-_-vv..-..-...-V.L “\‘Wfqm‘y

THE DETECTION AND IDENTIFICATION OF
COMPREHENSIVE PROBLEM SOLVING STRATEGIES USED '
BY SELECTED FOURTH GRADE STUDENTS

Dissertation for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
JOSEPH JENNINGS SHIELDS

1976

 

 

 

This is to certify that the

thesis entitled
THE DETECTION AND IDENTIFICATION OF COMPREHENSIVE
PROBLEM SOLVING STRATEGIES USED BY SELECTED
FOURTH GRADE STUDENTS
presented by

Joseph Jennings Shields

has been accepted towards fulfillment

of the requirements for .
Mathematlcs

Ph.D . degree in Education

 

 

7 51/61sz 41,.) @-(b

Major professor

 

Date April 19, 1976

0-7639

ABSTRACT

THE DETECTION AND IDENTIFICATION OF COMPREHENSIVE
PROBLEM SOLVING STRATEGIES USED BY SELECTED
FOURTH GRADE STUDENTS
BY

Joseph Jennings Shields

For the past seventy-five years many persons, within
and without the academic community, have called for more
real life problem solving in the elementary schools.

During the past ten years some programs have been developed
to teach problem solving for specific subject areas and, in
a few cases, to teach problem solving of an interdisciplin-
ary nature. However, these programs have, in general,
assumed a model of problem solving which is approPriate for
adult subjects. The purpose of this study is to examine
the problem solving behavior of selected fourth grade
youngsters to detect what strategies they use in the process
of solving Open ended problems, and to determine if the
process they utilize coincides with that of the model ap-
propriate for adult subjects.

A popular view of human problem solving is based on

an informational processing approach to problems. The

Joseph J. Shields

solver processes the problem in four steps: (1) clearly
define the goal of the problem, (2) generate a number of
alternatives, (3) synthesize information concerning these
alternatives, and (4) select the solution which is most
consistent with the goal.

Once the problem is defined, one or more search strat-
egies can be employed to aid the solver in reaching the
optimal solution. These strategies include: (a) making
inferences, (b) looking at related problems, (c) identify-
ing subgoals, (d) use of contradiction, and (e) working

backwards.

Procedures

 

Three male and three female subjects were randomly
selected from each of two fourth grade classes, the classes
being denoted the experimental and control classes. The
particular classes were chosen by the researcher because
of the students openness and independence shown in self-
initiated activities.

The study involved five interviews for each of the
twelve subjects, and was conducted at regular intervals
over a four month period. The purpose of these interviews
was to gather information concerning the manner in which
the students solved comprehensive problems.

During the study period the experimental class re-
ceived some problem solving training. The results of these

interviews were analyzed to suggest answers to the

Joseph J. Shields

following questions:

a) Is there an identifiable process which elementary
school children use in solving open ended compre-
hensive problems?

b) Which heuristics are used by elementary school
children engaged in solving these comprehensive
problems?

c) Does problem solving training affect the process
or strategies used by the successful fourth
grade problem solver?

d) Do youngsters who are engaged in regular problem
solving activities become more mature in their
view of the nature of problems, solutions, and the
means of attaining these solutions?

Problems and Results

 

For each of the five interviews conducted, the same
four step problem solving process was apparent in those
cases where the subject reached meaningful solutions.

The first interview involved finding a solution to an
evaluation type of problem: Determine which of five given
notebooks is the best for a school district to buy for all
fourth graders. The generalized problem solving strategy
of making inferences was used by all of the successful solv-
ers. Designing a classroom suitable for fourth grade young-
sters was the focus of the problem in the second interview.
Subjects used contradiction, made inferences, identified
subgoals, and looked at related problems in reaching solu—
tions to this problem. These strategies were also evident
in the solution to the third challenge of planning an ice

skating party for twenty-five children, and in the last

Joseph J. Shields

interview, concerned with planning and laying out a
playground.

The fourth interview centered on the solution to a
describing pe0ple problem: Determine which characteris-
tics are most useful in describing a person. Due to the
students inexperience with a sampling type of problem,
subjects made little progress in reaching partial solutions,

and did not employ generalized strategies in this problem.

Conclusions and Recommendations

The researcher concluded that fourth grade children
do employ an identifiable process in addressing comprehen-
sive problems, and this process sometimes involves the use
of one or more problem solving strategies. There were no
differences in the quality of solutions given by either the
experimental or control class. The solving process and
strategies became more refined as the study progressed, and
indicates a growth in problem solving ability of these
subjects. I

Based in part on these conclusions, the researcher rec-
ommends that more problem solving activities be carried out
at all grade levels. Secondly, new and existing programs
in problem solving training should be structured to build on

the natural processes and strategies of youngsters.

THE DETECTION AND IDENTIFICATION OF COMPREHENSIVE
PROBLEM SOLVING STRATEGIES USED BY SELECTED

FOURTH GRADE STUDENTS

BY

Joseph Jennings Shields

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Elementary Education

1976

® COpyright by
JOSEPH JENNINGS SHIELDS

1976

Dedicated to,

My Father and Mother,
Joseph and Mary Shields, Sr.

My Wife,
Kathryn

My Children,

Joseph III, my son
Christine and Ann, my daughters.

ii

ACKNOWLEDGMENTS

A number of people have taken an active interest in
the successful completion of this dissertation, and the
author wishes to extend a special thanks to several of them:
Professor William Cole, my program and research advisor, for
his guidance and continual support and encouragement that
this program would be completed; Professor William Fitzgerald
who was the first to teach me about USMES and encourage
me to undertake this type of study; Professor Bruce Mitchell,
the first person I chose for my doctoral committee, for his
friendship and research suggestions: Professor William
Schmidt for his aid in formulating the research proposal;
Professor John Masterson, who joined my committee so my
research could be completed and defended this year. To each
of these faculty members I eXtend a heartfelt thanks. Thanks
must also go to my father, Joseph Shields, Sr., for the many
references and ideas he supplied during this research.

A special thanks must go to those members of the
Ypsilanti School District who c00perated in this research:
Mr. Paul Kacanek, the Coordinator/Evaluator for the district,
for his help in finding suitable classes and for information
concerning the district; Mr. George Eaton, the principal of

Erickson Elementary School, who provided a location for the

iii

interviews; Mrs. Larson, and Mrs. Britton, the classroom
teachers, who gave freely of their time and provided
excellent students for this study.

I wish to thank Lawrence Institute of Technology for
partial financial support and continual cooperation during
my graduate work. My deepest thanks go to Cynthia Welsch,
who prepared some of the test materials used in this study
as well as corrected and typed the drafts of this thesis.

I thank each member of my family for their under-
standing and love which gave me the courage to continue
when I knew they needed my time. Finally, I want to thank
my two closest friends during these years at MSU, Dr. Frank
Fishell, and Mr. Curtis Walker, for the perspective they

gave my work.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . .

Chapter

I.

II.

THE PROBLEM . . . . . . . . .

Background and Need for the Study . .

Purpose of the Study . . . .
Definition of Terms . . . .
Procedures Lo 0 o o o o o o o

Assumptions and Limitations of the

THEORY AND SUPPORTING RESEARCH
Introduction . . . . . . . .
PART ONE . . . . . . . . . . .

A Theory of Problem Solving
Components of a Problem . .

Sequence of Events Involved in

Solving . . . . . . . . .
Presentation of the Problem
Problem Definition . . . . .
Devising a Plan . .'. . . .
Inference . . . . . . . . .
Identification of Subgoals .
Examining Related Problems .
Producing Solutions . . . .
Contradiction . . . . . . .
Working Backward . . . . . .
Deciding Among Solutions . .

PART TWO 0 I C O O O O O O O 0

Problem

Review of the Related Literature . . .

General Problem Solving . .

Creativity and Problem Solving . . . .
Problem Solving as Related to Other

Variables . . . . . . . .

viii

ix

III. IMPLEMENTATION or THE STUDY . . . . . .

Introduction . . . . . . . . . . . . .
Population and Classroom Selection . .
Descriptions of Classes and Sample . .
Description of the Problems Used in the
Interviews . . . . . . . . . . . . .
Interviewing Procedures . . . . . . .

IV. STUDENT RESPONSES TO THE PROBLEMS . . .

Introduction . . . . . . . . ... . . .
The Notebook Problem . . .'. . . . . .
Responses of the Experimental Class
Responses of the Control Class . . .
Identification of the Process and
Strategies Used by the Students .
Classroom Design Problem . . . . . . .
Responses of the Experimental Class
Responses of the Control Class . . .
Identification of the Strategies
Used by the Students . . . . . . .
Planning a Party Problem . . . . . . .
Responses of the Experimental Class
Responses of the Control Class . . .
Identification of Strategies Used by
the Students . . . . . . . . . . .
Describing People Problem . . . . . .
Responses of the Experimental Class
ReSponses of the Control Class . . .
Identification of the Strategies Used
by the Students . . . . . . . . .
The Playground Design Problem . . . .
Responses of the Experimental Class
Responses of the Control Class . . .
Identification of the Strategies Used
by the Students . . . . . . . . .

V. DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS

Contrast Between Classes and Individual
Growth . . . . . . . . . . . . . . .

Identification of Specific Strategies
Used by Subjects . . . . . . . . . .

Recommendations . . . . . . . . . . .
Research Recommendations . . . . . .
Curriculum Recommendations . . . . .

vi

62

62
63
67

71
77

79

79
79
81

85
87
88
90

92
96
98
101

104
109
110
113

116
118
119
121

123
128

128

131
134
134
137

APPENDICES . . . . .

A. Teacher Questionnaire

B. Teacher's Guide

C. Classroom Outline
D. Neighborhood Map Indicatin

Rinks

E. Playground Outline .

SELECTED BIBLIOGRAPHY .

vii

9 Ice Skating

139
139
144
202

203
204

205

1.

LI ST OF TABLES

Elapsed Time to Solutions . . . .

'viii

81

1.

2.

Ordered Subgoals

Unordered Subgoals

LIST OF FIGURES

ix

30
31

CHAPTER I
THE PROBLEM

Background and Need for the Study

The cry to make "real world" problems a prominent part
of the curriculum has been heard at various times during the
past seventy-five years. At the turn of the century the
”Perry Movement" in this country received great impetus from
E.H. Moore in his presidential address1 before the American
Mathematical Society at its annual meeting in 1902. During
this address he stated: "The fundamental problem (in the
pedogogy of elementary mathematics) is that of the unifica-
tion of pure and applied mathematics." Later, the Progres-
sive Education Association report of the Committee on the
Function of Mathematics in General Education2 in 1938 includ-
ed a chapter on some concepts basic to problem solving. The
Committee felt very strongly that "the major role of

mathematics in deve10ping desirable characteristics lies in

 

lE.H. Moore's presidential address before the American
Mathematical Society at its ninth annual meeting, December
29, 1902. The address was printed first in Science n.s. l7
(March,'1903).

2Progressive Education Association, Mathematics in
General Education: A Report of the Committee on the Function
gngathematics in General Education. 1938. In Readings In
The History of Mathematics Education, NCTM (Washington D.C.:
1970) , pp. 534-566.

 

 

 

 

 

1

2

the contribution it can make to growth in the abilities in-
volved in . . . problem solving."

To teach students to solve "real world" problems,
educators have emphasized mathematical skills and problem
solving techniques which could be learned by solving
simplified models of "real" problems. This distinction
between problems which are personal to the student and exer-
cises presented in class has produced varied meanings to
"problem solving."

Written material on the methods of problem solving has
appeared at various times during the last three score of
years. Perhaps the most widely read book on this subject

is How To Solve It by G. Polyal which presented the basic

 

concepts of problem solving and strategies for teaching
problem solving. During the sixth decade of this century
the twenty-first yearbook of the National Council of
Teachers of Mathematics contained a chapter on "Problem
Solving in Mathematics."2 This chapter discussed the theory
of problem solving and the implications of this theory for

the classroom.

 

1G. Polya, How to Solve It, (2nd ed., Garden City, N.Y.,
Doubleday and Company, Inc., 1957).

2Kenneth B. Henderson and Robert E. Pingry, "Problem
Solving in Mathematics," The Learning of Mathematics: Its
Theory and Practice, Twenty-first yearbook of the NCTM
(Boston, Houghton Mifflin Co., 1953).

 

 

 

General educators and learning theorists such as
Thorndike and Ausubel1 have investigated problem solving;
they state the need for problem solving practice cannot be
overemphasized. O'Brien and Shapiro2 interpret Piaget as
saying that the child is liberated from his egocentric view
of reality by social interaction and by confronting problems
in his daily experience. Hence it is "problem (solving) . .
that is a central vehicle for cognitive growth."3 Finally,
mathematics educators in the post "New Math" years continue
to support and encourage problem solving activities. For
example, Johnson and Rising4 state, "learning to solve pro-
blems is the most significant learning in every mathematics
class. . .”

The impact of this emphasis on problem solving, by

virtually every major reform suggestion of this century, is

 

1Robert L. Thorndike, "How Children Learn the Principles
and Techniques of Problem Solving," Learning and Instruction,
Forty-ninth yearbook of the National Society for the study»
of Education, (Chicago: University of Chicago Press, 1950),
pp. 191-216; David P. Ausubel, "Learning by Discovery,"
Educational Leadership, XX (January, 1969), p. 117.

2Thomas C. O'Brien and Bernard J. Shapiro, ”Problem
Solving and the DevelOpment of Cognitive Structures," The
Arithmetic Teacher, XVI (January, 1969), p. 11.

3

 

 

Ibid., p. 12.

4Donovan A. Johnson and Gerald Rising, Guidelines for
Teachinngathematics, (Belmont, California: Wadworth
Publishing Company, 1967), p. 104.

 

 

apparent in the Cambridge report of 1963 and in the School
Mathematics Study Group (SMSG) report of a Conference on
Secondary School Mathematics, 1966. The first of these re-
ports states, ". . . the problem material (in textbooks)
should get at least half of the time and attention of the

1

authors." And in the March, 1966 meeting of the SMSG

Conference, the committee on problem solving stated,
". . . the activity of constructing and analyzing mathematic-
al models of scientific and life situations is, apart from
technical questions, the principle link between mathematics
and the rest of civilization."2

The SMSG Secondary School Mathematics Units were
developed during the late sixties to ". . . make clear to
all students that mathematics is indeed useful, that it can
help us in understanding the world we live in and in solving

3 The fourteen units in

some of the problems that face us."
this program are an integrated plan for high school mathema-

tics containing substantial material on problem solving. The
committee on problem solving wanted "understandable remarks

and digressions on models and modeling and their various

aspects (to) be liberally . . . sprinkled throughout the

 

1Goals for School Mathematics, The Report of the
Cambridge Conference on School Mathematics, (Boston:
Houghton Mifflin Co., 1963), p. 28.

2SMSG: A Report of a Conference on Secondary School
Mathematics, March 14-16, 1966.

3The Panel on Secondary School Mathematics of the
School Mathematics Study Group, SMSG: Secondary School
Mathematics, Leland Stanford Junior University, 1971, page i.

 

 

 

mathematical education process."1 Problem solving material
appears in nine of the twenty eight chapters and deals with
problem formation, use of diagrams, tables, guesses, and
deductive reasoning.

Other mathematics programs, developed during the
sixties, utilized problem solving as an integral part of
the teaching and learning process. For-example,the Minneso-
ta School Mathematics and Science Teaching Project
(MINNEMAST) and the Madison Project both challenged the
student to discover solutions to problems through experi-
mentations, collecting data and making conjectures. However,
neither SMSG, MINNEMAST, nor the Madison Project had
comprehensive problem solving as a major theme. Indeed, the
recent conference on the K-12 mathematics curriculum held in
Snowmass, Colorado during June, 1973 concluded, "the process
of solving problems of any description has not been given
due attention in curriculum material."2 Throughout the
conference, the topic of problem solving was a continual and
pervasive subject of discussion. The conference report

contains the following suggestions:

 

lIbid.

2The Report of the Conference on the K-12 Mathematics
Curriculum, Snowmass, Colorado, Mathematics Education
Development Center, Indiana University, 1973, p. 33.

 

1. There is a need for a center or centers for basic
research in how children (students) solve problems.

2. There is a need for this same center to develop
curriculum material for both school students as
well as teachers.

There have been programs which attempted to cultivate
problem solving skills and creativity in the learner. For
example, the Creative Education Foundation at the State
University of New York at Buffalo has concentrated on the
improvement of adult creative problem solving through special
courses.2 This foundation published a student workbook,
instructor's manual and visual aids for use in teaching
creative problem solving courses for business, adult educa-
tion, military and university groups.3 A program developed
at the University of Minnesotatx>nurture creativity in
Elementary School children was titled "Invitation to Think—
ing and Doing" and was under the direction of R. E. Myers

and E. Paul Torrance.4 Currently, there are several

universities such as University of California at Los Angeles

 

1The Repprt of the Conference on the K-12 Mathematics
Curriculum, p. 34.

2Alex F. Osborn, Applied Imagination: Principles and
Procedures of Creative Problem Solving, (3rd Rev. ed.),
New York: Charles SchbnerTs Sons, 1963.

3S. J. Parnes, Creative Behavior Guidebook, New York:
Charles Scribner's Sons, 1967.

4R. E. Myers and E. Paul Torrance, Invitation to
Thinking and Doing. (Bureau of Educational Research, Univer-
sity of Minnesota, Perceptive Publishing).

 

 

 

 

 

 

 

and Massachusetts Institute of Technology which offer
courses in problem solving to undergraduates.

During the academic years 1974-1975 and 1975-1976 the
American Association for the Advancement of Science, with
support from the National Science Foundation, conducted
several short courses for college teachers entitled "Patterns
of Problem Solving."1 This course is designed to provide a
new dimension in higher education that crosses boundary
lines between disciplines. It is designed to provide the
attitudes and skills necessary in dealing with complex
problems, from the statement of the problem to its creative
solution. The National Science Foundation also supported
a new program, located in centers at Indiana University,

The University of Northern Iowa, and The Oakland Inter-
mediate School District, Oakland County, Michigan. This

new program is developing curriculum material for the teach-
ing of problem solving to elementary school pupils, using
the hand held calculator.

At the elementary level, the only current program of
widespread implementation whose major theme is problem
solving is the Unified Science and Mathematics for Elementary
School (USMES) program. The formulation of the USMES project

was in response to the recommendation of the 1967 Cambridge

 

1Announcement: NSF Chautauqua - Type Short Courses for
College Teachers, Academic Year 1974;1975’(Of?IEe of Science
Education, Washington, D.C.).

 

 

Conference on the Correlation of Mathematics and Science
in the Elementary School.1 Since its inception in 1970,
USMES has developed units which center on long range in-
vestigation of real and practical problems taken from the

local school and environment of the community.2

Purpose of the Study

The purpose of this study is to collect information
about the problem solving behavior of selected fourth grade
students, and analyze this information to detect and
identify the strategies used by those youngsters in
addressing comprehensive challenges. This case study of
five interviews with twelve subjects, randomly selected from
two classes, seeks to find regularities in their solving
process. From these regularities the researcher will
suggest answers to the following questions:

a) Is there an identifiable process which elementary
school children use in solving open ended
comprehensive problems?

b) Which heuristics are used by elementary school
children engaged in solving these comprehensive

problems?

 

1Goals for the Correlation of Elementary Science
and Mathematics, Houghton Mifflin Co., I966.

2USMES Guide, Education Development Center, Inc.,
Boston, June 1973.

 

 

 

c) Does problem solving training affect the process
or strategies used by the successful fourth grade
problem solver?

d) Do youngsters who are engaged in regular problem
solving activities become more mature in their
view of the nature of problems, solutions, and

the means of attaining those solutions?

Definition of Terms

 

No single definition of a meaningful problem and
problem solving as a process is agreed upon by mathematics
educators. For example, at the Snowmass Conference the
meaning of problem ranged from "the simple arithmetical
examples usually found in elementary texts, to the more
formal mathematical problem sets associated with abstract
mathematics."1 Polya defines what he thinks it means to
have a problem by saying that the individual searches
". . . consciously, for some action apprOpriate to attain
a clearly conceived, but not immediately attainable, aim."2

Dodson defines a problem by suggesting that it has the

following characteristics:

 

1The Report of the Conference on the Ke12 Mathematics

 

Curriculum, Snowmass, Colorado, p. 33.

 

2George Polya, Mathematical Discovery: On Under-
standing, Learning, and Teaching ProEIem Solvipg, (John
Wiley & Sons, Inc., 19727] p. 117.

 

 

10

1) It is a situation or question for which the
student does not already possess a solution, or
answer, or a method of obtaining the solution.

2) It is solvable using previous learning.

3) It cannot be solved by simple recall from memory
or the standard use of a computational algorithm.

4) Its solution does not depend on a special trick.1
Robinson, Tickle, and Brison define a comprehensive problem
as "a question that leads to alternatives, and where the
solver must synthesize additional information to decide
which alternative is the best answer to his question."2
Johnson and Rising define problem solving as "finding the
appropriate response to a situation which is unique and

3 All of these definitions

novel to the problem solver."
agree that the subject has a goal, or aim, which is tempo-
rarily unattainable or blocked. Non-mathematical writers

such as Vinacke and Williams4 agree on this characterization

of a problem.

 

1Joseph Dodson, Characteristics of a Successful In-
sightful Problem Solver, (NLSMA Report No. 31, School
Mathematics Study Group, 1972), p. 9.

2F.G. Robinson, J. Tickle, and D.W. Brison, Inguiry
Training: Fusing Theory and Practice (Ontario Institute
for Studies In Education, Toronto, Ontario, 1972). p. 4.

3Donovan A. Johnson, and Gerald Rising, Guideline for
Teaching Mathematics.

4Edger W. Vinacke, The Psychology of Thinkipg, (N.Y.:
McGraw-Hill Book Co., 1952), p. 160: Frank Williams,
Foundations of Creative Problem Solving: Principles and

_—‘_‘-

Applications, (Ann Arbor: Edwards Bros., Inc., 1960), P. 20.

 

 

 

 

 

 

 

11

In this study, the researcher shall use problem
solving to mean that activity the student engages in when-
ever he or she consciously looks for answers to a problem
which:

1. Presents a situation or challenge for which the
student desires a solution and to which there is
no immediate correct answer.

2. The student can arrive at a solution compatible
with his understanding of the problem and his
maturation.

3. Solutions cannot be found by employing special
tricks or a simple sequence of algorithms, i.e.,
it is complex.

This definition is global enough to include a wide variety
of problems, and implies that each problem has some condition
or conditions which are given, as well as transformations
which can be used on the given to reach the goal or aim.

It is assumed that the student desires a solution either

as an end in itself or as a means to an end, since problem
solving is an active cognitive process. The student who
already possesses a solution or does not desire a solution
will not act on the givens in a problem to arrive at a
solution. Furthermore, comprehensive problems may have more
than one "correct" solution, and thus, the choice of a
solution is dependent on the solvers understanding of the
problem and desired goal. Finally, the problem must be
complex and require the solver to work at the higher cogni-
tive levels. This third condition eliminates most of those

problems referred to as exercises, in many texts, which can

be solved by simple recall or the use of an algorithm

12

presented in class. Each of the five problems solved by
the subjects in this study satisfy the conditions listed
above.

The solution of a problem is a complete plan which
can be carried out to arrive at the precise goal the
solver desired. Note that a problem has reached a solution
when the learner is aware of the correct course of action
leading to the goal, irrespective of whether the plan is
implemented. Therefore, to evaluate the "correctness" of a
solution it is necessary that the researcher have the solver
report a solution, or indicate the plan which will lead to
the solution. This definition of asolution is equivalent
to the operational definition which Wickelgren gives,
". . . a solution is a sequence of allowable actions that
produces a completely specified goal expression."l Polya
also agrees with this characterization of a solution as a
plan leading to the goal or aim.2

To reach a solution, the solver will often use pro-

cedures, called strategies, which aid in organizing data

and eliminating alternatives. Kilpatrick3 calls these

 

1Wayne A. Wickelgren, How To Solve Problems: Elements
of a Theory offProblems and Problem Solvipg, TW.H. Freeman
and Co., San Francisco, 1974), p. 16.

 

 

2George Polya, Mathematical Discovepy: On Understand-
ing, Learning, and Teaching Problem Solepg, p. IIB.

3Jeremy Kilpatrick, Analyzing the Solution of Word
Problems in Mathematics, UnpuBlIEhed'doctoraI’dissertation,
Standord University, 1967, p. 19.

 

 

 

 

13

strategies, heuristics, and defines them ". . . as any rule,
technique, rule of thumb, etc., that improves problem solv-
ing performance." Covington1 lists thirty-eight of these
rules of thumb for problem solving, and McEver,2 in
Strategy Notebook: Tools for Change, provides fifty-nine
strategies for youngsters to practice.

In this study a generalized problem solving strategy,
or simply a search strategy, will mean one of five general
procedures which can be used in open ended problem solving.
These five are: (1) inference, (2) using a related problem,
(3) identifying subgoals, (4) contradiction, and (5) working

backwards.

Procedures‘

 

Two fourth grade classes, from schools near a large
metropolitan area, were selected based on the following
criterion:

1. Students learned in an environment where indivi-

dual feeling, creativity, and intellectual

activity were highly valued{ff

2. Students could accurately express their Opinion
when questioned.

3. Students had never worked on, or participated in,
any problem solving program.

 

lMartin V. Covington, "An Experimental Program for

Increasing Ingenuity in Visual Problem Solving," University
of California, Berkley, 1968.

2Catherine McEver, Strategy Notebook: Tools for
Change, (Interaction Associates, Berkley, CaIiforfiia, 1969).

14

4. Students were male and female, and from diverse
socio—economic backgrounds.

Three males and three females were selected randomly from
each of the classes described above. A two-way analysis of
variance was computed to help determine whether the classes
were similar in problem solving ability, using scores from
the Purdue Elementary Problem Solving Inventory as the
independent variable. This analysis supported the hypothe-
sis that there was no difference in the problem solving
ability of the two classes, or between sexes, at the onset
of the study.

One class was differentiated from the other by having
the teacher integrate problem solving activities into the
usual classroom work during the duration of the study.
These activities took one of two forms: 1) a comprehensive
challenge of the USMES type, or 2) a short puzzle type
problem.

The study involved tracing the behavior of the twelve
students during five interviews, distributed over a four
month period, in which they attempted to solve open ended
comprehensive problems. All of the interviews were conduct-
ed in a comfortable classroom of the school building, and’
during each interview only one subject and the researcher
were in the room. The interviews were recorded on a
cassett recorder, and observations made by the interviewer
were transcribed. These sixty interviews were analyzed to

find evidence to support or reject the claim that youngsters

15

use an identifiable process and specific strategies when
solving open ended problems.

The use of the clinical paradigm was appropriate for
this study, as the problem solving process used by the
subjects had to be analyzed in great detail to identify
strategies, and because any variation in the process or

strategies was of interest.

Assumptions and Limitations of the Study
For purposes of this study, the following assumptions
were applied:
a) That the comprehensive problems used during the
interviews were appropriate and interesting to

the subjects.

b) That all of the subjects wanted to participate in
the study.

c) That variations between the subjects have a ran-
dom effect on the results, and do not produce
erroneous conclusions.

d) That the setting and population in which the study
was conducted was not so unusual that the outcomes,
within limitation, could not be used to generate
hypotheses for similar populations.

This study was designed and undertaken with the following
restriction: Only students in the sample were in the study,
and any generalization of the results is limited to popu-

lation similar to the experimental group.

CHAPTER II

THEORY AND SUPPORTING RESEARCH

Introduction

 

The theory and research related to comprehensive
problem solving shall be presented in two parts. Part one
deals with a theory of problems and problem solving and is
used as a rationale for those strategies which may be useful
to elementary school children in solving problems. Part two
reviews the literature on problem solving and creativity and

those variables which may influence problem solving skill.
PART ONE

A Theorypof Problem Solving

Learning theorists have provided a good deal of
material on which to build sound hypotheses for problem
solving. In 1956 Bloom1 and his associates prepared a com-
prehensive model descriptive of the possible cognitive
levels. Bloom's taxonomy classified the various thinking
processes, from simple recognition and recall to the most
general process of Open search, into levels: Knowledge,

Comprehension, Application, Analysis, and Synthesis.

 

13.8. Bloom, et a1., Taxonomy of Educational Objec~
piyes: Handpook 1: Cognitive DomiIn, (New York: McKay
Publishing Co., I936Y:

16

17

Problem solving is included in the highest cognitive level
by Bloom. Avital and Shettleworth1 also defined a taxonomy,
somewhat simpler in appearance, analogous to Blooms and
again insisted that solving Open ended problems involved the
highest cognitive processes. Gagne considers problem solving
as the highest form of learning wherein the solver does not
apply previously learned rules but must "discOver a combina-
tion of previously learned rules that he can apply to
achieve a solution."2

Although problem solving is the highest form of learn-
ing for Gagne, he asserts that it is possible to teach at
this level provided requisite rules have been learned.
According to Gagne, persons who engage in problem solving
develop a collection of "higher-order rules, which are
usually called strategies"3 and which provide guidance to
the thinking process. Bruner agrees that it is possible
to teach children problem solving and states, ”there are
certain general attitudes or approaches towards subjects
that can be taught in earlier grades that would have con-

siderable relevance for later learning."4 He continues by

 

1Shmuel M. Avital and Sara J. Shettleworth, Objectives

for Mathematics Learning: Some Ideas for the Teacher,
(Bulletin No. 3, 1968, The Ontario Institute for Studies in
Education).

2Robert M. Gagne, The Conditions of Learning, (2nd.
edition, Holt, Rinehart and Winston, Inc., N.Y.,‘1970),p.214.

3

 

 

Ibid., p. 215.

4Jerome S. Bruner, The Process of Edugation, (Vintage
Books Edition, August, 1963, Random House, Inc., N.Y.),P. 43.

18

saying, "it might be wise to assess . . . (which) heuristic
devices are most pervasive and useful, and that an effort
should be made to teach children a rudimentary version of
them that might be further refined as they progress through
school." The identification and teaching of strategies
which are useful in solving open ended problems depends on

ones conception of the various components Of‘a problem.

Components of a Problem

All problems can be considered to be composed of
three types of information: first, information concerning
the givens: second, information concerning Operations which
can be used to transform the givens, and third, information
concerning goals.1 The givens are those expressions which
are present when the problem is posed. In problems such as
*finding the distance from a point S(2,3) to the line given
by 3x + y - 2 = 0, the givens are the point and line in the
plane as well as assumptions, definitions and theorems from
Analytic Geometry. In puzzle problems, such as 'Instant
Insanity,‘ the givens are the four cubes, each of which has
one of four colors on each side. In comprehensive problems
the givens are all those objects, materials, data, axioms,
and definitions, stated or implied which are known to exist
after the statement of the problem. Few problems explicitly

specify all of the givens. However, it is usually the case

 

1Wayne A. Winkelgren, How To Solvegroblems: Ele-
ments of a Theoryyof Problems and Problem Svaing, (W.H.
Freeman and Co., San Francisco, 1974). p. 10.

 

 

19

that some of the givens are implied, for example, a know-
ledge of elementary Calculus is at one's disposal in solv-
ing physics problems even though this fact is not generally
explicitly stated.

An operation is an action which you are permitted to
perform on the givens which can transform the givens into
'expressions containing the goal or a subgoal.’ For example,
in a chess game the problem is to find a way to Checkmate
your Opponent as quickly as possible and the allowable
Operations are the ways each piece can move. For comprehen-
sive problems the allowable Operations permitted on the
givens are often limited only by the solvers imagination.
For example, in a problem to find ways to increase profit in
a student business, students can examine past profits
statistically, survey prospective consumers for possible
improvements, and examine past marketing and manufacturing
techniques, as possible transformations on the givens.

The goal is the terminal expression which the solver
wishes to exist in the world of the problem. Note that the
goal state may not be completely specified in the statement
of the problem. This is the case in problems where the
solver is "to find" a quantity. For example, 'find the

3 + 2x2 - x - 2 = 0'

value of x which satisfies the equation x
does not specify the numerical value of the solution and

yet the goal is clear to the solver. To achieve the goal
state the solver may have to pass through one or more inter-

mediate problem states. A problem state is the set of all

20

the expressions that exist in the world of the problem at a
particular time. A search strategy for solving a problem is
a plan which directs the learner from initial representation,
through any necessary intermediate problem states, to a

final goal state.

Sequence of Events Involved in Problem Solving

The process of solving problems has been divided into
stages by various authors. Dewey, Polya, Johnson and
Gagne1 generally agree that these stages include interpret-
ing the problem, devising a plan, producing solutions, and
deciding among solutions. Because of the vast differences
in types of problems Gagne suggests that this model is not
applicable in all situations nor is the solver totally aware
of the occurrences of such stages in actual problem solving.
Nonetheless, this sequence model is useful in organizing
a discussion Of problem solving and in identifying the
strategies which will aid the solver in reaching the goal

state.

Presentation of the Problem

 

Problems may arise as a result of some external
stimuli such as finding a parking place as close to the

theater as possible yet in a place free Of parking

 

lJ. Dewey, How We:$hink, (Boston: D.C. Heath and Co.,
1933); G. Polya, How To Solve It, (2nd, edition, Doubleday
and Co., 1957); D.M. Johnson, The Psycholo of Thgught and
Judgement, (New York: Harper & Row, 1955); R.M. Gagne, SEE.
Conditions of Learning, p. 217.

 

 

 

 

 

 

21

restrictions. On the other hand, the problem may arise as

a result of some internal thought process as in the case of
a good friend who adopts a moral view radically different
from our own, making it necessary to reassess the friend-
ship. Whether the problem is externally presented or not,
the solver becomes aware that there is a problem which he
wants to solve. Parnes refers to this first presentation of
a situation requiring action as the "fuzzy problem"1 which

needs careful definition.

Problem Definition
The first step in problem solving involves carefully
defining the problem in precise terms. Stating a problem

forces some kind of representation. Posner2

states, "the
initial representation of a problem may be the most crucial
single factor governing the likelihood Of problem solution."
Although the initial definition of the problem may be less
important in a comprehensive problem, many books on problem

solving such as the ones by Polya and Osborn,3 emphasize

that the initial representation should not be made too

 

1Sidney J. Parnes, Student WOrkbook for Creative
Problem Solving Courses and Institutes, (3rd. Rev., I963,
State University of New York at Buffalo).

2Michael I. Posner, Cognition: An Intrpduction, (Scott,
Foresman Series.Scott, Foresman and Company, Glenview, Ill.,
1973), p. 149.

3G.'Polya, How To Solve IE; Alex F. Osborn, Applied
Imagination, (Rev. ed., N.Y.: Charles Scribner's Sons,
19575.

 

22

quickly. Parnes1 suggests that the solver ask himself
various questions to aid in viewing the problem from dif-
ferent perspectives. For example, suppose that a group of
students are asked to solve the following problem:

On one shool bus, an eight-year-old boy struck and

bit other children and kept the entire busload in

a noisy, Confused state. Due to the disturbances,

the driver nearly had several accidents. He pointed

this out, but the boy jeered and continued behaving
the same way. When the boy's victims complained,

the school principal tried to punish the rowdy by

keeping him in at recess. This had no effect, nor

did it appeal to his parents. The driver reported

that he was as bad as ever and was cautioned by the

principal that anyone who spanked him might be sued

or prosecuted. What should the bus driver do?

To aid the solver in viewing this problem from several
vantage points Parnes suggests that the solver should carry
out some ”fact—finding" before attempting to define the
problem. The students may wish to know: How does the boy
act in the school building? What are his special interests?
Does he have personal or family problems?

There is evidence to suggest that the initial repre-
sentation of a problem is influenced both by what is in the
problem and by what is in the problem solver. Humphrey and
DeGroot2 both concluded that the difference between average

chess players and chess masters was the way the masters

initially represented a problem in determining the next move

 

1Sidney, J. Parnes, Student Workbook for Creative
Problem-Solving Courses.

2B. Humphrey, Directed Thinking, (New York: Dodd,
Mead & Co., 1948): A. D. DeGroot, Thought and Choice in
Chess, (The Hague: Mouton & Co., 1965).

 

 

 

 

23

in a chess game. The less gifted players reasoned in the
same way as the masters after the initial representation
was reached. However, the time to this initial representa-
tion was considerably longer in the average player. This
suggests that experience in solving problems Of one type
aids in initially representing problems of that type.

The factors which influence the way a problem is
coded by the individual has been studied by several indivi-
duals. Duncker1 concluded that if an object had one
established use in a given situation, it was difficult to
use the Object in another way. He refers to this condition
as functional fixity. Glucksburg, Weisburg and Danksz
investigated this phenomenon for Objects which were labeled
and initially coded with that label. They concluded that
subjects who coded the object in memory, by the label, were
at a disadvantage in solving problems which called for uses
not associated with the label.

This evidence suggests that for a period prior to
careful definition of the problem the solver should consider

different uses for the givens and view the problem from

 

1K. Duncker, "On Problem Solving," Psychological
Monographs, 1945, 58.

2S. Glucksberg and J. H. Danks, "Effects of discrimi-
native labels and of nonsense labels upon availability of
novel function," Journa1_of Verbal Learning and Verbal
Behavior, 1968, 7, pp. 72-76; S. Glucksberg and R. W.
Weisberg, "Verbal behavior and problem solving: Some effects
on labeling in a functional fixedness problem," Journal
Of Experimental Psychology, 1966, 71, pp. 659-664.

 

 

 

 

24

various perspectives. After this initial encounter with the
problem the solver should choose that formulation of the
problem which most closely describes the precise goal

the solver wishes to reach. For example, in the school
bus driver problem, the students may decide that the best
way to fOrmulate the problem is: How might the bus driver
achieve peace on the bus? This particular choice may come
after several, less general, formulations of the problem
such as: How might the bus driver settle this boy down?
What might the bus driver do to isolate "problem children"
from the others? What might the bus driver do to obtain a
better understanding of the boy? How might the driver
provide a safe and sane ride?

The first step in the problem solving process, inter-
preting the problem, is embodied in the definition of the
problem. Any limitations on the givens and on possible
transformations should be stated or implied in this
definition. Since the definition of the problem describes
the goal state, it is essential that any evaluation of the
correctness of a solution be carried out in reference to

the stated goal and its meaning to the solver.

Devising A Plan

 

The second step in the problem solving sequence is
devising a plan or search strategy. Such plans are similar
to hypotheses and tell the solver where to look in memory

or what to examine in the external world in order to

25

advance towards a solution. Posnerl suggests that we "know
very little about the development of these plans except that
they appear to be a function Of consciousness" and that the
solver can often report them verbally. The development of

a search strategy is at the heart of the problem solving
process. At this stage of problem solving there are three
strategies which will aid in formulating a plan of action:
they are inference, identification of subgoals, and examin-

ing relations between problems.

Inference

 

All problems, clearly defined, present some of the
relevant information in implicit, rather than explicit, form.
This information is Often critical in solving the problem.
When presented with a clearly defined problem the solver
must make inferences and draw conclusions from the given
information which will render explicit statements from
those implied by the givens. For example, consider the
following problem from Analytic Geometry?

Give an equation which describes the locus of

points, C such that the line segment joining

(-3,0) to any point Of C is perpendicular to the

line segment from that point of C to the point

(3 I O) o
The solver could infer from the given information, 'the two
line segments are perpendicular,‘ that the slopes of the

tWO segments are the negative reciprocals of each other.

Inferences can be drawn from implicit or explicit statements

X

1Michael I. Posner, Cognition: An Introduction, p.162.

 

26

of the givens or the goal state. The solver expands the
goal or givens by bringing to bear all of the knowledge he
has concerning the problem which is stored in memory.

Polyal suggests that the solver be guided in this
search for relevant information by looking for conditions
which relate the givens and the goal. Wickelgren states,
"Drawing inferences (more generally, making transformations
of the goal or the givens) is probably the first problems
isolving method . . ."2 the solver should employ. He conti-
nues by stating that the solver should continue to draw
conclusions which satisfy one, or both, of the following
criterion:

a) The inferences have frequently been made in the
past from the same type of information;

b) The inferences are concerned with properties
(variables, terms, expressions, and so on) that
appear in the goal, the givens, or inferences
from the goal or the givens;

until it is difficult to draw new conclusions which seem to
have any likelihood of being useful. Inferences from
presented information also includes explicit symbolic or
diagrammatic representation of the information that appears
implicitly in the problem.

One method of generating new information from the

Givens or goal state is "brainstorming" the problem. This

Procedure which is espoused by the Creative Education

\

1G. Polya, How To Solve It.

 

2Wayne A. Wickelgren, How To Solve Problems, p. 23.
3

 

Ibid., p. 28.

27

Foundation, involves the deliberate stimulation of an
individual or groups divergent thinking ability. During
this procedure the flow of ideas is enhanced by withholding
all criticism in the early stage so that solvers will feel
free to propose ideas which may be judged as far off the
subject. By deferring judgment, it is assumed that even
"far out" ideas may suggest a realistic, creative solution
or plan. As Osborn, the founder and chairman of Creative
Education Foundation, states, "It is almost axiomatic that
quantity breeds quality in ideation."1

Brainstorming is a generalization of the inference
method appropriate for comprehensive problem solving. For
example, in the problem of keeping peace on the school bus,
brainstorming may lead to conclusions, inferred from the
desired goal Of peace, such as:

Don't let the boy take the bus. Keep the boy sepa-

rated from the other children. Arrange the route

so the boy is picked up last and dropped off first.

Arrange activities (on the bus) to keep the children

busy. Provide books and magazines of interest.

Equip the bus with TV.
Parnes suggests using "idea-spurring questions" such as;

Can I modify, magnify, minify, rearrange or combine any

 

1Alex F. Osborn, Applied imagination: Principles and
Procedures of Creative Problem SolGing, (3rd'rev. ed.
Chafles Scribner's Sons, New YorkTil963), p. 131.

 

 

28

existing information, or suggestion, to produce new
information which may be useful in solving the problem?1

Inferences can be made about the operations or
transformations which are in a problem. Indeed many
practical problems require the solver to think of a type of
operation that will solve the problem. One example of this
type of problem is the well known radiation problem of
Duncker:

Given a human being with an inOperable stomach tumor,

and rays which destroy organic tissues at sufficient

intensity, by what procedure can one free him of

the tumor by these rays and at the same time avoid

destroying the healthy tissue which surrounds it?2
The solver may infer several possible operations: avoid
contact between rays and healthy tissue, desensitize the
healthy tissue, or lower the intensity of the rays on
their way through healthy tissue. As a second example,
consider the one—heavy-coin problem:

You have a pile of 24 coins, twenty-three of these

coins have the same weight, and one is heavier

than the others. Your task is to determine which

coin is heavier and to do so in the minimum number

of weighings. You are given a beam balance, which

will compare the weights of any two sets of coins.
If the solver infers correctly that the beam balance
actually has three different outcomes, then the solver is

in a position to realize that the balance can provide him

with the answer to which of three subsets contains the

 

1Sidney J. Parnes, Student Workbook for Creative
Problem Solving Courses.

 

2K. Duncker, "On Problem Solving," Psychological
Monographs, 1945, 58.

 

 

29

heavier coin. Given two equal subsets of coins the left
hand pan of the balance is either above, even with or
below the right hand pan.

Inferences can be made from the givens, the goal or
the operations and practice in drawing conclusions from

these sources should be provided.

Identification Of Subgoals

A second strategy useful in devising a plan to solve
a problem is to identify one or more subgoals or subproblems.
In essence, the purpose of this strategy is to replace a
single difficult problem with two Or more simpler problems.
For example, in the Analytic Geometry problem posed in the
last section a reasonable subgoal would be to find the slope
of the segment from (x,y), a point of the locus, to the
points (-3,0) and (3,0). A second subgoal would be to
equate the negative reciprocal of one of these slopes to
the other.

The subgoal method is advantageous for attacking
problems that require a sequence of more than two or three
actions to solve--which is the case in comprehensive
problems. When a solver defines two or more subgoals to be
achieved in getting from the given state to the goal he
must decide whether one of these subgoals must be achieved
before the others. In some problems it is clear that one
of the subgoals(SGl) is closer to the givens than a second

(8G2), which is closer to the goal. In such a case the

 

30

solver should work to achieve SG1 first. When this
logical distinction can be made the problem contains
ordered subgoals.l A diagram of a problem containing

ordered subgoals is given in Figure One.

\~
\\,/ v

Figure l. Ordered Subgoals

 

 

 

 

The locus problem is an example of an ordered subgoal
problem. A second example is provided by the following
problem:

Nine men and two boys want to cross a river, using

an inflatable raft that will carry either one man

or the two boys. How many times must the boat

cross the river in order to accomplish this goal?
An excellent subgoal is to find the minimum number of trips
necessary tO get one man on the other side and get the boat
back to the starting side. Since there are nine men, the
solver can simply multiply this minimum number by nine and
subtract one to achieve the goal.

In comprehensive problems the solver usually defines
several subgoals which are logically independent. These

"subproblems" may often be broken into further sub-subpro-

blems which lead to the goal. If it is known a priori that

 

1Wayne A. Wickelgren, How To Solve Problems.

 

 

31

either of two sequences of actions, through two equivalent
subgoal chains, will lead to the desired solution the
solver is free to choose to begin work on that chain which
he feels will yield the goal more quickly. The problem is
said to contain unordered subgoals. Refer to Figure Two

for a schematic drawing of a problem containing unordered

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

subgoals.
Givens
J
I J F I
Subgoal l Subgoal 2 Subgoal 3 Subgoal 4
' T
F T _I

 

Sub-subgoal (2,1) Sub-subgoal (2,2) Sub-subgoal(2,3)

 

 

 

 

 

 

 

 

 

 

Figure 2. Unordered Subgoals

More often the solver does not know if a particular
sequence of subgoals will lead to the most "correct"
answer and should choose to work on that subgoal which
contains the fewest subgoals to be solved.

Most often in comprehensive problem solving the care-
ful statement of the problem, producing the problem
definition, is quite broad. For example, consider the
problem, ”How might students increase sales in a student
owned and operated store?" The solver might list ideas
such as, "Obtain more customers" or "step up advertising."

These are not solutions to the problem, rather subproblems

32

which will lead to an increase of sales in the store. The
solver would choose one of these ”approaches" at a time,
and attempt to probe for more specific sub-subgoals. For
example, in the subgoal 'Obtain more customers,‘ the

solver may identify sub-subgoals such as: How can we get
more customers in the store during the morning and early
afternoon hours? How can we get customers into departments
with sagging sales? Once a number of subgoals have been
identified, the solver chooses to work on that subgoal
which he feels will produce the most rapid, or most dramatic,
increase in sales.

The identification of subgoals may be one of the
most useful problem solving strategies for the elementary
school child, since it can be used throughout the school
years to solve arithmetic, algebra and geometry word
problems. For example, consider the following problem:

Each day, Abe either walks to work and rides his

bicycle home orrides his bicycle to work and

walks home. Either way, the round trip takes

one hour. If he were to ride both ways, it would

take 30 minutes. How long would a round trip take,

if Abe walked both ways.
The first subgoal the solver might define is to determine
how long it takes Abe to ride one way. The solver can then

determine, as a second subgoal, how long it takes to walk

one way.

33

Wickelgren, Parnes and Polyaleach.identify the subgoal
method as a generalized problem solving strategy which

should be a part of problem solving training.

Examining Related Problems

A third generalized strategy for devising a problem-
solving plan is to examine a related problem. Two problems
can be related in the following ways: one is equivalent
or completely analogous to the second, one may be similar
, in certain elements to the second or one may be a general-
ization of a second, that is, the second is a special case
of the first.

If two problems are equivalent, they are the same
except for the naming of the elements involved, that is,
they are isomorphic. For example, if in a checker game the
checker board is replaced with a chess board and the red
checkers with quarters and the black checkers with pennies,
without changing the Operations or goal, the two problems
(games) are equivalent. RecOgnizing that a problem is
equivalent to another problem is usually a simple matter,
and will only aid the solver in achieving the desired goal
if the isomorphic copy has been solved. However, the solver
who realizes that his problem is equivalent to another

Often gains new perspectives which aid in making inferences.

 

1Wayne A. Wickelgren, How To Solve groblems: Sidney
J. Parnes, Student Workpgok for Creative Problem-Solving
Courses: G. Polya, How To Solve It.

 

 

 

 

34

Two problems which are similar share some but not all
of their essential characteristics. For example, the linear
game of nim, involving the removal of counters from a
line,is similar to the two dimensional game of nim, wherein
the players remove all counters above and to the right of
a selected counter. Two similar problems may be of equiva-
lent difficulty, with the first simpler-than the second or
the first more complex than the second. For example, the
"missionaries—and-cannibals" problem is as difficult as
the similar "man, fox, goose and corn" problem. As in the
case of equivalent problems, knowing how to solve either
aids the solver in reaching the goal for the other.

Looking for a similar, simpler problem is a strategy

Polya stresses throughout his book, How To Solve It.1

 

A very common technique used to find a simpler problem
(though not the only way) is to reduce the number of
variables involved. For example, the one-heavy-coin problem
presented earlier involved twenty-four coins, and is solved
by weighing a subset of eight coins against any other subset
containing eight coins. The solver who is unable to make
the needed inferences when confronted with this problem
might try the same problem with fewer coins. Note that
examination Of a simpler problem will only produce the
insight needed to work the more difficult one if the simpli-

fied problem still contains the essential feature on which

——._

1G. Polya, How To Solve It.

 

35

the solution to the difficult problem depends. For this
reason, simplifying the problem may not aid the solver if
his simplification eliminates that facet Of the problem
which made it difficult. However, as Wickelgren states,
". . . it might still be good strategy to pose and
solve . . . a simpler problem before you attempt to solve a
more complex problem."1
Examination of special cases is especially useful in
proof problems or in problems where the solver must make
inferences concerning a more general problem. Mathematical
induction involves looking at special cases. Special cases
are simplified problems which, taken collectively, sometimes
aid the solver by providing data for the intuition to work
on. For example, determining the number of moves necessary
to complete the "Tower of Hanoi" problem with k disks is
most often approached by examining the problem with two,
then three, then four disks and recognizing a pattern.
Teachers report that students at all level of instruc-
tion use the method of related problems to solve word
Problems. For example, in solving "age problems" in elemen-
tary algebra the learner is taught to mimic the procedures
demonstrated by the teacher. In solving comprehensive
Problems, the student should be guided by previous experience

gained from solving similar problems. This applies to

making inferences which were helpful in similar problems,

g

1Wayne A. Wickelgren, How To Solve Problems, p. 173.

 

36

identifying subgoals which have proved fruitful in similar
situations, and gathering and interpreting data pertinent
to the problem. For example, students who have learned to
represent the mean time necessary for six different classes
to cross a street by using bar graphs can use this method
in the equivalent problem to represent the mean time
necessary for these classes to pass through the lunch line.
Indeed, Polya suggests that at each step of the solving
process the student ask himself, "Do you know any problem
with the same unknown?, . . . with a similar unknown?, . . .

related to yours and solved before?"1

Producing Solutions

 

Once the search strategy has been devised, the solver
follows the plan to its conclusion--the goal state. In
comprehensive problem solving the solver must accumulate
evidence which will support a decision concerning the
problem. Indeed, Posner suggests, ". . . many problems can
be viewed as involving the cOllection and evaluation of
evidence."2 For example, consider the simple problem of
"Should I take a raincoat to work this morning?" In order
to decide between the two hypotheses, "It will rain," and
"It will not rain," the individual could consider whether

it is raining at the present time. Does it look like it
1G. Polya, How To Solve It, p. 11.
2

 

M. I. Posner, Cognition: An Introduction, p. 176.

 

37

may rain? What is the weather report, the conditions of the
wind, a barometer reading, the amount of rain during the
past few days? Gathering and interpreting data is a neces-
sary part of producing solutions.

The solver may find that, after working on a problem
for some time that he is no closer to the goal state than
he was soon after the initial representation. The solver
must then concentrate on ways to redirect the search for a
solution. Evidence suggests that once a particular
strategy is chosen, it continues to direct the search in a
direction which may not lead to the goal. Luchingsl found
that subjects continued to use a formula which had worked
on previous problems even when a more direct method of
solution was available. This tendency to repeat a solution
once Obtained is called set, or Einstellung. Tresselt and
Leeds2 demonstrated that, unlike functional fixity,
Einstellung does not diminish even when the problem solving
process is interrupted for a period of days. Ways of redi-
recting the search have been suggested by several individuals.

DeBonO3 suggested the solver use a method called lateral

‘—

1A. S. Luchings, "Mechanization in Problem Solving:
The Effect of Einstellung," Psychometric Monographs, 1942,
54 (6), p. 95.

2M. E. Tresselt, and D. S. Leeds, "The Einstellung
Effect in Immediate and Delayed Problem-Solving," Journal
pf General Psychology, 1953, 49, pp. 87—95.

3E. DeBonO, New Think, The Use of Lateral Thinking in
the Generation of New Ideas, (New York} Basic Books, Inc.,
1968).

 

38

thinking, in which the solver looks at the problem from
various standpoints. The Creative Education Foundation1
Suggests that the solver go back and repeat the brain-
storming process. Crovitz2 suggests thinking of action
words which can be inserted-into the problem to break old
habits and produce new organizations. Since these methods
are not mutually exclusive, there have been few attempts to
compare them.

Wickelgren3 suggests two methods which are useful in
redirecting the search, working backwards, and the method of
contradiction. These strategies provide more specific ways
to reach a solution than the broad suggestions of DeBono,

Crovitz or the Creative Education Foundation.

Contradiction

 

The method of contradiction involves drawing conclu-
sions which contradict some piece of given information,
and so proves that a goal can not possibly be obtained
from the givens. This strategy gives us negative information
which may imply the desired result. The indirect proof
Sometimes used in mathematics problems requires the solver
to use contradiction. For example, the proof that /2 is

irrational utilizes the indirect proof.

‘—

lA. Osborn, Applied Imagination: Principles of
Qreative Problem Solving.

2H. F. Crovitz, Galton's Walk, (New York: Harper &
Row, Publishers, 1970).

3Wayne A. Wickelgren, How To Solve Problems, pp. 109—

 

 

 

 

151.

39

Another use of contradiction involves the elimination
of alternatives in a small search space. Such elimination
is useful in answering multiple choice problems:

"Which of the following is a solution to the equation

x3 + 4x2 - 7x - 10 = 0? x equals: (A) -2, (B) -s, (e) 4,
(D) 3, (E) none of these." The solver can determine whether
or not each of these numbers satisfies the equation more
quickly than he could solve it by a direct approach. This
particular strategy is used to solve logic problems such as
the "brakeman, fireman and engineer problem," wherein the
solver is asked to determine which of three trainmen is the
engineer (given some information about the trainmen' and
three passengers on the train). Logic problems using
contradiction in a small search space may provide practice
for elementary school students in using techniques which
could prove useful in later studies of algebra and geometry.

Solving open ended problems often involves a large
number of alternative specific goals making it impractical,
or impossible, to contradict them one at a time. In such a
case, it is sometimes possible to eliminate large subgroups
of alternative goals by contradiction. For example, in the
school bus driver problem, several possible alternative
hypotheses such as, provide TV on the bus, provide new games
and puzzles regularly, provide entertainment for the
children, provide movies or storytellers, can be eliminated
because to assume such a solution contradicts the cost

constraint the school board would impose. By drawing

40

conclusions from an assumed goal, and contradicting some
given information, the solver can gain considerable insight

into the nature of a plausible solution.

Working Backward

The method of working backwards is a strategy for
redirecting the solving process by focusing attention on the
goal as the starting point. Unlike contradiction, which
attempts to draw inferences from an assumed goal, the method
of working backwards involves trying to guess what step must
immediately precede the goal state. This method is helpful
'particularly when there is a unique goal and several given
statements. In problems where there is a unique goal,
Newell, Shaw and Simon1 have demonstrated the superiority
of working backward. In problems where there are a large
number of givens and a unique goal, the givens may have a
disjunctive relationship to one another. Hence, the
solver, using a direct approach, would have to use a great
deal of time trying various starting points (givens), some
of which need not be related.

Students who attempt to solve comprehensive problems
often come up with a large number of givens, or inferences
from the givens, in the initial stages of problem solving.

Because of the unique starting point used in working

k

1A. Newell, J. C. Shaw and H. A. Simon, "The Processes
of Creative Thinking," Contempprary Approaches to Creative
Thinking, Eds. H. E. Gruber, G. Terrell, M. Wertheimer,
(New York: Atherton Press, 1962).

 

41

backwards, namely the goal, the student is directed to
just those aspects of the given information that are
relevant to the solution.

The method of working backwards is useful to students
in less general problem solving contexts such as:

Three people play a game in which one person

loses and two people win each game. The one who

loses must double the amount of money that each

of the other two players has at the time. The

three players agree to play three games. At the

end of the three games, each player has lost once

and each person has $8. What was the original

stake of each player?
In this problem, the operation of exchanging money is
destructive, that is, changes the givens at each stage of
the problem. For this reason, by beginning with the goal
and using the inverse operation at each step the solver
can arrive at the solution that one player had $13, the
second had $7, and the third had $4 at the beginning of the
game. The problem of determining the optimal strategy for
winning the game of "nim" is another example of a problem
with a destructive Operation, and hence can be worked
using the method of working backwards from a unique goal.

The method of working backwards is extremely useful
in redirecting the search strategy and it may be that letting
Children work with games or puzzles which involve this
method will aid in their understanding of this procedure

and encourage them to use it more often. As Polya states

"Working backwards is a common-sense procedure within the

42

greeach of everybody. . ."l which should be cultivated into a

;>c>werfu1 tool in problem solving.

Deciding Among Solutions

 

In comprehensive problem solving the solver must deal
with a situation in which there is no optimal solution or
.111 ‘which the solution is unknown. Individuals generally do
:nc>t: attempt to find an optimal solution even when it can be
computed. Simon2 found that persons Often find solutions
Short of optimality which will be good enough or acceptable.
““163 subjects tended to avoid complex calculations in favor
(*5 ashort procedures on a fraction of the data. Posner
SWJSIQQests that this is the case because "limitations of
operational memory make it difficult to deal with a long
SeJ':ies of interrelated steps."3

When the solver has a great deal of data on which to
base a decision concerning solutions, it has been found that
th ese decisions are not made in an optimal or consistent

4

V'Eizy. Two studies, the first by Edwards and the second by

\

1G. POlya, How To Solve It, p. 230.

2H. A. Simon, The Sciences of the Artificial,(Cambridge,
b’lass.: MIT Press, 1968).

3

 

 

M. I. Posner, Cognition: An Introduction, p. 175.

 

4W. Edwards, "Dynamic Decision Theory and Probabilistic
Information Processing," Human Factors, 1962, 4, p. 59-73.

 

43

Slovic and Lichtensteinl indicate that persons often under-
weigh very strong evidence towards a particular hypothesis,
and overweigh evidence which shows only weak support of a
hypothesis. Furthermore, evidence which comes slowly is
more heavily weighted than evidence which comes quickly and

in great quantity. Peterson and DuCharme2

showed that
evidence introduced early will have greater impact on the
solver than evidence which comes late in the solving process.
These studies seem to suggest that the choice of the best
solution is heavily influenced by how it was produced.

To provide the solver with a less subjective rationale
for judging the correctness of a sOlution, Parnes3 suggests
that the solver list the evaluation criterion or "yardsticks"
by which the solver can mentally test the effectiveness of
each solution. The solver then grades each of his tentative
solutions: Good, Fair, Poor, or Doesn't Pertain, for each of
the criterion listed. The solution with the highest over-

all grade is selected as the most acceptable or satisfactory.

This method of judging the quality of a solution is

 

1P. Slovic and S. Lichtenstein, "Comparison of Bayesian
and Regression Approaches to the Study Of Information
Processing in Judgment," Organic Behavior and Human Perform-
ance, 1971, 6, p. 649-744.

2L. R. Peterson and W. M. DuCharme,-"A Primacy Effect
in Subjective Probability Revision, Journal of Experimental
Psychology, 1967, 73, p. 61-65.

3Sidney J. Parnes, Student Workbook for Creative
Problem Solving Courses.

 

 

 

 

44

essentially a check to see how well it meets the goals set
forth by the problem.

Another means Of evaluating the correctness of a
solution is to judge whether the solution produces insight
or the "Eureka" experience. Wertheirmer1 suggests that
one measure of insight is the ability to transfer the learn-
ing on a problem to new situations. This may depend on the
problem solution's ability to be represented in operational
memory at one time. For example, a student claims to have
insight into a mathematical proof when he is able to
condense the steps in such a way that they are present at
least implicitly in a representation that he can grasp all
at once. Kohler2 suggests that without this insight the
solver will be less convinced of the correctness of a solu-

tion even when evidence supports this conclusion.
PART TWO

Review of the Related Literature

 

A search Of the literature yielded many studies which
related problem solving ability to other variables. Rele-
vant material will be presented under one of the following
classifications: (a) general problem solving, (b) creativ-

ity and problem solving ability, (c) problem solving as

 

lM. Wertheirmer, Productive Thinkipg, (New York:
Harper 8 Row Publishers, 1945).

2W. Kohler, The Task Of Gestalt Psychology, (Princeton,
N.J.: Princeton University Press, l96_3.

 

 

45

related to other variables. In order to insure a broad
survey of the literature, the researcher drew from the
following sources: Educational Resource Information Center
(ERIC), Dissertation Abstracts, Psychological Abstracts,
books and articles suggested by the subject matter of
problem solving.

The ERIC data base is composed of most Of the educa—
tion related material published during the past ten years,
and is stored on computer tapes by System Development Cor-
poration of Santa Monica, California. A computer search
through the ERIC data base was completed to find all
material which was related to teaching elementary school
children methods of comprehensive problem solving. The
researcher requested all material which contained the
following three areas: (1) problem solving, productive
thinking, convergent thinking, divergent thinking, creative
thinking or creative activities; (2) elementary education,
primary education, elementary grades, elementary school
children, elementary school curriculum, primary grades or
early childhood education; (3) teaching, creative teaching,
instruction, concept teaching, teaching methods, teaching
procedures or teaching techniques. Of the publications in
the ERIC data base only 116 contained material involving
all three of these areas. Of these 116, only 12 were
actually related to teaching problem solving to children.

To complement the ERIC search, a search through Dissertation

46

Abstracts and Psychological Abstracts was carried out to
find studies which may not have appeared in the ERIC

material.

General Problem Solving

The nature of the problem solving process is discussed
in most general psychology and learning theory texts, and
is usually presented as a procedure which generally involves
four stages: Preparation, Definition, Hypthesis formation,
and Solution selection. The early work of Maier1 implied
that the solver is usually not aware of intermediate stages
in the solving process, that is, the perception of a
solution is most Often sudden. Later, Patrick2 concluded
that four stages Of problem solving are apparent: l) prep-
aration, 2) incubation, 3) illumination, 4) verification.
She agreed, however, with Maier that the creative solution
most Often appears as a whole. Simon and Newell3 reviewed
past and present theories of problem solving and currently

regard problem solving asamiinformation processing procedure

 

1N. R. Maier, "Reasoning in Humans,The Solution of a
Problem and its Appearance in Consciousness," Journal of
Comparative and Physiological ngchology, Vol. XI, 1931,
pp. 181-194.

2Catherine Patrick, "Whole and Part Relationship in
Creative Thought," American Journal of Psychology, Vol. LIV,
1941, pp. 128-131.

3H. Simon and A. Newell, "Human Problem Solving: The
State of the Theory in 1970," American Psychologist, 1971,
(Feb.), Vol. 26 (2): pp. 145-159.

 

 

 

47

wherein the solver (processor) operates on the data
(information) to arrive at a solution new to the solver.
In this theory, the information processing system, the
problem solver is confronted by a task, and the task is
defined objectively in terms of a task environment or
problem space. A search of the problem space in promising
directions, avoiding the less promising ones, will lead

to a solution. This is essentially the point of view that
Wickelgrenl takes in suggesting strategies or heuristics
which can be employed to aid in the search through the
problem space.

Simon and Newell2 suggest that the current view of
problem solving as information processing has enabled
theorists to pin down terms like "memory" and "symbol
structure" prevalent in prior problem solving theories.
Previous theories depended, to a great extent, on the struc-
ture of the problem space in arriving at solutions. Miller
and Slebodnick3 demonstrated the superiority of training
in strategies, over training in structure, in trouble

shooting electronic equipment. This result was again

 

lWickelgren, How To Solve Problems.

2H. Simon and A. Newell, "Human Problem Solving: The
State of the Theory."

3R. B. Miller and E. B. Slebodnick, "Research for
experimental investigation of Transferable skills in
electronics maintenance," Lackland AFB, Texas. Air Force
Publication: TRC TR - 2, 1958.

 

 

48

demonstrated by Rundquist, et al.,1 who showed that a strat-
egy trained group did better than a structure trained group
in terms of time to solution and the number of correct
initial hypothesis for this type of electronic equipment
problem.

There are some problem solving programs which have
been developed and which emphasize teaching the solver
strategies. The Creative Education Foundation program to
train for problem solving processes emphasizes a process
which relies heavily on a single heuristic—brainstorming.
Osborn2 suggests that research at the State University at
Buffalo indicates a significant improvement in idea-
producing skill. This improvement is due to the "defer
judgment" aspect of brainstorming which provides the solver
with a variety of tentative solutions.

The Covington, Crutchfield, Olton Project at the
center for Productive Thinking at Berkley produced a
program to teach problem solving. The program uses

detective stories to teach strategies for problem solving

 

1E. A. Rundquist, et al., "Preparation for Problem
Solving: Structure vs. Strategy Pre-Training," San Diego:
U.S. Naval Research Technical Bulletin 5: TB 66 - 1, July,
1966.

 

-2Alex Osborn, The Creative Education Movement, (Crea-
tive Education Foundation, Buffalo, New York, 1964).

 

49

to fifth and sixth graders through programmed learning.1
The program has been very successful in teaching children
to solve these types of problems, but the amount of transfer
to problems of a comprehensive nature has been limited. A
third program which has been developed by Interaction
Associates of Berkley, California is entitled "Tools for
Change." This group has develOped teacher manuals,2 for
primary and higher grades, and a strategy notebook3 for
student usage. "Tools for Change" is a problem solving
course consisting of a series of units each dealing with

a heuristic process. This program has identified thirty-
eight strategies which are classified under one of the

eight headings, such as strategies for manipulating informa-
tion, strategies for information retrieval, and master
strategies. The authors of this program have not developed
problem solving tests which can provide quantitative results
on which to judge the success of this program in teaching
open ended problem solving. 'Since this program uses games
and exercises to teach the heuristics, the amount of trans-
fer in using these strategies to problems which are compre-

hensive, may be limited.

 

.1COvington, Crutchfield, and Olton, The Productive
Thinking Program for Problem Solvigfll (Berkley, California,
1970). i

2Lucinda L. Kindred, Tools for Change, Primary Version,
(Second Ed. Interaction Association, Berkley, California,
1971).

3Catherine McEver, Strategnyotebook: Togls for Change,
(Interaction Associates, Berkley, CaIifornia, 1971).

 

50

The Unified Science and Mathematics for Elementary
Schools (USMES) is a federally funded program to teach
children to solve open ended problems. Students work with
their classroom teacher several hours per week, throughout
the year, on challenges such as weather prediction, design-
ing for human proportions, and planning a safe school cros-
sing.l Since its inception, USMES has used a variety of
evaluation techniques such as teacher logs, classroom
Observations, and standardized tests. In 1971 Shapiro2
studied the success of the USMES program in teaching stu-
dents methods for attacking problems. From two to six
students from thirty-one USMES classes and twenty—two
control classes were tested using ”The Notebook Problem:"
choose the best Of three notebooks for students to use as
a science and mathematics notebook. Students were scored
more favorably for responses which indicated consideration
of measurable dimensions (e.g., size-volume, weight, quan-
tity of paper, and cost). Shapiro concluded that USMES
students made significantly more measurable responses than
those in the control group. The USMES program neither
identifies nor teaches specific strategies, but rather
provides an environment in which the student may discover a

general.pattern of problem solving.

 

1
1973.

USMES Guide, Education Development Center, Boston,

 

2Bernard J. Shapiro, The Notebook Problem: Report on
Observations of Problem Solving Activity, Education DeveIOp-
ment Center, Boston, 1973.

 

51

The Inquiry Training Project, begun in 1969, for
sixth, seventh, and eight graders originated in Ontario,
Canada under the direction of the Ontario Institute for
Studies in Education. This project attempts to teach
children a generalized problem solving process which in-
volves defining the problem, finding alternatives, gathering
information, synthesizing information, and reaching a con-
clusion.1 This model is applied to six problem-solving
areas: 1) logical inquiry, 2) physical science experiments,
3) experiments involving the principle of randomization,

4) correlational analysis, 5) case studies and 6) real life
problems. Classes which are part of the project solve a
problem Of each type, using a variation of the generalized
problem solving process indicated above.

Evaluation of the first two years of the project have
involved both the subjective judgments of the teachers of
the program and objective tests to insure that there is an
effect and that it can be reasonably attributed to the new
program. The authors of the program feel that the most
reasonable interpretation of the results of these evalua-
tions is that the program does have a large effect on cer—
tain types of problem-solving abilities.

There have also been attempts to teach problem solving

within the framework of specific subject areas. For example,

 

lF. Robinson, J. Tickle, and D. W. Brison, In uir
Trainimg:lvFusing Theory and Practice, (The Ontario Insti-
tute for Studiesiin EducatiOn, Toronto, Ontario, Canada,
1972).

 

52

Schmiess studied success of a program to teach sixth grade
students to solve scientific problems.1 He concluded that
by teaching students the methods of scientific investigation
a significant gain can be made in the ability to solve new
problems of a scientific nature. In another area, Schwab
and Ambrose studied specific strategies for teaching
decision making in the social sciences.2 Although they
found no significant difference between the treatment and
control groups, the researchers suggested that the students
were aware of the heuristics but that they were not
permitted sufficient practice within the program to bring
about the desired change.

In summary, the evidence seems to suggest that an
informational processing theory of problem solving is most
widely accepted, and the most consistent with, a strategy
approach to problem solving. Within the past five years,
some programs have been developed which have attempted to
teach the skills involved in problem solving. Some of these
programs have limited themselves to certain types of
problems to be considered, some have limited success in

teaching comprehensive problem solving, and some have not

 

lElmer G. Schmiess, An Investigation Approach to Ele-
mentary School Science Teaching, (Ed. D. DissertaEiOn, Uni-
versity of North Dakota, University Microfilms, Ann Arbor,
Michigan, 1970).

 

 

2Lynne Schwab and A. Ambrose, The Interaction of
Decision-Making Style, Teaching Strategy, and Decision-Making
Content Material in Social Studies, (American Educational
Research Association, Washington, D.C., 1970).

 

 

 

53

identified specific strategies which can be strengthened in
the high grades. The two most promising programs appear to

be USMES, and the Inquiry Training Project.

Creativity and Problem Solvipg

Creativity is used in literature to mean a variety of
thinking processes. Vernon1 states, "; . . to.the psycholo-
gist, creative thinking is merely one of the many kinds of
thinking which range from autistic fantasy and dreaming to
logical reasoning. . ." In general, American psychologists
such as Gallagher and Wilson2 agree with Guilford3who uses
creativity, as a term, to describe divergent-productive
thinking. Torrance views creativity as:

A process of becoming sensitive to problems . . .;

identifying the difficulty; searching for

solutions . . .; testing these hypotheses;
finally communicating the results.

 

1P. E. Vernon (Ed.), Creativity, (Baltimore: Penguin
Books, 1970).

2J. J. Gallagher, Teaching the Gifted Child, (Boston:
Allyn Bacon, 1964); R. C. Wilson,_wThe Structure of the
Intellect," Productive Thinking in Education, Ed. M. J.
Aschner and C. E. Bish, (Washington D.C.: National Education
Association, 1965).

3J. P. Guilford, "Traits of Creativity," Creativity
and its Cultivation, Ed. H. H. Anderson (New York: Harper
and Row, 1959).

4E. P. Torrance, Torrance Tests of Creative Thinking:
Norms-Technical Manual (Princeton: Personell Press, 1966).

 

 

 

 

54

Thurston1 agrees that creativity involves reaching solutions
that are unique to the individual. Fleigler suggests that
the creative individual ". . . manipulates external symbols
or objects to produce an unusual event uncommon to himself
and/or his environment."2

Each of these definitions of creativity involves novel
combinations or an unusual association of ideas. Some
authorities require that these combinations be useful, that
is to say that they solve some social or theoretical problem.
Others merely require that the creative product is "new"
to the individual doing the creating. These definitions
suggest a very close relationship between problem solving
ability and creativity.

Since creativity and problem solving are closely
related, the characteristics of creative persons and ways
Ofrmrturing creativity are oftentimes examined. Drews3
found that creative adolescents were sensitive to experi-
ences both within and without themselves, and were open and

searching in considering alternatives. A number of studies

 

1L. L. Thurston, "Creative Talent," Application of
Psychology, Ed. L. L. Thurston, (New York: Harper and Row,
1952).

2L. A. Fleigler, "Levels of Creativity," Educational
Technology, 1959, 9 (21).

3E. M. Drews, "Profile of Creativity," NEA Journal,
1963, 52 (1), PP. 26-28.

 

 

 

 

55

were carried out by MacKinnon1 and his colleagues during the
sixties to determine the nature of a creative person. They
found that creative individuals were above average in
intelligence, more fluent, more alert, more independent in
thought and action, and relatively free from conventional
restraints. Guilford2 identified aptitude traits which may
have a direct bearing on creativity. These include the
ability to see problems, fluency, flexibility, originality,
redefinition (the ability to improvise), and elaboration.

Torrance and Razik3

also summarized the traits of a
creative person. They indicated that the creative person:
(a) was less repressed, less inhibited, less formal, less
conventional, (b) produced novel and unconventional solu-
tions to problems, (c) was more intuitive and perceptive

with an Open, searching behavior, (d) more dedicated when
solving problems, (e) showed flexibility in his tolerance

for ambiguity, and (f) is not afraid of failure or being

laughed at. Torrance4 listed five principles which he

 

1D. W. MacKinnon, "Personality Correlates of Creativity/'
Productive Thinking in Education, Ed. M. J. Aschner and E.
Bish (Washington D.C.: NatiOnSI Education Association, 1965).

2J. P. Guilford, "Traits of Creativity," Creativityfiand
Its Cultivation, Ed. H. Anderson (New York: Harper and Row,
EFF).

3T. A. Razik, "Recent Findings and Developments in
Creative Studies," Theory into Practice, 1966, 5, pp. 160-
165; E. P. Torrance, Rewarding Creative Behavior (Englewood
Cliffs, N.J.: Prentice Hall Publishers, I965).

4E. P. Torrance, "Give the Devil His Dues. . . ,"
Gifted Child Quarterly, 1961, 5, pp. 115-120.

 

 

 

56

believed to be important in developing creative thinking:
(1) be respectful of unusual questions, (2) be respectful
Of unusual ideas of children, (3) show children that their
ideas have value, (4) provide oppOrtunities for self-
initiated learning, and (5) provide for periods of non-
evaluated practice or learning. EnochsI examined the
effects of these principles and found that creative
thinking can be nurtured by applying these principles.
Hallman, Durr, and Hughes2 also suggest ways to nurture
creativity which are analogous to those of Torrance. Each
of these researchers agree that the classroom environment
should be free from authoritarian control, encourage self-
initiated learning and place a high value on the indivi-
dual's feelings, creativity, and intellectual activity. In
general, research during the past fifteen years, concerning
creativity, supports the hypotheses that a suitable class-
room and teacher environment can favorably affect measures
of creativity and it is reaSOned that this type of environ-
ment would be most conducive in producing an increase in

problem solving ability.

 

1P. D. Enochs, An Experimental Study of a Method'for
Developing Creative Thinkingiin Fifth Grade Children, (Un-
published Ph.D. Dissertation, University of Missouri, Dis-
sertation Abstracts, 1965).

2R. J. Hallman, "Techniques of Creative Thinking,"
Journal of Creative Behavior, 1967, 1, pp. 325-330; W. K.
Durr, The Gifted Child, (New York: Oxford University Press,
1964); H. K. Hughes, “The Enhancement of Creativity,"
Journal of Creative BehaviOr, 1969, 2, pp. 73-83.

 

 

 

57

Problem Solvipg as Related to Other Variables

There have been a few studies which have examined
strategies for comprehensive problem solving. Most of the
correlated information on variables involved in problem
solving has been gathered in studies of problem solving
in a particular subject area, such as mathematics or science,
or problem solving as a cognitive activity.l' These experi-
ments most often limited their definition of a problem to
those challenges which a student could work on and arrive at
a solution for within a short period of time. Nonetheless,
these studies provide a good deal of information about some
variables which seem to influence problem solving ability.

The literature yields many studies which indicate a
strong relationship exists between intelligence and problem
solving ability. Max Engelhart2 concluded that there is a
strong correlation between intelligence and arithmetical

problem solving ability. Studies by McCuen3 and Erickson,4

 

1P. C. Wason and P. N. Johnson-Laird (eds.), Thinkin
and Reasoning, (Penguin Modern Psychology Readings, Penguin
Books, Ltd., Baltimore, Maryland, 1968).

2M. D. Engelhart, "The Relative Contribution of Cer-
tain Factors to Individual Differences in Arithmetical
Problem Solving Ability," Journal of Experimental Psyghology,
I (September, 1932), p. 25.

3T. L. McCuen, "Predicting Success in Algebra,"
Journal of Educational Research, XXI (January, 1930), pp.
72-74.

4L. H. Erickson, ”Certain Ability Factors and their
Effect on Arithmetic Achievement," The Arithmetic Teacher,
V (Sept., 1958), p. 291.

 

 

 

58

although some twenty eight years apart, showed a positive
correlation between intelligence and arithmetic and algebra
problem solving ability of .54 and .58 respectively. One
of the most comprehensive studies dealing with the
characteristics of an insightful mathematical problem
solver was carried out by Dodsonl in 1969 in which he
analyzed the School Mathematics Study Group "Y" population.
He concluded that particular measures of intelligence and
thinking were strongly related to problem solving ability.
Houtz, et al.,2 also found that general problem solving
ability was significantly correlated to several cognitive
variables including intelligence. This high correlation
between intelligence and problem solving ability could also
be inferred from the assumed relationship between problem
solving and creativity, as it is highly correlated to
intelligence in itself.

Vos analyzed the problem solving heuristics of senior
high school mathematics students and concluded that

. . . mathematical maturity was a definite factor in problem

solving ability."3 Maturation, in general, seems to affect

 

1J. Dodson, Characteristics of Successful Insightful
Problem Solvers, NLSMA Report No. 31, Schbol Mafhematics
Study Group, 1972.

2J. C. Houtz, S. Ringenbach, J. F. Feldhusen, "Rela-
tionship of Problem Solving to Other Cognitive Variables,"
Psychological Reports, 1973, Vol. 33 (2), pp. 389-390.

3Kenneth E. Vos, "The Effect of Three Instructional
Strategies on Problem Solving Behaviors in Secondary School
Mathematics." Paper presented at the National Council of
Teachers of Mathematics 53rd annual meeting, Denver, Colorado,
April, 1975.

 

 

 

59

problem solving ability as interpreted by Magkaev and
Rimoldi. Magkaev1 concluded that, after studying 490 school
children solving four types of reasoning problems (grades
1-4), the amount of planning used to Obtain a solution
increased with the subject's age. Rimoldi2 found that the
thought process during problem solving becomes more logical
as the subject's age increased. P.

A third related variable is the perception time, that
is the length of time necessary for the solver to encode
the problem in memory and reach the first possible hypothe-
sis. An early study by Rokeach3 suggested that taking more
time at the perception stage of Solving enables the solver
to think more abstractly concerning the problem and reduces
his rigidity in suggesting possible solutions. Kennedy4

also found a strong relationship between perception and

 

1V. Magkaev, "An Experimental Study of the Planning
Function of Thinking in Young School Children," Vo ros
Psikhologii, 1974 (Sept.-Oct.), No. 5, 98-106. Eninsh
summary in Psychological Abstracts, Vol. 54, No. 4, October,
1975.

 

 

2H. J. Rimoldi, "Language and Thought Processes,"
Rivista de Psicologia General y Aplicado, 1974 (May-June)
Vol. 29 (128), 499-514. English'summary in Psychological
Abstracts, Vol. 53, 1975.

3M. Rokeach, "The Effect of Perception Time Upon
Rigidity and Concreteness of Thinking," Journal of Experi-
mental Psychology, XL, 1950, p. 206.

4M. M. Kennedy, Effect of Perception On and Informa-
tion Abgpt Organization on Problem Solvipg,(Ph.D. DiSserta-
EiOn, M.S.U., 1973).

 

 

 

 

 

 

 

60

success at solving problems similar to the "fireman, brake-
man, and engineer problem" (Chapter 2, Part One). This
suggests that the student should not be in a hurry to begin
using any heuristic in solving problems.

Many of the studies done on the mathematical problem
solving ability of students have included the sex of the
student as a factor in the design. The results of these
studies which investigated the differences between the
abilities of problem solving in boys and girls are non-
directional. Some researchers such as Neill1 and Sheehan2
concluded that boys were better problem solvers. In other
studies, such as one done by Cleveland and Bosworth,3 no
significant differences were found between their problem
solving abilities. Schuner4 found that boys do better on
algebra problems, but that girls do better on geometry
problems. Each of these studies recommends that more re-

search be done on the question of the effect of sex on

problem solving ability.

 

1R. D. Neill, The Effects of Perceptign on and
Information about Organisstion on Probiem SoIving, (Ph.D.
DissertatiOn, M.S.U., I973Y.

2T. J. Sheehan, "Patterns of Sex Differences in
Learning Mathematics Problem Solving," The Journal of
Experimental Education, XXXVI, p. 86.

3G. A. Cleveland and D. L. Bosworth, "A Study of Cer-
tain Psychological and Sociological Charactersitics as Relat—
ed to Arithmetic Achievement," The Arithmetic Teacher, XIV,
p. 387.

4J. Schunert, "The Association of Mathematical Achieve-
ment with Certain Factors Resident in the Teacher, in the
Teaching, in the Pupil, and in the School," Journal of Ex-
perimental Education, XIX, p. 231.

 

 

 

 

 

61

Another variable which has an impact on problem
solving is the solver's emotional state at the time the

1 found that if the

process is undertaken. Combs and Taylor
solver feels a mild degree of personal threat when problem
solving, the time necessary to complete the problem will be
increased. Cowen2 concluded that reducing the stress in a
problem solving situation by praising the solver during pre-
vious problem solving attempts resulted in significantly
less rigid responses in suggesting possible solutions.
Mohsin3 studied the effects of frustration on problem solv-
ing behavior. He concluded that frustration affects each
solver differently, and that the stronger the frustration
the higher the interference with problem solving success.
These studies suggest that the classroom environment, both
during the presentation of the strategies and the practice

using them to solve problems, he student oriented and free

from authoritarian control.

 

1A. Combs and c. Taylor, "The Effect of the Perception
of Mild Degrees of Threat on Performance," Journal of Abnor-
mal and Social Psychology: XLII, p. 420.

2E. L. Cowen, "Stress Reduction and Problem Solving
Rigidity," Journal of Consulting Psychology, Vol. XVI, 1952,
pp. 425-428.

3S. M. Mohsin, "Effect of Frustration on Problem
Solving Behavior," Journal of Abnormal and Social
Psychology, Vol. XLI, 1954, pp. 152.

 

 

 

 

 

CHAPTER III

IMPLEMENTATION OF THE STUDY

Introduction

 

The purpose of this research was to gather information
concerning the problem solving process and the strategies
used by elementary school children in arriving at meaningful
solution to open end comprehensive problems.

A secondary purpose of this study was to compare the
problem solving behaviors of children not engaged in a pro-
gram Of problem solving practice with children who regularly
worked on comprehensive problem solving tasks. Based on
this information compiled from a case study of twelve fourth
grade youngsters, hypotheses for future experimental studies
can be formulated. The use of this clinical paradigm was
apprOpriate, as the problem solving process used by the
subjects had to be analyzed in great detail in order to
identify strategies and because any variation in the
process or the strategies was of interest.

This study involved analyzing five interviews, con-
ducted Over a period of fourteen weeks, for each of twelve
subjects. Students were selected from two fourth grade
classes. The manner and rationale for choosing these parti-

cular classes is described in the first section of this

62

63

chapter. Group description and individual characteristics
of the students in this study is the subject of the second
section of this chapter. The major factor which differen—
tiated the students in the experimental class from students
in the control class was the problem solving experience the
experimental class received from their classroom teachers
during the experiment. The nature of this practice in
solving problems is discussed in the third section of this
chapter.

Five interviews, each based upon a different compre-
hensive problem, were conducted by the researcher. A
description of the comprehensive problems used in these
interviews, and a description of the setting and inter-
viewing procedure, is provided in sections four and five of

this chapter.

Population and Classroom Selection

 

This experiment took place in one of the nine elemen-
tary schools in the Ypsilanti School District, Ypsilanti,
Michigan. Located west of the Detroit metropolitan area,
Ypsilanti is a community of about 70,000 persons, reflect-
ing a variety of intellectual, socioeconomic, and racial
representations. The district educates approximately 3730
elementary school age youngsters, of which 491 are fourth

graders.1 From this fourth grade population, twelve

 

1These figures are based on information received from
the office of the Evaluator/Coordinator of the Ypsilanti
School District.

64

students constitute the sample of this study.

Fourth grade students were chosen for this study since
the researcher felt that at this grade level the students
possessed minimal arithmetical, reading, and writing
Skills as are necessary to solve comprehensive problems.
Furthermore, Silverman and Stone1 showed that at the third
grade level about half of the students were non-conservers,
and that conservers prevailed over non-conservers2 during
interaction in problem solving gibups.

Since three of the problems used in this study involve
linear and area measurements, the researcher felt that the
conservation of the properties Of length and area were
necessary to solve these challenges in a meaningful way.
Wadsworth3 suggests that children conserve both length and
area by nine years of age. The researcher therefore conclud-
ed that by the fourth grade most of the students would be
conservers, and less likely to acquiesce to the views of
another student because of insufficient cognitive growth.

On the other hand, since many individuals and organizations

(e.g., Polya, Ausubel, and the Cambridge Conference on

 

1W. Silverman and J. M. Stone, "Modifying Cognitive
Function Through Participation in Problem Solving Groups,”
Journal of Educational Psychology) 1972, Vol. 63 (6),
pp. 603-608.

2A child is said to be a conserver when he understands
that a property (e.g., length, mass, area) of a body is
preserved when the form or position of the body is altered.

3Barry J. Wadsworth, Piaget's Theory of Cogpitive

Development, New York: McKay Company, Inc., 1970, p. 31.

 

 

 

65

School Mathematics)1 emphasize teaching problem solving at
all levels, it seemed advisable to investigate the problem
solving process as early as possible in the students school-
ing. Moreover, Torrance2 reports that by the fourth grade
much of the studenttscreativity has been lost due to the
educational system itself. Vernon and Razik3 agree that
the wild questions and off-beat ideas of highly creative
students are discouraged in favor of more conventional
behaviors. These findings suggested that creative problem
solving should be initiated as soon as possible, with the
result being a reversal of this trend.

Pilot interviews conducted by the researcher four
months prior to the beginning of this experiment suggested
that the maximum number of students who could effectively
respond to a challenge during an afternoon was six.) If more
than six students were chosen from one class, the interviews
would take at least two days, and permit student discussion
of the problem outside the interview period. Therefore, six
students were chosen from each of two classes to provide the

twelve subjects of this study.

 

1G. Polya, How To Solve It (2nd edition, Garden City,

New York: Doubleday and Co., Inc., 1957); David P. Ausubel,
"Learning by Discovery," Educational Leadership XX (January
1969); Goals for School Mathematics, the Report of the Cam-
bridge Conference on SchOol Mathematics (Boston: Houghton
Mifflin Co., 1963).

2E. P. Torrance, ”Adventuring in Creativity,” Childhood
Education, NO. 20, 1962, p. 83.

3P. E. Vernon, editor, Creativigy (Baltimore: Penguin
Books, 1970), T.A. Razik, "Creativity, Theory into Practice,
No. 22, 1966, pp. 147-149.

 

 

 

 

 

 

66

Shapiro1 concluded after examining children from
seventy-four elementary school classes, grades 2 through 6,
that youngsters who did not have experience in solving
comprehensive problems made significantly fewer measureable
responses in attacking a problem than students who had this
experience. Shapiro's conclusion suggests that this study
should investigate the problem solving processes used by
both trained and untrained subjects. Therefore for the
duration of the study, one class received some problem
solving training, in addition to their usual school activi-
ties, from their classroom teacher.

The two classes used in this research were selected
to meet the following criterion:

a) Students learned in an environment where indivi-

dual feelings, creativity, and intellectual

activity were highly valued.

b) Students could accurately express their Opinion
when questioned.

c) Students had never worked on, or participated in,
any problem solving prOgram.

d) Students were male and female, and from diverse
socioeconomic backgrounds.

The researcher consulted the Coordinator/Evaluator
of the Ypsilanti School System to obtain the names of teach—
ers of classes which met the stated criterion. Principals

were contacted and appointments made with those teachers

 

1B. J. Shapiro, The Notebook Problem; Rspprt on
Observations_pf Problem SOlVing AEtiVity, EducatiOn Develop-
ment Center, Boston, 1973.

 

 

67

who expressed an interest in problem solving behaviors.

Two fourth grade teachers were selected based on their
interviews with the researcher, the suggestions of their
principal, and observations of their interaction with their
classes.

To provide a less subjective rationale for selecting
these teachers and their classes, the researcher asked each
teacher to complete a questionnaire. The questionnaire,
which appears in Appendix A, contained requests for personal
data and attitude assessment questions. The results of the
survey indicated that the selected teachers were similar in
attitudes cOncerning classroom management, student initia-

tive, and their ability to teach problem solving.

Descriptions of Classes and Sample

The two selected fourth grade classes were from the
same all white school with a student population from varied
socioeconomic backgrounds. One of the teachers of the
selected classes volunteered to provide problem solving ex-
perience for her class in addition to their usual activities.
The class having the additional problem solving experience
is designated as the experimental class throughout this
study, while the other is referred to as the control class.
The experimental class was composed of twelve male, and
eleven female students, with a mean age of nine years, three
months. The control class consisted of ten males and twelve

females with a mean age of nine years, five months.

68

The classes used in this study were two of the three
fourth grades in a school which ranked first and third
respectively in mathematics and reading among the nine
schools of the Ypsilanti School District on the State of
Michigan Assessment tests. These tests were administered
six weeks prior to the commencement Of this study.

From each of the two classes described above, a
random sample Of three males and three females was drawn.
Permission to include these students in the experiment was
secured from their parents or guardians.

(In an effort to show thatthe sample was homogeneous in
problem solving ability at the onset of this study, the re-
searcher attempted to assess this ability prior to the first
interview. The only instrument available which is consistent
with the theory outlined in Chapter Two and tests problem
solving ability is the Purdue Problem Solving Inventory.1

The Purdue Problem Solving Inventory was developed to
assess the specific behaviors which the developers felt were
involved in the problem solving process. These behaviOrs-
are: (l) sensing that a problem exists, (2) defining the
problem, (3) clarifying the goal, (4) asking questions,

(5) guessing causes, (6) judging if more information is
needed, (7) noticing relevant details, (8) using familiar

objects in unfamiliar ways, (9) seeing implications,

 

1J. Feldhusen, J. C. Houtz, and S. Ringenbach, "The
Purdue Elementary Problem Solving Inventory," Psyghology
Reports, 1972, 31, pp. 891-901.

 

69

(10) solving single-solution problems, (11) solving multiple-
solution problems, and (12) verifying solutions. The test
presents problem situations which were judged to be inter-
esting to students in various formats by teachers who had
used these materials. The abstract form Of the Inventory,
requiring the student to make a multiple choice response to

a problem presented in written form, was recommended for

this study by the developer of the instrument in a personal
correspondence.

The developers validated the Purdue Inventory by
administering it to black, white and Spanish speaking second,
fourth, and sixth graders from advantaged, and disadvantaged,
backgrounds. Total scores and scores on the subclasses were
obtained for 1073 children and a three way analysis of
variance was computed. The ethnic factor accounted for only
3 percent of the total variance, social status accounted
for 5 percent, and grade factor accounted for 3 percent. A
principle factor analysis established that there are
separate problem solving factors. Finally, the test retest
reliability of the inventory1 is .70 for advantaged students
on the abstract form.

A week before the interviews began, all twelve sub—
jects completed the abstract form of the Inventory under the

supervision of the researcher. A two-way fixed design, sex

 

1J. Feldhusen, J. C. Houtz, and S. Ringenbach, "An
Abstract Test of Problem Solving Ability," A paper presented
at a joint NCME-AERA session, New Orleans, 1973.

70

by treatment, Analysis of Variance was computed to detect
differences between classes and/or sex. The results of the
Analysis showed no significant differences between sexes and
classes, and no significant interaction between variables,
and therefore suggests that the two classes do not appear

to be different in problem solving abilities.

As indicated earlier in this section, the treatment
class received practice in problem solving, while the control
class did not. The nature of this practice involved a
combination of USMES activities and worksheets provided
by the researcher. The twenty-two worksheets appear in the
Teacher's Guide which can be found in Appendix B. The
worksheets were problems modified from those found in puzzle
books appropriate for nine year olds,1 and embody one or
more of the generalized problem solving strategies identi-
fied in Chapter Two. Approximately two of these worksheets
were to be completed by the students each week of the experi-
ment, under the direction of their teacher, as time
permitted.

In addition to these problem puzzles, the entire class
also worked on some comprehensive, teacher-supervised pro-
blems during the fourteen weeks of the study. The USMES
challenge "Consumer Research-Product Testing" and a design

problem planned by the researcher entitled "Design an

 

1For example, Jack C. Crawford, Hooray for Plgy,
(Doubleday & Co., 1958), Joan C. Weatherby, 100 Hours of Fun,
(Doubleday & Co., 1962).

 

 

71

Underwater Habitat" were investigated by the experimental
class during this period. The class spent eight weeks in
product testing; selecting the "best" paper towel, glue,

and notebook paper. The remaining six weeks were spent
designing the underwater structure suitable for three

persons to live in for seven days. In order that the teacher
effectively guide the students through these activities,

the researcher provided an in-service orientation to the
comprehensive problems. The class was to devote about

forty-five minutes twice a week to these activities.

Description of the Problems Used in the Interviews
Five interviews were conducted with each subject, at

approximately three week intervals, to determine which
problem solvingéstrategies, if any, were employed in
addressing five different challenges. The problems used in
the interviews were taken from the Unified Science and
Mathematics for Elementary Schools (USMES) materials since
teachers using this program report that students find the
material interesting and challenging. The particular pro-
blems were selected to represent four different types of
problems: (1) product evaluation, (2) design, (3) event
planning, and (4) description. The seOOnd and fifth inter-
views were design problems. The following paragraphs

describe each of the five problems.

72

The Notebook Problem

The first problem, which was widely used to assess

the USMES program, was to determine which of five notebooks

would be best for use as a fourth grade mathematics and

science notebook.

The students were shown five spiral note-

books, each of which differed from the others in at least

two ways. The notebooks had the following characteristics:

Notebook One -

Notebook Two -

Notebook Three -

Notebook Four -

This notebook, denoted the "red" notebook,

measured 10.1/2” by 8" and contained forty

(40) pages of white, wide-ruled paper. The

front cover, which was red, had an attached
pocket inside the notebook facing the first
page. This five-hole notebook cost 58¢.

The second notebook, labeled the "pattern"
notebook, had a multicolored floral design
centered on the front cover which had a
yellow border. This notebook contained
one hundred and four 10.1/2" x 8" white,
wide-ruled sheets. The pages were separat-
ed into four equal sections by unlined
blue pages. The cost of this five-hole
notebook was 97¢. The pages were bound
together with a double wire binding which
was heavier than the wire used in the
others.

The third notebook, referred to as the
"yellow" notebook, contained forty-five
sheets of 10.1/2" x 8" yellow paper with
wide-ruled red lines. The yellow cover
had a two-toned picture of two young
adults, looking at a flower, with the
words "Good Vibrations" below. This five-
hole notebook cost 68¢.

The "green" notebook measured 11" x 8.1/2"
and contained one hundred (100) sheets of
white, narrow ruled paper. The front cover
of this three-hole notebook was green and
the cost Of this book was 97¢.

73

Notebook Five - This notebook, which is denoted the "light
red" notebook, contained sixty-four sheets
of,10.l/2" x 8" wide ruled white paper.
The front cover of this five-hole notebook
was light red and had a name space. The
cost of this notebook was 67¢.

Each subject was seated at a desk with the five note-
books in front Of him/her and given the following intro-
duction and explanation to the problem:

Suppose the city of Ypsilanti planned to buy each

fourth grade student a notebook to be used for math

and science work. There are more than 450 fourth
graders in this city and the school board wants to
leave the final selection to the students. I want
you to tell me which of these notebooks they should
buy, that is which is the "best" notebook to be
purchased for all fourth graders in this city.

The student was also provided with pencil, pen, string,

ruler, crayons, and scratch paper which they were told they

could use in making their determination.

Classroom Design Problem

During the second interview, the students were
challenged to design a classroom which would provide for
all those activities that should go on there. The student
was seated at a desk with a large poster board on it. The
poster board had a scale drawing of a large 30' x 40' empty
classroom which had windows located on two walls. A reduced
copy of the classroom drawing is given in Appendix C. They
were also provided with scale cutouts of desks for the
students and the teacher. Blackboards, rugs, tables, chairs,
bookcases, and shelves of various shapes and number were

also provided, as well as a sink cutout and blanks which

74

they could use to represent any other item they wanted to
utilize in the room. Each student was given the following
instructions:

The city of Farmington is building a new elementary

school and wants future students to like it. I

want you to tell me what should go into the class-

room and how the room should be arranged. Using any

of these cutouts or others that we can make, plan a

classroom to provide enough space and which con-

tains the necessary furnishings to carry out all the
activities which must go on there.

The interviewer explained the scale drawing of the
classroom, indicating windows and doors, until the student
said he understood the layout and the scale to which it was
drawn. The interviewer asked and responded to questions
until the student said he or she understood each ready—
made cutout and that various sizes of certain pieces were

available or could be cut out to suit the needs of the

designer.

Plan an Ice Skating Party

The third problem of planning an ice skaing party for
fourth graders is a variation Of the "plan a picnic"
challenge used by USMES as a part of their on-going assess-
ment of the USMES program. The student was seated at a
desk with pencil and paper. In front of the student were
two large poster boards. On the first was a drawing of an
aerial view of a school and three neighborhood ice rinkS.
A reduced COpy of this.drawing of the neighborhood is given
in Appendix D. The rinks differed in various characteris-

tics, and are described as follows:

75

Area One - This area represented a park, with an ice rink,
six blocks northeast of the school. Within
the park, two blocks from the rink, was a rest
area containing a shelter, restrooms, and a
concession stand where food could be purchased.

Area Two - Due west of the school two blocks was a small
park with an ice rink slightly larger than that
in area one. The park provided a place near
the ice for a bonfire and had a concession
stand.

Area Three - Eight blocks north of the school was an ice
rink which had a 50¢ admission charge. Adja-
cent tO this, the smallest of the three rinks,
was a heated, enclosed building with rest-
rooms, concession stand, and rest area.

The second poster board contained pictures of food which
was available at each of the area's concession stands. In—
cluded on the menu were six types of sandwiches, pizza, soup,
drinks, snacks, and desserts, with the price of each item
listed next to it. The student was provided with the
following background explanation and instruction:

A school near my house has a fourth grade class

of twenty-five students which has saved fifty

dollars from bake sales they have sponsored.

They would like to use the money for a skating

party. They plan to go sometimes in January to

one of these three areas and perhaps spend some

of their money for something to eat. If you were

on the planning committee how would you plan the

party? Consider everything you think is important,

like where and when to go, what to do, and whatever

else they should think about.
The interviewer asked if the subject understood the drawing
and pointed out the different aspects of each area. When
the student indicated that he understood the problem, the

interview began.

76

Describing People

The fourth interview challenged the Youngsters to
determine what information would be best included in the
description of a person. After the student was seated in
the interview room, he was shown a photograph Of a person
unknown to him/her. The interviewer asked how to describe
the person in the photograph so that the interviewer's
brother would recognize that person at the airport. After
discussing the many describable attributes of a human
being with the subject each was asked:

Determine what is the best information to put

in a description so that a person can be quickly

and easily identified.
As the student responded to the problem, the interviewer

interrupted only to ask for clarification of a tentative

solution (e.g., "Why did you choose that characteristic?").

Playground Design

The last interview for each subject concerned the
solution of the playground design problem: Design a play-
ground suitable for an elementary school.> The subject was
provided with a large drawing of a play area of a school
printed on poster board. A reduced copy of the scale
drawing can be found in Appendix E. A second poster board
contained pictures of various pieces Of playground equipment,
both conventional and free form items. The price of each
unit was included with the picture. Finally, the student

was given scale drawn cutouts of this equipment in each of

77

the various available sizes, and poster board which could
be cut to any size to represent playground equipment of the
subjects own design. In addition, pencils, paper, scissors,
and a ruler were available for the student to use.

After the explanation of the scale cutouts, the
drawing of the playground and the catalog poster board was
acknowledged as understood by the student, the following
information and statement of the problem was given:

An elementary school in Ann Arbor is planning

a new playground for students from the first

through the fifth grade to use. The school

board has given the school $4,500 to be used for

this purpose. Suppose you were helping to plan

this playground. What equipment would you in-

clude, and where would you position it in the

play area?

The only interjections made by the interviewer were to ask

the students for clarification of what they were doing and

why, or to answer student questions.

Interviewing Procedures

 

Beginning the week prior to the problem solving
activities of the experimental group, and continuing at
three week intervals thereafter, the researcher conducted
interviews designed to detect patterns and strategies used
by the subjects in solving the problems described in the
Previous section. While it is clear that the interpretation
of the behaviors which were Observed could be a function of
Idle person making the inferences, the researcher felt that

reporting these behaviors and specific strategies could be

most effectively done by the researcher. The interviews

r-..

 

78

were conducted during the afternoon of that day of the

week which the classroom teacher felt would be least
disruptive to learning activities. Each teacher wanted the
interviews to take place in the afternoon.

All of the interviews and the pretest took place in a
classroom of the school building containing approximately
ten student desks of various sizes. The room was well
lighted, comfortably warm, and free from outside distractions.
The students were taken to the interview room by the research-
er, one at a time, and returned to their classrooms at the
end of each discussion. During the interview the only
people in the room were the subject and the researcher.

For the first, second, and fourth interview the
researcher sat facing the student and about five feet in
front of the solver. During the third and fifth interviews
the subject had a poster hanging on a blackboard in front
Of him, and the interviewer sat at the right side of the
respondent.

Each interview was recorded on a cassette recorder,
and Observations made by the researcher were transcribed.
Paper used by the student for computations were duplicated
and added to the interview material. A description of

these interviews is found in Chapter Four.

CHAPTER IV
STUDENT RESPONSES TO THE PROBLEMS

Introduction

This chapter contains a description of the problem
solving behaviors demonstrated by the students during each
of the five interviews. This material is organized, by
problem, into five sections. Each section is subdivided
into the following:

a) Setting and introduction to the problem

b) Responses of the experimental class

c) Responses of the control class

d) Identification of the strategies used by
each class

Throughout this chapter, the students are referred

to by first name, and a class designation; (E) for members

of the experimental class, and (C) for members of the control

class.

The Notebook Problem
The first interview took place during the second week
Of November, with the experimental class responding on
tPuesday afternoon, and the control class on Thursday. Up
Luntil the time of this interview, neither class had been

1“volved in any special problem solving activities. In

79

 

80

order that the students become acquainted with the inter-
viewer, they met with him twice, in their classrooms,

during the last week of October. The Purdue Elementary
Problem Solving Inventory was administered by the researcher
to all twelve students, in the room to be used for the
interviews, during the week prior to this first interview.

In the first interview each student was asked to
respond to the following challenge:

The school board is planning to purchase one thou—

sand of one of these five notebooks for the use

of each fourth grader in the district. Which

notebook do you think is the best to buy?

While the interviewer gave this statement of the challenge,
the student was shown five notebooks (described in Chapter
Three), each of which differed in at least two attributes
from the remaining four.

Meaningful solutions could only be reached by those
students who were able to define "best" as it applies to
notebooks, understand the aspects involved in the problem,
examine the alternatives, and select the alternative most
consistent with his/her definition.

After the problem was stated, it took an average of
twenty-six seconds for the students in the experimental
class, and twenty-nine seconds for the students of the
control class, to reach a conclusion. The elapsed time to
sOlution to the nearest five seconds, for each of the twelve
students, is given in Table l. The researcher feels that

tI‘lie‘short period of elapsed time before each subject arrived

git: their solution indicates a myoptic view of the problem.

81

Table l. Elapsed Time to Solutions

 

 

 

 

 

 

Experimental Class Control Class
Number Number
Student of seconds Student Of seconds
Trina 15 Shari 15
Jackie 30 Susan I 30
Denise 30 Linda 25
Todd 20 John 30
Michael 30 Mark A 30
Joey 3O . James 40
Average 26 Average 28

 

 

 

 

 

 

Responses Of the Emperimental Class

 

After the challenge to select the best notebook was
made, each student in the experimental class made a cursory
examination of the notebooks before announcing his/her
decision. The selection of each student, given in parenthe-
ses, is indicated below. Their choice is followed by their
response to the question, "Why do you feel that is the best
notebook?"

Jacquelyn (E): The one with yellow paper (yellow
notebook). Has yellow paper and you might
need it sometime.

Denise (E): This one (green notebook). Almost the
most sheets and the biggest sheets. This
one (pattern) has four more, but this one

(pattern) the sheets are smaller.

Trina (E): This one (light red notebook). Has a
space for your name.

82

Michael (E): That one, its more sturdy (pattern
notebook). I narrowed it down to these two
first (indicating the pattern and the green
.notebooks), then decided on this one.
(Interviewer requested that Mike expand his
answer.) First of all this one has yellow
paper, and I don't think you need that (yellow
notebook). This one has only sixty-four pages
(light red notebook) and this one only has
forty pages (dark red notebook). The binding
is better (on the pattern notebook). The
other (green notebook) may catch on something.

Todd (E): (dark red notebook) This one. You might
have to do your work over if you lost it,
and you won't lose it in here if she tells
you to keep it.

Joey indicated his choice of the green notebook by
pointing to it, but could give no explanation as to why it
was the "best" notebook for the school district to buy.

The interviewer then asked questions to discover
whether the students had considered any other aspects of
the problem. Three youngsters were unable to expand on
their answers [Joey (E), Jacquelyn (E), and Trina (E)].

Joey (E) was almost completely non—verbal during the
interview, shaking his head or shrugging his shoulders to
answer the questions which were directed at him. Jacquelyn
(E) and Trina (E) investigated the problem superficially,

as both seemed to evaluate the proposed challenge based
solely on visual characteristics unique to one notebook so
that an answer could be given. Neither could give a defense
of their choice, nor in answer to further questions had
timey considered attributes such as cost, number of sheets,

0:: ruling of the pages. Trina (E) did not respond to any

flilrlz‘ther questioning, and the interview terminated five

 

83

\

minutes after the statement Of the problem. Jacquelyn (E)
changed her mind four times during questioning, with each
selection reflecting the direction given by the interviewer:

Interviewer: What about the number of pages in the
notebooks?

Jacquelyn (E): I would pick this one (Indicating the
green notebook after spending thirty seconds
reviewing them a second time).

Interviewer: How about the cost? Is that important?

Jacquelyn (E): Very important.

Interviewer: If you had to choose between the yellow
notebook and the one which cost the least,
which one would be the best?

Jacquelyn (E): The one that cost the least.

Interviewer: How about the number of pages?

Jacquelyn (E) again changed her mind. The interview

was concluded with Jacquelyn (E) defending the notebook
which cost the least. Her rationale for this decision was
based on repeated attempts at finding the actual saving
involved in purchasing the least expensive notebook as
compared to the most expensive. Her frustration in

finding the answer is reflected in her reply:

Jacquelyn (E): Because, if you are going to get one
thousand, you would save more than a dollar.

The other three respondents were able to expand on
their initial answers. Michael (E) chose the pattern note-
.book based on several considerations: number of pages,
5Strength of the binding, color of the sheets. Todd (E)
ekplained why he felt that a pocket in the cover was impor-

tant and Denise explained that she wanted to maximize the

84

amount of paper in the notebook.

Responses of the Control Class

Each subject within the control class perused the
five notebooks before arriving at a decision as to what he/
she considered the "best." Their selections, and the
rationale for their choices, are as follows:

Shari (C): (green notebook) This one. Its got a lot
of pages. ’I like the big size. I have one
this size (referring to the dimensions of the
page).

Linda (C): (pattern notebook was indicated by holding
it up.) It has more paper, and it has two
different kinds (color) in it.

Susan (C): (pattern notebook) I pick this one. It
has dividers, and they could tell which sub-
jects and stuff. It would be a lot easier,
and they wouldn't have to have as many note-
books. (This notebook has four blue pages
in it which could serve as effective dividers.)

James (C): (dark red notebook) I think this one.
When you are finished with your math you
could put your work in the folder. It cost
less.

Mark (C): (pattern notebook) This one, because you
could keep different subjects separate.
Its a thicker book.
John (C): (pattern notebook) I'd say this one. I
think so because its got a lot of paper and
it also has a lot of colors on it and its
got some blue paper to write on.
When compared to the responses of the experimental
Class, the initial replies of the students from the control
<=1ass reflect a consideration of a greater number of aspects

<>f the notebook challenge.

85

Identification of the Process and Stratsgies
Used by the Students

 

Nine of the students, three experimental and six
control, used an identifiable process in reaching a
meaningful solution to this challenge.

In every instance, the student defined the problem
by establishing priorities as to those attributes he
deemed necessary in his choice of the "best" notebook.

After the definition stage, each student considered
the different notebooks and eliminated those which were
inconsistent with this ordering. This process resulted in
a choice which the individual felt was the optimal selection.

Based on Observation by the interviewer, it seemed
that the elimination of alternatives was carried out based
purely on perception. For example, to find the notebook
with the maximum number of pages, seven students (four
control and three experimental) chose one that was thick.
Comparisons were made by laying or standing two notebooks
side by side. Denise, for example, wanted the notebook with
the maximum amount of paper, that is, the greatest area for
writing. She didn't attempt to compare the total writing
area available in each notebook analytically, but rather
she held them up to check the dimensions of the pages
Visually. None of the youngsters attempted to count the
lTumber of pages, and only two of them, Michael (E) and

IDenise (E), gave evidence that they noticed that the number

of pages was written on each notebook.

86

Except for James (C) none of the twelve students
considered the price of each notebook until the interviewer
directed their attention to this aspect of the problem.
After such direction, five students (two from the experi-
mental class and three from the control class) compared
prices, but did not proceed to use this information to
support or reject their original choice. When the inter-
viewer asked about the cost of their selections, two
typical responses were those of Denise (E) and Susan (C):
Denise (B): (After noting the cost of the notebook)
This one (green) is still better, because
this one has more pages, and these two (green
and pattern) have the same amount of pages.
(The interviewer then challenged the subject
to explain why she chose the green notebook
over the less expensive ones.) Cause they
don't have as many (pages) and they might
not go through the whole year.

Susan (C): They are both the same (pattern and green
notebooks). You get more for your money.
(The interviewer pointed out that any of the
three notebooks cost less than either Of the
notebooks she was considering.) I just picked
this one.

This first problem also provided evidence that
children can and do make inferences. The problem solving
strategy of making inferences involves drawing conclusions
from the goal or givens. Each of the subjects who chose
a notebook after defining "best" to themselves made conclu-
sions based on the attributes (givens) of the notebooks.

John (C) felt that the best notebook should last as

long as possible. He reasoned from this goal that maximiz-

ing the lifespan of a notebook implies that the student

87

must not abuse it. John (C) inferred that students were
not as likely to misuse a notebook which they liked, hence
he wanted the ”best" notebook to be attractive. Michael
(E) inferred from the goal of finding a durable notebook
that the binding must be sturdy.

None of the twelve subjects considered the ruling
of the pages, the number of holes (three or five), or the
strength of the individual sheets. It was not clear from
the interviews whether the subjects avoided experiments of
testing the strength of the paper, or the visibility of
marks on a sheet, because they had not thought of them, or
because they were inhibited since they realized that the
notebooks were to be used by other youngsters.

In this first interview, half of the experimental class
gave meaningful responses, while each of the control class
demonstrated a process which 1ed_to valid solutions. The
researcher concludes that this variation is due to indivi-
dual persOnality differences, that is members of the control

class were more extroverted and willing to express opinions.

Classroom Design Problem
The second interview took place during the first
week of December, and concerned the planning and arranging
of a classroom. Since the first interview, the control class
had engaged in their usual classroom activities. The
experimental class had two periods in which they worked with

the Consumer Research-Product Testing unit of USMES. In

88

addition, this class had completed two of the problem
solving worksheets during the last two weeks of November.

During the second interview the students were asked
to respond to the following question:

How would you plan and arrange a classroom to

provide adequate space for all of the activities

that might be carried on there?

Each student was shown an outline of an empty class-
room and given a number of cutouts, drawn to scale,
representing various items which might be found in a class-
room. These materials were to be used by the student in
exhibiting his/her proposed layout of the classroom.

In the researcher's judgment, this challenge
provided an opportunity for the subjects to use several
strategies: inference, contradiction, and the identifica-
tion of subgoals. The nature of the problem allowed

students to investigate and reach conclusions without much

verbal response.

Responses of the Experimental Class

After the initial familiarization with the materials
to be used in seeking a solution, every student began by
choosing objects for the room without stating or requesting
the number of students for whom the room was being planned.
The interviewer interrupted the solving process to ask each
student how many children would use their classroom. Their

.replies are as follows:

89

Jacquelyn (E): About twenty.

Denise (E): Twenty to twenty-five seemed like the
right number.

Trina (E): Twenty-three. That's what most everybody
elses class have.

Joey (E): Twenty-two. Our class has twenty-three.
Michael (E): Twenty—four. That's the number of
students in our room, and every other room

in this school.

Todd (E): Twenty-five. Most classes around here
have about that many.

These students are from a fourth grade class which
contains twenty-three students.1 Their responses seem to
reflect these students' experience with classroom sizes.

Every experimental child began the solving process
by putting the desired number of student desks (or a desk
for the teacher) into the classroom. The researcher posed
the question: "Why did you begin with desks?" All of their
replies were essentially the same--desks are vital.

This evidence seems to support the claim that these
students did, in fact, establish an ordering of the compo-
nent tasks involved in filling a classroom. That is, they
were using ordered subgoals to reach a solution.

Jacquelyn (E), Denise (E), and Michael (E) arranged
the desks in U-shaped formation. This is the same arrange-

ment of desks found in the experimental classes' room.

 

1The principal of this school, in a discussion with the
researcher, indicated that the average class size had de-
creased during the past two years. The third grade classes
the year prior to this study averaged twenty-four students.

90

Trina (E) placed her desks in six rows, four desks to a row.
Todd (E) segregated the class into four rows of six desks
each, and put them in two rows on each side of the room.
Joey's (E) arrangement began with the teacher's desk in a
corner of the room, and three rows of student desks, in
semi-circles, facing that corner. This floorplan reminded
the interviewer of the way seats face the stage in an
auditorium. Joey (E) was asked to tell the interviewer
why he chose this way to set up the class, but he could
give no reply and simply shrugged his shoulders. I

A variety of items (tables and chairs, rugs, book-
cases, sink) were added to the classroom after the desks
had been placed. Each of the six students rearranged some
of the furniture to admit the placement of one or more
additional pieces. However, Denise (E) and Trina (E) ignored
obvious contradictory situations by placing chalkboards in
front of windows, and thus prevented light from entering
the room. Although both of these girls understood where
the windows were in the drawing at the onset of the challenge,
they must not have realized their location when they placed

the chalkboards ten or fifteen minutes later.

Responses of the Control Class

Immediately after the statement of the challenge each
member of the control group began putting items into the
classroom outline. The interviewer interrupted the solving

process to ask each student how many children would use

91

the classroom they were designing. Their answers and
rationales are indicated below:
Shari (C): I don't know. I'm going to try to get

in pretty many. (Her final arrangement
contained twenty-four desks.) Because
where I was we always had more than we have
here. (Shari had transferred into the school
during May of her third grade year.)

Linda (C): About twenty. Every room I've been in
had at least twenty.

Susan (C): Twenty-eight. I don't know (why I chose
that number).

James (C): Twelve. For two rows. We had it a
couple of years ago.

Mark (C): Twenty-two. I don't know (why I chose
that number).

John (C): Thirty. It's a good number. They could
teach more students in one class.

Except for James (C) each student used between twenty and
thirty student desks. John (C), James (C) and Susan (C)
planned for a class containing a significantly different
number of desks than found in the control classroom (twenty-
one).

Of the six students from the control class,
only Linda (C) and Mark (C) began by placing anything
other than student desks in the room. After placing two
items in the classroom, both of these students started put-
ting desks into the outlined classroom area. The inter-
viewer asked each of the six students why they began as
they did. Linda (C) could give no explanation for starting
‘with a rug and a bookcase. The remaining five felt that

desks were the most important furniture for the classroom,

92

and should go into the room as soon as possible. For
example, Shari (C) responded, "Well, you've got to have
desks for the students."

Four students [Shari (C), James (C), Mark (C), and
John (C)] arranged the desks in a row by column array.
Initially, Susan (C) and Linda (C) grouped the desks into
sets of four, however Linda (C) altered her arrangement to
include two sets of two rows of desks. This use of rectan-
gular array arrangement is interesting, since in the control
class the position of desks were currently arranged in
clusters (circular, square, rectangular), and had been for
the six weeks prior to this interview. None of the students
could explain why they thought their arrangements were the
best possible.

After the desks had been placed, the students added
items which were found in their control classroom (work—
tables, rugs, bookcases, sink). Each member included a rug
and a bookcase in one corner of the room, an arrangement
which coincides with that found in the control classroom.

Rearrangements of the furnishings were necessary when
space became limited or when the position of an item con-
tradicted its usage (e.g., a chalkboard which nobody could

see).

Identification of the Strategies Used py the Students
The first strategy which became apparent during the

interview was the identification and ordering of subgoals

93

and subtasks which were to be completed. Six experimental
and five control students clearly wanted to complete

the task of arranging desks before planning any other aspect
of the layout. This component of the design was therefore
given preference before undertaking any other component
task, such as planning a reading or art area.

A second strategy was detected as each student placed
more and more things into the room outline. When placement
of an item conflicted or interfered with the position or
purpose of another cutout which had been previously situated,
subjects would generally rearrange one or more of the pieces.
For example, Mike (E), when he added the chalkboards,
realized that his initial placement contradicted his goal
that "every student could see the board," and so proceeded
to rearrange the chalkboard location and some of the
earlier set desks.

Further examples of the use of contradiction were
apparent in the layouts of Mark (C) and Linda (C). Mark (C)
moved the teacher's desk to permit youngsters a better view
of the chalkboard. Linda (C) changed the complete arrange-
ment of students'desks to avoid contradicting her goal of
"not wasting space." Seven of the twelve students in this
study (three control and four experimental) made some adjust-
ments of the arrangements in their layouts as they encoun-
tered situations Or objects (e.g., walls, windows) which
precluded the attainment of a goal. The remaining five

students did not rearrange the cutouts and it was not clear

94

to the researcher whether these students used contradiction

in planning their classroom design.

Since there was very little verbal response on the
part of the subjects except when asked direct questions
with respect to this second challenge, the interviewer was

unable to detect behaviors which could only result from

inferences made by the subject. However, the interviewer

had the impression that some of the youngsters in this study
did use inference, in conjunction with other strategies,

in arriving at a solution. For example, Mark (C) placed

his desks in rows so that students would not be seated
behind each other. When asked why he did so, he replied,

"So everyone cansee." Clearly, his goal of providing

an equal opportunity for all students to see what was going
on implied that the line of sight from each desk (to the
te<‘=1<211er's desk and to the chalkboard) must be unobstructed.
Except in a few similar situations, the students worked
WitI‘Out commenting on his or her reason for a particular
placement, and hence the researcher can not suggest how
much ‘- inference was being utilized.

Another problem solving strategy became apparent as
the students completed their arrangements. All twelve of
tha planned layouts contained those objects which were

ECDund in their fourth grade classroom, that is bookcases,
cabinets, tables, and a sink were found in every floorplan.

The interviewer concluded that the subjects were recalling

the solution (arrangement) to the equivalent problem of

95

equipping the classrooms of their school when they planned

the arrangement of the proposed new room. To support this

conclusion, the interviewer asked each subject, after they

had completed their classroom design, if there was anything

else they wished to have included in the room. Three

students, Joey (E), Jacquelyn (E) and Linda (C), answered

no and their interviews were terminated. The other nine

children began naming things as they looked around the class-

room in which the interview took place. For example, Todd

(C) suggested "two wastebaskets, flag, clock, thermometer

(he meant thermostat), and closets," as he noticed these

items were also found situated within the interview room.

The researcher concluded that these students were using a

related problem, to which they knew the answer, (to respond
to the present challenge of classroom format.

The solutions to this challenge also exemplified the
lack of creativity demonstrated by the members of this study.
AS pointed out earlier, Joey (E) was the only student to
gin an arrangement of desks which was somewhat original.
MOrSover, no subject suggested items for the floorplan

wh ich were not found in either the interview room or in

th‘eir fourth grade classrooms. This condition existed even

though the researcher showed them templates, poster board,

scissors, and rulers that were available, and also explained

that scale cutouts of almost anything they suggested could

be provided. The interviewer has the impression that the

Subjects tried to give solutions which were like an existing

 

96

classroom or which was consistent with one they presumed the
interviewer had in mind. Indeed, Jacquelyn (E) and John (C)
asked ”Is this okay?" when they had completed their solu-
tions. Pointing to a specific cutout, Shari (C) was told
twice, after she began the problem, that she could put what-
ever she wanted into the classroom after she asked "Can I
use this?"

A second aspect of this challenge which was ignored or
poorly handled by the children is the understanding and use
of "scale" in the positioning of items within the classroom
boundaries. Each subject utilized the entire floorspace
of the scale drawing. There was no evidence that the
placement of any item was determined a priori by considering
the actual distance between fixed and movable objects. For
example, Joey (E) put seven chairs around a table which was
four feet in diameter. Denise (E) put a rectangular work
table and three chairs between the teacher's desk and the
b1ackboard--a distance of seven feet. Scale drawings are
generally encountered during the post elementary school
years, and neither the experimental nor the control class

had completed a unit on this topic.

Planning a Party Problem
The third interview took place on Tuesday and Friday
afternoons of the first week of January, four days after
returning to school from a two week Christmas vacation.

During the three class weeks of December, the control class

97

was involved in their usual classroom activities. The
experimental class worked on the USMES unit (Product Testing)
once, and with the problem solving worksheets three times
during these same three weeks. i

The third challenge involved planning an ice skating
party for twenty-five fourth grade youngsters, and assumed
that $50.00 could be used to cover any costs incurred.
The student was shown a map of the neighborhood surrounding
a school on which three recreational areas, each containing
an ice rink, were indicated. Each area differed from the
other two in at least two ways (e.g., proximity to the school,
enclosed shelter, restrooms). The student was instructed
that food service was available at each of the areas. A
large menu poster, listing and picturing food items which
could be purchased at each area, was positioned in front of
the subject after the introduction to the problem. The
students were allowed to work in silence except when it
was necessary for the researcher to interrupt and seek
clarification of what was being planned. For example, the
interviewer might ask, "Why did you select Area Two for the
party?" The solution of this third problem involves answers
to several subproblems which are essentially independent.
In other words, the subjects must answer many questions
before the plans for the ice skating party can be considered
complete. For example, the questions of where and when the
party will take place and what kind of snack or meal should

be planned can be construed as unordered subgoals. These

98

unordered subgoals must be resolved before a total solution

to the challenge is achieved.

Re3ponses of the Experimental Class

 

After the initial explanation of the problem by the
interviewer, the students were asked to name some items which
should be considered in planning a party. Jacquelyn (E) and
Todd (E) and Michael (E) responded with answers which
included when and where to go and what to do. Joey (E),
Trina (E) and Denise (E) made no response to the inter-
viewer's inquiry, and after thirty seconds the interviewer
suggested a few things which they might consider.

After the student listed or was told at least three
things to be considered in planning a party, the interviewer
asked, "What would you like to plan first?" Denise (E),
Trina (E), and Joey (E) did not answer, and after about
twenty-five seconds the interviewer suggested a direction:
"Where should we go?” The following responses were given:

Denise (E): Area Two because its closest, and its

got a big ice skating rink, and its got a
place for a bonfire.

Trina (E): Area Two because its closest. It doesn't
cost anything-~has a bonfire and food.

Joey (E): Area Three because its bigger.

The interviewer had the impression that although these
youngsters understood that there were several things to be
planned, they were unable or unwilling to select one of

these subproblems on which to commence work.

99

Jacquelyn (E), Todd (E), and Michael (E) were able to
suggest where they would begin working to plan the party.
Jacquelyn (E) began by solving the menu issue; Todd's (E)
first task was the selection of the location for the party,
and Michael (E) began by deciding the time aspect of the
problem. It was not clear to the interviewer whether each
of these children selected their answer because it was the
first subgoal to occur to them, or because they realized
that any of several equally valid starting points might be
used. After some solution was found for their initial sub-
problem, each of these six students directed their
attention to another subtask. Thus, even those three
students who needed some direction initially were aware of
and chose a second aspect of the problem on which to work.

With guidance, all of the students were able to reach
conclusions for several of the questions which make up this
challenge. The location selected and rationale for those
choices are given below:

Jacquelyn (E): Area Two. The bonfire would keep
you just as warm as three. It's closest.

Denise (E): Area Two because it's closest, and it's
got a big ice skating rink and it's got
a place for a bonfire. If you went to the
other ones, that one (Area One) would be fur-
ther away, and that one (Area Three) cost
fifty cents to get in.

Trina (E): Area Two because it's close, it doesn't
cost anything, and has a bonfire and food.

Joey (E): Area Three because it's bigger. (After
being asked if there is any other reason)--Has
restrooms and it's enclosed.

100

Michael (E): I think I'd go to the Area One because
it has the smallest rink and if you're trying
to find the kids it would be easier. You
would save some money.

Todd (E): Area One because you could just go there
and ice-skate. Area Two doesn't have rest-
rooms and Area Three cost 50¢ to get in and
you probably wouldn't have enough money to
get a real lunch.

Three students chose Area Two as the best place for
the party based primarily on its proximity to the school.
Trina (E) changed her mind when she noticed that there were
no restrooms in Area Two. Denise (E) planned to stay
"a couple of hours" and concluded that restrooms were not
necessary. When the interviewer pointed out the fact that
she planned to stay at Area Two for six hours, and that
there were no restrooms available, Jacquelyn (E) replied
that they "could go behind the bushes." Clearly what is a
problem for so many people is not necessarily a problem for
all fourth graders.

The selections of the best time for the party varied
widely and are listed as follows:

Jacquelyn (E): Afternoon, 1:00 - 3:00 P.M. You could

have twenty minutes to clean up. (School is
dismissed at 3:20 P.M.)

Denise (E): Maybe at dinner time. Sometimes it's

better to go at night because there is not

a whole lot of people.

Trina (E): All day. Maybe until 1:00. You could
come back and talk about it.

Joey (E): 5:30 - 7:00 P.M. (Joey did not justify
his answer.)

101

Michael (E): Sometimes in the afternoon, because
it's not as cold then. About two hours. If
you stayed one half an hour you wouldn't
have any time to skate. If you stayed longer,
you might get cold.

Todd (E): From 11:00 A.M. to 1:30 P.M. Because
when I go roller skating (which is easier),

I can't roller skate for that amount of time.
In the middle of the day you are rested.
Menus were planned by each youngster and all included
hot chocolate and a dessert. Except for Joey (E), the
students included some kind of hot sandwich or soup to
provide a balanced warm meal.

Four of the students, Jacquelyn (E), Denise (E),

Todd (E), and Michael (E), suggested some things to do while
at the recreational area. The interviewer asked if there
was anything else to consider in planning the ice skating

party. Michael (E) was the only one to make a suggestion:

"How are they going to get there?”

Responses of the Control Class

After the statement of the problem, the interviewer
asked each student to list some things to consider when
planning a party. Susan (C), Linda (C), and John (C) each
named three or four things that needed to be explored.
Shari (C), Mark (C), and James (C) did not respond within
half a minute, and the interviewer made suggestions as he
had with members of the experimental class.

Following this introductory discussion, the inter-
viewer asked, "What would you like to plan first?" Each of

the six members of the control class were able to suggest a

102

starting point. John (C), Linda (C), and James (C) commenced
work on the problem of deciding the location of the party.
Susan (C), Mark (C), and Shari (C) planned for the optimal
time before attacking another subtask.

With assistance, the six students worked through
additional subproblems (e.g., what to do, what to eat, how
long should the party last). The question of where to go
for the party was resolved by each student, and their
answers and rationale are as follows:

Shari (C): Area Two. It's shorter and it's about
the right size. (Lack of restrooms was point-
ed out to her.) Yeah, I just noticed.
(After some thought about the length of
time the party should last, Shari changed her
mind to Area Three.) Because they have bath-
rooms; could buy food and go inside.

Linda (C): Area One. It has restrooms. It has a
covered shelter. No shelter here (indicated
Area Two). Not Area Three because it cost
fifty cents to get in.

Susan (C): Area Three. It sounds better. It has
a nice size rink, has an enclosed building
in case it rains or something. (When asked
why she rejected the others, she replied)

I don't know. I still like Area Three.

James (C): Area Three because it's enclosed, heated,
and if you buy something to eat it would keep
warm.

Mark (C): Area Three. It's got restrooms. Because
it has a building and benches and you get warm-
er faster.

John (C): Area Three because it's enclosed. No one
could bust in and steal the money.

The question of when to go was settled by each of the stu-
dents. As in the case of the experimental class, the replies

were varied and, when justified, considered only one aspect

103

of the question of determining the optimal time. Consider
the responses below:
Shari (C): From a quarter of nine until twelve.
Linda (C): 8:00 P.M. I would stay for two hours.
(When asked if that was too late for some of
the students, Linda replied) For some people.
(She didn't alter her choice of the optimal
time for the party.)
Susan (C): At 1:00 in the afternoon to 3:00.
James (C): Morning. When school started until 12:00.

Mark (C): 4:00 P.M. - 6:00 P.M. Not too late so
they could see.

John (C): Lunch time. I'd stay until 3:30 because
I like to ice skate.

The dietary plans of each of the students included
three or four items. Each of them selected a hot sandwich
or pizza; Susan (C) was the only member of the control class
to choose soft drinks for the menu; the remaining five
students selected hot chocolate. All of the menus included
a dessert of fruit or candy bars.

Five students made suggestions when asked what
additional things should be considered in planning a party:

Shari (C): Make some rules. Don't leave the area,
all eat together.

Linda (C): Choose partners.

Susan (C): Permission slips. Make sure everyone has
skates and dressed warm.

James (C): What to bring: warm clothing,an extra
pair of socks.

Mark (C): Have some help in case someone doesn't
know how to skate.

104

The interviewer could not determine why five students of
the control class suggested additional questions to be
resolved and only one member of the experimental class.
The classroom teachers reported that the control class had
made a field trip during the fall, but the experimental

class had not.

Identification of Strategies Used by the Students

 

This challenge provided an opportunity for the students
to solve a problem containing several subproblems, many of
which were equally important. Six students, three from
each class, were able to identify three or four of those
subproblems. Three of the students from the experimental
class, and six students from the control class, were able
to select a subgoal to be achieved first. Hence, youngsters
can recognize problems involving several component parts,
and in some cases they can select a single aspect on Which
to begin work. None of the twelve students in this study
were able to work through three successive subproblems with-
out direction on the part of the interviewer, and the re-
searcher concluded that this was because these subjects
found it difficult to store several aspects of the challenge
in immediate memory while working on a single component
part. I

With guidance, all of the subjects were able to reach
conclusions for several of the questions which make up this

challenge. In working through these questions, three problem

105

solving strategies were demonstrated: inference, using a
related problem, and contradiction.

Inference from a desired goal was used repeatedly
in solving the subproblems of this challenge. For example,
Michael (E) wanted to minimize the discomfort of the class,
so he chose to have the party "sometime in the afternoon
because it's not too cold then." Mark (C) planned the
party to occur after school but "not too late, so they
could see." He inferred from these constraints that the
party should be held between 4:00 P.M. and 6:00 P.M.

In planning menus for the party-goers, two subjects made
inferences from the goal of providing good nutrition.
Jacquelyn (E) chose hot soup, fruit, and hot chocolate
because she wanted "something from the four food groups."
John (C) also wanted "one from each group, because that
means it's not a junky lunch." He selected a menu of sloppy
joes, hot chocolate, fruit, and pudding. James (C) also
included fruit in his dietary plans because it is "better
for you." Finally, eleven youngsters inferred that hot cho-
colate or soup would be advantageous in reaching the goal

of keeping warm on a winter's afternoon.

The choice of location for the ice skating party was
made using inferences from the student's definition of the
"best" skating area. The researcher construed from the
responses that "best" meant many things to the students, such
as warm, secure, having restrooms, and being close to school.

Some typical responses are listed below:

106

Denise (E): Area Two because it's closest and it's
got a big ice rink and it's got a place for
a bonfire. The others are further away and
that one (Area Three) cost fifty cents to
get in.

Michael (E): I think I'd go to Area One because it
has the smallest rink and if you were trying
to find the kids it would be easier.

John (C): Area Three because it's enclosed. No one
could bust in and steal the money.

Mark (C): Area Three because it's got restrooms,
benches, and you get warmer.

As reported in the discussion of the notebook problem
(challenge one) the subjects tended to focus on a single
dimension of the problem. This myoptic view is evident in
the solutions to several subproblems encountered in this
challenge. Often, this leads them into a situation
with conflicting alternatives. This conflict is resolved
using the strategy of contradiction. For example, Shari (C)
and Trina (E) had to reassesstheir solution to the location
problem when their choice contradicted the goal of providing
for the usual needs of the students. The following two
examples are additional support of the claim that subjects
use contradiction to eliminate alternative solutions to
subproblems encountered in this challenge:

After Susan (C) selected Area Three as the place to
have a skating party during the afternoon hours, she began
planning a snack break:

Susan (C): Pizza, cookies, soft drink, and a candy bar.

Interviewer: Would you have enough money for all of
those things after paying the admission charge?

107

Susan (C): (After finding the total cost) Yes.

Susan (C) realized that she hadn't used all of the available
money, and considered adding to the list. When she dis-
covered that the price of additional food would contradict
the cost constraintcf’$2.00 per person, she rejected the
alternative of increasing the snacks. Todd (E) also used
contradiction in planning his menu for the party. As he
selected food for the skaters, he kept a running total of
the price of these edibles. His first four items cost

$1.30 per person, and thereafter he tried alternatives and
added or rejected them based on whether or not the total
contradicted the $2.00 limit. ‘Todd (E) was the only subject
who considered the cost of the food as he was planning the
menu. The remaining eleven subjects focused on this aspect
of the problem only after the interviewer directed their
attention to the question of whether or not they had enough
money.

During Todd's (E) interview, another problem solving
strategy became apparent-—that of looking at a related pro-
blem. When Todd (E) planned the length of time the students
would stay at the rink, he made inferences based on his
solution to a similar skating problem:

Todd (E): From 11:00 to 1:30 because when I go roller
Skating, which is easier since you have four
rollers, I can't roller skate for that
amount of time.

An upper bound of two and half hours for the ice skating

party is established because it represents the maximum amount

108

of time Todd (E) could spend on roller skates.

A more common use of referring to a related problem
occurred when the students appeared satisfied that they had
a fairly complete solution to the challenge. Just before
the interview was terminated, in each instance, the inter-
viewer asked if there was anything else to consider in
planning the ice skating party. Six interviewees, five from
the control class and one from the experimental class,
responded postively. These subjects were asked why their
considerations were important, and each responded that
these arose when they had gone on other trips. Clearly
each one was thinking of the solution to some previous
planning problem which they had seen (experienced) on some
class field trip, scouting outing, or family vacation.

This was the first of the five challenges, used in
this study, which required the students to use simple arith-
metic to insure the accuracy of a partial solution. Eleven
of the students had to be asked if they had enough money to
purchase the food they had planned for the party. Todd (E)
was the only student who monitored his costs as he
proceeded with the pkuming. Two students, John (C) and
Trina (E), responded, "I don't know" to the interviewer's
inquiry, and needed direction before they totaled the
prices. Five students, Denise (E), Mark (C), James (C),
Susan (C), and Linda (C), attempted to add the cost without
the aid of pencil and paper, with mixed results. Two

students, Joey (E), and Michael (E), were able to accurately

109

estimate the cost by rounding off the individual prices to
quarters, and adding. The final two participants in this
study, Shari (C) and Jacquelyn (E), went directly to the
paper provided them to add the prices of the items they were
choosing. Of the eleven who received direction, five stu—
dents (two from the experimental class, and three from the
control) were unable to add the costs and arrive at a
correct total. Since these youngsters are part of the
classes which did fairly well on the state assessment test
in mathematics, the researcher concluded that children,

in general, are not given an ample opportunity to utilize

mathematics in comprehensive problem solving situations.

Describing People Problem

 

The fourth interview took place on the last Tuesday
and Friday of January. During this interview the students
encountered the problem of describing themselves, and a
person in a photograph, and to determine the best charac—
teristics to use in describing any person. Since the last
interview the control class had continued their usual class-
room activities. During the month of January the experi-
mental class completed four periods engaged in working on
the design of an underwater habitat, and six class periods
of work on the problem solving worksheets.

This interview began with the student describing
himself. Following some discussion on whether they had

ever tried to find someone using a description supplied by

110

another person, the interviewer showed the subject a black
and white photograph of an Air Force pilot getting into his
plane. Each student was told that it was necessary for
someone unfamiliar with the pilot to locate him at the air-
port, and that the student was needed to provide an accurate
description of the pilot in the picture. This preliminary
dialogue led to the statement of the general problem:

Which characteristics are the most useful in describ-
ing a person? Unlike the previous three challenges, this
problem was posed, and partial solutions derived, without
physical objects for the subjects to view and handle other

than the photograph.

Responses of the Experimental Class

 

The initial interaction between the students and the
interviewer involved their descriptions of themselves, and
they were sketchy and only moderately accurate. The
descriptions used most often frequently included color of
hair, color of eyes, and what was being worn. The following
listing gives some typical descriptions as initially
provided by the students:

Jacquelyn (E): Brown hair, blue eyes, black boots,
curly hair, bit off fingernails. (Jacquelyn
has straight brown hair and green eyes.)

Trina (E): I don't know. Possibly brownish hair,
brown or green eyes. (After a pause she

added) I can't think of anything.

Todd (E): Brown eyes, brown hair, about four feet ten,
yellow sweatshirt, and green pants.

I‘M ll

‘Ila titl’l III)

111

Each of the students was able to give some information
concerning his/her appearance. The two least outgoing
students, Trina (E) and Joey (E), gave the poorest descrip-
tions. None of the students supplied descriptions which
accounted for more than four attributes of their appearance.

The interviewer guided the discussion toward supplying

the description of the pilot in the photograph. A collection
of the student responses are as follows:

Todd (E): Air Force clothes, brown eyes, straight
teeth, dark hair, balding. Sort of a baby-
face. I mean chubby.

Michael (E): Tall, maybe overweight, five feet two
maybe. Blue uniform, short hair, rather long
legs.

Joey (E): Black hair, blue suit, brown eyes, I think.

Jacquelyn (E): He smiles a lot, short hair, dark
(hair). What color eyes does he got?

Denise (E) and Trina (E) did not describe the pilot in the
picture, but rather gave an indication of how they would
go about giving such descriptions:
Denise (E): Color hair, glasses or not, shape of
face, smiling or not, his clothes. How tall,
if he is tall or short, the way he wears his

hair, how old he is.

Trina (E): Tell what kind of hair he has, what he is
wearing, what his name is.

After each student discussed his description of the
pilot with the interviewer, the generalized problem was
posed: "Which characteristics are most useful in describing
a person?” To insure that each student was aware that there

were many possible characteristics which might be used to

112

describe a person, the interviewer and the student took
turns suggesting attributes of a persons' appearance. After
six or more characteristics were mentioned the challenge to
determine which details were best, was issued.

The responses of the experimental class were given
immediately and are listed below:

Trina (E): His name, clothes, color of his hair.

Denise (E): Probably color of hair, how tall he is.

Jacquelyn (E): Short or fat, what they wear.

Michael (E): How tall he is.

Todd (E): The face is best.
Joey (E) did not verbalize a response, rather shrugged his
shoulders to suggest, "I don't know." The remaining five
students were asked how they knew these were the "best"
characteristics. Todd (E) was the only student who answered
in a non-negative manner: "If it works, its a good way."

Each student was asked to suggest a test to verify
that their descriptors were the best. Denise (E) recommend—
ed the following experiment: "Give one person four charac-
teristics of an individual.' Give a second person four
characteristics of the same person, and see who could find
him first." She added that she had not thought to test her
choices before the interviewer asked her to prove she was
(correct in her selections of the best descriptors.

To determine whether the students used the solution
to the same problem which might be supplied by someone

else (e.g., family, teacher, police), the interviewer asked

113

if the students knew anyone who describes people well.
Jacquelyn (E) was the only youngster to respond affirmative-
ly, "My friend is a good describer and she gives the same
thing" (short or fat and what they wear). Under the inter—
viewer's direction, each student suggested what policemen
use in a description, however this information was not used
to support or reject their claim that they had given the

best descriptors.

Responses of the Control Class

 

The control class began the interview by providing
descriptions of themselves which involved four or five
attributes and turned out to be moderately accurate.

Shari (C): Girl, bluish-green eyes, strawberry-blonde
hair, lots of freckles,long hair.

Linda (C): Sparkle teeth, bug eyes, and I usually
wear my boots. (Linda is the only youngster
in her class who wears glasses and has braces.)

Susan (C): Blonde hair, light skin, about seventy—
five pounds, about four feet eight inches tall.

James (C): A boy, brown hair.

Mark (C): Skinny, brown eyes, red-blonde hair, blue
shirt, nine years old.

John (C): About five feet tall, seventy-five pounds,
wearing a coat and a yellow hat. (John is
the shortest male member of his class. The
interviewer estimated his height to be
approximately four feet, four inches. He
periodically wears his hat in the classroom.)

Johns's (E) inaccurate description may reflect his inability
to estimate, or perhaps suggests that he described himself

the way he would like to appear. Linda and Mark (C) provided

114

accurate descriptions of themselves by including negative
descriptors (e.g., bug eyes, skinny).

After a discussion of their descriptions, the inter-
viewer showed the students the photograph of the pilot,
and asked (after explaining the problem): "How would you
describe this person?" Each student's response is listed
below:

Shari (C): I can't see his eyes. Would he be wear-
ing that suit? Give his name and what he
does, and ask somebody.

Linda (C): Black boots, green suit. (Interviewer
asked what would happen if he were not wearing
the flight suit.) .Color of their hair, how
big they are, how old they would be, color
eyes. '

Susan (C): Short hair, about five feet ten inches,
and weighs about one hundred and sixty-five
pounds and white skin. (Interviewer asked
how she determined this accurate description.)
He just looks like it.

James (C): How many bags he has, rings, age, clothes.

Mark (C): This clothes or others, (pause) that's the
problem. Short hair. Maybe twenty-one.

John (C): Boots, flying suit, about six feet three
inches, dark hair. (John was asked how he
determined the height of the pilot.) Most
people are about seven feet five inches tall.

Only two of the members of the control class, John (C) and
Linda (C), used the changeable attribute of clothing in their
descriptions. Moreover, Mark (C) and Shari (C) gave
descriptions in which they explicitly noted the possibility
that the pilot's clothing could change. James (C) suggested

descriptors (number of bags, rings) which, though changeable,

could provide an accurate description.

115

In contrast to the three members of the experimental
class, Linda was the only student in the control class to
mention color in her description. Of all of the twelve
students included in this study, Susan provided the descrip-
tion which was most consistent with the true appearance of
the pilot.

Each student was then asked to determine which general
characteristics were the best descriptors of a person.
Shari (C) did not respond within a minute and was asked
to name several characteristics which could be used in
describing a person:

Shari (C): How tall, do they usually wear a happy

face,how he walks, color of hair, freckles

or not, color of eyes, what he is wearing--

a watch.
The interviewer again asked for the best descriptors, but
Shari (C) did not reply. When asked how descriptions were
given on TV shows, she answered: "On Emergency they tell
how old and what he is wearing." Shari made no further
responses to the challenge of determining the "best"
descriptors.

The remaining five students suggested what they felt
were the best characteristics for describing a person imme-

diately after the problem was posed:

Linda (C): How tall, fat or skinny and what kind of
luggage.

Susan (C): How he dresses, height, and location of
the person.

James (C): Name,age, color of hair (pause). Is
that right?

116

Mark (C): Slim, average, or husky, how tall, and
what color of eyes.

John (C): Give a picture. (If you don't have a
picture) Look for a uniform, check at desk,
and see if he is in.

Linda (C) and John (C) were clearly trying to give the best
descriptors to the person going to the airport. John's (C)
suggestion of using a picture skirts the issue of determin-
ing the best characteristics. None of the students could
suggest a way of proving their choices were the best

descriptors to use, and the interviews were terminated

shortly thereafter.

Identification of the Strategies Used by_the Students

 

The initial challenges of providing self descriptions
and a description of the pilot in the photograph were
posed to the students to give a setting for the generalized
problem of determining the best descriptors.

None of the twelve students of this study asked
questions to clarify the problem. Every student responded
based on their perception of the evidence found in the photo-
graph without considering how long ago the picture had been
taken, or whether the man was still in the Air Force. These
responses support earlier conclusions concerning the inabili—
ty of the subjects to consider several dimensions of a
problem.

The inclusion of color by four of the students (three
from the experimental group and one from the control) in

their descriptions of the pilot suggests that they ascribed

117

color to details they thought might be there. Nine students
(five from the experimental class, and four from the control
class) mentioned what the pilot was wearing as a part of
their descriptions, and yet they failed to consider the
possibility that this detail might change.

After the interviewer asked the question, "Determine
which general characteristics are the best to be used in
describing a person," eleven students gave immediate
responses. There was no evidence that any students did,
indeed, reach their conclusions by defining best, consider
the alternatives, and select descriptors most consistent
with the definition. None of the twelve students could
amplify their previous answers. Eleven said they could
suggest no test to verify that their descriptors were the
best.

In order to provide some direction, the interviewer
asked each student what the police use in describing a per-
son, and although each supplied a correct answer, it was
not used to support or reject their claim that they had given
the best descriptors. The researCher could find no evidence
that any subject had given, or intentionally referred to a
solution of this challenge which could be given by a police-
man, fireman, teacher, or parent. In other words, they did
not consider how others had solved related problems.

This challenge was apparently very difficult for all
of the subjects. The students appeared uncomfortable during

the interview, fidgeting in their seats and playing with

118

things in their hands. The responses to the generalized
problem showed no evidence that a problem solving strategy
or process was employed in arriving at their solutions.

The marked difference between the quality of solutions
to the previOus three challenges and this problem suggests
to the interviewer that students must have some experience
with at least some aspects of the problem they asked to
solve before they can be expected to find meaningful
solutions. Moreover, physical manifestations of the problem
should be available for their consideration, a conclusion
which is certainly consistent with Piaget's1 interpretation
of the concrete operational youngster. Furthermore, the
inability to generalize from the specific problem of
describing a particular person in this challenge may also
be due, at least in part, to the lack of experience the

youngster has in generalizing in problem solving situations.

The Playground Design Problem

 

The last interview occurred during the last week in
February, and traced the behavior of the twelve students in
solving the problem: k

Plan and layout a playground, costing less than)
$2500.00, suitable for an elementary school.

During February the control class continued with their usual
classroom activities. The experimental class completed four

periods working on USMES challenges (Product Testing and

 

1Barry J. Wadsworth, Piaget's Theory of Cognitive
Development, (New York: David McKay Co., 1970), p. 67.

 

119

Designing for Human Proportions) and six periods solving
puzzle sheets during the last month of this study. This
final problem involved several aspects of the previous
four problems and was utilized to determine whether growth
in problem solving ability had occurred.

The students were shown a drawing of a rectangular
play area, adjacent to a school, which was vacant except
for a few small trees. A poster board with pictures and
prices of playground equipment was positioned in front of
the students. Scale drawn cutouts representing each item on
the catalog poster were available for placement within the
designated play area. The problem was posed only after
the materials were explained and the students had the op-
portunity to investigate each piece. The subjects were
also informed that other playground catalogs were available,
and cutouts of any item which they wished to incorporate
into their plan would be made.

After the initial period of familiarization and expla-

nation, the students began to work immediately.

Responses of the Experimental Class

 

Michael's (E) initial response was unique. After he
noted the cost constraint on the worksheet, he began adding
the price of every item on the catalog poster in his head.
After several seconds, he asked, "Did you add up everything
on here?” suggesting that it might be possible to include

everything and hence avoid the problem of selecting and

120

monitoring cost. With help, he discovered he could not
purchase everything with $2500.00. He then proceeded to
select equipment for the playground without checking the
total cost until asked to do so by the interviewer. When
Michael (E) was asked how much the selected equipment would
cost, he totaled the prices and exceeded the $2500 limit.
Michael's (E) original sum was $3300, and realizing that he
had exceeded the permissible maximum with that figure, he
removed three picnic tables costing $120.00 each. He
requested the help of the interviewer in finding his new
total, $2940. Michael (E) remarked "If I take away some-
thing for $500;" he paused, and he then removed the $500.00
merry-go-round from the play area.

The remaining five students either kept a running
total of their expenses as they made their selections for
the playground or noted the prices on their worksheet and
totaled the cost after four or five items had been chosen.
This awareness of the financial aspect of this challenge is
very much different from the way the budget was handled in
the menu planning portion of the third challenge.

Each student in the experimental class except Todd (E)
used the entire play area in positioning the equipment they
chose. -Todd grouped five of his six pieces safely together
using a third of the available space because he didn't want
the equipment "too far apart. No use in running a mile."

Todd (E) and Michael (E) were the only students to

select equipment which was not represented on the poster

121

catalog. Michael (E) wanted picnic tables, grills, and a
basketball backboard and hoop. He commented that he ”wanted
more of a park, not just a playground." Todd (E) included
a basketball court in his design and needed to adjust the
size of the court twice in order to stay within the cost
constraint established.

The other four students made use Of the items listed
on the poster. The equipment which was selected most often
were those pieces the youngsters stated they enjoyed most or

thought other children would enjoy.

Responses of the Control Class

 

The members of the control class began selecting
equipment for the playground immediately after the problem
was stated. Linda (C) and Susan (C) selected playground
items without noting prices. Susan, however, was clearly
aware of the cost limitation since she asked, "How much is
that, or am I supposed to add it?" after she had chosen five
items of equipment. The other four students monitored
their costs as the students had done in the experimental
class. .Linda (C) incorrectly added the first time, and
arrived at a sum of $18.85. Convinced she was correct, she
began adding equipment to the play area, and although the
interviewer would call out the price of each item as she
chose it, she continued to write them with the decimal point
misplaced. When she reached a total of $45.38 she realized

that the school yard had become congested and announced,

122

”That's enough." Linda's teacher later remarked that Linda
was able to handle money problems given her in class, which
suggests to the interviewer that she misread the prices,
expecting to see a decimal point between the second and
third digits. After the first five prices were totaled,

a problem solving "set"l became apparent as she continued to
make the same mistake repeatedly.

Susan (C) and James (C) added the price of the first
few items (which totaled less than $2500.00) and began
adding pieces, one at a time, checking after each addition
to insure their cost total had not surpassed the limiting
value. James (C) added his expenses and announced "$450.00
to spend yet" (his total at that point being $2150). He
then looked at the catalog for something costing less than
or equal to the available cash. His choice of a $400.00
castle climber permitted him to say, "That's it! I have
$15.00 extra!" John's (C) initial cost total exceeded the
$2500 limit and it was necessary for him to delete items
from his playground until his costs were within that
constraint.

All of the students except Shari (C) used the entire
play area in planning the playground. Shari (C) left a
region, elliptical in shape, measuring approximately fifty
feet by thirty-five feet, empty and commented, "I left this

area open for playing kickball or something."

 

1During the problem solving process the solver may be
unable to redirect his thinking and continues to use a strat-
egy which has proven fruitless on previous attempts.

123

Linda (C) was the only youngster in the control group
who included playground equipment which was not represented
on the catalog poster. She requested the price and cutouts
for five bounce spring animals which she included "for the
younger kids." The remaining members of the control class
chose equipment for which cutouts were readily available,

and devoted their time to the positioning of these items.

Identification of the Strategies Used by the Students

In this problem, as in the Plan a Party Challenge,
several students made arithmetical errors in computing the
cost of the proposed expenditures. Those students who kept
a running total of what was being spent made little use of
estimation, but rather they treated the addition of each
item as a trial and error procedure. This suggests that
more practice in estimating answers would be helpful, and
should be provided in the arithmetic classes.

There were instances where the subjects'comments
suggest the problem had been defined, and from this defini-
tion of a goal inferences were made in choosing play things.
For example, Linda (C) chose two sets of swings, "one for
each age group." She also asked to see an additional
catalog to find the cost of five bounce spring animals
"for the younger kids." Michael (E) began with a climber
and a swing, then requested an additional brochure and
chose six picnic tables and three grills for the school yard

to make it "more of a park, not just a playground."

124

Denise (E), Shari (C) and James (C) wanted benches so that
children and teachers could "rest during the recess if

they wanted." The remaining seven students in this study
did not give reasons for their selections (four did comment
that they thought a particular item would be fun), and con-
clusions concerning the use of a particular strategy can-
not be made. Todd (E) and Shari (C) also used inference

in establishing the position of the equipment which they
chose. Todd (E) grouped items so that they wouldn't be
"too far apart. No use in running a mile," and Shari (C)
left an Open area "for playing kickball or something."
Clearly these two were planning the school yard with a goal
in mind, from which they inferred the general location of
the particular playthings they chose.

Further examples of the use of inference from the
goal are provided by the six subjects who said they wanted
a safe playground. These six placed equipment far enough
apart to insure that the play action area of any item would
not interfere with that of another fixed object. Some
typical responses are provided by the following:

Susan (C): If its crowded you can't run around. Not
safe.

James (C): So nobody gets hurt.

Shari (C): So nothing hits anything else.
Todd (E) positioned his slide so that "If there was a line,
it could go off this way," indicating the line would form

away from the playing radius of the other equipment.

125

Michael (E) and Denise (E) did not state their concern over
safety, but gave evidence that they wanted a safe playground
by measuring distances and re-positioning equipment which
they judged too close together. In all six of these inter-
views, the researcher concluded that the subjects inferred
that a safe playground is an uncongested one.

Besides Michael (E) and Denise (E), James (C) and
Mark (C) also used the ruler supplied them and the indicated
scale to position objects and determine distances on the
playground. These four children had not used the scale in
the classroom design problem solved three months prior to
the undertaking of this playground design challenge. The
reason for this change in behavior is not clear. Perhaps
congested classrooms don't seem as unsafe to these students
as does a crowded playground area. One might conjecture
that because there were comparitively fewer items which
could be placed on the playground the subjects had more time
to consider the position of each piece.

The use of contradiction was detected in the students
control of the cost of the playground equipment which they
chose. For example, Michael (E), Trina (E) and John (C)
exceeded the $2500.00 limit on their first addition of
their expenditures. Since this contradicted the amount
they were given to spend, each of these students began
subtracting pieces until a total less than $2500 was reached.

The student's use of a related problem and the identi-

fication of subgoals was not as evident as contradiction and

126

inference in solving this final challenge. The researcher
presumed that the subjects thought of their own school

yard and local playgrounds in the planning of their own
solutions to this problem. Supporting this assumption is
Shari's (E) reference to another playground which she used
in justifying the choice she made of benches for the school
yard: "Because at our park they have to sit on stumps."
Three of the students interviewed remarked that they had
never seen some of the free-form playground equipment items
pictured on the catalog poster before. Michael (E) wanted
a particular kind of basketball backboard because "that's
the kind we've got." The question of whether subgoals were
established cannot be answered, since there is no evidence
that the subjects viewed the problem as a combination of
ordered or equivalent subproblems.

The subjects of this study spent more time in solving
this challenge than any of the previous four presented to
them. Perhaps this was because it was more real to them,
or that they were more comfortable solving Open ended
problems at the conclusion of the experiment than they
were earlier in the study. Michael (E) and Todd (E)
supplied the school with basketball equipment. As noted
earlier, Linda (C) wanted equipment for young children that
was not listed on the poster catalog, and benches were
added by four students. Michael (E) wanted picnic tables

and grills. Blacktopping was requested three times. None

127

of these items were listed on the available board, nor were
cutouts available to them at the onset of the problem.

The interviewer had the impression that this was an
enjoyable challenge, one which the subjects felt was
reasonable. Mark (C) noted the similarity and differences
between the classroom and playground design problems; in the
classroom challenge the student "didn't have to worry about
money." Mark also mentioned later, in his classroom, that
this was easier to solve "because there were fewer things
to move." The time taken in selecting pieces, the care
with which they were positioned, and the facial expressions

of concentration convinced the interviewer that the subjects

wanted to find a good solution to this problem.

CHAPTER V
DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS

Contrast Between Classes and Individual Growth

The purpose of this study was to detect (through an
examination of problem solving behavior) the process and
strategies used by subjects taken from two similar fourth
grade classes. A secondary purpose of this research was to
compare the problem solving behaviors of children who
practiced comprehensive problem solving in the classroom
with children who received no such practice. The experi-
mental class was differentiated from the control class by
having their teacher integrate problem solving activities
into the usual classroom work during the duration of the
study. These activities took one of two forms: 1) a
comprehensive challenge of the USMES type, or 2) a short
puzzle type problem. It was felt that differences between
the two classes in problem solving behavior might suggest
that certain classroom activities affect problem solving
ability.

The solutions and procedures utilized to obtain these
solutions, for the five problems involved in the interviews,
suggest that neither class was better at problem solving

than the other. The researcher concluded that any change in

128

129

problem solving ability demonstrated by the experimental
class was not due to the added problem solving practice they
received. The apparent ineffectiveness of the problem
solving worksheets and practice with USMES challenges may be
due, in part, to the inconsistant manner in which it was
applied. In order that the teacher not feel constrained
and to encourage spontaneity on the part of the students,
the researcher permitted a great deal of leeway in the A
selection of when and how often problem solving practice
would occur in the experimental classroom. The teacher
agreed that, if possible, she would spend two periods per
week dealing with comprehensive problems and the same
amount of time working on puzzle sheets. However, the
four month period during which this study occurred contained
several holidays and vacation days, and often other school
activities took precedent over problem solving. Moreover,
the experimental classroom teacher commented that she
didn't feel comfortable with the comprehensive challenges,
and she didn't like spending a great deal of time on them.
Due to these conflicting demands for classroom time, and
the lack of enthusiasm on the part of the teacher for USMES
work, problem solving activities occurred twenty-eight
times during the experimental period with varying regularity.
Seventeen of these periods were used to work on a worksheet
from the teachers guide.

Although there is no detectable difference between

solutions provided by members of either class, there is

130

strong evidence to support the claim that some subjects be-
came more mature problem solvers during the experiment. All
twelve students in this study became more at ease with

the interviewer as the experimental period progressed.

By the third problem each interviewee was willing to

spend considerably more time on the problems, speculate on
alternative solutions, and was candid in his replies to

the researcher's questions. This willingness to spend time
to find non-trivial solutions indicates a growth in the
appreciation of the nature of comprehensive problem solving.
Specifically, Joey (E), Trina (E), and Jackie (E) demon-
strated essentially no problem solving in the first inter-
view, but were able to silently arrive at a solution (a
classroom layout) to the classroom design problem, the
second challenge presented to them. Nine youngsters were
able to control the cost variable in problem five, while
these same subjects had ignored this dimension in the third
problem. Mike (E), Denise (E), James (C), and Mark (C)
displayed some growth in using scale drawing in the fifth
problem as compared to the second challenge. Using this
skill, which is not a generalized problem solving strategy,
indicates a maturing in attitude towards problem solving,
that is, these subjects recognized the many facets of the
problem and realized that good solutions could be achieved
by careful consideration of the component parts of the

problem.

131

The researcher had the impression that, by the end of
the study, all of the subjects understood that although
many solutions existed and were different, they could be
equally valid. By the completion of the study, no student
tried to give the answer the interviewer might have had in
mind or that his or her teacher would want as was often
the case in the early interviews. This indicates a
maturation in the concept of a solution to a comprehensive
problem. By the fifth problem, subjects were able to work
with a fluid definition of a problem which was modified
as the subject became aware of other aspects of the problem.
The researcher concluded that dealing with different
problems and arriving at solutions brings a familiarity with

the problem solving process which aids in cognitive growth.

Identification of Specific Strategies Used by Subjects

 

During the sixty interviews examined in this study
the subjects (except for the three students mentioned in the
previous section who failed to "solve" the notebook problem
and the general failure to answer the "describing people"
challenge) were able to reach some partial solutions to each
problem. The process exhibited in reaching these goals
followed the four steps outlined in Chapter Two. The
solvers defined the problem in some way, generated alterna-
tive solutions, evaluated these alternatives, and selected
the optimal solution. The steps were not clearly differen-

tiated from one another, and the youngsters were not aware

132

that they passed through these steps in solving these
problems, but nonetheless these steps are identifiable.
Furthermore, subjects used these steps in subtasks which
were identified and contributed to the total solution.

During the second and third steps in the solving
process the subjects utilized some generalized problem
solving strategies. The strategies which were used are
listed below by problem:

Notebook Problem: Inference from the goal and from
the givens (notebooks) was widely used by each solver in
this challenge.

Classroom Design: Inference from a goal; use of
contradiction to eliminate alternatives; looking at a
related (equivalent) problem for tentative answers; use of
ordered subgoals to complete solutions.

Ice Skating Party: Inference from a goal; use of a
related problem to suggest alternatives; use of contradic-
tion to eliminate unacceptable solutions; identification of
unordered subgoals.

Describing People: Solutions to the generalized
problem were not found and no specific strategy was employed
in an attempt to find such a solution.

Playground Design: Contradiction, inference, and
looking at related problems were used.

The degree to which any of these strategies were used
varied between problems and from student to student.

Theeastrategies often appeared in primitive form,

133

unnurtured by any training, but their presenceis undeniable.
The manifestation of these heuristics is most evident in

the subjects manipulation of concrete materials. This kind
of reasoning, using a physical model, is consistant with

the cognitive development of a concrete operational
youngster.

There was no evidence to support the claim that fourth
graders can use the method of working backwards to solve
problems. There are two obvious reasons why this evidence
was not detected. Firstly, this strategy may require a
higher level of cognitive development than found at the
fourth grade level. A second reason working backwards may
not have been used by the subjects in this study is that
the problems did not lend themselves to utilization of
this stategy.

The failure of every student to find a legitimate
solution to the describing people challenge implies that
the problem lacked meaning to the subjects. The subjects'
inability to reach some meaningful conclusions suggests
that experiencing the need to accurately describe a person
is necessary for the development of schema which permits
precise definition of the problem. This problem also
lacked the physical embodiments which are necessary before
the concrete operational youngster can act. The researcher
concluded that this problem was inapprOpriate for the study
and should not be used to evaluate the problem solving

ability of any subject.

134

Recommendations

 

The author is aware that the nature of this type of
case study permits conclusions which reflect the bias of
the researcher. The following recommendations are made
based on conclusions from the evidence presented in the
previous chapter, and on the authors interpretation of stu-
dents'feeling during the interviews. The second set of
recommendations are intended for administrators, curriculum
planners, and teachers so that they can suggest activities

for the elementary school classroom.

Research Recommendations:‘ Continuing research is

 

essential to every empirical issue in mathematical education.
This is especially true for the issue of Problem Solving;

its nature, instruction, and learning. The following
suggestions are by no means an inclusive listing, but

rather represent areas which appeared notable during the
implimentation and interpretation of this study.

The amount of time required to conduct a thorough
clinical interview can be lengthy and therefore the research-
er recommends that similar future studies be planned to
allow at least an hour per interview. None of the sixty
interviews reported in this study required more than
thirty-five minutes to complete. However, based on the
responses given during the last three interviews, the
researcher feels that fourth grade children could work on

a problem for a longer period of time and find more complete

135

solutions without becoming tired or losing interest. To
plan interviews lasting an hour or mere implies that the
number of students interviewed during one day must be
limited. The researcher suggests that studies requiring
extensive interviews should include no more than four
students from a single class.

In order that concrete operational solvers reach
meaningful solutions, the problems used during the inter-
views must be challenging comprehensive situations which
have physical embodiments for the subjects to consider and
consult. Interest must be stimulated quickly and maintained
throughout the interview. Based on the responses to the
classroom and playground design problems and the ice
skating party problem, which contained such embodiments,
the researcher suggests that future studies use large,
colorful models. These can then be manipulated by the
solver during the interview and can be used to exhibit
solutions.

As noted earlier the students involved in this study
became more verbal and willing to speculate concerning
alternatives as the experimental period progressed. The
researcher concluded that this willingness to search for
meaningful solutions during an interview situation implies
that familiarity with the interviewer provides a more
comfortable environment for the solver. Therefore, the
researcher suggests that future clinical studies be

planned to permit each student three meetings with the

136

interviewer before the students are asked to solve any
problem. These meetings could be used to discuss the
student's home, school, friends, interests, and hobbies.

A clinical study conducted over a four month period
severely limits the opportunities to detect growth in
problem solving ability. This research should be duplicated
over a nine month period with a follow up interview conduct-
ed one year after the onset of the study. Although a long
term study can present a great deal of difficulties,
results from such a study could produce the most convincing
evidence of growth in the problem solving process.

This study investigated the behaviors of youngsters
who practice problem solving in the classroom as well as
students not involved in any past or present problem solving
program. As indicated elsewhere in this chapter, the re-
searcher could detect no difference in the problem solving
heuristics used by the two groups. To provide information
concerning students'experiences in the problem solving
process, this study should be replicated using subjects
who have been actively involved in USMES for at least one
year.

The researcher found several instances wherein the
students failed to reach meaningful solutions because of
weaknesses in mathematics and reading. The researcher was
unable to suggest to what extent these limitations inhibited
progress towards a solution. Therefore, studies should be

carried out to determine the relationship between mathematics

137

and reading ability and the problem solving strategies used
by students.

Finally, the manner in which students at all grade
levels solve problems is, for the most part, an unanswered
question. Case studies to determine how students solve
comprehensive problems should be conducted at each grade
level, K-12. Collectively, such studies may suggest a
pattern of growth in problem solving ability which would
have implications in the instruction of problem solving

skills.

Curriculum Recommendations: Worthwhile research in

 

education must ultimately be responsible to the child in the
classroom and provide partial answers to the questions of
what and how things should be set up to provide that child
with the "best" education. The following recommendations
are intended to provide guidance to the teacher who wishes
to improve the problem solving ability of his or her
students. These recommendations are based on the student's
responses to the challenges used in this study, supporting
research on student achievement and creativity and the
impressions the researcher had of the student's interest
and enthusiasm. Some of these suggestions also reflect the
feeling and experience of the researcher during three
years of work with USMES classes.

Classroom activities should include at least three

periods per week which are used for classroom investigation

138.

of Open ended problems.

Provide short problem solving puzzles for students
periodically and explore with them the manner in which
solutions are found.

Encourage independent and creative work by every
member of your class.

Design instructional strategies so that students
will be practiced at using mathematics and reading in
problem solving contexts.

Finally, allow each student to work on a comprehensive
challenge, by himself, sometime during the year and report
the results to an interested adult. Verbalizing results
will increase the insight into the nature of a problem and
its solution.

Problem solving is more than an enrichment activity
for the schools or part of the subject matter of mathematics.
It is an attitude and experience; the degree to which stu-
dents master these learning experiences dictates the degree

to which they will survive and grow in the future.

APPENDICES

APPENDIX A

TEACHER QUESTIONNAIRE

APPENDIX A

INFORMATION SHEET

 

Name Phone number

 

Address

 

City State ' Zip

 

 

 

Undergraduate College or University attended

 

Major Minor

 

 

Number of years of full-time teaching experience

 

Number of years teaching in Ypsilanti

 

Grade level you are teaching

 

How many years have you taught this grade?

 

How many hours do you have past your BachelOr's degree?

Do you have a Master's degree?

 

What area?

 

139

140

HOW DO YOU FEEL?

The theoretical portion of my research has presumed

certain attitudes in teachers are necessary in order that
the child will want to solve problems which are posed.

The following questions are presented to support

claims I have made concerning the similarity of attitudes
between you and your colleague in this experiment.

PART 1:

Circle one of the following:

SD-strongly disagree, D-disagree, A-agree, SA-strongly agree

1.

Almost every student could probably improve his or her
class performance if he/she really wanted to.

SD D A SA
It is unrealistic to expect students to show the same

enthusiasm for their school activities as for their
leisure activities.

SD D A SA

Some form of grading system is necessary in schools
because students are too immature to work for any higher
motivation.

SD D A SA

The teacher must at all times work to maintain order in
the classroom. If you let up for one moment, students
will misbehave and learning cannot take place.

SD D A SA
Even though they like to talk about it, most students

don't like to make decisions on their own. It is
hard to get them to assume any real responsibility.

SD D A SA
Being tough with students will usually get them to do
what you want.

SD D A SA
A good way to get students to do more work is to crack
down on them once in a while.

SD D A SA

10.

ll.

12.

13.

14.

15.

141

Even when given encouragement by the teacher, very few
students show a real desire to improve themselves in
class.

SD D A - SA

It weakens a teacher's prestige whenever he or she has
to admit that a student has been right (and he or she
has been wrong), or that a student has a better approach
to a problem than the teacher.

SD D A SA

The most effective teacher is one who gets results,
regardless of the methods used in handling students.

SD D A SA

It is too much to expect that students will try to do a
good job without being prodded by their teachers.

SD D A SA

The teacher who expects students to set their own
standards for superior performance will probably find
that they don't set them very high.

SD D A SA

If students don't use much imagination and ingenuity in
their classwork, it is probably because relatively few
people have much of either.

SD . D A SA

One problem in asking for the ideas of students is that
their perspective is too limited for their suggestions
to be of much practical value.

SD D A SA
It is only human nature for students to try to do as
little as they can get away with.

SD D A SA

142

PART 2:

Rate each item in questions 16 and 17 using the follow-
ing scale:

l-strongly disagree, 2-disagree, 3-not sure, 4-agree,
5-strongly agree

16. Teaching problem solving/reasoning is:

something I do well with most children .
something I find difficult to do with children
of experientially deprived backgrounds
something I find difficult to do with children
who lack concept acquisition

something I find difficult to do with children
who lack comprehension skills

something I find difficult to do with children
who have had a history of failure

 

 

 

17. I thfiflcthe factors that most frequently appear to con-
tribute to a students failure to make satisfactory
progress in school are:

a)
b)
c)

Circle one of the following in answer to questions 18 through
22:

VW-very weak, NI-needs improvement, A-adequate, G-good
E-excellent

18. My understanding of how non-institutional factors
(such as home background, family, socioeconomic back-
ground, environment, etc.) affect a student's achieve-
ment in school is:

VW NI A G E

19. My ability to increase or decrease the affects of
non-institutional factors on my students' achievement
in school is:

VW NI A G E

20. How do you assess your ability to help pupils apply
the mathematics you teach them to daily out-of-school
mathematical tasks?

vw . NI A G E

21.

22.

23.

24.

25.

143

How do you assess your ability to deVelOp instructional
strategies for teaching concepts?

VW NI ' A G B

How do you assess your ability to develop instructional
strategies for teaching problem solving/reasoning?

VW NI A G E

Which subjects do you enjoy teaching the most?
The least? ‘

How do you feel about your ability to get your pupils
to work independently?

VW NI A G E

How do you feel about your ability to provide appro-
priate materials and activities for the pupils who
will work independently?

VW NI A G E

APPENDIX B

TEACHER'S GUIDE

APPENDIX B

STRATEGIES FOR COMPREHENSIVE
PROBLEM SOLVING

A Teacher's Guide

J.J. Shields
Fall, 1975

PREFACE

The Strategies for Comprehensive Problem
Solving is a part of the author's doctoral research
program at Michigan State University. This portion
of the program involves several worksheets which
introduce or reinforce strategies useful in compre-
hensive problem solving. It is hOped that you and
your students will find these worksheets both enjoy-
able and beneficial. Your help in implementing this
program in your classroom, and your cooperation in
taking the time for the experimental evaluation to
measure its success, is deeply appreciated.

J.J.S.

ii

TO THE TEACHER

The purpose of this guide is to aid you in planning
and incorporating problem solving worksheets into your on-
going classroom work. It is to be used in conjunction with
the Unified Science and Mathematics for Elementary Schools

(USMES) Teacher's Resource Books.

 

Since problem solving is such a diverse topic and in-
volves many possible approaches, the strategies identified
in each worksheet overview represents the attitude of the
author as to which strategies children use in solving pro-
blems. In addition, the order in which these worksheets ap-
pear in this guide reflects the authors view of the problem
solving process. Feel free to rearrange them, in whatever
way you feel is most advantageous to your students' growth

develOpment, if necessary.
Finally, the introductory material is presented to

provide you with some background in a theory of problem

solving, and is not intended for your students.

iii

PROLOGUE

Introduction to the Program

 

The Strategies for Comprehensive Problem Solving is a
program which combines USMES challenges, found in their
resource materials, with supplementary worksheets contained
herein. It is designed for use with elementary school
children in an effort to introduce them to strategies which
they use to solve problems of a comprehensive nature. The
program assumes an informational processing approach to pro-
blem solving, and suggests a series of steps which the
solver passes through to arrive at a solution.

A Theorypof Problem Solving

 

Every problem consists of three parts: the givens, the
goal, and an indication of the methods we are permitted to
use in getting from the first to the second. Problems arise
everyday in a students life; they are situations in which
the student wishes to affect some change in a present condi-
tion of things. When a person becomes aware of a situation
which needs changing, he is in a position to clearly define
the problems-what is given, what the goal is, and exactly
what operations can be used to get from those givens to the
goal. Defining the problem is the first step in problem
solving. Secondly, a plan must be devised which can be
followed and will lead the solver to the desired goal. At
this stage of the solving process, the solver uses certain
strategies which will aid in the formulation of a plan of
approach. A strategy is a heuristic, or "rule of thumb,"

which has been useful in solving similar problems in the

past. The strategies which youngsters most often use are:

1) making inferences, 2) identifying subgoals, 3) examining
related problems, 4) using contradiction, and 5) working
backwards. In the third step of the solving process, the
plan is carried out to arrive at a tentative solution (or so-
1utions.flfmore than one is possible). The solver selects

the best solution from those found in step three. These

steps are summarized in Figure 1 below.

 

Awareness of a Problem

Step One: Define the problem precisely.

Step Two: Devise a plan to find possible
solutions to the problem.

Step Three: Carry out your plan.

Step Four: Select the optimum solution of

those possible.

 

 

 

Figure 1: THE FOUR STEPS IN THE PROBLEM
SOLVING PROCESS

The Strategies for Comprehensive Problem Solving program
will take about seven weeks to complete, dependent on your
scheduling within the framework of general classroom
activities. Each worksheet in this guide contains a puzzle
or problem which can be solved using a particular strategy
or emphasizes one particular aspect of the solving process.

The USMES activities provide an opportunity for your students

-2-

to eXamine a comprehensive problem; they make observations,

collect data, and draw conclusions.

Classroom Environment

 

Research suggest that the best problem solvers are those
individuals who practice problem solving. The emotional
and physical setting for this practice is very important,
because the solver must feel free to suggest "wild" alterna-
tive approaches to a problem. These alternatives may stimu-
late another's imagination or be modified into a highly
creative solution.

Choose a time during the day when the students are
free from constraints such as the need to get math done (or
science, reading, etc.); a time when all of your students
can work together. Since each worksheet contains an activity
which does not depend on any other worksheet, you can assign
these sheets at any time during the day. Moreover, this
time can be varied from day to day. There are enough work-
sheets so that you can give your class two or three different
problem sheets during each week of the experiment.

In the classroom, as in real life problem solving, the
solver has access to certain facts or techniques as he
recognizes the need for them. Therefore, many of your
students may ask for help with some of the worksheet problems.
Do not hesitate, after a child has spent sufficient time on
a problem, to answer specific questions he or she may pose.
Also, it is important to permit students to interact with

-3-

one another after each has toyed with the problem for a

while.

A student's success in this program might be judged by

his interest and his progress in finding some partial

solution to any problem situation. To obtain these goals,

the teacher should provide a classroom atmosphere in which

each student can, in his own way, search out some solution

to the problem or puzzle.

In summary, the following suggestions are made to the

teacher:

1.

Introduce the problem or puzzle situation in a
way that allows your students to enjoy the work-
sheet.

Be patient, and let your students make their own
mistakes and find their own way.

Assist individuals or groups of students as they
investigate the problem by asking questions,
making suggestions, etc.

Provide an opportunity for students to exchange
ideas and perhaps draw conclusions concerning the
particular strategy used in the worksheet.

THE PROBLEMS

The following pages contain a collection of challenges
for your students. The problems are arranged so that con—
secutive problems are not similar, or workable, using the
same strategy. If you feel that some other order is neces-
sary (because of time limitations, for example) select any
puzzle sheet you desire, since they are, for the most part,

independent of each other.

Some of the problems or puzzles require the student to
cut out figures, and move them around to find a solution.
A few should be done orally, in which case you will have to
take a leadership role in the discussion. Most of the work-
sheets can be done individually, and your students will gain
the greatest insight into the solving process if they attempt

them alone.

A brief introduction to each of the problems (including
their solution as well as a mention of the type of strategy
that will be used) is given before the individual worksheets

of this guide.

Worksheet One: GETTING STARTED ON THE USMES CHALLENGE

The first worksheet is a very simple one for most
students and can be done in a variety of ways. You might
decide to read each one of the quotes from a TV advertise-
ment to your class and have them guess the answer aloud or
you may elect to have them work individually and try to be
the first to identify all twenty-four (24) products.

No matter how you do this worksheet, it can lead to a
discussion of fair advertising and the claims made by
sponsors concerning their products. You can ask your
students how they would choose the best pencil, or paper
towel, etc. in a store, or how they would test the claims
of a sponsor. After such a discussion, the USMES challenge,

Consumer Research - Product Testipg, will provide further

 

opportunity to test various products and to learn something

about solving comprehensive problems.

from

1.

5.
6.

7.

What product is associated with each of these lines

a TV advertisement?

Melts in your mouth,
not in your hands.

Gets out the dirt kids
get into.

Double your pleasure,
double your fun.

You deserve a break
today.

See the light.

Breakfast of champions.-

Silly rabbit,
are for kids.

If it isn't fresh, I'm

outta business.
M-mrmrm good.

Look up America.
Baked by elves.

No more tears.

13.

14.

15.

16.

17.
18.

19.

20.

21.

22.

23.

24.

I can be very friendly.

Your common sense
cereal.

Baseball, hot dOgs,
apple pie and

__‘

Don't fiddle with the
middle.

Fly the friendly skies.
Finger lickin' good.
Squeezably soft.

You can't beat

for fighting caVIties.

Good to the very last
drop.

Puts sex appeal in a
tube.

Scrubbing bubbles.

How's your love life?

Worksheet Two: PATTERNS

The following worksheet is designed to challenge
your students to examine given information and make the
necessary inferences that will enable them to find a
pattern in a sequence of geometric figures.

1 The problem solving strategy of making inferences is
used more often than any other. Although inference alone
can be used to solve puzzles or problems, more often it is
used with other strategies in comprehensive problem solving.
The USMES activities will provide many opportunities for
your students to make inferences.

This worksheet utilizes the strategy of making infer-
ences from the given to find a geometric figure that fits
into the pattern. A solution to the patterns is indicated
on your worksheet. The last three patterns are a little
more difficult than the first four, and you might have to
give some direction or clues as to what to look for in these

series

Find the pattern in each series of pictures and draw

the next one or two pictures in the series.

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MAZE WORKSHEET

Starting with the goal in a problem and determining
what problem state must have existed before the goal is
reached is the strategy called working backwards. Whenever
a solver has a problem in which there are several places to
begin, each of which may lead to a desired unique goal, the
method of working backwards is very useful. For example,
solving maze problems is often accomplished by starting at
the "finish" and working back to the "start." This worksheet
provides the student with two maze-type problems on separate
pieces of paper.

Have each student try the first maze puzzle without
the help of any other student. Wait until almost every
student has finished and then ask how many students have
found the correct answer by working backwards. More than
half of your students probably began with the fish and
traced the line back to the correct answer--boy number 2.
Ask some of these students why they decided to work back-
wards. Some may conclude that since there were many places
to begin (boys) and only one goal (fish) it was easier to
begin with the fish.

The second puzzle can be used to reinforce this idea.
Divide your class into two groups. One group should be
directed to begin at the money and the second should begin

with one of the six boys. At your signal, tell all students

-10-

to begin. Most of those beginning at the goal should
complete the task before a majority of the other group.

In this problem, Robert will find the money.

-11-

Here are five boys sitting on the edge of a stream.
Though they don't know it, their fishing lines have
become tangled. One of them has caught a fish.

Can you tell which one?

 

-12-

Dan

Mr. Adams, the sixth grade teacher, has lost the
money for the class party coming up, and he has asked
six boys to help him find it.

Can you tell who will find the money?

Bill

 

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Worksheet: CREATIVITY

The process of problem solving is closely related to
creativity. Research suggest that the highly creative
person is usually a better problem solver than the less
creative person. In this worksheet (as in some of the
others) the student can generate many possible responses,
each as good as any other.

Give each student a worksheet, and explain that each
circle contained on it should be transformed to something
within a specific time period; say two minutes.

Some may automatically decide to draw as many faces as
they can think of, while others may try to approximate a
series of drawing which are made of the same shape (a base-
ball, an orange, a bowling ball, etc.). Some of your
students might draw pictures of objects which are all
different yet have a circular component. Finally, some
students might use a group of circles together to form a
train, an insect, etc. There is no single correct response,
and hence each student can accomplish the task with complete
accuracy.

Give your students the opportunity to share their res-
ponses with other members of the class, and by so doing new

ideas or uses for the circles may be suggested.

-14-

CLASSROOM EXERCISE

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-15-

Worksheet Five: USING CONTRADICTION

The method of utilizing contradiction is a useful one
in problems where alternative solutions are readily available.
The individual assumes one of these tentative solutions and
checks to see whether such a solution contradicts some
piece of given information. If the solVer reaches a con-
tradiction, then that solution is discarded and another al-
ternative is considered.

This worksheet contains five overlapping circles which
enclose ten regions. The object is to place a number from
l-lO in each region so that the sum of the numbers in each
circle is 14. Three numbers are given, and the student
should simply try various alternatives until he or she dis-
covers the correct way the numbers can be inserted into the
regions.

A better way would be to note that the top circle,
which contains a 10, can only use a l and a 3 in adjacent
regions. The circle with the 8 in it can only have a 2 and
a 4 in adjacent regions. Now the number of possibilities
have been reduced substantially, and the method of contra-
diction will yield a solution quickly.

The correct solution is indicated on your worksheet.

-l6-

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-17-

Worksheet Six: LOOK CAREFULLY

This sixth worksheet is a very short activity and you
can insert it into your day whenever you have a few minutes.
Problem solving often involves looking at data very
carefully and then drawing conclusions concerning the nature

of the given information. In this problem, your students
are asked to find the two bees in the group of eighteen
which look alike. The two which most closely resemble each
other are the second one in the second row, and the first
one in the fourth row. Each has clear wings and abdomen,

but blackened head.

-13-

There are eighteen bees in the picture below.
They all look similar. Sixteen are different from one
another in some way, but two of them are exactly alike.
Can you find the two that are identical?

 

-19-

'Worksheet Seven: CONSTRUCTING A SQUARE

This worksheet, found on page 21 of your guide, involves
arranging four pieces into a square and a triangle. Since
there are only four pieces, the student can use the
method of contradiction (page 16) to eliminate configurations
which can't lead to the desired shapes.

The worksheet involves cutting out four shapes and then
moving them around to arrive at the solution and will take
a few minutes to get started. Hence, it is best to have at
least twenty minutes set aside for this activity, or you may
give it out near the end of the day and allow your students

to take it home to finish. The solutions are given below.

 

 

 

 

 

-20-

Cut out the four pieces below and arrange them so
that they fit together to form a square. They
can also be arranged to form a triangle.

-21-

PERISCOPE WORKSHEET

Here is a worksheet that will test your students
ability and provide an opportunity to learn some natural
science. The sheet contains six cross section drawings of
periscopes in which your students are to place mirrors in
order that the viewer will see out the indicated slot.
Since some of your students may be unfamiliar with how a
periscope functions, the first drawing is completed to act
as a sample. Moreover, you may have to remind them that
light travels in a straight line, and so to "bend" the
light rays they will need a mirror. The correct arrange-

ments of mirrors appear on your worksheet (page 23).

-22-

A periscope is a devise by which you can see in one
direction by looking in another direction. Each box below
represents a different kind of perisc0pe. The eye looks
into the box through one hole and sees things outside
the box through the other hole. Draw an arrangement of
mirrors inside of each box to show how the box works.

SAMPLE :

 

 

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-23-

Worksheet Nine: DROODLES

"Droodles" are drawings which are made up of a few
simple lines and suggest different things to different people.
Each student is given a copy of the worksheet and is to
respond with "It looks like . . ." Droodles is more than a
child's game. It is an activity which can stimulate your
students imagination and give him or her important practice
in recognizing more than one point of view in problem
situations.

The object of this worksheet is to introduce your
students to the techniques of "Brainstorming.” Take each
picture and have your students give their opinion as to what
each Droodle is. Try to generate as many ideas as possible.
One student may see one thing and his ideas may stimulate
another child to see many more things in these pictures.
Some possible things to see are indicated under the pictures

of your worksheet.

-24-

"Droodles" are drawings which are made up of a few
simple lines and suggest different things to different
people. Look at each of these pictures and decide what you
think is pictured. You and your classmates may see
different things in the same picture.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-25-

Worksheet Ten: BREAKING THE CODE

This worksheet contains two puzzles. The first is easy
enough that most students will solve it with minimal assis-
tance. The second is more difficult, and will only be
solved by students who work at it for a long time and
receive your help.

The challenge is to determine digits which represent
the given letters. Each letter represents exactly one digit.
The strategies used to work this type of problem are infer-
ence and contradiction.

The solutions are given on your worksheet. To obtain
these solutions you might reason as follows:

1) the subtraction problem

Since A - A = B we know that B must equal 0. Since B
equals 0 (and in checking we find that C + A = AB in column
two; that is a two digit number ending in 0) we can conclude
that A = 1. Finally, since 10 + CA = 101, C must equal 9.

2) the addition problem

All letters represent distinct digits and no column
can add up to more than 26 (assuming column two has A =‘9 and
B = 8). Now A is a one digit number and the most which can
be carried from column two is 2, we can conclude that DE must
be 10. .Hence, D = l, E = O, and A = 9 or A = 8. Consider
column three: the sum C + B + A ends in an A so B + C = 10.
Try various alternatives for B and C such as B = 8 and C = 2,
or B = 7 and C = 3 until you arrive at the complete solution.

-25-

In the following subtraction problem, each digit has been
replaced by a certain letter. Can you reconstruct the
original problem?

 

A B A
- C A
A B

In the following addition problem, each digit has been
replaced by a certain letter. Can you determine which
digit each letter represents, and reconstruct the ori-
ginal problem?

 

A B A
A B

+ AA
D E E A

-27-

PROBLEM SENSITIVITY WORKSHEE'I‘

This worksheet is presented to provide practice in draw-
ing inferences from given information or situations. Your
students are to give as many answers as are possible to the
question, "What could happen in each of the following situa-
tions? What is the problem and what are some possible
solutions?"

Give each student one of the worksheets and have them
consider each cartoon. For example, in the first cartoon,
the two peOple in the picture could have a problem deciding
what TV program to watch. There are various suggestions your
students could make to the question "What could happen in
this situation?" They might suggest that one or the other
will give in and watch whatever is onEKLthe moment. They
may suggest that the persons could flip a coin to see who
gets their choice, or they might ask a third person to decide
who should get to choose a television program. Your students
might suggest many different feelings are possible for the
people in the cartoon; for example, kindness and considera-
tion, or anger and hostility.

The objective of this activity is to aid the student
in becoming aware that real world situations most often in-
volve the interaction of many people, places, and things.
Some students may begin to realize that solving a comprehen-
sive problem does not imply finding a unique correct solu-
tion, but rather involves selecting a solution which will be

-23-

the best solution "most of the time." Your concern should be
to get most of your students involved, and get them to view

the situation creatively from various points of view.

-29-

What could happen in the following situations? What is
the problem, and what are some possible solutions?

Mm: ,.
.rfig ”\r432?;

  

 

 

 

 

-30-

Worksheet Twelve: GETTING TO THE FAIR

In this problem your students are to find a way to get
a farmer, his goat, a wolf, and a cabbage across a river in
a boat which is only large enough for any two of them. Have
the students cut out the four rectangles representing the
farmer, the goat, the wolf, and the cabbage and try to solve
the dilema. We know: Only the farmer can row the boat.

The wolf and the goat can never be left alone. The goat and
the cabbage can never be left alone.

Your students can work out the problem using contradic-
tion to arrive at the following solution:

Firstly, row the goat across and return. Farmer
Reynolds could not take the cabbage across first since that
would leave the goat and wolf together, nor could he row
the wolf across since the goat would eat the cabbage.
Secondly, row the wolf across the river and row the goat back.
Thirdly, row the cabbage across and return. Finally, row the

goat across and continue to the county fair.

-31-

On his way to the Dutchess County Fair with his pet goat,
his tame wolf and his prize head of cabbage, Farmer Reynolds
came to a river that had to be crossed.

On the river was a small boat just large enough to transport
the farmer and one of his possessions. This presented a
serious problem. If the farmer took the cabbage and left
the wolf and the goat, the wolf would eat the goat. If he
took the wolf and left the goat with the cabbage, the

goat would eat the cabbage. For these same reasons, the
farmer had to plan ahead so that these unsatisfactory combi-
nations would not be left together on the other side of the
river.

Can you help Farmer Reynolds to get his possessions across
the river and to the fair?

(Cut out the pictures at the bottom of the page and try to
get all four across the river.)

‘ I

 

 

RIVER I ?

 

 

 

 

 

 

 

-32-

 

 

FARMER
REYNOLDS GOAT

 

 

 

 

 

 

WOLF CABBAGE

 

 

 

-33..

SUBDIVIDING THE FARM

The next puzzle (page 35) involves dividing an area
into four equal parts, each of which contain two circles
representing trees. Your students might eliminate various
possibilities by trial and error. The most common mistake
in working this type of problem is in allowing oneself to
assume that the two trees in a region must be in the same
straight line. If some of your students have trouble in
getting started, you might suggest that they try putting
one tree from each line into some region. The answer is

indicated on your worksheet.

-34-

Let us suppose a farmer has a piece of land shaped
like the drawing below. It's a good, high section of land
with eight (8) valuable shade trees. A company offers to
buy it from him for a housing development, providing it is
possible to divide the property into four equal portions,
each to contain two of the shade trees.

Can you show the farmer how to draw the lines so
that he can sell the land?

 

 

 

-35—

THE GREATEST WOMAN ATHLETE

This problem is a short puzzle which is solved by
using the strategy of working backwards. The problem pro-
vides some intuitive work with fractions.

We are told that Babe Zaharias' proportions were per-
fect: waist size in correct relationship to her neck, her
neck in proportion to her wrist, and her wrists to her ankles.
If we know her ankle measurement was 9 inches, we can work
backwards to find the size of her waist.

a) Once around her ankle is equal to one and half
times around her wrist. Hence,-wrist measurement is 6"

(one and a half of 6 is equal to 9).

b) Since her neck was twice around the wrist we know
her neck was 12".

c) Finally, since her waist size is twice the size of
her neck, she had a 24" waist.

This puzzle provides an excellent example of a problem
in which the solver must proceed in steps, that is find sub-
goals which will lead to the ultimate goal of finding her

waist measurement.

_36—

Babe Zaharias was voted the greatest woman athlete
of the twentieth century. A sport writer once claimed that
she had the perfect figure for sports. The prOper propor-
tions for a woman are: waist = twice around the neck: neck
is twice around the wrist; and once and a half around the
wrist is the same as once around the ankle.

If Babe had the perfect figure, and her ankle measure-
ment was 9", can you determine the following?

ANKLE

 

WRIST

 

NECK

 

WAIST

 

-37-

PATTERNS AGAIN

This worksheet contains ten series of numbers and
asks the student to find the next few terms of the sequence.
As in the earlier pattern worksheet, the solver must make
inferences concerning the data and formulate a hypothesis
which will enable him to predict the next few terms. The
first several patterns are easily recognized, but series
five through nine may require some hint. The solutions are

indicated on your worksheet.

-33-

Find the patterns, and then write the next four terms of
each of the following series:

1. 0 , 3 , 6 , 9 , 0 , 3 , 6 , 9 , 0 , 3 , 6 , 9 , . . .

2. 5 , 4 , 10 , 9 , 15 , 14 , 20 , 19 , . . .

3. 1 , 4 , 9 , 16 , 25 , . . .

4. 2 , 6 , 12 , 20 , 30 , 42 , 56 , . . .

5. 1 , l , 2 , 4 , 3 , 9 , 4 , 16 , 5 , 25 , . . .

6. l , 1 , 2 , 3 , 5 , 8 , 13 , 21 , . . .

7. 2 , 4 , 8 , 16 , 32 , . . .

8. 1 , 4 , 6 , 7 , 10 , 12 , 13 , l6 , 18 , l9 , . . .
9. 1 , 3 , 2 , 5 , 7 , 4 , 9 , ll , 6 , . . .
10. l , 3 , 5 , 7 , 9 , 11 , . . .

-39-

Worksheet Sixteen: PAINTED SOLIDS

Pictured in this worksheet are two painted solids
which are cut along the indicated lines. The task is finding
the number of sections of the cut up solids which have three,
two or one side painted. Using the strategy of making
inferences, we note that the only places where three sides
will be painted are at the corners. We then conclude that
to have two painted sides a section would have to be along
an edge, but not at a corner. Finally, we can infer that
the only sections with one painted side would have to be the
non-edge parts of the solid faces.

For the cube, there are eight sections with three
sides painted. Twelve sections have two sides painted, and
there are six sections with exactly one side painted. For
the triangular block, there are six sections which have paint
on three sides, eleven sections have two sides painted, and

six sections have exactly one side painted.

-40-

A large cube contains 27 small cubes as shown below.
The top and the bottom, and the four sides of the large
cube are painted grey.

How many small cubes have three grey sides? Two
grey sides? One grey side?

 

 

 

 

 

////
///,

 

 

 

 

 

 

 

A large triangular block consists of 24 small pieces,
as shown in the drawing below. Two sides, the bottom, and
the front and back of the large triangular block are paint-
ed with blue paint.

How many sections have three blue sides? Two blue
sides? One blue side?

 

 

 

 

 

 

 

 

-41-

MADDER ' S LADDER

This problem involves finding the number of rungs on
a ladder. To solve this question, the student must ignore
the first half of the problem and work backwards to answer
the first question. That answer can, in turn, be used to
solve the second part of the problem. There are 17 rungs on
the ladder. Don't forget that you do not count the rung you
are standing on when you start up or down.

An excellent way to solve this problem is to draw
lines representing rungs on the ladder, carry out the
perscribed instructions, and note the total number of

lines drawn.

-42-

When I first saw Mr. Madder, he was part way up a
ladder. He went up four rungs, down seven rungs, and up
ten. That put Mr. Madder at the top of the ladder.

Then he went down nhmarungs, up three rungs, and
down ten. That put him at the bottom of the ladder with

his feet on the ground.
How many rungs has Madder's ladder? On what rung

was Madder when I first saw him?

 

-43-

Worksheet Eighteen: NO SWIMMING

The problem of getting nine men and two boys across
a river in an inflatable boat is similar to the earlier
problem of getting the farmer, and his possessions, across
the river. Your students can use a similar procedure to
the one they utilized in that situation.

Since only one man could be in the boat at any
one time, the number nine just means that there will be a
number of trips necessary to complete the crossing. For
students who need some direction, you might suggest that
they first find a way to get one man across the river and
the boys and the boat back to the original side. Once
this is accomplished, the same procedure is repeated eight
more times.

To get one man across the river and the boys back
to the starting point: Two boys cross the river and one
boy rows back (that is two crossings). Now a man alone rows
across and the other boy rows back to the original shore.
There are now eight crossings.

Repeat this procedure eight additional times and
the two boys will be on the original shore (after 36 cros-
sings). One more trip will get the two boys across, and we
have therefore found that it takes 37 times across the river

to get all nine men and the two boys to the far shore.

-44-

Nine men and two boys want to cross a river using an
inflatable raft. The raft will carry one man or the
two boys.

How many times must the boat cross the river to accom-
plish this goal?
(A round trip equals two crossing.)

WW

RIVER

 

 

 

 

 

 

 

 

BOAT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

boy

 

 

-46..

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

WOrksheet Nineteen: WHO PLAYS WHERE

WOrksheet nineteen contains two puzzles which can be
solved using contradiction. The first puzzle involves match-
ing three people to three different positions on a basketball
team. The second one requires that your students play de-
tective to determine who broke a vase.

Give each of your students a worksheet, and ask them
to solve the first problem without talking to their friends.
After they have worked on it for a while, you might ask all
students with the correct answer to give clues to those who
have not arrived at a complete solution. The usual approach
to this type of problem is to draw a diagram showing the
people and their jobs. Assume one person plays one position,
for example suppose Bill plays guard, then check to see if
this assumption will contradict some given piece of informa-

tion. If it does, then try Lew at guard and see what happens.

Bill Lew Tom

 

Center

 

Guard

 

Forward

 

 

 

 

 

-47-

Note that Tom can't be the guard because of statement
four. Likewise, Bill is not an only child and therefore
he can't be the forward (statement three). Finally, Tom .
can't be the forward since he has a higher free-throw
average than the guard and the forward has the lowest free-
throw average.

If no one gets the correct answer, show the class
who the center is (Tom) and see if they can get the other

two boys in the correct positions.

-43-

WHO PLAYS WHERE?

 

On a certain class basketball team, the positions of
center, guard, and forward are held by Bill, Lew, and Tom
(though not necessarily in that order).

The forward is an only child.
The forward has the lowest free-throw average.

Tom's best friend is Bill's'brother.

Tom has a higher free throw average than the
guard.

What position does each student play?

WHO BROKE THE VASE?

 

The Foremans have gone out for the evening, leaving
their four children with a babysitter, Nancy Wiggens.
Among the many instructions the Foremans gave Nancy before
they left was that three of their children were consistent
liars and only one of them consistently told the truth, but
they didn't tell her which one of them was truthful. As
she was preparing dinner for the children, one of them
broke a vase in the next room. Nancy rushed in and asked
who broke the vase. These were the children's statements:

Betty: "Steve broke the vase."

Steve: "John broke it."

Laura: "I didn't break it."

John: ”Steve lied when he said that I broke it."

Knowing that only one of these statements was true, Nancy
quickly determined which child broke the vase.

Who was it?

-49-

COIN COLLECTION PUZZLE

This puzzle requires your students to cut out twelve
squares, each representing an ancient coin, and arrange
them in a six by six grid. It can be solved by simply
trying various possibilities and eliminating those which lead
to a contradiction of the constraint that exactly two coins
be in any one row. To aid some students in getting started,
you might put the first one in for them, later adding a
second, then a third, and so on until they can finish the
problem themselves. The solution is indicated on your work-

sheet.

-50-

Mr. Clifford Bradshaw,_a wealthy coin collector, owned
twelve ancient coins. At a coin collectors meeting, Mr.
Bradshaw wanted to display his twelve coins in a velvet
lined case. The case had thirty-six (36) compartments,
as shown below. He ordered his assistant to arrange the
display so that there would be two coins in each row,

column, and diagonal, and not more than two coins in the
same straight line.

How did the assistant carry out his employer's instructions?
(Cut out the coins and try to find the answer.)

 

 

 

 

 

 

 

 

 

 

 

 

 

-51-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(WIRE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-52...

 

 

 

RECONSTRUCTING THE SQUARE-REVISITED

The last two worksheets are simdlar and both involve
some cutting out of pieces and fitting them together to form
the shape of a square. The seven piece Tangram is a popular
commercially produced puzzle which may require a great deal
of time to complete. Adults may spend an hour working on
the problem so it is probably wise to give your students
a hint after they have it cut out and are familiar with
the pieces. After a time, tell them that the two large
triangles fit together to form half of the square. The
solution is given below in Figure 1.

Similarly, the reconstruction of the checker board is
a difficult problem and many of your students will lose
interest quickly if you don't give continued assistance. It

is included to challenge your best student.

 

 

 

 

 

 

 

 

 

/
Z

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-53-

Tangrams

wo Fung, a very rich man, had a beautiful square mosaic-
tile which he always carried with him. Upon waking one day
he picked up the tile to admire it. Alas! the tile slipped
from his hands and fell to the marble floor, breaking into

seven (7) pieces. He tried and tried, but he couldn't put
it back toqether again.

Can you help Wo Fung by reconstructing the square?
(Cut out the pieces and try to find the answer.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-54-

ht (8) pieces below and arrange them so

that they form an 8 x 8 checkerboard.

Cut out the eig

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-55-

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APPENDIX C

 

 

 

 

 

 

202

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APPENDIX D

  

203

 

 

APPENDIX E

 

 

 

 

SCHOOL

 

 

 

204

 

SELECTED BIBLIOGRAPHY

SELECTED BIBLIOGRAPHY

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205

206

Guilford, J.P. "Traits of Creativity." In Creativity and
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Torrance, E.P. Torrance Tests of Creative Thinking: Norms-
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