Individual Differences in Arithmetic

351 Pages·2005·2.01 MB·English

by  Ann Dowker

Individual Differences in Arithmetic
Standards in numeracy are a constant concern to educational policy makers. However,
why are differences in arithmetical performance so marked? In Individual Differences in
Arithmetic, Ann Dowker seeks to provide a better understanding of why these differences
in ability exist, encouraging a more informed approach to tackling numeracy difficulties.
This book reviews existing research by the author and by others on the subject of
arithmetical ability and presents strong evidence to support a componential view of
arithmetic. Focusing primarily on children, but including discussion of arithmetical
cognition in healthy adult and neuro-psychological patients, each of the central
components of arithmetic is covered. Within this volume, findings from developmental,
educational, cognitive and neuropsychological studies are integrated in a unique
approach. This book covers subjects such as:
• Counting and the importance of individual differences.
• Arithmetic facts, procedures and different forms of memory.
• Causes of, and interventions with, mathematical difficulties.
• The effects of culture, language and experience.
The educational implications of these findings are discussed in detail, revealing original
insights that will be of great interest to those studying or researching in the areas of
education, neuroscience and developmental and cognitive psychology.
Ann Dowker is a University Research Lecturer in the Department of Experimental
Psychology, University of Oxford.

Individual Differences in
Arithmetic
Implications for psychology, neuroscience and education
Ann Dowker

HOVE AND NEW YORK
First published 2005 by Psychology Press 27 Church Road, Hove, East Sussex BN3 2FA
This edition published in the Taylor & Francis e-Library, 2005.
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Simultaneously published in the USA and Canada by Psychology Press 270 Madison Avenue,
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Psychology Press is part of the Taylor & Francis Group
Copyright © 2005 Psychology Press
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All rights reserved. No part of this book may be reprinted or reproduced or utilized 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.
This publication has been produced with paper manufactured to strict environmental standards and
with pulp derived from sustainable forests.
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 Dowker, Ann. Individual differences in
arithmetic: implications for psychology, neuroscience, and education/Ann Dowker. p. cm.
Includes bibliographical references and index. ISBN 1-84169-235-2 1. Mathematical ability. 2.
Number concept. I. Title. BF456.N7D69 2005 513′.01′9–dc22 2005009940
ISBN 0-203-32489-7 Master e-book ISBN
ISBN 1-84169-235-2 (hbk)
Dedication
To the memories of Bob Hiorns and Neil O’Connor

Contents

Acknowledgements viii

Introduction 1
1 Children, adults; males, females: Weaknesses and talents 5
2 There is no such thing as arithmetical ability—only arithmetical abilities 23
3 Relationships between arithmetic and other abilities 48
4 Counting and after: The importance of individual differences 61
5 Is arithmetic a foreign language? Representing numbers and arithmetical 86

problems in different forms and translating between them
6 Derived fact strategies 109
7 A good guess: Estimation and individual differences 132
8 Arithmetic facts, procedures and different forms of memory 152
9 Effects of culture, language and experience 172
10 The brain and individual differences in arithmetic 196
11 ‘Maths doesn’t like me anymore’: Role of attitudes and emotions 211
12 Implications for helping children with their arithmetical difficulties 229
Conclusions 256

References 259
Author index 297
Subject index 318
Acknowledgements
I am very grateful to all the staff and children in the schools and nursery schools involved
in the various studies reported here, for all their help and co-operation over the years. I
also thank the following Oxfordshire primary schools: Barton Village Primary School,
Charlbury Primary School, Cutteslowe Primary School, Edward Field Primary School,
New Hinksey Primary School, New Marston Primary School, St Aloysius’ Primary
School, St Barnabas’ Primary School, St Ebbes’ First School, St Francis’ Primary School,
St Mary and St John Primary School, St Michael’s Primary School, St Nicholas’ Primary
School, St John Fisher Primary School, St Philip and St James’ Primary School,
Windmill Primary School and Wood Farm Primary School.
I also thank the following Oxford nursery schools and day nurseries: Bartlemas
Nursery School, Elms Road Nursery School, Julia Durbin Day Nursery, Kiddies Korner
Day Nursery, Sandfield Day Nursery, St Anne’s College Nursery, St Thomas’ Day
Nursery, Slade Nursery School and Summertown Nursery School.
In addition, my students and collaborators have been helped by numerous other
schools in the United Kingdom and overseas; although I cannot list all these schools by
name, I am extremely grateful to all of them.
I would also like to thank all the adults who have taken part in my studies.
The book, and the research that led up to it, have benefited from valuable discussions,
at various stages, with many people, including Daniel Ansari, Sir Christopher Ball, Brian
Butterworth, Yi-Ping Chen, Tim Coulson, Richard Cowan, Stanislas Dehaene, Margarete
Delazer, Chris Donlan, Karen Fuson, Herbert Ginsburg, Silke Goebel, Ati Hermelin,
Nancy Jordan, Liane Kaufmann, Deborah King, Tim Miles, Alison Price, Sonia Sciama,
Kathy Sylva, Ian Thompson, Anne Watson, Bob Wright and all members of the
Numeracy Intervention email discussion group. The opinions—and the errors—in this
book are, of course, my own.
Elizabeth Warrington kindly provided me with her Dot Counting test and permitted
me to use it.
I am grateful to all members of the SRU for their valuable assistance with the
research.
I am grateful to the British Academy, the ESRC, and the Esmee Fairbairn Charitable
Trust for financial assistance at various stages.
Arithmetic by Carl Sandburg
Arithmetic is where numbers fly like pigeons in and out of your head.
Arithmetic tells you how many you lose or win if you know how many you had before
you lost or won.
Arithmetic is seven eleven all good children go to heaven—or five six bundle of
sticks.
Arithmetic is numbers you squeeze from your head to your hand to your pencil to your
paper till you get the answer.
Arithmetic is where the answer is right and everything is nice and you can look out of
the window and see the blue sky—or the answer is wrong and you have to start all over
and try again and see how it comes out this time.
If you take a number and double it and double it again and then double it a few more
times, the number gets bigger and bigger and goes higher and higher and only arithmetic
can tell you what the number is when you decide to quit doubling.
Arithmetic is where you have to multiply—and you carry the multiplication table in
your head and hope you won’t lose it.
If you have two animal crackers, one good and one bad, and you eat one and a striped
zebra with streaks all over him eats the other, how many animal crackers will you have if
somebody offers you five six seven and you say No no no and you say Nay nay nay and
you say Nix nix nix?
If you ask your mother for one fried egg for breakfast and she gives you two fried eggs
and you eat both of them, who is better in arithmetic, you or your mother? (‘Arithmetic’
from The complete poems of Carl Sandberg. Copyright © 1970, 1969 by Lilian Steichen
Sandberg, Trustee, reprinted by permission of Harcourt, Inc.)