Table Of ContentChildren, Teachers and Learning Series
General Editor: Cedric CulKngford
Children's Mathematical Thinking in the
Primary Years
Titles in the Children, Teachers and Learning series:
J. Backhouse, L. Haggarty, S. Pirie and J. Stratton Improving the Learnin
Mathematics
M. Bonnett Children's Thinking
J. Campion Working with Vulnerable Young Children
C. Cullingford Children and Society
C. Cullingford The Inner World of the School
C. Cullingford The Mature of Learning
K. Gentle Teaching Painting in the Primary School
J. Glover and S. Ward Children's Music
D. Hartley Understanding the Nursery School
T. Jarvis Children and Primary Science
B. May all Negotiating Health
R. Stevenson and J. Palmer Learning: Principles, Processes and Practices
P. Wiegand Children and Primary Geography
J. Wilson and B. Cowell Children and Discipline
Children's Mathematical
Thinking in the Primary
Years
Perspectives on Children's Learning
Edited by Julia Anghileri
continuum
Continuum
The Tower Building 80 Maiden Lane
11 York Road Suite 704
London SE1 7NX New York NY 10038
www.continuumbooks.com
© Julia Anghileri and the contributors 1995
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, including
photocopying, recording, or any information storage or retrieval system,
without permission in writing from the publishers.
First published 1995
Reprinted 1999,2003,2005,2006
British library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library.
ISBN 0-8264-7911-1 (paperback)
Typeset by Create Publishing Services Ltd, Bath, Avon
Printed and bound in Great Britain by
Biddies Ltd, King's Lynn, Norfolk
Contents
List of Contributors vi
Foreword ix
Introduction xi
1 Focus on Thinking Julia Anghileri 1
2 Emergent Mathematics or How to Help Young Children
Become Confident Mathematicians David Whitebread 11
3 Investigational Starting Points and Children's Thinking
Don Mackay 41
4 Between the Lines: The Languages of Mathematics Tim
Rowland 54
5 Making Sense of Symbols Julia Anghileri 74
6 CAN Calculators Make a Difference? Laurie Rousham 92
7 What is your Favourite Colour? Libby Jared and Anne
Thwaites 110
8 In Search of the Unknown: A Review of Primary Algebra
Annie Owen 124
9 Re-solving Problem Solving Lyndon Baker 148
Name Index 169
Subject Index 171
V
List of Contributors
JULIA ANGHILERI is Head of the Mathematics Department at Homerton
College, University of Cambridge involved in the initial and in-service
training of teachers as well as research in Mathematics Education. She
has published a number of articles in professional and research
journals relating to children learning mathematics. She has an inter-
national reputation for the work she has done on children's under-
standing of multiplication and associated aspects of mathematical
language.
Other authors all contribute to Mathematics Education courses in
undergraduate and postgraduate initial training and to in-service
programmes for mathematics at Homerton College. The following
brief notes identify some of their expertise.
LYNDON BAKER is a freelance advisor for Mathematics Teaching, a
nical author, and developer of curriculum materials for mathematics
teaching. He is an active member of the Association of Teachers of
Mathematics and is currently involved in a range of initial and
in-service training courses, both school and college based.
LIBBY JARED was Head of Mathematics at a secondary school and is
now Senior Lecturer and Homerton College Information Technology
(IT) Co-ordinator. Libby has produced IT study packs for initial
training students covering all aspects of IT in primary schools and is
involved in the induction of students and teachers into the applications
of computing in children's learning.
DON MACKAY teaches and co-ordinates mathematics at Parkway
Middle School in Suffolk. As well as being a part-time member of the
teaching team at Homerton College, he has been invited as a teacher
advisor to contribute to working groups involved with the introduction
of the Mathematics National Curriculum.
vi
list of Contributors
ANNIE OWEN has teaching experience in both primary and secondary
schools in the UK and abroad. She is Senior Lecturer in Mathematics
Education and College Go-ordinator for mathematics curriculum
courses. Annie contributes regularly to the journal Junior Education
has written several books for teachers.
LAURIE ROUSHAM was Head Teacher of a primary school involved in
the CAN Curriculum Project under the direction of Hilary Shuard.
He is Senior Lecturer in Mathematics Education and is currently
involved in research based at Kings College in London. As a member
of the National Curriculum Council B Committee he contributed to
discussion and documentation accompanying the introduction of the
England and Wales - National Curriculum Statutory Orders.
TIM ROWLAND is Senior Lecturer in Mathematics and Mathematics
Education and is the Director of INSET at Homerton College. As
well as being an active and enthusiastic mathematician, he has a
well-established reputation in initial and in-service training of primary
teachers and is currently engaged in research into children's
mathematical thinking. Tim is a regular contributor to journals in
Mathematics Education.
ANNE THWAITES is Senior Lecturer in Mathematics and Mathematic
Education involved in many aspects of the initial and in-service
training of primary teachers. Her expertise with IT applications in the
teaching of mathematics ranges from infant and primary school
children to mathematics undergraduates and number theory. Anne is
Director of Curriculum and Professional Studies for the BEd degree.
DAVID WHITEBREAD is an ex-primary school teacher and is Senior
Lecturer in Psychology at Homerton College. His experiences with
postgraduate and initial training students as well as his involvement
with local infant schools offer a balanced expertise in the theory and
practice of education for young children.
vii
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Foreword
The books in this series stem from the conviction that all those who are
concerned with education should have a deep interest in the nature of
children's learning. Teaching and policy decisions ultimately depend
on an understanding of individual personalities accumulated through
experience, observation and research. Too often in recent years
decisions on the management of education have had little to do with
the realities of children's lives, and too often the interest shown in the
performance of teachers, or in the content of the curriculum, has not
been balanced by an interest in how children respond to either. The
books in this series are based on the conviction that children are not
fundamentally different from adults, and that we understand ourselves
better by our insight into the nature of children.
The books are designed to appeal to all those who are interested in
education and who take it as axiomatic that anyone concerned with
human nature, culture or the future of civilization is interested in
education - in the individual process of learning, as well as what can
be done to help it. While each book draws on recent findings in
research and is aware of the latest developments in policy, each is
written in a style that is clear, readable and free from the jargon that
has undermined much scholarly writing, especially in such a relatively
new field of study.
Although the audience to be addressed includes all those concerned
with education, the most important section of the audience is made up
of professional teachers, the teachers who continue to learn and grow
and who need both support and stimulation. Teachers are very busy
people, whose energies are taken up in coping with difficult circum-
stances. They deserve material that is stimulating, useful and free of
jargon and is in tune with the practical realities of classrooms.
Each book is based on the principle that the study of education is a
discipline in its own right. There was a time when the study of the
principles of learning and the individual's response to his or her
environment was a collection of parts of other disciplines - history,
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