Construction of Arithmetical Meanings and Strategies

361 Pages·1988·16.94 MB·English

by  Leslie P. Steffe

Recent Research in Psychology
Leslie P. Steffe Paul Cobb
With the Collaboration of Ernst von Glasersfeld
Construction of
Arithmetical Meanings
and Strategies
Foreword by Hermine Sinclair
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Leslie P. Steffe
Department of Mathematics Education
The University of Georgia
Athens, GA 30601. USA
Paul Cobb
Department of Education
Purdue University
West Lafayette, IN 47907. USA
Ernst von Glasersfcld
Professor Emeritus
Department of Psychology
The University of Georgia
Athens, GA 30601, USA
With 9 Figures
Library of Congress Cataloging-in-Publication Data
Steffe, Leslie P.
Construction of arithmetical meanings and strategies.
(Recent research in psychology)
Bibliography: p.
I. Number concept in children~Case studies.
I. Cobb, Paul. II. Glasersfeld, Ernst von. III. Title.
IV. Series.
BF723.N8S74 1988 153.1 '5 87-32297
ISBN-13: 978-0-387-96688-5 e-ISBN -13: 978-1-4612-3844-7
001: 10.1007/978-1-4612-3844-7
© 1988 by Springer-Verlag New York Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York. NY 10010. USA).
except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with
any form of information storage and retrieval. electronic adaptation. computer software. or by similar
or dissimilar methodology now known or hereafter developed is forbidden.
The use of general description names. trade names, trademarks. etc. in this publication. even if the
former are not especially identified. is not to be taken as a sign that such names. as understood by
the Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone.
Camera-ready copy prepared by the authors using Microsoftword.
9 8 7 654 3 :1
Foreword
The studies presented in this book should be of interest to anybody
concerned with the teaching of arithmetic to young children or with
cognitive development in general. The 'eaching experiment· was carried
out with half a dozen children entering first grade over two years in
biweekly sessions. Methodologically the authors' research is original. It is
a longitudinal but not a naturalistic study, since the experimenter-teachers
directed their interaction with each individual child with a view to his or her
possible progress. It is experimental in the sense that two groups of
subjects were selected according to criteria derived from an earlier study
(Steffe, von Glasersfeld, Richards & Cobb, 1983) and that the problems
proposed were comparable, though far from identical across the subjects;
but unlike more rigid and shorter "learning" or ''training" studies it does not
include pre- and posttests, or predetermined procedures. Theoretically,
the authors subscribe to Piagefs constructivism: numbers are made by
children, not found (as they may find some pretty rocks, for example) or
accepted from adults (as they may accept and use a toy). The authors
interpret changes in the children's counting behaviors in terms of
constructivist concepts such as assimilation, accommodation, and
reflective abstraction, and certain excerpts from protocols provide on-line
examples of such proc'0es ses at work. They also subscribe to Vygotsky's
proposal for teachers utilize the zone of proximal development and to
lead the child to what he (can) not yet do· (1965, p. 104). The child's
actual level of development is determined by the type of problem he or
she can solve without help; the zone of proximal development is
determined by the dynamic possibilities of each child and evidenced by
the kind of problems he or she can solve in collaboration with an adult:
two children at the same actual level of development may have very
different zones of proximal development. The authors also seem to agree
with Inhelder who, in the context of a different type of learning study,
states that 'raining procedures should steer the subjects in the right
direction, even if this results temporarily in incorrect reasoning. Variations
are possible and it is certainly not true that for each acquisition there is
only one predetermined construction process· (Inhelder, Sinclair, & Bovet,
1974, p. 25). Vygotsky, by contrast, seems to see the teacher's role
mostly as that of a direct guide towards the adulfs conceptual
construction of the new problems to be mastered.
The book provides an extremely detailed account of the different
types of counting behavior of half a dozen children over two years
v
(interview sessions were videotaped, an absolute necessity for the study
of counting behavior, where gestures are as important as utterances).
The data allowed the authors to reach their main goal: to document the
many subtle changes in children's counting and to interpret them
theoretically. At the same time, the results of their intensive study lead the
authors to affirm that a major shift in the arithmetic curriculum is
necessary: they have cogently demonstrated that many of the
widespread presuppositions about what young children know and what
they do not know are erroneous, and that better insight into how children
come to ·do mathematics· should greatly improve the teaching of
arithmetic.
In the constructivist theory of knowledge it Is assumed that new
knowledge arises from new ways of asking questions and from the raising
of new problems rather than from the construction of more sophisticated
means of dealing with known difficulties, and that confrontation and
discussion are essential for progress. From this point of view the present
book is certainly constructivist: it will give rise to new problems and
hypotheses, and to approbation as well as controversy. I hope that it will
incite others to conduct teaching experiments along the lines pursued by
the authors.
Hermine Sinclair
University of Geneva
vi
Preface
In this book, we present the results of a teaching experiment that
was carried out during the academic years 1980-1981 and 1981-1982.
The term "teaching experiment" does not mean an investigation of
teaching a predetermined or accepted way of operating. Instead, it is
primarily an exploratory tool, derived from Piaget's clinical interview and
aimed at discovering what might go on in children's heads. Because it
also Involves experimentation with the ways and means of Influencing
children's knowledge, the teaching experiment is more than a clinical
interview. Whereas the clinical interview is aimed at establishing where
children are, the teaching experiment is directed toward understanding
the progress children make over extended periods of time. We used the
methodology to investigate children's construction of (1) counting
schemes, (2) uniting operations and their systems, (3) lexical and
syntactic meanings of number words, 1 and (4) thinking strategies.
Theoretical Assumptions
The constructivist teaching experiment is based on two theoretical
assumptions. The first Is' that children, when faced with problematic
arithmetical situations, can develop their own solution methods. The
second assumption Is that any knowledge that involves carrying out
actions or operations cannot be instilled ready-made into children but
must, quite literally, be actively built up by them. These assumptions are
based on Piaget's (1970) analysis of knowledge:
Human knowledge is essentially active. To know is to
assimilate reality into systems of transformations. . .. I find
myself opposed to the view of knowledge as a copy, a
passive copy of reality. (p. 15)
A recent report on education expresses a similar view:
1 By "lexical meaning" we intend whatever meaning is associated with a
single word, and by ·syntactic meaning", those meanings that arise out of
the combination of words.
vii
All genuine learning is active, not passive. It involves the
use of the mind, not just memory. It is a process of
discovery, in which the student is the main agent, not the
teacher. (Adler, 1982, p. 50)
On the other hand, where mathematics teaching is concerned, we agree
with Menchinskaya (1969): "Neither scientific nor everyday concepts
spring forth spontaneously; both are formed under the influence of adult
teaching" (p. 79). From the perspective of the two theoretical premises
we have mentioned, the process of teaching in the teaching experiment
took on the form of guiding the students conceptual construction rather
than attempting to impart the "correct", adult way of doing things.
There is no reason to assume that a child will interpret a given
situation in the way that seems "obvious" to an adult or a teacher, nor can
one assume that a child will necessarily "see" that a particular way of
proceeding must reSUlt in the more standard adults solution. We believe
it is crucial to adopt this attitude for teaching mathematics in early
childhood, because little commonality can be assumed between the
teacher's and the children's conceptual structures. Piaget has
demonstrated the difficulty of constructing operations that manifest the
reversibility that is critical to an understanding of arithmetic. However, it is
easy to forget that most children, when they enter school, have not yet
constructed operations of this kind, and explanations or demonstrations
alone are not sufficient to make the children do so.
From our perspective, the essential pedagogical task is not to instill
"correct ways of doing" but rather to guide children's constructive
activities until they eventually "find" viable techniques. Such guidance
must necessarily start from points that are accessible to the children; in
order to establish these starting-points we must first gain insight into the
children's conceptual structures and methods, no matter how wayward or
ineffective they might seem to us, as adults. If there is any virtue in the
notion that children as cognitive organisms assimilate their experiences
using structures they already possess, and accommodate these
structures when their use does not lead to the expected result, it follows
that the teacher will be far more successful in precipitating
accommodations if he or she has gained some notion of what the child's
present structures and ways of operating are.
Purposes of the Teaching Experiment
The main purposes of the teaching experiment, based on the above
considerations, were to build a model of cognitive changes in children's
initial, informal "number sequences" and then workable means of
influencing their conceptualizations and activities. These informal number
sequences took the form of counting schemes; the driving force of the
viii
teaching experiment was to Investigate changes In the counting schemes
of six prenumerlcal children In teaching episodes during their first and
second grades In school.
We made no attempts to provide the participating children with an
alternative to their regular school mathematics program because we used
the teaching episodes as an investigative, experimental tool. The intent
was to create a working model of each child's current counting scheme
and then to formulate and test hypotheses about the models. We were
most interested In how various components of those models might
change In the context of solving problem situations.
Teaching Episodes
In a teaching episode, no simple cause-effect relationship is
assumed between the teacher's acts and what the child does because
experience, qua experience, is the individual's own. As teachers, we had
the tasks of (1) interpreting what we saw the children doing and (2)
attempting the ultimate act of decentering by imagining of our actions
from the children's perspective. In this sense, the course of a teaching
episode was also determined by the child. Each problem that we used in
a teaching episode was designed for a particular child; our model of that
child and its possible modifications constituted the rationale for each
problem. We teachers were responsible for devising Impromptu problems
during a teaching episode after we had interpreted the child's solutions to
prior problems. In fact, these spontaneously designed problems
represented a major modus operandi of the teaching experiment.
However, the focus was not on the problems as the adults conceived of
them, but rather as they were constructed and solved by the children.
Our primary objective was to give the children opportunities to abstract
patterns or regularities from their own sensory-motor and conceptual
activities. Guided by our current model of each child's counting scheme,
we hypothesized certain patterns or regularities that it might be possible
for a particular child to abstract. Activities were then initiated in the hope
that the child would reflect on and abstract those patterns or regularities
from his or her experiences.
As a matter of course, the teaching episodes were videotaped and
this record was used for various purposes. In this book, we are
concerned with the retrospective analyses at the end of the teaching
experiment.
Retrospective Analyses
We have found that there is no one "besr way to present the
extensive interactions among participants in a teaching experiment.
Choices have to be made, based on the goals of the investigators. One
ix
choice, the children's mathematical behavior and our interpretations of it,
essentially eliminates the extensive discussions that went on while the
teaching experiment was in progress and the almost daily decisions we
made concerning our teaching strategies and tasks. Our intention--to
analyze the children's constructions rather than how we adults might have
influenced them--was to study mathematical learning rather than
mathematical teaching. Nevertheless, mathematical learning was
observed in contexts where we teachers intervened to influence learning,
examine its limits, and examine the children's generative power and
flexibility.
The book contains the results of retrospective analyses that were
conducted after the teaching experiment was concluded in June 1982.
We had approximately 64 videotape 1S-to-30-minute segments for each
child, which we first analyzed in the sequence in which they were taped.
During the course of the teaching experiment, we had learned a
considerable amount about each child and had documented points of
progress. While these proved invaluable, our interpretations often
changed dramatically in the retrospective analyses.
Overview of the Book
Chapter I contains a theoretical analysis of the children's
construction of the counting scheme and the major hypotheses we
investigate in the next three chapters. Chapters II, III, and IV contain the
results of our study of the children's construction of their counting
schemes. Chapters II and III concern the three children who began the
teaching experiment in a perceptual period, and Chapter IV concerns th~
three children who began the teaching experiment in a motor period.
Case material is presented for each child. From the extensive videotaped
data, we have selected what we believe are critical protocols to document
our observations and interpretations. These protocols are "snapshots" of
the dynamiC interactions among teachers and children over 2 years.
In Chapters V, VI, VII, and VIII, we present independent analyses of
the changing meanings and strategies of the children and correlate these
changes with those in the children's counting schemes. Because these
analyses were conducted separately from the analyses of the children's
counting schemes and independently of each other, they lend credibility
to the relationships that emerged.
2 The phrases "perceptual period" and "motor period" refer to the type of
items the children could create and count. "Perceptual unit items" refer to
those items created from visual, auditory, or tactile signals and "motor unit
items" refer to those items .created from kinesthetic signals.
x
In the chapters on meaning, the presentation of protocols
established In the earfier chapters Is continued. For the chapters on
strategies, the focus changes from documenting the bases for our
interpretations to presenting our interpretations; the emphasis on the
individual children, however, is maintained. This change in style of
presentation is warranted, because we refer to the protocols presented in
the preceding detailed analyses of lexical and syntactical meanings.
Chapter IX contains an analysis of the cognitive changes in the
counting schemes of the six children over the duration of the teaching
experiment. Different types of accommodations the children made are
identified, and the role of experience is clarified for each type. A new,
ontogenetic model for the children's construction of the number sequence
is correlated with a phylogenetic analysis of the development of the
number sequence.
Any reader who has conducted videotape analyses can appreciate
the amount of time involved in playing a videotape, replaying segments,
transcribing protocols, correlating current observations with previous
ones, and searching for behavioral indicators of certain concepts or
operations. We invite readers to conduct their own analyses of the
protocols, which should be stimulating and interesting and perhaps give
rise to plausible competing hypotheses.
The Participating Children
The 6 participating children were selected from 20 first-graders who
were interviewed in October and November of 1980. The children were
from two classrooms in an Athens, Georgia, elementary school.
Interviews were conducted to establish the quality of the children's
number word sequences, the types of unit items they could count, and
patterns they could recognize, re-present, and coordinate with counting.
We selected three children in their perceptual periods--Brenda, Shawn,
and Tarus. Unfortunately, Shawn left the Athens area in February 1981,
and James, another child in his perceptual period, was chosen as
Shawn's replacement. Three children in their motor periods--Tyrone,
Jason, and Scenetra-were also selected. We taught each participant
twice a week for approximately 16 weeks during the school years 1980-
1981 and 1981-1982 beginning in December of each year.
xi