Developing Primary Mathematics Teaching: Reflecting on Practice with the Knowledge Quartet
 Chapters
 Front Matter
 Subject Index

 Chapter 1: Inside Naomi's Classroom
 Chapter 2: Knowledge for Teaching Mathematics: Introducing the Knowledge Quartet Framework
 Chapter 3: Transformation: Using and Understanding Representations in Mathematics Teaching
 Chapter 4: Transformation: Using Examples in Mathematics Teaching
 Chapter 5: Making Connections in Mathematics Teaching
 Chapter 6: Contingency: Tales of the Unexpected!
 Chapter 7: Foundation Knowledge for Teaching Mathematics
 Chapter 8: Using the Knowledge Quartet to Reflect on Mathematics Teaching
 References

Copyright
© Tim Rowland, Fay Turner, Anne Thwaites and Peter Huckstep 2009
First published 2009
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may be reproduced, stored or transmitted in any form, or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction, in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.
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List of Figures
 Figure 1.1 Naomi's representation of the frogs 5
 Figure 1.2 A note about Naomi's lesson 9
 Figure 3.1 Placevalue or arrow cards 46
 Figure 3.2 Spike abacus and base10 apparatus 46
 Figure 3.3 Placevalue or Gattegno chart 47
 Figure 3.4 Birthday card number line 49
 Figure 3.5 Empty number line representation of 36 + 27 50
 Figure 3.6 Empty number line representation of 36 + 27 using a compensation strategy 50
 Figure 3.7 Empty number line representation of ‘62 take away 43’ 51
 Figure 3.8 Empty number line representation of ‘the difference between 62 and 43’ – counting backwards 51
 Figure 3.9 Empty number line representation of ‘the difference between 62 and 43’ – counting forwards 51
 Figure 3.10 Empty number line representation of 3 + 2 52
 Figure 3.11 The 1–100 number grid 53
 Figure 3.12 The 0–99 number grid 54
 Figure 3.13 Using the 1–100 number grid to calculate 38 + 9 56
 Figure 3.14 A calendar: 23rd September and subtract 9 61
 Figure 3.15 Numdrum 63
 Figure 3.16 Moves on a hundred grid for adding or subtracting 9 or 11 to or from 56 64
 Figure 4.1 Examples of line symmetry in the environment 68
 Figure 4.2 An example of a rectangle, and another, and another… 69
 Figure 4.3 Simon finds 9 × 37 using the grid method 76
 Figure 4.4 Laura demonstrates the elaborated column multiplication method 76
 Figure 4.5 Translating and reflecting a triangle 85
 Figure 4.6 Subtracting 21 from 53 on an empty number line 86
 Figure 4.7 Subtracting 19 from 53 by subtracting 20 and adding 1 86
 Figure 4.8 Naomi's representation of the comparison subtraction 88
 Figure 4.9 Kate pairs the factors of 36, apart from 6 96
 Figure 4.10 Showing that 1+3 + 5 + 7 = 4 × 4 98
 Figure 5.1 A shape divided into quarters 105
 Figure 5.2 A collection of objects divided into two fifths and three fifths 105
 Figure 5.3 Fractions on a number line 106
 Figure 5.4 Comparing two cars in size 106
 Figure 5.5 Caroline's flower representation 110
 Figure 5.6 Halving the flower representation 111
 Figure 5.7 Halving the spider 111
 Figure 5.8 A representation of Haylock's model 113
 Figure 5.9 How can you represent 115
 Figure 5.10 A concept map about quadrilaterals 116
 Figure 5.11 Simon finds 9 × 37 using the grid method 119
 Figure 5.12 Laura demonstrates the elaborated column multiplication method 119
 Figure 5.13 The grid method for 56 × 24 120
 Figure 5.14 A ‘compact’ layout for 56 × 24 121
 Figure 5.15 The contracted layout for 56 × 24 122
 Figure 5.16 The contracted grid method for 56 × 24 122
 Figure 6.1 The congruence of the two halves of a rectangle 128
 Figure 6.2 Quarters of a rectangle? 128
 Figure 6.3 Parts of a rectangle – do they have the same area? 129
 Figure 6.4 Different sets of coins to represent 5p in value 133
 Figure 6.5 Perimeter and area of rectangles: a child's idea 136
 Figure 6.6 Two Ushapes made from squares 138
 Figure 6.7 Sean's attempt at 27 × 9 144
 Figure 6.8 Leroy's column multiplication 145
 Figure 6.9 Using fingers to count on from 2 to 6 146
 Figure 7.1 Jumping back on an empty number line to show 85 − 47 165
 Figure 7.2 The numbers 85 and 47 as locations on an empty number line 166
 Figure 7.3 Finding the difference between 47 and 85 on a number line 166
 Figure 7.4 Sharing 28 sweets between 4 children 172
 Figure 7.5 Putting 28 sweets into groups of 4 172
 Figure 7.6 Number trio – 28, 7 and 4 174
 Figure 7.7 Repeated subtraction to calculate 28 ÷ 4 on a number line 175
 Figure 7.8 Thinking about 469 ÷ 7 178
 Figure 7.9 Using chunking to calculate 319 ÷ 42 180
 Figure 7.10 Reflection, rotation and translation 181
 Figure 7.11 The path between object and image for a translation and for a rotation 182
 Figure 7.12 A collection of congruent shapes 183
 Figure 7.13 Translation of a triangle 184
 Figure 7.14 Translation of a square? 185
 Figure 7.15 A key pattern 185
 Figure 7.16 Producing a butterfly by reflection 186
 Figure 7.17 Building a reflection pattern on a square grid 186
 Figure 7.18 Rotation of a triangle, centre C, 90° clockwise 188
 Figure 7.19 Rotation of a square, centre C, 90° clockwise 188
 Figure 7.20 Some examples of logos or patterns with symmetry 189
 Figure 7.21 Wheel hubcaps 189
 Figure 7.22 Axes of symmetry on a hubcap 190
 Figure 7.23 Rotations of a hubcap 190
 Figure 8.1 Tens and units board 200
 Figure 8.2 Jamie's tens and units board 201
 Figure 8.3 Interactive whiteboard screen showing 16÷2 205
 Figure 8.4 Child's recording of 16 ÷ 2 205
 Figure 8.5 Interactive whiteboard screen showing 20 ÷ 5 206
 Figure 8.6 Lindsay's ‘Top Tips’ 210
List of Tables
 Table 2.1 Some suggestions for improving your knowledge for teaching mathematics 25
 Table 2.2 The codes of the Knowledge Quartet 29
 Table 2.3 Guidelines for observing, supporting and assessing the level of trainee teachers teaching mathematics 35
 Table 3.1 Resources for mathematics teaching 43
 Table 7.1 The different structures of subtraction 166
 Table 8.1 The lessons 197
 Table 8.2 Knowledge Quartet lesson reflection proforma 225
Index of Teachers and Lessons
Acknowledgements
We acknowledge with thanks and gratitude the assistance of the many people who have contributed to the development and production of this book. In particular we thank Jane Warwick for her contribution to the research which is at the heart of the book, David Thwaites for expert help with video editing, and Jamie Turner for his imaginative work on the cover photographs. The book would never have come into being without the collaboration of the many teacherparticipants whose lessons are featured in it. Their willingness to offer their practice for others to reflect on, warts and all, demonstrates their commitment to the improvement of mathematics teaching – their own, and that of the readers of this book.
Introduction
This book differs, in some significant ways, from other books on primary mathematics teaching. A short explanation of who the book was written for, what it is intended to do, and how best to use it, may therefore be helpful.
The book is for primary mathematics teachers and those who support their professional development in schools, local authorities and universities. It will be especially useful to student teachers (sometimes called ‘trainees’ in England) and earlycareer teachers, for whom the information in the book is likely to be novel. But the heart of this book is not so much information, as a process of structured reflection whereby teachers – at any stage of their career – can take control of the development of their expertise in teaching mathematics. The reflective process is the outcome of five years of research at the University of Cambridge. It is built on a framework which enables teachers and teacher educators to engage critically with actual lessons and teaching episodes – their own, or others' – with the aim of learning from teachinginaction. We call our framework The Knowledge Quartet: because it is in four parts, and because it is a way of building up professional knowledge for mathematics teaching. A full explanation is given in Chapter 2.
Theory and practice are interwoven throughout the book. In each chapter you will find:
 justification for the focus of the chapter;
 some exposition related to the topic, or some illuminating aspect of it;
 tasks – things for you to think about and discuss, where possible, with colleagues;
 descriptions of actual lessons, or episodes from lessons, for you to consider, discuss and analyse. Some of these episodes can be viewed as video clips on the book's companion website;
 our own reflections and analytical comments on these lessons and episodes.
The final chapter is devoted to accounts, tasks and analyses relating to a wide range of classroom scenarios, all taken from actual lessons taught by beginning or earlycareer teachers.
Each chapter can be read on its own, but you will gain most from the book if you read Chapters 1 and 2 first. Chapter 2, in particular, explains the Knowledge Quartet, the framework for reflection which features throughout Chapters 3 to 8.
Of course, the book can be read at home, or on the bus or the train. It is also recommended as a stimulus for group work and plenary discussion in teacher education settings and in continuing professional development. The Knowledge Quartet framework is also ideally suited for use in the improvement of teaching through lesson observation, both within initial teacher education and ongoing teaching development. In these various settings, the observer could be a mentor, tutor, colleague, subject coordinator or school manager of some kind. We would add, however, that we wrote this book so that teachers at different career stages might be encouraged and supported, and not as a means for them to be judged. The key to this positive, critical support is to allow time – if only 15 minutes – for postlesson review and discussion, shared by teacher and observer, and structured by the Knowledge Quartet.
We hope that readers will find the book enjoyable, useful and informative: we welcome any feedback on the content, and on the ways that the book is being used.
, , , .Cambridge, 2008 
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