Table Of ContentTRANSFORMING PRIMARY
MATHEMATICS
Fully updated to reflect the new national curriculum, this revised edition of Transforming
Primary Mathematics sets out key theories and cutting-edge research in the field to enable
teachers to take a fresh look at how they teach mathematics.
The book encourages teachers to reflect on their own beliefs and values about mathemat-
ics, asking them to question whether their current methods meet the needs of all learners and
the challenge of having high expectations for all. It provides clear, practical approaches to help
implement fundamental change in classroom environments and offers motivational teaching
styles to ensure meaningful mathematics learning.
Chapters take an inspiring, sometimes controversial, and often unconventional look at
the subject of mathematics, by:
• endorsing the use of a ‘new mathematics’—one based on problem solving, modelling,
inquiry and reasoning, not on abstract rules, memorising and regurgitation
• arguing that there is more to maths teaching than ‘death by a thousand worksheets’
• challenging norms, such as the practice of sorting children into sets based on their
perceived mathematical ability
• asking whether mathematical ability is innate or a result of social practices
• examining what a ‘mastery’ approach might entail
• highlighting the role of variation in supporting learning
• advocating an environment where teachers are encouraged to take risks.
Transforming Primary Mathematics is for all primary school teachers who want to make
mathematics welcoming, engaging, inclusive and successful.
Mike Askew is Professor of Mathematics Education at Wits University, Johannesburg, South
Africa, having also been a Professor at King’s College London University, UK, and Monash
University, Melbourne, Australia.
TRANSFORMING
PRIMARY MATHEMATICS
Understanding classroom tasks,
tools and talk
Updated and Revised Edition
Mike Askew
First published 2016
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
and by Routledge
711 Third Avenue, New York, NY 10017
Routledge is an imprint of the Taylor & Francis Group, an informa business
2016 Mike Askew
The right of Mike Askew to be identified as author of this work has been asserted by him in
accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form
or by any electronic, mechanical, or other means, now known or hereafter invented, including
photocopying and recording, or in any information storage or retrieval system, without permission
in writing from the publishers.
Trademark notice: Product or corporate names may be trademarks or registered trademarks, and
are used only for identification and explanation without intent to infringe.
First edition published by Routledge, 2011
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
Askew, Mike.
Transforming primary mathematics: understanding classroom tasks, tools, and talk /
Mike Askew. — Updated and revised edition.
pages cm
1. Mathematics—Study and teaching (Elementary) 2. Mathematics teachers—Training of. I. Title.
QA135.6.A85 2016
372.72—dc23
2015025086
ISBN: 978-1-138-95359-8 (hbk)
ISBN: 978-1-138-95360-4 (pbk)
ISBN: 978-1-315-66725-6 (ebk)
Typeset in Interstate
by Swales & Willis Ltd, Exeter, Devon, UK
CONTENTS
List of figures vi
List of tables viii
Preface ix
1 Introduction 1
PART I
Thinking through mathematics 11
2 Thinking about learning 13
3 Thinking about curriculum 31
4 Thinking about teaching 43
PART II
Transforming primary mathematics 55
5 Mathematical activity: mindful or fluent? 57
6 Variation theory 75
7 Transforming the learner 89
8 Building mathematical community 101
PART III
Teaching tripod 113
9 Tasks 115
10 Tools 129
11 Talk 147
Coda 157
References 158
Index 164
FIGURES
1.1 Two semi-circles joined at a point 5
1.2 A large third semi-circle joining the original two 5
1.3 A small third semi-circle joining the original two 6
2.1 Why 25 + 10 = 15 + 10 + 10 24
2.2 Amy and Ali’s working 25
2.3 Ben and Beth’s working 26
4.1 Josh’s sorting of the bears 52
5.1 Is this a cube? 60
5.2 A child’s drawing of a cube 60
5.3 What quantity is represented by the unshaded part? 61
5.4 What fraction? 62
5.5 Joe’s solutions 65
5.6 Dylan’s solution 65
5.7 Meg’s solution 66
6.1 120 – 90 as difference 82
6.2 122 – 92 as constant difference 83
6.3 119 – 89 as constant difference 84
6.4 235 – 180 as difference 84
6.5 237 – 182 as constant difference 84
6.6 Two dots, two regions 86
6.7 Five dots, sixteen regions 87
9.1 What fraction is shaded? 117
9.2 What fraction – ½, ¼ or ⅛? 19
9.3 A less obvious fraction 119
9.4 Solution 1 to the pizza problem 121
9.5 Solution 2 to the pizza problem 122
9.6 Equivalence 122
9.7 Area and perimeter 126
10.1 Decomposition 133
10.2 Taking away (1) 134
10.3 Taking away (2) 134
10.4 Compensation 135
10.5 Finding the difference 135
10.6 Constant difference (1) 135
Figures vii
10.7 Constant difference (2) 135
10.8 How many tickets are still on sale? (1) 138
10.9 How many tickets are still on sale? (2) 138
10.10 How many tickets are still on sale? (3) 139
10.11 Sally’s workings 140
10.12 Sally’s invented method 140
10.13 Standard grid method 141
10.14 Modelling 9 × 16 141
10.15 Tim’s working 142
10.16 Torn grids (1) 143
10.17 Torn grids (2) 143
10.18 Torn grids (3) 144
10.19 Using arrays to solve division problems 145
11.1 Solver and recorder 152
TABLES
2.1 Fruit, bowls and left-overs 20
6.1 Two sets of subtraction calculations 81
6.2 Numbers of dots and regions 87
9.1 Example what is/what if not? activities 125
PREFACE
‘Look behind you!’ As a child a highlight of each year was our family trip to the pantomime.
I loved the theatre and got to know the various pantomime plots off by heart. Every
pantomime revolves around the big ‘transformation’ scene. Cinderella’s humble pumpkin is
transformed into a glittering coach, Aladdin’s shabby laundry into a splendid palace, and
Jack’s few beans into a giant beanstalk. With drumrolls and flashing lights, these scenes
realize the magic of the theatre. But as well as the spectacle of worldly transformation, these
moments mark a turning point, a transformation, for the lead characters. Cinderella shall go
to the ball, Aladdin can woo a princess, and Jack gets to kill giants. They are not the people
they were before.
Transforming Primary Mathematics I hope invokes these two senses of transformation.
Mathematics can perform magic on the world; letting us see patterns and regularity where
previously we might only have noticed chaos and confusion. And, if we let it, the magic of
mathematics can transform us too.
Some background
Like our pantomime hero setting out on a journey, this book, in part, is a personal journey.
Let me set the context by telling something of my own story. I enjoyed mathematics at
school and was ‘good’ at it, although I realize now that that largely meant I was able to
remember rules. It seemed inevitable that I should do a mathematics degree, so I did, but
half-way through realized I actually understood rather little of what was going on in class.
Like many others, I fell out of love with mathematics. Leaving university, I had no career in
mind but landed a year working at the Open University on a history of mathematics project.
Two things happened that year that shaped everything I’ve done since. First, I got involved
in an after-school project working with seven- to eleven-year-olds. I was – still am – struck
by the energy and enthusiasm for learning that children of this age display and, although
becoming a teacher had not been something I had thought I might want to do, I applied
for and was accepted onto the primary teacher education course at Goldsmiths’ College in
London. The second thing was that in the summer before I went to Goldsmiths’ the Open
University gave me the chance to teach at their Mathematics Foundation course summer
school. There, John Mason introduced me to a new mathematics – one based on problem
solving, modelling and inquiry, not on abstract rules, memorizing, and regurgitation. Thanks
to John, I began to fall back in love with mathematics.
