Table Of ContentSpringer Series in Cognitive Development
Series Editor
Charles 1. Brainerd
Springer Series in Cognitive Development
Series Editor: Charles J. Brainerd
(recent titles)
Adult Cognition: An Experimental Psychology of Human Aging
Timothy A. Salthouse
Recent Advances in Cognitive-Developmental Theory: Progress in Cognitive
Development Research
Charles J. Brainerd (Ed.)
Learning in Children: Progress in Cognitive Development Research
Jeffrey Bisanz/Gay L. Bisanz/Robert Kail (Eds.)
Cognitive Strategy Research: Psychological Foundations
Michael PressleylJoel R. Levin (Eds.)
Cognitive Strategy Research: Educational Applications
Michael PressleylJoel R. Levin (Eds.)
Equilibrium in the Balance: A Study of Psychological Explanation
Sophie Haroutunian
Crib Speech and Language Play
Stan A. Kuczaj. II
Discourse Development: Progress in Cognitive Development Research
Stan A. Kuczaj, II (Ed.)
Cognitive Development in Atypical Children: Progress in Cognitive Development
Research
Linda S. Siegel/Frederick J. Morrison (Eds.)
Basic Processes in Memory Development: Progress in Cognitive Development Research
Charles J. Brainerd/Michael Pressley (Eds.)
Cognitive Learning and Memory in Children: Progress in Cognitive Development
Research
Michael Pressley/Charles J. Brainerd (Eds.)
The Development of Word Meaning
Stan A. Kuczaj, IIIMartyn D. Barrett (Eds.)
Fonnal Methods in Developmental Psychology: Progress in Cognitive Development
Research
Jeffrey Bisanz/Charles J. Brainerd/Robert Kail (Eds.)
Children's Counting and Concepts of Number
Karen C. Fuson
Karen C. Fuson
Children's Counting and
Concepts of Number
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Karen C. Fuson
School of Education and Social Policy
Nonhwestern University
Evanston, Illinois 60208
USA
Series EdilOr
Charles J, Brainerd
Program in Educational Psychology
University of Arizona
Tuscon, Arizona 85715
USA
With 7 lIIustrlllions
Library of Congress Cataloging-in-Publication Data
Fuson, Karen C.
Children's counting and concepts of number.
(Springer series in cognitive dcvelopment)
Bibliography: p.
Includes index.
I. Counting 2. Number concepts. I. Tille
II. Series.
QAII3.F87 1987 513'.2 87-16460
ISUN-13; 978-1-4612-8335-5 e-JSUN-IJ; 978-1-4612-37S4-9
1)01; 1O.IOO71978-1-4612-J7S4-9
© 1988 by Springer-Verlag New York Inc.
All rights reserved. This work may not be translated or copied in whole or in pan without the written per·
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9 8 7 6 5 4 J 2 I
To my family
Series Preface
For some time now, the study of cognitive development has been far and away the
most active discipline within developmental psychology. Although there would be
much disagreement as to the exact proportion of papers published in developmental
journals that could be considered cognitive, 50% seems like a conservative estimate.
Hence, a series of scholary books to be devoted to work in cognitive development
is especially appropriate at this time.
The Springer Series in Cognitive Development contains two basic types of books,
namely, edited collections of original chapters by several authors, and original
volumes written by one author or a small group of authors. The flagship for the
Springer Series is a serial publication of the "advances" type, carrying the subtitle
Progress in Cognitive Development Research. Volumes in the Progress sequence are
strongly thematic, in that each is limited to some well-defined domain of cognitive
developmental research (e.g., logical and mathematical development, semantic
development). All Progress volumes are edited collections. Editors of such books,
upon consultation with the Series Editor, may elect to have their works published
either as contributions to the Progress sequence or as separate volumes. All books
written by one author or a small group of authors will be published as separate
volumes within the series.
A fairly broad definition of cognitive development is being used in the selec
tion of books for this series. The classic topics of concept development, children's
thinking and reasoning, the development of learning, language development, and
memory development will, of course be included. So, however, will newer areas
such as social-cognitive development, educational applications, formal modeling,
viii Series Preface
and philosophical implications of cognitive-developmental theory. Although it is
anticipated that most books in the series will be empirical in orientation, theoretical
and philosophical works are also welcome. With books of the latter sort, heter
ogeneity of theoretical perspective is encouraged, and no attempt will be made to
foster some specific theoretical perspective at the expense of others (e. g., Piagetian
versus behavioral or behavioral versus information processing).
C.J. Brainerd
Preface
My earliest number research focused on the activity of counting objects by
preschoolers and on the more efficient and sophisticated counting procedures
primary school children spontaneously invent and use to solve addition problems.
It quickly became clear that children's use of the sequence of number words changed
radically over this age span. The sequence changed from being a rote list of words
paired with objects to being a representational tool used in flexible and gener
ative ways to solve addition and subtraction situations. This change depended
upon changes in children's ability to relate counting and concepts of cardinality.
Counting and cardinality first had to become related for children so that counting
produced cardinal knowledge. Counting of objects was then used for a considerable
period of time to solve addition and subtraction problems. Gradually, however,
this object counting became abbreviated and abstracted from the objects so that
the number words themselves became the "objects" that represented the addition or
subtraction situation. Thus, over this age span, the sequence, counting, and cardinal
meanings of number words became increasingly closely related and eventually
became integrated.
A primary purpose of this book is to trace these changes across the age span of
2 through 8. My work has necessarily concentrated in certain areas, so parts of this
age span can be described only sketchily. Work by others is similarly concentrated
in some areas, so the reviews of related literature can only fill in certain spots. An
attempt is made nevertheless to describe changes in children's number-word
sequences and in relationships between counting and cardinal meanings from single
set situations through fairly complex cardinal operations (multidigit addition and
subtraction) and cardinal relations (Piagetian operational equivalence and order
x Preface
relations). These developmental frameworks hopefully will provide a working
context within which research on the sketchier parts can proceed but be easily
related to past results and to concurrent efforts by other investigators. I hope that
readers will leave the book both understanding a considerable amount about some
specific aspects of children's counting and numerical thinking and with some notion
of the overall kinds of changes children's thinking and performance undergo over the
years 2 through 8.
Number concepts from age 2 through 8 provide a very interesting research arena.
The mathematical concepts and behaviors are fairly well-defined, so one can con
centrate on trying to understand how children are thinking in these numerical situa
tions. The concepts are important, for they are formative concepts in mathematical
thinking. The educational centrality of these concepts is widely accepted, so
research results may lead fairly readily to instructional research. Development of
these concepts is related both to linguistic development and to several aspects of
general cognitive development. This book then may be of interest to readers from
a wide range of backgrounds: cognitive development, education, and linguistics.
Chapter summaries have been written to enable readers with varying backgrounds
to obtain an overview of each chapter. Because the book was already so long, discus
sions of extensions to instruction and to many cognitive developmental issues have
had to be curtailed. However, many readers will be able to make these extensions
fairly readily.
This book was written with the intellectual, practical, and psychological support
of many individuals. I feel fortunate that this area has had and continues to have such
competent and professional people working in it. My own thinking and writing has
benefited enormously over the years from the work of, and conversations with,
many other researchers in this area. Some of these graciously agreed to react to
earlier versions of one or more chapters of this book: Arthur Baroody, Charles
Brainerd, Diane Briars, Tom Carpenter, Paul Cobb, Douglas Frye, Herbert Gins
burg, James Hall, Kevin Miller, John Richards, Walter Secada, Robert Siegler, and
James Stigler. The book was improved in major ways by revisions stemming from
their comments. Several graduate students have stimulated and contributed to my
thinking over the years: Diane Briars, Birch Burghardt, Youngshim Kwon, Barb
Lyons, Gerry Pergament, and Walter Secada. Many analyses reported here were
carefully and intelligently carried out by Diane Briars, Walter Secada, and Gordon
Willis. Work-study students for many years carefully scored, coded, and entered
data of various kinds with meticulous care and continuing enthusiasm for the vari
ous projects in which we were engaged; these included Richard Ames, Tom Ciaglo,
Thue Hoang, Fred Karr, Tracy Klein, Lloyd Kohler, Norman Lao, Krista Peterson,
Arlene Siavelis, and Luann Troxel. Many undergraduates collected data from chil
dren and/or coded videotaped data, taking independent studies and being rewarded
only by their learning about children: Kathy Amoroso, Holly Arnowitz, Patty
Bloom, Steve Cieslewicz, Chris Daley, Del Flaherty, Sue Ivanov, Pui Ling Lee, Lisa
Montgomery, Steven Myers, Joanne Murabito, Virginia Neal, Sharon Nussbaum,
Debbie Rubins, Karen Simon, Sherri Weinstein, and Toya Wyatt. Pat Gaul, Joan
LeBuhn, and Pat Terando patiently and capably typed tables, typed revisions, and
Preface Xl
checked references. And of course, special thanks to my cheerful Macintosh, who
allowed me to type in all sorts of positions and places (even outdoors) and always
greeted me with a smile no matter how early or how late work began.
Many schools and preschools in Chicago, Evanston, Glenview, Wilmette, and
Highland Park have participated over the years in the studies reported or summa
rized here; understanding preschool directors and school principals, patient and
supportive teachers, and children willing to share their mathematical thinking with
us have made most of this research productive and enjoyable. Special thanks go to
Betty Weeks and the National College of Education Demonstration School for con
tinual interest and support of our efforts, to the Walt Disney Magnet School for sup
porting counting, conservation, and addition research over two years, and to the
wonderful teachers on the primary teams at Lincoln and Orrington Schools in Evan
ston for three years of intensive involvement in addition and subtraction research.
The school-based research could not have been carried out without the sensitive and
organized field liaison work of Maureen Hanrahan and the intelligent and careful
interviewing and data analysis of Gordon Willis.
Financial support for some of the research reported here was provided by the
National Science Foundation and the National Institute of Education under Grant
No. SED 78-22048 and Grant No. SED 78-17365, the AMOCO Foundation, a
Spencer Foundation grant, and a Spencer National Academy of Education Fellow
ship. Any opinions, findings, and conclusions or recommendations expressed in this
publication are those of the author and do not necessarily reflect the views of the
National Science Foundation, the National Institute of Education, the AMOCO
Foundation, or the Spencer Foundation.
Finally, I coul~ not have completed the research, let alone write this book, without
the continual understanding of my family over the years. My daughters Adrienne
and Erica revealed many aspects of children's thinking to me and have consistently
provided warm and interesting interactions, enabling me to return to my work
renewed. Their childhoods have enriched my life and provided some sense of
balance when work threatened to become all. The research on which the book is
based and the writing of the book benefited enormously from the intellectual and
psychological support of my husband, James Hall. He forced me to struggle for clar
ity of design, thought, and expressiOlf;-patiently put up with very long hours of work
on my part; and cooked at least 4,000 meals to keep all of us going. His humor has
lightened our days, helping the six of us to become a 70s "blended" family (his two,
my two, we two): the older sisters Julie and Adrienne, the blonde youngers Dan and
Erica, and los padres. Of course ultimate thanks go to my own parents, for I would
not have been prepared to tackle these issues without their early direction and sup
port. My father encouraged my mathematical interests and helped me apply to and
attend National Science Foundation summer institutes for high school students, my
mother found Oberlin for me, and together they provided a wonderful supportive
home.
Evanston, Illinois Karen C. Fuson
