Problem solving and creativity

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Problem solving and creativity multiple solution methods in a cross-cultural study in middle level mathematics
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vii, 140 leaves : ill. ; 29 cm.
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Fouche, Katheryn Kirk
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Subjects / Keywords:
Problem solving -- Study and teaching -- Psychological aspects -- Cross-cultural studies   ( lcsh )
Creative ability -- Study and teaching -- Psychological aspects -- Cross-cultural studies   ( lcsh )
Mathematics -- Study and teaching (Elementary) -- Psychological aspects -- Cross-cultural studies   ( lcsh )
Middle school students -- United States   ( lcsh )
Middle school students -- Japan   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 94-101).
Statement of Responsibility:
by Katheryn Kirk Fouche.
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Typescript.
General Note:
Vita.

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University of Florida
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ocm30899873
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Full Text









PROBLEM SOLVING AND CREATIVITY: MULTIPLE
SOLUTION METHODS IN A
CROSS-CULTURAL STUDY IN MIDDLE LEVEL MATHEMATICS












BY


KATHERYN KIRK FOUCHE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA

1993













ACKNOWLEDGMENTS


The success of my graduate years was due to the support of many
people. Thanks in great part to my committee, my experiences were

positive and productive. It was an honor to have been taught by the

masters. Dr. Mary Grace Kantowski, my major advisor, mentor, and
friend, shared her expertise, resources, and art. She challenged me

to become a better problem solver, teacher, and researcher. While

department chair, Dr. Eugene Todd recruited me and provided

encouragement. Dr. Elroy Bolduc always had time to lend advice and a
sympathetic ear. Dr. Nelson contributed support from the Department

of Mathematics "across the street," and Dr. David Miller always found

time to design the next phase of the study.
The support and love of God and my family continues to be my

source of energy and inspiration. Thanks to Kevin and David who may

deservedly claim part of this degree in exchange for their good nature,
flexibility, and understanding. To the rest of my family, Mother, Bob,

Barbara, Chuck, Lindy, Chad, Joe, Betsy, Megan, Katie, and, Lindy,
thanks for believing in me and for your encouragement.
Thanks to my friends in Japan, Professor Toshiakira Fujii and

Professor Tadao Ishida, who arranged for the Japan-U.S. comparisons

and to Christina Carter for her hours spent coding the Japanese data.
Thanks to the panel of experts who ranked the problem solving

methods, Dr. Carolyn Ehr of Fort Hays State University, Dr. Peggy








House of the University of Minnesota, Dr. Mary Grace Kantowski of the

University of Florida, Dr. William Moulds of Towson State University

and Dr. James Wilson of the University of Georgia. Thanks to Dr. Jerry

Becker of Southern Illinois University for his help and resources.

Thanks to Alisan Hardman who graded each of the Torrance Tests of

Creativity and to Sally Scudder for helping me code the original

marble problem. Thanks to my friends, Cindy King, Donna Otzel, Luke
Reckamp, Joan Donnelly, and their students at P. K. Yonge, Vanguard

High, and Osceola Middle for welcoming me into their classrooms,

and thanks to the teachers and students at Gainesville High, Ft. Clarke
Middle, Ft. King Middle, and Westwood Middle who shared their

solution methods.
A special thanks is reserved for Dr. John Zbikowski who

coached, edited, and consoled. Thanks to Stephanie Robinson and

Donna Otzel, my friends and fellow graduate students, to Susan Starks

who stayed with me until the wee hours during that final copying
session, and to the Norman Hall gang.
This dissertation is dedicated to my dad, Joseph L. Kirk, Jr. He

would have been so proud.













TABLE OF CONTENTS


ACKNOW LEDGM ENTS............................................................................................ iia

ABSTRACT .............................................................................. .................................. vi

CHAPTERS

I INIRODUCTION......................................................... .......................... 1

Purpose of Study...................................................... ............................. 2
Rationale ........................................................ .......................................... 4
Research Questions ............................................................................. 6
Definition of Key Term s ....................................................................... 7
Summ ary..................................................................................................... 8

II REVIEW OF RELATED LITERATURE................................................ 10

Problem Solving....................................................................................... 10
Summary of Related Problem Solving Research................. 11
Constructivist Theory................................................................... 12
Teaching and Learning from a Constructivist
Perspective ................................................... .............................. 20
Creativity.................................................................... .............................. 24
M them atics Education in Japan......................................................... 30
Sum mary......................................................... ......................................... 34

III PROCEDURES ....................................................... ................................ 36

Hypotheses........................................................... .................................. 36
The Main Study....................................................................................... 39
The Subjects .................................................................................... 39
Instrum entation for the M ain Study........................................ 40
The M arble Problem ..................................................................... 40
Rationale for Reexamination of the Marble Problem.......... 44
Adm inistration of Instrum ents................................................. 46
Group Assignm ent ..................................................................... 47
Treatm ent Procedures................................................................. 48
Analysis of Creativity Data for American Students............ 51
Weight Assignment for Solution Methods............................. 52








The U.S.-Japan Comparisons.............................................................. 53
Summary................................................................................................... 55

IV DATA ANALYSIS AND RESULTS OF THE STUDY...................... 56

Review of the Design and Data Collection Procedures.............. 56
Results of Hypotheses Testing............................................................ 59
Number of Solution Methods .................................................... 59
Complexity of Solution Methods.............................................. 63
Pretest/Posttest Comparisons .................................................. 66
U.S.-Japan Comparisons.............................................................. 69
Other Results............................................................................................ 71
Summary........................................ ........................................................ 73

V CONCLUSIONS AND IMPLICATIONS.............................................. 77

Overview of Study.................................................... ............................. 77
Discussion of Results.......................................................................... 80
Implications .............................................................................................. 85
Suggestions for Future Research ...................................................... 88
Limitations ................................................................................................. 90
Conclusions................................................................................................ 92

LIST OF REFERENCES..................................................................................... 94

APPENDICES

A PROBLEMS FOR TREATMENT .......................................................... 102

B THE MARBLE PROBLEM ................................................................... 106

C PANEL TO DETERMINE COMPLEXITY WEIGHTS......................113

D DESCRIPTION OF SOLUTION METHODS....................................1... 19

E PARENTAL CONSENT FORMS ........................................................... 122

F THE PILOT MARBLE PROBLEM ........................................................ 135

BIOGRAPHICAL SKETCH .................................................................................... 140













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


PROBLEM SOLVING AND CREATIVITY: MULTIPLE
SOLUTION METHODS IN A
CROSS-CULTURAL STUDY IN MIDDLE LEVEL MATHEMATICS



Katheryn Kirk Fouche

August, 1993



Chairman: Eleanore L. Kantowski
Major Department: Instruction and Curriculum

The purpose of this study was to investigate the solution
methods employed by middle level students engaged in solving
nonroutine mathematical problems and the role of creativity during
the reflection process. The solution methods of students who
participated in problem solving sessions designed to elicit multiple
solution methods for nonroutine mathematical problems were
compared to methods employed by students who received no such
deliberate practice. The solution methods were compared by number
of different methods and attained level of complexity. Complexity was
determined by the level of understanding indicated by the solution
method and its generalizability.








Additionally, this study included an international perspective.
The solution methods of students in Japan were compared to those of
American students. Comparisons of both the number of alternative
methods employed and the attained level of complexity were made
between nations and between grades.
The major findings from the study suggest that both the number
of solution methods students were able to generate for a single
nonroutine mathematical problem and the level of complexity of those
methods were significantly increased when students were given the
opportunity to engage in reflective problem solving activities. Also,
the initial solution method tended to be unsophisticated with the level
of complexity increasing with each new method generated. The
number of solution methods students were able to generate for a
single nonroutine mathematical problem was positively related to their
measure of creativity, but there was no significant relationship
between attained level of complexity and measure of creativity.
For the U.S.-Japan comparisons, American eighth grade students
were able to produce a greater number of alternative methods than
either the Japanese sixth or eighth grade students or the American
sixth grade sample. Although the American eighth grade students
were able to generate more alternative solution methods than their
Japanese counterparts, there were no differences between attained
level of complexity.













CHAPTER I
INTRODUCTION
In 1989 the National Council of Teachers of Mathematics
(NCTM) published its Curriculum and Evaluation Standards for School
Mathematics, which outlined recommendations for the next phase of
moving mathematics into the Information Age of the 21st century.
The goals for all students, as expressed in the Standards, include
providing opportunities to "become mathematically literate. This term
denotes an individual's ability to explore, to conjecture, and to reason
logically, as well as to use a variety of mathematical methods effectively
to solve problems" (p. 6). In 1991 NCTM's Professional Standards for
Teaching Mathematics recommended several ways to promote
problem solving. The ideal mathematics classroom community is
described as one in which students rely on mathematical reasoning,
not just memorization; solve problems, not just mechanically find
answers; and connect mathematics instead of treating mathematics as
a body of isolated concepts and procedures. The challenges presented
in the curriculum Standards and the teaching Standards eloquently
blend the goals of an international host of visionaries involved in
mathematics education.
Although a seemingly novel idea for teachers who persist in
using the lecture-practice method, the idea of encouraging multiple
solution methods for a single problem is not a recent development. In
1942 Brownell suggested that







to be most fruitful, practice in problem solving should not
consist of repeated experiences in solving the same problems
with the same techniques, but should consist of the solution of
different problems by the same techniques and the application
of different techniques to the same problems. (p. 439)
Teachers are still being encouraged to model and discuss with their
students a variety of strategies that can be used to solve a single
problem (NCTM, 1991). Providing an opportunity to reflect on a
problem, after attempting a solution, in order to find additional
methods is part of the looking back phase of problem solving
described by Polya (1957). Although the practice of finding multiple
solution methods for a single problem has been suggested as a
technique for teaching problem solving, as a method of alternative
assessment, and as a way to foster development of the creativity that is
a component of problem solving (House et al., 1983; NCTM, 1989,
1991; Thompson, 1991), there is little evidence to suggest that the
looking back phase of problem solving is a regular procedure in the
American mathematics classroom (Stigler & Perry, 1988).
When students are required to look for alternative solution
methods for a single problem, what characterizes their methods? Is
there a difference in the methods employed by students who have
practiced reflecting during problem solving and those who have not?
What characteristics of the problem solver are correlated with the
complexity and number of solution methods a student is able to find?

Purpose of Study

The purpose of this study was to investigate solution methods
employed by students in problem solving and the role of creativity
during the looking back phase of problem solving. Multiple solution







methods for a single nonroutine mathematical problem were
compared by number of different solution methods and level of
complexity. These comparisons involved students who participated in
a treatment designed to practice searching for multiple solution
methods for a single problem and those students who received no
such deliberate practice. Additionally, an international comparison
was conducted in which the solution methods of Japanese students
were compared to those of U.S. students. Teaching students to reflect
on the problem solving process is a routine part of the Japanese
elementary mathematics classroom (Azuma & Hess, 1991; Becker et
al., 1990; Nagasaki & Becker, 1993; Stigler & Stevenson, 1991).
Reportedly, students in Japan often spend an entire classroom period
on a single problem, while American teachers too often strive to
complete as many problems as possible. Since this study will
document underlying mechanisms of problem solving as well as
differences in achievement such as level of complexity represented in
the method, cross-cultural comparative research was especially useful
for this research agenda.
Both Standards suggest that students be challenged by tasks that
promote divergent yet sound mathematical thinking: tasks that allow
creative ideas to flourish. The possibility of a correlation between
creativity in problem solving and solution methods was explored for
the United States sample. Creativity was defined in terms of fluency,
originality, abstractness, elaboration, and resistance to closure,
categories in the Torrance Test of Creative Thinking (Torrance et al.,
1992).








Rationale

In her address at the New Orleans NCTM Annual Conference
(April, 1991), Iris Carl, NCTM President, described the 1990s as the
most exciting time in history for mathematics education. The
challenges outlined by NCTM in Problem Solving in School
Mathematics (1980) and The Agenda in Action (1983) have moved
into the next phase of reform. According to Carl, the curriculum
Standards and the teaching Standards outline a more extensive
educational reform than ever suggested in history. The major thrust of
the reform movement as defined by both Standards involves fostering
the students' ability to think, reason, and communicate mathematically
in order to facilitate a deeper understanding. With the
implementation of the current reform movement still in its early
stages, there exists a need for research that adds to the body of
knowledge concerning how students solve problems.
Silver (1985) outlined the strides in the area of mathematical
problem solving performance to that date and summarized suggestions
for the research agenda of this decade. One of the most prevalent
themes in the document involves the need to view problem solving
and problem solving activity within a broader context than traditionally
found in schools. This broader context would provide experiences for
students "to do mathematics rather than having it done to them" (p.
276). That is, students need to engage in generative mathematical
inquiry and activity so that mathematics might be more stimulating.
Additionally, there exists a need to research some specific areas
of problem solving. The establishment of a positive correlation








between some measure of creativity and the ability to generate
alternative solution methods for a mathematical problem would
provide impetus to the theory that teaching should tap and develop
the creative potential of every learner.
Examining methods of solution for a nonroutine mathematical
problem and the outcomes when alternative methods are sought
involves problem solving assessment. Silver (1985) insists that there
is overwhelming agreement concerning the need to investigate the
lesser-known area of assessing problem solving. Focusing on the
process of problem solving, as in this study, shifts the evaluation
emphasis from the product to the process used to obtain the solution.
Silver and Kilpatrick (1985) note that although the assessment of
problem solving should provide testers with information needed to
make instructional decisions, assessment activity has not received
attention proportionate to need. After two decades of research on
how students solve problems, they argue, very little research has had
direct influence on problem solving assessment.
Finally, this study was enriched by the unique opportunity to
add a cross-cultural dimension. Research involving comparisons to
other cultures has been suggested as a method of improving
understanding of education in one's own culture. According to Sowder
(1989),

American researchers are habituated to American classrooms
and remain unaware of some of their most obvious aspects.
Examining classrooms and educational systems of other cultures
can give us a fresh set of lenses to use in viewing American
classrooms.
Of the few cross-cultural studies that have been
undertaken, the results often document differences in
achievement across cultures without tracking the underlying
mechanisms that may produce the differences. Cross-cultural








research is difficult and costly to conduct, yet it provides unique
opportunities to add to our basic understanding of the teaching
and learning of mathematics. (p. 35)
Having visited Japanese classrooms and having assisted with the U.S.-
Japan Cross-national Research on Students' Problem Solving
Behaviors, this researcher was afforded a rare opportunity to conduct
problem solving research via an established network of
communication.

Research Questions

This study addressed the following questions:
1. What is the relationship between the number and complexity
of methods employed in the solution of a nonroutine mathematical
problem and training specifically designed to elicit multiple solution
methods?
2. Is there a relationship between the number and complexity of
methods employed in the solution of a nonroutine mathematical
problem and a student's measure of creativity?
3. When comparing the responses of Japanese students to those
of American students to a nonroutine mathematical problem, is there a
difference in the number or complexity of methods found?
4. When comparing the responses of Japanese students to those
of American students who have spent about the same amount of time
studying mathematics in school, is there a difference in the number or
complexity of methods found?








Definition of Key Terms

Creativity is possessing the flexibility to formulate different
representations for the same problem and the originality to combine
elements of a problem in novel ways.
Creativity measure is the mean of standardized scores for
fluency, originality, elaboration, abstractness of titles, and resistance to
premature closure as determined by the Torrance Test of Creative
Thinking: Figural A (Torrance, 1990).
Generalizability is the degree to which a solution method for a
specific case of a problem solution can be extended to solve the
general case.
Looking back is the stage of problem solving described by Polya
(1957) as a period of reflection when one might check a solution,
relate the solution method to other problems, extend the problem, or
look for alternative solution methods. This research concentrated on
the aspect of looking back that involves the search for alternative
solution methods.
A method of solution is the process employed by a student to
resolve a problem.
A nonroutine problem is a situation for which the individual who
confronts it has no algorithm that will guarantee a solution. That
person's relevant knowledge must be put together in a new way to
solve the problem (Kantowski, 1974).
Problem solving is "(a) the process, or set of behaviors or
activities that direct the search for the solution, and (b) the product,
or the actual solution" (Kantowski, 1977, p. 163).








Sophistication or complexity of a solution method is an
indication of the level of intellect engaged in the solution of a problem
and the generalizability of the method.

Summary

The curriculum Standards (1989) describe the ideal
mathematics classroom as one where mathematically literate students

deepen their understanding of mathematics by exploring,
conjecturing, reasoning logically, and using a variety of mathematical
methods effectively to solve problems. This research explored the

nature of these important activities. Although it is impossible to
isolate these activities, this research focused on one observable
process associated with deeper understanding and reasoning, the
process of finding alternative solution methods for a given problem.
The solution methods of students who practiced the activity of

searching for alternative methods were compared with the methods of
students who received no such deliberate practice. In addition to the

number of different solution methods a student was able to find, the
complexity or sophistication of the methods was appraised. The
degree of sophistication or complexity was based on a solution
method's generalizability and the level of understanding represented

by the method. In order to explore a possible correlation between
creativity and solution methods, a measure of a student's creativity was
compared to the number and complexity of solution methods
employed. An international perspective was added to this study by the
inclusion of the solution methods of Japanese students in the

comparisons.








When asked to look for another way to solve a given problem,
students and even teachers often respond, why? The attitude that one
correct answer is sufficient may seem logical to many. Is a student
who can find four different ways to solve a problem a better problem
solver than the student who can only think of one method? When
solving a problem should students be encouraged to use only the most
efficient method or have their choice of any method that employs
sound mathematical procedures? The motivation to incorporate
creative problem solving into an overcrowded curriculum must come
from sound research that provides evidence that it is a necessity. How
do teachers stimulate the students' creative potential, and what effect
does that stimulation have on the solution methods employed during
problem solving? These questions were addressed in this research
project.













CHAPTER II
REVIEW OF RELATED LITERATURE
The review of literature presented in this chapter will cover the
following topics related to the present study: (a) problem solving, (b)
creativity, and (c) a comparison of mathematics education in Japan
and the U.S. These lines of research are related to the present study
in the following manner. Problem solving has been the focus of
numerous important research activities, but there exist gaps in the
research that connect problem solving instruction to learning. One
aspect of problem solving that is of particular interest in the present
study is that of creativity and its role in the search for alternative
solution methods. Finally, the investigation of alternative solution
methods was enhanced in the present study by U.S.-Japan
comparisons.

Problem Solving

The purpose of this section is threefold. First, a summary of
related problem solving research to date will be presented. The
summary indicates that the teaching and learning of mathematical
problem solving have generally been studied as separate topics. There
Is a need for additional research that combines both of these
classroom elements as did the present study. Secondly, the
constructivist theory, which provided the framework for the present
study, will be discussed. Finally, support for the treatment format








implemented in the present study will be examined through a review

of teaching and learning research from a constructive perspective.

Summary of Related Problem Solving Research

By 1980 when NCTM recommended that problem solving should
be the focus of mathematics education for that decade, several
important research questions had been explored. A generalized
summary of problem solving research through that date by Suydam
(1982) suggests that
1. Problem solving strategies or heuristics can be specifically
taught, and when they are, they are used more and students achieve
correct solutions more frequently.
2. There is no optimal strategy for solving problems.
3. Students should be a) faced with problems in which the
approach to solving the problem is not apparent and b) encouraged to
generate and test many alternative approaches.
4. Teaching children strategies for problem solving provides
them with a repertoire from which they can draw as they meet the
wide variety of problems that exist.
5. Students choose to employ some strategies more frequently
than others, with various strategies used at different stages of the
problem solving process.
6. A child's problem solving achievement is related to
development level.
7. Problem solving skills are improved by incorporating them
throughout the curriculum.







The bulk of this research focused on the teaching of problem solving
heuristics.
During the 1980s researchers intensified their investigation of
how students solve problems. Romberg (1992) contends that the
intense controversy and reflection that took place during this period
provided the impetus for the radical reform period in which we are
now engaged. The focus of research, however, needs to be more
unified. Although substantial knowledge has been gained, Romberg
and Carpenter (1986) point out that most research on problem solving
can be classified as either research on teaching or research on
learning. Research on teaching has been focused on instruction and
on the group as a whole with less emphasis on what Is taught, whereas
research on learning has primarily been concerned with individual
students and their cognitions, not with how learning occurs.
Carpenter and Fennema (1991) suggest a new research paradigm "that
blends what is known about students' learning, thinking, and problem
solving with what is known about teachers as active, thoughtful
professionals" (p. 7). Such a paradigm would make it possible to
integrate research on teaching and learning by accounting for
teachers' cognition, students' cognition, and the learning-teaching
process within a seamless framework (Cobb et al., 1991). A
description of the learning and instruction research on which the
present study was based follows.

Constructivist Theory

Most mathematics curricula of the 1950s and 1960s were based
on the stimulus-response theory (Davis, 1990). The mental processes








of the students were thought to be unscientific. Therefore, the
dominant teaching strategy involved showing or telling students what
to do, supervising their practice, and evaluating their ability to
regurgitate facts and imitate rituals. Since World War II there have
been various projects and reform movements based on the notion that
more effective models of teaching and learning mathematics exist.
Davis contends that encouraging students to think creatively about
mathematics and helping them understand what they are doing have
been common goals for all of these reform movements. Although the
origins of the constructivist theory is uncertain, it remains, according
to Davis, central to those involved in the major reform efforts in
mathematics education since World War II.
Noddings (1990) has noted that although there exist conceptual
differences in the constructivist theories currently influencing
mathematics education, the following themes are generally accepted:

1. All knowledge is constructed. Mathematical knowledge is
constructed, at least in part, through a process of reflective
abstraction.
2. There exist cognitive structures that are activated in the
processes of construction. These structures account for the
construction; that is, they explain the result of cognitive activity
in roughly the way a computer program accounts for the output
of a computer.
3. Cognitive structures are under continual development.
Purposive activity induces transformation of existing structures.
The environment presses the organism to adapt.
4. Acknowledgment of constructivism as a cognitive position
leads to the adoption of methodological constructivism.
a Methodological constructivism in research develops methods
of study consonant with the assumption of cognitive
constructivism.
b. Pedagogical constructivism suggests methods of teaching
consonant with cognitive constructivism. (p. 10)
About the time that the evaluation Standards were published, the
National Research Council encouraged a strong mathematics program








for each student in its document Everybody Counts (1989). One of its
more profound statements suggests that

In reality, no one can teach mathematics. Effective teachers are
those who can stimulate students to learn mathematics. ....
students learn mathematics well only when they construct their
own mathematical understanding. (p. 58)
In a recent review of literature, Cobb et al., (1991) concluded, 'The
notion that students learn mathematics by actively reorganizing their
own experiences is almost universally accepted by the mathematics
research community" (p. 84). Teaching and learning theories under
the umbrella of this notion have been termed constructivism
(Lochhead, 1985). The idea was born long before the label existed.
Dewey (1956) suggested that people cannot convey ideas from one to
another. He contends that when someone shares an idea, it is to the
receiver a fact. The fact only becomes an idea when the receiver
wrestles with the conditions of the problem by thinking it out
individually. Piaget (1973) believed that the goal of education is not to
promote the memorization of ready-made facts, but instead to help
students build their own set of ideas through activities designed for
inquiry and discovery. Richards and von Glasersfeld (1980) felt that
the act of searching for alternative solution methods to a problem
reflects the essence of constructivism inherent in Piaget's model of
adaptation and accommodation.
It is likely that few mathematics educators would argue with the
notion that actively constructing knowledge is preferable to passively
receiving knowledge from external sources. Von Glasersfeld (1990)
extended constructivism to the extreme when he described radical
constructivism. His theory of knowledge is based on the belief that







one can only construct knowledge within one's own world of
experiences and, therefore, has no way to know the truth that exists
in the reality of the outside world. Kilpatrick (1987) warned that the
radical constructivist view operates on a negative feedback or blind
view toward the real world. If the only reality that one can come to
know excludes the reality outside of one's self, then one can only learn
about the world's constraints, what does not work. With neither
knowledge nor information flowing in or out, one cannot communicate
meanings, therefore knowledge is subjective. Kilpatrick argued that
the radical constructivist cannot make inferences about what the
student is thinking and must rely on overt responses as clues about
the restraints controlling the student's internal processes. While
learning from errors can be effective, students cannot be expected to
respond to constant negative feedback that is precipitated by failure.
As suggested by Kilpatrick, the constructivist theory adapted for this
research does not put "sanitizing quotation marks" around terms such
as problem solving, but instead builds upon the basic principles of the
constructivist theory that are inherent in the mathematics education
reform movement presently under way,
Vygotsky (1978) identified the range of cognitive activities
within which learning occurs as the "zone of proximal development."
Vygotsky defines this zone as "the distance between the actual
developmental level as determined by independent problem solving
and the level of potential development as determined through
problem solving under adult guidance or in collaboration with more
capable peers" (p. 86). Abilities that are in the embryonic state today
are the abilities that will mature and internalize tomorrow. Vygotsky's








zone of proximal development represents a helpful model for problem
solving instruction.

Vygotsky's theory places the teacher in the important position of
decision maker and model. Activities that only tap a student's actual
developmental abilities do not require reconstruction and are
exercises, not problems. In order to balance motivation, self-
confidence and frustration, teachers must choose problems that are
within each student's zone and then carefully decide when and how to
model the problem solving process. Choosing good problems to
challenge yet not overwhelm each student could prove to be
impossible, especially for a class with diverse abilities or experiences.
One remedy lies in the practice of requiring students to provide as
many alternative solution methods as possible for each problem. Thus,
students could begin within their actual developmental level and then
explore their zone through cooperative learning or modeling by the
teacher or peers.
Newell and Simon (1972) described problem solving as a search
through a "problem solving space" until a solution is found. The
problem space is where a student internally represents the initial
situation and the desired goal situation. Trial and error occurs when
the search is random as opposed to a search generated from some
mathematical principle. Trial and error is often the search method
employed by inexperienced problem solvers. Experienced problem
solvers tend to use heuristics, or more sophisticated rules for
selecting search paths. It is possible, according to Newell and Simon,
for one to become engaged in a mind "set." When the mind becomes
fixed in a particular path, the problem solver can find no other way to








"think" about the problem. One's mind could become set in an
incorrect path and thus the problem would remain unsolved, or the
mind set could occur when attempting an alternative solution method.
When additional solution methods lie within students' levels of
development, on which type of solution methods do they tend to
become set? This question was explored in this research.
Newell and Simon (1972) described the mental processing of
most problem solving activity as serial rather than parallel steps
limited by the problem solver's capacity for processing. Thus,
different steps in the problem solving process compete for the solver's
limited mental resources. Bruner et al. (1956) see this competition as
a cause for a conflict of goals. One goal is to complete the task
efficiently and the other goal is to minimize cognitive strain. The ideal
solution method provides an efficient strategy with minimal cognitive
strain. However, a problem solver faced with a conflict might choose a
less efficient solution method in order to keep cognitive strain within
acceptable bounds.
A problem solver always begins with an initial representation of
the problem. This initial representation may be constructed by the
individual or be the result of how the problem was presented. Simon
and Hayes (1976) indicated that the initial representation of the
problem has a strong effect on the solution methods chosen by the
solver. They suggest that the problem solver will change
representations that place too heavy a burden on short-term memory.
The case may be, therefore, that a mind set could be the result of a
conscious or unconscious avoidance of cognitive strain.







As a result of the belief that students learn mathematics well
only when they construct their own mathematical understandings,
both the curriculum and evaluation Standards (1989) and the teaching
Standards (1991) are filled with verbs such as explore, justify,
represent, solve, construct, discuss, use, investigate, describe,
develop, and predict. The student is not considered a passive receiver
of information, but as an active participant both mentally and
physically.
This active involvement of the student during problem solving is
reflected in the constructivist theory of learning and teaching. With
each new experience students reformulate their individual picture of
the world by adjusting and adapting to some changes while initiating
others (Confrey, 1990). The depth of students' understandings relies
to a great extent on their ability to defend their position. Confrey
emphasizes the importance of flexibility in this process. To assist
students in constructing their own understanding teachers must show
interest in and value each student's response. Teachers must
approach even the foreign or unexpected response with interest while
probing for possible implications. When a student adopts a solution
method based on an inadequate or incorrect knowledge base, the
teacher must first understand the student's way of thinking before
attempting to assist the student In restructuring those views.
Reflection is necessary for the construction process.
According to Confrey (1990), the most fundamental quality of a
student's construction is reflected in the student's own belief in the
construction. Constructivists contend that students must believe in
order to know. Other clues exist for teachers to judge the power of







students' constructions. Among the 10 qualities of a powerful
construction listed by Confrey are the following:


1. A convergence among multiple forms and contexts of
representations;
2. An ability to be reflected on and described;
3. A potential to act as a tool for further constructions;
4. An ability to be justified and defended. (p. 111-112)
Teachers in a constructivist classroom must be committed to helping
students create powerful constructions.
As a facilitator, teachers use three levels of questioning during
problem solving sessions to develop students' reflective process
(Confrey, 1990). At the first level teachers use questioning to evaluate
and increase students' level of understanding of the problem. Asking
students to restate the problem gives clues about their understanding.
Time spent at this level can be time curtailed in developing a plan. At
the second level teachers use questions to help students deepen their
own understanding of the cognitive strategies. Instead of "filling in
the blanks" for students who are not able to describe what they are
doing, through proper questioning the teacher can request more
precision. After a student understands the problem and can describe
the solution method, teachers' third level of questioning should
require the student to defend the solution. The rigor of questioning is
based on the teacher's knowledge of the students and the complexity
of the method being employed. These three levels of questioning
dealing with the student's (a) interpretation of the problem, (b)
cognitive strategies, and (c) justification of the strategies become the
teacher's tools for developing each student's reflective process.







Teaching and Learning from a Constructivist Perspective

Given that each student in a classroom has a unique combination
of mathematical experiences and abilities, how does the teacher
provide opportunities for each student to individually create
knowledge and understanding? Carpenter and Fennema (1991)
suggest that teaching is problem solving and that both the teacher and
each student enter the classroom with a unique knowledge base.
Therefore, instead of following an outline of "prescribed" procedures,
it is up to the teacher to make informed decisions during the
instruction-thinking-learning process. To cultivate good problem
solvers, teachers must challenge students with carefully chosen
activities that encourage students to create and test new rules, to
reconstruct their own mental schemas.
Summarizing research on teaching mathematics for
understanding and on students' understanding of mathematics,
Lampert (1991) has suggested that two of the teacher's most
important roles involve (a) choosing and posing good problems and (b)
developing tools for communication between teachers and learners. A
good problem, according to Lampert, is one that creates a learning
environment conducive to students expressing their mathematical
thinking. Given that each student enters the classroom with a unique
combination of mathematical knowledge and experiences, a good
problem is one for which a range of alternative solution methods exist,
at least one of which lies within each student's ability to solve but also
admits of solution methods that extend and deepen the student's
understanding. Good problems allow for communication between








students and teachers and thus establish communication tools that can
be used to connect a student's familiar ways of solving problems with
other less familiar methods. Students who prepare a discourse of
defense for their own solutions exhibit more mathematical
understanding than those who accept mathematical principles by
virtue of the teacher's authority. Finding multiple solution methods
for good nonroutine problems has the potential of placing students in
control of their mathematical thinking and focuses attention on the
process rather than the product. Knowledge of the mathematical
assumptions that underlie each strategy employed by students
becomes an integral part of the teacher's evaluation and decision
making.
The present study focused on encouraging students to search for
additional solution methods for nonroutine mathematical problems. If
students are to take charge of their learning through active
involvement, how should problem solving sessions be structured in
order to maximize results? Kantowski (1980) suggested that a
teacher's role in problem solving instruction should be based on the
student's developmental level. For a novice problem solver, the
teacher assumes the role of a model. As a student progresses in his or
her understanding and problem solving skills, the teacher becomes
more of a crutch, then a problem provider, and finally, a facilitator. At
each level it is the responsibility of the teacher to choose good
problems or evaluate those posed by the students.
It is assumed that spending time on reflection about a problem
solution method allows a student the opportunity to gain a deeper
understanding of the problem. As a result of an increased level of








understanding can the teacher expect to see evidence of a higher level
of sophistication in the solution methods?
Vygotsky (1978) suggested that the zone where learning takes
place is bounded by what one can master alone and what one can
accomplish with the assistance of an adult or more capable peer.
While grouping students for cooperative learning has been promoted
as one method for helping students become actively involved in their
own learning, the debate about how students should be grouped is
ongoing. Tudge (1990) conducted a study to determine the effects of
various grouping arrangements on students' solution methods to a
nonroutine problem. He was particularly interested in investigating
what happened to the more capable or equally capable peer when
students were grouped for cooperative learning. Does the zone of
proximal learning allow for regression as well as advancement?
Each student in the study was given an individual pretest to
establish which of the six methods of solution each subject would
employ. The methods were assigned a level of sophistication
according to the complexity of the method. Acting on the advice that
students should be paired with peers who are within their own zone of
proximal development (Mugny & Doise, 1978), Tudge assigned each
subject to a treatment group identified as follows: (a) those who
continued to work individually, (b) those who were paired with a
student who had chosen a solution method at an equal level of
sophistication, (c) those who were paired with a student working at a
level of sophistication one or two below, or (d) those who were paired
with a student working at a level of sophistication one or two above.
Students were asked to predict the results of several scenarios








involving weights being placed on a balance scale. Students arrived at
their predictions through discussion with their partner without the
benefit of testing their conjectures.
The results indicated regression for over one-third of the
subjects whose partners used the same method and almost half of
those whose partners had used a less sophisticated method.
Additionally, the students who regressed to a lower method than the
one used on the pretest continued to use that less sophisticated
method in subsequent individual testing. Only subjects who worked
with more capable peers benefited from the cooperative learning
problem solving activity.
Tudge (1990) conducted a follow-up study to determine what
factors are most critical in determining whether students adopt a
method of a higher or a lower level of sophistication. He discovered
that a student's degree of confidence was very influential during
interaction. Methods of less sophistication could often be used more
confidently than more complex rules, therefore students often
abandoned methods that were generated through a deeper
understanding of the problem.
In subsequent research Tudge (1990) repeated his former study
with the addition of informative feedback. In this study students were
encouraged to test their conjectures using the weights and balance
scale. The results of this study highlighted the importance of
feedback for each student in the group. Feedback not only benefited
students in the selection of a method identified as more sophisticated,
it overshadowed the effects of discussion with a partner. Dewey
(1956) emphasizes the importance of time for students to interpret,







infer, share observations, and reflect. He sees this sharing time as
"the social clearing-house, where experiences and ideas are
exchanged and subjected to criticism, where misconceptions are
corrected, and new lines of thought and inquiry are set up" (p. 55).
Dewey cautioned that this sharing time should not be a time when a
student is forced to say something, but instead a time offered for the
student who has something to say. For the present study feedback was
offered not only during cooperative learning time, but also each
session concluded with a time for students to share solution methods
with the class.
Problem solving research should blend research on student
cognition and instructional methods of problem solving. The present
study focused on not only student cognitions as represented by their
solution methods but also the instructional process of asking students
to search for alternative methods after having solved the problem.

Creativity

The notion of creativity encompasses a vast array of definitions
and theories. Creativity can be thought of in terms of individuals who
possess certain talents, or in terms of the actual resulting product.
Although differences exist in interpretations, all of the definitions have
in common the notion of originality.
Beyer (1987) separated critical thinking from creative thinking.
He claims that critical thinking is a skill involving examining and
breaking down reality in order to understand a situation; whereas
creative thinking is a talent that requires the combining of elements of
reality in novel ways to formulate new understandings. Parker (1963)







defined creativity as the "art of seeking out, trying out and combining
knowledge in new ways" (p. 170). Creativity has been identified as
occasions when one combines previously unrelated structures in such
a way that the whole is greater than its parts (Koestler, 1964; Parnes
et al., 1977) In their review of creativity, Mumford and Gustafson
(1988) concentrated on the product of creativity by focusing on
studies that involved the production of a creative outcome. Hocevar
and Bachelor (1989) agreed but were quick to point out that studying
creative thinking and creative personalities is valid not only because of
their interest, but also because there is evidence to suggest that they
are potential causes of creative productions. Mayer (1989) suggested
that creativity is the production of something that is new to the
individual. This definition is problematic; it could be used to
characterize all learning as creative. For this study, a good definition
of creativity in problem solving could be used to identify those who
possess the flexibility to formulate different representations for the
same problem and the originality to combine elements of a problem in
novel ways. A creative solution method reflects thinking that is
determined to be unusual for the sample under consideration.
In summary of a wide range of theories, Brown (1989) listed four
general views of creativity. In these four views creativity is (a) an
associative process; (b) an aspect of intelligence; (c) a largely
unconscious process; and (d) an aspect of problem solving. These
perspectives will be reviewed briefly below.
The associative view was held by several theorists who based
many of their ideas on the Spearman Principle (Brown, 1989).
Spearman's model "involves an active process in which associations








with an initial idea can be freed from their relation to it and thus lead
to something wholly new" (p. 5). This theory was the basis for
important subsequent studies including those of divergent thought.
Haensly and Reynolds (1989) reviewed the debate concerning
the connection between intelligence and creativity that has evolved
over the past half-century and concluded that the most prevalent
current view is that "creativity is a distinct category of mental
functioning that has limited overlap with intelligence, both in the
processes used and in the characteristics of individuals who exhibit
them" (p. 111). Haensly and Reynolds believed that trying to
determine how much intelligence is associated with how much
creativity would result in a simplistic view of a vast array of mental
capabilities. They suggest that an individual's most intelligent
responses could still be considered ordinary. The creative response
extends into the realm of the extraordinary. Guilford (1959)
emphasized a curriculum that promotes creative thinking as a means
to prepare our students for a changing world.
The debate over creativity and intelligence would not be relevant
in mathematics education with a curriculum based on the ideals
outlined in the curriculum and evaluation Standards and teaching
Standards. The framework for such a curriculum would reflect
Guilford's (1965) attitude that "creativity is not a special gift of the
select few. It is instead a property shared by all humanity, to a greater
or smaller degree" (p. 7). All students need to be provided with
opportunities and the motivation to think. Although helping students
learn to be good problem solvers appears first in the list of goals for







mathematics education (NCTM, 1989), the ultimate goal is engaging
students in creative teaching and learning.
The third view of creativity, that it is unconscious thought,
might be extracted from a model described by Wallas in 1926. The
stages of the model are (a) preparation, which includes the gathering
of requisite knowledge and skills, (b) incubation, the phase in which
retrospective thought is unconscious (c) illumination, the occurrence
of insight, and (d) verification, the process of correcting or revising an
idea. Armbruster (1989) supported the use of the model because it
has been an implied part of so many respected studies over the years
and because it is a useful means of organizing a discussion of creativity.
Armbruster warned, however, that Wallas' writing implies that creative
thought is linear, whereas research suggests that creative thought is
more interactive and iterative, with communication among the stages.
No one knows exactly what occurs during the stage of incubation, but
Armbruster concluded that flexible knowledge that has been acquired
during the preparation stage is restructured into new mental
structures. Though incubation proceeds unconsciously, the creative
individual may have a metacognitive skill that allows for an efficient
and effective control over the reconstruction of their schemas.
The stages of creative thinking described by Wallas (1926)
closely parallel the phases of problem solving detailed by Polya (1957):
understanding the problem, devising a plan, carrying out the plan, and
looking back. During the first phase, understanding the problem, a
problem solver prepares to solve the problem by identifying as much
information about the problem, known and unknown, as possible. The
more information a solver can establish about the problem, the deeper







the understanding. Spiro et al. (1987) believed that the creative
individual can think independently while gathering knowledge and
representing it in a flexible and productive schema. Spiro and Myers
(1984) further suggested that the ability to maintain flexibility in
representation might be attained by consciously considering the same
information using many different models. In summary, it is during
this important phase of understanding and preparation that a problem
solver determines the probability of a correct, and possibly creative,
solution.
Polya's second phase of problem solving, devising a plan, is
related to but different from incubation, the second phase of creativity.
Whereas incubation is unconscious, devising a plan suggests a
conscious search of one's knowledge base in order to find or
reconstruct a solution method that will eventually work. There is no
way to estimate how much time an individual might require for
incubation, but time is a necessary ingredient for successful
incubation. Polya stressed the importance of allowing students
adequate time to think about a problem.
The joy that accompanies finding the "answer" during Polya's
third stage of problem solving could be equated to the excitement that
coexists with the third stage of creativity, illumination. Although
illumination is often described as a point when an idea spontaneously
springs from the unconscious to the conscious. Spiro et al. (1987)
believe it is the result of unconscious flexible representation.
Polya's (1957) fourth stage of problem solving, looking back, is
similar to the verification phase of creativity. These reflective phases
of problem solving and creativity require the student to relate the








present situation to previous situations, think in terms of "what if,"
and look at the situation from a variety of viewpoints. These reflective
phases of problem solving and creativity also require a student to shift
from the newly acquired sense of relief to additional mental strain
(Armbruster, 1989). This task is often overlooked or consciously
avoided by students and teachers. Even students who have been
trained to look back at a problem do not systematically do so
(Kantowski, 1977).
With creativity and problem solving so closely paralleled what, if
anything, can be done to improve a student's creativity? Torrance and
Torrance (1973) collected evidence to support their belief that
although creativity is a natural process, teaching can make a
difference. They identified skills involved in creative problem solving

that require practice and can be enhanced by teaching. Although
Torrance and Torrance issued no guarantee with even the best
teaching situations, they promote a classroom that provides for the
deliberate teaching of skills involved in creative problem solving in
order to increase the probability that creative development would
occur. According to Torrance, to develop these skills, which involve
both cognitive and emotional functioning, one must be provided with

adequate structure, motivation, involvement, practice, and interaction
with teachers and other students. If, in fact, everyone possesses

some degree of creativity that needs to be fostered, decision makers
must insist upon a curriculum that provides each student with
appropriate activities and experiences.
In summary, the topic of creativity has been an interest for
numerous researchers over the past century. Creativity, whether








identified as a personality trait or a product, is considered a valuable
asset in business, industry (Scott, 1992), leisure activities, and
parenting. Thus, it becomes the duty of educational systems to
produce educated, creative graduates. The educator's task of
nurturing the creative potential possessed to some degree by each
individual has been aided by Wallas's description of identifiable stages
that interact during creativity. The strong similarities between
Wallas's stages of creativity and Polya's phases of problem solving give
justification to research focused on their relationship. Knowing the
processes through which a creative problem solver passes, and
recognizing the products of creativity help educators plan for, manage,
and evaluate the problem solving curriculum outlined in the
curriculum and evaluation Standards.
The present study investigated the possibility that encouraging
students to represent a problem in as many different schemas as
possible taps their potential for creative problem solving. The present
study compared the solution methods selected by students identified
as creative with those who show less of an inclination for the type of
creativity surveyed. Polya's suggestion that students need time to
think about a problem was incorporated into the present study. The
possibility that the pursuit of several alternative representations for a
given problem leads to more sophisticated solution methods was also
examined.

Mathematics Education in Japan

Achievements of the Japanese in business and education have
captured worldwide attention. On the Second International








Mathematics Study (SIMS) (McKnight et al., 1987), Japanese eighth
grade students ranked first among 20 countries in all computational
and noncomputational categories for that age group. Ironically, many
of the educational reform ideas that have brought Japan to its present
competitive status were ideas borrowed from other countries
including the United States (Shimahara, 1992; U.S. Department of
Education, 1987). Japan's accomplishments in mathematics
education made the prospect of including their students in a study of
problem solving very intriguing.
The fact that Japanese students far outperform students in the
United States on international tests of mathematics achievement is
well documented (Husen, 1967; McKnight et al., 1987; Stevenson et
al., 1990). While American educators Investigate numerous variables
that might explain the discrepancy in scores, Japanese educators are
carefully analyzing American schools (Burstein & Hawkins, 1992).
Americans are understandably fascinated by the educational
achievements of Japanese students, but one may be puzzled by
Japanese educators' interest in American instructional practices.
Matthew Prophet, superintendent of the Portland School District,
where 1 of 5 Japanese immersion programs in the U.S. is offered,
thinks that "the Japanese are trying to be more like us" (quoted in
Graves, 1991, p. 14). One of the reasons for Japanese interest in U.S.
schools is that American classrooms are viewed as more conducive to
creative and assertive behavior. Ikuo Idaka, a Japanese consul for
cultural affairs and education in New York City, stated that Japanese
educators are coming to the United States "to find out how the
individual student character is being respected" (quoted in Graves,








1991, p. 14). A Japanese commission established to evaluate Japan's
educational system reported,

Despite its merit, the main thrust of this country's education has
been to have students memorize information and facts. The
development of the ability to think and judge on one's own and
the development of creativity have been hampered. Too many
stereotyped persons without marked individuality have been
produced. (Haberman, 1985, p. Fl)
Japanese mathematics educators, members of a virtually single-culture
nation, are curious about how the diverse American students solve
problems (Becker, 1992; Nohda, 1989). Partly due to an interest in
U.S. mathematics education, a team of Japanese researchers assisted
in the design of the present study and two of the team's mathematics
educators invested their own time to manage the data collection
process in Tokyo and Hiroshima.
Many factors could contribute to the disparity in the
mathematics achievement of the U.S. and Japan. Contrasts in the
home environment and other environmental differences (Chen, 1991;
Stevenson et al., 1986), amount of instruction in mathematics (Stigler
et al., 1987), textbooks (Stigler et al., 1982), and other factors have
been shown to influence achievement. While the effects are difficult to
isolate, they are each correlated to some degree with differences in
achievement. Lynn (1982) even tried to show that the gap was due to
a significant difference in the cognitive abilities between Asian and
American children. Lynn's results were later refuted (Stevenson et al.,
1985).
Although no systematic relationship between time spent on
mathematics and achievement could be established when considering
all 20 countries in the SIMS study (McKnight et al., 1987), when







comparing U.S. and Japan scores specifically, time spent on
mathematics was identified as a major contributing factor (Stigler et
al., 1987). American students spend less time each year in school,
less time each day in classes, less time each day in mathematics class,
and less time in the mathematics classroom receiving instruction than
Japanese students. American students, according to this study, spend
more time working alone at their seats on material that they do not
understand well, engage more often in irrelevant activities, and spend
more time in transition from one activity to another. The fact that
Japanese seventh grade students have spent 40% more time studying
mathematics (an equivalent of 2.4 more years) than seventh grade
students in the United States was also established as a contributing
factor to Japanese students' superior performance in relation to U.S.
counterparts by Iben (1988). Since solution methods employ skills
that exist in a student's knowledge base, a student who has had the
equivalent of between two and three years of extra time spent on
mathematics might be expected to produce more sophisticated
solution methods. Results of a U.S.-Japan Cross-national Research on
Students' Problem Solving Behaviors (Becker, 1992) suggest that
although students in the United States may use similar solution
methods for nonroutine problems, Japanese students tend to employ
more mathematically sophisticated methods at an earlier age.
Although data analyzed qualitatively give some support to the
hypothesis that the complexity and sophistication of solution methods
of Japanese students could be equated with American students two
grades their senior, to date there is still little statistical evidence to
support such a claim. With mathematical computation scores that







rank at the top globally, the opportunity to examine the solution
methods of Japanese students for nonroutine problems added a timely
and valuable dimension to problem solving research.
The present study was partially based on one portion of a former
project, the U.S.-Japan Cross-national Research on Students' Problem
Solving Behaviors, henceforth referred to as the U.S.-Japan. Cross-
national Research. (Becker, 1992; Becker & Miwa. 1987; Nohda,
1989). The objectives of the research dealt with various aspects of
problem solving that could be compared between the two cultures.

Summary

Mathematics education has been the target of many reform
movements since World War II; none as radical as the changes now
underway with the learning and teaching of mathematics being
brought together as a single focus. One of the common themes of each
of the major reform movements is that students should be given the
opportunity to learn mathematics by actively constructing their own
meaning. In the role of facilitator, the teacher inherits the task of
choosing problems so that each student is challenged but not
overwhelmed, problems for which a variety of solution methods lie
within each student's zone of proximal development, and problems for
which the reflective process can be guided through questioning
techniques. Constructivist teachers encourage students to think
creatively about mathematics and help them understand what they are
doing. Students are being challenged to not only solve problems, but
to be creative problem solvers. While much attention has been
awarded international tests of computational skills, there is interest in





35

how students in the U.S. and Japan compare on methods used to solve
nonroutine mathematical problems.
This chapter contains a summary of problem solving research to
date, a review of learning and teaching theories adopted by those who
come under the umbrella of constructivists, and a comparison of
mathematics education in the U.S. and Japan. The next chapter will
describe the treatment format for students in the main study and the

procedures used to test the hypotheses.













CHAPTER III
PROCEDURES

This chapter includes a description of the data collection
techniques, the treatment, and method of analysis for (a) the main
study which examined the number and complexity of solution methods
students employed for a nonroutine mathematical problem and (b) the
U.S.-Japan comparisons that served as a supplement to the main
research..

Hypotheses

The purpose of the present study was to investigate methods of
solution used to solve a nonroutine mathematical problem and the role
of creativity in the reflective phase of problem solving. Problem
solving methods of students who participated in sessions designed to
encourage the construction of alternative solution methods were
compared to the methods employed by students who had received no

such deliberate practice. The study was enhanced by the inclusion of
Japanese students for comparisons. Methods of solution for a
nonroutine, pattern finding problem were compared and contrasted in
order to subject the following hypotheses to statistical tests:
Hol: There is no significant relationship between the number of

different solution methods employed for a nonroutine
mathematical problem and the treatment received.







Ho2: There is no significant relationship between the number of

different solution methods employed for a nonroutine
mathematical problem and receiving the pretest.
H03: There is no significant relationship between the number of

different solution methods employed for a nonroutine
mathematical problem and the measure of creativity.
H04: The relationship between the number of different solution

methods employed for a nonroutine mathematical problem
and treatment received does not differ by pretest.
Ho5: The relationship between the number of different solution

methods employed for a nonroutine mathematical problem
and the measure of creativity does not differ by pretest.
Ho6: The relationship between the number of different solution

methods employed for a nonroutine mathematical problem
and measure of creativity does not differ by treatment.
Ho7: There is no significant relationship between the

complexity of different solution methods employed for a
nonroutine mathematical problem and the treatment
received.
Ho8: There is no significant relationship between the

complexity of different solution methods employed for a
nonroutine mathematical problem and receiving the
pretest.
Ho9: There is no significant relationship between the

complexity of different solution methods employed for a
nonroutine mathematical problem and the measure of
creativity.








Ho10: The relationship between the complexity of different

solution methods employed for a nonroutine mathematical
problem and treatment received does not differ by pretest.
Holl: The relationship between the complexity of different

solution methods employed for a nonroutine mathematical
problem and the measure of creativity does not differ by
pretest.
Ho12: The relationship between the complexity of different

solution methods employed for a nonroutine mathematical
problem and measure of creativity does not differ by
treatment.
Ho13: There is no significant difference between the number of

different solution methods employed for a nonroutine
mathematical problem on the premeasure and on the
postmeasure.
Ho14: There is no significant difference between the complexity

of different solution methods employed for a nonroutine
mathematical problem on the premeasure and on the
postmeasure.
H015: There is no significant relationship between the number of

different solution methods employed for a nonroutine
mathematical problem and grade level.
Ho16: There is no significant relationship between the number of

different solution methods employed for a nonroutine
mathematical problem and nationality.
Ho17: There is no significant relationship between the

complexity of different solution methods employed for a








nonroutine mathematical problem and grade level.
Ho18: There is no significant relationship between the

complexity of different solution methods employed for a
nonroutine mathematical problem and nationality.
Analysis of variance was used to test the hypotheses. The alpha level
set for rejection of the null hypotheses was 0.05.

The Main Study

Following the description of the design of the main study will be
a description of the U.S.-Japan comparisons that served as a
supplement to the main study.

The Subjects

A total of 595 students were involved in one or both parts of the
present study. The survey sample of the main study consisted of 217
eighth grade American students from four algebra and four prealgebra
intact classes. The eight classes were from four different middle
schools located in Central Florida. Most of the students in three of the
schools were from middle class backgrounds, whereas the fourth
school maintains the same racial and SES balance as the state of
Florida. Only algebra and prealgebra students, who typically are more
skilled than students in general mathematics classes, were included in
the present study since it was assumed that students who possess the
greater number of mathematical skills have more potential for
providing multiple solution methods than would those students whose
pool of mathematical resources is not as rich. Also, algebra and
prealgebra students were asked to participate because 171 students








from the main portion of this study were also included in the U.S.-
Japan comparisons of the present study. Since all eighth grade
students in Japan take algebra, any differences in solution methods
might be masked by an imbalance of computational skills between
Japanese students, who have all practiced these skills, and American
students who do not take algebra and who therefore cannot be
assumed to have practiced the same computational skills as Japanese
students.

Instrumentation for the Main Study

The two instruments used in the present study include a revised
version (figures 3-1 and 3-2) of the marble problem which was part of
a former study, the U.S.-Japan Cross-national Research (Appendix F)
and the Figural Booklet A version of the Torrance Test of Creative
Thinking: Thinking Creatively with Pictures (Torrance et al., 1992).

The Marble Problem

The purpose of the U.S.-Japan Cross-national Research (Becker,
1992) was to compare and contrast how students in each country
solve nonroutine mathematical problems. Analysis of student's
performance on one of the five mathematical problems included in the
research, the marble problem, was used as a basis for part of the
present study. The U.S. and Japanese mathematics educators who
conducted the research considered a wide range of problems before
deciding on the five to be included in their complete survey.
Problems selected were nonroutine: the solution would not be
immediately obvious to the solver but would be within the solver's








ability range. Also, each of the selected problems afforded a variety of
alternative solution methods. The problems chosen were judged to be
the best for the two samples in terms of simplicity, understandability,
and the inclusion of the necessary skills for solving the problems in
the curriculums of both groups. The marble problem, figures 3-1 and
3-2, is a nonroutine, pattern-finding problem. The marble problem
was chosen because the solution, although simple, can be found by
students of varying mathematical abilities using numerous different
methods. Beginning with the simple case shown in Part 1, the
problem becomes progressively more generalized in Parts 2 and 3.
The marble problem was administered to sixth, eighth, and eleventh
grade students in the United States and sixth and eighth grade
students in Japan. It was the only problem of the five that was
included at three different grade levels.
Having been assigned the task of assisting in the analysis of the
student's solutions of the marble problem for the U.S.-Japan Cross-
national Research, this researcher began by studying the problem and
recording solution methods that might be expected from the students.
When the researcher's list was exhausted, the student responses were
skimmed first and then classified according to which method was
used. Methods generally included enumeration, the identification of
several different patterns, various grouping techniques, and a formula.
The list of identified solution methods was subsequently extended to
include additional creative methods supplied by the students. Each
identified method of solution was evaluated independently.
To provide an objective check on coding, the researcher refined
a list of solution methods and the characteristics that could be used to












Part 1:

Marbles are arranged as follows:

first second third fourth
place place place place







Do not erase anything you write down, just draw a line through
anything you feel is in error.

(1) If you were to continue building marble structures, how
many marbles would there be in the fourth place?

FIND THE NUMBER OF MARBLES USING AS MANY DIFFERENT
METHODS AS YOU CAN. Show your method and the number of
marbles.


FIGURE 3-1
Part 1 of the Revised Marble Problem









first second third fourth
place place place place

*

* .



Part 2:

(2) How many marbles would there be in the sixteenth (16th)
place? Show one method of solution and the number of
marbles.







Number of marbles
Part 3:-------------------------------
(3) How many marbles would there be in the one hundredth
(100th) place?
Show one method of solution and the number of marbles.







Number of marbles


FIGURE 3-2
Parts 2 and 3 of the Revised Marble Problem.







distinguish each method. Results of the marble problem data analysis
suggested that Japanese students found methods of solution that,
although of the same type, were identified as higher level more often
than did their American counterparts (Fouche & Kantowski, 1992).

Rationale for Reexamination of the Marble Problem

One of the stated objectives of the U.S.-Japan Cross-national
Research was to compare several aspects of how students in each
culture solve nonroutine mathematical problems. The examination of
the Japanese marble problem data, as reported by Ishida (1991),
combined with the analysis of the American sample, provided insights
that elicited interesting speculations and generated additional
questions. As for using the marble problem data as a basis for
comparisons between countries, only the few general observations
could be documented.
Additionally, the compatibility of the analysis techniques
employed by each team member and the channel of communication
available between team members was a factor in the comparison
process. In the case of the marble problem, cross-national data
comparisons were limited because coding, analysis, and reporting
techniques were not uniform between countries. Determination of

significant differences was unachievable since Ishida (1991) did not
report any statistical analysis of the Japanese marble problem data.
The frequency with which each identified strategy was employed
could be obtained from the Japanese report but there was no
indication of the students' mean number of solution methods. There
was not a consistent opinion of what constitutes a generalizable







solution method and comparisons of a student's initial solution
method, to a subsequent method differed between reports.
In addition, the present tense wording on the American survey
for Part 1 which read, "How many marbles are there in the fourth
place?", proved to be confusing for many of the American sixth grade
students (See Appendix F). Responses such as "There are no marbles
in the fourth place." were not uncommon. Others indicated that the
five dashes in line with four marbles from Stage 3 were related to the
five marbles that would go on line two of that stage. Imprecise
wording of the question reduced the validity of even basic comparisons
such as the percent of students obtaining the correct solution.
However, the marble problem data did provide interesting
results that warranted further investigation. While computational
skills have been the focus of many cross-cultural studies, there have
been few studies designed to compare the problem solving methods
for a nonroutine mathematical problem. With the nation involved in a
radical reform movement in mathematics education, valuable insights
for curriculum, teaching, and evaluation could be found in research
focused on creative problem solving.
An indication of the need to revise the original marble problem
was suggested by results of the U.S.-Japan Cross-national Research.
The present tense wording of the Part 1 question, "How many marbles
are in the fourth place?" was changed to, "If you were to continue
building marble structures, how many marbles would there be in the
fourth place?." Also, to avoid possible confusion, the dashes in the
fourth place in the original problem were replaced with a question
mark. To ensure that students did not spend all of their allotted time








on Part 1, students were allowed 8 minutes for Part 1 and 8 minutes
for Parts 2 and 3 combined. To eliminate the need for turning back to
Part 1 to refer to the problem, the problem and diagram were
repeated for Parts 2 and 3. The text of the revised version of the
marble problem appears in figures 3-1 and 3-2 with the entire
problem booklet included in Appendix B.

Administration of Instruments

The revised marble problem was used as both a pre and posttest
measure. This researcher began each pretest by reading a script
(Appendix E) to the class that further explained the purpose of the
research and the importance of their participation. The survey
booklets were then distributed, the problem and procedures
explained, and the survey administered. The students were
monitored throughout the survey to maintain an atmosphere
conducive to concentration and to discourage the sharing of ideas.
The entire process for the marble problem took about 20-25 minutes
of the first half of a class period. As a posttest, the revised marble
problem was administered in the same manner.
To obtain a measure of students' creativity, the Torrance Tests of
Creativity Thinking: Figural A (Torrance et al., 1992) was selected for
this study. Given that creativity has been defined to include a wide
range of personality traits and resulting products and given that much
of the creative process is unconscious, trying to measure creativity is
problematic. However, the Torrance Tests of Creative Thinking
(TTCT) have shown significant validity and reliability in the areas of
fluency, flexibility, originality, and elaboration (Torrance, 1990).







Although there remains a question as to how much originality the
Torrance or any other test can actually measure (Cooper, 1991), the
TTCT has played an important part in research and group assessment
for over two decades. Norms for that test were developed using a
representative sample of students in the U.S. in grades K-12+ in all
major regions of the country. Rater-reliability coefficients for grade
eight with the streamline scoring procedure for fluency, originality,
abstractness of titles, elaboration, and resistance to premature closure
consistently stay well above .90 (Torrance, 1990).
The Figural A version of the TTCT was administered according
to the author's guidelines to every subject included in the main study.
To provide for uniformity in the recording instrument used and to
motivate students, the researcher gave a new, sharpened cedar wood
pencil to each student. Students were timed for ten minutes on each
of three sections with instructions for each section delivered verbally
as well as included in the test booklet.

Group Assignment

The major factor that effected group placement of the eight
classes was teacher cooperation and interest in the study. The
teachers who were the most enthusiastic about the study were those
who were convinced that the 10-11 class periods that would be
devoted to the study would provide valuable learning experiences for
their students and would promote the problem solving skills
incorporated in their existing curriculum. The five teachers who
volunteered their students for the treatment group were themselves
interested in problem solving and in how their students would







respond to the lessons. The eight classes were in schools that service
a similar population of students and parents. Therefore, the teacher
was the deciding factor, not the students. As the result of a project

just prior to the time of this study, this researcher had worked closely
with three of the five teachers who offered their students as subjects
in the treatment group. The four treatment classes were therefore
chosen from those taught by these three teachers.

Treatment Procedures

For the investigation of creativity the eight classes were divided
into four groups with one algebra and one prealgebra class in each
group. A quasi-experimental, Solomon four-group design was
employed. Group identification is shown in Table 3-1.

TABLE 3-1
Treatment

yes no

Group 1 Group 2 yes Pretest

Group 3 Group 4 no


The two treatment groups received eight full class periods of
problem solving instruction over a period of ten weeks during the
spring of 1992. The objective of the problem solving instruction and
planned practice was to elicit multiple solution methods for each of
the eight problems. The problems used in this study appear in
Appendix A. Classes were planned and taught by the researcher with
occasional assistance. All instruments were administered by the
researcher.







For each problem solving class the instructor served as a
facilitator and the procedures were as follows.
5 minutes Introduction to the problem of the day
10 minutes Students work individually to produce as many

solution methods as possible
20 minutes Students work cooperatively in small groups to

produce additional solution methods
15 minutes Solution methods shared with the class by

individual students via an overhead transparency
on which they had recorded their solution
method.
Deviations from this routine occurred with the 'Ten People in a
Boat", the "Beans, Toothpicks, and Cubes", and the "Kids With Beans
and Kids With Figs" problems. Since students could not be expected,
in a single class, to solve the 'Ten People in a Boat" problem and
practice the correct moves with enough accuracy to recognize
patterns in their moves, students were introduced to the problem the
week before it was to be used as an in-class problem solving activity.
Thus, any student who had not mastered the correct sequence of
moves required by the problem was given directions and an
opportunity to practice during the time regularly devoted to the
introduction to the problem.

The "Kids With Beans and Kids With Figs" problems were
designed as cooperative learning activities (Erickson, 1989) so
students did not have the ten minutes to work alone on the problem.
Instead, each student was given a clue card and the four students in
the group had to use the four clues to solve the problem. The rules for








the activity included directions that no student could show their clue
card to another group member. If someone wanted to have a
particular clue repeated, the owner of the clue was to reread it to the
group, not pass it around. Once the group found the solution they
were asked to think of other ways to find the answer. In addition to
alternative solution methods, the "Kids with Figs" problem had four
possible solutions. Instead of sharing with the class, each group
showed the researcher their method before looking for an additional
method. When they had found all the possible solution methods with
or without hints, they were given the second problem. A third
problem was planned, but no group was able to exhaust more than two
problems.
The third deviation was not scheduled prior to the study as were
the other deviations. After administering the "Beans, Toothpicks, and
Cubes" problem to the two classes in the pretest-treatment group, it
was determined that the problem was too difficult for the majority of
students to solve in more than one way during a single class period.
Therefore, for the no pretest-treatment group the problem was
introduced and practiced in class and then the students were given
four days to work on the problem outside of class before sharing
solution methods.
The eight problems chosen for this study shown in Appendix A
were selected because each was considered to be (a) motivational, (b)
problems for which several alternative solution methods existed
within the ability range of the subjects, and (c) problems that were
familiar to the researcher who had successfully used them with other








groups. The order in which the problems were administered is as
follows:
Week 1: Horses and Ducks
Introduction to Ten People in a Boat
Week 2: Ten People in a Boat
Week 3: Diagonals in a Polygon
Week 4: Dots on the Side of a Triangle
Week 5: Telephone Lines
Week 6: Triangular Numbers
Week 7: Kids With Beans and Kids With Figs
Week 8: Beans, Toothpicks, and Cubes
Students' work was collected each class period but was not
graded since students were encouraged to work together and the
groups were not necessarily the same each week. Additionally,
without grades it was hoped that students would be more free to think
about the process instead of the product.

Analysis of Creativity Data for American Students

Given students who perform at approximately the same
mathematical skill level, those who possess the higher measure of
creativity might be expected to exhibit greater problem solving
potential than those who demonstrate a less promising capacity for
creativity. In order to test this conjecture, students in the United
States sample were administered the Figural A version of the TTCT:
Thinking Creatively with Pictures. All tests were graded by one
graduate student who had previously been trained to use the TTCT.
Standardized scores for fluency, originality, elaboration, abstractness







of titles, and resistance to premature closure were averaged to obtain
the mean measure of creativity variable. A relationship between
students' measure of creativity and the number and complexity of
different methods of solution found on the marble problem was then
investigated.

Weight Assignment for Solution Methods

Each student response was coded as zero if Irrelevant
computations that did not relate to the problem were shown, the
solution was incorrect, or there was no response. Correct solution
methods were given a weighted decimal value from one to five
depending on the level of sophistication. Responses were recorded
sequentially for Part 1. No credit was given for a response that
repeated a previously used method.
To assist the researcher in assigning a sophistication value to
each solution method, a panel of five experts in the field of problem
solving in mathematics education independently ranked each method.
A copy of the letter requesting this assistance and a list of those who
responded are included in Appendix C. The weighted value of each
response represents its ranking according to its predetermined level
of sophistication. Sophistication rankings for solution methods are
based on the degree of intellectual engagement indicated and on
generalizability.
Detailed descriptions of each method of solution appear in the
results section and in Appendix D. Methods of solution fall into the
general categories of enumeration, pattern recognition, grouping, or a
formula.








The U.S.-Japan Comparisons

For the U.S.-Japan comparisons, results of 100 sixth and 100
eighth grade participating Japanese students were included in the
study. Because the Japanese educational system does not promote
ability grouping, no instructions were given to proctors concerning
the selection of classes for the survey. The two schools chosen, one
school in Hiroshima and one in Tokyo, are both public schools and
therefore represent the Japanese population more closely than would
a private school with very selective entrance requirements. Although
no classes identified as "gifted" were part of the survey, it is assumed
that students from both countries had experienced some problem
solving activities. In addition to the 217 American eighth grade
students who were included in the sample for the creativity portion of
the study, 96 eleventh grade and 94 sixth grade students were
included in the sample of U.S. students for the U.S.-Japan
comparisons. The sixth and eleventh grade students were from one
additional local middle school and one high school; both with students
of comparable backgrounds to the eighth grade American subjects
The eleventh grade students were enrolled in a trigonometry class
while the sixth grade subjects were in a regular sixth grade

heterogeneous mathematics class.
The data collection for the Japanese sample was administrated
by two colleagues of the researcher, Professor Toshiakira Fujii of the
University of Yamanashi and Professor Tadao Ishida of Hiroshima
University, both members of the U.S.-Japan Cross-national Research
(Becker, 1992). These Japanese professors and the researcher








worked together on the collaborative research project during their
visit to the United States and this researcher's visit to Japan in 1990.
As a result of the above mentioned collaborative research project when
Japanese researchers visited U.S. classrooms and participated in joint
discussions on problem solving, a communication network was
formulated and friendships established. Professors Fujii and Ishida
each collected one half of the Japanese data.
The Japanese sample included 100 sixth and 100 eighth grade
students equally divided between Tokyo and Hiroshima. Three intact
classes from each location were actually tested and papers were
chosen at random by the coder, Christina Carter, for inclusion in the
study. Mrs. Carter, personal friend of the researcher and University of
Florida graduate student in mathematics education and a resident of
Hiroshima during the data collection process, has special expertise in
Japanese education. For the past three years, Mrs. Carter has been
very involved with the Japanese elementary school that two of her
children attended and is fluent enough in Japanese to communicate
both orally and in writing.
Both the number of solution methods employed for the
nonroutine mathematical problem and the complexity of those
solution methods were compared between the U.S. and Japanese sixth
and eighth grade data. Since middle level students in Japan have
spent the equivalent of approximately two to three more years of
classroom time studying mathematics than their American
counterparts, different grade comparisons were investigated. In
particular, the sixth grade Japanese results were compared to the







eighth grade U.S. results and the eighth grade Japanese results to the
eleventh grade data from the U.S.


Summary

In the spring of 1992 data were collected from 217 American
eighth grade students to investigate solution methods for a nonroutine
mathematical problem. A relationship between a measure of creativity
as determined by the Torrance Test of Creative Thinking: Figural A,
and the number and complexity of solution methods employed was the
focus of the main study. A Solomon four-group design was used with
one prealgebra and one algebra class in each group. The two
treatment groups each received eight full sessions of problem solving
designed to elicit multiple alternative solution methods for the
treatment problems. The marble problem (Appendix B) was used as
both a pre and posttest measure.
The main study was enhanced by a cross-national comparison of
solution methods employed for the marble problem. In addition to the
marble problem results collected for the main study, 96 eleventh
grade and 94 sixth grade American students and 100 sixth and 100
eighth grade Japanese students were administered the marble
problem. Comparisons of number and complexity of solution methods
were not only made between nations but also same-grade and
different-grade comparisons were investigated.













CHAPTER IV
DATA ANALYSIS AND RESULTS OF THE STUDY
The purpose of the main study was to investigate a possible
relationship between a student's measure of creativity and both the
number and complexity of solution methods employed for a
nonroutine mathematical problem. Comparisons of solution methods
of students in the United States and those of Japanese students were
also made. This chapter begins with a review of the data collection
procedures for both the main study and the US-Japan comparisons,
followed by a report of the analyses.

Review of the Design and Data Collection Procedures

A quasi-experimental Solomon four-group design was used for
the main study involving 217 subjects. Each of the four groups
consisted of one algebra and one prealgebra class of eighth grade
students from four middle schools in two Central Florida counties.
The number of subjects assigned to pretest and/or treatment groups is
shown in Table 4-1.


TABLE 4-1
Number of Subjects in Each Group

Pretest No Pretest
Treatment 64 46

No Treatment 54 53








The protests and posttests were identical and consisted of one
nonroutine mathematical problem. The marble problem, shown in
Appendix B, was chosen because, even though the answer can be easily
obtained, it can be solved using numerous different methods that vary
in level of sophistication. This problem was also one of five problems
chosen for inclusion in an earlier study; the U.S.-Japan Cross-national
Research (Becker, 1992).

The weighted level of sophistication for each solution method
was determined by a panel of five experts in the area of problem
solving who independently ranked each solution method on a decimal
scale from one to five according to the degree of understanding
represented by the method and its generalizability. The mean of the
five rankings became the method's level of sophistication or
complexity. The solution methods generally fall into the categories of
pattern finding, grouping, and a formula. The 13 different solution
methods for Part 1 appear in Appendix D along with the
corresponding level of sophistication and a description of each
method.

To obtain a measure of creativity, the Torrance Test of Creative
Thinking: Figural A was administered to each student in the main
study. The creativity measure was computed by calculating the mean
of each student's standardized scores for fluency, originality, flexibility,
abstractness of titles, and resistance to closure.

The treatment consisted of eight class periods of problem
solving in which one problem (Appendix A) was solved during each
session using as many different solution methods as possible. Students
spent time working individually and in small groups to devise and








carry out their solution plans and then took turns presenting
representative methods to the class via overhead transparencies.
For the US-Japan comparisons results of 496 subjects were
analyzed. The number of subjects in each grade by country is shown in
Table 4-2. For the eighth grade American sample, the pretest results
of the 118 students in the pretest groups of the main study were used.
The Japanese sample consisted of 100 sixth and 100 eighth grade
students from public schools in Hiroshima and Tokyo. Data collected
in Japan were used for comparisons only since it was not possible for
this researcher to obtain a measure of creativity from the Japanese
subjects and a matching variable was not available.


TABLE 4-2
Number of Subjects in the U.S.-Japan Comparisons

Country Grade Subjects

U.S. 6 91

U.S. 8 118

U.S. 11 87

Japan 6 100

Japan 8 100


In the analyses that follows, each hypothesis is stated in null
form followed by the decision to reject or not reject and a description
of the results. An alpha level of 0.05 was used for all tests. The results
section divides the hypotheses into four parts, (a) hypotheses 1 6
were used to analyze the number of solution methods employed, (b)
hypotheses 7-12 were used to analyze the complexity of the solution








methods, (c) a repeated measure was used to test hypotheses 13-14

which addressed the changes in mean number and complexity of

solution methods over time, and (d) hypotheses 15-18 concerned the
U.S.-Japan comparisons by nation and grade for both number and
complexity of the pretest solution methods.


Results of Hypotheses Testing


Number of Solution Methods

An analysis of variance was used in analyzing the three main
effects (treatment, pretest, and creativity) on the total number of

solution methods generated on the posttest. The results shown in

Table 4-3 confirm that the overall model accounts for a significant
amount of variation in the mean number of solution methods. At least

one of the main effects was significantly related to the differences in

mean number of solution methods between the four groups.


TABLE 4-3
__ANOVA Model of Main Effects

Source df SS MS F

Between Groups 3 29.72 9.91 9.17 **

Within Groups 213 230.21 1.08

Iotal 216 259.93
p<.05
*p<.01


The first three hypotheses were used to examine each of the
three main effects. The results are shown in Table 4-4.








TABLE 4-4
Summary of ANOVA Posttest Number of Solution Methods

Source DF Type III SS F

Treatment 1 20.89 19.33 **

Pretest 1 2.79 2.58

Creativity 1 9.11 8.43 **

Error 213
*p<.05
**p<.01


Hol: There is no significant relationship between the number of

different solution methods employed for a nonroutine mathematical

problem and the treatment received.

This hypothesis was rejected. The results shown in Table 4-4
suggest that the mean number of solution methods found on the
posttest was significantly related to the treatment effect. The mean
number of solution methods employed on the posttest by the

nontreatment groups was 2.00 with a standard deviation of 1.07, while

the treatment groups generated a mean of 2.60 solution methods with

a standard deviation of 1.04.
Ho2: There is no significant relationship between the number of

different solution methods employed for a nonroutine mathematical

problem and receiving the pretest.

This hypothesis was not rejected. As shown in Table 4-4,
receiving the pretest did not significantly effect the outcome of the
posttest. The mean number of solution methods generated by each

group on the posttest is shown by group in Table 4-5.








TABLE 4-5
Mean Number of Solution Methods on Pretest and Posttest

Group n Pretest SD Posttest SD

Pretest-Treatment 64 2.05 0.84 2.61 0.99

Pretest-No Treatment 54 2.06 1.16 2.12 1.08

No Pretest-Treatment 46 2.59 1.13

No Pretest-No Treatment 53 1.87 1.06


H03: There is no significant relationship between the number of

different solution methods employed for a nonroutine mathematical

problem and the measure of creativity.

This hypothesis was rejected. Table 4-4 shows that the mean
number of solution methods found on the posttest was significantly

related to the student's measure of creativity. A test for curvilinearity

was also conducted but no such relationship was found. Table 4-6
shows the mean creativity score for each total number of solution

methods ranging from zero to five.

TABLE 4-6
Mean of Creativity Measure for Posttest Number of Solution Methods
number creativity standard
of methods mean deviation

0 100.50 19.63

1 99.02 14.82

2 99.19 14.40

3 101.36 14.30

4 105.04 13.00

5 115.33 11.85








The results of the next three hypotheses ruled out the possibility
of two-way interactions among the three main effects. None of the

three hypotheses were rejected.
H04: The relationship between the number of different solution

methods employed for a nonroutine mathematical problem and

treatment received does not differ by pretest.
Ho5: The relationship between the number of different solution

methods employed for a nonroutine mathematical problem and the

measure of creativity does not differ by pretest.
Ho6: The relationship between the number of different solution

methods employed for a nonroutine mathematical problem and
measure of creativity does not differ by treatment.


TABLE 4-7
Summary of ANOVA Posttest Number of Solution Methods
Two-Way Interactions

Source DF Type III SS F

Treatment*Pretest 1 0.31 0.29

Creativity*Pretest 1 0.67 0.62

Creativity*Treatment 1 2.60 2.41

Error 210
*p<.05
**p<.01


The mean number of solution methods generated on the posttest was
not significantly effected by the interaction of pretest with treatment,

the interaction of the measure of creativity with the experience of

taking the pretest, or the interaction of creativity and the treatment.

These results are shown in Table 4-7.










Complexity of Solution Methods

An analysis of variance was used in analyzing the three main

effects (treatment, pretest, and creativity) on the level of complexity
attained on the posttest solution methods. The results shown in Table

4-8 confirm that the overall model accounts for a significant amount of

variation in the level of complexity attained on the posttest solution

methods. At least one of the main effects was significantly related to

the differences in the mean level of complexity attained by each group

on the posttest solution methods.

TABLE 4-8
ANOVA Model of Main Effects on the Complexity of Solution Methods

Source df SS MS F

Between Groups 3 44.54 14.85 11.77 **

Within Groups 213 268.69 1.26

total 216 313.23
p<.05
*p<.01


Hypotheses 7-9 were used to examine the relationship between each

of the three main effects on the level of complexity attained on the

posttest solution methods. The results are shown in Table 4-9.
H07: There is no significant relationship between the

complexity of different solution methods employed for a nonroutine
mathematical problem and the treatment received.








TABLE 4-9
Summary of ANOVA Posttest Complexity of Solution Methods

Source DF Type III SS F

Treatment 1 33.54 26.59**

Pretest 1 8.30 6.58*

Creativity 1 4.08 3.23

Error 213_
*p<.05
** p < .01

This hypothesis was rejected. Table 4-9 indicates that the level
of complexity attained on the posttest solution methods was
significantly related to the experience of receiving the treatment. The
mean level of complexity attained by the treatment groups (on a scale
of zero to five) was 3.36 (standard deviation of 1.11), while the mean
for the nontreatment groups was 2.56 (standard deviation of 1.17).
H08: There is no significant relationship between the

complexity of different solution methods employed for a nonroutine
mathematical problem and receiving the pretest.

This hypothesis was rejected. The experience of taking the
pretest was significantly related to the level of complexity attained on

the posttest solution methods. The mean level of complexity attained
for those who took the pretest was 3.15 (standard deviation of 1.18),
while the mean of those who did not take the pretest was 2.75
(standard deviation of 1.21).
H09: There is no significant relationship between the

complexity of different solution methods employed for a nonroutine
mathematical problem and the measure of creativity.








This hypothesis was not rejected. The mean level of complexity

attained on the posttest solution methods was not significantly related

to the individual's measure of creativity.

The results of hypotheses 10-12 ruled out the possibility of two-

way interactions among the three main effects. None of these

hypotheses were rejected.
Ho10: The relationship between the complexity of different

solution methods employed for a nonroutine mathematical problem
and treatment received does not differ by pretest.
Holl: The relationship between the complexity of different

solution methods employed for a nonroutine mathematical problem

and the measure of creativity does not differ by pretest.
Ho12: The relationship between the complexity of different

solution methods employed for a nonroutine mathematical problem

and measure of creativity does not differ by treatment.


TABLE 4-10
Summary of ANOVA Posttest Complexity Level of Solution Methods
Two-Way Interactions
Source DF Type III SS F

Treatment*Pretest 1 0.18 0.14

Creativity*Pretest 1 0.02 0.02

Creativity*Treatment 1 0.61 0.48

Error 210
p < .05
** p < .01

The mean level of complexity attained on the posttest solution

methods was not significantly related to the interaction of pretest with
treatment, the interaction of the creativity measure with the








experience of taking the pretest, or the interaction of creativity and

the treatment. These results are shown in Table 4-10.


Pretest/Posttest Comparisons


Hypotheses 13-14 were used to compare the number and level of

complexity of solution methods on the pretest and the corresponding

number and level of complexity on the posttest. Both were tested

using an analysis of variance with repeated measures.
Ho13: There is no significant difference between the number of

different solution methods employed for a nonroutine mathematical

problem on the premeasure and on the postmeasure.

This hypothesis was rejected. While the mean number of

solution methods on the pretest was approximately the same for

subjects in both the treatment and nontreatment groups, when

averaged across the pretest and the posttest (F = 2.56, df =1, 116),

there was a significant interaction found between time and treatment.

The graph of the of the two pre/post lines shown in Figure 4-1

illustrates the significantly different slopes for treatment and control

groups.

treatment





no treatment


pI~t~t poattest


pretest


posttest








FIGURE 4-1
Interaction of Time and Treatment on Number of Solution Methods


The change in the number of solution methods found on the pretest
and the number generated on the posttest was different depending on
whether the student experienced the treatment (F = 4.42, df = 1,
116, p < 0.05). The treatment and control groups were almost equal
on mean number of solution methods found on the pretest, but after
the experience of the treatment, the treatment group showed
dramatic gains while the control group remained approximately the
same. Table 4-11 shows the mean number of solution methods for the
treatment groups and the nontreatment groups on both the pre and
posttest measure.

TABLE 4-11
Mean Number of Solution Methods Over Time By Treatment

Means Treatment SD Control SD

Pretest Number of Methods 2.05 0.84 2.06 1.16

Posttest Number of Methods 2.61 0.99 2.13 1.08


Ho14: There is no significant difference between the complexity

of solution methods employed for a nonroutine mathematical problem
on the premeasure and on the postmeasure.

This hypothesis was rejected. The graph of the of the two
pre/post lines shown in Figure 4-2 illustrates the absence of
significant interaction for treatment and control groups over time.



















pretest posttest

FIGURE 4-2
Time and Treatment on Complexity of Solution Methods


However, both main effects, treatment and the pretest, significantly

affected the mean level of complexity attained on posttest solution

methods. The linear model showed that the effect of the treatment

was not the same for treatment and nontreatment group (F = 20.20, df

= 1, 116, p < 0.01). While there was a significant difference within

subjects on the complexity attained on the pretest and on the posttest

(F = 5.98, df = 1, 116. p < 0.05). the significance of the interaction

found between time and treatment showed that within subjects, this

difference did not significantly depend on whether the student

received the treatment (F = 0.90, df = 1, 116). Table 4-12 shows the


TABLE 4-12
Mean Complexity Level Over Time By Treatment

Means Treatment SD No Treatment SD

Pretest Complexity Level 3.07 0.92 2.52 1.19

Posttest Complexity Level 3.51 1.12 2.71 1.10


treatment




no treatment








mean complexity level of solution methods for the treatment groups
and the nontreatment groups on both the pre and posttest measure.

US-Japan Comparisons

The last four hypotheses were tested using the pretest results of
the U.S. eighth grade students who were in one of the main study
pretest groups, a sample of U.S. sixth grade students, and the results
from 100 Japanese students in each of the grades six and eight. Since
U.S. students, who by their middle to high school grades, have spent
approximately the same amount of time studying mathematics as have
Japanese students two to three grades their juniors, comparisons
were made between both the mean number and complexity level of
solution methods for grades six and eight.
HO15: There is no significant relationship between the number of

different solution methods employed for a nonroutine mathematical
problem and grade level.
This hypothesis was not rejected (F = 1.08, df = 1, 406). There
were no significant differences when comparing all eighth grade
students to all sixth grade students. When compared by grade and by
nation, the U.S. eighth grade students found significantly more
solution methods than did the Japanese eighth grade students but not
more than either sixth grade sample. The means are shown in table
4-13.
H016: There is no significant relationship between the number of

different solution methods employed for a nonroutine mathematical
problem and nationality.








TABLE 4-13
Mean Number of Solution Methods By Grade and Nation

Country Grade n Mean SD

U.S. 6 91 2.01 0.91

U.S. 8 118 2.05 0.99

Japan 6 100 1.90 1.02

Japan 8 100 1.66 0.84


This hypothesis was rejected (F = 7.62, df = 1, 406, p < 0.01).
The U.S. sixth and eighth grade students found significantly more
solution methods than did students in the Japanese sample. The
mean number of solutions for the combined U.S. sixth and eighth
grade students was 2.03 (standard deviation of 0.96), while the
Japanese mean was 1.70 (standard deviation of 0.94). Results of
comparisons by grade and nation showed that the eighth grade U.S.
students found significantly more solutions than did the eighth grade
Japanese students, but not more than the American or Japanese sixth
grade students. The means are shown in Table 4-13.
Ho17: There is no significant relationship between the

complexity of different solution methods employed for a nonroutine
mathematical problem and grade level.
This hypothesis was not rejected (F = 1.44, df=l, 406). There
were no significant differences when comparing all eighth grade
students to all sixth grade students. Follow-up comparisons of sixth
and eighth grade students yielded no significant differences in
complexity level for either grade by nation. The means are shown in
table 4-14.









TABLE 4-14
Mean Complexity Level of Solution Methods By Grade and Nation

Country Grade n Mean SD

U.S. 6 91 2.78 0.97

U.S. 8 118 2.81 1.09

Japan 6 100 2.86 1.13

Japan 8 100 2.56 1.15


Ho18: There Is no significant relationship between the

complexity of different solution methods employed for a nonroutine
mathematical problem and nationality.

This hypothesis was not rejected (F = 0.81, df=l, 406). There
was no significant difference in the level of complexity attained on the
solution methods between the U.S. sample of students and the
Japanese sample. The means are shown in Table 4-14.


Other Results

During the coding of the marble problem results and the
treatment experience it was observed that students tended to initially
solve the marble problem with a solution method that was less
complex than subsequent methods. The conjecture was made that the
complexity level of student's solution methods tends to increase with
the generation of additional methods. To test this conjecture the
mean complexity level of the first solution method for all students who
generated at least one method on the pretest was computed. Then
the mean level was computed on the second solution method for all








students who generated at least two methods, etc. An ANOVA,
designed so that independence of groups was not violated, was used to
test for significant differences in the means for each individual

student. The results indicated that as the number of solution methods
increased, the mean level of complexity was significantly different (F
= 9.58, df = 4, 127, p < 0.01). The mean level of complexity for each
solution in the order generated on the pretest is shown in Table 4-15.
As students generated additional solution methods, the complexity of
their methods tended to increase.


TABLE 4-15
Mean Level of Solution Method Complexity in the Order Generated on
Pretest

Ordinal Position n Mean SD

1st Solution 111 2.04 0.93

2nd Solution 89 2.69 0.94

3rd Solution 32 2.85 1.08

4th Solution 8 3.10 1.16

5th Solution 2 3.73 0.04


Since this trend was discovered on the pretest, the question was
then directed to the treatment. Would this trend still hold for
students who had experienced the treatment? To answer this
question the same ANOVA was applied to the posttest results for those
students who participated in the treatment. The results again
indicated that as the number of solution methods increased, the mean
level of complexity was significantly different (F = 15.41, df = 4, 175,

p < 0.01). The mean level of complexity for each solution in the order








generated on the posttest is shown in Table 4-16. As students
generated additional solution methods, the complexity of their
methods tended to increase. The increase was more pronounced for
those who had participated in the treatment than for those in the
control group.


TABLE 4-16
Mean Level of Solution Method Complexity in the Order Generated on
Posttest
Ordinal Position n Mean SD

1st Solution 107 1.92 1.09

2nd Solution 97 2.80 1.11

3rd Solution 58 2.95 1.09

4th Solution 21 3.61 0.79

5th Solution 3 3.13 0.29


Summary

The purpose of this research was to investigate solution methods
used by students to solve a nonroutine mathematical problem. The

main study investigated a possible relationship between creativity and
both the number and complexity of solution methods employed by
eighth grade students. Additionally, comparisons of solution methods
of students in the U.S. and those of Japanese students were made.

A quasi-experimental Solomon four-group design was used for
the main study. The four groups consisted of a pretest-treatment
group, a pretest-no treatment group, a no pretest-treatment group,
and a no pretest-no treatment group. The treatment consisted of
eight class periods of problem solving designed to elicit multiple








solution methods for a single nonroutine mathematical problem. The
protests and the posttests were identical and consisted of one
nonroutine mathematical problem. Each test was evaluated for the
total number of different solution methods employed and the level of
complexity attained.
Hypotheses 1-6 were used to investigate the number of solution
methods employed in relation to three independent variables
(treatment, pretest, and creativity) and any possible two-way
interactions. Results showed that while there was no significant
pretest effect or interactions, the number of solution methods
generated on the posttest by students in the treatment groups (2.07)
was significantly greater than those who did not experience the
treatment (2.00).
The posttest number of solution methods was also positively
related to the student's creativity as measured by the Torrance Test of
Creative Thinking: Figural A. The mean creativity measure for those
students who found more than two solution methods was significantly

greater than the mean of those who found two or fewer methods.
Hypotheses 7-12 were used to investigate the level of complexity
attained in the solution methods in relation to the three independent
variables (treatment, pretest, and creativity) and any possible two-way
interactions. Results showed that there was a significant pretest and
treatment effect, but no significant relationship between a student's
measure of creativity and the level of complexity attained. There were
no two-way Interactions.
The level of complexity attained on the posttest solution
methods was significantly related to the experience of receiving the








treatment. The mean level of complexity attained by the treatment

groups (on a scale of zero to five) was 3.36, while the mean for the
nontreatment group was 2.56. There was a significant pretest effect
for the complexity level attained on the posttest. The experience of
taking the pretest significantly increased the level of complexity
attained on the posttest.

A repeated measures procedure was employed for hypotheses 13
and 14 to test the treatment effects over time on both the number and
complexity of solution methods. The change in the number of solution

methods found on the pretest and the number generated on the
posttest was different depending on whether the student experienced
the treatment. The treatment and control groups were almost equal
on mean number of solution methods found on the pretest, but after

the experience of the treatment, the treatment group showed
dramatic gains while the control group remained approximately the
same.

The results of the repeated measure for level of complexity was
quiet different. While both the treatment and the pretest significantly

affected the level of complexity attained on posttest solution methods,
the effect of the treatment was not the same for all students who
received the treatment. While there was a significant difference
within subjects on the complexity attained on the pretest and on the
posttest, the lack of significant interaction between time and

treatment showed that within subjects, this difference did not
significantly depend on whether the student received the treatment.

Hypotheses 15-18 were used to compare by grade and by nation
the number of solution methods and level of complexity attained by








students in the U.S. with students in Japan. The U.S. sixth and eighth
grade students found significantly more solution methods than did the
sixth and eighth grade Japanese students. Follow-up comparisons
revealed that the U.S. eighth grade students found significantly more
solution methods than did the Japanese eighth grade sample, but not
more than either sixth grade group. There were no significant
differences in attained level of complexity when the means were
compared by grade or by nation.

Additionally, an important finding was made concerning the
complexity of students' solution methods. A student's initial solution
method tended to be less sophisticated than the other methods
generated. As the number of solution methods increased, the mean
level of complexity tended to increase for subsequent methods. This
trend existed on the pretest and was even more pronounced on the
posttest for students in the treatment group.













CHAPTER V
CONCLUSIONS AND IMPLICATIONS

Overview of Study

There have been a number of reform movements in mathematics
education in the last 40 years but none so radical and encompassing as
the reforms outlined in NCTM's Curriculum and Evaluation Standards
for School Mathematics (1989), and Professional Standards for
Teaching Mathematics (1991). In the ideal classroom every student
is guaranteed the opportunity to construct mathematical concepts and
develop his or her power to communicate mathematically, reason

mathematically, make mathematical connections, and solve problems.
This study incorporated each of these strands by focusing on the
teaching and learning of reflective problem solving.
The purpose of this study was to investigate solution methods
employed by middle level students engaged in solving nonroutine
mathematical problems and the role of creativity during the reflection
process. The solution methods of students who participated in
problem solving sessions designed to elicit multiple solution methods
for nonroutine mathematical problems were compared to methods
employed by students who received no such deliberate practice. The
solution methods were compared by number of different methods and
attained level of complexity. Additionally, this study included an
international perspective. The solution methods of students in Japan







in grades six and eight were compared to those of American students.

Comparisons of both the number of alternative methods employed and
the attained level of complexity were made between nations and
between grades. The solution methods of Japanese students for
nonroutine mathematical problems provided for a particularly
interesting comparison because Japanese achievement scores have
received worldwide attention and because of Japanese elementary
educators' emphasis on the looking back phase of problem solving.
The opportunity to conduct cross-national problem solving research
with the Japanese provided a rare and timely opportunity to view the
teaching and learning of problem solving through a different set of
lenses.
For the main study the comparison groups were composed of
intact eighth grade algebra and prealgebra classes arranged in a quasi-
experimental, Solomon four-group design. The 217 students were all
enrolled in middle level schools in Central Florida. Each of the four
groups was composed of one algebra and one prealgebra class.
Students in the treatment groups participated in eight problem
solving sessions spanning a 10-week period. The premeasure and
postmeasure consisted of a single nonroutine mathematical problem
for which each student was to generate as many solution methods as
possible in a given time frame. The problem was one of five included
in a previous international study, the U.S.-Japan Cross-national
Research (Becker, 1992).
The format of the problem solving treatment sessions was based
on the constructivist theory that supports the claim ". .. students learn

mathematics well only when they construct their own mathematical








understanding" (National Research Council, 1989, p. 58). In the
constructive classroom the teacher does not attempt to dispense
knowledge but instead acts as a model and facilitator. The key
element in the problem solving sessions was the significance awarded
time spent on reflection. Thus, attention was shifted from the
product to the process. Since some students generate solution
methods that can be classified as innovative when compared to
methods employed by their peers, the role of creativity during
problem solving was addressed. Each student in the main study was
assigned a measure of creativity as determined by the Torrance Test of
Creative Thinking: Figural A. Standardized scores for each of the five
categories (fluency, flexibility, originality, elaboration, and resistance
to premature closure) were used to compute a mean standardized
creativity score.
The 13 different solution methods generated by the survey
sample were ranked according to sophistication or complexity. The
rank, based on the level of intellect engaged in the solution method
and the generalizability of the method, represents the mean of
rankings offered by five mathematics educators considered experts in
the field of problem solving.
For the U.S.-Japan comparisons premeasure results of 100
eighth and 100 sixth grade students were chosen at random from the
intact classes who participated in the study. Each of the schools
involved, one in Tokyo and one in Hiroshima, are public schools,
therefore they represented the Japanese population more closely than
would a private school with very selective entrance requirements.








Discussion of Results

The major findings that resulted from the data analyses were as
follows:
1. Both the number of solution methods students were able to
generate for a single nonroutine mathematical problem and the level
of complexity of those methods were significantly increased when the
eighth grade students in the main study were given the opportunity to
engage in reflective problem solving activities.
The constructivists agree that students need opportunities to
construct their own knowledge. The theme of the curriculum and
teaching Standards is that students should be involved in actively
"doing" mathematics. As a result of incorporating these standards in
the treatment, students were not only able to generate a greater
number of alternative solution methods but were also able to move to a
more generalizable, thus a more sophisticated, method. Even though
the treatment problems did not require a formula, some students
progressed to that high form of generalization during the reflection
process. The number of students who could generate a formula
continued to increase throughout the treatment period. This increase
could be attributed to a deeper understanding of the problem which
developed during reflection as well as to feedback. At the end of each
problem solving session students shared individually or in groups the
full range of solution methods generated during the session. This
communication time was intentionally arranged so that methods would
be shared by order of increasing complexity. Thus students were able
to witness the extension of many solution methods to the most general







case; a formula. While the percentage of students who generated a
formula on the pretest and posttest remained at about two percent for
the nontreatment group, the percentage of students in the treatment
group went from three percent on the pretest to seventeen percent
on the posttest.
However, once a student obtained a formula, it was often difficult
for that student to generate less sophisticated solution methods.
During the treatment it was observed that the algebra students were
more likely to produce a formula early in the problem solving process,
then loose their flexibility to a mind set. The prealgebra students, on
the other hand, remained flexible for a longer period and were actually
able to generate a greater number of less complex solution methods.
For example, this was especially evident during the "horses and ducks
" problem (Appendix A) which states that, among the farmers

collection of horses and ducks there are 9 heads and 26 feet.
Students were to find the number of horses and ducks in the
collection. While the algebra students who had just completed a unit
on simultaneous equations had difficulty finding a method other than
applying their newly acquired skill, the prealgebra students who had
not been introduced to this algorithm, generated many creative
methods. To the students a formula represented the ultimate solution
method. Thus, after finding a formula, they seemed to lose their
flexibility and motivation for continuing the search. The algebra
students were also more reluctant to recognize unsophisticated
methods such as counting or drawing a picture as bonafide. Those
who used the less complex solution methods often wished to remain
anonymous and only began to look for unsophisticated methods when







they realized that such methods would "count." While the percentage
of students in the control groups who used counting as a method of
solution remained at approximately 48% on both the pre and posttest
measures, the percentage of students in the treatment groups who
used counting as a method of solution went from approximately 30%
on the pretest to 68% on the posttest. Algebra students in search of a
formula to represent a problem could gain valuable insights to the
problem by constructing a less sophisticated solution method which
might then provide the framework for the desired formula. Results
from this research suggested that even when algebra students are
unable to produce a formula, they do not systematically look for or
value less sophisticated solution methods. Results also indicated that
students can be trained to employ alternative solution methods as a
means for producing a more generalizable solution.
2. Students' initial solution methods tended to be of low
complexity with the level of complexity increasing with each new
method generated.
Although the task of searching for alternative solution methods
for a single problem is not a new suggestion (Brownell, 1942; Polya,
1957), justification for investing the time required for such an activity
is not apparent in many American classrooms where the objective is
"answer-getting." Results of this study indicated that with the

exception of a few students of algebra, when students are asked to find
the answer to a nonroutine mathematical problem, initially they tend
to employ a solution method rooted in a basic level of understanding of
the problem. When required to search for additional methods, these
students are given the opportunity to increase their understanding of







the problem's mathematical principles. A deeper understanding of
the problem could promote the generation of more complex and thus
more generalizable solution methods.
Results of this study showed that for the marble problem there
was an upward trend in level of complexity with each additional
method generated. This phenomenon occurred on the pretest and
was even more pronounced on the posttest. Additionally it was
observed during the treatment sessions that when students were given
problems for which there exists a variety of solution methods within
their zone of proximal development, they tended to initially employ
solution methods that only required a basic understanding of the
problem's mathematical structure. While probing for additional
methods students deepened their understanding of the problem and
constructed more complex solution methods.
3. The number of solution methods students were able to
generate for a single nonroutine mathematical problem was positively
related to their measure of creativity but there was no significant
relationship between attained level of complexity and measure of
creativity.
With a shift from the procedural/rote learning model of
mathematics education to the constructive model, there is increased
opportunity to exercise mathematical creativity. Krutetskit (1976)
insists that students must not only master mathematical skills but
must also demonstrate mathematical creativity. While Krutetskii tends
to associate mathematical creativity with mathematical giftedness, the
challenge put forth in today's classroom is that every student be given
the opportunity to exercise his or her mathematical creativity.








Research conducted by Torrance (1973) demonstrated that creativity
can be improved through training.
Since younger students have been observed inventing more
sophisticated methods of counting on their own (Groen & Resnick,
1977), older creative students might be expected to also invent more
sophisticated methods of solving problems. While creativity was
positively related to the number of solution methods students were
able to generate and the level of complexity tended to increase with
each additional method, for this study there was no significant
relationship between creativity and attained level of complexity. Since
creativity allowed students the flexibility necessary to approach
problems from different perspectives, why did creativity not account
for a significant amount of the increase in complexity?
The attained level of complexity in students' solution methods
may be effected more by their opportunity to engage in reflective
problem solving than their measure of creativity. This claim was
supported by the fact that the level of complexity attained on the
posttest was significantly effected by whether the student took the
pretest. The increase in level of complexity that resulted from the
pretest experience possibly masked differences that may have been
related to creativity. It may also indicate that no matter how creative,
the sophistication of students' solution methods Is limited by their
level of mathematical expertise. Creativity alone cannot substitute for
sound mathematical principles and the opportunity to engage in
reflective problem solving.
4. When the number of solution methods generated by students
in Japan and America were compared on the pretest, American eighth








grade students were able to produce a greater number of alternative
methods than either the Japanese sixth or eighth grade students or
the American sixth grade sample.
5. Although the American eighth grade students were able to
generate more alternative solution methods than their Japanese
counterparts, there were no differences between the level of
complexity attained in the methods.
There was no significant difference in the level of complexity attained
by any sixth or eighth grade group in the entire study sample.

Implications

This research adds support for the suggestions outlined in the
curriculum and evaluation and the teaching Standards; suggestions
that form the basis for the most aggressive reform movement in the
history of mathematics education in the United States. Research on
teaching and research on learning were brought together as a single
focus with ramifications for practitioners at all levels and researchers.
This research demonstrated how the role of the teacher must
change in order for the students to "do mathematics instead of having
it done to them." The format of each lesson allowed students the
opportunity to reason mathematically, to make mathematical
connections, to communicate mathematically, and to reflect on their
problem solving methods. The teacher whose only method of
instruction involves an attempt to dispense knowledge in a direct
teaching mode is not allowing students the opportunity to construct
their own meaningful mathematics. The advances of technology and
the rapid growth of information awaiting students who will be entering







the work force of the twenty-first century make it imperative that our
students be equipped with more than a vast array of computational
skills.
One of the most important tasks for teachers in this role of
model, guide, and coach is to provide good problems. Good
nonroutine mathematical problems are those that can be solved using
many alternative methods, at least one of which is within each
student's zone of proximal development. Why should a student who
has successfully solved a problem be prompted to formulate alternative
methods? Results of this research suggested that generally a student's
initial solution method was of low complexity. The complexity level
tended to increase as additional methods were generated. Therefore,
each student can begin solving the problem at his or her individual
level of understanding and then progress to more complex solution
methods by constructing his or her own mathematical meaning.
In addition to becoming a better problem solver, each student
must be given the opportunity to develop his or her creative potential.
Like any other talent, creativity needs to be nurtured. In a long-range
study conducted by Torrance (1972), creative achievement differences
between the more creative and less creative subjects tended to widen
as time elapsed. In an era when employers are placing an increased
emphasis on creative thinking and ideas are, in many cases, as
important as technical skills, students must be offered a curriculum
designed to exercise their creative potential. Problem solving
provides a challenging outlet for creative construction of mathematical
principles. Asking students to generate multiple solution methods for







good problems allows them to use their creativity to probe for a
deeper understanding of the problem's mathematical structure.

Headlines have painted a discouraging picture of mathematics
education in the United States. However, it is important that
educators not make hasty decisions in response to these reports. In
his testimony before the U.S. House of Representatives, Huelskamp
(1993), for example, warned the Committee on Education and Labor
about the importance of decisions based on sound data. His report
concluded that "the low opinion educators hold of themselves and the
poor public perception of teachers are based on misinterpretations of
simplistic data, such as average SAT scores and international
comparisons. This unfortunate cycle of low self-esteem, followed by
unfounded criticism from the public, raises the specter of a downward
spiral in future educational quality" (p. 720). Berliner (1993) has also
been concerned that administrative decisions, based on the negative
publicity of the failures of American schools, are potentially dangerous.
His evaluation of our educational system sheds a much brighter light
with claims that the American system has logged remarkable
successes for many of our students and parents. While he is able to
defend against many of the unfair comparisons that have spawned an
attitude of panic, he also concludes that our students are capable of
learning more mathematics at an earlier age. If opportunities for
students to exercise creative problem solving and critical thinking
talents are being postponed until the skills deemed necessary for such
activities are internalized, then valuable time for construction of
meaningful mathematics is possibly being exchanged for fragmented
memorization of rules and computational skills; computation that








could often be accomplished with a simple calculator. With
differences that exist in individual creative talent widening as children
mature (Torrance, 1972), it is imperative that each student be given
opportunity to become a creative problem solver. This study provides
evidence suggesting that when American students are given the
opportunity to unleash their creative talents in reflective problem
solving activities, they are able to construct impressive mathematical
understanding.

Suggestions For Future Research

This study not only provided valuable insights about reflective
problem solving, it also generated many questions for further research.
Since the U.S. sample included only algebra and prealgebra students,
the study could be repeated with heterogeneous classes and with
classes at different grade levels. Qualitative studies that focus on
student-student, student-teacher interactions and individual
constructions would provide additional clues about how sessions
should be structured to maximize creative problem solving behaviors.
Although the pre and postmeasure for this research included
sections which extended the marble problem to the sixteenth and one
hundredth place, those results were not included in the analyses.
Questions remain about how students generalize solutions. The
treatment problems began with the simplest case and progressed to
the more general. When students are directed initially to the general
case do they loose the flexibility to generate alternative solution
methods? What long range results might we expect if algebra students
were systematically trained to solve a simpler case before attempting








to find a formula. Would the outcomes be the same for the more
capable algebra students and the less able learner?
What are the attitudes of students who are required to exercise
their creative talents? Students in this study were excited about their
problem solving sessions. It was difficult to sort the origins of that
attitude but it seemed to be a combination of many factors. Middle
level students, characterized by their energy and quick minds, usually
welcome a change in their routine no matter how rewarding that
routine might be. The subjects in this study felt that our sessions
were something other than a mathematics class. This was probably
due in part to the fact that there were no grades assigned but they
were also aware of the absence of their mathematics book and
worksheets.
When the creative potential of students is unleashed it becomes
the teacher's responsibility to channel the talents in the proper
direction. The students needed little encouragement to begin their
search for solution methods but required coaching to sustain the
effort. Their attitudes about problem solving showed marked
Improvement over the course of the treatment with the exception of
the sharing sessions. While they were able to realize the importance
of sharing methods, they too often found that time boring. They
competed for the opportunity to share their own methods but were
impatient when it was their classmates' opportunity. This may have
been due to the change of pace that occurred during sharing.
Students often had difficulty expressing their findings. They did not
have the rhetorical skills or the mathematical vocabulary necessary to
translate their methods into coherent explanations. Researchers








might further explore methods for training students to articulate their

mathematical ideas.
There continues to be a need for joint international research.
The goal should not be to model ourselves after someone else or to
impose our system on another culture. Instead, cooperative efforts
establish "what is" so that each country can blend a workable model for
"what can be." With each possibility there are informed choices to be

made. Cooperative research must remain open and honest.
The United States and Japan should continue their cooperative
research in mathematics education. Each country has much to learn
from each other's successes and failures. Duke (1986) believes that
the economic growth in Japan has depended extensively on the role of
the school. He suggests that Japan could offer the U.S. lessons about
perseverance, high expectations for all students, and commitment to
the group without forfeiture of individual creativity or freedoms. For
lessons Japan might learn from the U.S., Duke points to our class size,
diverse teaching methods, flexibility, and alternative evaluation
methods.

Before we can incorporate lessons from Japan or any other
country, researchers must carefully analyze what changes are feasible,
make informed predictions of the expected results, and construct

workable plans of action. The Standards have built our foundation but
there are many unanswered questions for the researcher.


Limitations

The use of intact classrooms was a limitation in this study. Even
thought the students were from similar socioeconomic-economic







backgrounds and shared similar mathematics curriculums and skill
levels, there were no common standardized tests that could be used
for matching students on their mathematics proficiency. A better
cross-section of students on SES and mathematical ability would have
increased validity and reliability for this research.
Time was also a limitation in this study. While each treatment
session was awarded an entire period of instruction, time for students
to think about the problem and incubate their ideas was limited. How
might the results have been different if students were introduced to a
nonroutine problem and then encouraged to think about it over a
longer period? Illumination often comes after sleep or at unexpected
moments. Creative thinking cannot always be scheduled.
The time for treatment was also limited. Ideally, students
should be creatively challenged on a daily basis and encouraged to
regularly engage in reflective problem solving. While significant
differences resulted after only eight problem solving sessions, the
results were not intended to imply that problem solving should be
scheduled for a specific time frame. The hypotheses tested in this
research could be the basis for a longitudinal study.
Although interesting, results of the international comparisons in
this study were limited in statistical merit since there was no way to
match subjects. Effort was made to choose students who could be
compared fairly but there was no instrument to determine the degree
of compatibility. No measure of the Japanese student's creative talent
was available. The international component of this research was only
possible because of the communication and cooperation of those who
assisted from Japan. The lack of controls compounded by distance







and language barriers limited the validity of the cross-national
findings.


Conclusions

The purpose of this research was to investigate methods
employed by middle level students in the solution of a nonroutine
mathematical problem. The number of alternative methods generated
and the level of complexity attained was compared between students
who participated in a treatment and those who received no such
deliberate practice. Since some students are able to generate solution
methods that are unusual when compared to those of their peers, the
role of creativity was investigated. An international perspective was
added to this research with the inclusion of data from students in
Japan.
During the eight treatment sessions designed to encourage the
generation of multiple solution methods, students not only
significantly increased the number of alternative solutions they were
able to generate, but also the level of complexity in their solution
methods. While a student's measure of creativity was positively related
to the number of methods generated, a relationship between attained
level of complexity and creativity could not be established. Complexity
level was significantly related to the experience of taking the pretest.
With the exception of a few algebra students who often began
their solution search with an attempt to set up a formula, a student's
initial solution method tended to be less complex than their
subsequent methods. Thus, when students do not reflect on a
problem with the intention of finding alternative solution methods,







their single solution method likely represents only a basic
understanding of the problem's mathematical principles. The level of
understanding represented and the generalizability of the method
tended to increase with each new method generated. This
phenomenon was established on the pretest and was even more
evident on the posttest results.
Even though middle level students in Japan have spent about the
same amount of time studying mathematics as American students two
to three grades their senior, the results of this research did not
support the hypothesis that Japanese sixth grade students and
American eighth grade students demonstrate approximately the same
level of problem solving ability. American eleventh grade students
found significantly more solution methods and attained a higher level
of complexity in their methods than did any other group in the study.
At a time when problems of American mathematics education
are a frequent theme of the headlines, we need to evaluate what is
positive about our system and capitalize on our discoveries. This
research reinforced the belief that students are capable of
constructing their own mathematical meanings. Their creativity
allowed them the flexibility to consider problems from a variety of
perspectives and generate more generalizable solution methods
through a deeper understanding of the problem. While creativity
cannot replace sound mathematical principles, it can be harnessed to
personalize mathematics comprehension.