Quantitative problem solving processes in children.

313 Pages·2017·13.67 MB·English

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Doctoral Dissertations 1896 - February 2014
1-1-1977
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John J. Clement
University of Massachusetts Amherst
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QUANTITATIVE PROBLEM SOLVING
PROCESSES IN CHILDREN
A Dissertation Presented
By
JOHN J. CLEMENT
Submitted to the Graduate School of the
University of Massachusetts in partial fulfillment
of the requirements for the degree of
DOCTOR OF EDUCATION
May 1977
Education
(c ) John J. Clement 1977
All Rights Reserved
QUANTITATIVE PROBLEM SOLVING
PROCESSES IN CHILDREN
A Dissertation Presented
By
John J. Clement
Approved as to style and content by:
—
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As
Howard Feelle, Chairperson
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William Masalski, Member
Edward Riseman, Member
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Klaus Schultz, Member^
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Mario Fantini Dean
,
School of Education
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Acknowledgment
s
I would like to express my deep appreciation to the many people
from whom I have received advice and encouragement. First, I would
like to thank my advisor, Howard Peelle, for his energetic discussions
and criticisms and for all he has done to ensure that this study was
completed. I am grateful to the members of the committee. Bill
Masalski Klaus Schultz, and Ed Riseman, and to the Dean's repre-
,
sentative, Dick Konicek, for their helpful comments, guidance and
support
I would like to thank Jack Easley and Klaus Witz, whose approach
to cognition has influenced me deeply, for the intellectual stimula-
tion and generous advice I received during the year and a half I spent
as a visiting scholar at the Committee on Culture and Cognition, Uni-
versity of Illinois.
Lastly I would like to thank those who participated in the study:
the children, for their ideas and good cheer, and the teachers, for
their patient cooperation and help.
ABSTRACT
QUANTITATIVE PROBLEM SOLVING PROCESSES IN CHILDREN
May, 1977
John J. Clement, A.B., Harvard University
Ed.D. University of Massachusetts
,
Directed by: Professor Howard Peelle
Exploratory clinical interviews were conducted with third and
fourth grade students as they attempted to solve quantitative story
problems. Models of individual students' cognitive processes are de-
veloped that account for many of the diverse phenomena observed in
the interview tapes.
Observed phenomena include: acted-out solutions; counting-based
solutions; solutions via. a number sentence; immediate solutions; solu-
tions via written symbol algorithms; use of drawings; and spontaneous
activities related to inverse, commutative, associative, and distribu-
tive principles. Although some of the students' solution approaches
seem related to standard methods taught in school, others do not.
Several kinds of intuitive solutions to story problems ordinarily
solved via division or multiplication are reported from students who
have never studied these operations in school. These solutions involve
practical actions and suggest that the students possess a practical know-
ledge base which could be tapped as a foundation for learning arithmetic
concepts in school. The study also examines spontaneous occurrences
of solution approaches that are often referred to as heuristics, such
as: solving a problem in pieces; using more than one approach to attack
vi
a problem; generating related problems; and using a convergent trial
and error approach.
Some general features of the models of cognitive processes proposed
to account for these phenomena can be summarized as follows: Piagetian
and neo-Piagetian concepts such as assimilation, disequilibrium, intern-
alized actions, cognitive structures, and parallel structure activity
are utilized in these models and are related to specific instances of
observed behavior. Other concepts utilized include: competition for
dominance, external and internal assimilation, chaining, and recursion.
Several types of cognitive structures are discussed, including written
symbol algorithm structures, counting-based structures, and practical
action structures. These last structures organize actions such as shar-
ing objects or cutting an object in half. A method of diagramming is
used which allows one to model cognitive structure interactions in a
student as they occur in time.
The protocol analyses suggest that these children have knowledge
structures which are active and semi-autonomous in the sense that their
structures aggressively assimilate problem situations, generate related
cases, dominate other structures, drive explanations, and influence per-
ceptions. Many of the reasoning processes modelled take the form of
structure interactions that are spontaneous rather than being governed
by established, hierarchical procedures. Structures are shown interact-
ing in a manner similar to the way different species interact in an eco
system—conflicting with each other; cooperating with each other; and
interacting with the environment.
It is suggested that the intuitive knowledge structures and reason-
vii
processes discussed can be tapped as starting points for building
mathematical ideas in the classroom. This approach may help students
develop a knowledge of mathematics that is applicable to real-world
problems as opposed to merely being an isolated set of rules for
manipulating symbols on paper.
) )
viii
CONTENTS
Page
Acknowledgments
Abstract
v
List of Figures xiii
Chapter
I. INTRODUCTION AND RELATED RESEARCH 1
Introduction
1
.
Issues discussed (2); An initial model (3); De-
velopments in cognitive theory (4); Character-
istics of the study
(5 )
Related Clinical Research 6
'
Key Piagetian Concepts 13
A biological model for cognition (13); Patterns
in play and the origins of knowledge (15);
Structures (17); Assimilation (l8); Accommo-
dation and disequilibrium (l8); Action-oriented
knowledge (20); Internalized action (2l); The
symbolic or semiotic function (21
Neo-Piagetian Research 22
Microanalysis (23); Elementary, action-oriented
knowledge structures (24); Informal vs. formal
knowledge (2?); Structure activity and parallel
processing 28
( )
II. METHODOLOGY I 33
Studying the Conceptions of the Child 33
The Exploratory Clinical Interview 36
Analysis of the Protocols 4l
The Setting of the Study 43
43
The Students
Mathematics Program in the Students' Classrooms . 45
46
Transcript Notation
HI. INITIAL PROTOCOL ANALYSES 48
48
A Range of Solutions
Observations and theory (49); Use of the terms
'knowledge' and 'reasoning' (50