Table Of ContentThe Child's
Understanding of Number
The Child's
Understanding of Number
Rochel Gelman and C. R. Gallistel
HARVARD UNIVERSITY PRESS
CAMBRIDGE, MASSACHUSETTS, AND LONDON. ENGLAND
To Adam, who counts
Copyright © 1978, 1986 by the
President and Fellows of Harvard College
All rights reserved
Printed in the United States of America
10 9 8 7 6 5 4 3
Library of Congress Cataloging in Publication Data
Gelman, Rochel.
The child's understanding of number.
Bibliography: p.
Includes index.
1. Number concept. I. Gallistel, C. R. II. Title.
[QAI41.15.G44 1986] 372.7'2'044 85-27506
ISBN 0-674-11636-4 (cloth)
ISBN 0-674-11637-2 (paper)
Preface, 1986
This book was written with two audiences in mind: those interested in
cognitive development in general and those interested in the psycho
logical foundations of mathematical thought in particular. Hence.
it can be used in courses in cognitive development. the psychology
of mathematics. and early mathematics education.
Chapters 1,2. and 3 highlight the problems of standard accounts of
early cognitive development and introduce the view that preschoolers
have greater cognitive capacities than previously thought. Especially
suitable for a course on cognitive development. these chapters focus
on the difficulty of interpreting data that characterize preschoolers
as unable to pass the same tasks as older children. We argue that the
immediate success of appropriately designed training studies implies
the existence of some prior understanding of basic concepts in the
domains being trained. Otherwise the child would not be able to as
similate the training. However, such indirect arguments are just that.
Chapter 3 surveys studies that were designed with the young child in
mind and that demonstrate directly an early understanding of some
fundamental concepts in the domains of communication and classifi
cation. The next seven chapters may be read as a case study in early
cognitive development, using number concepts as the case. The
advantage of this program is that it lends itself to a precise analysis of
the variables which contribute to competence.
Chapters 11 and 12 focus on the theoretical implications of the
findings regarding what preschoolers can do in the numerical do
main, with Chapter 11 most relevant to courses stressing formal
models of cognitive development. The rest of the book fits well into
courses on early cognitive development, the role of structural con
straints on learning, and theories of development.
Appropriate for a psychology of mathematics course, Chapters
v
VI I Preface, 1986
5-12 serve to organize questions about the order in which arithmetic
concepts develop; the role of counting, as opposed to logic, in the
young child's ability to represent the numerical value of sets and
reason arithmetically; the way mathematics informally learned by
children naturally differs from mathematics learned in formal set
tings; and what children have to learn in order to expand their
knowledge of arithmetic. The book can be used in conjunction with
any of several recent, edited volumes on mathematical learning and
knowledge in school-aged children (see the references at the end of
this preface), such as Carpenter, Moser, and Romberg (l 982);
Ginsburg (l 983); Heibert (1986); the Resnick and Ford (1981) text;
and other research monographs, including Ginsburg'S (1977), as well
as Piaget (1952).
Finally, it is our experience that teachers and students of preschool
curriculum find many useful ideas in Chapters 1-3 and S -1 O. The
emphasis on what preschoolers are capable of doing and knowing
fits well with their own views that the very young actively engage
themselves in the many different learning environments they are
exposed to.
The work recounted in this book is brought up to date in the fol
lowing papers: Gelman (l982a) develops the argument that the
kinds of numerical abilities described in this book are universal. This
paper, like the book, concentrates on preschool children's conceptual
competence, those principles which constitute the core of their under
standing of the domain of numbers. Greeno, Riley, and Gelman
(1984) and Gelman and Meck (1986) turn to the question of the rela
tion between this domain-specific conceptual competence and two
other nqnspecific kinds of competence-that at building up com
plex procedures from subprocedures (procedural competence) and
that at assessing the general and idiosyncratic requirements of tasks
(utilization competence). In both studies the authors argue that suc
cessful performance depends on the integration of these three dis
tinct types of competence. In addition, Gelman and Meck illustrate
how one can account for the rclation between domain-specific con
straints on cognitive development and the acquisition of general skills
that span domains, thus providing a way to integrate two different
theoretical approaches to cognitive development.
Our book argues that the conceptual competence of the young
child consists of two sets of principles that guide behavior in the
numeri(~al domain. One set of principles defines and interrelates the
operations of addition and subtraction and the numerical order and
Preface, 1986 / Vll
equivalence relations; the other set defines counting for the purpose
of representing the numerosity of a set. The principles that define
counting are the how-to principles discussed in Chapters 8 and 9. The
abstraction and order-irrelevance "principles" have a different
status. Whereas the how-to principles specify the properties of a
proper count sequence and define what it means to count, the order
irrelevance and abstraction principles do not function this way; they
are not an active part of the child's conception of number. Rather,
they emphasize the absence of two widely presupposed constraints
on the child's definition of counting. Young children do not restrict
their counts to perceptually alike items. Nor do they use only those
counting procedures for which they may have been reinforced. The
absence of these constraints makes the how-to principles more general
and powerful than they otherwise would be.
The only argument in the book we would want to change-as
opposed to develop-in the light of subsequent work, is the one in
Chapter 12, explaining why preschoolers fail the number conserva
tion task. We proposed that, despite preschoolers' prowess in the
numerical domain, they are unable to use a principle of one to one
correspondence to establish and reason about equivalence. Gelman
(l982b) tested this hypothesis and given her results, it is clear that
there are some conditions under which children will recognize
numerical equivalence on the basis of one to one correspondence
(and hence conserve). Still, they clearly prefer to establish numerical
equivalence through counting.
Recent References
Carpenter, T. P., J. M. Moser, and T. A. Romberg, eds. 1982. Addition and
subtraction: a cognitive perspective. Hillsdale, New Jersey: Erlbaum.
Gelman, R. 1982a. Basic number abilities. In Advances in the psychology of intel
ligence, vol. I, ed. R. J. Sternberg, pp. 182-205. Hillsdale, New Jersey:
Erlbaum.
Gelman, R. 1982b. Accessing one to one correspondence: still another paper
about conservation. British Journal of Psychology 73 :209-220.
Gelman, R., and E. Meek. 1986. The notion of principle: the case of counting.
In Conceptual and procedural knowledge: the case of mathematics, eel. J. Hiebert.
Hillsdale, New Jersey: Erlbaum.
Ginsburg, H. P. 1977. Children's arithmetic: the learning process. New York:
Van Nostrand.
Ginsburg, H. P., ed. 1983. The development of mathematical thinking. New York:
Academic Press.
I Preface, 1986
VIII
Greeno, j. G., M. S. Riley, and R. Gelman. 1984. Conceptual competence and
children's counting. Cognitive Psychology 16:4-134.
Hiebert, j., ed. 1986. Conceptual and procedural knowledge: the case of mathe
matics. Hillsdale, New Jersey: Erlbaum.
Resnick, L. B., and W. F. Ford. 1981. The psychology of mathematics for instruc
tion. Hillsdale, New Jersey: Erlbaum.
Preface to the First Edition
This book deals primarily with the preschool child's conception of
number and how that conception develops. This central concern is,
however, embedded in a larger framework, namely, an overriding in
terest in early cognitive capacities and their relationship to subsequent
capacities. The initial chapters develop this broader framework.
The account of the young child's number concepts presented in the
central chapters grew out of research by Gelman and her students on
the cQgnitive abilities of preschool children. The first few chapters
constitute a general preamble to the specific topic, the nature and de
velopment of number concepts. The preamble motivates our presen
tation of the experimental work on number, which focuses on the task
of uncovering what young children do have in the way of numerical
concepts. After reviewing previous work on the young child's use and
understanding of number and highlighting the need to investigate
counting abilities in children of preschool age, we present a model of
counting abilities. The model is used to account for the data we have
collected on the way young children count. We then turn to a consid
eration of how young children reason about number, treating count
ing as an algorithm that creates the representations of numerosity
employed in reasoning. Having covered the way preschoolers repre
sent and reason about number, we compare and contrast formal de
scriptions of arithmetic with our description of what preschoolers do
and do not do in dealing with number. This comparison sets the stage
for our discussion of the what and how of development.
The theory of the preschooler's understanding of number that
emerges from our work is designed to account for the considerable
range of things that the child can do as well as for the equally con
siderable range of things the young child cannot do. We aim to con
vince the reader that number concepts are a rewarding area for devel-
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