernment of critical aspects 
is essential for learning. 

Furthermore Gibson and Gibson argue that learning to differentiate objects is a 
matter of discovering and seeing differences between the objects and identifying 
their properties. Knowing what something is implies knowing what it is not. Seeing 
something (a situation, a concept, a principle and so on) in the light of its contrast 
brings out the characteristic properties of the object in question. Earlier 
educationalists have also emphasised the importance of contrast and variation. For 
instance, Maria Montessori stressed that the training of the senses is done by means 
of systematic variation in different sense modalities against a background that is 
invariant (Marton & Signert, 2005). A similar rationale was proposed by Dienes 
(1960) in his theory of mathematics learning. A concept must be taught by varying 
essential features of the concept, he argues. The same idea is found in Chinese 
mathematics education. The "bian shi" method is based on variation and 
invariance. When a pattern of variation and invariance is created, learners are given 
the opportunity to discern various aspects of, for instance, a concept. 

The idea of variation and invariance is fundamental to variation theory. 
Discernment is a function of an experienced variation. That is, we attend to or 
discern certain aspects of a situation if they stand out. "Things tend to stand out 
when they change or vary against a stable background or when something stays 
unchanged against a changing background" (Lo et al., 2006, p. 19). For instance, if 
the same example (invariant) is solved by different methods (variant) it is likely 



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LEARNING TO DESIGN FOR LEARNING 

that the learner will notice that the method could be different for the same example, 
thus the method will be discerned. However, if the case is the opposite (the same 
method is used to solve different examples) it is not opened up for possibilities to 
discern an alternative method. Or, in other words, the possibility to use different 
methods is not promoted. 

Studying a learning situation from the point of view of what varies and what is 
"invariant" is an efficient way to describe the promoted space of learning. It is 
possible to study classrooms where the same topic is taught and the same teaching 
arrangements are used to identify differences in learning possibilities. To give an 
example, Runesson and Mok, (2005) studied how fractions were taught in two 
different classrooms. The aim was to describe and analyse how this topic was 
handled and what was possible to learn, that is the focus was on the enacted object 
of learning. In both classrooms the topic taught was comparison of fractions with 
different denominators. By using manipulatives, as well as applying the method of 
amplification, the students were able to learn, for instance, that 5/8>l/2. However, 
in addition to this, in one of the classrooms, the students worked with the task, "Is 
it possible for 5/8 to be smaller than 1/2? Discuss with classmates and explain the 
results in words and diagrams ". This task was critical for what was possible to 
learn about the nature of fractions. After having established the fact that 5/8>l/2, 
the learners were immediately confronted with the opposite idea, that it could be 
the reverse (if we imagine that we are talking about 1/2 of a big family pizza and 
5/8 of a small pizza). Contrasting one situation where 5/8>l/2 with another where 
5/8<l/2 created an opportunity to discern a critical aspect of fractions; that the size 
of the fractional part is relative; it is the related to the whole. It was then possible to 
simultaneously discern the part-part (i.e., 1/2 and 5/8) and the whole-whole relation 
(small and big pizza) at the same time. This did not happen in the other classroom. 
So, in one classroom the students were given the opportunity to gain a more 
complex or broad understanding (in terms of being able to discern more aspects) of 
the object of learning than was the case in the other. 

It has been concluded that different patterns of variation, in different 
combinations and structures, create different learning opportunities. Do such subtle 
differences in how the object of learning is handled matter? Do they have an 
impact on students' learning outcomes? I do not claim that the handling of the 
content is the one and only factor that is significant for learning. Neither do I assert 
that there is a one-to-one correspondence between teaching and learning in terms 
of 'what is possible to learn is the thing learned'. A classroom situation is complex 
and, although one aspect may remain invariant in the lesson, the students 
themselves can open dimensions of variation on their own, or a dimension may be 
opened but still not discerned. What can be said is whether it is possible or not to 
discern certain critical feature. However, differences in the pattern of variation and 
invariance are reflected in student learning outcomes immediately after the lesson, 
but can also be noted in the long run. For instance, Holmqvist, Gustavsson, and 
Weinberg (2007) have demonstrated that the different nature of the enacted object 
of learning was observable even four weeks after the (single) lesson. It seems as if 
a particular pattern of variation and invariance generated new learning long after 



157 



ULLA RUNESSON 

the initial learning situation itself. 

LEARNING STUDY - AS A SYSTEMATIC AND ITERATIVE ENQUIRY INTO THE 
PROCESS OF TEACHING AND LEARNING 

In the two classrooms described by Runesson and Mok (2005) one can assume that 
the two teachers handled the topic differently because of variation in their 
awareness of this particular aspect and its importance for a more advanced 
understanding of fractions differed. If teachers differ to the extent they are aware 
of critical aspects of the object of learning, would it not be possible to collectively 
investigate what the critical aspects are? Furthermore, if a pattern of variation and 
invariance is possible to identify in every lesson, would it not be possible to make 
teachers more aware of how they could use variation in a more thoughtful and 
systematic way to bring out significant features and thus promote possibilities for 
student learning? These questions were the point of departure for initiating a joint 
research project on a model of learning that first started in Hong Kong in 2000 and 
was taken up a couple of years later in Sweden. This initiative was named the 
learning study and was inspired by design experiment (e.g., Cobb, Confrey, 
diSessa, Lehrer, & Schauble., 2003) and the tradition of Japanese and Chinese 
teachers, who do in-depth and systematic analysis of their lessons (Ma, 1999; 
Stigler & Hiebert, 1 999; Yoshida, 1 999). 

A learning study has points of similarity to a lesson study, particularly in the 
collective inquiry into how the intended goals are manifested in the 'research 
lesson'. A learning study is a cyclic process of planning and revision (see 
Figure 1). 




1 . Choose an object of learning 


4 


2. Pre-test the students 
i 


» 


3. Co-plan the lesson 
i 


* 


4. Conduct the lesson 


* 


5. Post-test students 




6. Teachers watch the video 
recorded lesson and revise it 



Figure I. The learning study cycle. 

The process starts with a group of teachers choosing and deciding about the object 
of learning - a capability or a value to be developed. (I) With the background of 



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LEARNING TO DESIGN FOR LEARNING 

their previous teaching experiences and information from research findings, they 
design a pre-test (a written test or an interview) and give it to the students. (2) The 
teachers can study the results of the test to find out about the particular learning 
difficulties their students may have. This may lead to reconsideration or a 
delimitation of the object of learning. (3) They plan and one of the teachers in the 
group implements the research lesson. (4) The lesson is documented by video 
recording and after the lessons a post-test is given to the students. (5) The teachers 
meet again in a post-lesson session to analyse the recorded lesson and the results of 
the post-test. They reflect on students' performances and the enactments of the 
lesson. (6) If needed, they revise the lesson plan and another teacher implements 
this in her class (new students!). This continues in a number of cycles until all 
teachers in the group have conducted one lesson. 

In the next section I illustrate the learning study process with an example from a 
learning study in a Swedish comprehensive school. My aim is to offer a deeper and 
more elaborated description of the learning study process, its stepwise and 
systematic character and the potential for teachers' learning in and from their 
practice. 

LEARNING ABOUT TEACHING AND LEARNING ALGEBRAIC 
REPRESENTATIONS 

Usually a learning study group consists of a group of three to five teachers teaching 
students of the same age. In the study reported here three teachers and their three 
mixed age-group classes (11-12 years old) were involved. The teachers' teaching 
experience varied; one of them had worked for more than 30 years as a teacher 
(Tl), the other two had worked for 5 and 10 years, respectively. Since this learning 
study was included in a research project (Kullberg & Runesson, 2006) a researcher 
from the university worked together with the group as a tutor. This was the third 
learning study with this group of teachers. They were familiar with the principles 
of the guiding theoretical framework, both from their previous experience of 
learning study and from a one day expert seminar about variation theory and its 
application in the research of teaching and learning. 

The analysis here draws on data from audio-recordings from the pre- and post 
sessions with the teachers, video recordings from the lesson and test-results from 
the pre- and post tests. The focus is on the teachers' reflections and comments on 
the recorded lesson, what they noticed and learned about how the content was 
handled, the students' reactions, and how this affected the planning and revision of 
the lesson as well as the manifested lesson. 

First Planning Step: Choose an Object of Learning 

The teachers met twice in sessions lasting 90 minutes to decide about the object of 
learning. Initially a particular topic, which the students found problematic to learn 
and the teachers thought was difficult to teach was identified. Since the duration of 
the research was restricted to one, or sometimes, two lessons they chose a specific 



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capability as the object of learning. In this case the teachers had previously 
experienced that students often failed on particular test items in the Swedish 
National Assessment in Mathematics on converting algebraic expressions to 
sentences describing relationships, so they decided to do something about this. The 
students in question had not been taught about variables and letters as symbols in 
mathematics before, so it would be the first time they encountered this topic. 

Much time of the pre-lesson sessions were devoted to constructing a pre-test. 
Inspiration came from literature, but mainly from their own experience. In learning 
study the pre-test serves two aims. It should 'scan' what the students know about 
the particular topic and related concepts, but also make it possible find out what the 
learning obstacles may be. 

Second Planning Step: Pre-test the Students 

In order to ascertain students' prior understanding it is advisable to combine the 
pre-test with a student interview; however, this was not the case in this study. In 
order to further define the object of learning and what specific capabilities to 
develop, the results on the pre-tests were analysed in a qualitative way, that is, it 
was not the total test-score that was of interest, but the different questions and how 
the students responded to these. 

Third Planning Step: Co-planning of the Lesson 

What was revealed from the test-results led to a shaping of the object of learning 
chosen; they wanted the students to be able to write algebraic expressions for a 
given example of an additive relation, that is, to find a general formula for an 
example with only one numerical value given (e.g., a chocolate bar costs 5 kronor 
more than a toffee) and vice versa. In the planning of the lesson, as well as in other 
steps of the study, the role of the researcher was merely supportive; for instance, 
she provided the group with literature or brought in ideas from her experience as a 
researcher. The researcher encouraged the teachers to come up with ideas of their 
own and try them. So, the way the lessons in this study were designed and enacted 
were principally based on the teachers' own ideas. 

When planning the first lesson the teachers discussed possible explanations to 
students' failures on the pre-test. For example, they assumed that "X and Y are 
frightening" to the students, and "letters" would be less confusing if the notations 
were closer to the example, thus choosing a letter close to the variable would 
facilitate the understanding. For instance, Daddy's age would be represented by the 
letter D. They decided to start with saying. "Daddy is three years older than 
Mummy", after that writing "Daddy's age - Mummy's age = 3 years" and next, 
shorten this into "D-M=3". They planned for different patterns of variation; for 
instance, that the example (and thus the variable, for example, age, price, weight) 
must vary just as the positions of the symbols. 

When planning the first lesson they also decided about the post-lesson test. A 
new assignment (compared to the pre-test) was chosen; the students were asked to 

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find different expressions to the same example. At this point of the study, the 
teachers had met five times in total. Now the first lesson was ready to be 
conducted. 

First Cycle: Conducting Lesson I 

The lesson shifted between plenary sessions (whole class) and student pairs with 
the focus on how the same example could be represented by different equations. 
The task was to find equations for the given example, or vice versa. In the 
discussion about the example (a chocolate bar costs 5 kronor more than a toffee), 
the following was written on the board: 

C-T = 5 

C-5 = T 

T + 5 = C 

C = T + 5 

T = C-5 

C-5 = T 

5+T = C 
The equations are all permutations of T +5=C, that is, the position of the symbols, 
and thus the operation, changes. The letters chosen to represent the variables are 
the initials for (the price of) a chocolate bar (C) and toffee (T). In all the examples 
taken in the lesson, no other letters but the above initials were used. The lesson was 
conducted in accordance with the plan. It was clearly demonstrated that the same 
example could be represented by different algebraic expression and that the 
variable is represented by a letter. 

First Cycle: Post-session Meeting after Lesson 1 

After being informed about the learning outcomes of the post-test, the teachers 
watched the video-recorded lesson and commented upon it. The teachers had 
conducted two learning study cycles before, so they were familiar with focusing on 
how the content was taught and the students' reactions, rather than on organisation 
and individual teacher behavior. This could be illustrated by the following 
example. Henry, one of the teachers (T2), commented on the learners' 
understanding when he said: 

T2: The task was not 'write different formulas ...', still they do so. They turn 
the letters around. Hm ... interesting! 

The teachers were satisfied with that the lesson was in accordance with what they 
had planned; that the same equation could be represented differently. However, 
when watching the recorded lesson 1, they noticed the way the teacher commented 
on the examples given to a specific equation by the students. She repeatedly said: 
"Does that [example] correspond with the formula?" and "Is it correct?" even 
though all the examples were correct. Teacher 1 (i.e., the teacher who had 
conducted the lesson) became aware of that and said: 

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ULLA RUNESSON 

Tl: Here [in this situation] I could have given an example that was incorrect. 
Yes, I could have come up with that, one that was wrong! 

Tutor: Yes, writing an incorrect one! 

Tl Yes, I could. They should have had [an incorrect] one. That's why they 
failed on that [test-] item [in the post-test]. That's why they haven't got any 
further. 

T2: Yes, when going through the first example you could have chosen... 

Tl ...I could had given the example '1+7=9 [and asked] is that correct'? 
Some minutes later Teacher 2 said: 

T2: If you had taken an incorrect one there...! 

Tl: I think that is the point. You see they turn... they do that for this one 
[example] too, they [do] turn C+T=4 and T-4=C. This in is line with what I 
can see on the post-test. 

T2: Wow. That's great! 

They noticed that the expression C+T=4 is varied in the lesson and Tl also saw 
how that corresponds to the outcomes of the post-test, and what the students have 
learned. One of the tasks was to write different expressions for the example 'an 
apple and a cucumber cost 15 crowns'. This was in line with the intension of the 
lesson, with the pattern of variation and invariance that was planned and how the 
lesson was enacted. This is also what the students learned; those who solved the 
task correctly all used the initial letter of the variable but changed the positions 
(see Table I). 

When Tl observed herself teaching she discerned an alternative; that things 
could have been done differently and how she could have been able to increase the 
scope for learning. She commented on the question "Is it correct?" and probably 
realised that the question could be the key to the opening of another pattern of 
variation: "I could have given an example that was incorrect. Yes, I could have 
come up with that, one that was wrong!" thus, that it would have been possible to 
show an expression that did not represent the example. The idea of coming up with 
an incorrect expression was caught by the rest of the group. T2 said: "If you had 
taken an incorrect one there...!" This insight was new to them; that a variation of 
correct and incorrect expressions could be a potential for learning. Being familiar 
with variation theory, which states that that which is varied is likely to be 
discerned, they realised that in order to see what a correct expression was, one 
needed to know what was incorrect. So contrasting correct with incorrect 
expressions was added to the planning of lesson two. 

The teachers also reflected upon whether the lesson opened up possibilities for 



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understanding of the concepts of variable and constant. For instance, was it 
possible for the students to understand that the difference in age between two 
persons is the same independent of their age? They decided to bring up the idea of 
variable (i.e., something that varies) by substituting and varying values for the 
letters. In this way, they anticipated that the learners would learn that the relation 
between the variables is the same independent of the values taken. 

Second Cycle: Conducting Lesson 2 

To the original lesson plan some more dimensions of variation were added. So, 
besides varying the algebraic representation for the same example, it was planned 
to contrast correct and incorrect expressions and to substitute the letters with 
values (i.e., to keep the constant "in-variant" and varying the values of the 
variables). The lesson was similar to lesson 1 with respect to the shift between 
plenary and pair work and conducted in accordance with the planning. Just like in 
lesson 1 the tasks given were to find formulas for different examples, and vice 
versa (the example was "invariant" and the representation varied). For the example 
"My cousin John is 1 years younger than me, Henry", permutations of J+10=H 
were suggested. Just as was planned, when this and similar examples were given, 
the teacher himself suggested an incorrect equation (e.g., H+10=J). In accordance 
with the planning, it was demonstrated that the letters could be substituted by 
numerical values. The teacher showed that no matter what values are used, all 
variations of J+10=H correspond to the example, (except for H+10=J). It could be 
noted that one of the students commented on this by saying: "it could correspond to 
another example". The teacher agreed, but without following up on this. Instead, he 
continued to explain that no matter how old Henry and John are the difference in 
age will be the same. So the students were clearly told that the different formulas 
correspond to the same example independent of the values of H and J. This aspect, 
the idea of a variable, was never brought out in lesson 1. Thus, in that respect the 
promoted space of learning was more complex than in lesson 1 . 

The remark from the student saying that H+10=J could correspond to another 
example is one example of a noticeable feature of lesson 2; that the students 
opened up for alternative interpretations and variation. Another example of this is 
when "My cousin John is 10 years younger than me, Henry" was written on the 
board and the teacher asked: "What do we use when we are doing maths?" Apart 
from replies like "numbers", "minus", "ruler and rubber", one student answered: 
"X" and "Y". This was probably not anticipated by the teacher and he did not 
comment on this at all. The teacher reacted in the same way some minutes later 
when he asked for a formula for the example and another student suggested: 

S: You can write X minus ten equals Y! 

T2 writes X-10=Y on the board. 

T2: Explain. How did you work this out? 



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S: Mm, X that's you [your age] minus ten years and Y is John. 

This suggestion from the student reveals a new aspect of the object of learning; that 
there are other possible ways of representing the variable than using the initials. 
However, the teacher does not pick this up at all. Instead he says: 

T: Okay, So you changed our names to X and Y instead. Is it possible to 
change my name to something else besides X and Y? Elsie? 

E: Er,..to letters. 

T2: Yeah, what letters? 

E: A or B, C. 

T2: Yes, what else? 

S?: (inaudible) 

T2: Yes. Fiona? 

F: If you knew the age you could take John's age plus yours equals ten. 

The teacher agrees, but wants to wait a while with that. He wants them to come up 
with letters. He says: 

T2: Right. If we knew how old they are. All right we'll get to that later on, I 
think. Can you use anything else? William has used X and Y and then Elsie 
and Minimi said other things too. Could we use other letters? What letters? 
Roger? 

Roger: A and B, maybe? 

T2: Oh yes, we could use that. Albert? 

Albert: H and J. 

T2: Why? 

Albert: Because your names are Henry and John. 

T2: Aha! Yes we could actually use those, H and J, because they are symbols 
for Henry and John. Instead of writing the names out fully, you can write H 
and J instead. 

This excerpt shows that the students suggest other letters than the initials. Although 
the teacher does not disapprove of using A and B or X and Y, my interpretation is 

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LEARNING TO DESIGN FOR LEARNING 

that he gives prominence to the letters H and J. His comment: "Aha! Yes we could 
actually use that" is said in an encouraging way, and supports my interpretation. It 
is likely that he prefers the initials (H and J), which can be seen in the following 
also. One student suggested: 

S: Circle plus 10 equals square. 

T2: Ah. So instead of J and H you put in square and circle. Okay. We could 
have that, but let's stick to H and J so you all can follow me. We'll work with 
H and J, these abbreviations of John and Henry. 

He accepts the suggestion "square and circle", but points out that H and J are the 
symbols he wants them to use and permutations of J+10=H are written on the 
board. 

From this it appears that the students want to vary the symbols, whereas the 
teacher would like to see a variation of J+10=H. That is, he keeps the letters 
invariant but varies the algebraic expression. Probably he had a particular intention 
with that, since he later, after having substituted numbers for the letters in several 
expressions (J+I0=H, H-10=J and so on), draws the students' attention to the 
formula H+10=J. This is supposed to be a counter-example, that is an incorrect 
expression for the example given, actually implying that "Henry is 10 years older 
than John", just as the teachers' group had planned. 

Several similar examples were talked about in the lesson. Again the students 
suggested other letters than the initials. Although these were accepted by the 
teacher; the teacher himself constantly used the initials to represent the variables. 
Furthermore, for each and every example, he gave a counter-example, hence an in- 
correct formula, and asked if it correspond to the example; "Is it correct?" So, 
again we can see that the pattern of variation put forward by the teacher is different 
from the one put forward by the students. The teacher varies the algebraic J+10=H 
(correct as well as incorrect permutations) while keeping the letters invariant. The 
students varies the letters used (e.g., Y and X), and the algebraic expression 
(however, they never gave an incorrect expression). It was the students who 
suggested other letters than initials of the variable in the algebraic expressions-not 
the teacher. The teacher presented one pattern of variation and invariance, the 
students another. You could say that the students in the lesson opened a dimension 
of variation which was not planned by the teachers. 

Second Cycle: Post-lesson Session 2 

What features of the lesson caught the teachers' attention in the post-lesson session 
when watching the recorded lesson and what did they learn from lesson 2? When 
watching the recorded lesson the tutor noticed that one of the students suggested 
using X and Y: 

Tutor: Nice comments from your students. It doesn't necessarily have to be H 
and J. 



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T2: That one should have been taken up! 

When the tutor commented on this, teacher 2 (who conducted the lesson) 
immediately realised that the variation suggested by the students - using X and Y 
as letters for the variable - could have been useful and that he could have elicited 
that more. So, when observing the recorded lesson, the teachers became aware of 
the possibility of using other letters or other symbols as well. For instance, Tl 
expressed her awareness by asking: 

Tl : Did you write X and Y there? 

T2: No, 1 didn't. I wanted to get to the abbreviations H and J. Therefore I 
didn't want to confuse them with X and Y. I didn't want to use that. 

T2 confirms that he did not support the suggestion "X and Y"; he wanted to keep 
to the initials. He also explains why. When "circle and square" were suggested by 
one of the students in the lesson recording, T2 commented on this by saying: "I 
was guided by the manuscript". My interpretation is that since the group had much 
focus on contrasting correct and incorrect representations in the planning, in the 
lesson, the teacher was concentrated on that and not "tuned" to the opening of 
variation of symbols. So, it was not until the post-lesson session that T2 realised 
that the students' suggestion of using other symbols than the initials could create 
potential for learning. My interpretation is that, by carefully analysing the lesson 
and being exposed to a variation (the letters used could be others than the initials), 
he (and his colleagues who had planned the lesson together) became aware that by 
restricting the symbols to the initials of the variable, one significant aspect of the 
object of learning-how you choose symbols for the variables is arbitrary-was never 
brought out. This could be traced to the early planning sessions in the learning 
study. To, facilitate learning and not confuse the students with X and Y which they 
assumed were "frightening" and strange to the students, they decided to use letters 
as abbreviation. 

That the teacher in the lesson was not tuned to the variation opened by the 
student may be an effect of the well planned lesson (cf., "I was guided by the 
manuscript" above). That is, T2 was focused on coming up with an algebraic 
expression that was incorrect, specifically a counter example (which was decided 
in the first post-lesson session), so that he was inhibited from giving his whole 
attention to what the students were coming up with. However, a new insight came 
from carefully watching the lesson recording. But above all, when they were 
exposed to a dimension of variation revealed by the students in the lesson, they 
experienced a critical aspect of the object of learning: that the symbols representing 
the variable are arbitrary, was never presented in the lesson. This pattern of 
variation created by the students afforded a learning space for the teachers in the 
post-lesson session. 

This insight prompted them to revise the plan for the third lesson. They decided 
to deliberately introduce a variation of symbols. If the students did not come up 
with other letters and symbols than the initials, the teacher herself (T3) would take 



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the initiative. Thus to the previous dimensions of variation proposed by the 
teachers yet another one was added. They also planned a change of assignments; to 
match each example with different (both correct and incorrect) expressions. 

Third Cycle: Conducting Lesson 3 

Initially two examples were chosen, "My friend Anna is 5 years older than me, 
Joan" and "A chocolate bar costs five crowns more than a toffee". Besides 
varying the positions of the variables (and thus the operation), for instance, T+5=C, 
they also varied the symbols used. So the letters T and C (shorthand for toffee and 
chocolate) as variables in the expression were written on the board simultaneously 
with expressions with X and Y and other symbols (e.g., Dand □). Next, just as was 
planned, the assignment to match an example and different algebraic expressions 
was worked on in student pairs. The example given was the following: "Martha 
has five more marbles than Colin". The expressions given were mixed; some of 
them corresponded to the example, others did not. After the pair work, when this 
example was discussed in the whole class, something happened that was probably 
not foreseen by the teacher. When the teacher asked if M=C-8 was a correct way to 
represent the example, most of the students said it was incorrect. Willie objected to 
this; the equation could represent the example provided that the number of marbles 
Martha had was represented by C, and Colin's marbles by M, he argued. 

W: It could correspond if C were Martha and M were Colin. 

T3: Listen to Willie. 

Willie's suggestion was noticed by the teacher, who called for the students' 
attention. Several students disagreed. The students who had taken for granted that 
if a letter (other than X and Y) was chosen to represent the variable, it had to be the 
initial, probably found this idea provocative. 

SI : No, it cannot, since Martha is M. 

S2: Right. It cannot. 

S3: That's not correct. 

T3: But you could just as well write 'banana' here, or 'strawberries'. So 
M=C-5, is correct if M=C and C=M. 

Hence, besides opening the planned variation of the position of the symbols (and 
thus the operation) and the symbols (abbreviations/X and Y/Dand □), it was 
demonstrated that it was possible to vary the symbols chosen by swapping them. 
My interpretation is that this pattern of variation and invariance brought out the 
arbitrariness of the symbols in an equation even more; it is possible to choose any 
symbol, provided it is defined. 

In lesson 3 a more complex pattern of variation appeared compared to the 

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previous two lessons in the cycle, the potential space of learning was expanded by 
the opening of more dimensions of variation in lesson 3. This implied that more 
and critical aspects of the object of learning were revealed in lesson 3 just as lesson 
2 was "richer" in that respect compared to lesson I . 

It could be suggested that during the learning study cycle the lesson plan and the 
enacted lesson developed in terms of critical aspects exposed to the learners. 
Taking their point of departure in the recorded lessons that is, the way they 
experienced how the object of learning was handled and the students' learning 
outcomes, the teachers successively became aware of aspects that they anticipated 
were critical and necessary for student learning and revised the plan accordingly. 
The awareness was sometimes a result of inputs from the students, who introduced 
a variation which was not planned by the teacher. For instance, in lesson 2 the 
students brought up a significant aspect by suggesting a variation of the 
representation of the variable. As was shown above, Teacher 2 did not pay much 
attention to that in the lesson. However, when watching the recorded lesson, this 
variation was experienced by the teachers and made them discern an aspect they 
had taken for granted. Due to this judgment, the teachers decided to revise the last 
lesson in the cycle, so this aspect would be elicited. 

Teachers ' Learning and Student Learning - A Parallel Process 

The case described above suggests that it is possible for teachers to learn about the 
object of learning from their teaching and from the learners. My interpretation is 
that they successively developed their understanding about learning and teaching 
how to write algebraic expressions for examples with an additive structure, thus 
about the object of learning. Initially they learned that it is necessary to 
demonstrate different representations for the same example. When they planned the 
first lesson, they assumed that using other letters than the initials would be 
confusing; however, after lesson 2, when watching the recorded lesson, they 
realised what effect this assumption had on students learning possibilities, 
remarkably, by a variation suggested by the students. The new insight affected how 
the next lessons were planned and implemented. In the same way, they became 
aware that contrasting correct an incorrect representations would possibly bring out 
other learning affordances. From having planned and conducted a rather simple 
pattern of variation, in terms of bringing out few critical aspects in lesson 1, they 
became more and more aware of, and could discern, additional critical aspects as 
the cycle progressed. They also managed to implement the plans in a way that 
made it possible for the learners to discern these aspects. It is clear that the teachers 
went through a process in which their learning about their students' learning and 
what it takes to learn about algebraic expressions was developed. 

If we take a look at students' learning outcomes of the post-test, we find a 
parallel process among the learners during the cycle (Kullberg & Runesson, 2006). 
One of the assignments on the post-test was to find different expressions for the 
example, "An apple and a cucumber cost 15 kronor". The results are shown in 
Table I . 



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LEARNING TO DESIGN FOR LEARNING 

Table 1. Percentage (number) of students who changed the positions of the symbols on the 

post-test 

The same letters, Varying letters, Incorrect or no 

varying position the same position answer 

'Lesson 1 (N=23) 87% (20) 0%(0) 13%(3) 

Lesson 2 (N=26) 73% (19) 19% (5) 8% (2) 

Lesson 3 (N=27) 59% (16) 30% (8) 11% (3) 

*NOTE: each lesson was conducted with a different class of students. 

From Table 1 it can be seen that the majority in the three classes were able to write 
two different equations. Only 13%, 8% and 11 %, respectively, failed on this. 
However, the rate of students who changed the letters was greater after lesson 2 
and 3 (19% and 30 %). Hence, it was more common that students, for instance, 
gave the answer A+C=15 and X+B=15. This was not found in any case after lesson 
1 . Besides, on the post-test after lesson 3 two students varied both the order and the 
chosen symbols (e.g., A+C=15 and 15-X=C). Our conclusion is that the post- test 
to some extent reflects the learning possibilities in the three lessons, where the last 
lesson afforded a more complex understanding of how to write an algebraic 
expression for a given example and how to choose symbols representing the 
variables. In my interpretation, the idea that the representations chosen are arbitrary 
was brought out in a more clear and distinct way in lesson 3. In this lesson it was 
pointed out that you could use any representation, but the symbol had to be defined 
as representing a specific variable. In this case, it is interesting that this condition 
was brought out by the students. How could that happen? It seems that the more 
complex pattern of variation presented invited the learners to explore other and 
more complex variations. Furthermore, this development of learning possibilities 
was parallel to a learning process among the teachers. 

This parallel process is also interesting in that the awareness and the suggestions 
for variation from the teachers afforded an opening from the students just as if the 
variation opened up by the teachers invited the students to find other variations 
themselves. For instance, the more complex pattern of variation that was planned 
and constituted in lesson 3 seems to have invited the discovery of other critical 
aspects of the arbitrariness of the symbols on the initiative of one of the learners. 

DISCUSSION AND REFLECTIONS 

To summarise, in learning study the aim is to enhance student learning, not 
primarily to organise learning in general, nor implement a new curriculum or 
teaching arrangements. However, this could be the effect of the learning study, 
although it is not the point of departure. In a learning study the aim is to help the 
students to learn something specific. We ask: What are the necessary conditions for 
learning something and how can these be met in the learning situation? If students 
do not learn what we expected, we do not seek the answers to their failure in 
inadequacy of the student; neither do we seek them in the teaching arrangements or 

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methods used. Instead we focus on the students' learning-what their difficulties 
are-and on how the content must be handled in the lesson in order to overcome the 
learning obstacles. We try to find out what it takes to know something; what kind 
of mathematical structure the specific capability has, what aspects of the object of 
learning are critical and, hence, are necessary to discern in order to learn. Once we 
think we know about these, we try to draw the students' awareness of them by 
planning the learning situation so it will open up variation in those particular 
dimensions. 

In a learning study teachers get the opportunity to observe colleagues teach the 
same thing. This is one of the features of a learning study that makes it appropriate 
to mathematics teacher education. However, we have not been able to introduce 
learning study in a larger scale in prospective teachers training, but the approach 
seems to have several advantages in that respect. For instance, the prospective 
teachers could learn an inquiry approach to their teaching in which they 
systematically plan, enact and evaluate teaching and learning, guided by some 
theoretical principles. The theoretical framework does not tell us what the critical 
aspects are or what to do, but gives an analytical tool to understand learning 
obstacles from the point of view of the learner and how the object of learning must 
be enacted in order to overcome these. It could be assumed that learning study 
offers an opportunity to enhance the (prospective) teacher's subject knowledge as 
well as the pedagogical content knowledge. In learning study teachers receive 
immediate feed-back on their interactions with learners. It argued that the most 
effective place to improve teaching is to do something in the context of the 
classroom (Stigler & Hiebert, 1999). This is exactly what happens in learning 
study. 

A key question is whether it is possible for teachers to work with learning study 
on regular basis. In learning study one can go deeply into a specific topic for a long 
period of time; sometimes for a whole semester. Teachers have competing 
demands, and operate under time constraints. However, our experience in working 
with teachers suggests that, by deeply investigating the particular, one can gain 
knowledge that is of general character also. So in that sense the benefit from a 
learning study might have effect on teachers' work in general. In learning study the 
teachers get insights into the nature of powerful instances of teaching and learning. 
Therefore, it may not be necessary to be engaged in learning studies all the time. 
One learning study a year or less might be sufficient. 

There are other challenges with implementing the learning study approach in 
regular practice. In learning study one must be open for considering existing 
practice; thus the approach could be perceived as a threat. For example, a teacher 
might think, "my existing practice and that which I take-for-granted could be 
questioned, it may be necessary to combine my ideas with the ideas of my 
colleagues". Our experiences suggest that the collaborative nature of learning study 
creates a shared ownership of the lessons. A collective inquiry takes place which 
results in a collective responsibility for the planned lesson and, foremost, for 
students' learning (for more on collective teacher inquiry, see Chapters 14, 15, 
Volume 3). 



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LEARNING TO DESIGN FOR LEARNING 

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Marton, F., & Tsui, A. B. M. (Eds.) (2004). Classroom discourse and the space of learning. Mahwah: 

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Vila Runesson 
University of Gothenburg 
Sweden 



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8. BELIEFS ABOUT MATHEMATICS AND 
MATHEMATICS TEACHING 



In this chapter we trace the growing research interest in teachers ' beliefs and describe the 
multiple ways the term belief is used. We also summarise and critique research on teachers ' 
beliefs pertinent to the learning and teaching of mathematics. We focussed on articles 
published between 1997 and 2006 in influential and readily available journals. The work 
examined is clustered under various headings including: beliefs about pedagogy and 
learning, beliefs about the nature of mathematics and mathematics content areas, beliefs 
about technology, about gender and equity issues, and beliefs about aspects of mathematics 
achievement. Strengths and weaknesses of the research studies surveyed are discussed, 
methodological issues are highlighted, and some theoretical issues about teachers ' beliefs 
are fore grounded. In the final section we indicate areas in which research consensus has 
been achieved and point to others still plagued by ambiguity and worthy of further, well 
planned, and focussed investigation. 

INTRODUCTION 

Studies on beliefs about mathematics and mathematics teaching and learning have 
been a relatively new addition to the mathematics education research agenda. 
"Indeed, in a comprehensive review of American and Canadian research in 
mathematics education reporting on research conducted primarily in the 1960s and 
1970s, the word "belief does not even appear in the index for this nearly 500-page 
volume" (Lester, 2002, p. 346). Nor does belief feature among the entries in the 
index of Wittrock's (1986) Handbook of research on teaching. However, more 
recent reviews of the mathematics research literature reveal a vastly different 
picture. For example, writing in a special journal issue on "affect and mathematics 
education" Leder and Grootenboer (2005) noted "Since 1995, MERGA [the 
Mathematics Education Research Group of Australasia] publications have 
invariably included a significant number [...] of articles and papers related to 
aspects of affect in mathematics teaching and learning. [...] Over the ten years 
(since 1995) beliefs have been a popular concern and the focus of between half to 
two-thirds of the papers within this subgroup in any given year" (p. 3). 

Inspection of recent Handbooks concerned with mathematics and teacher 
education also confirms the burgeoning research interests on beliefs and 
mathematics teaching and learning. The Second International Handbook of 
Mathematics Education (Bishop, Clements, Keitel, Kilpatrick, & Leung, 2003) 
contains 30 entries under the index heading of belief; its predecessor The 
International Handbook of Mathematics Education (Bishop, Clements, Keitel, 



P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 173-192. 

© 2008 Sense Publishers. All rights reserved. 



HELEN FORGASZ AND GILAH LEDER 

Kilpatrick, & Laborde, 1996), none. The Handbook of International Research in 
Mathematics Education (English, 2002) has 13 such entries; the Handbook of 
Research on the Psychology of Mathematics Education (Gutierrez & Boero, 2006) 
features 38. Since some entries cover multiple references to beliefs on the same 
page, and others to multiple page sections in which beliefs and aspects of 
mathematics education are discussed, index entries provide, of course, only a crude 
measure of research interests in a particular area. This is illustrated clearly when 
belief index entries of the first (Grouws, 1992) and second edition of the Handbook 
of Research on Mathematics Teaching and Learning (Lester, 2007) are compared. 
The former has 19 index entries; the latter, surprisingly, just one although a full 
chapter is devoted to "Teachers' beliefs and affect". 

More general handbooks such as the Handbook of Educational Psychology 
(Alexander & Winne, 2006) also deal in some depth with teacher beliefs. Topics 
discussed in a chapter devoted to "teacher knowledge and beliefs" include a 
veritable alphabet range of topics linked to teachers' beliefs. These, according to 
index entries, include the beliefs of teachers - from K to 1 2 - and: 

Adolescence and Assessment; Basic skills; Culture and Curriculum 
resources; Diversity and Democracy; Effort and Explicit knowledge; Family 
structure; Gender stereotypes; High stakes tests; Instructional decisions and 
Implicit knowledge; Journal data; Knowledge; Learning to teach; 
Mainstreaming and Mathematics knowledge; Nature of intelligence; 
Observational methodologies and Outcomes; Parental involvement and 
Problem behaviours; Quantitative and Qualitative methods; Race and 
Relating to students; Self efficacy and Student achievement; Tests and 
Teaching identity; Urban teacher; Values; "Whole" of teachers' mental lives; 
Young teachers. 

These diverse entries evocatively convey the varied theoretical lenses through 
which affect and beliefs have been explored in the broader research literature: 
political science, history, anthropology, sociology, and psychology. A more 
restrictive set of theoretical avenues, it will be seen from this review chapter, has 
been taken in the exploration of teachers' beliefs. 

Mason (2004) has creatively captured the many terms used as virtual synonyms 
for beliefs. His diverse list, which ranges from "A is for attitudes, affect, aptitude, 
and aims" through "E is for emotions, empathies, and expectations" and "J is for 
justifications and judgements" to "S is for sympathies and sensations" and "Z is for 
Zeitgeist and zeal" (p. 347), highlights the importance of delineating the scope of 
this chapter. 

As noted by Bar-Tal (1990), it is "especially social psychologists, who have 
devoted much effort into studying the acquisition and change of beliefs, their 
structure, their contents, and their effects mainly on individuals' affect and 
behaviour" (p. 12). Leder's (2007) inspection of papers presented at two recent 
major mathematics education research conferences, those of the International 



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BELIEFS ABOUT MATHEMATICS AND MATHEMATICS TEACHING 

Group for the Psychology of Mathematics Education [PME] 1 and of the 
Mathematics Education Research Group of Australasia [MERGA] confirms Bar- 
Tal's observation. 

Knowing that a range of definitions of beliefs have been adopted (see, for 
example, Leder & Forgasz, 2002a), we initially had difficulty deciding what to 
include in this review. To this end, Wilson and Cooney's (2002) observation with 
respect to teachers provided broad descriptors to assist in our literature searches, 
and permitted some degree of judicial choice in what to include and exclude: 

[TJhere does not appear to be a consensus about what constitutes beliefs or 
whether they include or simply reflect behaviour. Generally speaking, neither 
does there seem to be agreement about the notions of teachers' conceptions 
or teachers' cognitions. However, regardless of whether one calls teacher 
thinking beliefs, knowledge, conceptions, cognitions, views, or orientations, 
with all the subtlety these terms imply or how they are assessed [...,] the 
evidence is clear that teacher thinking influences what happens in classrooms, 
what teachers communicate to students, and what students ultimately learn, 
(p. 144) 

Wilson and Cooney (2002) also identify a relationship between teachers' beliefs 
and students' learning. This suggests that researchers should not ignore teachers' 
beliefs as influencing learning outcomes. As noted above, researchers and 
curriculum developers use a range of terms interchangeably when discussing the 
affective domain. Three decades earlier than Wilson and Cooney (2002), Cockcroft 
(1982) had highlighted the effects of teachers' beliefs (attitudes) on students. In his 
influential publication, Mathematics counts, Cockcroft (1982) wrote: 

It is to be expected that most teachers will attach considerable importance to 
the development of good attitudes among the pupils whom they teach [...] 
Attitudes are derived from teachers' attitudes [...] and to an extent from 
parents' attitudes. [...] Attitude to mathematics is correlated [...] with the 
peer-group's attitude, (p. 61) 

The Programme for International Student Assessment [PISA] "measures student 
performance in reading, mathematics and science literacy and also asks students 
about their motivations, beliefs about themselves and learning strategies" (OECD, 
nd). Again, as an acknowledgment of the interaction between affective factors and 
the learning of mathematics, those planning the PISA project explained: 

Mathematics related attitudes and emotions such as self-confidence, 
curiosity, feelings of interest and relevance, and the desire to do or 



1 The 31" annual PME conference held in Seoul, Korea, July 8-13, 2007. In recent years the goals of 
PME have been broadened beyond psychological to include "other aspects of teaching and learning 
mathematics and the implications thereof. 

2 The 30"' annual MERGA conference held in Hobart, Tasmania, Australia, July 2-6, 2007. 



175 



HELEN FORGASZANDGILAH LEDER 

understand things [...] are important contributors to it (i.e., mathematical 
literacy) [...] The importance of these attitudes and emotions as correlates of 
mathematical literacy is recognised. (OECD, 2004, p. 26) 

The influential National Council of Teachers of Mathematics [NCTM] (2000) 
similarly argued: 

Students' understanding of mathematics, their ability to use it to solve 
problems, and their confidence in, and disposition toward, mathematics are 
all shaped by the teaching they encounter in school, (p. 16, emphasis added) 

Thus, there is explicit recognition that concerns with teachers' beliefs goes 
beyond theoretical interests alone, and encompasses practical classroom-based 
issues. 

Structure of This Chapter 

In planning this chapter we took on board the advice seemingly given by Berliner 
and Calfee (1996) to the authors of their Handbook and reproduced by Winne and 
Alexander (2006). 

(Authors were charged) to prepare chapters that build on the history of a 
particular domain, lay out seminal issues and questions, and survey the major 
results and puzzlements, illuminating each with [...] descriptions of 
particular findings, and connecting the domain with important questions that 
warrant further investigation [...]. (p. xi) 

Focussing on the years 1997 to 2006 the following databases were trawled for 
pertinent literature on the beliefs of mathematics teachers: A+ Education, The 
Australian Education Index [AEI], ProQtiest, ERIC, and Google Scholar. Three 
high impact mathematics education journals were also examined: Educational 
Studies in Mathematics, the Journal for Research in Mathematics Education, and 
the Journal of Mathematics Teacher Education. The references found were 
analysed, and we concluded that the review could most conveniently be grouped 
under a number of headings: beliefs about pedagogy and learning, with separate 
discussions on teachers at different grade levels: primary, middle years, and 
secondary; beliefs about mathematics and mathematics content areas; beliefs about 
technology; gender, equity and beliefs; achievement and beliefs; and theoretical 
discussions and issues related to teachers' beliefs. It should be noted that in some 
research studies there was overlap in these dimensions. Thus, some studies are 
discussed in more than one section of the chapter with suitable links made between 
them. 



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BELIEFS ABOUT MATHEMATICS AND MATHEMATICS TEACHING 
THE LITERATURE ON TEACHERS' BELIEFS 

Beliefs about Pedagogy and Learning 

Teachers' beliefs about pedagogy are closely interwoven with their beliefs about 
the ways in which their students learn. In some articles this link can be inferred; in 
others it is made explicitly. 

Primary teachers. In many of the articles in which primary teachers' pedagogical 
beliefs were explored findings were based on intensive scrutiny of a small number 
of case studies or investigations involving a small number of teachers. Semi 
structured interviews, classroom observations - in some research projects carried 
out by a participant observer; in others by a neutral recorder - lesson plans, and 
journals completed by the participating teachers and the researchers themselves 
were techniques frequently used for data gathering. In some studies, interviews 
were also held with parents and with colleagues of those involved in the study. 
Data obtained from surveys and questionnaires containing items with Likert-type 
response formats were reported in the relatively few large scale studies identified 
by our search of the literature. The duration of many of the studies, and in 
particular those aimed at changing teachers' pedagogical beliefs and any impact of 
this on instructional practices, spanned a substantial slice of the school year. 
Aspects of some longitudinal studies have also been reported. For example Steele 
(2001) and Wood and Sellers (1997) drew on data from longitudinal studies which 
respectively spanned four and two years. 

The conceptualisation of mathematics and instructional practices as advocated 
by the mathematics reform movement featured prominently in a number of studies. 
The extent to which teachers' beliefs and practices reflected the aims of the 
movement and the curriculum content and instructional approaches promoted by it 
was identified in some studies (e.g., Archer, 2000). In others (e.g., Clarke, 1 997; 
Senger, 1998; Steele, 2001), attempts were made to gauge whether the strategies 
advocated were actually adopted. Opportunities to reflect on their in-class practices 
seemed to enhance not only changes in teachers' beliefs but also to promote a 
stronger congruence between teachers' (changed) beliefs and their practices (e.g., 
Clarke, 1997; Raymond, 1997; Senger, 1998; Steele, 2001). Obstacles in the way 
of changing teachers' beliefs and their practices included preconceived ideas about 
students' (socioeconomic background related) needs (Sztajn, 2003), emphasis on 
practical constraints preventing the translation of beliefs into practice (Quinn & 
Wilson, 1997), teachers' limited mathematics background knowledge (Halai, 1998; 
Raymond, 1997; Steele, 2001), students' behaviours, and teachers' own school 
experiences. Indeed, prospective teachers' professional preparation courses, it was 
reported in several studies (e.g., Archer, 2000; Raymond, 1997), had less influence 
on teachers' beliefs and practices than teachers' own classroom experiences and 
their experiences as students of mathematics. Before turning to the next section, 
Sztajn's (2003) provocative caution on the link between teachers' beliefs and 



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HELEN FORGASZ AND GILAH LEDER 

practices is worth noting. "Beliefs and practice", she argued, "are consistent - if in 
a study we find they are not, then I think we asked the wrong questions" (p. 74). 

Prospective primary teachers. Prospective teachers are readily available to serve, 
within appropriate ethical guidelines, as subjects in research projects. It is therefore 
not surprising that this group, overwhelmingly comprised of females, was 
frequently targeted by those wishing to explore beliefs about mathematics and 
pedagogy. Some of the trends noted in the review of research on the mathematics 
related beliefs of primary teachers were also apparent when work involving 
prospective primary teachers was reviewed. Small sample studies again 
predominated, though this now typically comprised a relatively small group of 
students (e.g., the group taking a particular course) rather than just two or three 
individuals. Much emphasis was again given to the impact on participants' beliefs, 
and possibly their (intended) instructional practices, or exposure to reform-oriented 
and constructivist approaches during their professional training. Interviews, written 
materials, and group discussion were used for data collection in many of the 
studies surveyed (e.g., Ambrose, 2004; Langford & Huntley, 1999; Mewborn, 
1999); in others heavy reliance was placed on survey responses. Some of the latter 
studies (e.g., Ambrose, 2004; Seaman, Szydlik, Szydlik, & Beam, 2005; Vacc & 
Bright, 1999; Wilkins & Brand, 2004) used previously constructed questionnaires; 
others - including Hart (2002) and Spielman and Lloyd (2004) relied on 
information obtained from surveys specifically constructed for the investigation 
described. 

A strong theme that emerged from our survey of relevant articles was the 
positive effect on participants' beliefs of practicum and structured field work 
experiences, preferably coupled with times for reflection (e.g., Ambrose, 2004; 
Mewborn, 1999; Vacc & Bright, 1999). Ambrose's (2004) conclusion is 
representative of such findings. 

Providing prospective teachers with intense experiences that involve them 
intimately with children poses a promising avenue for belief change. 
Coupling these experiences with reflection allows the beliefs that arise from 
these experiences to be examined and refined, (p. 1 17) 

The benefits of guided field work experiences are further reinforced by 
Langford and Huntley's (1999) study in which prospective teachers were given the 
opportunity to do a summer internship in a variety of settings: business, industry, 
scientific institutions, informal educational settings including museums and zoos, 
as well as in the more formal educational setting of a regular classroom. Coupled 
with the internship were extended collaborations with relevant professionals, for 
example, mathematicians, scientists or educators. 

It appears that the interns, on the heels of the challenge of their summer 
internship, had directed their thinking from what they did during the summer 
to what they hoped to do as teachers. They dared to take on challenging tasks; 
they envisioned themselves as risk-taking teachers who intended to question 



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BELIEFS ABOUT MATHEMATICS AND MATHEMATICS TEACHING 

and pursue understanding alongside their students [...]; they envisioned 
themselves as encouraging curiosity in their students [...]; they hoped to 
encourage their future students to take an active role in their own learning. 
(Langford & Huntley, 1999, p. 294, emphasis in the original) 

Various "tools" to effect change were used in the research reports we identified. 
A strong emphasis on field experiences, including field work early in the course, 
has already been noted. Others found useful included using children's literature to 
identify ideological positions which in turn were reflected in instructional 
strategies (Cotti & Schiro, 2004) and, most commonly, course content carefully 
structured to capture child-centred, reform-movement advocated, and constructivist 
approaches (Anhalt, Ward, & Vinson, 2006; Quinn & Wilson, 1997; Scott, 2005; 
Spielman & Lloyd, 2004; Timmerman, 2004; Vacc & Bright, 1999). Authors 
typically concluded that appropriate course content could lead to changes in the 
beliefs of the prospective teachers. For example: 

After one semester, teachers in the (innovative, reform oriented) curriculum 
materials section [...] placed significantly more importance on classroom 
group work and discussions, less on instructor lectures and explanations, and 
less on textbooks having practice problems, examples and explanations. They 
valued student exploration over practice. (Spielman & Lloyd, 2004, p. 32) 

In contrast, for the group in which the courses relied heavily on a traditional 
textbook "there was little change in the teachers' beliefs, in which practice was 
valued over exploration" (Spielman & Lloyd, 2004, p. 32). 

A cautious note about changes in beliefs captured during the study on which 
they were reporting, and not necessarily reflected in classroom practices, was 
injected by several researchers. Time constraints during prospective teacher 
education courses often prevented optimum practices from being implemented. 
"Considerable personal reflection on one's beliefs and behaviour would seem to be 
necessary for one to develop a coherent pedagogy... It is not clear whether pre- 
service teacher education programmes can structurally accommodate these needed 
'reflection events'" (Vacc & Bright, 1999, p. 107). Ambrose (2004) concluded that 
new beliefs were often added to, rather than fully replacing, previously held 
beliefs. Scott (2005) indicated that when the prospective teachers in her study tried 
to reconcile conflicting theory and practice, they most frequently turned to 
practising teachers who did not necessarily share their beliefs about constructivist 
practices. Discrepancies between espoused beliefs and observed or intended 
practices were reported by Quinn and Wilson (1997) and Timmerman (2004). In 
many of the reports which contained positive accounts of functional changes in the 
prospective teachers' beliefs it was nevertheless concluded that the extent to which 
these changes would eventually be translated into practice in classrooms could only 
be a matter of speculation. 

Middle school teachers and prospective teachers. Issues already identified in 
earlier sections were also found in work involving middle school teachers. Britt, 



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HELEN FORGASZ ANDGILAH LEDER 

Irwin, and Ritchie (2001), for example, reported that the poor mathematical content 
knowledge of the middle school teachers involved in their two year professional 
development programme prevented them from making the changes in beliefs and 
practices towards reform-oriented teaching promoted by the programme. In 
contrast, experienced, mathematically knowledgeable secondary school teachers' 
beliefs and practices changed in the desired direction. Middle school teachers' lack 
of confidence to teach mathematics was also identified by Beswick, Watson, and 
Brown (2006) as they sought baseline information at the beginning of a three year 
study aimed at assisting teachers to improve student performance in mathematics. 
Given the decade long emphasis in many countries, including their own 
(Australia), on constructivist, student-centered teaching their finding that "in spite 
of progressive beliefs many teachers and their students work in quite traditional 
classrooms" (p. 75) suggests that the dichotomy between theory and practice, that 
is, the schism between espoused beliefs and actual practice, is remarkably 
persistent. Teachers, as already noted, are quite likely to have adopted reform 
oriented rhetoric without altering their instructional practices. Zevenbergen (2003) 
interviewed 20 middle school teachers on the implementation of a unit of work 
based on the New Basics framework (a reform-based curriculum introduced into 
Queensland, Australia). Three groups of teachers were identified: the 
conservatives, the pragmatists, and the contemporary. The three groups of teachers 
held different beliefs about mathematics pedagogy, curriculum, and assessment. 
Identifying beliefs in these areas, Zevenbergen (2003) maintained, might be a 
useful first step in affecting change. 

The report of an action research collaboration between a middle school 
mathematics teacher and a mathematics teacher educator (Edwards & Hensien, 
1999), aimed at making the former's instructional practices congruent with reform 
advocated instructional practices, confirmed just how difficult the road to change 
can be. "The interplay between teachers' beliefs and their instructional practices 
seems to be more dynamic, interactive, and cyclic than a simple linear cause-and- 
effect relationship", Edwards and Hensien (1999, p. 189) argued. Change in 
instructional practice consistent with changes in belief was gradually achieved 
through an intensive, long term (multiple years), collaborative partnership 
requiring levels of sustained commitment, time, mutual trust and understanding - 
features rarely available to busy practitioners with multiple demands on their time 
and services. 

Secondary school teachers and prospective teachers. The data gathering methods 
described in the sections above were also used in many of the studies in which the 
beliefs of secondary school teachers about mathematics and pedagogy were 
examined. In studies in which surveys were used to gauge beliefs of larger groups 
of teachers, the dominance of females reported for studies involving prospective or 
practising primary teachers, changed to one of male dominance (e.g., Barkatsas & 
Malone, 2005). Secondary teachers, at least in those studies in which this 
information was sought, generally believed themselves to be competent 
mathematically (Britt, Irwin, & Ritchie, 2001; Doerr & Zanger, 2000) - a factor 



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BELIEFS ABOUT MATHEMATICS AND MATHEMATICS TEACHING 

which influenced the extent to which they adopted or moved towards reform- 
oriented teaching. Whether it was access to, or professional development on 
innovative materials or problem-solving focussed text books that led to 
mathematics classroom practices consistent with the strategies advocated by the 
reform movement, seemed to depend heavily on the teachers' beliefs about their 
students' mathematical ability (Arbaugh, Lannin, Jones, & Park-Rogers, 2006; 
Quinn & Wilson, 1997), their beliefs about the nature of mathematics, and their 
broader social and cultural beliefs (Archer, 2000; Barkatsis & Malone, 2005). The 
importance of an individual's values in the formation of beliefs was also explored 
in some depth by Cooney, Shealy, and Arvold (1998) in their detailed study of a 
small group of prospective secondary mathematics teachers. 

The limited "uptake" of mathematics reform movement sanctioned practices in 
the secondary mathematics classroom severely hampers, according to Frykholm 
(1999), their implementation by beginning teachers, including those fully familiar 
with, and accepting of, those principles. Frykholm's (1999) six cohorts of 
secondary mathematics student teachers, collectively spanning an investigative 
period of three years, contrasted the "large doses" of theory to which they were 
exposed during their training with the limited practical advice they received and 
experience they gained in how to implement the reform movement advocated 
strategies. In addition to the teacher dominated lessons they experienced during 
their own schooling, the bulk of the lessons they observed during their (semester 
long) teaching internship were again traditional, and teacher dominated. Thus 
although the prospective secondary mathematics teachers were "eager to gain 
knowledge of reform" and searched for "new models of instruction to emulate 
them [...] they recognised the ways in which their emerging beliefs often run 
counter to their teaching practices" (p. 102). Ensor (2001), too, described 
discrepancies between the participants' beliefs and practices as they began their 
career as secondary mathematics teachers. Her interpretation of this is reminiscent 
of the view expressed by Sztajn (2003) who hypothesised, as noted earlier, that 
when beliefs and practice are found to be inconsistent, then it is possible that the 
wrong questions may have been asked. Ensor (2001) similarly noted: 

The apparent discrepancy between Mary's words and her deeds [...] only 
emerged if the professional argot was taken to embody a fixed meaning for 
both teacher educators and student teachers. Once it was perceived to mean 
something different from Mary's perspective, she could be seen to be acting 
consistently with what she said. (p. 317) 

In the studies reviewed so far, interest has centred on teachers' beliefs about 
pedagogy and learning. Their beliefs about mathematics and mathematics content 
areas are considered next. 

Beliefs about Mathematics and Mathematics Content Areas 

Since the number of studies of practising teachers' and prospective teachers' 
beliefs about mathematics and mathematics content areas was relatively small, the 

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HELEN FORGASZ ANDGILAH LEDER 

views of practising mathematics teachers and prospective mathematics teachers at 
all levels - primary, middle years, and secondary - are presented together. 

Beliefs about the nature of mathematics Teachers' and prospective primary 
teachers' beliefs about the nature of mathematics have been the focus of several 
studies. Based on interview data from 17 primary and 10 secondary teachers, 
Archer (2000) reported marked differences in the views of primary and secondary 
teachers with respect to the nature of mathematics and its place in the curriculum. 
While primary teachers related mathematics to the everyday life experiences of 
students and recognised its inter-relationship with other dimensions of the primary 
curriculum, secondary teachers considered mathematics to be self-contained and 
not strongly linked to students' lives. The primary teachers' views were considered 
consistent with the holistic approach of primary education, and the secondary 
teachers' with the organisation of secondary schools. The primary teachers also 
identified differences in how mathematics had been taught to them and how they 
now taught. Seaman, Szydlik, Szydlik, and Beam (2005) administered the same 
instrument used by Collier (1972) to a cohort of prospective primary teachers in 
1998 and found that compared to Collier's 1968 cohort, the beliefs of the 1998 
cohort were more informal, that is, better aligned with constructivist principles. 
They argued that beliefs about mathematics and mathematics teaching are shaped 
by early educational experiences, a finding also reported by Lindgren (2000), and 
suggested that the differences between the two cohorts reflected changes over time 
in their learning experiences. 

Beliefs and their relationship to other learning- related factors and outcomes 
have also been examined. Perry, Way, Southwell, White, and Pattison (2005) 
found a negative correlation between prospective primary teachers' achievement 
scores and their beliefs about the importance of computation and correct answers in 
mathematics. In a study exploring how nine primary teachers responded to 
observations of reform-minded teaching, Grant, Hiebert and Wearne (1998) found 
that reactions depended on the views of mathematics held by the observing 
teachers. Compared to teachers who believed in the learning of concepts and 
processes, those who considered mathematics to be a set of skills and algorithms to 
be taught did not implement what they had seen into their own practice as intended 
by the programme developers. McDonough and Clarke (2005) claimed that the 
Early Numeracy Research Project [ENRP], a professional development 
programme, had served as a catalyst to change primary teachers' beliefs about the 
nature of mathematics. 

Gathering pre- and post-data from prospective primary teachers was a common 
means of identifying if changes in beliefs had taken place as a consequence of 
intervening experiences. In the Langford and Huntley (1999) study, discussed 
earlier, it was found that the summer internship experiences had challenged the 
students' conceptions of and beliefs about the nature and processes of mathematics 
and science. The researchers claimed that internships held promise for teacher 
reform in that those who had participated "intend to bring a holistic, conceptually 



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oriented view of mathematics and science into their classrooms" (p. 296). Szydlik, 
Szydlik, and Benson (2003) and Timmerman (2004) gathered data on beliefs about 
the nature of mathematics and its teaching, at the beginning and end of particular 
pre-service courses. After the courses, Timmerman reported stronger disagreement 
with tradition-oriented beliefs, and Szydlik et al. (2003) found that students' beliefs 
were more supportive of autonomous behaviours. 

Beswick (2005) and Barkatsas and Malone (2005) reported findings from large 
scale survey data on secondary teachers' beliefs about the nature of mathematics 
and the teaching and learning of mathematics; the former gathered data in 
Australia, the latter in Greece. Barkatsas and Malone (2005) indicated that 
"mathematics teachers' beliefs about mathematics could not be separated from 
their beliefs about teaching and learning mathematics" (p. 80) and data from one 
case study revealed that a very experienced teacher's espoused beliefs were less 
traditional than observed classroom practices would suggest. Beswick (2005) 
found that very few teachers held beliefs about the nature of mathematics that were 
solely consistent with Ernest's problem solving view; the extent to which 
classrooms could be characterised as constructivist was, however, related to how 
strongly teachers held this view. Based on data from one teacher who held a 
problem-solving view of mathematics and a constructivist view of mathematics 
learning, Beswick (2004) demonstrated that other beliefs about students and their 
abilities can affect teachers' practices in different contexts. 

Beliefs about Mathematics Content Areas 

There were surprisingly few studies which specifically focused on beliefs about 
mathematics content areas. However, teachers' and prospective teachers' beliefs 
about the mathematics curriculum in general, and the more specific content areas 
of problem-solving, modeling, proofs, and numeracy have been examined. Lloyd 
(2002) presented data from two case studies and discussed how experiences with 
innovative curriculum materials could change teachers' beliefs about mathematics 
and the content of the curriculum. Findings from one part of a survey administered 
to 42 middle years teachers (grades 5-8) by Beswick, Watson, & Brown (2006) 
indicated that the teachers were "convinced of the importance of being numerate" 
(p. 72), and that at least two-thirds of them believed that it was important to 
understand fractions, decimals, and percentage. Among other aims, Anderson, 
White and Sullivan (2005) used a mixed methods approach to examine primary 
teachers' beliefs about problem solving. Survey data indicated that many agreed 
that problem-solving was an important dimension of mathematics learning; open- 
ended and unfamiliar questions were considered by many to be more appropriate 
for high achieving students. Brown (2002) reported that International 
Baccalaureate examiners and assessors held a range of beliefs about the nature of 
mathematical modeling that were consistent with those reported by Kyleve and 
Williams (1996). Mingus and Grassl (1999) found that prospective primary and 
practising secondary teachers believed that proofs should be experienced fairly 
early in schooling. The secondary prospective teachers had more sophisticated 



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beliefs about the nature of proofs; beliefs about the role of proofs in mathematics 
learning were related to these beliefs. 

Beliefs about Technology 

In this section, beliefs about calculators, computers, and other forms of technology 
that can be used for the teaching and learning of mathematics are reported. Again, 
since the number of studies was quite small, findings for practising and prospective 
teachers at all levels are presented together. 

While not directly focusing on beliefs about calculators in a study of one 
secondary teacher's implementation of graphing calculators into her teaching, 
Doerr and Zangor (2000) claimed that "the role, knowledge and beliefs of the 
teacher influenced the emergence of [...] rich usage of the graphing calculator" (p. 
161). Based on questionnaire findings about their beliefs about graphics calculators 
- positive, neutral, and negative - Honey and Graham (2003) selected three 
prospective secondary teachers as participants in their study. Despite the 
differences in their beliefs, there were various reasons why each was generally 
reluctant to use the calculators during the teaching practicum. Forgasz and Griffith 
(2006) reported on teachers' beliefs about the effects of the imminent introduction 
of Computer Algebra Systems [CAS] calculators as compulsory tools into the final 
year of schooling mathematics courses in Victoria (Australia). It was found that the 
mathematics teachers were generally optimistic about the effects that CAS 
calculators would have on their teaching, on student learning, and on the 
curriculum. While there were a few dissenting voices, there was little difference in 
the views of male and female teachers. 

Findings from a three-year study in which data were gathered on secondary 
teachers' and their students' views on whether computers assisted students' 
understanding of mathematics were described by Forgasz (2005) and Forgasz, 
Griffith, and Tan (2006). The students (30%) were less convinced than their 
teachers (60%) of the positive impact of computers on their learning. Another 
dimension of the study involved interviews with a small number of teachers and 
observations of their lessons in which students used computers. Forgasz (2006) 
summarised the differences in the teachers' beliefs about how boys and girls work 
differently with computers in the classroom: 

[...] boys' competence, confidence, and interest in computers generally, 
appear to advantage them over girls when computers are used in the 
mathematics classroom. It seems that teachers feel the need to focus boys' 
attention to the task at hand and encourage and support girls to engage with 
the technology. It would appear that without positive intervention with girls, 
it is more likely that boys will gain more from their interactions with 
computers in the mathematics classroom, (pp. 459—460) 



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BELIEFS ABOUT MATHEMATICS AND MATHEMATICS TEACHING 

Gender, Equity and Beliefs 

Studies in which data were analysed by gender but in which gender was not the 
major focus of the research are not reported here. However, it should be noted that 
the findings from such studies reflect those of the work reviewed here. Practising 
teachers' and prospective teachers' gendered beliefs about boys and girls have been 
examined at both the primary and secondary levels. The patterns of beliefs found 
were generally consistent with those reported in earlier times (see, for example, 
Fennema & Leder, 1990). Based on quantitative data from 52 primary teachers in 
Germany, Tiedemann (2000, 2002), for example, found that the teachers held 
biased views that were more likely to disadvantage girls than boys. Tiedemann 
(2000) summarised the results as follows: 

Teachers rated mathematics as more difficult for average girls than for 
equally achieving boys. Teachers thought that average achieving girls were 
less logical than equally achieving boys. Girls were thought to profit less than 
boys from additional effort and to exert relatively more effort to achieve the 
level of actual performance in mathematics. With regard to girls, teachers 
attributed unexpected failure more to low ability and less to lack of effort 
than with boys. Nonetheless, teachers were aware of the girls' lower self- 
concept of mathematical ability. In summary an image emerges that, in the 
view of these teachers, girls, especially those of average or low achievement, 
must exert relatively more effort than boys to achieve a certain level of 
mathematical performance, (p. 204) 

An instrument designed to tap beliefs about mathematics as a male domain 
devised by Leder and Forgasz (see Leder & Forgasz, 2002b) was administered to 
secondary prospective teachers in Australia and the U.S. (Forgasz, 2001a). The 
teachers were asked to respond to the items as they believed secondary school 
students would respond. The same instrument had previously been administered to 
large samples of secondary students in both countries and the results indicated that 
their beliefs about mathematics as a male domain were inconsistent with previous 
findings in the field (Forgasz, 2001b). The prospective teachers' views of the 
students' beliefs were more in-line with earlier research, that is, that mathematics 
was perceived to be a male domain. There was a remarkable similarity in the 
patterns of the results from Australia and the USA. Soro (2000) reported that 
Finnish lower secondary school teachers held different beliefs about boys and girls 
as mathematics learners. Boys were considered more likely than girls to succeed in 
mathematical tasks demanding high cognitive abilities, and some teachers had a 
tendency to stereotype mathematics as a male domain. 

There appears to be a dearth of research in recent times on mathematics 
teachers' beliefs about other equity dimensions including race/ethnicity /culture and 
socio-economic backgrounds. Arguably, students with special needs can be 
considered under the equity umbrella. In this area the study by DeSimone and 
Parmar (2006) was somewhat unique. They surveyed a large group of middle 
school teachers to determine their beliefs about coping, in the regular classroom, 



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HELEN FORGASZ AND GILAH LEDER 

with students with learning disabilities. It was found that many teachers considered 
their teacher education course preparation to have been insufficient to prepare them 
for the realities faced in the classroom with these students. 

Achievement and Beliefs 

Yun-peng, Chi-chung, and Ngai-ying (2006) argued that China is a country where 
there is great pressure to perform on public examinations, including the national 
mathematics Olympiad. While the Olympiad results had no effect on students' 
future opportunities, teachers' pay and promotion prospects were related to 
students' performance levels. In a case study of two schools, one urban and one 
rural, the authors claimed that the teachers' beliefs about what students needed to 
learn, the goals of learning, teaching approaches, and the perceived difficulty level 
of the textbook affected performance levels. The findings appear consistent with 
those of Arbaugh et al. (2006) and Quinn and Wilson (1977) with respect to the 
likelihood that lack of access to resources had contributed to differences in the 
teachers' beliefs about students' abilities, and that geographic location had served 
as a factor contributing to inequitable outcomes. 

Using the TIMSS data from the U.S. and hierarchical regression modeling, the 
relationship between constructivist teachers' beliefs and practices and students' 
achievements were examined by Gales and Yan (2001). The relationship was also 
examined for behaviorist teachers. It was found that lower student achievement 
was related to the behaviorist teachers' belief that diversity in the classroom had a 
negative impact on achievement. A negative relationship was also found between 
the constructivist teachers' belief that mathematics is a practical, structured, and 
formal guide for addressing real world situations and student achievement. The 
latter finding may be indicative of the type of questions being asked in the TIMSS 
study, that is, the questions were inconsistent with this view of mathematics held 
by the constructivist teachers. 

Theoretical Discussions and Issues Related to Teachers ' Beliefs 

In a reflective piece, Cooney (1999) wrote of prospective teachers' unsophisticated 
understanding of school mathematics, the need to match their mathematical 
experiences with what would be expected of the reflective and adaptive teacher, 
and theoretical considerations for the conceptualisation of teachers' belief 
structures. Gates (2006) described two teachers with strongly held, ideologically- 
based, opposing views on whether students should be in homogenous or 
heterogeneous achievement groupings for mathematics learning. He used these 
teachers as examples to illustrate why so many reforms in mathematics education 
may have failed. He argued that: 

What is required is not an approach based on hoping such difference can be 
resolved, but one based on recognising ideological diversity and social 
conflict, focusing on the way social structures influence meaning rather than 



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BELIEFS ABOUT MATHEMATICS AND MATHEMATICS TEACHING 

negotiation between individuals within different power structures and cultural 
communities, (p. 365) 

Cooney's (1999) and Gates' (2006) views were implicit in many of the studies 
reviewed on teachers' beliefs about pedagogy and learning. Leatham (2006) 
challenged researchers to conceptualise teachers' beliefs as inherently sensible and 
to explore and explain apparent inconsistencies in beliefs and practices rather than 
simply pointing them out; similar ideas were expressed by Ensor (2001) and Sztajn 
(2003). 

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 

In conducting this review of the literature for the period 1997-2006 on 
mathematics teachers' beliefs, we noted that extensive research had been 
undertaken in some areas but that there has been much less focus on others. For 
example, there was much more work on primary teachers' beliefs about pedagogy 
and learning with less attention being paid to the views of middle years and 
secondary level teachers. There was more research on teachers' beliefs about 
pedagogy and learning than on their views about mathematics and its content areas, 
mathematics achievement, or technology for mathematics learning. With the 
exception of a few studies in which teachers' gender-related beliefs were 
examined, there was a dearth of work on teachers' beliefs related to other equity 
dimensions or students with special needs. 

We noted that the range of methodological approaches adopted in the studies 
discussed above was fairly limited. Larger scale quantitative data were generally 
gathered from surveys or questionnaires and, with few exceptions, the analyses 
were restricted to descriptive statistics. Small scale qualitative studies dominated 
the research on primary teachers' beliefs in particular. Although they were 
carefully conducted and analysed, the limited generalisability of the findings 
remains a challenge in the field. 

It was found that in many studies there was no theoretical discussion of the 
construct "beliefs", let alone any definition offered prior to "beliefs" being 
measured or inferences about beliefs from interview data or observation of 
classroom practices being drawn. This observed methodological weakness lends 
further support to Mason's (2004) somewhat cynical discussion of the A-Z of 
synonyms used for the construct, and Wilson and Cooney's (2002) observation 
about what might constitute teachers' beliefs, cited earlier. 

Based on the review of the literature presented, the following conclusions can be 
drawn: 

- The beliefs about the teaching and learning of mathematics of teachers at all 
levels are affected by a range of factors and can be context and student 
dependent; 

- Circumstances dictate whether teachers' beliefs about the nature of mathematics 
and their beliefs about the teaching and learning of mathematics are clearly 
related; 



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HELEN FORGASZ ANDGILAH LEDER 

- There are many obstacles to overcome in attempting to change beliefs to be in 
line with contemporary understandings of good mathematics pedagogy; 

- Appropriate practicum and field-related experiences can impact strongly on 
prospective teachers' beliefs about the nature of mathematics and its teaching 
and learning; 

- Teachers' generally hold positive beliefs about the effects of technology on 
students' mathematics learning; 

- Teachers' beliefs about boys and girls and mathematics learning remain gender- 
stereotyped. 

The conclusions drawn from this review are generally consistent with those 
reported by Philipp (2007), with the exception of the more positive beliefs reported 
here of teachers' views about technology and mathematics learning. Da Ponte and 
Chapman (2006) were critical of continued efforts in researching teachers' beliefs 
without simultaneously exploring aspects of teachers' practices. There is some 
merit in this argument if the sole aim is to understand better why findings on 
espoused and enacted beliefs continue to reveal inconsistencies. However, as noted 
above, there is still a lack of knowledge on teachers' beliefs about a range of issues 
related to the teaching and learning of mathematics which can be explored without 
reference to practice even if there may be a direct bearing on practice. We 
advocate, however, that researchers carefully define what they mean by "beliefs" 
and that a breadth of methodological approaches is adopted in exploring them. 

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Helen J. Forgasz 
Faculty of Education 
Monash University, Australia 

Gilah C. Leder 
Faculty of Education 
Monash University, Australia 



192 



SECTION 3 

KNOWLEDGE OF EQUITY, DIVERSITY AND 
CULTURE IN TEACHING MATHEMATICS 



JILL ADLER AND DANIELLE HUILLET 



9. THE SOCIAL PRODUCTION OF MATHEMATICS 
FOR TEACHING 



This chapter aims to show the impact of culture on the learning of mathematics 
and consequently that studies of mathematics for teaching require strong 
theoretical frameworks that foreground the relationship between culture and 
pedagogy. For this purpose, we describe two different research projects in 
Southern Africa, each focused on the notion of mathematics for teaching. The first 
study analyses teacher learning of the mathematical concept of limits of functions 
through participation in a research community in Mozambique, and is framed by 
Chevallard's anthropological theory of didactics. The second, the QUANTUM 
project, studies what and how mathematics is produced in and across selected 
mathematics and mathematics education courses in in-service mathematics 
teacher education programmes in South Africa, and is shaped by Berstein 's theory 
of pedagogic discourse. We argue that separately and together these two studies 
demonstrate that mathematics for teaching can only be grasped through a 
language that positions it as structured by, and structuring of the pedagogic 
discourse (in Bernstein 's terms) or the institution (in Chevallard's terms) in which 
it 'lives'. 

INTRODUCTION 

Shulman (1986, 1987) posited the notions of subject matter knowledge (SMK), 
pedagogical content knowledge (PCK) and curriculum knowledge (CK), as critical 
categories in the professional knowledge base of teaching. In so doing, he 
foregrounded the centrality of disciplinary or subject knowledge, and its 
integration with knowledge of teaching and learning, for successful teaching. The 
past two decades have witnessed a range of studies related to SMK and an 
emphasis of research on PCK, many focused on mathematics (e.g., Ball, Bass & 
Hill, 2004; Ball, Thames & Phelps, 2007). As a consequence, a new discourse is 
emerging attempting to mark out mathematics for teaching as a distinctive or 
specialised form of mathematical knowledge produced and used in the practice of 
teaching. As noted in Adler and Davis (2006), this discourse is fledgling. 

In this chapter we describe two different research projects in Southern Africa 
each focused on the notion of mathematics for teaching. We foreground the social 
epistemologies that informed and shaped these studies: Chevallard's 
anthropological theory of didactics (Chevallard, 1992, 1999) and Bernstein's 
theory of pedagogic discourse (Bernstein, 1996, 2000), and illuminate their critical 



P. Sullivan and T. Wood(eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 19S-221. 

© 2008 Sense Publishers. All rights reserved. 



JILL ADLER AND DANIELLE HUILLET 

role in each study. We argue that separately and together these two research 
projects demonstrate that mathematics for teaching can only be grasped through a 
language that positions it as structured by, and structuring of, the pedagogic 
discourse (in Bernstein's terms) or the institution (in Chevallard's terms) in which 
it 'lives'. From this perspective, mathematics is learned for some purpose, and 
within teacher education, this would be for mathematics teaching, and/or becoming 
a mathematics teacher. There are thus limits to the appropriateness of the use of 
general categories like PCK and SMK, as well as the distinctions between them. 

MOZAMBICAN TEACHERS RESEARCHING THE LIMIT CONCEPT 

We begin with a study of teacher learning through participation in a research 
community in Mozambique, motivated by the desire to impact on teachers' 
knowledge of advanced mathematical concepts. The study drew inspiration for its 
questions from Chevallard's notions of personal and institutional relations to 
concepts, and from his elaborated anthropological theory of didactics for framing 
and interrogating the notion of mathematics for teaching in this study. It shows 
■ how the institutional relation to the mathematical concept of limits of functions, as 
well as each teacher's position within the new institution influenced the 
development of a new personal relation to this concept. However, the weight of 
strong institutions such as Mozambican secondary school and Pedagogical 
University hindered the development of a more elaborated relation to the 
mathematical concept of limit that allows the challenging of these two strong 
institutions' relation to this concept. This opens up questions about both SMK and 
PCK and their inter-relation, particularly in teacher education practice. 

Starting Point of the Study 

In Mozambican didactic institutions, the teaching of limits of functions typically 
has two components: a formal component, the e-5 definition, derived from within 
mathematics that students are sometimes asked to memorise; and a procedural 
component, the calculation of limits using algebraic transformations. Mozambican 
teachers study the limit concept in these didactic institutions, secondary schools 
and university. As a consequence, their mathematical knowledge of limits is 
reduced to these two aspects (Huillet & Mutemba, 2000). Their teaching mirrors 
the way they have been taught as students, and thus, the secondary schools' routine 
for teaching limits. This study started from reflection on the conditions for 
changing these institutional routines. Considering that teachers are the main actors 
in the didactical relation in the classroom, teaching limits in a more elaborated way 
would only be possible if teachers develop their mathematical knowledge of this 
concept. This led to the following questions: 

- How could limits of functions be taught in Mozambican secondary schools so 
that students not only learn to calculate limits but also give meaning to this 
concept? 

196 



THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

- What kind of knowledge does a teacher need to teach limits in schools in that 
way? 

- How could Mozambican secondary school teachers acquire this knowledge? 

These questions have been addressed in this study through the lens of the 
Chevallard's anthropological theory of didactics. 

Chevallard's Anthropological Theory of Didactics 

The anthropological theory of didactics (ATD) locates mathematical activity, as 
well as the activity of studying mathematics, within the set of human activities and 
social institutions (Chevallard, 1992). It considers that "everything is an object" 
and that an object exists if at least one person or institution relates to this object. To 
each institution is associated a set of "institutional objects" for which an 
institutional relation, with stable elements, is established. 

An individual establishes a personal relation to some object of knowledge if s/he 
has been in contact with one or several institutions where this object of knowledge 
is found. S/he is a "good" subject of an institution relative to some object of 
knowledge if his/her personal relation to this object is judged to be consistent with 
the institutional relation. For example, in this study, the relation that Mozambican 
mathematics teachers established with the limit concept was shaped by the 
relationship to this concept in the institutions in which they learned it. For most 
teachers, this contact occurred in Mozambican institutions (secondary school as 
students, university as students, and as secondary school teachers). The institutional 
relation to an object of knowledge can be analysed through the social practices 
involving this object inside the institution. Chevallard (1999) elaborates a model to 
describe and analyse these institutional practices, using the notion of praxeological 
organisation or, in the case of mathematics, mathematical organisation. The first 
assumption of this model is that any human activity can be subsumed as a system 
of tasks (Chevallard, 1999; Bosch and Chevallard, 1999). Mathematics, as a human 
activity, can therefore be analysed as the study of given kinds of problematic tasks. 

The second assumption of this theory is that, inside a given institution, there is 
generally one technique or a few techniques recognised by the institution to solve 
each kind of task. Each kind of task and the associated technique form the practical 
block (or know-how) of a mathematical organisation (MO). For example, in 
Mozambican secondary schools, students are taught to calculate limits using algebraic 
transformations. A specific algebraic transformation is associated to each kind of 
limit, constituting the practical block of a specific MO. Other kinds of tasks could be: 
to read limits from a graph, to sketch the graph of a function using its limits, to 
demonstrate the limit of a function using the definition, etc. These kinds of tasks are 
hardly used in Mozambican secondary schools, but can be found in other institutions, 
for example in secondary schools or universities in other countries. Students are then 
expected to solve each of these tasks using a specific technique. The institutional 
relation to an object is shaped by the set of tasks to be performed, using specific 
techniques, by the subjects holding a specific position inside the institution. In an 

197 



JILL ADLER AND DANIELLE HU1LLET 

institution, a specific kind of task T is usually solved using only one technique x. Most 
of the tasks become part of a routine, the task/technique practical blocks [T, x] 
appearing to be natural inside this institution. 

The third assumption of the theory of mathematical organisations is that there is 
an ecological constraint to the existence of a technique inside an institution: it must 
appear to be understandable and justified (Bosch & Chevallard, 1999). This is done 
by the technology 9, which is a rational discourse to describe and justify the 
technique. This constraint can be interpreted at two levels. At the students' level, it 
means that students should be able to understand the technique. At the mathematics 
level, we must ensure that the technique is "mathematically correct" with reference 
to scholarly knowledge. 1 These ecological constraints can sometimes lead to a 
contradiction, given that the ability of students to understand will be constrained by 
their development and previous knowledge. It can be difficult for a technique to be 
both understandable and justified at the same time. 

The technology 9 itself is justified by a theory 0, which is a higher level of 
justification, explanation and production of techniques. Technology and theory 
constitute the knowledge block [9,0] of a MO. According to Chevallard (1999), the 
technology-theory block is usually identified with knowledge [un savoir], while the 
task-technique block is considered as know-how [un savoir-faire]. 

The two components of a MO are summarised in the diagram below. 



Mathematical Organisation 



/ 



Practical Block 

(Kinds of tasks 

and techniques) 

[T,x] 



\ 



Theoretical Block 

(Kinds of technology 

and theory) 

[9,0] 



Figure I. Mathematical organisation. 

A MO around a particular kind of task in a certain institution is specific. For 
example, calculating the limit of a rational function when x goes to infinity by 
factorisation and cancellation is a specific MO. The corresponding technology 
would be, for example, the theorems about limits, and the corresponding theory the 
demonstration of these theorems using the e-8 definition. The integration of several 
specific MOs around a specific technology gives rise to a local MO. For example, 
calculating several kinds of limits using algebraic transformation constitutes a local 



We mark out mathematical knowledge Intentionally in order to signal that what counts as scholarly 
mathematical knowledge is not unproblematic 



198 



THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

MO. In the same way, the integration of several local MOs around the same theory 
gives rise to a regional or global mathematical organisation. 

In order to teach a mathematical organisation, a teacher must build a didactical 
organisation 2 (Chevallard, 2002). To analyse how a didactical organisation enables 
the set up of a mathematical organisation, we can first look at the way the different 
moments of the study of this MO are settled in the classroom. Chevallard (2002) 
presents a model of six moments of study. They are the following: first encounter 
with the MO, exploration of the task and emergence of the technique, construction 
of the technological-theoretical bloc, institutionalisation, work with the MO 
(particularly the technique), and evaluation 3 . The order of these moments is not a 
fixed one. Depending on the kind of didactical organisation, some of these 
moments can appear in a different order, but all will probably occur. For example 
the study of mathematical organisations at university level is often divided in 
theoretical classes and tutorials. The theoretical block is presented to students in 
lectures, as already produced and organised knowledge, and tasks are solved using 
some techniques (practical block) during tutorials. In that way there is a 
disconnection between the theoretical component of the organisation and its 
applications. This is what happens in Mozambique with limits of functions. At the 
university level, the e-5 definition and the theorems about limits and their 
demonstrations using this definition are usually taught in theoretical classes, while 
tutorials are dedicated to calculating limits using algebraic transformations 
(Huillet, 2007a). In that case, the reasons why the theory exists gets lost. And so 
we see the institution as structuring of, and being structured by, the particular 
mathematics in focus. 

The Use of Anthropological Theory of Didactics in this Study 

The anthropological theory of didactics (ATD) has been used in this study to 
analyse the teachers' personal relation to this concept and its evolution through 
their work within a new institution, using different aspects of mathematics for 
teaching limits. 

In the first place, ATD has been used as a tool for analysing the institutional 
relation of Mozambican didactic institutions to the limit concept, in particular the 
secondary school institution and the Pedagogical University where most 
mathematics teachers are trained. For each of these institutions, the practical block 
and the knowledge block of the mathematical organisation related to limits of 
functions have been analysed through the examination of the syllabus, the national 
examinations (secondary school), worksheets used in secondary schools (there is 
no textbook for this level in Mozambique), textbooks used at the Pedagogical 
University and the exercise book of a student. This analysis highlighted a 



2 We note here that didactical organisations are specific to certain topics or contents in mathematics. 

3 There is an interesting similarity between these moments of study and the interpretation of Hegelian 
moments of judgement in pedagogic discourse as described by Davis (2001 ), and referred in Adler and 
Pillay (2O07). 



199 



JILL ADLER AND DANIELLE HUILLET 

dichotomy between two regional mathematical organisations: the algebra of limits 
based on the e-5 definition, and the existence of limits, based on algebraic 
transformations to evaluate limits. This dichotomy, which also exists in other 
secondary schools in other countries (Barbe, Bosch, Espinoza, & Gascon, 2005) 
and is explained by the nature of the limit concept, seems to be exacerbated in the 
Mozambican case. This can explain the limited personal relation to limits of 
Mozambican teachers (Huillet, 2005a). 

Secondly, anthropological theory of didactics was used to design the research 
methodology. Considering the institutional relation previously described and how 
it strongly shaped teachers' personal relations to this concept, this personal relation 
could only evolve if teachers were in contact with this concept through a new 
institution where this concept lived in a more elaborated way. 4 Other institutional 
or personal constraints could influence the usual way of teaching limits in schools. 
The argument was that their personal relation did not allow them to challenge the 
institutional routines. The evolution of their knowledge was a necessary, although 
not sufficient, condition for any change of the way of teaching limits in 
Mozambican secondary schools. Consequently a new institution was set up, where 
four final-year student-teachers from the Pedagogical University researched some 
aspects of the limit concept and shared their findings in periodic seminars. The 
researcher was both supervisor of the teachers' individual research and facilitator 
of the seminars. 

In the third place, ADT was used to analyse a mathematics teacher's task(s) 
when planning a didactical organisation, using Cheval lard's model of the moments 
of study (Chevallard, 2002). This allowed the development of a general framework 
for describing the knowledge needed by a teacher to perform these tasks. It 
includes scholarly mathematical knowledge of the MO, knowledge about the social 
justification for teaching this MO, how to organise students' first encounter with 
this MO, knowledge about the practical block (tasks and techniques) using 
different representations, knowledge on how to construct the theoretical block 
according to learners' age and previous knowledge, and knowledge about students' 
conceptions and difficulties when studying this MO. This description of 
mathematics for teaching 5 was used to define research topics for the teachers 
involved in the research group. In line with the overall approach in this study, 
within these aspects of mathematics for teaching limits, the boundary between 
SMK and PCK as developed by Shulman is blurred. Rather, and this is elaborated 
further below, each aspect has two components, a mathematical and a pedagogical 
component. Some aspects are more mathematical, some others more pedagogical, 
but they are necessarily merged in the human activity of mathematics teaching. 



4 Obviously, it cannot be claimed that a change in teachers' personal relation would automatically result 
in a change of their way of teaching limits at school. 

5 The expression mathematics for leaching to design the knowledge needed by a mathematics teacher is 
the same as defined by Ball et al. (2004) and Adler and Davis (2006). 



200 



THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

The Evolution of Teachers' Knowledge through the New Institution 

The new institution set up for this study was a research group, where four teachers, 
honours students at the Pedagogical University, researched different specific 
aspects of the limit concept and shared their findings in periodical seminars. The 
researcher was their supervisor and the facilitator of the seminars. The evolution of 
these teachers' knowledge through the new institution was analysed in detail for 
five aspects, or sub-aspects, of mathematics for teaching limits in schools: how to 
organise students' first encounter with limits of functions, the social justification 
for teaching limits in secondary schools, the essential features of the limit concept 
(part of the scholarly mathematical knowledge), the graphical register (part of the 
practical block) and the e-8 definition (also part of the scholarly mathematical 
knowledge). For each of these aspects, categories were defined both for teachers' 
mathematical knowledge (ranked in several degrees from "knowing less" to 
"knowing more") and for teachers' ideas about teaching, related to this aspect 
(ranked again in several degrees from "being close to the secondary school 
institutional relation to limits" to "challenging this institutional relation"). An 
example of each of these can be found in Appendix 1, where the co-presence of 
both mathematical and teaching knowledge is evident. These further illuminate that 
learning of mathematics through the range of tasks in this research institution was 
for the purposes of teaching mathematics. 

In this chapter we will not detail the methodology used to collect and analyse 
data in the study. We only present some results that help understand the role of the 
new institution in the development of a new personal relation to limits and discuss 
the weight of this institution in comparison with strong Mozambican didactic 
institutions as are the Secondary School and the Pedagogical University. We focus 
on two teachers, selected because they represent two extreme situations in relation 
to their teaching experience and to their position within the group: Abel, 6 an 
experienced teacher who had taught limits in school for years; and David, the 
youngest teacher in the group, with very little teaching experience. The evolution 
of these two teachers' personal relation to limits during the research process, for 
the five aspects selected for the study, is presented in Table 1 below. 

This table shows that while both teachers' personal relation to limits evolved, 
this was uneven, particularly for the two last aspects of the limit concept: the use of 
the graphical register and the e-8 definition. In this chapter we focus on these two 
aspects. 



' These are pseudonyms. 

201 



JILL ADLER AND DANIELLE HUILLET 

Table I. Evolution of Abel's and David's personal relation to the limit concept according 
to five aspects of mathematics for teaching limits 





Categories 


Abel 


David 


First Encounter 


FE-MK1 toFE-MK2 


FE-MKI->FE-MK2 


FE-MK1 -+FE-MK2 




FE-TI toFE-T6 


FE-T2 -> FE-T4 


FE-TI ->FE-T5 


Social Justification 


SJ-MK1 toSJ-MK4 


SJ-MK2->SJ-MK4 


SJ-MKI ->SJ-MK4 




SJ-T1 to SJ-T3 


SJ-TI -► SJ-TI 


SJ-TI ->SJ-T3 


Essential Features 


EF-MKI toEF-MK4 


EF-MK1-»EF-MK4 


EF-MK2-> EF-MK4 




EF-TI to EF-T2 


EF-TI->EF-T2 


EF-T1-»EF-T2 


Graphical Register 


GRRRtoGRR6 


GRRI ->GRR3 


GRR2-»GRR5 




GRRS to GRS6 


GRS2->GRS2 


GRS3->GRS6 




GR-TI toGR-T3 


GR-TI ->GR-T2 


GR-TI ->GR-T3 


£-6 Definition 


D-MKI toD-MK4 


D-MKI ->D-MK3 


D-MKI ->D-MK2 




D-Tl to D-T4 


D-Tl ->D-T3 


D-T4 -> D-T2 



FE-MK Mathematical knowledge about the first encounter with the limit concept. 

FE-T Ideas about teaching related to the first encounter with the limit concept. 

SJ-MK Mathematical knowledge about the social justification. 

SJ-T Ideas about teaching related to the social justification. 

EF-MK Mathematical knowledge about essential features. 

EF- T Ideas about teaching related to essential features. 

GRRR Knowledge about how to read limits from graphs. 

GRRS Knowledge about how to represent a limit on a graph. 

GR-T Ideas about teaching related to the graphical register. 

D-MK Mathematical knowledge about the definition. 

D-T Ideas about teaching related to the definition. 



In the wider study, Huillet (2007b) shows the limited evolution of all four teachers' 
knowledge about the graphical register, and explained this in relation to the general 
difficulty that the teachers had in working with graphs. The teachers in this study 
did not display deep understanding of basic mathematical knowledge such as the 
concept of function, and the use and interpretation of graphs in general. 
Nevertheless, the evolution of Abel's and David's knowledge about the use of the 
graphical register for studying limits was very different from each other, with 
David's knowledge about this aspect evolving more than Abel's (as well as the 
other teachers in the study). We suggest two explanations for this uneven outcome. 
Firstly, this aspect was directly linked to David's research topic (Applications of 
the limit concept in mathematics and in other sciences). Secondly, and this 
explanation is more speculative, David used the interviews as a means for learning. 



202 



THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

By positioning himself more as a student than as a teacher, David was able to take 
advantage of each opportunity for learning, by asking questions and attempting to 
solve more tasks. In contrast, Abel did not try to solve many graphical tasks. He 
assumed more of a teacher's position and thus one who should already know. As a 
consequence, he did not engage in the interviews in ways that could have enabled 
his knowledge about the use of graphs for teaching limits to evolve. 

The e-5 definition belongs to the scholarly mathematical knowledge and, like 
the graphical register, requires a deep understanding of basic mathematical 
concepts. Furthermore, it is intrinsically difficult (Huillet, 2005b) and it is part of 
the syllabus of Mozambican secondary schools. At the beginning of the research 
process, none of the teachers could explain this definition, and their understanding 
of it evolved slightly during the study. Again, Abel's and David's ideas about 
teaching this definition in secondary schools evolved differently, curiously in 
opposite directions. At the beginning of the research process, Abel, the 
experienced teacher who had taught this definition in schools, argued that it was 
right that it be taught in school. By the end of the study, he had reached the 
conclusion that it was not appropriate to teach this definition at secondary school 
level: students were not able to understand it in that form. In contrast, David was 
initially inclined not to teach the definition. At the end, however, he was willing to 
teach it while acknowledging students' difficulties in understanding this definition. 

Let's analyse the evolution of how these two teachers positioned 7 themselves 
within the new institution, that is, within the research group and so with the 
possibilities for their relations to the limit to evolve. At the beginning of our work 
together, Abel positioned himself as an experienced teacher. During the seminars, 
he volunteered to explain some aspects of limits to his colleagues, particularly the 
e-5 definition, trying to show that he had mastered this topic. During the first 
interview, he constantly referred to what was done in schools when teaching this 
concept, showing that he knew the syllabus and the way limits are usually taught in 
secondary schools. However, he faced difficulties during his explanations to his 
colleagues and felt 'ashamed' about it, as he told the researcher during the second 
interview. He then faced difficulties during his research study - an experiment in a 
secondary school. He experienced these difficulties as his failure as a teacher, and 
not as the result of the research, or as a researcher. He also became aware that he 
had been teaching limits in school, in particular the e-5 definition and L'Hopital's 
Rule, in ways that were problematic for his students. This reflection on his 
practice, although very hard for him, offers an explanation as to why, at the end of 
the research process, Abel said that the e-5 definition should not be part of the 
secondary school syllabus. 

In contrast, David's initial position within the research group was of learner- 
teacher, a university student teaching as he completed his studies. During the first 
interview, he analysed the way limits are usually taught in Mozambican secondary 



What we mean by 'positioned' here is the way in which this particular teacher related to both the 
researcher and others in the group. 



203 



JILL ADLER AND DANIELLE HUILLET 

schools as a student who did not understand the e-8 definition. He did not 
participate much in the discussion during the first seminars, giving way to his more 
experienced colleagues. However, he was able to argue with them in the last 
seminars. The end of the research process coincided with the conclusion of his 
teacher training course, and it seems that at that point, he then positioned himself 
more as a teacher than as a student. He was thus anticipating the institution where 
he was going to teach limits i.e., the secondary school institution. He knew that 
secondary school students were not able to understand this definition, but at the 
same time that it is part of the Mozambican Grade 12 syllabus. He remembered 
studying it in that grade. It is arguable, that as a prospective teacher, the weight of 
the institution he was moving to became more influential in his thinking. 

The analysis of the evolution of these two teachers' personal relation to limits of 
functions related to two critical aspects of this concept shows that these teachers 
experienced the weight of the two institutions in different ways, particularly where 
relations to the limit concept was in conflict. 

Institutional Strengths and Weaknesses 

Chevallard's anthropological theory of didactics points out the importance of 
institutional relations to an object of knowledge and how an individual's personal 
relation to this object is shaped by the institutions' relations where this individual 
met this object. The study of the personal relation of four Mozambican student- 
teachers, who had mainly* been in contact with the limit concept through 
Mozambican institutions, showed how their personal relation to limits at the start 
of the study was consistent with the Mozambican Secondary School's and the 
Pedagogical University's institutional relations to this concept. It also showed that 
this personal relation evolved during their contact with this object of knowledge 
through another institution, the research group, holding a different institutional 
relation. However, this study also pointed out some limitations in the evolution of 
this knowledge. 

These results lead to the following questions: 

- Why, at the end of the work within the research group, were the teachers' 
personal relations to limits of function not fully consistent with the relation of 
the new institution? 

- How could the new institution be modified so as to enable teachers to learn 
more about limits, according to the expected (and more elaborated) personal 
relation? 

Elements of answers to these questions have been given in the previous section. 
One of them is the lack of basic understanding of some mathematical concepts that 
hindered the evolution of teachers' knowledge, especially the mathematical 
components of this knowledge. It seems that, in these cases, more direct 
engagement with these aspects of the limit, supported by explanations and 



One of the teachers also studied in a university in East Germany. 
204 



THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

systematic solution of tasks was necessary for teachers to overcome their 
difficulties. This is what happened with David. During the third interview he tried 
to solve many graphical tasks, asking questions and drawing on the researcher's 
explanations to solve them. He thus engaged with the limit concept in various and 
new ways. 

The mathematical component in the new institution was apparently not strong 
enough. In the first case, the researcher did not anticipate the extent of the 
weakness of the teachers' knowledge of basic mathematical concepts. For 
example, she knew that the teachers were not used to using graphs in the study of 
limits, but she did not imagine that they would have so many difficulties working 
with graphs in general. For example, they sometimes confused x-values and y- 
values (or the two axes), or a limit with the maximum of the function (Huillet, 
2007b). Secondly, the researcher was reluctant to play the role of a teacher within 
the group, because she wanted to observe how the teachers' personal relation to 
limits evolved through research. She felt that teaching them would influence the 
results of her research. 

With regard to the more pedagogical component of the teachers' personal 
relations to limit, we already saw that, during the research process, all teachers 
changed their ideas about teaching the e-5 definition in secondary schools. While 
the experienced teacher said that he would not like to teach the definition any 
more, the other three teachers argued that this definition should be taught in 
schools. This evolution was explained by the weight of the secondary school 
institutional relation to limits: they were now positioned as teachers and not as 
students, as at the beginning of the research process. This suggests another 
weakness of the research group as an institution. Relative to well established 
institutions such as the Mozambican Secondary School and the Pedagogical 
University, institutions with a strong tradition of teaching and well established 
routines, the research group appears as a very weak institution. That this institution 
enabled the teachers to become aware of strong gaps in the teaching of limits in 
secondary schools and at university does not necessarily imply that they will be 
able to stand up against strongly institutionalised routines. Organising students' 
first encounter with limits in a different way, introducing different kinds of tasks, 
for example graphical tasks or tasks to link limits with other mathematical or other 
sciences concepts, do not mean going against the secondary school syllabus, but 
adding something to it. Not to teach the e-5 definition, even knowing that the 
students will not understand it, is a bigger step to take because this definition is 
part of the syllabus. It can be seen as an act of rebellion against the institution. 
Elsewhere, Huillet (2007a) has argued that these research outcomes emerging as 
they are from the Mozambican context and through a study that placed 
mathematics at its centre open up important questions about the literature in 
mathematics education on teachers-as-researchers. In the "teachers as researchers" 
movement, teachers usually studied some pedagogical aspect of their teaching, 
taking the mathematical content for granted; this did not allow them to challenge 
the content of their teaching. This can also be seen as the result of the dichotomy 
between mathematics and pedagogy in teacher education. This dichotomy is 



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JILL ADLER AND DANIELLE HUILLET 

reproduced in Shulman's distinction between SMK and PCK, as if SMK were 
some kind of 'universal mathematical knowledge' and PCK mathematical 
knowledge specific for teaching. However, his description of SMK's substantive 
and syntactic structures contradicts this separation. 

The substantive structures are the variety of ways in which the basic concepts 
and principles of the discipline are organised to incorporate its facts. The 
syntactic structure of a discipline is the set of ways in which truth or 
falsehood, validity or invalidity, are established. (Shulman, 1986, p.9) 

The syntactic structure of the discipline is important for teachers to engage in 
mathematics in a way that enables the construction of new didactical organisations. 
This study shows that the teachers involved in the research group did not grasp the 
syntactic structures of the limit concept during their training, but were only able to 
lead with substantive structure. From our perspective, this is not sufficient to teach 
limits in a way that challenges institutional routines, so making it comprehensible 
to students. We can then ask the question: does the syntactic structure of 
mathematics belong to SMK or PCK? We argue that this distinction is not 
appropriate and that, in teacher education, mathematics should live in a way that 
enables reflection at the same time on the mathematical and pedagogical aspects of 
the content to be taught. Where, when and how, then, in teacher education 
(particularly in professional development) practice, are teachers to have 
opportunities for further engagement with both syntactic and substantive aspects of 
the limit function. Could a research institution be strengthened so as to offer 
teachers further possibilities for elaborating their knowledge of limits of functions, 
and if so, how? The institution of mathematics teacher education itself- its objects 
and tasks, in Cheval lard's terms - are in focus in the QUANTUM research project. 
In the next section we describe aspects of QUANTUM, foregrounding the 
theoretical resources drawn on to enable us to 'see' this "inner logic of pedagogic 
discourse and its practices" (Bernstein, 1996, p. 18), and specifically how it comes 
to shape mathematics for teaching in teacher education practice. 

THE QUANTUM RESEARCH PROJECT IN SOUTH AFRICA 

QUANTUM is the name given to a research and development project on quality 
mathematical education for teachers in South Africa. The development arm of 
QUANTUM focused on qualifications for teachers underqualified in mathematics 
(hence the name) and completed its tasks in 2003; QUANTUM continues as a 
collaborative research project. Between 2003 and 2006, the QUANTUM project 
has studied selected mathematics and mathematics education courses offered in 
higher education institutions as part of formalised (i.e., accredited) mathematics 
teacher education programmes for practising teachers in South Africa. Our analysis 
of these courses led to deeper insights into and understanding of what and how 
mathematics for teaching comes to 'live' in such programmes. We drew on, and 
elaborated, a set of theoretical resources from Bernstein in our study. These are our 
focus in this section of the chapter. 

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THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

Starting Point of the Study 

An underlying assumption in QUANTUM is that mathematics teacher education is 
distinguished by its dual, yet thoroughly interwoven, objects: teaching (i.e., 
learning to teach mathematics) and mathematics (i.e., learning mathematics for 
teaching). It is these dual objects that lead to what is often described as the subject- 
method tension. Others describe this as one of the dilemmas in teacher education 
(Adler, 2002; Graven, 2005). The inter-relation of mathematics and teaching is 
writ large in in-service teacher education (INSET) programmes (elsewhere referred 
to as professional development for practising teachers) where new and/or different 
ways of knowing and doing school mathematics, new curricula, combine with new 
and/or different contexts for teaching. Such are the conditions of continuing 
professional development for practising teachers in South Africa. The past ten 
years saw a mushrooming of formalised programmes for practising teachers across 
higher education institutions in South Africa, in particular, Advanced Certificates 
in Education (ACE) programmes. 9 The ACE qualification explicitly addresses the 
inequities produced in apartheid teacher education, where black teachers only had 
access to a three-year diploma qualification. As a result, most ACE programmes 
are geared to black teachers, at both the primary and secondary levels. Many of 
these are focused on the content of mathematics and constituted by a combination 
of mathematics and mathematics education courses. Debate continues as to 
whether and how these programmes should integrate or separate out opportunities 
for teachers to (re)learn mathematics and to (re)learn how to teach. 

A consequent assumption in QUANTUM is that however the combinations are 
accomplished, both mathematics and teaching as activities and/or discourses are 
always simultaneously present in all components of such programmes. Moreover, 
their interaction within pedagogic practice will have effects. This latter assumption 
is derived from a social epistemological approach to knowledge (re)production in 
pedagogic practice, and motivated by the work of Basil Bernstein, specifically how 
he deals with the conversion or translation of knowledges into pedagogic 
communication. And it is this orientation that leads us to reframe the broad 
problematic discussed as the following research question: What is constituted as 
mathematics for teaching in formalised practising mathematics teacher education 
practice in South Africa, and how is it so constituted? 10 



' The ACE (formerly called a Further Diploma in Education - FDE) is a diploma that enables teachers 
to upgrade their three-year teaching diploma to a four-year diploma. The goal is to provide teachers 
with a qualification regarded as equivalent with a four-year undergraduate degree. 
10 In Chevellard's terms, the question would be: what is the institutional relation to mathematics that is 
set up and how does it function? 



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JILL ADLER AND DANIELLE HUILLET 

Bernstein's Theory of the Pedagogic Device and Related Orientations to 
Knowledges 

In the introduction to this chapter, we noted that Chevallard and Bernstein share a 
social orientation to knowledge. Both hold that rigour in educational (or didactics) 
research is a function of coherence between an overarching theoretical orientation, 
research questions and methodology. For Chevallard, a didactic organisation needs 
to be built to teach a mathematical organisation. The reciprocal effects of this are 
inevitable. Bernstein too sees knowledges in school, or any pedagogic context, as 
structured by pedagogic communication. His theory of the pedagogic device 
describes a set of principles and rules that regulate this structuring. It is these that 
we have brought to bear on our investigation into mathematics for teaching in 
teacher education practice. 

For Bernstein, the principles of the transformation of knowledge in pedagogic 
practice are described in terms of the 'pedagogic device' (Bernstein, 2000). The 
pedagogic device is an assemblage of rules or procedures via which knowledges 
are converted into pedagogic communication." It is this communication (within 
the pedagogic site) that acts on meaning potential. That is, pedagogic discourse 
itself shapes possibilities for making meaning, in this case of mathematics for 
teaching. The pedagogic device is the intrinsic grammar (in a metaphoric sense) of 
pedagogic discourse, and works through three sets of hierarchical rules. 

Distributive rules regulate power relations between social groups, distributing 
different forms of knowledge and constituting different orientations to meaning 
(Bernstein refers to pedagogic identities). In simpler terms, the regulation of power 
relations in pedagogic practice effects who learns what. Whereas for Chevallard, 
orientations to meaning lie in 'institutional and personal relations' to a concept, the 
distributive rule brings social structuring effects to the fore, a function of 
Bernstein's concern with educational inequality and its social (re)production. 

Recontextualisation rules regulate the formation of specific pedagogic 
discourse. In any pedagogic practice knowledges are delocated, relocated and 
refocused, so becoming something other. In the context of QUANTUM, the 
recontextualising rule at work regulates how mathematics and teaching, as a 
discipline and a field respectively, are co-constituted in particular teacher 
education practices. Here there is further resonance with didactic transposition, and 
with Chevallard's notion of institutionalisation, particularly the effects of strong 
and weak institutions on changing practices. The recontextualising rule is possibly 
the most well known and used element of Bernstein's work, and elaborated 
through the concepts of classification and framing. Classification refers to "the 
relations between categories" (2000, p. 6),' 2 and how strong or weak are the 
boundaries between categories (e.g., discourses or subject areas in the secondary 
school) in a pedagogic practice. Framing refers to social relations in pedagogic 



" In Chevallard's terms, this transformation occurs in the setting up of the didactical organisation. 

12 For Bernstein, boundary maintenance is through power and changing or weakening the insulation 

between categories will reveal power relations - and so be contested (p.7). 



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THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

practice, and who in the pedagogic relation controls what (2000). For our purposes 
in this chapter, the issue is whether and how mathematics and teaching as two 
domains are insulated from each other or integrated and then through what 
principles. The way knowledges are classified and framed, in any educational 
practice, the varying strength or weakness of the insulations, will constitute a range 
of pedagogic modalities 13 and shape what comes to be transmitted. 14 In particular, 
they will impact on what comes to be mathematics for teaching. 

Acquisition, in Bernstein's terms, is elaborated by what he refers to as 
'recognition' and 'realisation'. In any pedagogic setting, learners need to recognise 
what it is they are to be learning, and further, they need to be able to demonstrate 
this by producing (realising) what is required - what he refers to as a 'legitimate 
text'. 15 Recognition and realisation link with the third set of rules operating within 
the pedagogic device. Evaluative rules constitute specific practices - regulating 
what counts as valid knowledge. For Bernstein, any pedagogic practice "transmits 
criteria" (indeed this is its major purpose). Evaluation condenses the meaning of 
the whole device (2000), so acting (hence the hierarchy of the rules) on 
recontextualisation (the shape of the discourse) that in turn acts on distribution 
(who gets what). What comes to be constituted as mathematics for teaching (i.e., as 
opportunities for learning mathematics for teaching) will be reflected through 
evaluation and how criteria come to work. 

Despite the significance of evaluation in this theory, and in contrast to 
recontextualising rules, Bernstein's evaluative rules are not elaborated. Much of 
the pedagogical research on teacher education that has worked with Bernstein's 
framework focused on his rules for the transformation of knowledge into 
pedagogic communication, and particularly the distributive and recontextualising 
rules of the pedagogic device (e.g., Ensor, 2001, 2004; Morais, 2002). These 
studies foreground an analysis of classification and framing in a particular 
pedagogic modality, and related recognition and realisation rules that come to play. 
Ensor's study of mathematics prospective teacher education and its 
recontextualisation in the first year of teaching has advanced our understanding of 
the what, how and why of recontextualisation across sites of practice (university 
and school). The study argues that the 'gap' between what is taught in a 
programme for prospective teachers, and the practice adopted by teachers in their 
first year of teaching is not simply a function of teacher beliefs on the one hand, or 



" Bernstein describes two contrasting educational codes - ideal types - formed by strong and weak 

classification. A collection code has strong classification and strong framing; in contrast, an integrated 

code has weak classification and framing. In the latter boundaries between contents and between social 

relations are both weak. 

" As Graven (2002) explains, "in educational terms, Bernstein's use of the terms 'transmitter' and 

'acquirer' may seem pejorative. However, he uses them throughout various pedagogic models and they 

are merely sociological labels for descriptive purposes. They should therefore not be interpreted to 

imply transmission pedagogies" (Ch. 2, p.28). 

13 In Chevallard's terms, when learners are able to produce the legitimate text, they show that their 

personal relation fits the institutional relation (that they are "good subjects" of the institution). 



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JILL ADLER AND DANIELLE HUILLET 

constraints in schools on the other. The gap is explained through the principle of 
recontextualisation. The privileged pedagogy enacted in the teacher education 
programme was unevenly accessed by the teachers in her study. Ensor, drawing on 
Bernstein, shows how this distribution was a function of what and how criteria for 
the privileged practice were marked out, and so what teachers were or were not 
able to recognise as valued mathematical practice, and then realise this in their 
school classrooms. Morais' work focused on primary science, and tackles the 
phenomenon of primary teachers not being subject specialists. She argues that 
because of their weaker science knowledge base, the pedagogic modality in their 
teacher education should combine strong classification with weakened framing. In 
this way primary science teachers can be offered an enabling set of social relations 
within which to engage further learning of science. Science content needs to be 
clearly bounded and visible (i.e., strongly classified), and structured by sequencing 
and pacing to suit primary teacher interests and needs (weakly framed). 

However, as Davis has argued (Davis, 2005) a focus on classification and 
framing, while productive, backgrounds the special features of the content to be 
acquired. Even in Morais' study, the specificity of the science to be learned by 
primary teachers remains in the background. The concern with mathematical 
production in teacher education has thus led the QUANTUM project to focus 
instead on evaluation, and on the criteria for the production of legitimate 
(mathematical) texts. This required elaboration of the evaluative rule. Before we 
describe the methodology we have developed and used in QUANTUM to 
foreground the content to be acquired, one additional aspect of Bernstein's work 
requires discussion. 

Mathematics and Teaching as Differing Domains of Knowledge. 

Bernstein (2000) provides conceptual tools to distinguish different forms of 
knowledge and so to interrogate mathematics and teaching. In the first instance, he 
distinguished vertical and horizontal discourses, the criteria for which are forms of 
knowledge, and the most significant of which is whether knowledge is organised 
hierarchically or segmental ly (2000). But, as he argues, this broad distinction does 
little to assist with understanding discourses in education, and ensuing issues of 
pedagogy. Education (and so too mathematics education) invokes a wide range of 
vertical discourses. He thus developed further distinctions, insisting that while 
these were accompanied by additional conceptual apparatus, they were important 
analytically. 

For Bernstein, within vertical discourses we can distinguish between 
hierarchical and horizontal knowledge structures; and within the latter, between 
strong and weak grammars. Different domains of knowledge are differently 
structured and have different grammars. The natural sciences have hierarchical 
knowledge structures and strong grammars. They have "explicit conceptual 
syntax" and so recognition of what is and is not physics, for example, is apparent. 
Development is seen as "the development of theory which is more general and 
more integrating than previous theory" (2000, p. 162). The social sciences (hence 

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THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

education), in contrast, have horizontal knowledge structures, where development 
proceeds through the introduction of "new languages" that "accumulate" rather 
than integrate. Within the social sciences, some have relatively strong grammars 
i.e., their conceptual syntax enables "relatively precise empirical descriptions" 
(e.g., linguistics, economics); while others have weak grammars (e.g., sociology, 
education). Bernstein describes mathematics as a horizontal knowledge structure as 
it "consists of a set of discrete languages for particular problems", with a strong 
grammar. 16 While mathematics largely does not have empirical referents, there is 
little dispute as to what is and is not mathematics from the point of view of the 
kinds of terms used, and the ways they are connected and presented. Education, in 
contrast, is horizontally structured but with a weak grammar. Empirical 
descriptions of educational phenomena vary widely, a function of the multiple 
languages used to describe these, many of which lack precision. 

What then might be the effects of weakening of knowledge boundaries between 
two domains of knowledge when one has a weak and the other a strong grammar? 
There has been a great deal of contestation over curriculum policy in South Africa 
and elsewhere that has advocated weakening the boundary between mathematics 
and everyday knowledge. The motivation for this move lies in the view that 
horizontal discourses of 'realistic' or 'relevant' settings for mathematics provide 
access to and meaning for abstract mathematical ideas. The critique of this arises 
within a Bernstein framework, and posits that 'realistic' or 'relevant' settings can 
work instead to background mathematical principles, and so result in denying 
access, particularly to those already disadvantaged by dominant discourses. 17 In a 
similar vein, Taylor, Muller and Vinjevold (2003), for example, draw on Bernstein 
to argue that integration in teacher education through the weakening of boundaries 
around content knowledge can result in methods of teaching dominating pedagogic 
discourse in teacher education at the expense of content development of teachers. 
Here too, effects will then be skewed against the already disadvantaged. They see a 
danger for teachers in greatest need for access to further content knowledge being 
subjected to an education dominated instead by examples of supposed good 
practice. In QUANTUM we share this concern. However, our assumptions are 
different. In our analysis, forms of integration are inevitable in contemporary 
educational practice. The issue is how these are and can be accomplished without 
damage to agents. Hence the need to understand varying practices and thus our 
focus in QUANTUM: what is at work in mathematics teacher education in South 



16 We are not convinced of the distinction drawn here between mathematics as horizontal and physics as 
vertical. Physics too has discrete languages. The inter-related distinctions here are, in our view, 
questionable, but full discussion beyond the scope of this chapter. The significant distinction, which in 
our view is productive and illuminating, is between week and strong grammars. Physics and 
mathematics both have strong grammars. 

17 It is beyond the scope of this chapter to detail this debate. It is well known in the field of mathematics 
education. Interested readers might find the work of Cooper and Dunne (1998) interesting, as well as 
the debate between Jo Boaler (2002) and Sarah Lubienski (2000) in the Journal for Research in 
Mathematics Education. 



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JILL ADLER AND DANIELLE HU1LLET 

Africa when there is more or less integration of mathematics and 
teaching/education in these programmes? 

Putting the Evaluative Rule to Work in the Study 

One major difficulty that arises in an integrated educational code (or when here is a 
weakening of classification in a curriculum), is what is to be assessed and the form 
of assessment. 18 Criteria must be worked out. Whether or not this is explicitly 
done, criteria will emerge and be transmitted. In QUANTUM'S terms, implicit 
criteria can be rendered visible because any act of evaluation has to appeal to some 
or other authorising ground in order to justify the selection of criteria. In Adler and 
Davis (2006) and Davis, Adler and Parker (2007), we describe in more detail how 
we have worked from the proposition that the authorising grounds at work in 
teacher education pedagogic practice illuminate what comes to be privileged (in 
terms of knowledges and their integration). 19 Given the complexity of teaching and 
more so teacher education, we started from the assumption, and related concern, 
that what comes to be taken as the grounds for evaluation is likely to vary 
substantially within and across sites of pedagogic practice in teacher education. 
Our methodology and language of description have allowed us to examine the 
diverse ways in which mathematics and teaching come to be co-produced in 
mathematics teacher education practice. 

The QUANTUM research project began in 2003 with a survey of 1 1 higher 
education institutions offering formalised (i.e., accredited) mathematics teacher 
education programmes in South Africa. We collected information on courses 
taught including formal assessments. Phase 1 of the overall study focused on 
formal assessment carried out across courses in our data archive. We focused on 
actual assessment tasks, examining what and how mathematics and teaching 
competence were expected to be demonstrated in these tasks. We developed an 
analytic tool, using the notion of "unpacking" (Ball, Bass, & Hill, 2004), but 
redescribing it in line with our methodology. For Ball et al., "unpacking" captures 
the specificity of mathematical know-how required in the practice of teaching. We 
were particularly interested in whether assessment tasks demanded some form of 
'unpacking' of mathematics. 

A full account of this phase of QUANTUM'S work is provided in Adler and 
Davis (2006). We started from the assumption that there are (in the main) two 
specialised knowledges to be (re)produced in mathematics teacher education: 
mathematics and mathematics teaching. That these knowledges are specialised 
implies there is some degree of internal coherence and consistency. However, in 
line with the discussion on knowledge structures and grammars above, the ways in 
which coherence and consistency are established in mathematics and mathematics 



18 See Moore (2000) for an interesting discussion of the challenges of disciplinary integration in a 
university foundation course in South Africa, and how these manifested in assessment practices. 

19 See Davis and Johnson (2007) for further development of 'grounds' at work in school mathematics 
classrooms. 



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THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

teaching differ. In mathematics a strong internal "grammar" allows for a degree of 
relatively unambiguous evaluation of that which is offered as mathematical 
knowledge; in mathematics teaching the ambiguity is greatly increased because the 
field is populated by academic, professional, bureaucratic, political and even 
popular discourses. However, we asserted that despite those differences, where the 
knowledge to be reproduced is relatively coherent and consistent, justifications can 
be structured in a manner that conforms to the formal features of syllogistic 
reasoning. Whether or not explicit coherent reasoning (be it mathematical 
reasoning or reasoning about teaching mathematics) was required by tasks thus 
provided the analytic resource we needed to identify "unpacking" consistently 
across different tasks. 

Our examination of each task involved identifying the primary and secondary 
objects (mathematics and/or teaching) of the task, and then whether an 
understanding of the logical chains (explicit coherent reasoning) relevant to the 
knowledge to be reproduced was explicitly demanded. As these tasks arise in 
mathematics teacher education, we expected that their objects may well be both 
teaching and mathematics and that they could vary in their demands for unpacking. 
Our analysis of tasks across formal evaluations in our data set was very interesting. 
Simply, we found that the kind of mathematical work required in teaching was 
infrequently assessed, with assessment tasks in mathematics focused 
predominantly on the reproduction of some mathematical content or skill. There 
was evidence, though limited and infrequent, of assessment of 'unpacking' of 
mathematical ideas - what specific mathematics teachers need to know and know 
how to use in practice to make mathematics learnable in school. There was thus a 
disjuncture between what is valued at the level of intention, and what comes to 
count as legitimate and valued knowledge in mathematics teacher education. Of 
course, this analysis did not provide any insight into the pedagogical practice of 
which these assessments were but a part. 

In phase 2 of our study, we focused on in-depth study of selected courses. This, 
in turn, required an elaboration of the language we had developed so far, the details 
of which are in Davis, Adler and Parker (2007). Pedagogic practice functions over 
time, unlike static assessment tasks. The unit of analysis thus required rethinking. 
As already noted, we accepted as axiomatic that pedagogic practice entails 
continuous evaluation, the purpose of which is to transmit criteria for the 
production of legitimate texts. Further, any evaluative act, implicitly or explicitly, 
has to appeal to some or other authorising ground in order to justify the selection of 
criteria. Our unit of analysis became what we call an evaluative event, that is, a 
teaching-learning sequence that can be recognised as focused on the 
'pedagogising' of particular mathematics and/or teaching content. In other words, 
an evaluative event is an evaluative sequence aimed at the constitution of a 
particular mathematics/teaching object. 

Each course, all its contact sessions and related materials, were analysed, and 
chunked into evaluative events. Following on from Phase I, after identifying 
starting and endpoints of each event or sub-event, we first coded whether the 
object of attention was mathematical and/or teaching, and then whether elements 



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JILL ADLER AND DANIELLE HUILLET 

of the object(s) were the focus of study (and therefore coded as M and/or T) or 
were assumed background knowledge (and then coded either m or t). We worked 
with the idea that in pedagogic practice, in order for some content to be learned it 
has to be represented as an object available for semiotic mediation in pedagogic 
interactions between teacher and learner. The semiotic mediation that follows 
involves moments of pedagogic reflection that in turn involve (following Davis, 
2001) pedagogic judgement. All judgement, however, hence all evaluation, 
necessarily appeals to some or other locus of legitimation to ground itself, even if 
only implicitly. Legitimating appeals can be thought of as qualifying reflection in 
attempts to fix meaning. We therefore examined what was appealed to and how 
appeals were made over time and in each course, in order to deliver up insights 
into the constitution of MfT in mathematics teacher education. 

Given the complexity of teaching, and more so of teacher education, as 
previously intimated, we expected that what came to be taken as the grounds for 
evaluation was likely to vary substantially within and across the courses we were 
studying. Indeed, through interaction with the data, we eventually described the 
grounds appealed to across the three courses in terms of six ideal-typical 
categories: (I) mathematics, (2) mathematics education, (3) the everyday, (4) 
experience of teaching, (5) the official school curriculum, and (6) the authority of 
the adept. In each course we found differences in what was appealed to and how, 
differences that point to very different opportunities for teachers to (re)learn 
mathematics for teaching. 

In one course (focused on teaching and learning algebra) mathematics was 
integrated with methods for teaching mathematics. In this course, the grounding of 
objects reflected on during class sessions was predominantly in what we called 
empirical mathematics (particular examples). In a course that focused on teaching 
and learning mathematical reasoning, the emphasis was in the domain of 
mathematics education, and so specific mathematics backgrounded. As could then 
be expected, a substantial grounding of objects reflected on during class sessions in 
this course was in mathematics education, particularly texts reporting research 
related to teaching and learning mathematics. Interestingly, when mathematical 
objects were in focus, and this occurred through the class, grounding for these was 
both empirical (with examples) and principled (discussion was expected to 
conform to demands of mathematical discourse). 

There are many reasons to explain why these two courses differed as such. Our 
interest, however, was that very different forms of mathematics for teaching were 
constituted in these courses, offering very different opportunities for learning. 
Mathematics teachers, whatever level, in our view, need to grasp mathematics in 
principled ways if they are themselves to enable mathematical learning in their 
classrooms. In-service mathematics teacher education should offer opportunities 
for engaging with mathematics as a principled activity. Of course, this is not to say 
that these two courses each capture the mathematics for teaching in the overall 
programmes of which they were each but a part. In Davis et al. (2007) our 
discussion of these courses is elaborated through a further analysis of how in each, 
the way teaching is modelled appears to link with what and how mathematics and 



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THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

teaching are integrated, and then too with how mathematics for teaching is 
constituted. We argue there that modelling the practice is a necessary feature of all 
teacher education. 20 There needs to be some demonstration/experience (real or 
virtual) of the valued practice; that is, of some image of what mathematics teaching 
performances should look like. In the Algebra course, the model was located in the 
performance of the lecturer whose concern (stated repeatedly through the course) 
was that the teachers themselves experience particular ways of learning 
mathematics. This experiential base was believed to be necessary if they were to 
enable others to learn in the same way. The mathematical examples and activities 
in the course thus mirrored those the teachers were to use in their Grades 7-9 
algebra class. In the Reasoning course the model of teaching was externalised from 
both the lecturer and the teacher-students themselves, and located in images and 
records of the practice of teaching: particularly in videotapes of local teachers 
teaching mathematical reasoning, and related transcripts and copies of learner 
work. The externalising was supported by what we have called discursive 
resources (texts explaining, arguing, describing practice in systematic ways). 

Our findings in both phases of the study need to be understood as a result of a 
particular lens, a lens that we believe has enabled a systematic description of what 
is going on 'inside' teacher education practice at two inter-related levels. The first 
level is 'what' comes to be the content of mathematics for teaching, i.e., the 
mathematical content and practices offered in these courses. We are calling this 
MfT. It is not an idealised or advocated set of contents or practices, but rather a 
description of 'what' is recognised through our gaze. Some aspects of MfT here 
can be seen as closer to SMK, and others to PCK in Shulman's terms. However, 
each of these categories is limiting in describing 'what' mathematics is offered in 
these courses. At the second level, is the 'how'. This content is structured by a 
particular pedagogic discourse; and a key component in the 'how' that has 
emerged in the study, is the projection and modelling of the activity of teaching 
itself. In Bernstein's terms we have seen, through an examination of evaluation at 
work and of how images of teaching are projected, that different MfT is offered to 
teachers in these programmes. The research we have done thus suggests in 
addition, that developing descriptions of what does or should constitute maths for 
teaching outside of a conception of how teaching is modelled is only half the 
story. 21 

MATHEMATICS FOR TEACHING: A SOCIAL PRODUCTION 

We stated in the introduction to this chapter that studies and developments related 
to mathematics for teaching have their roots in Shulman's seminal work in the 



20 This further supports our assumption that forms of integration are internal to pedagogic practice in 
teacher education. 

21 We note here that a similar point is made in Margolinas, Coulange and Bessot (2005) pointing further 
to resonance between the orientation to knowledge for teaching in QUANTUM and didactical theory as 
developed and used in studies in France. 



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JILL ADLER AND DANIELLE HU1LLET 

1980s that placed disciplinary knowledge at the heart of the professional 
knowledge base of teaching. We also noted that while there has been considerable 
research, the discourse of mathematics for teaching is fledgling. Neither of the two 
studies drew directly on the categories of professional knowledge as posited by 
Shulman, despite being driven by the same concern: to develop or deepen 
mathematical knowledge as it is (or needs to be) used in teaching, and a starting 
point that the way mathematics is used in teaching has a specificity. From our 
perspective, all mathematical activity (and hence all mathematics wherever it is 
learned) is directed towards some purpose, and within teacher education, this 
would be for mathematics teaching, and/or becoming a mathematics teacher. While 
the notion of PCK in particular is compelling in teaching and teacher education - it 
emphasises that pedagogic reasoning in mathematics teaching is content-filled - it 
does not live outside of the institutions where it functions - and these are 
inherently social. There are limits to the appropriateness of general categories like 
PCK and SMK, as well as to the distinctions between them. 

Shulman's work spurred several studies (and continues to do so) attempting to 
build on his notions, particularly PCK, but as Ball, Thames and Phelps (2007) 
point out, these notions remain poorly defined. In this paper Ball et al. pull together 
the accumulation of their work over the past decade that has included (a) 
describing mathematics for teaching from close observation of a detailed archive 
of a year of mathematics teaching in a third-grade class taught by Ball in the 
United States and (b) developing measures of content knowledge for teaching. This 
research has led them to strengthen and elaborate Shulman's initial work by 
providing clear definitions and exemplars of distinctive categories within and 
across SMK and PCK. A particular move they make is to define two new 
categories within SMK - or content knowledge for teaching: what they call 
common content knowledge and specialised content knowledge. They argue that 
this distinction is necessary to capture the specificity of teachers' mathematical 
work - and that recognition of this specificity lies at the heart of effective 
mathematics teaching. In simple terms, teachers need to know aspects of 
mathematics that is not required by 'others' (i.e., in common use). But what is 
common use? From a social epistemological perspective, all mathematical activity 
is towards some purpose, and occurs within some or other (social) institution. The 
notion of 'common' content knowledge is thus problematic, and so too then, the 
marking out of specialised content knowledge. 

We nevertheless share with Ball and her colleagues a concern with mathematics 
for teaching. In the two studies we have presented here, we have shown how two 
different social epistemologies have been productive for studying mathematics for 
teaching. The Mozambique study described and explained teachers' evolving 
personal relations to the limit through their participation and engagement with this 
concept in a new institution. We showed that this evolution was uneven, across 
teachers, and then also in relation to different aspects of mathematics for teaching. 
We elaborated in particular, the relatively poor evolution of the teachers' grasp of 
graphical representations and the e-S definition. We argued that these outcomes 
were a function of the strength of the research institution relative to the dominant 



216 



THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

institutions of the secondary school and Pedagogical University. An important 
element of this argument was an interpretation of aspects of mathematics for 
teaching through an ATD lens that reflected the limits of the distinction posited by 
Shulman between SMK and PCK. The evolution of each of the teachers' personal 
relation to the limit concept was described in terms of both subject knowledge and 
knowledge of teaching. Emerging from this study is the observation that 
professional development programmes for practising teachers in a context like 
Mozambique need to provide opportunities for substantive engagement with the 
content of mathematics, opportunities that were not available in the research 
institution set up, despite placing mathematics at the centre. So where and how 
then, is this engagement with mathematics to function in mathematics teacher 
education? 

In the South African study of courses within formalised in-service mathematics 
teacher education programmes - where engagement with mathematics was a goal - 
we described the mathematics that came to be constituted (came to 'live') in and 
across different courses. By examining evaluation at work in the courses we were 
able to 'see' what and how mathematics and teaching are co-constituted through 
pedagogic discourse. We showed that different models of teaching combine with 
varying selections from mathematics, mathematics education and teaching practice 
to produce different kinds of opportunities for teachers in these courses to learn 
mathematics (for teaching). 

Separately and together these two studies demonstrate that mathematics for 
teaching, and its learning in any institutional setting can only be grasped through a 
language that positions mathematics for teaching as structured by, and structuring 
of, the pedagogic discourse (in Bernstein's terms) or the institution (in 
Chevallard's terms) in which it 'lives'. Both provide strong conceptual tools with 
which to interrogate how mathematics is recontextualised in pedagogic settings. 
Separately and together they contribute to the growing body of knowledge related 
to the what of mathematics teacher education, and particularly to subject 
knowledge for teaching. 



217 



JILL ADLER AND DANIELLE HUILLET 



APPENDIX 1 

Categories of teacher's mathematical knowledge about the graphical register 
(reading and sketching) 



GRRI 


The teacher is not able to read any limit from the graphs. 


GRR2 


The teacher is able to read some limits along a vertical or a horizontal 
asymptote (when the graph does not cross the asymptote). 


GRR3 


The teacher is able to read limits along a vertical or a horizontal asymptote 
(when the graph does not cross the asymptote), and infinite limits at infinity 
(jc— > oo, y— > °o). 


GRR4 


The teacher is able to read limits along a vertical or a horizontal asymptote 
(even when the graph crosses the asymptote), and infinite limits at infinity (x— * 

oo,_y— > oo). 


GRR5 


The teacher is able to read most limits but faces small difficulties. 


GRR6 


The teacher is able to read all kinds of limits. 


GRSI 


The teacher is not able to sketch any graph using limits or asymptotes. 


GRS2 


The teacher is not able to indicate any limit on axes. He is able to sketch a 
standard graph having two asymptotes, one vertical and one horizontal. 


GRS3 


The teacher indicates limits along a vertical or a horizontal asymptote as a 
whole branch. He does not acknowledge that drawing several branches may 
produce a graph that is not a function. 


GRS4 


The teacher indicates limits along a vertical or a horizontal asymptote as a 
whole branch. He acknowledges that the produced graph does not represent a 
function. 


GRS5 


The teacher indicates limits along a vertical or a horizontal asymptote as a local 
behaviour. 


GRS6 


The teacher is able to indicate any kind of limit on axes. 



Categories of teacher's ideas about the use of graphs to teach limits 



GR-TI 



The teacher would not use graphs when teaching limits. 



GR-T2 



The teacher acknowledges the importance of the graphical register in teaching 
limits. 



GR-T3 



The teacher acknowledges the importance of the graphical register and explains 
how he would use it or articulate it with other registers. 



218 



THE SOCIAL PRODUCTION OF MATHEMATICS FOR TEACHING 

ACKNOWLEDGEMENTS 

This chapter forms part of the QUANTUM research project on mathematics for 
teaching, directed by Jill Adler, at the University of the Witwatersrand. This 
material is based upon work supported by the National Research Foundation 
(NRF) under Grant number FA200603 1800003. Any opinion, findings and 
conclusions or recommendations expressed in this material are those of the 
author(s) and do not necessarily reflect the views of the NRF. Dr Huillet's study in 
Mozambique was supported by QUANTUM and the Marang Centre at the 
University of the Witwatersrand. 

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Sciences and Technology Education (pp. 5 1-65). Cape Town: SAARMSTE. 
Barbe, J., Bosch, M., Espinoza, L., & Gascon, J. (2005). Didactic restrictions on the teacher's practice. 

The case of limits of functions in Spanish high schools. Educational Studies in Mathematics, 59, 

235-268. 
Bernstein, B. (2000). Pedagogy, symbolic control and identity: Theory, research, critique (Revised 

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Selected papers (pp. 131-167). Grenoble: La Pensee Sauvage. 
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Davis, Z. (2005). Pleasure and pedagogic discourse in school mathematics: a case study of a problem- 
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Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational 

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South Africa. 



JillAdler 

Marang Centre for Mathematics and Science Education 

School of Education 

University of the Witwatersrand 

Johannesburg 

South Africa 

and 

Department of Education and Professional Studies 

King's College 

London 

United Kingdom 

Danielle Huillet 
Mathematics Department 
Faculty of Sciences 
Eduardo Mondlane University 
Maputo 
Mozambique 



221 



PAOLO BOERO AND ELDA GUALA 



10. DEVELOPMENT OF MATHEMATICAL 
KNOWLEDGE AND BELIEFS OF TEACHERS 

The Role of Cultural Analysis of the Content to Be Taught 



We claim that the cultural analysis of the content to be taught (CAC) is one of the 
most important components of teacher education, if we want to develop teachers' 
mathematical knowledge and at the same time call into question their beliefs 
about mathematics and mathematics teaching. In this chapter we discuss the 
importance of CAC and its place in mathematics teacher education. We also 
provide some criteria for choosing content, tasks, and methodology to develop 
CA C Some examples of CAC activities are presented that concern probability and 
statistics as well as conjecturing and proving. 

INTRODUCTION 

The first part of the title underlines the fact that teachers, when entering an 
education programme, already have some mathematical knowledge "to be 
developed". Implicitly, it suggests the need to clarify in what direction 
development must be oriented in order both to intervene on teachers' beliefs about 
mathematics in general and about specific mathematical subjects, and to provide 
them with useful knowledge for planning and analysing teaching. 

Consistent with our 30 years' experience with mathematics teacher education in 
different situations and current literature and trends in the field, in this chapter we 
stress the importance of the cultural analysis of the content to be taught (CAC) as 
one of the crucial components of mathematics teacher education. CAC adds to 
professional knowledge, usually considered in the literature as "subject matter 
knowledge", "pedagogical content knowledge," and "general pedagogical 
knowledge" (see Shulman, 1986), by including the understanding of how 
mathematics can be arranged in different ways according to different needs and 
historical or social circumstances, and how it enters human culture in interaction 
with other cultural domains (economics, physical sciences, philosophy, etc.). As 
such, CAC can lead teachers to radically question their beliefs concerning 
mathematics in general and specific subject matter in particular. 

The relevance of CAC in teachers' preparation is related to presenting 
mathematics as an evolving discipline, with different levels of rigour both at a 
specific moment in history (according to the cultural environment and specific 
needs), and across history, and as a domain of culture as a set of interrelated 



P. Sullivan and T Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 223-244. 

© 2008 Sense Publishers. All rights reserved. 



PAOLO BOERO AND ELDA GUALA 

cultural tools and social practices, which can be inherited over generations (see 
Hatano &Wertsch, 2001 ).' 

CAC is not intended to be taught through regular lectures about mathematics by 
a university professor but it can be used to open new windows for teachers as 
"decision makers" (see Sullivan & Mousley, 2001; Malara & Zan, 2002), provided 
that suitable mathematical subjects are selected and teachers are personally 
involved in suitable mathematical activities and related educational reflections. 

Consequently, in this chapter we present: the place of CAC in mathematics 
teacher education; a short discussion of methodological issues (related to the 
difficulty of providing evidence for our assumptions through ordinary comparative 
methods); some criteria for choosing mathematical subjects, tasks, and educational 
methodology for teacher education, in order to make CAC involving and profitable 
for teachers; and some examples of how CAC concerning two delicate subject 
areas (probability and statistics; and conjecturing and proving) can enter teacher 
education and affect teachers' beliefs about these subject areas, as well as 
mathematics in general. 

THE PLACE OF CAC IN MATHEMATICS TEACHER EDUCATION 

Taking the relevance of "subject matter knowledge" and "general pedagogical 
knowledge" for granted (see Boero, Dapueto, & Parenti, 1996), current 
mathematics education literature on teacher education stresses the importance of 
"pedagogical content knowledge" (PCK) as defined by Shulman (1986) as one 
possible direction for integration and development of teachers' professional 
knowledge. Following Shulman, for 

the most regularly taught topics, [...] the teacher should become acquainted 
with the most useful forms of representation of those ideas, the most 
powerful analogies, illustrations, examples, explanations, and 
demonstrations- in a word, the ways of representing the subject that make it 
comprehensible to others [...] what makes the learning of specific topics easy 



' In the last three decades, the nature of mathematics as a culture (or part of a culture) has been 
investigated by several scholars with reference to different definitions of "culture". In particular, Wilder 
(1986, p. 186) writes that "A culture is the collection of customs, rituals, beliefs, tools, mores, etc. 
which we may call cultural elements, possessed by a group of people", and applies this general 
definition to mathematics. 

We prefer to make reference to the definition of culture proposed by Hatano and Wertsch (2001), 
because it better accounts for the systemic character of mathematics (including mathematical activities), 
its "inheritance over generations", and the relevance of symbolic and social tools. 

"'Culture' means the special medium of human life consisting of a set of interrelated artefacts (Cole, 
19%), shared to some extent among members of the community and often inherited over generations. 
These artefacts include physical tools, common sense knowledge and beliefs, social organisations, and 
conventional patterns of behaviour associated with the physical, symbolic, and social tools" (Hatano & 
Wertsch, 2001, p. 80). 



224 



CULTURAL ANALYSIS OF CONTENT 

or difficult: the conceptions and preconceptions that students of different ages 
and backgrounds bring with them to learning. (1986, p. 9) 

Some studies in the educational sciences (see Hiebert, Gallimore, & Stigler, 
2002) and in the field of mathematics education (Mason, 1998) have discussed or 
further elaborated PCK, distinguishing between its different components and their 
effects on teachers' beliefs (see Torner, 2002, also Chapter 6, this volume). 
Focusing on PCK can help call into question students' and teachers' beliefs about 
mathematics, specific subject matter, and how to present it. However, this scarcely 
contributes to CAC, the cultural analysis of the content to be taught (its possible 
different axiomatic organisation, its relevance in mathematics, its links with other 
subjects, and so forth.). As we have seen, according to the original definition of 
Shulman, PCK concerns knowledge of the content that is directly related to 
teaching (ways of illustrating, exemplifying, explaining it), and students' content- 
related "conceptions and pre-conceptions". Connections between PCK and some 
aspects of CAC may be established only when dealing with different ways of 
representing a given topic or the relationships between students' conceptions and 
the social and historical roots of mathematical knowledge. However, neither 
mathematics as a culture (its possible organisations and evolution across history), 
nor the dynamic relationships between mathematics and other cultures (two key 
aspects of CAC) are taken into consideration in the construct pedagogical content 
knowledge. The following two examples are aimed at clarifying the above issues. 
More detailed examples will be presented in the second part of this chapter. 

The first example (relevant for primary school mathematics teacher education) 
concerns cardinal (Cantor) and ordinal (Peano) axiomatic organisations of the 
content "natural numbers", and their relationships with the roots of the concept of 
natural number in its uses in everyday life situations and in different past and 
present cultures. 

The second example (relevant in particular for high school mathematics teacher 
education) concerns rigour in mathematical proof: historical-epistemological 
evolution (from Euclid to Hilbert); relationships with rigour in ordinary arguing in 
different domains (law, natural sciences, philosophy); acceptable, semantic based 
rigour (mathematicians' ordinary proofs) versus formal rigour (proof as formal 
derivation, according to the logicians' models). 

In both cases the considered pieces of CAC knowledge escape the borders of 
PCK as defined by Schulman, and do not enter ordinary taught content knowledge 
(see later). 

In our opinion, CAC is relevant for different purposes: on one hand, to allow the 
teacher to consider the content not as a given object to be presented in an efficient 
way, but as a multifaceted cultural entity that can live different lives in the realm of 
mathematics, according to the context (mathematicians' community in different 
historical periods, school, professional contexts, etc.); on the other hand, to gain 
some distance from the content to be taught and evaluate its importance in the 
mathematics and scientific culture (thus becoming aware of the wider 
consequences of poor learning). CAC can also contribute to interpretations of 



225 



PAOLO BOERO AND ELDA GUALA 

students' learning difficulties and preconceptions; in particular, it can reveal the 
nature of some difficulties as related to didactical obstacles inherent in the ways of 
presenting a given content in school, or to epistemological obstacles inherent in its 
very nature (for a distinction between didactical obstacles and epistemological 
obstacles, see Brousseau, 1 997). 

Let us move now to possible relationships between CAC and the acquisition of 
content knowledge. When Shulman (1986, p. 9) defines "subject matter 
knowledge" as "the amount and organisation of the knowledge per se in the mind 
of the teacher", he claims that "to think properly about content knowledge requires 
[one] to go beyond knowledge of the facts or concepts of a domain. It requires 
understanding the structures of the subject matter." However, in Shulman's 
definition, "structures of the subject matter" are considered as given; the need to 
discuss them in relation to different social, cultural, and historical contexts in 
which the mathematics was and is used (a crucial component of CAC) is not taken 
into account. In another research perspective, the theoretical construct of Didactical 
Transposition (Chevallard, 1985) implies a need to analyse differences and 
relationships between mathematicians' mathematics and school mathematics and 
to consider different contexts for use of mathematics (with their constraints on the 
nature of mathematical activities) as possible occasions for "re-contextualisation" 
of mathematical knowledge. When mathematics teacher education is considered 
within this perspective, the role of CAC in the development of teachers' 
professional knowledge is not evident (Chevallard, 2000); instead the focus of 
investigations are on the mathematical, institutional, and didactic constraints that 
"determine the teacher's practice and ultimately the mathematical organisation 
actually taught" (Barbe, Bosch, Espinoza, & Gascon, 2005, p. 235). 

Properly speaking, at least some parts of CAC (those related to different 
axiomatic organisation of mathematical content, and to mathematical modeling) 
should be developed in the teaching of mathematics for all those who have to deal 
with mathematics in a critical way (not only mathematics teachers, but also 
physicists, economists, statisticians, etc.). This is not the case in most universities 
all over the world! 

Considering how teachers acquire content knowledge, we observe that CAC is 
not usually included in the ordinary mathematical curricula that provide 
prospective teachers with "subject matter knowledge." In particular, in Italy (like 
in many other countries) most prospective high school mathematics teachers are 
taught "subject matter knowledge" in university courses that are not specific for 
prospective teachers and are intended to provide students only with a strong 
technical background in mathematics. Even in papers concerning what content 
knowledge should be acquired by practising teachers and how, when possible, to 
organise specific mathematics courses for them, we find limited evidence of CAC. 
Attention is mainly given to the choice of tasks (cf, Zaslavsky, Chapman, and 
Leikin, 2003) and educational methodology, usually based on a problem-centred 
approach, in order that teachers learn mathematics and at the same time experience 
learning of mathematics in an educational environment suitable for transposition to 
their professional activities. Connections with CAC are implicitly evoked in a few 



226 



CULTURAL ANALYSIS OF CONTENT 

sources. In particular, Schifter (1993) describes a mathematics course for teachers 
in which the intended major goal was to enable participants to experience genuine 
mathematical activities and reconsider and broaden their understanding of what 
mathematics is. Even in that situation, the real focus was more on teachers' 
reflection on their mathematical practices than on the increase (and/or change) in 
their knowledge about what mathematics is with respect to different historical, 
social, and cultural contexts. The conclusion is that attention to CAC as a 
component of teachers' preparation is lacking both in theoretical perspectives and 
in practical teacher education. 

We add that it is not easy to find instructors or space in the curriculum for CAC 
in teacher education. Teacher educators coming from the educational sciences 
usually have poor competencies in CAC. Mathematics educators with a scientific 
background in teaching of mathematics and a technical background in mathematics 
(this is the case of most mathematics educators in Italy) usually concentrate on 
PCK and eventually on subject matter technical knowledge (when they realise that 
teachers do not know enough of the specific subjects they will teach). On the 
mathematicians' side, most of them (in Italy like in many other countries) think 
that teacher education should extend (in terms of quantity and/or technical depth) 
teachers' mathematical knowledge in those fields that are strictly related to what 
they teach, in order to make them more sure and comfortable with the content to be 
taught. Other mathematicians (a minority) insist that teachers' mathematical 
knowledge be updated (especially in the case of practising teacher education for 
the high school level), so that they learn new ways of thinking about traditional 
mathematical content and new directions of development for the mathematical 
sciences. This position theoretically provides a place and space for CAC. However, 
it usually results in lectures that go far beyond the content to be taught. Their focus 
is on presenting research perspectives in mathematics. In both cases, the 
underlying assumption is that "better" (i.e., deeper, broader, or updated) 
knowledge of mathematics translates into better teaching of mathematics. Both 
positions do not tackle the problem of teachers' beliefs concerning mathematics 
and related distortions; also, neither provides teachers with relevant knowledge and 
tools for the "didactical transposition" of mathematical knowledge. 

The previous analysis shows how it is difficult to locate CAC in existing 
theoretical perspectives and practical situations of mathematics teacher education; 
we aim to show that CAC can be a reasonable, feasible, and useful component in 
this endeavor. In general, we acknowledge the importance of PCK and we do not 
deny the necessity of integrating teachers' content knowledge (especially when 
teachers' previous preparation has not covered important content areas to be 
taught), but we would like to stress the importance of developing teachers' 
mathematical knowledge while at the same time integrating their content 
knowledge and calling into question their ways of thinking about mathematics. In 
order to attain this aim, we think that relevant tools can be derived from 
epistemology of mathematics and history of mathematics, in order to frame and 
substantiate the CAC component of teacher preparation. Psychology of learning 
and mathematics education can provide further tools in order to benefit from the 



227 



PAOLO BOERO AND ELDA GUALA 

CAC component and relate its contributions to educational choices (as we will 
illustrate in the examples presented below). 

As a final remark related to feasibility, we point out again that CAC is sterile if 
it enters mathematics teacher education only as the subject of lectures. Standard 
lectures are received by teachers as cultural contributions that in the best case 
contribute to what they say when they speak with their colleagues or with the 
mathematics teacher educator, not to what they do in the classroom (here the 
distinction between "knowing about" and "knowing to act", proposed by Mason & 
Spence, 1999, is appropriate). We think that mathematical content for teacher 
education, tasks related to that content (concerning the crucial role of tasks in 
teacher education, as in Zaslavsky, Chapman, and Leikin, 2003), and methodology 
of teacher education should be three interrelated components for mathematics 
teacher education on CAC. Together, these can contribute to the image of 
mathematics (and the image of mathematics teaching as well) conveyed to 
teachers. 

REMARKS ABOUT OUR RESEARCH METHODOLOGY 

Our contribution is based on more than 30 years of work with teachers at different 
school levels (from primary school to high school) in different situations: 

- prospective teacher preparation at the university (where many "students" are 
already practising teachers who need a university degree in order to get tenure at 
a school, or to move from one school level to another); 

- practising teacher education (sometimes promoted by the university, more 
frequently organised by schools, districts, or the ministry of education); 

- cooperative efforts with teachers that are engaged in educational research and 
didactic innovations promoted by university researchers; in this case, they act as 
"teacher-researchers" who share responsibilities with university researchers (see 
Arzarello& Bartolini Bussi, 1998; Malara& Zan, 2002). 

CAC is one of the common components of our mathematics teacher education 
interventions in all the abovementioned situations. We must say that its importance 
grew progressively in our courses during the past three decades; indeed, we 
observed how it can work (in well arranged teacher education activities) as the 
component of teacher development most suitable to challenge some of teachers' 
deep beliefs about mathematics and to raise doubts about the cultural foundation of 
many traditional educational choices and their effect on students. 

We cannot provide any results of comparative studies substantiating our above 
claim, although we have a lot of materials at our disposal (teachers' written texts 
and recorded oral interventions collected during teacher professional development 
activities, and follow-up of teacher education in terms of tasks for their students 
and analyses of their students' behaviors). The reasons for this serious limitation of 
our discourse are inherent: first, in the changes in the school system (programmes, 
commitment of teachers, and their ideological orientations) and in our expertise in 
managing teacher education over the last 30 years; second, in the impossibility of 
isolating the component "CAC" from the other components of teacher education, 



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and measuring (or at least evaluating in an objective way) its short- and long-term 
specific effect on teachers' professional performances. However, we think that 
there are some elements that can support our claim about the effects of CAC in 
teacher education: the analysis of CAC tasks and teachers' behaviors shows how 
teachers' beliefs are intentionally called into question as they answer the open- 
ended questions posed by the instructor; also, the qualitative analysis of the 
evolution of teachers' answers and their planning of didactical situations shows 
their increase in awareness about both the crucial cultural nodes of their 
educational choices and on what aspects of students' behaviour to focus. 

For this reason, our presentation of examples of tasks and related teachers' 
behaviours shows how tasks encouraged teachers to take the CAC component into 
account and resulted in changes in their cultural and educational perspectives 
(including an awareness of the importance of CAC), emerging both in their 
planning of didactical situations and in their texts at the end of the education 
programme. Concerning these teachers' written products, we have used two kinds 
of final evaluation tasks: in some cases, cultural analyses of the content are 
explicitly required for specific subjects; more frequently, tasks do not concern 
CAC in explicit terms but give the teachers an opportunity to demonstrate 
competencies in CAC either in presenting difficulties, doubts and questions related 
to a specific subject, or in planning didactical situations for it. These tasks can also 
reveal the effects of CAC education on the teachers' beliefs. 

THE CHOICE OF MATHEMATICAL CONTENT FOR CAC 

We think that not all mathematical content is suitable for developing teachers' 
awareness of the nature of mathematical knowledge and calling into question their 
beliefs. We also think that exemplary CAC activities on well-chosen topics can 
have an effect on other topics, if teacher education challenges teachers to reflect on 
those experiences and their cultural meaning beyond the specific content dealt with 
in the CAC perspective. 

In Italy, well-established content areas (e.g., in the case of high school: 
Euclidean geometry, analytic geometry, or rational numbers) are well-known by 
teachers (or, at least, they think they know them rather well). Reflective activities 
about those content areas can provide teachers with new ideas, but it is difficult to 
engage teachers in restructuring their knowledge with educational intentions. They 
know main definitions and theorems, they can solve ordinary problems, and 
current textbooks offer a variety of well-established guidelines of how to teach the 
subject matter. Tasks could be chosen in order to raise questions about deep 
aspects of teachers' knowledge, but those tasks would be removed from ordinary 
teaching practices, and teachers' difficulties would appear to them as artificially 
generated (with no impact on their professional duties). Historical and 
epistemological contributions by the instructor could enrich teachers' knowledge 
and cultural framing of the content to be taught (and possibly provide them with 
information to be conveyed to their students), but this would not put the core of 
their content knowledge and related images into question. For instance, the 



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transition from Euclid's geometry to Hilbert's axiomatic perspective can be 
presented to teachers in more or less detail (as we have experienced several times). 
The effect is that teachers stick to some anecdotal information and eventually 
present it to their high school students in order to illustrate the spatial referent-free 
validity of Hilbert's statements, but it remains detached from the students' actual 
geometrical activity in the classroom. Incidentally, we observe that even 
descriptions of students' learning processes and psychological interpretations of 
their difficulties in well-established content areas are insufficient to call into 
question ordinary teaching practices. 

In teacher education, we would like to challenge teachers' knowledge in order to 
help them to learn more about mathematics and mathematical activities (from the 
mathematical, epistemological, historical, psychological, and didactical point of 
view). At the same time, we would like to call into question some general views of 
teachers about mathematics (for instance, the idea that mathematical rigour is an 
absolute across history, or that rigour only depends on the use of mathematical 
formalisms, or that the crucial aim to attain in school is students' knowledge of 
mathematical objects and structures: definitions, statements with their proofs, 
algorithms, etc.). Having these aims in mind and taking previous considerations 
into account, we think that it is better for CAC activities to exploit "important" 
content areas where teachers' previous knowledge is lacking or rudimentary and 
"important" mathematical activities where teachers' performances are usually poor 
("important" depending on both present evolution of curricula and relevance in 
mathematics). Based on current literature and our direct knowledge of the Italian 
situation, we have selected (amongst the possible content areas and mathematical 
activities that can be chosen according to the previous criteria—as important 
mathematical subjects to develop CAC and put teachers' beliefs into question in an 
exemplary way): a) the domain of probability and statistics; and, b) the activities of 
conjecturing and proving (particularly in the domain of elementary arithmetic). 

With reasonable adaptation to fit teachers' different professional needs and 
mathematical background inherent in their previous curricula, we think that these 
subjects can work well for teacher education for all school levels (from primary 
school to the last grades of high school). In our teacher education activities, there 
are other subjects that are suitable for productive CAC interventions; in particular, 
mathematical modeling for high school teacher education and decimal numbers for 
primary school teacher education. The advantage of presenting examples in the 
areas a) and b) consists in the fact that we can see how CAC on the same subject 
can be adapted to teachers of different grade levels. 

Probability and Statistics 

In Italy, probability and statistics is still a marginal subject in high school and in 
university mathematical preparation of teachers. When probability is taught in 
depth (as happens sometimes at university, for teachers who are required to obtain 
a Mathematics degree), it is ordinarily taught at a formal, sophisticated level 
(substantially, as a "chapter" of measure theory, endowed with specific 

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terminology and results). When probability is taught in secondary school, it is 
taught as a set of rules to solve standard exercises. In both cases, the modeling 
aspect of probability is neglected (or reduced to rote exercises), and no room is left 
for a discussion about the cultural importance of probabilistic thinking. During the 
last two decades in Italy, pressure on schools has been exerted (by national 
"programmes" and prescriptions for "curricula") in order to develop teaching of 
probability and statistics as an important area of mathematical thought related to 
several applications and cultural aims (including scientific education of new 
generations). Thus, we are in the optimal situation of having at our disposal an 
"important" subject with no established teaching tradition, potentially rich from a 
cultural point of view, and possible to develop at different levels of symbolic 
treatment, with interesting problems accessible at each level. CAC activities in the 
field are potentially rich in connection with other cultural fields (e.g., natural 
sciences, social sciences.), and with teachers' and students' frequent specific 
misconceptions. The activities can also open a window on genuine mathematical 
modeling as one of the crucial aspects of mathematical culture. 

Conjecturing and Proving 

Conjecturing and proving in mathematics is recognised today as a major source of 
educational challenges all over the world, because it is one of the characterising 
features of mathematical culture. It is also a source of difficulty for students in high 
school and university mathematics courses (particularly as concerns the capacity of 
checking the validity of a statement, finding counter-examples, producing elements 
for a general justification, evaluating justifications). It depends on linguistic and 
logic skills that must be developed very early (beginning in kindergarten, 
according to the U.S. National Council of Teachers of Mathematics (NCTM, 2000) 
standards). 

Usually, secondary school mathematics teachers have learned (as high school 
and/or university students) to understand and repeat proofs for standard theorems 
in different areas (Euclidean geometry at the high school level; calculus, linear 
algebra, and so forth, at the university level). Primary school teachers have a poor 
preparation in proving (most of them have encountered only some Euclidean 
geometry theorems at the secondary school level). Almost no teacher has expertise 
in producing conjectures, finding counter-examples, and producing justifications 
for unknown statements. Thus, the area of genuine conjecturing and proving is a 
virgin field for almost all teachers! As we will see, the impact on teachers of CAC 
education in this area might be: first, to let them experience these aspects of 
mathematical activities; second, to induce them to distinguish between the 
development of productive processes, on one side, and the elaboration of their 
products (according to cultural constraints), on the other, as different sides of 
mathematical competency. 



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TASKS FOR CAC 

Detailed criteria for choosing tasks for CAC depend on the chosen content area of 
intervention in mathematics teacher education: for instance, tasks concerning 
algorithmic performances have more scope in arithmetic than in geometry. 
However, we can point out some general ideas about how to choose and shape 
tasks. 

First of all, tasks must be clearly related to crucial educational issues in the 
chosen area. According to our experience, other tasks are not attractive for most 
teachers and not suitable for challenging their ways of thinking. This criterion does 
not mean that tasks must be the same at a given school level; teachers willingly 
engage in other tasks provided that they see the connection with their professional 
work. For instance, to analyse a given problem situation (once the problem has 
been solved) in terms of prerequisites is an acceptable task, even if the level of the 
problem is not accessible for primary or high school students. And a "difficult" 
(out of the reach of students) problem can be acceptable if it serves to clarify how 
the difficulties met at an adult level need a long-lasting educational intervention to 
be managed in a successful way. 

Tasks suitable to an approach using CAC can be: 

- solutions of mathematical problems (problems that need new tools for teachers 
can be proposed as well.); 

- analysis of students' or colleagues' solutions, according to criteria of 
correctness, clarity, and efficiency; 

- analysis of tasks (in suitable cases, comparison can be proposed between a- 
priori analysis of tasks and a-posteriori analysis, according to produced 
solutions and difficulties met by solvers); 

- reconstruction of the conceptual hierarchy of a given subject (with the aim of 
producing its "conceptual map"); and 

- production of tasks for students, related to the main issues of a given task 
proposed at the adult level (with the aim to develop prerequisites, or to meet 
students' crucial difficulties, or to justify the introduction of further tools for 
students). 

This list of tasks does not include tasks that directly call into question teachers' 
beliefs (about the subject matter, or about mathematics in general), nor tasks that 
directly offer CAC elements (for instance, comparing different axiomatic 
treatments of the same subject). We stress the importance of teachers discovering 
the need to reconsider their own beliefs and to know more about the subject, 
starting from what the teachers easily recognise as a mathematical task or an 
educational task. It is the nature of tasks (related to the opportunities offered by the 
chosen subject) that create the interest for CAC perspectives and discussion of 
teachers' beliefs. The difficulties met by teachers on apparently "easy" questions, 
their different answers, their different evaluations of colleagues' and students' 
solutions and mistakes, and their different educational proposals can work as 
occasions in that direction. Here, the choice of suitable mathematical subjects (as 
discussed in the previous subsection) plays a crucial role - according to our 



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experience it is more difficult to provoke such surprises and puzzling diversity in 
the case of well established school mathematics areas. 

INCORPORATING CAC INTO MATHEMATICS TEACHER EDUCATION 

In mathematics teacher education, CAC activities on a given subject can be 
planned according to a plurality of individual and collective tasks, and instructors' 
interventions, conveniently organised in teaching routines. A complete, ideal 
routine can include individual problem-solving, followed by collective comparison 
and discussion of individual solutions (chosen by the instructor as representative of 
the whole set of solutions - possibly including mistakes), then individual analysis 
of the task in terms of prerequisites and difficulties, followed by collective 
discussion guided by the instructor, and systematic "injection" of CAC elements. 
In some cases, individual creation of tasks suitable for students, and related 
collective discussion of proposed tasks, can bring the sequence of activities to a 
satisfactory conclusion for teachers. 

As we see in the examples below, in some cases (especially in short education 
interventions, or when teachers are already accustomed to the previously described 
style of work) the same individual task can include some of the listed steps and 
guided discussion can be organised at the end, as a premise for the "injection" of 
CAC elements by the instructor. 

The "transfer" problem is one of the most delicate for mathematics teacher 
education on CAC. The very nature of CAC does not allow acquired knowledge 
(concerning historical, epistemological and socio-cultural aspects of the content) to 
be directly transferred to other domains. For instance, knowledge concerning the 
origin of calculus of probability in the XVII century and its multiple roots in the 
study of gambling, in rational philosophies, in demographic studies (see Hacking, 
1975) is a relevant CAC knowledge for teachers at all school levels. Indeed it 
suggests how deeply probabilistic thinking is related to different cultural domains. 
This knowledge cannot be transferred to other domains of mathematics. 

Also, the knowledge of different axiomatic treatments of a mathematical content 
and their historical evolution is "local" knowledge related to that specific content. 
Transferability can (and should) concern the teachers' attitude towards the content, 
"How did this notion, or method, or activity develop over history? How did its 
organisation and symbolic representation change?", 'What was, and is now its 
relevance in mathematics? And in the applications of mathematics? ", "What are 
the analogies with other domains of mathematics?" are crucial questions related to 
CAC that the teacher should learn to pose and (at least in some cases) to answer 
with the help of appropriate sources and or experts. The role of the instructor 
should be to demonstrate the general character of those questions when he/she 
deals with them in CAC education on a given content or activity, and to assist 
teachers to formulate them in suitable ways on further content or activity (see 
examples at the end of the next sections, where some teachers come to ask 
questions in the spirit of CAC that can open possibilities of effective transfer). 



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In general, the didactical organisation of mathematics teacher education 
activities on CAC should be as coherent as possible with the model of classroom 
teaching suggested in the other education activities (particularly those concerning 
PCK and general pedagogical knowledge). This parallelism cannot be maintained 
in a systematic way, especially in the case of primary school teacher education. In 
that case, teachers can be invited to discuss why the choice of the instructor at a 
given moment was contradictory with the proposed principles for classroom work, 
and the related consequences. 

We observe that the instructor's role in CAC education activities is not only that 
of a facilitator of the development of teachers' knowledge and a stimulator for their 
beliefs to be put into question. CAC involves specific knowledge that teachers do 
not have and cannot reconstruct on the basis of their culture. The problem for the 
instructor is to motivate (in the teachers' eyes) the need for further knowledge to 
better interpret difficulties, situate problems, and frame subject knowledge 
according to different needs. From the perspective of teachers' professional 
preparation, this example of a "mediating role" in action can be useful for their 
educational choices. 

MATHEMATICAL WORK AND ITS CULTURAL IMPACT: THE CASE OF 
PROBABILITY AND STATISTICS 

CAC in probability and statistics offers several opportunities to develop teachers' 
knowledge and call into question teachers' beliefs related to the specific field as 
well as to mathematics in general. We present some snapshots from our teacher 
education experience for primary school, lower secondary school and high school 
teachers. They concern some of the tasks that we consider crucial for our work 
with teachers in the field (independent from the number of hours at our disposal) 

First, let us consider the "classical" definition of probability as the ratio between 
the number of favourable outcomes and the number of all possible outcomes, 
provided that they are equally likely. Many teachers at all school levels are not 
aware of the crucial importance of the condition "provided that they are equally 
likely" and (if they are acquainted with it) of its possible epistemological limitation 
(the definition seems to depend on the notion of "equally likely", thus on the 
application of the defined notion). Depending on the school level, we use different 
tasks to provoke discussions on these issues. For instance, in primary school 
teacher education a suitable task is, "Let us throw two dice and consider the sum of 
the upper digits. Is it preferable to bet on odd or on even?" Many teachers 
(including more than one third of those who learned probability in high school) 
answer "even, because the even outcomes are 2, 4, 6, 8, 10, 12, while the odd 
outcomes are 3, 5, 7, 9, and 11: only five odd outcomes, against six even 
outcomes". The discussion of the produced solutions, conveniently guided by the 
instructor, usually brings teachers to the correct solution and to the discovery (or 
re-discovery) of the condition that outcomes must be "equally likely"; incidentally, 
we observe that the same task is suitable for approaching one of the crucial nodes 
of probability in primary school (see Consogno, Gazzolo, & Boero, 2006). The 



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ensuing discussion can be oriented to reveal the meaning and the status of the 
condition "equally likely outcomes" in the classical definition of probability, and to 
approach the issue of rigour in mathematics. This aspect became relevant at 
different levels for different reasons: on one hand, teachers consider as rigorous 
many definitions and proofs that are not rigorous at all; on the other hand, they 
pretend to require an "absolute" rigour from their students (according to their 
standard models of rigour), without taking into account the fact that rigour must be 
related to the needs inherent in the situation to be dealt with. 

For teachers of every school level, we consider it important that they can 
experience different kinds of knowledge organisation in the field of probability. In 
this field, it is easy to find problems that can be solved in substantially different 
ways, according to the theoretical tools available to the solver. For primary school 
teachers, we have chosen the task: 

"A box contains 8 counters with numbers 1, 2, 3, 4, 5, 6, 7, 8. They are drawn out 
without replacement. The first four drawn out numbers were 1, 2, 4, and 8. What is 
the probability that the next three drawn out numbers are odd? " 

An inelegant but safe solution consists in considering all possibilities: 3,5,7; 
3,7,5; 5,3,7; 5,7,3; 7,3,5; 7,5,3; 6,3,5; 6,3,7; ...; then evaluating the ratio between 
the 6 favourable outcomes and the 24 possible outcomes. Another strategy can be 
based on the notion of conditional probability and the theorem of compound 
probability: the probability of getting an odd number is 3/4 for the fifth counter 
drawn out, 2/3 for the sixth one, (provided that the previous number was odd), 1/2 
for the seventh one (provided that the previous two numbers were odd); thus the 
probability of getting three odd numbers is 3/4.2/3.1/2 = 1/4. One could arrive at 
the same result by considering the fact that 1/4 is the probability of getting the 
counter that carries the number 6 as the last counter drawn out. 

We have observed how many teachers with an extensive background in 
probability do not choose the second way of reasoning (even if the instructor 
invites the teachers to find "other solutions "). The reasons for this are sometimes 
inherent in a poor operational mastery of the necessary notions (Mason & Spence, 
1999) would say that those teachers know about them, but they do not know to act 
with them). Teachers typically comment, "1 have studied them six years ago, but 1 
am not confident with them. " Sometimes, reasons depend on a deeper doubt: "by 
counting all possible outcomes and those that are "favourable, " 1 am sure to 
control the situation; when applying theorems, 1 feel uncomfortable, because 
theorems and rules are like black boxes for me, and 1 do not know their exact 
boundaries of validity, or whether formal manipulations can introduce mistakes". 
These remarks are suitable (under the teachers' guidance) to open important 
windows on the ways of functioning of mathematics: how theorems represent an 
economy of thought; and how theory works as a mathematical model (according to 
Norman: see Dapueto & Parenti, 1999) for those situations to which theory is 
applied. 



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The last, easiest way of solving the problem is chosen by few teachers. During 
the comparison of solutions produced by teachers, is interesting to discuss the 
reasons for it: "/ have thought about it, but it seemed to me too easy "; "I was 
surprised when Norma presented her solution: she was able to understand that the 
complementary event was easier to manage". Again, teachers' comments can put 
important educational and cultural facts into evidence. The first comment above 
calls for the relevance of the didactical contract (Brousseau, 1997) and the nature 
of problem solving in school, which is strongly influenced by it. The second 
comment opens another window on some aspects of productive mathematical 
activities (particularly the need to achieve some distance from the problem 
situation and consider it from different points of view). 

In high school mathematics teacher education, occasion for reflection on the 
same issues comes with the proof of Bernoulli's theorem. Combinatorics offers 
useful tools for proving it (but the proof needs a lot of steps). Reasoning based on 
random variables allows one to get the solution quickly. Again, we find that some 
teachers think that reasoning based on random variables is too far from the nature 
of the problem and does not allow them to perform the step-by-step monitoring of 
reasoning necessary to prevent possible mistakes. "Reasoning is in another place, 
it is as if it does not concern the problem. And how to be sure that reasoning with 
random variables fits the problem in a sure way? " 

In our experience in teacher education, links between probability and statistics 
are a delicate subject. We think that it is necessary to deal with this subject because 
it can open windows on important cultural issues (e.g., why people are convinced 
that chance tends to equilibrate too frequently on outcomes of unlikely events? 
When can we derive sure conclusions from outcomes of random events?) and on 
the modeling potential inherent in probability theory. Also, the cultural value of 
mathematics as a component of scientific rationality can be demonstrated. 

For teachers of all school levels, we enter the subject through the "black bottle 
experiment". It can be organised as follows: the teachers are divided into groups of 
two or three. Each group has one closed bottle that contains the same unknown 
quantities of blue and red marbles (e.g., 6 red marbles and 4 blue marbles in each 
bottle). The colour is the only difference between marbles. It is not possible to see 
the contents of the bottle; when the bottle is inverted, it is only possible to see one 
marble. Each group must produce a sequence of 300 or 400 outcomes (requiring 
less than IS minutes), then they must draw a diagram of the cumulative 
frequencies (after 20, 40, 60, etc. outcomes). Finally, they answer some questions: 

- what are the features of the diagram? 

- is it possible to guess the number of red marbles? 

- what about the difference between the number of red marbles and the number 
of blue marbles? 

- what information can we derive from the collected data? 

The comparison of the diagrams and the answers to the previous questions, 
together with the construction of the diagram that represents the cumulative 
frequencies of the outcomes collected by all the groups, allow teachers to realise 
that: "we cannot get any information about the number of red marbles in the 



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bottle, we can only guess the ratio between the number of red marbles and the total 
number of marbles"; "the number of trials can influence the 'stabilisation' of the 
frequency diagram near to a value that suggests the ratio between the number of 
red marbles and the total number of marbles" ; "the difference between the number 
of red outcomes and the number of blue outcomes tends to increase, in spite of the 
fact that the ratio between red outcomes and blue outcomes tends to become 
stable ". 

The following step consists in individually answering to the following question: 
"In some diagrams produced by groups we see that, in spite of an initial 
prevalence of red or black outcomes, the tendency (when the number of trials 
increases) is to approach the same value. How can we explain this fact? " 

Here, we can observe how some teachers reflect common misconceptions: 
"Chance balances out the initial prevalence of red or blue marbles, " "After too 
many red marbles, chance makes more blue marbles come out, so equilibrium is 
reached". Other teachers explicitly refuse such explanations, but they are not able 
to explain what happens. Only a few teachers write that "The increasing number of 
trials has the initial prevalence of red marbles absorbed in the ratio". The 
comparison of different answers allows teachers to move towards a shared answer 
to the initial questions and reflect on the reasons why people think that "chance 
tends to balance irregular outcomes ". 

The final step usually consists of a collective discussion of the question: "How 
many trials should we perform, in order to be sure that the proportion of the red 
and blue marbles in the bottle is that suggested by the frequency diagram? " 

In primary school teacher education, the aim of this question is to encourage 
discussion of the notion of "sure answer". With 10-12 groups (each group 
producing one series of 300 or 400 trials) it is easy to find at least one diagram 
(derived from one series of trials) that suggests a hypothesis "rather different" from 
the other ones. There is not time to experience several series of 3000 or 4000 trials, 
but teachers can rely on their instructor's words and imagine that even a 
cumulative diagram derived from 3000 or 4000 trials could suggest a "relatively 
bad" hypothesis. Teachers can come to understand that the notion of "sure 
answer" cannot be absolute, but instead it depends on the "acceptable discrepancy" 
(inherent in the use of expressions like "rather different"), on the number of trials, 
and on the "acceptable risk" for the validity of the hypothesis. 

In secondary school teacher education, the aim is to motivate the introduction of 
theoretical tools (in our opinion, at least Bernoulli's theorem and Tchebichev's 
inequality must be presented) that allow one to measure the degree of uncertainty 
of a hypothesis derived from the analysis of frequencies. 

In both cases, the discussion should be guided to consider the importance of the 
posed question in different domains (physical sciences, medical sciences, polls, 
etc.). 

Traces of the impact of the CAC activities in the case of probability and 
statistics can be found in the planning of didactical situations concerning the 
introduction of such subject matter in school, in the analysis of students' products, 



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and in the answer to open ended questions concerning "discoveries, doubts, open 
problem. " Criteria to evaluate such traces are: 

- precision in the language to deal with topics in the field; 

- focus on relevant cultural issues; and especially (for answers to open ended 
questions) 

- quality of doubts and open problems that are posed. 

For instance, we consider important (in the CAC perspective and keeping 
teachers' current beliefs in mind) that a secondary teacher writes: "/ have 
understood that (differently from other domains of mathematics) in the field of 
probability and statistics the answers to many problems must be formulated in 
terms of 'it is more likely that', instead of 'it is true that'. But it seems to me that 
also in probability and statistics theory provides true statements: the statement 'the 
probability that... is less than... ' seems to me true like a normal statement in other 
fields. I would like to know more about this issue ". Statements like this one are not 
rare in our corpus of teachers' texts. On one hand, they show how teachers 
formulate questions in the CAC perspective; on the other hand, they can be 
exploited by the instructor to fuel further discussions aimed at transferring the 
CAC way of looking at mathematics to other mathematical domains. 

MATHEMATICAL PRODUCTS AND PRODUCTION PROCESSES: THE CASE OF 
CONJECTURING AND PROVING 

Let us consider the following individual task: 

"Evaluate whether the following texts, produced by Grade IX students that are just 
beginning to prove in Mathematics, are satisfactory to 'Provide a general 
justification, i.e., a mathematical proof for the statement: the sum of two 
consecutive odd numbers is divisible by four. ' Give reasons for your evaluation. " 

(note that d is the initial letter of dispari, odd in Italian, and p is the initial 
letter of pari, even in Italian) 

The following are three proofs that can be used as the basis of discussion. 

Proof I. d+d+2=2d+2=2+2d=4d; 4d is clearly a number divisible by 4. 

Proof II. By making some trials, like for instance 3+5, 15+17, 31+33, I 
realise that I always get sums made by the first odd number and by the same 
odd number increased by two, thus I get the double of an odd number plus 
two. This result is divisible by four because the sum of two equal odd 
numbers would be (alone) an even number divisible only by two, but if I add 
two I get the consecutive even number, which is divisible by four because 
even numbers follow each other with the rule that if one is divisible only by 
two, the following one is divisible by four (like: 2, 4; 6, 8; 22, 24; etc) 
because the multiples of four are four units far from each other. 



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Proof III. d=p+l, the following odd number is d+2=p+3; I must make the 
addition d+d+2 that makes p+p+4 because d+d+2=p+l+p+3=p+p+4. 

This task is systematically used as an introductory task in an 8 to 10 hour 
introduction of CAC elements in teacher education, which takes place in a 30-hour 
course of mathematics education mostly taken by practising primary school 
teachers. It has also been used with lower secondary school teachers and high 
school teachers (within a 15 to 21 hour course on conjecturing and proving). 

It is interesting to analyse how teachers react to this task and notice the 
differences between primary school teachers and high school teachers. Most 
primary and high school teachers write that Proof II is "a less mathematical proof 
than Proof I and Proof 111", and more than one half of high school teachers add 
that Proof II "is not a true mathematical proof because it uses examples. " 

The discussion after the individual task reveals that most teachers in both cases 
did not engage in understanding the text of Proof II; the presence of "algebraic 
proofs" I and III reveals the teachers' conception of the ideal mathematical proof 
as a formal algebraic derivation (even if high school proving in Euclidean 
geometry was a long lasting experience of verbal proving for all of them!). Seeing 
some examples in Proof II (without considering their real function within the text) 
induces the majority of high school teachers to think that "the student has not 
understood that the validity of mathematical proof cannot rely on examples". Few 
teachers (three out of a sample of 18, in a education course last year), after 
reconsidering Proof II under solicitation of the instructor, realise that "in this case 
examples have only a heuristic or illustration function; they are not good in the 
text of a standard mathematical proof but this text works substantially well as a 
general justification! ". 

In contrast, most primary school teachers do not consider examples as mistakes 
or inappropriate in a mathematical proof; when the instructor suggests they read 
the text again, they simply conclude that "it works, " although "it is less 
mathematical than the other ones. " 

Coming now to Proof I and Proof III: most teachers (even primary school 
teachers) recognise the mistake within Proof I. High school teachers explain that 
"the student does not know the rules of algebraic transformations. " Under the 
request of the instructor, they say that "There is nothing to save in that proof. " 
Primary school teachers are more indulgent; some of them say that (provided that 
the algebraic transformation would have been managed in a correct way) "4d 
would have brought the correct conclusion of divisibility by 4 ". 

Proof III usually receives different evaluations by high school teachers and by 
primary school teachers, and within both groups of teachers. The fact that the 
algebraic transformations work well causes about one half of high school teachers, 
and about three quarters of primary school teachers, to be satisfied with Proof III. 
Very few primary school teachers identify the lack of a crucial step (proving 
divisibility of p+p+4 by 4). In contrast, one third of high school teachers speak of 
"incomplete proof" (other teachers write that the student "had the intuition that it 
worked, but did not feel the need to make the last steps "). Invited by the instructor 



239 



PAOLO BOERO AND ELDA GUALA 

to consider again Proof II, most teachers of both school levels do not realise that 
the author of Proof II was able (in the substance) to move verbally from 
odd+odd+2 to the full justification of its divisibility by 4. 

According to this synthetic description of teachers' behaviour in an individual 
task and related collective discussion, it is clear how the task provides a rich source 
of occasions (for the instructor) to call into question many stereotyped conceptions 
teachers have about proving and, more generally, about mathematics and their role 
as mathematics teachers. In particular, the dominant idea of the necessity of 
mathematical formalism in proving, the lack of distinction between a valid 
justification and its conventional style of presentation, and more attention paid to 
the correctness of the product than to the quality of the process. 

Starting from the aforementioned task and depending on the time available and 
the school level, CAC education can be developed in different directions with 
different aims. For primary school teachers, one of the main final aims can be to 
discover the importance of specific logic-linguistic skills inherent in exhaustive, 
logical arguing; as well as reflecting on the fact that many statements in 
mathematics can be validated without a specialised symbolic apparatus. The same 
aims can be intermediate steps for education high school teachers, but in this case 
further goals can be considered: 

- to make clear why Proof III is not satisfactory from a mathematical point of 
view; 

- to make clear why Proof I, in spite of the student's mistake, contains some valid 
elements (if we consider the underlying process); 

- to identify and discuss the various functions of algebraic language in 
mathematics and in mathematical proving. 

In our experience, a second, common task for high and primary school teachers 
(in the domain of conjecturing and proving) is: "To produce a conjecture about the 
GCD of all the products of three consecutive integer numbers, and prove it ". 
Usually, some words must be added, in order that teachers understand what they 
must do (by itself, this fact offers an occasion for reflecting on characteristics of 
mathematical language). 

The analysis of the task in terms of CAC shows relevant potential in different 
directions, suitable for different developments according to the teachers' school 
level. Many high school teachers in the conjecturing phase, and most of them in 
the proving phase, try to solve the problem using algebraic language, which results 
in an impasse. Most primary school teachers prefer to explore the situation with 
numerical examples. They produce suitable conjectures, but when they try to prove 
them, many are not able to move from verification of examples to a general 
justification. High school teachers' conceptions about the privileged role of 
algebraic language in every non-geometrical activity come again to the foreground 
(in spite of the discussions following the previous task). Other conceptions that 
must be called into question (in the perspective of high school teaching) are 
revealed by the analysis of teachers' behaviours and the collective discussion on 
the second task: a limited use of examples (justified by saying that "whenever 
possible, when we deal with theorems we must avoid the use of examples", with a 



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CULTURAL ANALYSIS OF CONTENT 

clear extension to conjecturing of warnings concerning proving); when examples 
are used, the difficulty of exploring them in order to find the common "structure," 
in particular the difficulty in moving from the fact that 6 appears to be the GCD to 
the discovery of the reasons why it happens in all cases (teachers say that 
"examples only help to see, in some cases, if the statement is reasonably true "; the 
possibility of discovering structural regularities by exploring examples is ignored). 

After the first two tasks and related discussion guided by the instructor (4-5 
hours of work), the injection of elements of history and epistemology of 
mathematics by the instructor is usually a turning point in the teachers' ways of 
conceiving some aspects of conjecturing and proving: in particular, the analysis of 
Euclid's proofs of some arithmetic theorems legitimates the use of verbal language 
as a genuine tool for mathematical proving at all school levels; the comparison 
between different styles of presentation of proofs (Euclid, Lobacevsky, Hilbert) 
illustrates the historical non-linear evolution of the style of proofs; (for high school 
teachers) the discussion of some claims by Thurston (1994) raises the question of 
what is relevant in mathematics and in the communication of mathematical 
knowledge. 

For high school teacher education, one of the most useful tasks is the following 
(used also with prospective teachers): "Generalise the proposition: 'The sum of 
two consecutive odd numbers is divisible by four', and prove the generalised 
proposition" (see Boero, Douek, & Ferrari, 2002, for details about difficulties and 
strategies). Teachers' difficulties in understanding what "Generalise " could mean 
in such a situation, the plurality of possible generalisations, the difficulties in 
proving some of the conjectures (depending on the difficulties of translating them 
into suitable algebraic expressions) are all sources of reflection on important CAC 
aspects of mathematical activities and on teachers' related beliefs. In particular, the 
following question is frequently raised by teachers when they discover that many 
generalisations have been produced within their group: "What are the criteria to 
decide whether this is a meaningful generalisation? And who decides this in 
mathematics? " 

At the end of the activities, evidence in teachers' written texts and oral 
interventions reveals that their beliefs on conjecturing and proving, and related 
educational choices, have been at least partially re-oriented: under open-ended 
tasks (" Write down your discoveries, persisting difficulties, doubts, open 
questions"), teachers' texts show how the activities have put into question some 
previous firm convictions ("Now, I feel much more free to write in Italian my 
thoughts and solutions in mathematics; my teachers had always discouraged me 
from doing it, with the motivation that I had to learn the language of mathematics, 
and I was reproducing the same with my students") and have contributed to 
opening new windows for further learning ("It seems to me that well chosen 
numerical examples can work as generic examples in geometry, from the point of 
view of discovering structural facts: am I right?") in the perspective of 
transfer. 

On tasks concerning evaluation of samples of students' proofs (or justifications, 
at the primary school level), teachers not only take into account the distinction 



241 



PAOLO BOERO AND ELDA GUALA 

between correctness of the product and quality of the process (this might merely 
depend on the need of satisfying the instructor's requests), but also spontaneously 
move to personal considerations about the relationships between students' 
performances and their educational context: "Ivan's proof has some gaps, but 
they concern implications that the student probably thinks to be obvious; it 
would be necessary to know better the level of rigour that is requested by his 
teacher". 



CONCLUSION 

CAC, the cultural analysis of mathematical content, has been presented in this 
chapter as one of the most important components of teacher education, in spite of 
the fact that little room is devoted to it in teacher education and in current literature 
on teacher education. Through snapshots derived from our experiences in teacher 
education, we tried to show how CAC activities can be organised using suitable 
subject matters, tasks, and educational methods. We tried also to provide evidence 
for the impact of CAC activities on teachers' beliefs related to the specific field, 
and concerning mathematics in general. 

There is an unsolved problem that we would like to pose in this concluding 
section - that is the preparation of mathematics teacher educators to work from a 
CAC perspective. As remarked in the second section of this chapter, today 
mathematics teacher educators are mostly interested in developing teachers' 
competencies related to PCK, and also their preparation is oriented in the same 
direction (e.g., Zaslavsky & Leikin, 2004). Mathematicians who engage in teacher 
education do not seem interested in the CAC perspective, and their scientific career 
as mathematicians does not depend on CAC competencies. The CAC perspective 
needs competencies coming from epistemology of mathematics, history of 
mathematics, and philosophy of mathematics. However, experts in these fields do 
not intervene in teacher education, or, if they do, their involvement is not designed 
to call into question teachers' beliefs and the cultural orientation of their teaching - 
only standard lectures in those fields are offered to teachers. 

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Barb£, J., Bosch, M., Espinoza, L., & Gascon, J. (2005). Didactic restriction on the teacher's practice: 
The case of limits of function in Spanish high schools. Educational Studies in Mathematics, 59, 
235-268. 

Boero, P., Dapueto, C, & Parenti, L. (1996). Didactics of mathematics and the professional knowledge 
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International handbook of mathematics education (Vol. 2, pp. 1097-1122). Dordrecht, the 

Netherlands: Kluwer Academic Publishers. 
Boero, P., Douek, N., & Ferrari, P. L. (2002). Developing mastery of natural language: Approaches to 

theoretical aspects of mathematics. In L. English (Ed), Handbook of international research in 

mathematics education (pp. 241-268). Mahvvah, NJ: Lawrence Erlbaum Associates. 
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht, the Netherlands. 

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Pensee Sauvage. 
Chevallard, Y. (2000). La recherche en didactique et la formation des professeurs: Problematique, 

concepts, problemes. In M. Bailleul (Ed), Actes de la X-eme Ecolse d'Eti de didactique des 

mathematiques (pp. 98-1 12). Caen, France: 1UFM. 
Consogno, V., Gazzolo, T., & Boero, P. (2006). Developing probability thinking in primary school: A 

case study on the constructive role of natural language in classroom discussions. In J. Novotna, H. 

Moraova, M. Kratka, & N. Stehlkova (Eds.), Proceedings of the 50* Conference of the International 

Croup for the Psychology of Mathematics Education (Vol. 2, pp. 353-360). Charles University in 

Prague, Czech Republic. 
Dapueto, C, & Parenti, L. (1999). Contributions and obstacles of contexts in the developments of 

mathematics knowledge. Educational Studies in Mathematics, 39, 1-21. 
Hacking, I. (1975). The emergence of probability. Cambridge, UK: Cambridge University Press. 
Hatano, G., & Wertsch, J. V. (2001). Sociocultural approaches to cognitive development. The 

constitution of culture in mind. Human Development, 44, 77-83. 
Hiebert, J., Gallimore, R., & Stigler, J. (2002). A knowledge base for the teaching profession: What 

would it look like and how can we get one? Educational Researcher, 31, 3-15. 
Malara, N., & Zan, R. (2002). The problematic relationship between theory and practice. In L. English 

(Ed.), Handbook of international research in mathematics education (pp. 553-580). Mahwah, NJ: 

Lawrence Erlbaum Associates. 
Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of 

attention. Journal of Mathematics Teacher Education, 1, 243-267. 
Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics. The importance of knowing- 

to act in the moment. Educational Studies in Mathematics, 38, 135-161. 
Schifter, D. (1993). Mathematics process as mathematics content: A course for teachers. Journal of 

Mathematical Behavior, 12, 271-283. 
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 

15, 4-14. 
Sullivan, P., & Mousley, J. (2001). Thinking teaching: Seeing mathematics teachers as active decision 

makers. In F.-L.-Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 

33-52). Dordrecht, the Netherlands: Kluwer Academic Publishers. 
Tomer, G. (2002). Mathematical beliefs - A search for a common ground: Some theoretical 

considerations on structuring beliefs, some research questions, and some phenomenological 

observations. In G. Leder, E. Pehkonen, & G. Tomer (Eds.), Beliefs: A hidden variable in 

mathematics education? (pp. 73-94). Dordrecht, the Netherlands: Kluwer Academic Publishers. 
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical 

Society, 30, 161-177. 
Wilder, R. L. (1986). The cultural basis of mathematics. In T. Tymoczko (Ed.), New directions in the 

philosophy of mathematics (pp. 185-199). Boston, MA.: Birkhauser. 
Zaslavsky, O., Chapman, O., & Leikin, R. (2003). Professional development in mathematics education: 

Trends and tasks. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), 

Second international handbook of mathematics education (Vol. 2, pp. 877-915). Dordrecht, the 

Netherlands: Kluwer Academic Publishers. 



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Zaslavsky, O., & Leikin, R. (2004). Professional development of mathematics teacher educators: 
Growth through practice. Journal of Mathematics Teacher Education, 7, 5-32. 



Paolo Boero 

DMA 

Genoa University 

Italy 

Elda Guala 

DMA 

Genoa University 

Italy 



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SECTION 4 

ASSESSMENT OF, AND RESEARCH ON, 
TEACHER KNOWLEDGE 



ANNE D. COCKBURN 



11. ASSESSMENT OF MATHEMATICAL KNOWLEDGE 
OF PROSPECTIVE TEACHERS 



In this chapter I use four case studies - two from primary and two from secondary 
- to illustrate the key issues surrounding the assessment of prospective 
mathematics teachers ' knowledge. The examples are taken from cross-cultural 
settings and demonstrate the universal complexities involved in planning and 
assessing future teachers efficiently and effectively. Criteria for successful 
teaching — as defined by Ball, Bass and Hill (2004) - and students ' views on the 
assessment procedures in two of the cases provide an added dimension to the 
discussion. 

INTRODUCTION 

A hundred years ago teaching was seen as a highly transmissive activity: children 
were perceived as empty vessels to be filled with knowledge. During this same 
time, no formal qualifications for teaching were required. It was not until the 
middle of the last century that teachers were required to gain a formal qualification 
in teacher education before they could practise in government schools in the U.K. 
Prior to that, for example, a degree in mathematics was sufficient to teach the 
subject in secondary schools. Since then there have dramatic changes in the view 
of teaching and the view of students. The widespread acceptance of a more 
constructivist approach views teaching and learning as active and interactive 
processes. There is now an almost universal requirement that individuals need to 
successfully complete some recognised form of teacher education before they can 
practise regardless of the age they wish to teach. Teaching mathematics is no 
longer seen as passing on a series of formulae and procedures which need to be 
drummed into - often very reluctant - learners. Indeed, more than 20 years ago 
Shulman (1986) articulated the necessary content knowledge required for teaching 
in terms of three categories, which he describes as: 

• Subject matter content knowledge which goes beyond the basic facts to 
encompass teachers', [...] understanding of the subject matter and the ability 
to, [. . .] be able to explain why a particular proposition is deemed warranted, 
why it is worth knowing, and how it relates to other propositions, both within 
the discipline and without, both in theory and practice (p. 9). 

• Pedagogical content knowledge which, as the phrase implies, extends to 
subject knowledge for teaching or, '[...] the ways of representing and 



P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 247-272. 

© 2005 Sense Publishers. All rights reserved. 



ANNE COCK.BURN 

formulating the subject knowledge that make it comprehensible to others' 

(ibid, p 9). 
• Curricular knowledge which is a knowledge of the full range of topics which 

might be taught within a subject area together with an understanding as to if, 

when and how they might best be presented. 
More recently, Hill, Rowan, and Ball (2005) stressed that, "Effectiveness in 
teaching resides not simply in the knowledge a teacher has accrued but how this 
knowledge is used in classrooms" (pp. 375-376). Elsewhere Ball, Bass, and Hill 
(2004) discuss extensive research they have done with teachers to expand on 
Shulman's work and present some important insights for teacher education. Firstly, 
they demonstrate that, unlike the untrained mathematicians in the past who had 
been taught, "...to compress information into abstract and highly usable forms" 
(Ball et al., 2004, p. 10), teachers are required to "unpack" (Ball et al., 2004, p. 10) 
ideas and thus be able to both explain how mathematical concepts evolve but also 
be able to unravel a pupil's thinking processes when they have misunderstood 
some aspect of their work. 

Secondly, they stress that, "Teaching involves making connections across 
mathematical domains, helping students build links and coherence in their 
knowledge" (Ball et al., 2004, p 1 1). Thus, for example, a teacher might present a 3 
* 5 rectangle as shown in Figure I below and then rotate it through 90' to 
demonstrate that multiplication is commutative, that is 3 * 5 = 5 * 3. 



Figure I. A 3 x 5 rectangle which might be used to demonstrate 
that multiplication is commutative. 

Thirdly, they argue that teachers need to be aware of future mathematics their 
pupils will encounter to reduce the likelihood of misconceptions developing with 
regard to, for example, subtraction where, in the past, it was common to teach in 
the early grades that you always subtract the smaller number from the larger. 

Finally, they conclude that, "Knowing mathematics for teaching often entails 
making sense of methods and solutions different from one's own, and so learning 
to size up other methods, determine their adequacy, and compare them, is an 
essential mathematical skill for teaching [...]" (Ball et al., 2004, p. 15). These 



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ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

views - although not generally articulated in such detail - are apparent in some of 
the key practitioner documents available around the world. For example the six 
principles for school mathematics issued by the National Council of Teachers of 
Mathematics (2000) in the United States include statements such as, 

• Curriculum. A curriculum is more than a collection of activities: it must be 
coherent, focused on important mathematics, and well articulated across the 
grades. 

• Teaching. Effective mathematics teaching requires understanding what 
students know and need to learn and then challenging and supporting them to 
learn it well. 

• Learning. Students must learn mathematics with understanding, actively 
building new knowledge from experience and prior knowledge. 

While in England teachers are advised that, 

In good mathematics teaching the skills and knowledge that children are 
expected to learn are clearly defined and the teacher has mapped out how to 
lead the children to the mathematics. Children know that they can discuss, 
seek help and use resources as and when they need to. They like to be 
challenged and enjoy the opportunities to practise and apply their learning. 
[...] Children who need more support than others are identified quickly and 
receive early intervention to help them maintain their progress... The teacher 
or practitioner recognises that mathematics is a combination of concepts, 
facts, properties, rules, patterns and processes. Leading children's learning 
must take account of this and requires a broad repertoire of teaching and 
organisational approaches. [. . .] Good mathematics teaching requires a good 
knowledge of the subject, an understanding of the progression in the 
curriculum being taught and a recognition that some teaching approaches are 
better suited to promote particular learning and outcomes. (Department For 
Education and Skills, 2006, pp. 65-66) 

Reviewing the above it would appear that, over the past 100 years there has 
been a marked shift in several significant quarters as to how mathematics teaching 
and learning is perceived. Education is now seen as a much more dynamic process 
which requires high levels of knowledge and understanding with successful 
teachers being highly skilled in a range of pedagogical techniques. Although there 
might be fairly widespread agreement about that, however, when it comes to 
assessing teachers' practice, "[...] there is little agreement on what, whom, and 
how to measure, and for what purpose" (Hill, Sleep, Lewis, & Ball, 2007, p. 149). 

While this observation might create an interesting challenge, it becomes a major 
issue when the spotlight is turned away from practising teachers and focused on 
prospective teachers. As a profession we need to be convinced that they have 
reached the necessary requirements to become efficient and effective practitioners. 
No longer are mathematics examinations sufficient. What is required are far more 
sophisticated and wide ranging techniques which capture, as far as possible, the 



249 



ANNECOCKBURN 

subtleties and complexities involved in mathematics education for primary and 
secondary pupils. 

Thus this chapter considers how prospective mathematics teachers are assessed 
when undertaking their teacher education. It uses four case studies as catalysts for 
discussion. These are presented as in-depth examinations of experienced educators' 
thoughts and actions when planning and implementing assessment of prospective 
mathematics teachers. The choice of examples allows for comparisons and 
similarities to be made across countries (three were selected) and across age phases 
(e.g., a primary and secondary programme within the same institution are 
considered). Semi-structured interviews (Robson, 1993) were conducted to elicit 
data on common issues across programmes and yet ensure that the integrity and 
individuality of each course description was explored. A qualitative descriptive 
approach was adopted (Sandelowski, 2000). Key issues to emerge were the range 
of strategies adopted, the high levels of tutor involvement required for 
comprehensive assessments of prospective teachers' performance and the extent to 
which outside agencies are involved in the process. 

CASE I : ASSESSMENT OF ELEMENTARY TEACHERS IN A 
MIDDLE EASTERN CONTEXT 

The first case is a 3-year first degree course for prospective primary teachers 
undertaken at a teachers' training college in a middle-eastern context. To qualify 
for one of the 20 places in this undergraduate course applicants must pass 
examinations set by both their schools and the college in - among other subjects - 
mathematics. Once accepted they study units in mathematics, psychology, 
pedagogy and the language of instruction while also gaining practical teaching 
experience in local schools. On qualification the majority of these teachers will 
teach 6-12 year-olds. 

In each year of their three years of study, students do four mathematics courses 
which combine to result in eight hours of mathematics per week for 30 weeks per 
year. Typically each course comprises subject knowledge and how to teach 
mathematics in a ratio of 3:1. All the courses and their assessments are designed by 
the person/people who teach them. No external bodies are involved although every 
year the mathematics tutors from all the colleges in the country meet for a day to 
share their programmes as a way of stimulating new ideas rather than anything 
more formal such as moderation. 

When preparing her mathematics teaching, the course tutor said that, not only 
does she change her courses each year, but she also reviews how she assesses the 
students' work as part of the course planning process. For every course unit she 
teaches she employs two assessment strategies: 
I) examinations to assess key points. These tests last 90 minutes and are used to 

assess subject knowledge and its application. For an example of a typical 

problem see Figure 2. 



250 



ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

A child solved the following exercises as follows: 
10 + 4x2 = 28 
10x4 + 2 = 42 
10-4-2 = 4 
10-4:2 = 3 

a. Correct his work. 

b. Explain his mistakes. 

c. Give some more examples which help you to diagnose his knowledge. 
Write the expected answers in each case. 

d. What are the prerequisite skills required for the above? 

e. Suggest how you might teach this child. 

Figure 2. Example of an examination question for prospective primary teachers. 

2) exercises, which are set three or four times a semester, and require students to 
reflect on an issue in depth. For example students might be asked to read a 
research paper on the different forms of subtraction. They then discuss it as a 
group, formulate some questions on the topic which they use as a basis for an 
interview with children in school. They analyse the data collected and present a 
report of their findings to their seminar group. Another activity might be for 
each student to select a non-routine task which they have enjoyed completing in 
a seminar and try it with a class of primary pupils. They observe the children at 
work, analyse their actions with the expectation of recounting their 
observations and providing an in-depth account of their conclusions to their 
colleagues. 

In addition, every week the students are on teaching practice they are observed 
and assessed by a pedagogical supervisor from the college. This individual may not 
be a subject specialist but, particularly in cases of concern, they may then discuss a 
student's progress with colleagues in the mathematics department at the college. 

Towards the end of their third year the prospective teachers undertake a project 
for which they select a series of sessions on a specific topic - such as addition - to 
teach in school. They are required to plan these in the light of their knowledge and 
experience together with reference to the research literature. The prospective 
teachers then teach the unit in school with some of their lessons being observed in 
the manner described above. Following the series of lessons, the prospective 
teachers are required to do an oral presentation at college explaining what they 
prepared and why and how they adapted their planning in the light of their 
classroom experiences. These presentations are observed and commented on by 
several college tutors although it is only the course mathematics tutor who formally 
assesses them and assigns a mark. As an aside, the respondent said that she would 
like the students to do more such reading but added that this could be a problem as 
there are few appropriate native language texts and some students have difficulty in 
reading English. 



251 



ANNE COCKBURN 

If, at any time during the process of their teacher education, the tutor notices that 
a student's performance is weak, she explains, "I do my best to give all the help the 
students need. I spend a lot of time on this". As a result, she added that students 
seldom fail any aspects of the course. On the rare occasions that they do not 
perform to a sufficiently high standard, students are required to re-take the year 
they have completed unsatisfactorily. 

The tutor of the above programme does not assess all aspects of her 
mathematics courses and, indeed, she argues that it might be counter productive to 
do so. She does, however, adopt a rather unusual - and seemingly effective - 
strategy which she believes develops her students' confidence and understanding 
without the burden of assessment anxieties. Specifically, she sets a problem at the 
start of some of her sessions for students to complete in their own time. At the next 
meeting she asks for volunteers to discuss their solution. This practice is repeated 
throughout the year, and at no time are students put under pressure to explain their 
work unless they wish to do so. As a result of this strategy, over the years she 
discovered that students become self-motivated and those who had difficulty 
solving problems, set early in the year, research the topics in detail and, in so 
doing, gain confidence and a willingness to experiment in the classroom. 

First Commentary 

Clearly the design of the above course involved detailed thinking about the 
teaching, learning and assessment processes. Of paramount importance to the tutor 
seemed to be the need for her prospective teachers to know considerably more than 
mathematical facts. Their knowledge was assessed using regular examinations but 
the application of this knowledge was also monitored in a variety of ways to ensure 
that the prospective teachers had an in-depth appreciation of the facts and knew 
how to incorporate them into the educational process. This was done through the 
frequent exercises set each semester and the observation of practical teaching in a 
school environment. 

Reflecting on the above, it would appear that both Shulman's (1986) and Ball et 
al's. (2004) criteria for successful teaching are assessed and, in several cases - such 
as pedagogical content knowledge and knowing mathematics for teaching - on 
more than one occasion. Indeed, there are ample opportunities where students 
could potentially demonstrate Ball et al's criteria as summarised in Table I. This 
notwithstanding, one of the tutor's approaches to advancing the students' own 
mathematical knowledge, understanding and confidence was not formally assessed 
but appears to be a highly effective teaching and learning technique. 



252 



ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

Table 1. Potential opportunities for prospective teachers to demonstrate Ball et al. 's criteria 
for successful teaching in Case 1 



Assessment to demonstrate students' ability to. 



Casel 



'Unpack' information 



Make connections 



Anticipate pupils' future mathematical needs 



Evaluate and incorporate other methods/solutions 



1. examination 

2. exercises 

3. teaching 
practice 

4. project 

1. examination 

2. exercises 

3. teaching 
practice 

4. project 

2. exercises 

3. teaching 
practice 

4. project 

1. examination 

2. exercises 

3. teaching 
practice 

4. project 



CASE 2: ASSESSMENT OF ELEMENTARY TEACHERS IN THE U.K. CONTEXT 

The majority of prospective teachers in the first case - apart from those opting to 
teach 5- and 6-year-olds - will only teach mathematics once qualified. In contrast, 
those in the second case study are likely to teach the full range of primary 
curriculum subjects when they complete their initial teacher education course. 
When comparing and contrasting the two case studies therefore one should take 
cognisance of the fact that both groups of prospective teachers are being prepared 
to teach approximately the same age range of children and, although there may be 
some cultural variations, the range of mathematical topics they will teach 
elementary pupils will be broadly the same. 

Every year approximately 170 graduates embark on a 39-week course which 
prepares them to teach all subjects across the nursery and elementary sector. They 
will, however, have opted for particular age ranges making them most highly 
equipped to teach 3-7 year-olds or 5-11 year-olds. To qualify for this over- 
subscribed course all of the prospective teachers have gained a good honors degree 
(in any subject), have had experience working with children and have undergone a 
rigorous interview including a short presentation. The only entry requirement 
which specifically refers to mathematics is a pass at C grade or above (available 



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ANNE COCKBURN 

grades are A*, A - G) in GCSE mathematics - a public examination typically taken 
by 1 6 year-olds but which can be taken at any age. 

During the course of their initial teacher education all prospective teachers study 
the full range of curriculum subjects taught in primary schools. Their mathematics 
course comprises eight lectures on key mathematical topics - such as number, 
calculation, children's errors, curriculum structure - and twelve hours of 
workshops which focus on the practical aspects of mathematics teaching in areas 
such as shape and space, measurement, data handling. In addition, those studying 
to teach 3-7 year-olds have 20 hours input on mathematical development in the 
early years. Prospective teachers who fail their audit (see below) also attend up to 
five hours of mathematics clinics. Others, with a particular interest in the subject, 
may opt for a 10-hour project in planning and conducting mathematics in out-of- 
school contexts. The prospective teachers spend approximately half of their course 
working at the university and the other half in two schools where they develop 
their expertise in practical teaching. In order to be eligible for qualified teacher 
status at the end of their year's course all prospective teachers are required to: 

1) pass a mathematics audit prepared and marked by the course tutors but which, 
periodically, may be scrutinised by external examiners and government 
inspectors from the Office for Standards in Education (Ofsted). This audit is 
taken under examination conditions and may take up to two hours to complete. 
Every year questions, such as using informal methods to calculate a percentage, 
are designed to test mathematical knowledge while others are included to assess 
understanding and application. Typically the latter might be of the type, 
"Which of the following is true/false and give reasons why". An example of 
such a question is given in Figure 3. Prospective teachers who fail the first 
attempt at the audit attend the clinics described above 



A cuboid-shaped container has sides of lengths 0.5m, 0.8m and 
0.25m. It would need to be used 40 times to fill a tank with 4 
cubic metres of water. 

This is true/false, because... [delete as appropriate, and explain/show 
why] 



Figure 3. A typical question in a mathematics audit in Case 2. 

and then undertake a re-examination. Prospective teachers have a further 
chance if they fail again, but a second failure results in their not qualifying for a 
teaching certificate. 
2) work in groups of three students plan and deliver a 20-minute workshop for 
colleagues on a specific mathematical topic such as fractions. They present the 
material - including practical tasks and a display - explaining how the concept 
can be developed across the primary years of schooling. The work is then 
written up in 500 words by each member of the group. This, together with the 

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ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

presentation, is assessed. If students fail they are given one opportunity to re-do 
the work. 

3) successfully complete their teaching practices in two placement schools. To do 
so, for the mathematics component, prospective teachers need to demonstrate 
sufficient competence in the preparation, planning, teaching, assessment and 
evaluation of their practice. For all prospective teachers this assessment is done 
by their class teacher working in collaboration with a university tutor who may, 
or may not, be a mathematics specialist. Evidence used in making these 
judgments is collected from the teaching practice files which all prospective 
teachers are expected to prepare and maintain. These include records of the key 
written documents the prospective teachers use as part of their practice such as 
schemes of work, lesson plans, pupil assessment details and reflections on 
sessions taught. In addition prospective teachers are observed teaching on 
several occasions by both their class teacher and their tutor. At least one of 
these observations must be of a mathematics session. Should a student fail to 
meet the specified criteria (see below) they are usually given the opportunity to 
undertake a further teaching practice in the following academic year. 

Every year a sample of prospective teachers (i.e., about 10%) are also 
observed by an external examiner from another school or higher education 
institution by way of a moderation exercise. Additionally, when an institution is 
inspected by Ofsted - typically every 3-4 years - government inspectors 
observe a range of prospective teachers across the ability range working in 
schools. The inspectors' role is not to assess the prospective teachers per se but 
rather to judge the way in which the higher education institution monitors and 
works with individuals. 

4) the final form of assessment the prospective teachers are required to undergo is 
a centrally prepared online (QTS) test required by the government Department 
for Education and Skills. In essence it is a basic skills test in numeracy and it 
may be taken as many times as necessary. Individuals are not, however, 
allowed to work as fully qualified teachers until they have passed this exam. 
The tests include typical questions you might ask pupils - such as '46 x 8 
equals?' and 'A science teacher has £100 left in his budget. How many books at 
£ 6.95 each can he buy?' This notwithstanding, the focus is not so much on the 
teaching of mathematics, but rather on the numeracy of the prospective teacher 
and especially their competence to interpret professional data - for example, 
graphs and tables of examination results. 

The above four criteria are designed to determine whether the prospective 
teachers have fulfilled the professional standards for qualified teacher status as 
defined by the Training and Development Agency and set out in the Appendix. As 
discussed, prospective teachers are generally given the opportunity to re-take any 
aspect of the course they have failed. A second failure - with the exception of the 
QTS test which can be taken as often as necessary - means, however, that they are 
unable to graduate as a qualified primary teacher. 



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ANNE COCKBURN 

Second Commentary 

The aim of this chapter is not to compare and contrast the input in mathematics 
education the different groups of prospective teachers receive on their various 
courses but, in passing, it is interesting to note that the difference in contact time 
focusing specifically on mathematics is markedly different in Cases 1 and 2, 
although, ultimately, the prospective teachers will teach much the same 
mathematics curriculum in their respective countries. 

Another notable difference between the courses is the number of prospective 
teachers enrolled in each. Teaching arrangements are not central to this chapter 
although the idea of holding certain classes only for the weaker mathematicians - 
and thus reducing class size and increasing individualised attention - is 
noteworthy. For the purposes of this discussion, class size is relevant in so far as it 
impacts on assessment strategies. For example in both Cases 1 and 2, audits 
(termed 'examinations' in Case I) are taken of the prospective teachers' subject 
knowledge. Given that there are only two mathematics specialist tutors in Case 2, 
however, it would be impossible for them to assess three or four detailed exercises 
per semester for each of 170 individuals. The tutor in Case 2 emphasised that the 
audits included questions to, 'assess understanding and application', but, to an 
outsider, it seems unlikely that this could be as thorough as the regular tasks 
concentrating on specific issues in depth as described in Case I . 

In part, the practical teaching of mathematics was assessed in similar ways in 
the two cases with prospective teachers being observed working in the classroom 
by someone who may or may not be a mathematics specialist. Again, however, the 
larger cohort in Case 2, seems to restrict the possibility for a more in-depth 
approach to student assessment. Certainly the prospective teachers are observed 
teaching mathematics and their paperwork (e.g., long- and short-term planning, 
pupil assessment records, and self evaluation records) are thoroughly scrutinised 
but, unlike their counterparts in Case 1, there is only a limited requirement for 
them to provide detailed work on a topic such as in the form of a workshop rather 
than a formal presentation. The sheer number of prospective teachers - 170 versus 
20 - would make this an impossible task for the subject specialists in Case 2. Table 
2 provides a summary of opportunities where the prospective teachers in Case 2 
might potentially demonstrate their ability to meet Ball et al's (2004) criteria for 
successful teaching. As might be predicted there is less scope for them than in Case 
1. 



256 



ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

Table 2. Potential opportunities for prospective teachers to demonstrate Ball et al 's criteria 
for successful teaching in Case 2 

Assessment to demonstrate students' ability to.... Case 2 

'Unpack' information 1. audit 

2. group work 

3. teaching practice 
Make connections 2. group work 

3. teaching practice 

Anticipate pupils' future mathematical needs /. audit 

3. teaching practice 

Evaluate & incorporate other methods/solutions 3. teaching practice 



Another marked difference between the two cases for discussion at this point is 
the fact that the tutor in Case 1 appears to have full control over how her 
prospective teachers' work in mathematics is assessed. She involves her non- 
specialist colleagues on occasion and once a year she discusses her strategies with 
professionals at other institutions but the responsibility for the assessment of 
prospective teachers' work in mathematics clearly resides with her. In contrast, in 
Case 2 all of the prospective teachers have to take, and pass, a centrally set online 
test. Failure to pass it prevents prospective teachers from taking up a teaching post 
as a fully qualified practitioner. In addition, all of the assessment techniques used 
in Case 2 are open to scrutiny by two external parties. The first are external 
examiners who may or may not be mathematics specialists but who, in their 
capacity as expert practitioners in primary education, may voice an opinion as to 
the rigor and suitability of the assessment arrangements employed. They cannot 
insist that anything is altered but their opinions are usually highly regarded and 
therefore, on the rare occasions they make suggestions, their advice is generally 
heeded. The other group who periodically (i.e., every three or four years or so) 
take an interest in the course assessment are government inspectors from the Office 
for Standards in Education (Ofsted). Their views on the strategies used, together 
with their thoughts on the entire elementary education programme, determine how 
the course is rated nationally: since these gradings began over 10 years ago, Case 2 
has consistently been ranked as one of the top three primary teacher education 
courses in the country. This has had a significant impact on the number of student 
places allotted to the course (this is determined centrally) and the caliber of the 
applications made to the institution. Poor gradings can result in the ultimate 
closure of a course. 

To summarise the above in the light of Shulman (1986) and Ball et al.'s (2004) 
criteria, students in both cases seem to be assessed across all of the aspects listed 
although, as discussed below, there is a question over the extent that students' 
mathematical knowledge is thoroughly measured in Case 2. Finally, consider 
whether it is better to produce 20 fully prepared and thoroughly assessed 
elementary teachers whose primary task is to teach mathematics each year or to 



257 



ANNE COCKBURN 

produce 170 general ist teachers whose performance - although perhaps not so 
intensely assessed - is deemed to be of a recognised national standard? Despite the 
lack of a simple answer, the question raises some important issues and as the 
reader, your initial reaction to it might also be worthy of examination. 

CASE 3: ASSESSMENT OF PROSPECTIVE HIGH SCHOOL TEACHERS IN A 
EUROPEAN CONTEXT 

Case 3 comprises a 5-year university programme course which is available for 
those wishing to teach mathematics in secondary (high) schools. Every year there 
are 40 places on the course: generally about 15 are taken by those wishing to do 
mathematics with the remaining 25 going to individuals opting to teach 
mathematics and physics. Most of the prospective teachers on the course have 
spent three of the five years studying courses in mathematics although some will 
have undertaken broader programmes of study combining mathematics with, for 
example, components in physics, statistics or engineering. Prior to entry onto the 
final two years of the course prospective teachers are required by law to undertake 
firstly, a multiple choice written test to assess their knowledge of mathematical 
topics at university; secondly an oral test to assess how this knowledge might 
influence the mathematical topics taught in high schools. 

Over the 2-year teacher education period the prospective teachers take 12 
examinations and receive 1,200 hours teaching comprising: 

* 300 hours of educational studies (undertaken with secondary prospective 

teachers studying other subjects), 

* 300 hours on the didactics, history and epistemology of mathematics, 

* 300 hours of laboratory work on the didactics of mathematics, and 

* 300 hours on school placements. 

With the exception of the school placements, the assessment is decided at an 
institutional rather than a national level. As will be described below, school 
experience is assessed using a state examination set at regional level. 

All of the 270 prospective teachers undertaking secondary school teacher 
education courses, regardless of subject specification, take four written 
examinations as part of their educational studies course which focus on topics such 
as psychology and sociology. Prospective teachers take two courses in didactics 
and two in the history and epistemology of mathematics. At the end of each they 
take a written examination. Typically prospective teachers might be given a 
mathematical problem from a school textbook to solve and then be required to 
discuss the didactical (epistemological) issues surrounding it. The better 
prospective teachers not only solve the problems successfully but they also refer to 
appropriate literature on the subject and analyse the suitability of the text for 
teaching secondary school pupils. Each essay is marked by four university tutors. 
Should there be any disagreement in the grading, or should a student be borderline, 
an oral examination is arranged to further explore what the student has written and 
to ask them more general questions about their knowledge and understanding of 
the teaching of secondary school mathematics. 



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ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

The prospective teachers also attend two laboratory courses each semester and, 
working in self chosen groups of two or three individuals, decide on an area, for 
example, the teaching of parabolas in secondary school, to work on. Their task is to 
consider how the topic might best be treated from (a) a mathematical and (b) a 
didactical point of view. There is no word limit set but, through successive 
dialogues with their colleagues and with their tutor, each group has to produce a 
portfolio of their work by the end of the academic year. Each student is then 
interviewed by a panel of four (i.e., two school teachers and two university tutors) 
to determine their individual contribution to their group's work, their mathematical 
knowledge and understanding of the topic selected and their thinking on how the 
topic might best be taught. Each of these orals is specifically tailored to the student 
under consideration. Course attendance is deemed sufficient to gain the necessary 
credits to pass two second-year laboratory courses on the didactics of mathematics, 
however, formative assessment plays a key element in the teaching of these units. 

The final aspect of the mathematics prospective teachers' programme is their 
school placements. These are arranged by specially trained school teachers who 
are employed by the university as supervisors and who spend approximately half of 
their time working in school and the other half at the university. There are three 
components to a student's school experience. Firstly, they spend time observing 
how a school operates at a general level noting what is required of the school and 
how the organisation is managed. Secondly, they spend time observing at least two 
teachers. Finally, having planned mathematics sessions in conjunction with a 
teacher (to ensure that the pupils' programme is not interrupted) the prospective 
teachers then practise and refine their teaching skills. This third component of the 
school placement is assessed by both a practising school teacher and the student's 
supervisor in that they are asked to comment on the student's strengths and 
weaknesses. In addition, prospective teachers compile a report on the three aspects 
of their practical teaching experience. These are usually of the order of 100 -120 
pages but they have been known to be as long as 300 pages. The intention is that 
the report demonstrates a student's understanding of children's mathematical, 
social and emotional needs. Following their school experience prospective teachers 
undertake a 3-hour written test set by the State. In essence they select three topics 
at random from a bowl and from these topics they have to choose one, such as the 
introduction of derivatives, polynomials or matrices and determinants. In the 
assigned 3-hour period they then prepare what might best be termed medium term 
plans for the topic including the intended school type, the age group to be taught, 
the pre-requisite skills required for successful learning, assessment strategies and 
teaching techniques. On the following day each student then orally presents the 
same work in front of a panel of four teaching practice supervisors and four 
university tutors. On this occasion, however, the work is supplemented with a 
discussion of the resources which might be used and any reflections the 
prospective teachers might have on their presentation. This is followed by 
questions from the panel regarding the choices the student made when planning 
their scheme of work and their general thoughts on their teaching practice 
experience as documented in the report described above. 



259 



ANNECOCKBURN 

None of the assessment strategies employed include the involvement of external 
examiners. Nor is there any assessment of the prospective teachers' mathematical 
beliefs but, as in Cases 1 and 2, these may have been apparent in their practical 
teaching and written work. Prospective teachers who fail their first year 
assessments are required to repeat the year if they wish to continue their studies. 

Third Commentary 

Although it is not the focus of this chapter, it is clear that the above is a thorough 
course. Moreover, not only are a range of assessment strategies used, but there is a 
high commitment of specialist tutors' time for marking and moderation. The 
prospective teachers' level of mathematical knowledge is assessed prior to the 
course and then, less explicitly, throughout their teacher education programme 
when, for example, their ability to plan, teach and reflect on their practical teaching 
is monitored. It is interesting to note that the prospective teachers' understanding of 
students' mathematical, social and emotional needs is also considered to be 
sufficiently important to be assessed (see below). Thus, it would appear that 
prospective teachers are assessed in a very comprehensive manner which, as shown 
in Table 3, includes the criteria specified by Ball et al. (2004). 

Table 3. Potential opportunities for prospective teachers to demonstrate Ball et al 's criteria 
for successful leaching in Case 3 

Assessment to demonstrate students' ability Case 3 

to.... 

'Unpack' information 



Make connections 



/. 


examination 


2. 


lab task 


3. 


teaching practice + 




report 


1. 


examination 


2. 


lab task 


3. 


teaching practice + 




report 


2. 


lab task 


3. 


teaching practice + 




report 


3. 


teaching practice + 




report 



Anticipate pupils' future mathematical needs 



Evaluate and incorporate other 
methods/solutions 



CASE 4: ASSESSMENT OF PROSPECTIVE SECONDARY SCHOOL TEACHERS IN A 

UK CONTEXT 

Case 4 focuses on a secondary school teacher education course at the same 
university in England as the second case. To obtain a place on the I -year 
postgraduate 36-week (PGCE) course, the majority of applicants are required to 



260 



ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

have a first degree at least 50% of which is mathematics. This is a requirement of 
the national body, the Teacher Development Agency. At the discretion of the 
course tutor, however, individuals without such a strong mathematical background 
but who are otherwise deemed to be appropriate candidates for the course - for 
example because of extensive school experience - may be required to take a full 
time 6-month mathematics enhancement course or a 2-week subject booster course 
depending on their mathematical competence. In addition applicants are only 
accepted into the course if they have, or are about to gain prior to beginning the 
course, experience of working in school. Every year approximately 20 mathematics 
prospective teachers enroll on the programme which caters for 200 potential high 
school teachers in total. 

The course comprises 24-weeks practical teaching in two different schools and 
12 weeks university based work of which 120 hours is devoted to preparing 
prospective teachers to teach mathematics to pupils in the 1 1-18 age range and 108 
hours to general professional development. Subject knowledge per se is not taught 
although, as described below, it is assessed. 

The course tutor designs some aspects of the prospective teachers' assessment 
but others are determined by the broader university secondary school programme 
team and yet others were required by external agencies (see below). Being new to 
teacher education the tutor said of the assessments she designed, "I've kind of 
collected what other people do at other institutions". More specifically, 

1) During the course of their PGCE year all the prospective teachers complete and 
mark an audit of their subject knowledge. The audit comprises past 
examination papers for pupils aged 14-, 16- and 18-years-old and the intention 
is that the exercise helps student teachers identify gaps in their mathematical 
knowledge which they will endeavour to fill during the course of their training. 
The tutor takes no part in the process and says, "I don't know that they don't 
cheat". 

2) Also informally the tutor set, "[...] my own mini assignments to build up to the 
formal PGCE assignment and to build up the prospective teachers' written 
skills". So, for example, a student might be asked to prepare a written 
presentation on an article they have read on mathematics education which they 
then present to their peers. This is followed by a discussion and informal 
assessment by the tutor and other prospective teachers. Another task might be 
to teach their peers something that they considered that they taught well in 
school and ask for feedback. 

3) More formally, irrespective of the subject they are preparing to teach all of the 
prospective teachers on high school programmes at this university are required 
to write a curriculum assignment. They are all given the same topic and their 
work is assessed using common criteria. This assessment strategy has been 
widely discussed and developed by the course tutors over the past 10 years and 
has met the formal approval of professional partners (i.e., practitioners in 
schools), external examiners from other higher education establishments and 
government Ofsted inspectors (see below). The assignment (which should be 
approximately 5200 words in length) requires prospective teachers to describe 



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ANNE COCKBURN 

the selection of a scheme of work with the rationale for doing so, how they 
taught and assessed it, how they would modify it in the future giving particular 
attention to any two lessons in the scheme and what they learnt from the 
exercise. A common marking sheet is used by all secondary curriculum tutors 
and focuses on the professional and academic choices the prospective teachers 
made and their justifications for doing so (see Figure 4 for marking criteria - 
the possible ratings for each statement are "good", "satisfactory", "borderline", 
and "unsatisfactory"). Each assignment is marked by the appropriate 
curriculum tutor, a moderation meeting takes place and then a sample are 
selected for marking by another member of the course team - who might, for 
example, be a scientist or a geographer - to ensure equity across the secondary 
programme. Should a student fail this assignment they are given one 
opportunity to do it again. Should they fail again they fail the written 
component of their PGCE course although, in some instances - depending on 
the extent of the fail - they may be given the opportunity to re-submit the work 
the following academic year. 

PGCE Criteria for Curriculum Assignment 

1 Description and analysis of Scheme of Work A drawing on your personal 
experience 

2 Literature on pupils' learning and progression from one key stage to the next 
drawing on research literature and national assessment related report evidence. 

3 Rationale for the revised scheme of work describing how it improves on Scheme A 
and matches teaching methods to the learning objectives. 

4 Assessment in the revised scheme of work explaining how formative and 
summative assessment tasks allow you to assess pupils' learning. 

5 Detailed comment on two lessons in the scheme. 

6 Comment on your professional development making explicit how the new scheme 
improves upon the first, and how the process will inform your future practice. 

General criteria for coursework assignments 

7 Reference to a wide range of relevant literature 
integrated into your assignment 

8 Quality of the communication of your ideas 

(including fluency, structure, coherence, awareness of reader) 

9 Accuracy (spelling, punctuation, grammar) 

Referencing: Is the referencing system used clear and consistent? 
MA Criteria for Curriculum Assignment 

1 Has the writer related personal experience and professional practice to broader 
principles and to relevant literature? 

2 Does the writer provide evidence of having developed exposition and argument that 
goes beyond description and unsubstantiated opinion? 

3 Does the work show that personal insight has been gained through reflection and 
analysis? 

Figure 4: Common marking criteria used in Case 4. 
262 



ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

4) Additionally, in common with all the other high school student teachers (i.e., 
covering other subject specialisations), the mathematics prospective teachers 
undertake two other formal written assignments. The first is a learning 
assignment of 4000-6000 words which has to be completed in their first term. 
In essence it requires prospective teachers to reflect on pedagogical processes 
by comparing and contrasting sessions in their own and other subjects. The 
other task they are required to complete is of similar length and focuses on a 
whole-school issue of their choice albeit in consultation with a teacher in one of 
their schools. This has to be completed in their second term and might consider 
issues such as bullying, health education or home/school relationships. 

5) During their PGCE year prospective teachers undertake practical teaching in 
two schools. Following six-day visits they spend 34 days in 'school A' in their 
first term and then four separate days, followed by 70 days in 'school B' which 
they begin half way through their second term. In both cases the intention is 
that they are given, '[...] a broad range of practical experience' and 
'complementary advice, guidance and support of placement school staff and 
university staff (Secondary Team, 2006, p. 42). Although the mathematics 
tutor will visit every student during a practice, each school takes the lead 
responsibility for their student's formative and summative assessment. In 
essence this involves assessing prospective teachers': professional values and 
practice; their mathematical knowledge and understanding; their ability to plan 
and set appropriate targets for pupil learning; the quality of their monitoring 
and assessment; and their teaching and management skills. These requirements 
were set out by the Training and Development Agency (2007) and these 
standards must be met by all prospective teachers prior to qualification (see 
Appendix). Should there be any question as to whether a student is meeting the 
necessary criteria, a representative of the school is asked to inform the 
university with the result that the mathematics tutor will observe the student 
teach a range of classes and discuss the student's progress with the mathematics 
specialists in the school. A joint decision is then made on the student's 
readiness to enter the teaching profession. Should an individual be deemed 
unready to qualify, assuming they are making sufficient progress, they will fail 
the course but be given the opportunity to undertake an additional placement 
the following academic year. Each year a cross section of prospective teachers 
are visited by an External Examiner from either a secondary school or another 
university. This is not to determine whether a student should pass or fail the 
course but rather it is a form of moderation exercise to ensure that, as far as 
possible, individuals deemed to be of a certain standard in one institution would 
be of similar quality in another. In addition, Ofsted inspectors, as discussed in 
Case 2, observe a sample of prospective teachers every 3-4 years as a means of 
monitoring the quality of provision provided by an institution. 

6) Finally, as part of the Department of Education and Skills requirements, all the 
prospective teachers complete successfully the same centrally produced online 
numeracy test as described above for the primary prospective teachers in the 
same UK institution. This can be done at any time during their PGCE year and 



263 



ANNE COCKBURN 

may be taken as many times as necessary until the student has passed. Until 
they do so, however, they cannot be awarded Qualified Teacher Status. 

Fourth Commentary 

An unusual feature of the above programme is that the student teachers are 
responsible for auditing their own subject knowledge and taking remedial action as 
appropriate. Anyone who has tried to teach something they know but do not fully 
understand will appreciate that it can be both unwise and highly embarrassing to 
attempt to teach anybody anything unless you are well versed in the subject matter. 
Having said that, although they are enrolled on a professional course, the 
prospective teachers in this case might not have reached the realisation that 
acquiring a sound subject knowledge is in their best interest. They might also be 
anxious to pass their course and appear to do well casting doubt on the reliability of 
such an assessment strategy. 

The range of mini-assignments appears to be a good way to encourage 
prospective teachers to read and think about teaching from a range of perspectives 
and this idea was further developed in the task requiring them to compare and 
contrast the pedagogical processes involved in two different subject areas. 

Comparing assessment strategies on the two high school teacher education 
courses it is clear that in both cases, prospective teachers embark on their 
programme of study with sound mathematical knowledge and, with the exception 
of the mathematics audit undertaken by Case 4 prospective teachers, much of the 
assessment is based on the application of this knowledge within an educational 
context. 

As with Case 3 the prospective teachers had to plan a series of sessions, teach 
them and then evaluate their performance. In theory, prospective teachers in Case 
4 might have more autonomy when planning as those in Case 3 are required to plan 
with their teachers for the sake of curriculum continuity. That being said, the 
topics available to them are likely to be limited as most secondary schools in 
England follow a nationally prescribed curriculum and many pupils are in the 
process of preparing to sit national examinations. 

Prospective secondary school teachers in both cases are observed teaching by 
subject specialists and practising teachers although, only in Case 4 do the latter 
take the lead responsibility for the prospective teachers' final assessment of 
practical teaching. This will be discussed further below. 

Two final significant differences between the assessments of the two secondary 
school education courses surround the planning and preparation of schemes of 
work. As discussed above, in both cases prospective teachers plan and prepare 
work which they then teach and, presumably, in both cases they are given ample 
warning, resources and preparation time in which to do this. Additionally, 
however, prospective teachers in Case 3 are also expected to select one of three 
topics picked at random and, within a 3-hour period, produce hypothetical teaching 
plans as a result. In other words, unlike Case 4 prospective teachers, they are 
expected to demonstrate the facility to prepare one of a range of topics at short 

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ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

notice. Further, unlike their UK colleagues discussed above, they are then expected 
to present, discuss and justify their work in detail to a panel of experts. Although 
this must take a considerable amount of time, it is almost certainly thorough and 
appears to be a good measure of prospective teachers' potential to prepare and 
reflect on their planning. 

Thus, to conclude, in both cases students appeared to be assessed across the 
range of criteria set out by Shulman (1986) and Ball et al. (2004) but in Case 3 it 
was in a more thorough and interactive manner (see below). Additionally, in Case 
4 there was greater possibility that students could manipulate their assessment tasks 
- for example by selecting the topics they were most knowledgeable about or, 
even, cheating - so that they would be seen in the best possible light. With this 
proviso, Table 4 provides a summary of the potential opportunities for the 
prospective teachers in Case 4 to demonstrate Ball et al's. criteria for successful 
teaching. 

Table 4. Potential opportunities for prospective teachers to demonstrate Ball et al's criteria 
for successful teaching in Case 4 

Assessment to demonstrate students' ability to. . .. Case 4 

'Unpack' information I. audit 

2. mini assignment 

3. curriculum 
assignment 

4. teaching practice 
Make connections 3. curriculum 

assignment 
4. teaching practice 
Anticipate pupils' future mathematical needs 1. audit 

2. mini assignment 

3. curriculum 
assignment 

4. teaching practice 
Evaluate & incorporate other methods/solutions 4. teaching practice 



SOME REFLECTIONS FROM PROSPECTIVE TEACHERS 

Although it was not part of the original plan the opportunity arose to ask 
prospective teachers in Cases 2 and 4 to complete a short questionnaire outlining 
their thoughts on the assessment procedures they were undertaking as part of their 
courses. It was not intended to be a comprehensive survey. Nor, indeed, can the 
data necessarily be taken to be representative. Some of the information provided, 
however, was thought-provoking and therefore worthy of inclusion in such a 
discussion paper. 

In Case 2 80% of the prospective teachers and 72% of Case 4 prospective 
teachers considered the levels of assessment on their courses to be 'about right'. 



265 



ANNE COCKBURN 

6% and 16% respectively judged that they were over-assessed and, as one pointed 
out, all assessment is potentially stressful. This may, indeed, be the case for some 
or all of the cases described above but, as will be discussed later, it may be 
necessary to adopt a variety of approaches to ensure as broad and as fair an 
assessment programme as possible for all. What might be missing, however, is 
sufficient explanation to the prospective teachers as to why such detailed 
assessment is necessary. In part this should be part of the educational process but 
also, equally importantly, it seems appropriate that professional educators should 
share their thinking and expertise with future professional educators. Thus, for 
example, as implied in the discussion of Ball et al.'s work (2004) at the beginning 
of the chapter, future teachers of the very youngest school children need to have a 
thorough knowledge and understanding of the mathematics well beyond their 
chosen age range in order to reduce the likelihood of mathematical misconceptions 
arising as they teach. An example of this is graphically described in Iannone and 
Cockburn (2006) when, in response to a child, a grade 2 teacher finds it necessary 
to call upon her knowledge of negative numbers to explain subtraction. 

In Case 2, 14% of the prospective teachers and only one of the Case 4 
prospective teachers deemed that they were insufficiently assessed. As a future 
teacher of 3-5 year-olds revealed, "We are tested on mathematical ability but not 
on our understanding and ability to apply it to children except during an 
observation - which can of course be arranged to be in an area of strength!" Many 
of the respondents found that lesson observations were extremely helpful 
explaining, 

"It provided a rich tapestry of feedback and useful relevant advice." (Jack) 

"Being observed by mentor/curriculum tutor [...] (is most effective) [...] 
because the feedback can be very useful and encouraging." (Helen) 

Some found written assignments valuable, 

"Essays have been good for reflection but can be hard for some 
mathematicians to get the most out of." (Paul) 

"They (essays) enabled me to analyse my teaching and develop new 
strategies to improve." (Susie) 

However others questioned the worth of such exercises noting, 

"[...] Many very able trainees struggled to express their views on paper." 
(Jack) 

"Just because you can write an essay, it doesn't mean you can teach. 
Likewise if you struggle with essays, it doesn't mean you can't teach." 
(Rona) 



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ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

DISCUSSION 

The main aim of teacher education is not preparing the students to a sufficiently 
high standard so that they do well in their assessments; nor should it be. It would, 
however, be both irresponsible and naive not to assess their performances to ensure 
that, as far as possible, they are well equipped to enter the profession as effective 
teachers of mathematics. At the beginning of the last century, this was perceived to 
be a straight-forward process: students' mathematical knowledge was tested and, if 
deemed to be of a sufficiently high standard, they were considered fit to pass this 
knowledge on to the next generation. 

At the outset of this chapter the attributes associated with successful teachers by 
Shulman (1986) and Ball et al. (2004) were described. These are markedly 
different from those of over 100 years ago. Now successful teaching is considered 
to be a dynamic, interactive activity which requires considerable knowledge and 
understanding of children and adolescents, mathematics, and the process of 
education. Were the students in the above case studies assessed in terms of these 
more complex criteria? Moreover, could they be? Reviewing the information 
provided it appears that, on paper, all the courses measured the students' 
knowledge, understanding and professional expertise in such terms to a greater or 
lesser extent. Was this, however, the case in reality? 

The art of the assessment involves measuring the students' attainments as 
comprehensively and as accurately as possible while making efficient use of the 
available resources and, most crucially, not adversely affecting the education 
process itself. It is possible, for example, for a course to be so rigorously assessed 
that it takes an inordinate amount of time, worry and attention - on the part of both 
students and their tutors - and proves to be highly expensive. None of the courses 
described above appeared to fall into that category although, as mentioned above, 
the assessment of the students in Case 3 in particular was highly demanding of 
tutors' time and expertise. 

In 1979 Doyle memorably concluded that prospective teachers 'exchange 
performance for grades'. Whether the various tests and examinations given in 
these cases resulted in an exchange of performance for grades they had the desired 
effect on some, "The audit forced me to really develop my own maths skills and 
undertake personal study" (Rob) and "The maths audit gave me confidence. 1 
found I knew more than 1 thought and could explain how I found the answers" 
(Beth). That being said, someone remarked, that they revised [studied] for the 
audit, "[...] and forgot key concepts soon after". Earlier in the chapter, on 
discussing the prospective teachers' self-monitoring an audit of their subject 
knowledge in Case 4, it was noted that they were future professionals and that one 
would hope that they would take responsibility for filling the gaps in their learning 
rather than attempting to cheat the system by pretending that they performed well. 
Some of the prospective teachers gave the impression that passing the course was 
more important than securing knowledge commenting, for example, that test 
situations were the most effective form of assessment as you "can't cheat" (Paul) 
and any assessment involving others were ineffective as, "you only need to do a 



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ANNE COCKBURN 

little work" (Sam). Another student clearly missed the intended value of an 
assignment when she complained, 

For the last assignment (curriculum) we were told we needed to have 20-30 
references however other subjects were told only one would be enough 
(Rona). 

Another aspect of 'exchanging performance for grades' is that it has the 
potential for skewing attention. If, for example, an aspect of the course is not 
assessed will prospective teachers fully engage? Data from Case 1 - where the 
prospective teachers were invited (rather than expected) to discuss a problem they 
had been set - suggested that, certainly for some of them, focusing on assessment 
was not what drove them to succeed. Indeed, if they saw the value of the work and 
did not feel pressured, the absence of formal assessment enhanced their 
performance in the course. Further insight into this issue comes from an ongoing 
international study of mathematical misconceptions in the primary school funded 
by the British Academy'. The project involves practitioners discussing children's 
difficulties in mathematics across the 5-1 1 age range. A by-product of the work is 
that it has become clear that several of the teachers - all of whom are qualified to 
teach across the primary age range - were fairly selective in what they attended to 
when they were student teachers, 

Today has been useful from my point of view as a teacher. Now I'm going 
down to a younger year group, 1 have a better understanding of what I need to 
be really careful about. (Suzanne) 

You focus on your own group and forget what they are going on to [...] 
(Ivan) 

Although these data are anecdotal, they support the case for explicitly teaching 
and assessing student teachers on their knowledge and understanding as to how a 
mathematical concept develops across several years. This idea is further supported 
by findings from the same project which indicate that misconceptions may begin 
early in a child's schooling and yet not become fully manifest until several years 
later, as Ivan suddenly appreciated, 

I have done, "if you add two things it gets bigger and if you subtract gets 
smaller" [...] you don't tend to think what you do has bearing on what others 
do further up the school. (Ivan) 

Such data also demonstrate the value of continuing professional development 
(CPD) for teachers with particular reference to more advanced mathematical 
concepts which were perhaps not fully covered and assessed in some of the above 
courses. For teachers CPD serves as an opportunity to reflect on issues which were 
not necessarily apparent to them in their inexperienced days as a prospective 
teacher. For others CPD is a chance to fill in some of the gaps they chose to ignore 
during their teacher preparation course. More specifically, it would be naive not to 
acknowledge that there are invariably be some prospective teachers in every cohort 

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ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

who focus their learning and preparation predominantly on their perception of the 
most probable topics for assessment. It may be worth it for tutors - particularly in 
Cases 1 , 2 and 4 - to adopt a less predictable approach. For example, if their 
prospective teachers were asked to discuss a topic selected from a wide range of 
possibilities at random (as in Case 3), prospective teachers might broaden the scope 
of their attention during their teacher education programme. 

Finally it is important to consider the extent to which outside agencies were 
involved in the various assessment processes. To some - such as government 
agencies in the U.K. - they are seen as key for accountability purposes. They 
could, however, also be perceived as being both intrusive and a waste of time. In 
Case 1 there was no external involvement; in Case 3 it was only in the sense of 
setting - rather than marking - some of the assessment but in Cases 2 and 4 there 
was considerable scope for contributions from other interested parties. In both of 
these cases it was clear that having external examiners visit from other institutions 
was generally supportive and useful. The best examiners took on the role of 
critical friends and provided an objective commentary on the nature of the 
assessment tasks, procedures and outcomes. Some, being closely engaged in 
similar teacher education programmes in their own institutions, were also able to 
give suggestions on other possible assessment strategies and to compare the quality 
of prospective teachers with their own. Other external examiners were head 
teachers in primary and secondary schools and, as such, usually had an excellent 
understanding of the environment awaiting the prospective teachers on 
qualification. They therefore were able to comment on the trainees' readiness to 
enter the profession and confirm our judgments on such matters. Some of these 
benefits might be available less formally in Cases 1 and 3 for the tutor in the 
former met annually with national colleagues working in similar roles and, in both 
cases, there was considerable involvement and collaboration with practising 
mathematics teachers in schools. 

The other form of external intervention experienced in Cases 2 and 4 was not a 
significant part of the assessment process in the other institutions considered other 
than in setting part of the final assessment in Case 3. Every three or four years 
initial teacher education programmes in Cases 2 and 4 undergo an Ofsted 
inspection. The details of these inspections varies from time to time but, in 
essence, they involve a team of government appointees scrutinising vast amounts 
of course paperwork prior to coming to an institution for several days. These visits 
generally include meetings with the programme team, detailed interviews with 
individual tutors, examination of the prospective teachers' work and observations 
of a sample of prospective teachers (selected by the inspectors) teaching in school. 
There is no doubt that they create a tremendous amount of extra work and stress for 
all concerned. All the inspections in Cases 2 and 4 to date have had very 
successful outcomes. If, however, an inspection result is deemed less than 
satisfactory a programme is put under intense scrutiny and closely monitored every 
year. The number of prospective teachers assigned to the institution is reduced 
and, if there are not marked - and rapid - signs of improvement the course may 
well be closed. The process feels draconian but it is intended to maintain high 



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ANNECOCKBURN 

standards. Obviously future teachers should experience the best possible education 
but it would appear that market forces - rather than government inspectors - play 
that role in Cases 1 and 3 to equally good effect. 

CONCLUDING REMARKS 

The four cases presented in this chapter exemplify some of the carefully planned 
assessment strategies adopted by teacher education programmes in different 
settings at the beginning of 21 s1 century. Gone are the days when chalk-talk and a 
knowledge of mathematics sufficed for teaching in primary and secondary 
classrooms. Nowadays teaching is perceived as being more interactive, complex 
and intellectually challenging. It is not for the mediocre, the passive or those 
simply concerned with passing a few tests and securing a job. Recruiting and 
retaining mathematics teachers is a major concern (Cockburn & Haydn, 2004), 
however, not at any price. It is of paramount importance, therefore, that those 
involved in teacher education ensure that only those who understand and 
demonstrate the potential to be successful mathematics teachers succeed in 
entering the profession. 

APPENDIX 

Requirements for Newly Qualified Teachers as set out by the Training and 
Development Agency (2007) 

The standards specifically pertaining to mathematics in case studies two and four 
are that prospective teachers must demonstrate that they have: 

Q10 Have a knowledge and understanding of a range of teaching, learning and 
behaviour management strategies and know how to use and adapt them, 
including how to personalise learning and provide opportunities for all 
learners to achieve their potential. 

Oil Know the assessment requirements and arrangements for the 
subjects/curriculum areas they are trained to teach, including those relating to 
public examinations and qualifications. 

Q12 Know a range of approaches to assessment, including the importance of 
formative assessment. 

Q14 Have a secure knowledge and understanding of their subjects/curriculum 
areas and related pedagogy to enable them to teach effectively across the age 
and ability range for which they are trained. 



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ASSESSMENT OF MATHEMATICAL KNOWLEDGE 

Q15 Know and understand the relevant statutory and non-statutory curricula and 
frameworks, including those provided through the National Strategies, for 
their subjects/curriculum areas, and other relevant initiatives applicable to the 
age and ability range for which they are trained. 

Q16 Have passed the professional skills tests in numeracy, literacy and 
information and communications technology (ICT). 

QI7 Know how to use skills in literacy, numeracy and ICT to support their 
teaching and wider professional activities. 

ACKNOWLEDGEMENT 

The author gratefully acknowledges the financial support of the British Academy 
for the research study Mathematical Misconceptions in the Primary Years (Grant 
number: LRG - 42447). 



REFERENCES 

Ball, D. L., Bass, H., & Hill, H. C. (2004, April). Knowing and using mathematical 

knowledge in teaching: Learning what matters. Invited paper presented at the Southern 

African Association for Research in Mathematics, Science and Technology, Cape Town, 

South Africa 
Cockburn, A. D., & Haydn, T. (2004). Recruiting and retaining teachers. London: 

Routledge Falmer. 
Department for Education and Skills (2006). Primary framework for literacy and 

mathematics. London: Author. 
Doyle, W. (1979). Making managerial decisions in classrooms. In D. L. Duke (Ed.), 

Classroom management (pp. 42-74). Chicago, IL: University of Chicago Press. 
Hill, H. C, Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge 

for teaching on student achievement. American Educational Research Journal, 42(2), 

371-406. 
Hill, H.C., Sleep, L., Lewis, J.M., & Ball, D.L. (2007). Assessing teachers' mathematical 

knowledge. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics 

teaching and learning (pp. 111-155). Charlotte, NC: Information Age Publishers & 

National Council of Teachers of Mathematics. 
Iannone P., & Cockburn A. D. (2006): Fostering conceptual mathematical thinking in the 

early years: A case study. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), 

Proceedings 3(f h Conference of the International Group for the Psychology of 

Mathematics Education (Vol 3, pp. 329-336). Charles University in Prague, Czech 

Republic. 
National Council of Teachers of Mathematics (2000) Principles and standards for school 

ma/AeOTa//'«.http://standards.nctm.org/document/chapter2/index.htm (accessed 16/1 1/07) 
Robson, C. (1993). Real world research. Oxford: Blackwell Publishers. 
Sandelowski, M. (2000). Whatever happened to qualitative description? Research in 

Nursing and Health, 23, 334-340. 



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Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational 

Researcher, 75,4-14. 
Training and Development Agency (2007). Professional standards for teachers. London: 

Training and Development Agency for Schools. 



Anne Cockburn 

School of Education and Lifelong Learning 

University of East Anglia 

U.K. 



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12. RESEARCHING TEACHERS' MATHEMATICS 
DISCIPLINARY KNOWLEDGE 



At the substantive heart of this chapter is a framework for mathematics lesson 
observation. The framework was developed in research in the UK between 2002 
and 2004, and modified in important ways in 2007. The research which led to the 
development of the framework drew on videotapes of mathematics lessons 
prepared and conducted by prospective elementary teachers towards the end of 
their initial training. A grounded theory approach to data analysis led to the 
emergence of a framework - the 'Knowledge Quartet' - with four broad 
dimensions, through which the mathematics-related knowledge of these teachers 
could be observed in practice. This methodological paper describes in detail the 
process by which the Knowledge Quartet evolved, the chronology of that process, 
and some of the associated theoretical influences and pragmatic constraints. The 
chapter concludes with remarks on the ways that the research is being applied to 
teacher education, in the analysis and improvement of mathematics teaching, and 
also in its evaluation. 

INTRODUCTION 

In his 1985 presidential address to the American Educational Research 
Association, Lee Shulman proposed a taxonomy with seven categories that formed 
a knowledge-base for teaching. Three of these elements are 'discipline knowledge', 
being specific to the subject matter being taught. They are: subject matter 
knowledge, pedagogical content knowledge and curricular knowledge. Shulman 
(1986) notes that the ways of conceptualising subject matter knowledge (SMK) 
will be different for different subject matter (discipline) areas, but in his generic 
account he includes Schwab's (1978) notions of substantive knowledge (the key 
facts, concepts, principles and explanatory frameworks in a discipline) and 
syntactic knowledge (the nature of enquiry in the field, and how new knowledge is 
introduced and accepted in that community). For Shulman, pedagogical content 
knowledge (PCK) consists of "the ways of representing the subject which make it 
comprehensible to others [...] [it] also includes an understanding of what makes 
the learning of specific topics easy or difficult [...]" (Shulman, 1986, p. 9). The 
notion of PCK as a distinct domain has been disputed (e.g., McNamara, 1991), but 
Shulman's intention seemed essentially to be to reify the hitherto 'missing link' 
between knowing something for oneself and being able to enable others to know it. 



P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 273-298. 

© 2008 Sense Publishers. All rights reserved. 



TIM ROWLAND 

This chapter is set in the context of UK research on beginning teachers' 
mathematics disciplinary knowledge. It charts research undertaken since 1998 by 
teams in London and Cambridge. This account is mainly methodological, and 
references are given to reports elsewhere of substantive findings. The majority of 
this chapter concerns a research project which investigated how mathematics 
teachers' disciplinary knowledge is made visible in teaching itself. 

UK RESEARCH ON MATHEMATICS TEACHER KNOWLEDGE 

Recent government initiatives have been taken in a number of countries to enhance 
the mathematics knowledge (typically vaguely conceptualised in the rhetoric) of 
prospective and serving elementary teachers. The rather direct approach to a 
perceived 'problem' in England was captured by an edict in the first set of 
government 'standards' for Initial Teacher Training (ITT) issued in 1997: 

All providers of ITT must audit trainees' knowledge and understanding of the 
mathematics contained in the National Curriculum programmes of study for 
mathematics at KSI and KS2, 1 and that specified in paragraph 13 of this 
document. Where gaps in trainees' subject knowledge are identified, 
providers of ITT must make arrangements to ensure that trainees gain that 
knowledge during the course [...]. (Department for Education and 
Employment, 1997, p. 27) 

The process of audit and remediation of subject knowledge within primary ITT 
became a high profile issue following the introduction of these and subsequent 
government requirements (Department for Education and Employment, 1998). 
Within the teacher education community, few could be found to support the 
imposition of the 'audit and remediation' culture. Yet the introduction of this 
regime provoked a body of research in the UK on prospective elementary teachers' 
mathematics subject-matter knowledge (e.g., Goulding, Rowland, & Barber, 2002; 
Goulding & Suggate, 2001; Jones & Mooney, 2002; Morris, 2001; Rowland, 
Martyn, Barber, & Heal, 2000; Sanders & Morris, 2000). The proceedings of a 
symposium held in 2003 usefully drew together some of the threads of this 
research (BSRLM, 2003). In every case, the participants were prospective (so- 
called 'trainee') elementary school teachers, on three- or four-year undergraduate 
or one-year graduate teacher preparation courses. The methodology at the heart of 
these studies involved a questionnaire - a test - taken at some time before or 
during the course. The subject matter being assessed was typically determined by 
that specified by the Department for Education and Employment (1998), mainly 
related directly to the elementary school curriculum, but also including some topics 
in the school curriculum up to about Year 9 (pupil age 14). The logic of the 
situation pointed to auditing the trainees' mathematics knowledge at the beginning 



1 In England and Wales, Key Stage 1 (KSI) is the first phase of compulsory primary education, between 
the ages of 5 and 7. Similarly, KS2 covers ages 7 to 1 1 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

of their course, to establish where the 'gaps' were at the outset, and this was the 
case in most of the studies. 

An exception was that of Rowland, Martyn, Barber, and Heal (2000), in which 
the audit was administered four months into a one-year course, after the 
prospective teachers had encountered the mathematics content within the teaching 
methods course, "giving them maximum opportunity and professional motivation 
to recall those topics they had forgotten (for lack of use) since they did 
mathematics at school" (p. 4). The research of this London-based team included an 
investigation of the relation between the prospective teachers' SMK, as assessed by 
a 16-item written audit instrument, and their teaching competence. The audit was 
administered by the research team; the assessment of teaching, on a four-point 
scale based on criteria in regular use, was made jointly by their university-based 
tutor and a practising teacher-mentor in the participant's placement school. A chi- 
square test showed a significant association between audit score and an assessment 
of teaching competence. This finding turned out to be robust when we replicated it 
with a different cohort of prospective teachers (Rowland et al., 2001). Participants 
obtaining high (or even middle) scores on the audit were more likely to be assessed 
as strong numeracy teachers than those with low scores; whereas those with low 
audit scores were more likely than other participants to be assessed as weak 
numeracy teachers. 

INTO THE MATHEMATICS CLASSROOM 

Although this was interesting in itself, and attracted some media attention, we 
wanted to find out more about what was 'going on'. By now, several UK-based 
researchers had formed a consortium named SK1MA ('subject knowledge in 
mathematics') and a five-person Cambridge-based SK1MA subgroup took forward 
this new line of enquiry. 

The reasoning behind this move was, if superior SMK really does make a 
difference when teaching elementary mathematics, it ought somehow to be 
observable in the practice of the knowledgeable teacher. Conversely, the teacher 
with more limited SMK might be expected to misinform their pupils, or somehow 
to miss opportunities to teach mathematics 'well'. In a nutshell, we (the Cambridge 
team) wanted to identify, and to understand better, the ways in which elementary 
teachers' mathematics content 2 knowledge, or the lack of it, is evident in their 
teaching. Certain parallels can be drawn with recent studies of Deborah Ball and 
her colleagues that studies of the "work" of teaching with the goal of providing a 
"practice-based theory of knowledge for teaching" (Ball & Bass, 2003). The same 
description could be applied to our own study, but while parallels can be drawn 
between some of the outcomes, the two theories are very different. In particular, 
the theory that emerges from Ball et al. unravels and clarifies the formerly 
somewhat elusive and theoretically-undeveloped notions of SMK and PCK. 



2 By 'content' knowledge, we include any kind of disciplinary (in this case mathematics-related) 
knowledge. In particular, 'content' knowledge encompasses both SMK and PCK. 



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TIM ROWLAND 

Shulman's SMK is separated into 'common content knowledge' and 'specialised 
content knowledge', while his PCK is divided into 'knowledge of content and 
students' and 'knowledge of content and teaching' (Ball, Thames, & Phelps, 
submitted). In our theory, the distinction between different kinds of mathematical 
knowledge is of lesser significance than the classification of the situations in which 
mathematical knowledge for teaching surfaces in the classroom. In this sense, the 
two theories may each have useful perspectives to offer to the other. 

As a research group, we tended to think and talk about research goals more than 
research questions, but a key research question would be: in what ways is (novice 
elementary) teachers' mathematical knowledge made visible in their teaching? The 
parentheses capture the sense that while our interest in this question clearly focused 
on mathematics teaching, it extended beyond 'novice elementary' to all phases of 
teachers' careers and pupils' ages. The phrase "made visible" in the putative 
research question is intended to contain a strong element of "observable". From 
the outset, we envisaged this research as a classroom observation study. We were 
genuinely curious to know what knowledgeable teachers do in the classroom that 
might enhance their pupils' experience of learning mathematics, and what others 
did not, or could not, do. Our interest in whether teacher knowledge can be 
observed may well have been influenced by our previous and ongoing roles as 
tutors in initial teacher education. By 2002, the official guidance to teacher 
education 'providers' (mostly university education departments) on the assessment 
of trainee teachers' mathematics knowledge had shifted from some form of 
'testing' (cf, Department for Education and Employment, 1997) towards seeking 
evidence for this knowledge in the act of teaching itself, in particular within the 
prospective teachers' practicum placements. 

Evidence of secure subject knowledge and understanding is most likely to be 
found in trainee teachers' teaching, particularly in how they present complex 
ideas, communicate subject knowledge, correct pupils' errors and in how 
confidently they answer their subject-based questions. (Teacher Training 
Agency, 2002, p. 19) 

This is one of those statements about education that must seem self-evident, to its 
author 3 at least, but whose warrants are not at all clear to others. It seemed 
possible, however, that our intended classroom observation might go some way to 
finding out whether the claim, about evidence for teachers' subject knowledge 
being "found" in their teaching, was sustainable. 

From Questionnaire to Observation 

We were somewhat apprehensive as we set about planning for this project. Earlier 
SKIMA studies of trainee teachers' mathematical knowledge, located within the 



3 A culture of anonymity entered into English government documents of this kind more than a decade 
ago, so we are not privy to the identity of the person(s) who made this assertion or any other in the 
Guidance. 



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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

earlier 'audit' culture, had been more straightforward. A 90-minute written 
assessment consisting of 16 test items in mathematics had been administered to a 
whole PGCE cohort at the same time, in each of the SA7M4-participant 
universities. The scripts were collected, marked and the response to each question 
coded on a 5-point scale from (not attempted, no progress towards a solution) to 
4 (full solution with convincing and rigorous explanations). Each code 
corresponded to explicit descriptions of different responses developed from trials. 
The key boundary was between 2 and 3: a score of 3 or 4 indicated that the 
prospective teacher was 'secure' in this area of knowledge, in the sense that they 
appeared to know the topic adequately for their professional purposes. Scores of 0, 
1 and 2 did not offer this assurance, and guided study and peer tutoring was 
advised in such cases. This was followed by various quantitative and qualitative 
analyses of the written responses. This seemed a good deal more tangible than our 
proposal to look for evidence of teacher knowledge in their work in classrooms. 
We had, at best, a vague idea of what we might be looking for, and the content of 
the classroom scenarios we would be observing would be beyond our control. 

Participants and Ethical Desiderata 

In the UK, the majority of trainees follow a one-year, full-time course leading to a 
Postgraduate Certificate in Education (PGCE) in a university education department 
(or school, institute or faculty), about half the year being spent working in a school 
under the guidance of a school-based mentor. All primary trainees prepare to be 
generalist teachers of the whole primary curriculum. Later in their careers, most 
take on responsibility for leadership in one curriculum area (such as mathematics) 
in their school, but, almost without exception, they remain generalists, teaching the 
whole curriculum to one class. 

Most of the data collection for this study took place in 2002, in the context of a 
one-year PGCE course in which each of the 149 trainees followed a route focusing 
either on the 'lower primary (LP)' years (ages 3-8) or the 'upper primary (UP)' (7- 
11). In the first instance, we obtained mathematics content knowledge data for the 
whole cohort of prospective teachers, using the 16-item audit discussed above. This 
was administered four months into the course, for the reasons stated earlier. The 
next phase, the heart of our study, entailed observations of lessons taught by some 
of the 149 participants. Since we were interested in the relationship between 
mathematics content knowledge and classroom teaching, it seemed to us that the 
participants observed ought to represent a range of subject knowledge competence, 
as measured by the audit. For the purpose of this research, the total scores for each 
paper (maximum 64) were used to identify groups with 'high', 'medium' and 'low' 
scores. In practice, these vague notions were operationalised by defining a high 
score to be in the range 64-60, medium 59-48, and low 47 and below. Of course, 
these are crude measures, not least because these totals are aggregates of ordinal 
measures. These boundaries were further determined by the professional 
judgement by experienced course tutors about these prospective teachers' need for 
significant remedial support, modest support (or self-remediation), or none. The 

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TIM ROWLAND 

proportions of trainees in these three categories were not pre-ordained. In the 
event, the boundaries turned out to be close to the 20th and 70th percentiles in this 
cohort. In order to compare and contrast the teaching of trainees with different 
levels of audited content knowledge, we decided to observe equal numbers of 
participants from each of the three categories. Finally, because the curriculum 
would appear to make different demands on content knowledge in the lower 
primary and upper primary years, we also wanted these two phases to be equally 
represented. Our resources made it possible to devote the equivalent of one person 
full-time for two weeks to the observation; each classroom visit would take about 
half a day to observe and videotape one lesson. These factors and constraints 
eventually influenced the decision to identify 12 trainees for observation, and to 
observe each of them teaching a whole mathematics lesson on two separate 
occasions. The 12 trainees represented the intersection of each subject knowledge 
category with each of the two LP/UP age-phase groups, with two participants in 
each cell of the 3x2 grid of possibilities. This seemed a sensible compromise 
between the alternatives of observing 6 trainees (too few trainees) 4 times each, 
and observing 24 trainees just once (too few observations per person). Once the 
SMK/phase restrictions and geographical location of their school-based placements 
was taken into consideration, we had very little room to manoeuvre in selecting the 
12 participants, of which three were male, reflecting reasonably well (if it were 
relevant) the I in 6 proportion in this PGCE cohort as a whole. These 12 were 
invited to participate in the video study phase of our project, and all agreed. 

It was made clear to them, in writing, that in observing them, our role would be 
that of researcher, not university tutor (although four of the research team also 
fulfilled that role in other circumstances). They were therefore assured that our 
observations would play no part in the university's summative assessment of their 
teaching competence, or any other aspect of their certification. We sought and 
gained their permission to use data from the videotapes of their lessons in research 
papers, and to use short extracts from some of the tapes for research presentations, 
within future teacher education programmes, and at future 'partnership' meetings 
with cooperating schoolteachers ('mentors'). Later, it was also necessary to obtain 
similar permissions from mentors and headteachers in these prospective teachers' 
placement schools, and from the parents or carers of the children in their classes. In 
a few cases this parental permission was withheld, and practical arrangements 
(such as ensuring that certain children were off-camera, or supervised in an annexe 
to the class) were made to respect this choice without abandoning the observation 
altogether. 

Procedure 

Data collection. The two lesson observations took place in the 5th and 7th weeks 
of an 8-week final teaching placement. By this late stage of their training, the 
participants had completed the mathematics methods course at the university, and 
had taught in two schools for about 15 weeks in total. The school mid-term break 
occurred between the two observed lessons. Trainee-participants were asked to 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

provide a copy of their planning for the observed lesson. The focus and content of 
the lesson were chosen by the trainee teacher. The observer operated a tripod- 
mounted video camera from the back of the class, while the trainee-teacher being 
observed wore a clip-on radio microphone linked to the camera audio input. The 
camera tracked the trainee-teacher, but would occasionally capture some artefact in 
the classroom (such as a set of exercises displayed on a wall) which had some 
immediate relevance to the mathematics teaching being observed. When and if 
necessary, the observer made handwritten field notes about anything that interested 
her, or which might be particularly relevant to the focus of our enquiry: 
specifically, how the teacher's mathematics content knowledge is evidenced in the 
classroom. These included notes of any relevant aspects of the lesson that might 
not have been captured on the video recording, such as off-camera pupil comments 
or interactions. Pupils' spoken contributions were audible on the video-recording if 
they were picked up by the radio microphone: for the most part, this included 
pupils' remarks during the whole-class teaching portions of the lesson and during 
seatwork portions when the trainee-teacher was working closely with an individual 
pupil or a group. Our assessment of the obtrusiveness of the observer and the 
technology can be anecdotal at best, but in the video recordings it is rare to see any 
evidence of the pupils taking any interest in their 'visitor'. The demeanour of the 
trainee-teachers themselves appears much as it might be when we visit to observe 
them as tutors. None of this is to deny that our presence might have caused some 
perturbation in the 'usual' classroom processes. At the conclusion of the lesson, the 
observer took care not to give any feedback or evaluative comments on the lesson 
to the teacher- trainee, in order to emphasise that they were not there in their role as 
teacher educator. 

Data preparation. As soon as possible after the lesson (usually the same day) the 
observer/researcher wrote what we call a Descriptive Synopsis of the lesson. This 
was a brief (around 500 words) account of what happened in the lesson, so that a 
reader might immediately be able to contextualise subsequent discussion of any 
events within it. These descriptive synopses were typically written from memory 
and with the use of the field notes, with occasional reference to the videotape if 
necessary. 

Mason (2002) suggests that there are three ways in which data may be 'read': 
literally, interpretively and reflexively. She describes the first of these as 
documenting a 'literal' version of "what is there", but points to the difficulties for 
interpretive researchers in attempts to do this, since "what is there" is necessarily 
interpreted by the observer. Mason suggests that in most qualitative research a 
construction of what data means, or what can be inferred from it, will be made. 
This involves the reading of data interpretively. In a reflexive reading of data, the 
researcher considers the way in which their own experiences and concerns might 
influence their interpretation (Winter, 1989). 

The Cambridge research team subscribed to the view that every human account 
of events is an interpretation of the messenger/teller's experience, and every 
'reading' of such an account is the reader's interpretation of the message, which 



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itself is an artefact, a set of signs. Therefore no 'objective' account of a lesson can 
be written, and none can be read. However, we guarded against 'smuggling' 
interpretive and inferential passages into these descriptive synopses. With this in 
mind, in addition to what the observer believed to be their best efforts at 
straightforward description, different text styles were used to identify in the 
synopses (a) anything that the observer/researcher thought might turn out to be 
significant, or critical, moments or episodes with respect to the trainee-teacher's 
mathematics content knowledge, for consideration later by the research team (b) 
any evaluative comment within the descriptive synopsis; this was to allow 
occasional (in fact, quite rare) comments of the kind that one might write, as a 
tutor, on a lesson observation report (acclaim or criticism), yet which went beyond 
description. This was in part to alert others to the fact that the observer had 
imposed their own evaluation onto the description, but also in recognition that such 
remarks might be found to have some analytical value later. The descriptive 
account offered by the observer could be challenged by a member of the team for 
its 'objectivity', and if the challenge were conceded, one of these two formats 
would be applied to the hitherto 'normal' text in the account. 

A pilot lesson was videotaped and analysed in this way, as a kind of rehearsal of 
our intended means of 'capturing' the lesson. We also discussed aspects of the 
content of the lesson that caught our immediate interest with respect to our 
intended focus on teacher knowledge. In retrospect, that was an extraordinarily rich 
lesson. The teacher, Trudy, was an undergraduate primary teacher education 
student, and therefore not a participant in the main study. Trudy's university 
subject specialism was music. She had not studied mathematics since the age of 16, 
however, in our audit she scored a perfect 4 on every item. Any temptation we 
might have felt to include her lesson in the data was subsequently lifted because 
the videotape of Trudy's lesson was mislaid, and has never been found. However, 
this raises interesting questions about 'exciting' data from pilot studies, and what 
researchers could do and should not do with it. 

We note in passing, and with some regret, that we did not look to the trainee- 
teachers themselves as potential sources of interpretive data on the lesson that they 
taught. Our reasons were pragmatic; we appreciated the willingness of our 
participants to have their lessons observed and recorded, and we decided not to 
impose on their goodwill, or that of the personnel in their placement schools, any 
further. 4 



4 In a later, separate phase of the project, after the Knowledge Quartet had been developed as a 
conceptual tool, we interviewed the trainee-teacher at the end of the school day, when one team member 
met with him or her to view the videotape and to discuss some of the episodes. In the first part of this 
stimulated recall interview (Calderhead, 1981), the trainee was invited to 'think aloud' about any aspect 
of the mathematical content of the lesson. Then the interviewer drew the teacher's attention to key 
issues that had been identified by the team earlier in the day, and invited the trainee to comment and 
offer their own perspective on the relevant episodes (Thwaites et al., 2005). 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 



Coding 



After all the video tapes of the lessons were collected, the hard work of analysing 
these 24 lessons began. We took a grounded approach to the data for the purpose of 
generating theory (Glaser & Strauss, 1987). At the outset, we did not know what 
kind of 'theory' might emerge from our close scrutiny of the lesson videotapes. It 
might have been an explanatory theory of the kind "Because this teacher knew x, 
he or she did (or did not do) y in the lesson". Alternately, it might have been a 
'lens' type of theory - a new way of seeing classroom events from the perspective 
of teacher knowledge. In the event, the theory that materialised was more of the 
second kind. 

In our grounded theory approach to the videotapes of the lessons, we watched 
the tapes of all the lessons, usually in two and threes, sometimes as a whole team. 
We articulated and compared our interpretations of episodes from each of the 24 
videotaped lessons. We identified specific actions in the classroom that seemed to 
provide significant information about the trainee's mathematics content knowledge 
or their mathematical pedagogical knowledge. We reflected later that most of these 
actions related to various choices made by the trainee, in their planning or more 
spontaneously. In this way, we homed in on particular moments or episodes in the 
tapes. Each such moment or episode was assigned a preliminary code (or more 
than one if appropriate). We developed these codes as we went along (examples 
will be given later), and most of them recurred as we saw what looked like the 
same kind of phenomenon in different episodes within the same, or another, lesson. 
It soon became apparent that the majority of the salient moments and episodes, and 
the corresponding codes, related to issues of mathematics pedagogy 3 rather than 
knowledge of mathematics per se. Perhaps this is not surprising, since the subject- 
matter was elementary mathematics, and half of the lessons were with children 
aged 4-7. For example, the issue in introducing subtraction to young children is not 
whether an educated adult teacher can himself, or herself, subtract one small 
integer from another, but whether they know the fundamental subtraction structures 
or models, appropriate contexts for these structures and ways of representing them, 
and a range of relevant student mental strategies. 6 Nevertheless, pupils' 
spontaneous remarks and questions did, on occasion, tax the trainees' overt 
knowledge and understanding of mathematics, in unexpected ways (see Jason 
below). 

At first the identification of such moments, and accounts of their significance 
for our research, was in the form of proposals, or conjectures, for consideration by 
the team. They could be challenged or supported, and retained or rejected by 
consensus. The grounds for such a challenge included relevance and significance. 



What our colleagues in other parts of Europe would more likely call 'didactics'. 
6 We probably depart from Ball et al., (submitted) here, in that they seem to regard these aspects of 
mathematics content knowledge as aspects of specialised content (i.e., subject matter, for them) 
knowledge. This is unimportant, for the present purpose at least: we could re-frame our observation by 
saying that what Ball et al. call common content knowledge was rarely an issue, either for concern or 
celebration, in the 24 lessons we analysed. 



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TIM ROWLAND 

Relevance is subjective to a degree, but the coding was expected to be relevant to 
the focus of our research: the role of teachers' mathematics-related knowledge in 
mathematics teaching. For this reason, 'child demonstration' (CD), which was at 
one time one of 18 agreed codes in use, was challenged and discarded, leaving 17 
codes. Whilst on several occasions the teachers did invite a child to demonstrate 
something to the class, it was agreed that this could happen irrespective of the 
teacher's SMK or PCK. The notion of significance is also subjective: while we did 
not attempt to define what made an event sufficiently 'significant' to merit being 
coded, decisions by individuals or subgroups in the team were often challenged on 
the grounds of significance. For example, one of our codes was called 'overt 
subject knowledge' (OSK). It related to moments when the teacher displayed some 
aspect of their SMK. One such example was when Nathalie compared the 
probabilities of two particular outcomes when two dice are thrown, by identifying 
and listing the sample space and isolating the events. The label itself (OSK) could 
have been improved, but it stuck. On one occasion, another trainee had been 
teaching how to add 10 by adjusting the tens digit in the numeral (e.g. 47 -> 57). 
One of the team had provisionally coded OSK the prospective teacher's fluency 
with the strategy. This was successfully challenged, on the grounds that there was 
nothing special or unusual about that. It is the kind of knowledge that the 'average 
citizen' would be expected to have, whereas OSK was intended to mark knowledge 
that somehow went beyond the ordinary and everyday. This same significance 
criterion applied equally to codes that related to PCK more than SMK. In fact, 
there would be circumstances where the "how to add 10" might be coded, but with 
a code more related to PCK. 

It was important for us to keep in mind that our research focus was mathematics 
content knowledge, and not other more general kinds of pedagogical awareness or 
expertise. For example, ensuring that all children are positioned to be able to see, 
say, a demonstration of counting by one of their peers is undeniably important, but 
not within the scope of our current enquiry. There were also times when we had to 
remind ourselves that we were not in the role of 'partnership tutors' (school 
placement supervisors). In fact, our analyses took place after the outcome of the 
trainees' teaching placement had been decided. This was helpful to ourselves and 
to the participants, because our task was to look for issues relating to their 
mathematics knowledge for teaching, and not to make summative judgements 
about or high-stakes assessments of their competence as mathematics teachers. 

An initial 'long list' of codes was rationalised and reduced by negotiation and 
agreement in the research team. Typically this came about through identifying and 
unifying duplicate codes, and by eliminating the codes associated with events that 
were agreed to be weak in significance. 

Responding to Children 's Ideas 

By way of illustration of this coding process, we give here brief accounts of two 
episodes that we labelled with the code RESPONDING TO CHILDREN'S IDEAS 
(RCI). It will be seen that the contribution of a child, in each case, was unexpected. 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

Within the research team, this code name was understood to be potentially ironic, 
since the observed response of the teacher to a child's insight or suggestion was 
often to put it to one side rather than to deviate from the planned lesson 'script', 
even when the child offered further insight on the topic at hand. 

Illustrative episode I. Jason was teaching elementary fraction concepts to a Year 3 
(pupil age 7-8) class. The pupils each had a small oblong whiteboard and a dry- 
wipe pen. Jason asked them to "split" their individual whiteboards into two. Most 
of the children predictably drew a line through the centre of the oblong, parallel to 
one of the sides, but one boy, Elliott, drew a diagonal line. Jason praised him for 
his originality, and then asked the class to split their boards "into four". Again, 
most children drew two lines parallel to the sides, but Elliott drew the two 
diagonals. Jason's response was to bring Elliott's solution to the attention of the 
class, but to leave them to decide whether it is correct. He asks: 

Jason: What has Elliott done that is different to what Rebecca has 

done? 
Sophie: Because he's done the lines diagonally. 

Jason: Which one of these two has been split equally? [...] Sam, has 

Elliott split his board into quarters? 
Sam: Urn ... yes ... no ... 

Jason: Your challenge for this lesson is to think about what Elliott's 

done, and think if Elliott has split this into equal quarters. 
There you go Elliott. 

At that point, Jason returned the whiteboard to Elliott, and the question of whether 
it had been partitioned into quarters was not mentioned again. What makes this 
interesting mathematically is the fact that (i) the four parts of Elliott's board are not 
congruent, but (ii) they have equal areas (iii) this is not at all obvious. Furthermore, 
(iv) an elementary demonstration of (ii) is arguably even less obvious. This seemed 
to us a situation that posed very direct demands on Jason's SMK and arguably 
PCK too. It is not possible to infer whether Jason's "challenge" is motivated by a 
strategic decision to give the children some thinking time, or because he needs 
some himself. 

Illustrative episode 2. Naomi was introducing the subtraction 'comparison' 
structure (e.g., Carpenter & Moser, 1983) to a Year I class (pupil age 5-6). She set 
up various comparison (or "difference": see Rowland, 2006) problems, in the 
context of frogs in two ponds. Magnetic 'frogs' are lined up on a board, in two neat 
rows. In the first problem, Naomi says that her pond has four frogs, and her 
neighbour's pond has two. The class agreed that she had two more frogs than her 
neighbour. 7 Then Hugh offered the following thought: 

Hugh: You could both have three, if you give one to your neighbour. 



7 We reflected that this particular example, with equal subtrahend and difference, is pedagogically 
problematic (see e.g. Rowland, Thwaites, & Huckstep, 2003). 



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TIM ROWLAND 

Like Jason, Naomi acknowledged the child's idea, but in this case she dismissed 
any further consideration of the alternative avenues that it could lead down. 
Naomi: 1 could, that's a very good point, Hugh. I'm not going to do 

that today though. I'm just going to talk about the difference. 

Madeleine, if you had a pond, how many frogs would you like 

in it? 

One can readily sympathise with Naomi's response to Hugh's insight, which seems 
to deviate too far from the agenda that she had set for the lesson. Naomi 
acknowledges Hugh's observation, but refuses to be diverted from her course. With 
the benefit of hindsight, one can see that she had the option, if she were brave (or 
confident, or reckless) enough to choose it, to take Hugh's remark as the starting 
point of rather a nice enquiry. This would almost certainly have prompted 
investigation of the difference between two numbers, as intended, as well as 
halving, the concept of arithmetic mean, and the distinction between even and odd 
numbers. 

The identification of opportunities to respond to children's ideas (RC1), whether 
taken or sidestepped in the videotaped lessons, was not intended to suggest - and 
certainly not at the data analysis stage being described here - what the teacher 
(prospective teacher in this case) should or should not have done. The immediate 
context here is research, and the enlightenment of the research team. In a different 
context, that of mathematics teacher development, as I shall explain later, 
identification of such an opportunity by an observer raises the possibility for the 
teacher to reflect on it, to add to their knowledge for teaching, and to inform future 
action. I am not suggesting that every potential diversion should be pursued. 
Before deviating from their plan, the teacher must make a more-or-less 
instantaneous cost-benefit assessment of the outcome of doing so, whether they 
feel sufficiently confident to depart from their 'script', and whether the time 
available is sufficient to see the new venture through to a meaningful conclusion. 

Reflexive Interpretation 

We viewed and considered episodes like these without knowing in advance what 
we might 'find' in them, or what to expect. We had no theories in advance - at a 
level other than Grand Theory such as constructivist epistemology and grounded 
theory methodology - to shape the way that we looked at them. We were looking 
for ways in which these teachers' mathematical content knowledge 'played out' in 
their work in the classroom and, to the best of our knowledge, no existing 
framework existed to "organise the complexity" 8 of what we saw in the 24 lessons 
with that particular focus on teacher knowledge. We did not come to our analysis 
of the tapes tabula rasa however. Our reflexive interpretation (Winter, 1989) of the 
data was made explicit in subsequent 'analytical accounts' of the lessons (see 



* This phrase, in connection with the Knowledge Quartet, is due to Michael Neubrand (personal 
communique, May 2005). 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

below), in which we made reference to connections of various kinds that came to 
mind when we viewed the lessons. These might be, for example, to something we 
had witnessed in the past, something someone had said in a discussion, or in a 
lecture, or something we had read. An instance of a connection of the latter kind 
that we made with the episodes of Jason and Naomi (above) is the following. 

On more than one occasion, Bishop (e.g., 2001, p. 244) has recounted an 
anecdote about a class of 9- and 10-year-olds who were asked to give a fraction 
between Vi and %. One girl answered 2/3. When the teacher asked how she knows 
that it lies between the two given fractions she replied "Because 2 is between the 1 
and the 3, and on the bottom the 3 lies between the 2 and the 4". Bishop uses the 
anecdote to illustrate how a teacher's response is determined by the values that 
they espouse. Through the lens of our research question, we also note two things 
about the situation. First, the girl's answer is probably not one that the teacher had 
expected, even less one that s/he had hoped for. For me, it hints at the notion of the 
Farey Mean of two fractions, (sometimes called their 'mediant'), but this topic is 
not standard material for the school curriculum, in England at least. More to the 
point, it would be perfectly possible to have gained a first degree, or even a 
doctorate, in mathematics without having encountered it. Even to begin to entertain 
the girl's proposal would necessitate substantial deviation for the teachers' planned 
lesson script. The teacher could ignore or effectively dismiss the girl's proposal 
(saying, for example, "You couldn't be sure that way"). To take it seriously would 
probably involve probing her answer or instituting some investigation of the 
generality of her reasoning. It follows that responding to children's ideas (RCI) of 
this kind potentially entails recourse to mathematics content knowledge resources, 
either substantive or syntactic, or both (Schwab, 1978). 

This story (Bishops') is re-told here not because it comes from our data, but 
because it is an example of the kind of 'baggage' - our existing ways of thinking 
about the world in general and mathematics classrooms in particular - that shaped 
the 'reflexive' interpretation of our data. 

A Staging Post 

The inductive process described above generated an initial set of 1 8 codes which 
were reduced to 17 soon after. Next, each of us focused in detail on about five 
videotapes, and elaborated the Descriptive Synopsis into an Analytical Account of 
each lesson. In these accounts, significant moments and episodes were identified 
and coded, using the agreed set of codes and code-names, with appropriate 
justification and analysis concerning the role of the trainee's content knowledge in 
the identified passages, with links to relevant literature. Other members of the 
research team then re-visited and re-considered three or four of these codes in 
every analytical account. This intensive activity took place in the summer months, 
from June to August. In September, we presented some findings at a conference 
(Huckstep et al., 2002). Our presentation focused on three of the 18 codes: namely 
choice of examples, making connections and responding to children 's ideas. We 
illustrated our talk with clips from the videotapes. The presentation was well- 

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TIM ROWLAND 

received, and we envisaged making another five similar presentations before we 
had exhausted the codes. In the event, our thinking took a different turn, one we 
had not planned from the outset - though, arguably, we should have. 

THE EMERGENCE OF THE KNOWLEDGE QUARTET 

Perhaps as a consequence of feedback received at the conference presentation, 
around that time we began to realise the practical potential of what we had 'found' 
in our research study for our work, and for that of our colleagues, in teacher 
education. In the UK, a large part (typically a half to two-thirds) of the graduate 
initial training (PGCE) year is spent teaching in schools under the guidance of a 
school-based mentor. The proportion of time differs in other countries, but 
practicum placement is more-or-less universal. Placement lesson observation is 
normally followed by a review meeting between a school-based teacher-mentor 
and the prospective teacher. On occasion, a university-based tutor will participate 
in the observation and the review. Research shows that such meetings typically 
focus heavily on organisational features of the lesson, with little attention to 
mathematical aspects of mathematics lessons (Brown, McNamara, Jones, & 
Hanley, 1999; Strong & Baron, 2004). Our 'pure' research clearly offered a basis 
for us to develop an empirically-based conceptual framework for lesson reviews 
with a clearer focus on the mathematics content of the lesson and the role of the 
trainee teacher's mathematics SMK and PCK. Such a framework would need to 
capture a number of important ideas and factors about content knowledge within a 
small number of conceptual categories, with a set of easily-remembered labels for 
those categories. 

The identification of the 17 categories could be a stepping stone in the 
development of a framework for observing and reviewing mathematics teaching 
with prospective teachers, and potentially not only these novices. We (the 
Cambridge team) did not want a 17-point tick-list, like an annual car safety check, 
but a readily-understood scheme which would serve to frame an in-depth 
discussion between teacher and observer. Our codes were useful to the extent that 
we had a set of concepts and an associated vocabulary sufficient to identify and 
describe various ways in which mathematics content knowledge plays out in 
elementary mathematics teaching. In the autumn of 2003 we began to think about 
how we could group the codes into a smaller set of 'big ideas' for mathematics 
teaching. 

Eventually, we made a large paper label for each of the codes, spread these 
labels out on the floor, and began to separate them into sets. Each suggestion for 
putting two or more codes in the same subset had to be backed up with a reason of 
some sort. For example, someone put choice of examples and choice of 
representation together, saying that these were both ways that teachers use to make 
an abstract idea accessible when they are teaching it. In fact, this person said, these 
two codes are characteristic examples of the ways that teachers 'transform' their 
subject knowledge, as Shulman (1987) put it, in order to help others to learn it. 
Another grouping included the three codes decisions about sequencing; 



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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

anticipation of complexity, and recognition of conceptual appropriateness. The last 
two of these capture instances of planning, or of in-the-moment decision-making, 
that appeared to be informed by the teacher's awareness of the level of challenge 
and conceptual complexity entailed in the mathematical subject matter in hand. 
This is a well-documented topic in the research and professional literature, and 
much can be learned from experience too. Indeed, as remarked earlier, the original 
conception of PCK "includes an understanding of what makes the learning of 
specific topics easy or difficult ..." (Shulman, 1986, p. 9). These kinds of teacher 
knowledge contribute to decisions about the sequencing of instruction and student 
activity and the sequencing of exercises - considerations encompassed in the first 
of the three codes in this grouping. The crucial awareness and consensus, for us, 
was that these three codes contribute to students' sense of coherence within a 
lesson, and from lesson to lesson: that the shifts of focus and activity were by 
design, and not by chance. As we discussed and tried to capture this unifying 
quality with respect to these three codes, the words, 'coherence', 'cohesion' and 
'connection' came to mind. 

By an extended process of argument, debate and negotiation, we eventually 
agreed on grouping the 1 7 codes into four superordinate categories which, together, 
we later called the Knowledge Quartet. Each of the four categories is a unit, or 
dimension of the Knowledge Quartet. We have named the four dimensions: 
Foundation; Transformation; Connection; Contingency. These four units represent 
more comprehensive, higher-order concepts, in keeping with standard practice in 
grounded theory research (Strauss & Corbin, 1998, pp. 113-114), which involves 
first categorising the data ('open' coding), then connecting categories ('axial' 
coding') before finally proposing a core category (selective coding). "First analyse, 
then synthesise, and finally prioritise" (Dey, 1999, p. 98). This is certainly a way of 
describing what we had done in grouping the 17 codes into four categories, and 
conceiving the whole (the Quartet) as a tool for focused mathematics lesson 
observation. 

Not only the constituents, but also the names of these four Knowledge Quartet 
categories were in flux for more than a year. Indeed, notes of a research team 
meeting late in 2002 record five superordinate categories, with tentative names like 
Knowing (a precursor of Foundation) and Showing (a blend of elements of 
Transformation and Connection). A separate category had been created to group 
just two codes: adherence to textbooks (AT); concentration on procedures (CP). 
The thinking had been that these, in some way, marked instrumental attitudes and 
beliefs about the nature of mathematics and mathematics teaching. Eventually 
these fundamental affective considerations were subsumed by the Foundation 
dimension. Three months later, the components of the categories had more or less 
stabilised but their names were still not settled, being: theoretical background and 
beliefs; transformation, presentation and explanation; coherence; contingent action 
(Huckstep et al., 2003). 'Coherence' was originally chosen in preference to 
'Connection', a term which had gained popularity in the UK in connection with the 
Effective Teachers of Numeracy study (Askew et al., 1997). Our conceptualisation 
of the corresponding unit of the Knowledge Quartet was specific in its inclusion of 



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TIM ROWLAND 

codes related to the sequencing of instruction, and we wanted to distinguish our 
notion from the 'connectionist' beliefs orientation which headlines the Effective 
Teachers study. In the end, we decided that it would be sensible to 'go with the 
flow', and explain our nuanced use of 'connection' as and when necessary. 

We had reduced the names of all the dimensions to single words by the end of 
2003 (Rowland, Huckstep, & Thwaites, 2003). A further refinement of the codes 
was judged to be necessary in 2004 as we prepared the manuscript for a journal 
article (Rowland et al., 2005). As we listed the elements of each dimension, we 
could not avoid the feeling that a code that we had labelled 'making connections' 
(MC) now seemed rather limp as a component (admittedly one of four) of the 
dimension that we were calling Connection. Looking back at our data, we saw that 
the code 'making connections' had been used - in keeping with Askew et al. 
(1997) - to refer to the participant teachers' efforts (or missed opportunities) to 
make connections between concepts, or to show how two procedures might be 
related. Consequently, the rather tautologous MC was subdivided into two codes: 
making connections between concepts (MCC) and making connections between 
procedures (MCP). 1 should emphasise that there were instances of both of these 
new codes in our original data: it was not a case merely of imagining ways to make 
'making connections' more focused. Indeed, a clear instance of MCP can be seen 
in the case of Laura (Rowland et al., 2004). It could be argued that a code 'making 
connections between procedures and concepts' ought to be necessary, since it 
relates to the important notion of procept (Gray & Tall, 1994). However, there 
were no instances of these particular novice teachers making such a connection, or 
missing an opportunity to make one, in our data. In principle, the code could be 
added later - as we shall propose later in this chapter. 

In this way, our description and understanding of the Knowledge Quartet more- 
or-less stabilised a year after it was conceived. Since then, our experience of using 
the Knowledge Quartet has shown that it covers the various ways that the teacher's 
subject-specific knowledge comes into play in the classroom. It has also been 
shown to be a useful tool for thinking about mathematics teaching, and a 
framework for professional discussions of lesson planning and lesson review. 
Many moments or episodes within a lesson can be understood in terms of two or 
more of the four units; for example, a contingent response to a pupil's suggestion 
might helpfully connect with ideas considered earlier. Furthermore, the application 
of subject knowledge in the classroom always rests on foundational knowledge. 

Conceptualisation 

Since this chapter is envisaged, principally, as methodological, this is not the place 
to spell out the substantive conceptualisation of the four dimensions of the 
Knowledge Quartet at length. Such an account can be found in Rowland et al. 
(2005). However, a brief characterisation of each unit of the Knowledge Quartet is 
as follows. The first category ■, foundation, consists of trainees' knowledge, beliefs 
and understanding acquired 'in the academy', in preparation (intentionally or 
otherwise) for their role in the classroom. The key components of this theoretical 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

background are: knowledge and understanding of mathematics per se and 
knowledge of significant tracts of the literature on the teaching and learning of 
mathematics, together with beliefs concerning the nature of mathematical 
knowledge, the purposes of mathematics education, and the conditions under 
which pupils will best learn mathematics. 

The second category, transformation, concerns knowledge-in-action as 
demonstrated both in planning to teach and in the act of teaching itself. As 
Shulman indicates, the presentation of ideas to learners entails their re-presentation 
(our hyphen) in the form of analogies, illustrations, examples, explanations and 
demonstrations (Shulman, 1986, p. 9). Of particular importance is the trainees' 
choice and use of examples presented to pupils to assist their concept formation, 
language acquisition and to demonstrate procedures (Rowland, Thwaites, & 
Huckstep, 2003). 

The third category, connection, binds together certain choices and decisions that 
are made for the more or less discrete parts of mathematical content. In her 
discussion of "profound understanding of fundamental mathematics", Ma (1999, p. 
121) cites Duckworth's observation that intellectual 'depth' and 'breadth' "is a 
matter of making connections". Our conception of this coherence also includes the 
sequencing of material for instruction, and an awareness of the relative cognitive 
demands of different topics and tasks. 

Our fourth and final category, contingency, is witnessed in classroom events that 
are almost impossible to plan for. In commonplace language it is the ability to 
'think on one's feet'. As indicated earlier, in the comments on episodes with Jason, 
and Naomi, it includes the readiness to respond to children's ideas and a 
consequent preparedness, when appropriate, to deviate from an agenda set out 
when the lesson was prepared. The Knowledge Quartet framework is summarised 
in Table 1 . 

THEORETICAL SAMPLING: THE CASE OF MAIRE 

Since its initial development, the Knowledge Quartet has been put to the test as an 
instrument for mathematics lesson observation and analysis. This testing has taken 
a number of forms, including its application in varying degrees in three doctoral 
studies, including one longitudinal study of the knowledge, beliefs and practices of 
early career elementary teachers (Turner, 2007). Although our experience to date 
indicates that the Knowledge Quartet is comprehensive in its scope, we take the 
view that the details of its component codes, and the conceptualisation of each of 
its dimensions, are perpetually open to revision. This fallibilist position (Lakatos, 
1976) seems to us to be as appropriate for a theory of knowledge-in-mathematics- 
teaching as it is for mathematics itself. In grounded theory methodology, it is also 
inherent in the notion of 'theoretical sampling' (Glaser & Strauss, 1967), whereby 
the application of the theory exposes some shortcoming, and thereby lays it open to 
refinement, modification and possible improvement. 



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Table I. The Knowledge Quartet (adapted from Rowland et al., 2005) 

The Knowledge Quartet 

Prepositional knowledge and beliefs concerning: 

*the meanings and descriptions of relevant mathematical concepts, and of 
g relationships between them; 

'c *the multiple factors which research has shown to be significant in the 
"js teaching and learning of mathematics; 

g- *the ontological status of mathematics and the purposes of teaching it. 
w« Contributory codes: awareness of purpose; identifying errors; overt subject 

knowledge; theoretical underpinning of pedagogy; use of terminology; use of 

textbook; reliance on procedures. 

Knowledge-in-action as revealed in deliberation and choice in planning and 
teaching. The teacher's own meanings and descriptions are transformed and 
presented in ways designed to enable students to learn it. These ways include 

o the use of representations, analogies, illustrations, explanations and 

■§ demonstrations. The choice of examples made by the teacher is especially 

ij- visible: 

<■£, *for the optimal acquisition of mathematical concepts and procedures; 

§ *for confronting and resolving common misconceptions; 

£ *for the justification (by generic example) or refutation (by counter-example) 
of mathematical conjectures. 

Contributory codes: choice of representation; teacher demonstration; choice 
of examples. 

Knowledge-in-action as revealed in deliberation and choice in planning and 
teaching. Within a single lesson, or across a series of lessons, the teacher 
unifies the subject matter and draws out coherence with respect to: 

g 'connections between different meanings and descriptions of particular 

'H concepts or between alternative ways of representing concepts and carrying 
out procedures; 

§ *the relative complexity and cognitive demands of mathematical concepts and 

^ procedures, by attention to sequencing of the content. 

Contributory codes: making connections between procedures; making 
connections between concepts; anticipation of complexity; decisions about 
sequencing; recognition of conceptual appropriateness. 

Knowledge-in-/«/eraction as revealed by the ability of the teacher to 'think on 
^ her feet' and respond appropriately to the contributions made by her students 
c during a teaching episode. On occasion this can be seen in the teacher's 
60 willingness to deviate from her own agenda when to develop a student's 
'■s unanticipated contribution might be of special benefit to that pupil, or might 
.0 suggest a particularly fruitful avenue of enquiry. 



c 



O 



Contributory codes: responding to children 's ideas; use of opportunities; 
deviation from agenda; teacher realisation. 



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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

An illuminating instance of this incremental process is the case of Maire, one of 
the prospective teacher participants in the study by Dolores Corcoran, located in 
Ireland. Maire was observed teaching a lesson on whole-number division. The 
lesson was analysed at length, through the lens of the Knowledge Quartet, in 
Corcoran (2007) and also in Rowland (2007). The class was a mix of 3rd and 4th 
Class girls (age 9-10 years). Maire had written separate worksheets on division for 
each age group, both set in a fantasy Harry Potter 9 scenario. The first problem for 
each of the two groups was as follows: 

3rd Class: Ron has 18 Galleons 10 and a pack of cards costs 3 Galleons. How 
many packs can he buy? 

4th Class: Fred and George want to buy magic worms to put in everyone's 
bed. They had 44 Galleons, and each worm costs four Galleons. How many 
worms could they buy? [This question continues: if there are 30 beds, how 
many more worms would be needed?] 

One of the ways in which pupils make sense of mathematical operations, 
procedures or concepts is by appreciating the range of different situations to which 
they apply. The various problems and scenarios relevant to a particular operation 
can be grouped into a small number of categories, each with the same fundamental 
structure. In the case of division, there are at least two key problem structures (e.g. 
Vergnaud, 1983), variously called partition (or sharing) and quotition (or 
measurement, or grouping). In the partition version of 20 divided by 4, for 
example, we share 20 things among 4 people, and each receives the same quota of 
5 things; in the quotitive scenario, we distribute a pre-set quota of 4 things to 5 
people. 

In each of the two Harry Potter problems under discussion, the problem 
structure is quotition. They both begin with a certain supply of Galleons, and a 
fixed quota (3 Galleons, 4 Galleons), whereas the 'answer unit' is packs for one 
problem, worms for the other. However, in exploring how to resolve the problem, 
Maire drew on the language and concepts of partition. In the case of the 3rd class 
problem, for example, 

Maire had provided butter beans as manipulatives for the 3rd class, to represent 
the Galleons. One pupil, Rosin, read out the 'packs of cards' problem, while 
Megan volunteered to count out 18 butter beans. As Megan counted out the 18 
butter beans, Maire offered a few words of explanation about the "wizard money", 
then she asked: 

Maire: How many groups does she [Megan] need to break it into and 

can you tell me why? Hannah, what do you think? 

Hannah: Into three groups. 



9 The Harry Potter novels by J. K. Rowling are well-known in Ireland, and in most parts of the world for 
that matter. 

10 Galleons are the fictional currency in use at Hogwarts, which is Harry Potter's school. 



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TIM ROWLAND 

Maire: Into three groups. Well done, and why? You can read the 

question again if you want. 

As observed above, the problem structure here is quotition. However, Maire's 
query here to the number of groups and not to their size, points inappropriately to a 
partition structure, and this is picked up by Hannah. This is hardly surprising since, 
as Brown (1981, cited in Dickson et al., 1984) found, primary school children have 
a propensity to opt for partition in preference to quotition when asked to supply a 
problem for a 'bare' division sentence. Indeed, Ball (1990) found the same to be 
true of prospective teachers. Maire congratulates the child ("Well done") on her 
inappropriate suggestion. However, for some reason, Maire is then inspired to ask 
Hannah to explain ("and why?"). The interaction then takes a different direction. 

Hannah: Because there's three packs of cards. 

Maire: It's not that there's three packs of cards. But what is it about 

the cards? 

Hannah: It costs three galleons. 

It is interesting to note that the child's first justification ("there's three packs of 
cards") would be entirely appropriate for the partitive division model that she 
proposed earlier i.e., 18 Galleons, 3 packs of cards, therefore 6 Galleons for each 
pack. But, for reasons that we can only speculate, Maire is pulled up short at this 
point. She knows that there are not three packs of cards. Maire has inadvertently 
directed the pupils to the wrong division structure, she realises that this is so, and 
she resolves to find a way out: 

Maire: It costs three galleons. So if you share out your 3 galleons, you 

see how many packs of cards you're able to buy [...] Megan 
you're already ahead of us. You've got 18 and what are you 
doing? 

Maire is attempting to alter the direction of the discussion. Her use of the word 
'share' to point to quotition is perhaps unfortunate (and, indeed, 'grouping' earlier 
when she had partition in mind). The child who responds has not altered course: 

Child: Splitting them up into three groups ... 

Maire: Ahh ...? Into groups of three [she nods]. And how many 

groups do you have? 

Maire's response "Into groups of three" is a direct correction, and her language is 
now correctly aligned with quotition/grouping. The child replies: 

Child: Six. 

Maire: So how many packs of cards could Ron buy? 

Child: Six. 

Maire: He could buy six packs of cards. Can everybody follow that? 

What sentence would you write to explain what we just did? 

This ability to change course as a result of reflection in action was not observed in 
the lessons that were the data for our original study. Schon's (1983) term 
'reflective practitioner' conjures up the notion of teachers as professionals who 
learn from their own actions and those of others. Schon distinguished between two 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

kinds of reflection. Reflection on action refers to thinking back on our actions after 
the event, whereas reflection in action is a kind of monitoring and self-regulation 
of our actions as we perform them. We see an instance of reflection-in-action in 
this episode, and in what we would call a 'contingent moment'. Maire could not 
have prepared (in her planning) for what she did at that moment, but what she did 
say and do brought about a significant and pedagogically important shift in the 
discourse and the cognitive content of the lesson. This was possible because Maire 
seems to have experienced an insight of some kind, an 'aha' of a pedagogical kind. 
This insight of Maire's was significant in terms of our conceptualisation of the 
Contingency dimension of the Knowledge Quartet. This dimension was rooted, as 
it arose from the data in our original study, in the teacher's response to children's 
insights and misconceptions. In this instance we seem to have a moment where 
Maire herself suddenly realises that the problem, the child's suggestion, and her 
approval, simply do not 'stack up'. In analysing Maire's lesson through the lens of 
the Knowledge Quartet, we apply a theory derived from practice back to practice. 
Maire's moment of insight is an instance where theoretical sampling has found the 
current theory wanting, and caused it to be rethought and enhanced. Consequently, 
we have added an additional code to the three previously associated with 
Contingency. Provisionally, this new code is 'teacher realisation' (TR). In 
principle, just one significant instance of TR, and the expectation that others might 
subsequently be identified, would justify its inclusion in the conceptualisation of 
the theory. In fact, once the notion of TR was raised and discussed within the 
research group, one of the team was able to point to further instances in the many 
lessons observed in her longitudinal study (Turner, 2007). 

FINAL REMARKS 

From Observation to Evaluation 

In a paper presented to a meeting of teacher education researchers, Skott (2006) 
highlighted the changing nature of the relationship between theory and practice in 
teacher education. He observed that the theories brought to bear on the task of 
improving teaching increasingly derive from studies of teaching, and coined the 
term 'theoretical loop' to capture this dialectical relationship between theory and 
practice in teacher education. In the case of the research which has been the focus 
of this chapter, the Knowledge Quartet came about as the outcome of systematic 
analysis of mathematics teaching. Initially, we viewed it as a way of managing the 
complexity of describing the role of teachers' content knowledge in their teaching. 
In the spirit of Skott' s theoretical loop, we subsequently developed ways of using 
the Knowledge Quartet as a framework to facilitate analysis and discussion of 
mathematics teaching among prospective teachers, their mentors and teacher 
educators (see Rowland & Turner, 2006). 

The progression from observation of teaching to its description and analysis is 
clear, but, thus far, I have been less explicit about the evaluation of teaching. In the 
spirit of reflective practice, the most important evaluation must be that of the 



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TIM ROWLAND 

teacher him/herself. However, this self-evaluation is usefully provoked and assisted 
by a colleague or mentor. In a number of papers (e.g. Rowland et al., 2004; 
Rowland et al., 2004; Rowland & Turner, 2006) we have exemplified this 
provocation through the identification, using the Knowledge Quartet, of tightly- 
focused discussion points to be raised in a post-observation review. We have 
suggested that these points be framed in a relatively neutral way, such as "Could 
you tell me why you etc?" or "What were you thinking when etc?". It would be 
naive, however, or a kind of self-delusion, to suggest that the mentor, or teacher 
educator, makes no evaluation of what they observe. Indeed, the observer's 
evaluation is likely to be a key factor in the identification and prioritisation of the 
discussion points. In post-observation review, it is expected that the 'more 
knowledgeable other' will indicate what the novice did well, what they did not do 
and might have, and what they might have done differently. The Knowledge 
Quartet is a framework to organise such evaluative comments, and to identify ways 
of learning from them. 

Methodology 

In his China Lectures, Hans Freudenthal (1991) ranted against the (then) new breed 
of professional methodologists. His words capture my own experience of research 
'design' far better than I could express it myself: 

I don't remember when it happened but I do remember, as though it were 
yesterday, the bewilderment that struck me when I first heard that the training 
of future educationalists includes a course on "methodology". This is at any 
rate the custom in our country but, judging from the literature in general, this 
brain-washing policy is an international feature. Please imagine a student of 
mathematics, of physics, of- let me be cautious, as I am not sure how far this 
list extends - impregnated, in any other way than implicitly, with the 
methodology of the science that he sets out to study; in any other way than by 
having him act out the methodology that he has to learn! In no way do I 
object to a methodology as such - 1 have even stimulated the cultivation of it, 
but it should be the result of a posteriori reflecting on one's methods, rather 
than an a priori doctrine that has been imposed on the learner. (Freudenthal, 
1991, pp. 150-151) 

Many readers will have little sympathy for Freudenthal's suggestion that research 
methodology might be the outcome of reflecting on research action, arguing that 
educational research has reached new heights of scientific sophistication since 
Freudenthal composed this diatribe against "the pure methodologists, whose 
strength consists in knowing all about research and nothing about education" (pp. 
150-151). At the same time, Freudenthal's version of events agrees reasonably 
well with my own experience. Much of the account that 1 have given of the 
research processes that the SKIMA team followed in arriving at what we came to 
call the Knowledge Quartet has been possible with the benefit of hindsight, "the 
result of a posteriori reflecting on one's methods". I have done my best to be true 

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RESEARCHING TEACHERS' MATHEMATICS DISCIPLINARY KNOWLEDGE 

to our intellectual and practical experience, as documented and remembered, rather 
than to offer some idealised, even sanitised, version of events. 

Applications and Next Steps 

Research originally fuelled by curiosity about teacher knowledge and classroom 
practices led to the development of the Knowledge Quartet, a manageable 
framework within which to observe, analyse and discuss mathematics teaching 
from the perspective of teachers' mathematical content knowledge, both SMK and 
PCK. The framework is in use in teacher education programmes in Cambridge and 
elsewhere. Those who use it need to be acquainted with the details of its 
conceptualisation, because mere labels such as 'connection' may, for each 
individual, connote meanings other than those intended. Initial indications are that 
this development has been well-received by teacher-mentors, who appreciate the 
specific focus on mathematics content and pedagogy. They observe that it 
compares favourably with government guidance on mathematics lesson 
observation, which focuses on more generic issues such as "a crisp start, a well- 
planned middle and a rounded end. Time is used well. The teacher keeps up a 
suitable pace and spends very little time on class organisation, administration and 
control" (Department for Education and Employment, 2000, p. 11). 

It is all too easy for analysis of a lesson taught by a novice teacher to be (or to 
be perceived to be) gratuitously critical, and it is important to emphasise that the 
Knowledge Quartet is intended as a tool to support teacher development, with a 
sharp and structured focus on the impact of their SMK and PCK on teaching. The 
post-observation review meeting usefully focuses on a lesson fragment, and on 
only one or two dimensions of the Knowledge Quartet, to avoid overloading the 
trainee-teacher with action points. 

More recently, we have been analysing secondary mathematics lessons through 
the lens of the Knowledge Quartet. In a different development, colleagues working 
in English, science and modern foreign languages education found potential in the 
Knowledge Quartet for their own lesson observations and review meetings. What 
might the conceptualisations of the dimensions of the Knowledge Quartet look like 
in these and other disciplines? A more fundamental question would be: can a 
framework for knowledge-in-teaching developed in one subject discipline be 
legitimately adopted in another? 

ACKNOWLEDGEMENTS 

This research was supported by a grant from the University of Cambridge Faculty 
of Education Research Development Fund. Peter Huckstep, Anne Thwaites, Fay 
Turner and Jane Warwick were co-researchers in the project. 



295 



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Tim Rowland 
Faculty of Education 
University of Cambridge 
U.K. 



298 



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CHAPTERS 



JOHN MASON 



13. PCK AND BEYOND 



DEDICATED TO THE MEMORY OF LEONE BURTON (1936-2007) 

Having been invited to join Leone Burton in reflecting on the chapters in this 
volume of the handbook, I dedicate my remarks to her memory. She was 
tireless in promoting social justice, gender equity, and above all, enquiry at 
every level of engagement, by students, teachers, and teacher educators. I hope 
that my attempts are worthy of her vision. 



In this summary I look beyond the confines of PCK (pedagogic content knowledge) 
and its transformation into MJT (mathematics for teaching) and MiT (mathematics 
in teaching) in order to try to contact the essence of necessary and desirable 
preparation for effective mathematics teaching. I take an unreservedly eclectic and 
embracingly constructivist stance while at the same time being driven by an almost 
Aristotelian desire to probe the essence of what it is that informs and guides 
effective and efficient teaching involving mathematics at any level, aware that such 
essences say as much or more about my awareness as about teaching in general. 
At the same time I try to maintain a Heideggerian respect for 'being': in this case, 
being mathematical with and in front of learners. I suggest that there will be 
negative as well as positive consequences for trying to delineate and codify 
effective teaching acts, and make a plea for mathematics teaching to lead the way 
in promoting learning, teaching, and teaching teachers as essentially human 
activities which make integral use of the complexities of being human that cannot 
be set down in lists of attributes. 

SUMMARISING THE VOLUME 

I offer a brief summary of some of the points from the various chapters, as a 
foundation for suggesting directions of development of research and practice in 
professional development in mathematics education, and of PCK in particular. 

Anna Graeber and Dina Tirosh give a fine summary of Lee Shulman's 
articulation of the notion of pedagogic content knowledge (PCK) and related 
constructs. They also track some of the forces which gave rise to this notion, to 
which might be added a significant political component. As well as contributing to 
making more precise a research agenda by re-articulating and developing Dewey's 



P. Sullivan and T. Wood (eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching 

Development, 301-322. 

© 2008 Sense Publishers. All rights reserved. 



JOHN MASON 

notion of psychologising the subject matter (Dewey, 1902/1971, p. 22), Shulman 
can also be seen as justifying the existence of education departments in the face of 
criticism from other academics. Unfortunately his paper, and the many subsequent 
ones drawing on it for inspiration and support, have failed to quell the voices of 
critics who proclaim that knowing the subject is sufficient for teaching it. This is 
despite the fact that as long ago as Aristotle the term master (probably the origin of 
current 'masters' degrees and the use of 'master' to refer to teachers in some 
schools in the past) was used to distinguish between someone who had learned and 
someone who had gone beyond the mere learning of subject matter so as to be 
equipped to teach it. Although Shulman does not mention it explicitly, in order to 
psychologise the subject matter it is necessary to be up to date both with the subject 
matter and the psychology of current learners, so the assumption that ongoing 
study of and engagement in mathematical thinking is unnecessary for alert and 
effective teaching is just as short-sighted and impoverished as overlooking the 
difference between studying content and being informed about teaching. 

Paolo Boero and Elda Guala add to Shulman's categories the cultural analysis 
of content, which includes both awareness of how mathematics, being a cultural 
domain, can be arranged and presented according to epistemological, socio-cultural 
and psychological interests, and how it interacts with other cultural domains. Jill 
Adler and Danielle Huillet use discourses developed by Bernstein (1996, 2000) and 
Chevallard (1987, 1992) to provide a theoretical basis for re-formulating PCK and 
its related constructs in terms of an anthropological theory of individual and 
institutional relationships with practices and concepts, and contrasting types of 
discourse (hierarchical, strong and weak grammars). 

PCK and its extensions attempt to bridge a gap between knowing for oneself 
and supporting others in coming to know as well. This fits nicely alongside the 
Vygotsky articulation of development as converting 'acting in oneself (when 
triggered or cued) into 'acting for oneself (initiating an action oneself) (Valsiner, 
1988; van der Veer & Valsiner, 1991) which goes beyond Dewey's notion of 
making use of the current actions and concerns of children (Dewey 1913). What 
seems to be difficult, as the chapters in this volume show, is to put into words 
precisely what it is that guides and directs expert teacher behaviour, that is, how 
teachers come to act 'for themselves' rather than simply 'in themselves', expressed 
practically while remaining theoretically and philosophically well founded. 

Doug Clarke reminds us of the many slips between cup and lip concerning 
teaching. At one end of the spectrum spanned by the institution of education there 
is an ideal education imagined by policy makers. This is transformed into an 
intended or planned curriculum, and then by textbook authors into a presented 
curriculum. This in turn is implemented and enacted by teachers, which in turn is 
experienced by learners. Learners are then tested both by teachers and by external 
agencies (an achieved curriculum), the results often being mistaken for evidence 
about attainment of the ideal, the implemented or the experienced curriculum. 
Since there can be so much more that happens for learners than is revealed by what 
they are tested on, there is also a maturing or interiorised curriculum which 
develops over time, visible more often than not only through narrated memories, 



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because the more fully something is integrated into a person's functioning, the less 
visible it is likely to be, even when they are probed about the reasons for their 
choices. 

Tim Rowland concentrates on methodological issues arising when a team of 
researchers probes novice teachers' use of their mathematical understanding. Using 
video records of novice teachers' teaching, researchers attempt to locate some of 
the kinds of mathematical choices being made, to trace some of their origins, and 
to uncover major influences on those choices. There is an increasing interest 
internationally in this kind of research in order to validate assumptions about 
effective teacher education and professional development and to inform further 
research, viz. ICMI 15 th Study Conference (Ball & Even, 2005); Oberwolfach 
meeting (Ufer, 2007); ICMI Meeting (Ball & Grevholm, 2008) much of which is 
summarised by Graeber and Tirosh. 

Helen Foragasz and Gilah Leder review research concerning beliefs as a 
significant attribute of the psyche which mediates practices and modulates what is 
attended to. Beliefs about the nature of mathematics; about its purpose in society, 
in the curriculum, and in students' lives; about how it is done, and learned; about 
what supports and fosters mathematical thinking; about what students are capable 
of; about the person's own competencies, dispositions, self-efficacy and identity; 
and about the opinions of others (particularly peers) all appear to play a role in 
influencing classroom and homework activity. When alignments and 
misalignments between parents, teachers and learners are included, the whole 
becomes a highly complex domain of interactions. What is certain is that whatever 
the construct 'beliefs' is taken to include, they are dynamic and evolving, 
sometimes in relatively stable configurations, and sometimes highly unstable. 

Yeping Li, Yunpeng Ma and Jeonsgsuk Pang describe various practices in the 
preparation of prospective elementary teachers in China and Korea. This leads 
them to probe the nature of teacher PCK by comparing and contrasting both 
systems of teacher education and self-reports of sample participants in those 
systems. They highlight an ongoing divide between content courses and methods 
courses which is also a dominant feature of teacher education in North America 
and elsewhere, though not in England, Australia or New Zealand, among others 
(see Stacey this volume). They also reveal interesting self-reported weaknesses in 
teacher understanding of the published curriculum in both China and Korea, and in 
teacher knowledge about when and how topics are taught in earlier years. Their 
report distinguishes between teachers carrying out a mathematical calculation 
(comparing the result of dividing one fraction by two different fractions) and 
teachers articulating how they would explain to students by calling upon 
procedural or conceptual justification. They are led to challenge the assumption 
that most teachers in mainland China have a profound understanding of 
fundamental mathematics, at least as exposed in response to their probes. When 
combined with Rowland's chapter, fundamental problems arise in reliably 
researching the region between mathematical and didactical-pedagogical knowing. 

Shiqi Li, Rongjin Huang, and Hyunyong Shin describe practices in the 
preparation of secondary teachers in China and Korea. They note that in Confucian 



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JOHN MASON 

Cultures heavily influenced by Marxist elements of the Soviet Union the aim of 
education is "to transmit the most stable knowledge accumulated over the past 
thousands of years" to young generations. This is in line with the Chinese notion of 
teaching which is "to transmit, instruct, and disabuse". Recent political changes 
have brought about a transformation of education, with mathematics seen as 
contributing to economic progress as well as training the mind. Teacher education 
is dominated by the perceived need for a profound mathematical understanding, 
with pedagogy having a lower profile. In China, mathematics courses for teachers 
are being redesigned to integrate geometrical and algebraic thinking. In Korea, 
recent recognition of the value of integrating mathematics and pedagogy for 
prospective secondary teachers has been acknowledged. 

Anne Cockburn compares the ways in which four different teacher education 
programmes, one in a Middle Eastern context, two in the U.K. and one in a 
continental European context, set about providing opportunities for novice teachers 
to demonstrate and develop the various types of knowing and 'being' associated 
with PCK. A range of devices are used, from stimulus to explore and research 
beyond the institutionalised course material, self-audits, presentations to and with 
peers, essays, examinations and government tests. Despite the claim that 
'prospective teachers trade performance for grades' (Doyle, 1979), it seems that the 
assessment instruments did influence the perspectives if not the practices of some 
at least of the graduates of the programmes. Cockburn emphasises that the 
prospective teachers may sometimes exchange performance for grades, but that 
they are also driven by wanting to be successful as teachers. How prospective 
teachers can be suitably and reliably assessed for professionalism remains an on- 
going issue. 

In his analysis of the usefulness of the distinction between CK and PCK, Mike 
Askew notes that research evidence for a strong correlation between the two, 
certainly at primary level, is difficult to find. What people have mostly focused on 
is based on a deficit model to try to account for perceived deficits in learning. His 
research has led him to a different way of conceptualising the basis for effective 
pedagogic choices made by primary teachers, to do with sensibility - for rather 
than knowledge - of mathematics. He builds on the argument that not only is 
teaching a profession concerning care of and for learners, but also care of and for 
mathematics as well. Effective teachers have a relationship of curiosity and 
enquiry. Instead of 'death by a thousand bullet points' induced by lists of 'things 
teachers need to know', teachers could be best supported by re-awakening their 
curiosity and natural disposition to enquire, and to engage them in mathematical 
exploration. Put another way, awareness is not the same as the aggregation of a lot 
of facts and procedural fluencies, which in turn is redolent of the 18th century 
observation that 'a succession of experiences does not add up to an experience of 
that succession'. A different level or type of awareness is required (Mason, 1998). 

Kaye Stacey considers preparation of secondary teachers, noting that in 
common with the use of mathematics in other fields, what is required is a 
'productive interplay between what the teacher knows about mathematics [and] 
what the teacher knows about students and curriculum'. This includes task 



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analysis, example construction, student propensities and so on. She uses 
international data to compare and contrast different systems along many different 
dimensions. Four dimensions are distinguished: mathematical content and desirable 
qualities of this knowledge; experiencing mathematics in action, going back to 
Polya (1962); knowing about mathematics through its history, changing 
epistemologies and philosophies; and knowing how to learn mathematics through 
pedagogic strategies. Through a comprehensive analysis, Stacey reveals 
problematic aspects of each dimension in attempts at implementation, especially 
when there are teacher shortages. 

Ulla Runesson adopts Ference Marion's approach to the lived experience of 
learners, in which what is learned is the making of new distinctions by discerning 
variation, and becoming aware of new relationships and properties. So by 
clarifying precisely what it is that is to be learned, it is possible to provide learners 
with appropriate variation of significant features (dimensions of possible variation) 
within sufficient proximity in time and space so that the variation is likely to be 
detected. Runesson reports on comparison studies focusing on what is varied and in 
what ways, so that learners can detect variation and so assimilate the consequences. 
Her case studies provide evidence that where a feature is varied suitably, learners 
give evidence of assimilating that variation in their own behaviour subsequently. 

The emerging picture of mathematics in and for teaching is of an immensely 
complex domain involving the interaction of multiple factors. Mathematics itself as 
a disciplined mode of enquiry and as a collection of techniques and concepts, 
didactical tactics, pedagogic strategies, historical-socio-cultural and personal- 
psychological forces all contribute both core sensitivities and obstacles which have 
epistemological, affective and behavioural components. Institutionalisation, in the 
form of privileged discourses and relationships forms an ecological environment. 
As with many concepts in education taken up by mathematics educators, and as 
demonstrated in each of the chapters in this volume, PCK as a technical term does 
not draw a clear and definitive distinction. 

In the following sections I suggest that if the term PCK is used as a checklist of 
qualities, quantities and dimensions, it will only serve to obscure what is essential 
and central. It may even contribute to making the challenges of teacher education 
more difficult than they already are, by appearing to make things simpler, 
especially to policy makers. 

REFLECTION AND TRANSFORMATION 

Instead of seeing PCK or mathematics in and for teaching as a checklist, it can be 
used as a reminder, as a means of literally re-minding through bringing back to 
attention that knowing. Even knowing and doing mathematics does not in itself 
equip someone to teach effectively. Even knowing and doing mathematics and 
knowing about the pedagogic and historical-philosophical and socio-cultural 
influences is not enough to be an effective teacher. Just as learners need to be 
prompted to stand back from activity which arises from engaging in mathematical 
tasks if they are to become efficiently aware of the significance of that activity, so 



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JOHN MASON 

teachers similarly benefit in the effectiveness of their practices if they are prompted 
to draw back from the action of teaching and to reflect, contemplate, and analyse as 
part of preparing for the future. 

Drawing back, or reflecting, as it is often labelled, is itself a complex action. If 
it is to be at all effective, it has to be much more than revisiting the highlights, 
much more than regretting the low points, and much more than resolving to 'do 
better'. It has to include trying to get a bigger picture, trying to see the wood 
instead of the individual trees. For example, specific incidents can be turned into 
phenomena by linking them to other incidents from the past and to possible similar 
incidents in the future. This hind and fore sight can usefully be linked to actions 
that it would be desirable to have come to mind, whether something that worked 
recently, or an alternative to something that did not 'work'. Reflection is most 
effective when it turns into 'pro-flection', that is, imagining something similar 
happening in the future and acting in some desirable or preferred way. This is how 
change and development is fostered and sustained. Effective professional 
development in whatever sphere of activity involves recognition of specific actions 
which could be undertaken in the future, and it is enhanced when the fundamental 
power of imagination is used to prepare the way (Mason, 2002). Theories and 
principles are all very well, but remain as abstractions until teachers can imagine 
themselves putting them into action. 

Transformations 

Each transformation from the imagined curriculum to the interiorised curriculum 
involves people making choices of what to stress or fore-ground, and what 
consequently to back-ground or even ignore. Usually carried out as a top-down 
process, each stage involves developing more detail in order to exemplify 
articulated generalities until reaching the specificities of classroom practice and 
learner experience. Once the learner has been inveigled into activity, it is necessary 
to learn from that experience, a process which seems to require explicit action 
more often than not. In order to foster and promote learning from experience, it is 
necessary that teachers are sensitive to, or aware of, what it is they are trying to 
achieve, the 'object(s) of learning'. This is in contrast to, for example, teachers 
who have been heard to say "the author's job is to teach; my job is to make it 
enjoyable for students". 

Each stage of the transformation of curriculum also involves a problematic 
transformation not unlike the didactic transposition identified by Yves Chevallard 
(1985) in which expert awareness is transposed into instruction in behaviour 
(specifications of what to do). Nowhere is this more obvious than when teachers 
have enjoyed an experience of working on mathematics for themselves with 
colleagues. Because they want to share that experience with their learners, they 
construct worksheets which guide and direct learners through the 'exploration'. 
This is where a richly conceived PCK could come into play, but where a checklist 
perspective transposes the experience into something altogether different. For 
some reason it is difficult for teachers in these circumstances to bring themselves to 

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relax constraints and trust their learners to get on with mathematical exploration 
using their natural sense-making powers; instead they feel a need to control and 
direct. In other words, it is easy to think that what you did is what others need to do 
in order to have the same experience, even if you also acknowledge that others 
cannot have 'the same experience'. And so worksheets are devised which drive 
learners through actions which previously arose spontaneously for the teacher. 
Sometimes, despite the transposition, learners get a taste of mathematical thinking 
and mathematics in the making, but often they do not. 

The force of a collection of articulated requirements, such as in a curriculum, is 
to direct attention to checklists and away from being, from awareness. Instead of 
allowing themselves to 'be mathematical with and in front of their learners' 
teachers feel forced into controlling learner attention through short-term goals 
involving direct instruction and rehearsal of procedures. 

SENSITIVITY AND AWARENESSES 

My aim in this section is to try to probe beneath and beyond the PCK as a 
formulation of knowledge and beliefs, by adopting the language of sensitivities and 
awarenesses. I use the term 'awareness' in the sense of Caleb Gattegno (1987) to 
refer both to conscious and to unconscious actions integrated into psycho-somatic 
functioning. The section begins with an attempt to put this discourse in context, 
followed by some examples of what is being approached, and then some 
development of theoretical aspects arising from the other chapters in this volume. 

Acknowledging Others 

Many authors have tried to articulate the essence or core of topics, and the 
difficulty in doing so is reflected in the multitude of discourses which then arise. 
Different labels signal different foci of attention, stressing or foregrounding 
different aspects without denying others which are backgrounded. 

Several authors have made use of metaphors involving a journey involving 
continuity, movement and terrain, in trying to describe how teachers might 
envisage their learners' growth of mathematical experience and appreciation. For 
example, Marty Simon (1995) used the term 'hypothetical learning trajectories' as 
a metaphor for how teachers imagine students developing as they encounter and 
move through a mathematical domain. James Greeno (1991) referred to learning 
number as more like getting to know a landscape than a linear journey, and 
Catharine Fosnot and Maarten Dolk (2002) refer to 'landscapes of learning' in 
preference to learning trajectories (see Clements, 2002 for a review of various 
ways that the construct has been interpreted). Marian Small (2004) refers to 
'developmental continua' as a version of 'learning trajectories' which are used both 
to describe the process of development of mathematical ideas, and as hypothetical 
constructs to prompt teachers to plan sequences of lessons rather than working on a 
lesson by lesson basis. Using this metaphor, teachers need to be familiar with the 
landscape of each mathematical topic, being able, through listening to and 

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JOHN MASON 

watching learners in action, to recognise what aspects of the landscape might 
benefit from more attention. 

It is not completely clear that learning can be either continuous, or a smooth 
ride. The 'staircase' metaphor of carefully planned steps whereby the learner 
ascends the mountain of knowledge surely belongs to past eras rather than the 
present. Even so, as a metaphor it is preserved in the images evoked in the 
language of knowledge being 'accumulated' or 'acquired', and in the specification 
of levels through which learners are supposed to progress. Rather, at various points 
there are shifts of perspective and attention to be made, somewhat akin to phase 
transition in physics: increase in energy usually results in rise in temperature but 
there are phase transitions where no increase takes place while the 'state' changes. 
Barbara Clarke (this handbook) uses the term 'growth points' to refer to times 
when learners are on the edge of making a shift in mathematical perspective or 
appreciation. Marty Simon (2006) refers to 'key developmental understanding' to 
refer to the underlying core awarenesses around which competence, fluency, 
facility, understanding and appreciation coagulate. He and his colleagues (Simon et 
al., 2004; Simon & Tzur, 2004) challenge the simplistic version of pedagogical 
sequencing in which learners are 'taught what they do not yet know', by pointing 
out that what is required is that learners become aware of the relationship between 
their actions (prompted by tasks) and the effects of those actions, so as to 
accommodate new concepts through modification of actions in order to meet fresh 
challenges. 

In seeking to articulate the essence of a mathematical topic, Gattegno (1987) 
based his science of education on a complex notion of awareness (which need not 
be conscious). John Mason elaborated this into a three-fold framework based on 
the classical structure of the psyche (Mason, Volume 4) which was manifested in 
Griffin and Gates (1989) and re-articulated in Mason and Johnston-Wilder, 
2004/2006) as 'the structure of a topic'. This was designed to augment and extend 
the notion of 'concept images' (Tall & Vinner, 1981). It focuses specifically on 
features of individual topics around which the language of technical terms, 
techniques and their associated incantations, generating problems and contexts in 
which the topic appears, concept images, examples and obstacles all cluster. In this 
perspective, teachers are engaged in an ongoing refinement of their articulation of 
the key elements of each and every topic they teach, though it is not often that 
systematic records are kept so as to inform preparation and planning in the future. 
This is one place where mathematics education seen as a disciplined mode of 
enquiry could support the development of prospective and serving teachers by 
supporting them in 'collecting data' in a structured format. 

Dvora Peretz (2006) usefully distinguishes between latitudinal understanding 
which involves both contextualised and de-contextualised familiarity, and depth of 
understanding, which is commonly described in terms of levels. She also points out 
that understanding can be described as an ongoing process, and as an act, but she 
urges maintaining complexity rather than being drawn into ever more distinctions. 
Brent Davis and Elaine Simmt (2006) attempt to maintain the complexity of 
teaching mathematics, but nevertheless distinguish four embedded and recursively 



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PCK AND BEYOND 

similar dynamical aspects: mathematical objects, curriculum structures, classroom 
collectivity and subjective understanding. 

Liping Ma (1999) makes use of a simpler construct, namely 'knowledge 
packages': 'given a topic, a teacher tends to see other topics related to its learning' 
and such topics comprise the knowledge package for the topic to be taught 
constituting profound understanding of fundamental mathematics (Ma, 1999, p. 
118). Using this articulation, teachers need a profound understanding of 
fundamental mathematical concepts in order to make informed choices when 
planning and when interacting with learners. Here the emphasis appears to be on 
the mathematics but it includes knowledge of both didactic tactics and pedagogical 
strategies. 

Doug Clarke also draws attention to the notion of 'the big ideas' in various 
topics (Papert, 1980). These are the core or main ideas around which the topic is 
clustered. A related notion is that of 'mathematical habits of mind' (Cuoco, 
Goldenberg, & Mark, 1996) which meshes nicely with the five strands of 
understanding described by Kilpatrick et al. (2001). What these contribute to PCK 
is like an all embracing and sustaining 'field' of influence which can inform 
teacher choices both when planning and when interacting. 

In an attempt to delineate various aspects so as to be able both to inform policy 
making and to enable more focused research, PCK has been approached as a bridge 
between mathematics and teaching by considering common content knowledge 
(CCK), specialised content knowledge (SCK), knowledge of content and students 
(KCS) and knowledge of content and teaching, with the addition of knowledge of 
the mathematics horizon and knowledge of the curriculum (Ball & Bass, 2006); 
Hill, Rowan, Ball, & Bass 2005), as in the diagram: 



< \'A ••Cr^r 




Figure 1. Taken from Bass et al. (2007). 

Each technical term discerns and adds to the overall picture a subtle aspect 
which is marginalised or obscured by others. When PCK is perceived as 
dominantly psychological, it leads to a metaphor of teachers accumulating 
sufficient knowledge to have useful actions 'come to mind' when planning for and 
when running sessions with learners. When PCK is perceived as dominantly social, 
it leads to a metaphor of distributed cognition in which teachers draw upon 
knowledge distributed in the historical-cultural-social and institutional practices, in 
the texts, work-cards, apparatus and other materials available, and in the physical 



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JOHN MASON 

and affective milieu co-constructed with the learners. A more encompassing 
articulation may be more useful (Ruthven 2007). What comes to mind as possible 
actions is triggered by and resonated with a myriad of external factors in relation to 
personal sensitivities and awarenesses. These forces are not simply additive, but 
have complex interactions (Mason, Volume 4). 

Each technical term introduced into the literature promotes the discernment of a 
subtle aspect which may be marginalised or obscured by other terms, and so adds 
to an overall picture of increasing complexity. It should not be surprising that it is 
difficult to pin down the essence and scope of a topic, because what is sought lies 
somewhere on or over the border between the explicit and conscious and the 
implicit, tacit and unconscious. It is precisely in this border territory that creativity 
is possible, through metaphoric resonances and idiosyncratic metonymic 
associations. If every topic could be completely articulated and captured, learning 
and living would be entirely mechanical, merely a matter of checklists. It is the 
presence in human beings with will and intentions, factored through complex 
psyche (see Mason, Volume 4) which makes education so important as well as so 
difficult. 

Examples 

Mathematics education, in common with mathematics, makes most progress when 
paradigmatic examples are provided through which it is possible to re-generalise 
for oneself. Here then are some examples which could be thought of as knowledge 
and beliefs, as awarenesses and sensitivities, and as assumptions and principles. In 
the sense of Boero and Guala, these arise from cultural analysis of content. 

In respect of counting on: awareness that a counting word is also the 
cardinality of a collection; sensitivity to the proceptual nature of this 
awareness (Gray & Tall, 1994). 

In respect of linear equations: equations are a statement of a constraint on 
the coordinates of sets of points; sensitivity to the need for multiple 
exposures to the dual presentation of equations as symbolic objects and as 
graphical objects. 

In respect of algebraic expressions: expressions can be seen as a rule or 
formula for calculating an answer, as the answer, as an expression of 
generality, and as an expression involving as-yet-unknown quantities; 
sensitivity to the need for continued exposure to these multiple ways of 
perceiving expressions, and to the value of learners expressing for 
themselves. 

In respect of decimals: each 'decimal place' has an associated value, so 
decimal notation is a short-form for a sum; sensitivity to the obstacle of 
inappropriate language use within and without the classroom (naming 
0.12 simply as "oh point twelve"). 



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PCK AND BEYOND 

In respect of multiplication: it arises from at least ten different types of 
actions (Davis & Simmt, 2006; see also Vergnaud, 1983). 

In respect of mathematics generally: what learners say and do is rarely 
wilful and usually principled; sensitivity to 'buggy algorithms' (Brown & 
van Lehn, 1980; van Lehn, 1989). 

These examples are necessarily summary and selective. It is the role of 
mathematics educators not just to elaborate and communicate expanded versions of 
these, but to work with teachers on how they can educate their awareness of these 
sorts of aspects of mathematical topics. 

One common feature of these examples is that they do not involve technique, 
but rather are aimed at underlying awareness which may be manifested as 
'theorems-in-action' (Vergnaud, 1981; 1997), may be succinctly articulated, or 
may be on some spectrum in between. But they can still be interpreted from several 
different stances. For example, a predominant stance currently sees teaching as 
about procedures and concepts which are believed to be learned by learners 
generalising from examples (worked examples of procedures, examples of 
concepts). One implication is that teachers need to have access to a range of 
examples and be aware of different obstacles to trying to learn from examples (for 
example, using variation theory, Runesson, this volume). They also need to be 
aware of successive processes of generalisation that need to take place for learners, 
and ways to provoke these. A different stance could be that learners actually learn 
by construing phenomena, and that practice to mastery is efficiently achieved by 
promoting explorations in which learners naturally construct examples and as a by- 
product of specialising for themselves, use procedures they need to rehearse, in 
order to re-generalise for themselves. Another stance could be based on shifts in 
epistemology and in the structure of attention required in order to appreciate the 
mathematical topic. 

Although it is important for teachers to have a sense of what is available to be 
experienced and learned when they design, select, and present mathematical tasks, 
it is vital that they are aware that experience alone does not ensure learning, that 
learning is not like climbing a staircase step by step, as elaborated earlier. 
Furthermore, mathematics is most efficiently and effectively learned when students 
are provoked into using and developing their own powers of sense-making rather 
than having authors and teachers trying to do this for them. If teachers are aware 
that students often go through periods of mathematical 'babble' (Ainley, 1999; 
Malara, 2003; Berger, 2004) as they struggle to articulate and crystallise their 
experience, then they are more likely to arrange to 'keep the students immersed in 
the task long enough' (James & Mason, 1982) to enable them to reach some 
reasonably coherent articulation from which they can reconstruct the concepts and 
techniques when needed in the future. 



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JOHN MASON 

Seeking an Essential Core of Mathematical Topics 

In order to probe more deeply into the essential core of a mathematical topic, it is 
very tempting indeed to provide labels, to make distinctions and to add aspects. 
While aiding precision, it also contributes to the growth and profusion of 
discourses. An example of an often neglected aspect is raised by Doug Clarke who 
emphasises the need for realistic expectations of learners, both as individuals and 
as a group. Like many other awarenesses or sensitivities, this is a delicate matter. 
On the one hand there is no point in embarking on something that is going to 
alienate several or all of the group. On the other hand, the greatest obstacle to 
student learning is expectations, as manifested by teachers, by themselves and 
peers, and by their parents. If "didn't" (as in "I didn't get the answer") or 'won't' 
(as in "they won't be able to ...") is allowed to turn into "can't", it can be a long 
road back (Dweck, 2000). If students are not challenged to reach beyond what they 
can do comfortably, they will not undertake the modification of familiar actions 
which is necessary in order to develop further. As Robert Browning put it in his 
poem Andrea del Sarto (L 91), "Ah, but a man's reach should exceed his grasp, or 
what's a heaven for?" Where teachers' expectations are challenged and extended, 
possibilities both for teaching and for learning are opened up. As Clarke reports, 
expectations can be higher at the end of a period of collaborative professional 
development than before. 

Imagined actions influence articulation which reflects perception and orientation 
which privilege imaginable actions. For example, trying to 'ensure' or 'guarantee' 
learning is likely to draw upon a mechanistic metaphor involving goals and goal- 
seeking behaviour with an underlying assumption about rationality, reason, and 
consistency which is hard to credit to most human beings most of the time. This is 
compatible with checklists, levels, and staging points, and is nicely compatible 
with centralised control of teaching. Unfortunately there is a chasm between 
mechanistic delivery of curricula, and the lived experience of classrooms. The flip 
side is treating students and teachers as creative and trustworthy agents who will 
respond to opportunities to grow and develop, to meet challenges and to 'think 
outside the box' if they are respected and trusted. But the balance between these is 
delicate, and unstable: as soon as trust is abused, it is temptingly natural to impose 
control, and so to begin a slippery descent into more and more constraints; once 
control is established and power exercised, it is very hard to let go of, because 
letting go seems to lead to unstable conditions and so controls are re-imposed. The 
metaphor of a ball perched on an upturned salad bowl is apposite: any slight 
disturbance and the ball falls off, unable to return to its previous stability. 

Approaching essence from the point of view of the mathematical topic, each 
mathematical topic (at school and early university) arises because a class of 
problems has been resolved, and the methods used are deemed teachable. Thus to 
bring the core sensitivities to mind is to reconstruct a route from 

a collection of problems which prompted the original search for a 
solution; 



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through being aware of technical language especially where it uses 
otherwise familiar words but in possibly unfamiliar ways; 

through concept images (Tall & Vinner, 1981), examples and ways of 
thinking associated with technical terms introduced; 

through techniques and methods devised to resolve the class of problems; 

through being aware of obstacles encountered by students, both 
epistemological obstacles requiring a shift of perspective and attention, 
and pedagogical obstacles arising from previous and current teaching; 

to a sense of various other contexts in which the topic has arisen. 

Justifications for and elaboration of these way-markers can be found in (Griffin & 
Gates, 1988; see also Mason & Johnston-Wilder, 2004/2006, 2005). Each 
contributes an aspect of the topic related to a prominent part of the human psyche 
(see Mason, Volume 4). 

Approaching essence from the point of view of mathematical thinking, each 
task, interaction, and aspect of a topic can be related to the ways in which 
mathematical heuristics (Polya, 1962; Schoenfeld, 1985), habits of mind (Cuoco, 
1996), mathematical thinking (Mason et al., 1982) and mathematical themes 
(Mason & Johnston- Wilder, 2004) are brought to the surface and made use of by 
learners, not just by the teacher and text-author. 

Approaching essence from the point of view of didactic tactics, the choice of 
tasks, modes of (re)presentation, and metaphor-analogies draws upon experience of 
teaching and of learning the particular topic, the particular concepts and the 
particular techniques (Rowland this volume). 

Approaching essence from the point of view of mathematical pedagogy, various 
frameworks highlight useful distinctions which can be used to structure 
interactions with learners so as to enrich encounters with mathematical concepts 
and techniques as fully as possible (Mason & Johnston- Wilder, 2004/2006). 

Approaching essence from the point of view of the lived experience of learners, 
attention is directed to what is being varied and in what ways, and how the 
consequent shifts of attention can be supported so as to be accommodated and 
integrated into learner functioning (Runesson, this volume). Attention is also 
directed to how learners' powers are being evoked and provoked so as to amplify 
their disposition to engage with mathematics productively. 

Approaching essence from a historical-socio-cultural point of view, each topic is 
a manifestation of forces, investments and concerns (Boere & Guala, this volume). 

Essence, then, depends on what is stressed or privileged. Essence is not an 
atomic 'thing', a pearl to be extracted from the shell of mathematical topics. Rather 
it is an amorphous conglomeration of all of the above approaches. The 
'knowledge' and 'beliefs' referred to in the literature are, in my view at least, 
impoverished generalisations which, like the oyster shell, look innocuous and 
bland, whereas inside there is multi-coloured mother-of-pearl. The effects of 
basing behaviour on privileging mathematics, as in these approaches, which value 
and relate to the diversity of idiosyncratic variations of individuals, are in contrast 

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JOHN MASON 

with effects of privileging cultural and social organisation and interaction. Of 
course effective behaviour will make use of the power of both collaborative and 
competitive interaction, but organisation of interaction follows and arises from the 
mathematical core, whereas mathematics often fails to arise or follow from 
organised social interaction which is not also centred in mathematics. 

MECHANISM, ACTION AND CHOICE 

One of the avowed purposes of trying to detail teacher knowledge and beliefs is to 
inform the search for mechanisms whereby teachers manage to influence learners. 
Teaching consists of acts in which actions involving learners are initiated. If 
control of those actions is retained by the teacher, there is less opportunity for those 
actions to involve learners sufficiently to produce any lasting transformation in 
their perspective, cognition, affect or future behaviour. The many-hued 
'constructivist' discourse is centred on transformation of the individual and of the 
group, but if the teacher is unaware of the kinds of transformation which are 
possible, they may not even pursue or sustain actions, much less notice 
opportunities to initiate important actions. Thus Vygotskian discourse stresses the 
difference between scientific and natural knowledge, the former requiring specific 
teaching. Underpinning this 'knowledge' are shifts in the structure of attention and 
shifts in epistemological preferences. For example, shifting from basing validation 
of mathematics in experiences in the material world, to basing it on mathematical 
structural reasoning is non-trivial for many, especially when the material world is 
over-stressed for so long before the shift is initiated. These sorts of shifts, located 
through 'cultural analyses of content' (Boero & Guala, this volume) are likely to be 
central for the aim of teacher education and professional development. 

It may easily be that the very notion of 'mechanism' with its associated sense of 
'cause-and-effect' is completely inappropriate in the context of education, where 
the forces acting are products of human psyche, including will, attention, cognition, 
affect and behaviour, unlike machines. The metaphor of 'forces' which underpins 
much of the discourse in this area is itself open to question. Are social forces to be 
seen like forces in physics, adding up to some resultant force? If so, then what is 
needed is a delineation of the strength and direction of those forces so that the 
resultant can be predicted. But perhaps the 'forces' are more like chemical 
reagents. These actually transform and combine with each other to form new 
reagents. The combination is no simple addition but something much more 
complex; the final whole is not the sum of its original parts. Other metaphors are 
also possible, such as an environmental-biological one in which change is endemic 
and responses evolve over time. 

A mechanistic view is amplified by and a product of an approach to assessment 
of teacher expertise in terms of knowledge (as manifested in essays and 
examinations). Anne Cockburn quotes Doyle (1979) as describing what happens 
when assessment is allowed to dominate instruction, as learners "exchange 
performance for grades". In other words, the learners' task is to work out the 
institutionalised responses to institutional probes (in the discourse of Chevallard 



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PCK AND BEYOND 

from Adler & Huillet, this volume). Not only are learners impelled to develop 
facility in procedures, but novice teachers are impelled to gargle the discourse in 
the form of theoretical constructs privileged by their educators. 

A mechanistic view supports training teachers in behaviours to be manifested in 
front of children; a holistic view supports fostering in teachers awareness of their 
own mental and physical states so that they are empowered to choose to be 
mathematical both with and in front of their learners. A mechanistic view presumes 
a degree of rationality and uniformity that simply does not match lived experience. 
Choices are rarely made rationally, especially when encountered in the flow of 
teaching. The very notion of 'choice' is complicated. 

Choice Making 

How and when are choices made? A rationalist position is that choices are made in 
every moment by the T that dominates discourse. A phenomenological position 
observes that what look like choices or potential choices are either the working out 
of habits, that is, of choices made long in the past, or reactions triggered by 
associations and assumptions, most of which function below the level of 
consciousness. 

Teachers are constantly evaluating the situation as they perceive it, both in 
planning and in engaging with learners. As anyone who has observed a lesson 
being taught knows only too well, there are numerous moments when the observer 
notices opportunities that the teacher does not exploit. This leads to the assumption 
that the teacher is making choices moment by moment. Attempts to probe those 
moments more closely led Kathleen McNair (1978a, 1978b) to use what came to 
be called 'stimulated recall' by showing teachers a video of a lesson soon after the 
lesson, and inviting them to comment on what strikes them, and to try to elaborate 
on what they were thinking at certain moments of interest to the researcher. This 
technique continues in use in many projects currently both with teachers and with 
learners. 

In responding to these stimulated recall probes, teachers call upon various 
theories, abstractions, principles and discourses. For example, "they needed to 
spend more time manipulating" could be taken as a reference to three modes of 
representation (Bruner, 1966) and the principle that sufficient experience of 
physically manipulating objects is required before learners begin to internalise the 
action so that they can imagine it rather than doing it, before moving to symbols to 
(re)present that action. Taken at face value, the terms used and the principles 
invoked provide a rationale for their actions, and so indicate the basis for their 
choices. 

However the situation is rather more complicated. As responsible people, it is 
considered culturally essential to be able to respond (literally, to justify) with 
explanations of our actions. The assumption that it is necessary to be able to justify 
and explain actions by reference to principles or theories presupposes that the 
explanations correspond to what actually happens. However, human beings are 
'narrative animals' (Bruner, 1996): consciousness is an ongoing story told by 

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JOHN MASON 

individuals, drawing on narrative practices of communities in which they 
participate and are immersed. But the purpose of narratives is as much for self- 
calming and preservation of agency and identity (reaffirmation of the T) as 
anything else (Ouspensky, 1950). Thus it is felt to be essential to construct a story, 
especially when probed, yet that story may have little to do with the actual lived 
experience. Furthermore, continued pressure to tell stories to account for actions 
may actually lend credence to the mistaken assumption that such stories are 
actually descriptive. 

Part of the narrative arising from being 'responsible' is that behavioural 
practices are initiated, sustained and completed by some guiding principles. If only 
the principles can be located, together with the mechanism(s) whereby these 
determine behaviour, professional development and behaviour modification would 
be straightforward. Close inspection of experience suggests that many of our 
actions are in fact habitual and automatic, triggered by metonymic associations and 
metaphoric resonances. Indeed, it has been proposed that consciousness itself as a 
generator of behaviour is an illusion (Norretranders, 1998). Thus the search for 
mechanism only makes sense for situations in which people are acting 
mechanically, out of habit. Put another way, most of the time we are operating on 
automatic pilot, guided by pre-established patterns. In these circumstances, 
Skinner's stimulus-response theories and practices are both apposite and effective 
for making predictions. Of course it is necessary to function on automatic much of 
the time, as it is impossible to consider carefully multiple options and to make a 
reasoned choice at every moment. However, when will is factored in, acting as it 
does through dispositions with affective, enactive and cognitive dimensions, the 
metaphor of mechanism (presumably going back to Descartes' entrancement with 
the cuckoo clock), becomes at best suspect and at worst inappropriate for 
predicting the impulses acting when fresh choices are made, when a lesson 
switches from automatic working out of established patterns, to engagement with 
fundamental ideas, with ways of thinking and acting, and with core mathematical 
awarenesses. 

An alternative to being 'responsible' is to transform it into 'being responsible', 
with emphasis on the 'being'. Boero and Guala move in this direction by stressing 
mathematics as a cultural domain which is entered and absorbed, like any culture, 
through participation. Not only are institutional practices acquired, whether 
through peripheral participation or direct instruction and training, but that extra 
human element that is so hard to articulate has a chance to emerge, manifested in 
the 'being' of the teachers. The more that educators can 'be mathematical ly- 
pedagogically-didactically aware, with and in front of novice teachers' the more 
likely it is that those novices will develop in wisdom as well as in knowledge, in 
'being' as well as in expertise. To achieve this, it will be to their advantage to 
engage in 'cultural analysis of content'. 

For these reasons, trying to trap choice-making, and to establish the grounds for 
those choices is exceedingly difficult. As a first approximation only, the theories, 
principles and discourses drawn upon in narratives in order to explain actions after 
the fact can be taken as at best an indicator of what comes to mind when a response 



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is demanded, even if these may not be driving behaviour in the first place. Perhaps 
this goes some way to account for differences between the responses to probes 
recorded by Yeping Li and colleagues on the one hand, and Ma (1999) on the 
other. As consumers of research, educators also have theories and values which 
dispose them to accept or challenge the findings of colleagues. 

An alternative, phenomenological approach to choice-making is to stimulate 
teachers to work on developing their mathematical being: how they respond to 
being stuck on a problem; how they respond to incomplete, inappropriate, or 
incoherent conjectures and guesses from learners; how they work with others to get 
them to conjecture and convince, to stimulate them to use their natural powers of 
mathematical sense making; how they are alive to possible avenues of enquiry 
through extending, varying and generalising; and how they connect mathematical 
topics together through the use of heuristics and mathematical themes. Then, as 
complex beings, they can manifest that being with and in front of learners, 
sensitive to the ways in which learners can be stimulated to make use of their own 
powers both individually and collectively. 

Knowing 

Another way of casting these observations about the mechanisms of choice is in 
terms of knowing. People can 'know-that' the pedagogic and didactic literature 
concerning a mathematical topic has found certain results through research; they 
can 'know-how' to act in a theoretical sort of way; they can 'know-why' certain 
tasks might be more effective than others, or know-why certain ways of organising 
the classroom or interacting with learners might be more effective than others. All 
of these constitute 'knowing-about' and can be tested by essay writing or multiple- 
guess probes of one sort or another. But as all teacher education programmes 
demonstrate through the importance of practicums (school-based practice), these 
are at best preliminaries to 'knowing-to' act in the moment, to having a suitable 
idea come to mind when it might be appropriate, to being mathematically sensitive 
as Mike Askew puts it. 

One of the reasons that it is easier for someone observing a lesson to pick out 
possible 'choice-moments' than for the teacher is that observers are not under 
pressure to act. They do not have the tunnel vision which is integral to engaging in 
action, especially when in the role of initiating action. For example, it is easy to be 
aware of your breathing, of other people's postures, of reasons for acting in certain 
ways, of possible ways to proceed etc. when sitting watching or listening. It is 
much more difficult once you start talking, because the act of talking absorbs 
attention. 

The aim of teacher education is to prepare the ground so that novice teachers 
will find themselves increasingly sensitised to noticing possibilities for initiating, 
sustaining or completing actions which they might not previously have had come 
to mind. This brings us back to the basis for those choices: mathematics, 
mathematical didactics (tactics pertinent to a particular topic) and mathematical 



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JOHN MASON 

pedagogy (strategies pertinent to many different topics and to fostering and 
sustaining mathematical thinking generally). 

In order to support and sustain growth of mathematical being it is necessary that 
teachers develop as autonomous actors with sensitivity both to their own 
awarenesses and to the states of learners, mediated by such curricular forces as 
actually impinge on their practice. Guided by values and dispositions, it is vital to 
be neither so solipsistic as to treat everyone like themselves, nor so altruistic that 
they lose contact with their own appreciation of the structural necessities of 
mathematics. 

Where classroom organisation and learner interaction has been stressed in 
preference to mathematics, attention is likely to remain there. When asked why 
they acted in some way at some moment, teachers will tend to refer at first to 
organisational matters, whether of a general pedagogic nature, or, when pressed, of 
a mathematical pedagogical nature. Typical discourses which might be employed 
include generating discussion, promoting conjecturing, and perhaps, provoking 
proving. Where mathematical being has been stressed and become the generator of 
actions, teacher attention is more likely to be centred on the development of 
learners' mathematical powers and habits of mind, and use of mathematical 
heuristics and themes. There will be explicit concern about the mathematical object 
of learning, rather than vague and generalised passing reference to the 
mathematical topic. 

Put another way, where mathematical being is developing, attention will be on 
the specific didactical issues of specific topics, with pedagogical issues concerning 
classroom organisation and social interaction called upon to serve a greater aim. 

PROBING 

How can one probe beneath the surface of classroom practice to determine what 
influences the choices that appear to be being made? This is a non-trivial question. 
Although experienced teachers, as with experts in any field, are able to discern 
details and recognise relationships which are usually overlooked by less 
experienced observers, it is difficult to articulate what it is that is being noticed, 
even more difficult to describe the basis for that noticing, and even more difficult 
still to try to sensitise novices to make similar distinctions. The use of technical 
terms in any discipline, certainly in mathematics, and particularly in mathematics 
education, is a sign that the author makes distinctions which are considered to be 
significant. Whereas in mathematics the interplay between definition, examples 
and use in context is usually adequate to achieve consensus in the use of the 
technical term by a community, in mathematics education it is more difficult to 
achieve agreement as to the criteria for making distinctions. The notion of PCK is a 
case in point, as Graeber and Tirosh and other authors in this volume note. The 
'discipline' of mathematics education has not developed to the point of agreed and 
established modes of making, labelling, and negotiating the meaning or use of 
technical terms. Instead we resort to verbal explanations and occasionally to 



318 



PCK AND BEYOND 

examples, which often prove to be multiply interpretable and so less than definitive 
or paradigmatic. 

Why Probe Deeper? 

Apart from the political consideration of justifying the existence and enterprise of 
teacher education (both prospective and practising teacher professional 
development), another reason put forward for probing what it is that informs expert 
teacher behaviour is to improve research into teacher effectiveness, and 
consequently, teacher education itself. The thinking is that in order to study 
whether activities influence student learning, it is necessary to distinguish what it is 
that teachers might or might not be doing, and to correlate these with learner 
outcomes and performance. One immediate consequence of this would be to 
inform programmes of teacher development and enhancement. Naturally, one 
would expect teacher education and professional development not only to take into 
account the multiple layers of awareness and sensitivities being worked on, but to 
mirror that in not only be 'being mathematical with and in front of learners', but 
also 'being pedagogical and didactic with and in front of teachers'. 

However, anything with positive potential also has negative possibilities. 
Anything which sounds like test items for teacher knowledge could be added to the 
list of competency tests. This in turn would amplify the stance which converts 
words into checklists and lists into competencies, and which in turn leads to 
mechanistic training of corresponding behaviour without requisite attention to 
educating awareness. The 'checklist' culture which pervades education fails to 
recognise that education is about human beings not machines, and that awareness 
and affect are intricately interwoven with behaviour. Furthermore, this compound 
psyche is driven not by rational reasoning (or checklists of procedures), but by 
intention and will. People are not good at 'doing what they are told', nor is what 
they do very effective if they are merely following orders. This applies just as well 
to learners of mathematics as to teachers and educators. Where individuals exercise 
personal and professional judgement, guided by articulated values and dispositions, 
as well as by taken-as-shared aims and goals, creativity and flair prosper. 

Probing More Deeply Still 

For progress to be made, I am confident that it is necessary for learners, teachers 
and educators all to take a reflective stance. This means interrogating personal 
experience in order to elaborate on personal concept images and example spaces 
associated with different topics, while at the same time building on the literature of 
observed and theoretical obstacles to form a rich picture of the core awarenesses, 
the key sensitivities teachers need in order to provoke learners appropriately and to 
respond creatively and effectively to what they say and do. I am sure that there will 
be ongoing empirical enquiry to observe teachers and to delineate the actions they 
initiate and engage in, and then to probe them for what brought those actions to 
mind, and for the basis for making choices, including choices not to act in certain 

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JOHN MASON 

ways. 1 hope that the two approaches work together, and that we suffer neither the 
potential negative consequences of more testing nor the effects of assuming that 
responses to test items are evidence of sensitivity and awareness. 

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John Mason 

Department of Mathematics 

Open University 

and 

Department of Education, 

University of Oxford 

United Kingdom 



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