INFORMATION TO USERS
This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted.
The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction.
1.Thc sign or "target” for pages apparently lacking from the document photographed is "Missing Page(s)”. If it was possible to obtain the missing pagc(s) or section, they arc spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure you of complete continuity.
2. When an image on the film is obliterated with a round black mark it is an indication that the film inspector noticed cither blurred copy because of movement during exposure, or duplicate copy. Unless we meant to delete copyrighted materials that should not have been filmed, you will find a good image of the page in the adjacent frame. If copyrighted materials were deleted you will find a target note listing the pages in the adjacent frame.
3. When a map, drawing or chart, etc., is part of the material being photo graphed the photographer has followed a definite method in "sectioning" the material. It is customary to begin filming at the upper left hand corner of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again-beginning below the first row and continuing on until complete.
4. For any illustrations that cannot be reproduced satisfactorily by xerography, photographic prints can be purchased at additional cost and tipped into your xerographic copy. Requests can be made to our Dissertations Customer Services Department.
5. Some pages in any document may have indistinct print. In all cases we have filmed the best available copy.
University Microfilms international 300 N, ZEEB RD.. ANN ARBOR. Ml 4B106 8121772
B r ew er, K a t h l e e n * Ho f f m a n
A STUDY OF THE EFFECT OF PROBLEM SOLVING INSTRUCTION ON THE STUDENTS ABILITY TO SOLVE MATHEMATICAL VERBAL PROBLEMS
The Ohio State University PH.D. 1981
University Microfilms International 300 X. Zceb Road, Ann Atbor, M l 48106
Copyright 1981 by Brewer, Kathleen Hoffman All Rights Reserved PLEASE NOTE:
In alt cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark V ...
1. Glossy photographs or pages______
2. Colored illustrations, paper or print_____
3. Photographs with dark background_____
4. Illustrations are poor copy_____
5. Pages with black marks, not original copy_____ :
6. Print shows through as there Is text on both sides of page_____
7. Indistinct, broken or small print on several paqesVt
8. Print exceeds margin requirements______
9. Tightty bound copy with print lost In spine_____
10. Computer printout pages with Indistinct print_____
11. Page(s)______lacking when material received, and not available from school or author.
12. Page(s)______seem to be missing in numbering only as text follows.
13. Two pages numbered______. Text follows.
14. Curling and wrinkled pages______
15. Other______
University Microfilms International A STUDY OF THE EFFECT OF PROBLEM
SOLVING INSTRUCTION ON THE STUDENT'S ABILITY TO SOLVE
MATHEMATICAL VERbIl PROBLEMS
Dissertation Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the j Graduate School of The Ohio State University
by
Kathleen Hoffman Brewer, B.S., II.A.
•kicicfe
The Ohio State University
1981
Reading Committee : Approved by Dr. Lorren L. Stull Dr. C. Ray Williams Dr. Suzanne Damarin Dr. James Kerber * AdviseAdviser Early and Middle Childhood Education ACKNOWLEDGMENTS
X would like to express my sincere appreciation to
my adviser, Dr. Lorren L. Stull for his encouragement and
guidance In the planning and execution of this investi gation.
I am also grateful to Dr. C. Ray Williams who was
instrumental in my decision to pursue graduate studies and who encouraged and supported me every step of the way.
In addition, I am indebted to Dr. Suzanne Damarln for her helpful suggestions and guidance during the plan ning, data collection and composition stages of the re search. Her professional assistance was Invaluable.
Special mention must also be given to the principal, teachers and students of the participating school in this investigation. Their cooperation made this research pro ject both possible and an enjoyable experience.
I would also like to recognize the untiring efforts of my sister. Sister Margaret Hoffman, O.S.F., who spent hours critiquing, questioning and proofreading this dis sertation.
A final word of thanks to my husband, John, and my entire family for their loving support and understanding. ii VITA
March 4, 1948 ...... Born - Columbus, Ohio
1970...... B.S. The Ohio State Univer sity, Columbus, Ohio
1970-1978 ...... Elementary Teacher, Columbus, Ohio Public Schools
1978...... 11.A. The Ohio State Univer sity, Columbus, Ohio
1978-1980 ...... Teaching Associate, Early and Middle Childhood Education, The Ohio State University, Columbus, Ohio
1980-1981 ...... Mathematics Specialist Columbus Public Schools
FIELDS OF STUDY
Major Field: Early and Middle Childhood Education Studies in Early and Middle Childhood Education Dr. Lorren L. Stull, Dr. C. Ray Williams
Studies in Mathematics Education Dr. Suzanne Damarin
iii TABLE OF CONTENTS Page
ACKNOWLEDGMENTS...... ii
VITA...... Ill
LIST OF TABLES...... vi
CHAPTER
I. Introduction, Background and Statement of the Problem...... I
Introduction and Background...... 1 Statement of the Problem...... 11
II. Review of Research ...... 13
Theoretical Framework ...... 14 Heuristic Teaching...... 20 Characteristics Related to Students' Problem Solving Performance...... 26
III. Methodology...... 31
Subjects...... 31 Classroom Procedures...... 33 Description of Treatments ...... 34 Instruments...... 41 Analysis of the Data...... 46 Summary of the Two Methods of Instruction . . 48 Summary of Teacher Behavior in Experimental Situations...... 51 Schedule for the Study...... 56
IV. Analysis of the D a t a ...... 58
Statistical Findings for the Interviews . . . 58 Statistical Findings for the Written Test . . 76
APPENDICES
A. Procedure for Assignimg Groups ...... 103 Seating Chart ...... 106 iv B. A Sample of the Questions Posed in Treatment A. . 107
Examples of Problems Used Throughout the Study...... 110
C. Scoring of Interview Problems...... 113
Interview Checklist ...... 117 Sample Interview Transcripts...... 118 Interscorer Reliability Chart ...... 125
D. Iowa Problem Solving Tests and the Statistical Data for Each Test...... 127
BIBLIOGRAPHY...... 141
v LIST OF TABLES Table Page
1 Descriptive Statistics of Pre-Interview Scores and Pcfst-Interview Scores...... 60
2 Analysis of Covariance Table for Total Post-Interview Scores...... 62
3 Partial Sum of Squares for ANCOVA of Total Post-Interview Scores...... 62
4 Cell Means (Cell Sizes) of Total Pre-Interview Scores...... 63
5 Cell Means (Cell Sizes) of Total Post-Interview Scores...... 63
6 Analysis of Covariance Table of Post-Interview Scores on Criterion 1, "Understanding the Problem"......
7 Partial Sum of Squares for ANCOVA of Post- InterView Scores on Criterion 1, "Under standing the Problem"......
8 Cell Means (Cell Sizes) for Pre-Interview Scores on Criterion 1, "Understanding the Problem"......
9 Cell Means (Cell Sizes) for Post-Interview Scores on Criterion 1, "Understanding the Problem"......
10 Analysis of Covariance Table of Post-Interview Scores on Criterion 2, "Devising a Plan". . . . . 68
11 Partial Sum of Squares for ANCOVA of Fost- Interview Scores on Criterion 2, "Devising A Plan...... 68
12 Cell Means (Cell Sizes) for Pre-Interview Scores on Criterion 2, "Devising a Plan". . . . . 69 vi 13 Cell Means (Cell Sizes) for Post-Interview Scores on Criterion 2, "Devising a Plan". . . . 69
14 Analysis of Covariance Table of Post- InterV.iew Scores on Criterion 3, "Carrying Out the P l a n " ...... 71
15 Partial Sum of Squares of ANCOVA of Post- Xnterview Scores on Criterion 3, "Carrying Out the Plan...... 71
16 Cell Means (Cell Sizes) for Pre-Interview Scores on Criterion 3, "Carrying Out the Plan...... 72
17 Cell Means (Cell Sizes) for Post-Interview Scores on Criterion 3, "Carrying Out the Plan...... 72
18 Analysis of Covariance Table of Post- Interview Scores on Criterion 4, "Looking Back"...... 74
19 Partial Sum of Squares for ANCOVA of Post- Interview Scores on Criterion 4, "Looking Back"...... 74
20 Cell Means (Cell Sizes) for Pre-Interview Scores on Criterion 4, "Looking Back"...... 75
21 Cell Means (Cell Sizes) for Post-Interview Scores on Criterion 4, "Looking Back"...... 75
22 Descriptive Statistics of Written Pretest and Posttest Scores...... 77
23 Analysis of Covariance Table of the Overall Posttest Scores on the Written Test...... 79
24 Partial Sum of Squares of ANCOVA of the Overall Total Posttest Scores...... 79
25 Cell Means (Cell Sizes) for the Total Pretest
Scores on the Written Tests4...... 80 26 Cell Means (Cell Sizes) for the Total Posttest Scores on the Written Tests...... 80
27 Analysis of Covariance Table of Posttest Scores on Criterion 1, "Understanding the Problem". , . 82 vii 28 Partial Sum of Squares for ANCOVA of Posttest Scores on Criterion 1, "Understanding the Problem"...... 82 ♦# 29 Cell Means (Cell Sizes) for Pretest Scores on Criterion 1, "Understanding the Problem" . 83
30 Cell Means (Cell Sizes) for Posttest Scores on Criterion 1, "Understanding the Problem" . 83
31 Analysis of Covariance Table of Posttest Scores on Criterion 3, "Carrying Out the Plan"...... 85
32 Partial Sum of Squares for ANCOVA of Posttest Scores on Criterion 3, "Carrying Oiit the Plan"...... 85
33 Cell Means (Cell Sizes) for Pretest Scores on Criterion 3, "Carrying Out the Plan". . . . 86
34 Cell Means (Cell Sizes) for Posttest Scores on Criterion 3, "Carrying Out the Plan". . . . 86
35 Analysis of Covariance Table of Posttest Scores on Criterion 4, "Looking Back"...... 88
36 Partial Sum of Squares for ANCOVA of Post test Scores on Criterion 4, "Looking Back". , 83
37 Cell Means (Cell Sizes) for Pretest Scores on Criterion 4, "Looking Back"...... 89
38 Cell Means (Cell Sizes) for Posttest Scores on Criterion 4, "Looking Back"...... 89
viii CHAPTER I
INTRODUCTION, BACKGROUND AND
STATEMENT OF THE PROBLEM
Introduction and Background
The expression "problem solving" occurs in various professions and disciplines. It is a comprehensive term that has different meanings for many people. For example, !i when the researcher approached a principal about conduct ing a study concerning problem solving in his school, the principal immediately interjected, "Oh, I hope you are going to work with our fourth grade children for they have
j lots of them." The experimenter had in mind mathematical,
i verbal problems but the principal's misconception was a natural one. It speaks to the broader, long-range ambi- | tion of all educators to develop students who have the
! ability to solve problems of everyday living whether they are social, mathematical or economic. 9 ! This aim has been approached theoretically by many,
j • * including Newell and Simon who define a problem as a situa- j tion in which a person wants something and does not know immediately what series of actions he must perform to se cure it. Bourne, Ekstrand and Doraipowski (1971) see a 1 2 problem situation as one in which a person (1) is trying
to attain some goal; (2) his initial attempts fail to
accomplish this end, and (3) at least two, and commonly
a large number of alternative courses of action are pos
sible. According to the eminent George Polya, to have a
problem means: to search consciously for some action
appropriate to attain a clearly conceived, but not imme
diately attainable, aim. Life, regardless of one's age or financial status, presents dilemmas which require the
skills of problem solving. While these challenges create anxiety and frustration, they can also be a source of
satisfaction and mental stimulation.
Concentrating on these latter aspects, Dewey (1933),
Polya (1957) and others view problem solving as a teach able process consisting of general techniques that can be applied to a large variety of dissimilar problems, problem setting and subject matter. From this perspective problem solving becomes an activity wherein the focus shifts to the thinking, techniques and processes that one goes through to resolve a problem situation and produce an answer or solution. Attempts are made to develop problem solving behaviors such as conjecturing, predicting and drawing conclusions. In this researcher's opinion these behaviors, as well as others such as divergent thinking, anticipation of problems, resourcefulness, etc. must be encouraged if educators hope to nurture creative, self-
directed citizens capable of dealing with the growing
challenges of the future. This research project, however,
deals with only one phase of the described long-range goal;
for the purpose of this study, the term "problem solving" will denote the methods, procedures, strategies and heuristics that students use in solving verbally stated mathematical problems.
Computational proficiency, numerical concept devel opment and applied problem solving skills have long been recognized as the basic goals of elementary mathematics programs. Although mentioned in most curriculum descrip tions, problem solving has seldom been pursued with the same immediacy or emphasis as the other two goals. Lester, in agreement with numerous other educators, cites three factors that contribute to this situation. They are:
1) Problem solving is the most complex of all
intellectual activities and is consequently
the most difficult intellectual ability to
develop;
2) Elementary school mathematics text books
typically are deleterious rather than facil-
itative in developing problem solving skills
and processes in children; and
3) Elementary school teachers do not view problem solving as a key feature of their mathematics
programs.
Each of these statements is worthy of further examin ation. As human endeavors go, it is a complex task to' help someone else become a better problem solver. Yet teachers, parents and even children routinely engage in this task. It is still largely a mystery why certain ef forts with certain learners seem to produce lesser or greater results. Attempts to describe why or how a person solves a mathematical problem have resulted in rather shallow, incomplete conclusions (Hatfield, 1978). Even the most successful problem solvers have difficulty in identifying why they are successful, and even the best mathematics teachers are hard pressed to pinpoint what it is that causes their success ( Lester, 1978).
The second obstacle that stands in the way of promot ing problem solving in the elementary classroom is the
"deleterious" textbook. The greatest single influence on school mathematics instruction, with the possible excep tion of the teacher, is the mathematics textbook. Many mathe matics educators are very critical of the textbook treatment of problem solving. An overwhelming majority of the acti vities presented as problems in elementary mathematics texts, are actually little more than exercises designed for computational practice. Some texts even use headings such as "Some problems to test your multiplication skills".
Further, Suzanne Damarin points out the need for textbook
authors and teachers to refer to computational exercises
as "practice" or "exercises" rather than "problems".
Otherwise, Damarin notes, many children expect to solve
problems with the same mechanical efficiency and speed
that they associate with computations, and often feel de
feated when they cannot. Used accurately the word "problem"
refers to a situation in which previous experiences, know
ledge and intuition must be coordinated in an effort to
determine an outcome. There are word problems in current
textbooks, hundreds in fact, but virtually all of those
hundreds of word problems fit the Polya characterization of”one-rule-under-your-nose" (Polya, 1966). That is, in such problems students are explicitly directed to a single pre
scribed method.
A second criticism is that textbooks do not include
enough situations which involve real-world applications of mathematics.
If applied problem solving is our concern, such evidence as we have from assessments of the net effects of school instruction indicates that we are anything but effec tive. That isn't so very surprising since real problems with real data essentially do not exist in the textbooks that dominate school mathematics instruction. Hence, most children simply do not encounter applied problem solving in their school work and specific instruction in problem solving of any sort is probably rare(Bell, 1979, p.11). 6
(Note: Applied problem solving, according to Polya, involves "problems which might really occur in a sane and reasonable life.)
Bell analyzed word problems in a widely used K-6 textbook
series with respect to realism, among other aspects. His results indicate a decided lack of integration of the real world. Since such books largely determine the curriculum
in most classrooms, it is not very likely that applied
problem solving is taught in a meaningful way.
If, however, one considers the teacher the primary influence on school instruction rather than the textbook, another concern arises. Most elementary school teachers perceive mathematics to be a static and closed field of study. To them mathematics is more mechanics than ideas and involves very little independent or original thought.
They do not view problem solving as an important aspect of the mathematics curriculum. This attitude may be attributed to their own mathematical experiences; problem solving, more than likely, was not given priority status when they studied mathematics. As a result, many teachers do not feel confident as problem solvers themselves and are hesitant to risk failure in the classroom.
In,spite of this concern in regard to the teacher's role in promoting problem solving, however, mathematics educators seem to be united theoretically on the topic of problem solving and its place in the curriculum. The 7 National Council of Supervisors of Mathematics published a position paper on Basic Mathematical Skills (1976) in which they stated, "Learning to solve problems is the principal reason for studying mathematics." The National
Council of Teachers of Mathematics opens its recently published An Agenda for Action (1980) with the statement,
"The development of problem solving ability should direct the efforts of mathematical education through the next decade." This positive attitude is encouraging, however not unique*for in 1935 William Brownell stressed the im portance of providing "real and described quantitative situations" in mathematics instruction.
"...the ultimate purpose of arithmetic instruction is the development of the ability to think in quantitative situa tions. The word 'think' is used advised ly; the ability merely to perform certain operations mechanically and automatically is not enough. Children must be able to analyze real and described quantitative situations." (Brownell, 1935, p.28).
Also, reports from the Joint Commission of the Mathe matics Association of America and the National Council of
Teachers of Mathematics (1940), the Cambridge Conference on School Mathematics (1963) and the Snowmass Conference on the K-12 Mathematics Curriculum (1973) all emphasized the importance of problem solving. While mathematics educators have emphasized the need to teach problem solving in the elementary schools, research has provided the teacher with 8 very few guidelines for appropriate instructional tech niques. Bell (1979) found "the sheer amount of literature on problem solving to be nearly overwhelming, yet the yield for one interested in school Instruction is disap pointingly small."
Some suggestions for instructional techniques, how ever, have been offered. Henderson and Pingry (1953) stated:
From what we know about learning, there is only one way students can learn to solve problems...by solving problems and studying the process...Unless stu dents study the process of solving prob lems as an end in itself there is scant likelihood that they will learn the generalizations which will enable them to transfer their ability to new prob lems as they arise.
But perhaps the most influential educator on the topic of problem solving and the person that has been most attentive to the needs of the classroom teacher is George
Polya. Polya's book How to Solve It was published in 1945.
That and his subsequent work laid the foundations for the study of general strategies for problem solving in mathe matics. An article entitled "On Solving Mathematical Prob lems in High School" by Polya appeared in the latest
National Council of Teachers of Mathematics Yearbook (1980) with an editor's note that reads: 9
This article, although originally presented in the November 1949 issue of the California Mathematics Council Bulletin offers some thoughts about problem solving that are as current today as they must have been avant- garde then. It should be read by all teachers of mathematics, not just those who are teach ing high school mathematics.
In the article Polya points out that "if the teacher helps
his students just enough and unobtrusively, leaving them
some independence or at least some illusion of indepen
dence, they may experience the tension and enjoy the triumph
of discovery. Such experience may contribute decisively
to the mental development of the students." To Polya,
teaching problem solving involves considerable experience
in solving problems and serious study of the solution pro
cess. He encourages teachers to direct the students'
attention to certain key questions which correspond to
the mental operations used to solve problems. These ques
tions are organized around Polya's "four steps" and should
assist students in solving problems more efficiently and
in examining their own problem solving processes. The major questions are:
1. Understand the Problem
What is the unknown?
What are the data?
What is the condition? 10
2. Devising a Plan
Do you know a related problem?
Could you solve a simpler problem?
Could you restate the problem?
3. Carrying Out the Plan
Can you check each step?
4. Looking Back
Can you check the result?
Can you check the argument?
In addition to the examination of ones own problem solving processes, the careful analysis of other people's strate gies seems to facilitate problem solving. While working with college students, Bloom and Broder (1950) found that unsuccessful problem solvers showed improvement after considering protocols of successful problem solvers and comparing and contrasting these approaches with their own.
In light of such findings, the researcher will create a learning situation in which elementary school stu dents can discuss not only the methods employed in solv ing problems, but also the decisions involved in deter mining those methods. Such an environment should allow them to gain a better understanding of the problem solving process. 11
Statement of the Problem
The purpose of this study is to compare and contrast the changes in the problem solving behavior of the stu dents who receive instruction based on Polya's four steps
(Treatment A) with students who will not receive the in struction but will solve problems and share their results
(Treatment B). Answers will be sought to questions re garding the differential effects of instructional treat ment on the students' ability to:
1. understand the problem situation and the question
to be answered;
2. devise a plan to solve the problem, coordinate
information given in the problem with pre
viously learned information, use a visual
ization of the problem statement, etc.;
3. carry out correctly the selected plan;
4. evaluate both the solution and the solution pro
cess in light of the conditions of the problem.
The researcher will also' compare the scores of male and female students noting any differences in performance.
It is the aim of this study to determine the effec tiveness of the treatment; to indicate any achievement dif ferences of boys and girls; and to offer insights into the problem solving behavior of elementary children. The data collected and the conclusions drawn from this study should 12
provide valuable information for use in the development
of effective instructional techniques for improving
children's problem solving abilities.
This study will be conducted over a six week period
during which time the experimenter will mefet with students
three times a week for 35-40 minute sessions. A written
pretest will be administered to each of the 32 students
involved and followed by a comparable posttest at the end
of the study. The written pre- and posttest are designed
to evaluate three of Polya's four steps in his problem
solving model.
A second technique, aimed at analyzing students'
performance on all four subskills outlined in Polya's model, will involve individual interviews. The students will work two problems while "thinking aloud" both prior
to and following treatment.
Two groups of students will be compared in this study.
One group will receive specific instruction based on Polya's model and the opportunity to apply the strategies in
solving mathematical problems. The other group will solve problems without the benefit of instruction. The experi menter expects a difference in all students' pre- and post
test scores and a larger gain on the posttest scores of
those who will receive instruction in problem solving. CHAPTER II
REVIEW OF RESEARCH
The significance of improving the learner's problem
solving competence within school mathematics is well est ablished. Recommendations for emphasizing problem-solving
techniques can be found throughout history. It Is said that in ancient times Plato consistently gave Aristotle direc tion in seeking answers to problems rather than handing him information or solutions that others had secured through personal experience. More recently, in How We Think (1933),
John Dewey wrote of the importance of "reflective thinking;" this term is not synonymous with problem solving but is clearly an essential part of the process. In the chapter entitled "Why Reflective Thinking Must Be An Educational Aim,"
Dewey writes:
...thinking enables us to direct our activities with foresight and to plan according to ends-in-view or purposes of which we are aware. It enables us to act in a deliberate and intentional fashion to attain future objects or to come into command of what is now dis tant and lacking. It converts action that is merely appetitive, blind and impulsive into intelligent action.
Realizing the importance of analyzing man's thinking ability, philosophers, psychologists, social psychologists, mathematicians, and mathematics educators have studied problem
13 14
solving processes throughout the ages. This concentrated
effort and many still unanswered questions attest to the
complexity of the process. Still under examination are the
questions of what actually takes place when a person attempts
to solve a problem, what specific "heuristics" or techniques
seem to be helpful and what characteristics are associated with successful problem solvers.
The literature reviewed below deals with several as pects of this problem: 1) Theories of human problem solving;
2) Heuristic teaching as It relates to mathematical problem
solving; and 3) Characteristics of successful problem solvers.
Theoretical Framework
Although educators agree that problem solving is one, if not the most Important skill to learn, it is very dif ficult to help someone become a better problem solver.
Attempts to describe why or how a person solves a problem have often resulted in shallow, incomplete pictures. Explan ations of how problem-solving competence builds across a person's experiences are also thin. A person's success in this process seems to be dependent upon the experiences brought to the situation and upon a willingness both "to en dure suspence and to undergo the trouble of searching."
(Dewey, 1933). 15 Faced with a problem a person who is motivated to find a solution looks for any suggestion of an answer to his problem - the formation of some tentative plan. The data at hand cannot supply the solution, they can only suggest it. What then are the sources of the suggestion? According to Dewey (1933), they are past experiences and a fund of relevant knowledge at ones' command.
"If a person has had some acquaintance with similar situations, if he has dealt with materials of the same sort before, suggestions more or less apt and helpful will arise. But unless there has been some analagous experience confusion remains mere confusion. Even when a child (or a grown-up) has a problem, it is wholly futile to urge him to think when he has no prior experiences that involve some of the same conditions." (Dewey, 1933*p. 15).
In reference to the second criterion for problem solving success, "a willingness to endure suspence and undergo the trouble of searching," Dewey points out that to many persons, both suspence of judgment and intellectual search are dis agreeable. "They (the problem solvers) want to get them ended as soon as possible. They cultivate an over-positive and dogmatic habit of mind, or feel perhaps that a condition of doubt will be regarded as evidence of mental inferiority'.'
(Dewey, 1933). All this points to the idea that the problem solving process is very personal in nature and generalizations about problem solving are difficult to support.
Interest in human cognitive processes and representation of knowledge structures can be traced back to early European 16
psychologists such as Selz (1913, 1924) and Bortless
(1932). Later Gestalt psychologists, including Dunker
(1945), Kohler (1927), and Wertheimer (1945/1959), stressed
the importance of understanding in achievement of problem
solutions. They explained the problem solution as involving
a sudden insight into a situation and an integration of pre
viously learned responses in a novel way. Although their
contributions were noteworthy, they provided little direc
tion for those interested in the teaching or learning of
problem solving.
Because it is intrinsic to their discipline, a number
of mathematicians and mathematics educators have devised models which attempt to describe the problem solving as it
relates to mathematics. The eminent mathematician, George
Polya, presented his four-stage model in his book, How to
Solve It.(1957). According to Polya's model, the problem
solving process is four parts:
1) Understanding the Problem
2) Devising a Plan
3) Carrying Out the Plan
4) Looking Back
In the first phase, the problem solver must at least understand the question and want to answer it. He or she must recognize what is known (the data), what is unknown
(the goal), and what conditions are present. In the second 17 phase, a problem Solver might search his or her past exper
iences to think about a related problem that has already been
solved, or might tentatively try a number of attacks before
settling on one which seems promising. In the third phase, the problem solver carries through the plan to a solution, or reaching an impasse, returns to the planning phase. Fin ally, the problem solver checks the solution against the data and conditions presented in the problem.
Polya was definitely a pivotal figure in the area of problem solving. He analyzed and synthesized the human prob lem solving process and made specific suggestions for edu cators. He has had a noticeable impact on all those who have subsequently studied human cognition.
Two researchers who extended Polya's work in the area of analysis of human problem solving were Newell and Simon
(1972). Using the terminology of modern technology, they succeeded in describing how man possesses task-oriented symbolic information. Their work attests to the complexity of the human mind. Newell and Simon conceptualized problem * solution as the successful outcome of search processes, and provided a language, in effect, for expressing and opera tionalizing concepts central to a cognitive theory of prob lem solving.
After extensive research with computer simulations of humans playing chess, discovering proofs in logic, and 10 solving cryptarithmetic puzzles Newell and Simon proposed a theory of human problem solving. They began with five general propositions which were supported by their research findings:
1) Humans, when engaged in problem solving in the
kinds of tasks considered, are representable as
information-processing systems (IPS).
2) This representation can be carried to great de
tail with fidelity in any specific instance of
person and task.
3) Substantial subject differences exist in the way
information is processed which are not simply
non-essential variations but involve differences
of program structure, method, and content.
4) Substantial task differences exist in the way
information is processed, which also are not
simply non-essential variations but involve
differences of structure and content.
5) The task environment (the external situations that
define the problem) (plus the intelligence of
the problem solver) determines to a large extent
the behavior of the problem solver, independently
of the detailed internal structure of his infor
mation processing system (p.788). 19 According to Newell and Simon, the human IPS is a
serially organized system whose components are an active
processor, sensory input systems, motor output systems,'an
internal short-term memory (STM), an internal long-term memory (LTM), and external memories (EM). They postulate
that problem solving takes place by search in a "problem
space" which consists of:
1) a set of elements, each representing a state of
knowledge about the task;
2) a set of operators, which are information processes,
each producing new states of knowledge from
existing states of knowledge;
3) an initial state of knowledge, which is the know
ledge about the task that the problem solver has
at the 8tart of the problem solving;
4) a problem which is posed by specifying a set of final
desired states to be reached by applying operators;
5) the total knowledge available to a problem solver
when he is in a given knowledge state (p. 810).
Scandura (1974) also offers an information processing model but his is based on a hierarchical ordering of rule learning and executive control mechanisms. Scandura hypo thesizes that a subject will apply a rule to achieve a goal
(solve a problem) if the subject possesses that rule. If a procedure such as an algorithm or a rule for achieving a goal is not known, Scandura claims that a problem solver tries 20 to derive a procedure for solving a problem. Once the new procedure or higher order rule is derived the problem solver applies it to find a solution.
Heuristic Teaching
Although it is evident that human cognition is complex a number of strategies have been developed indicating general agreement that at least some aspects of the process can be learned. The term "heuristic" from the Greek word heuriskein meaning "to discover" or "to find" is applied to any efforts designed to facilitate problem solving.
Higgins (1971) defines heuristic teaching in mathe matics as "a category of instructional methods that make primary use of one or more problem-solving techniques in mathematics." It is evident from Higgins’ writing that he was greatly influenced by Polya (1957, 1962, 1965) who is a strong advocate of the teaching of heuristic. In his book, How To Solve It, Polya (1957) explains heuristic in the following way:
Modern heuristic endeavors to understand the process of solving problems, especially the mental operations typically useful in this process...A serious study of heuristic should take into account both the logical and psychological background...Experience in solving problems and experience in watch ing other people solving problems must be the basis on which heuristic is built... We should not neglect any sort of problem and should find out common features in the way of handling all sorts of problems; we should aim at general features, independent of the subject matter of the problem.(pp.129-130) 21
Polya's philosophy of heuristic teaching is best sunnned up in his own words, "I am trying, by all the means at my disposal, to entice the reader to do problems and to think about the means and methods he uses in doing them." (Polya,
1965, p.x ). Polya sees the teacher of mathematics as one who has the opportunity to give children a taste for and the means of independent thinking by providing appropriate problems which challenge their curiousity. The teacher can help the students by posing stimulating questions, and by focusing their attention on the methods used in the solution. Polya has collected a list of questions and sug gestions which are helpful in discussing problems with stu dents. The questions and suggestions are grouped into the four phases of his model, previously presented,and are characterized by their generality since they may be used with all sorts of problems, and by common sense in that they can occur naturally or obviously to the problem solver.
Polya's strategies have been applied in a number of studies. When ninth-grade algebra students were taught by a heuristic method which emphasized Polya's questions,
Ashton (1962) found that this method produced significantly better problem solvers than a conventional textbook method.
In the experimental groups pupils were taught to ask questions such as "What is the unknown? What are the data? What are the conditions?" In the textbook method the procedure for problems of a particular type was demonstrated and the pupils 22
were then assigned similar problems to solve.
Nicely (1976) conducted an exploratory study to deter
mine if the use of heuristics (a list derived from Polya)
would assist above-average sixth-grade students in solving
problems. Fifty students volunteered for the study which
lasted fifteen weeks with treatment sessions from one and
one-half to two hours each week. Students tried to solve
the problems at home and then during the following session
the problems were discussed and solved. In general, the
use of heuristics was evident on the post test, and while
the students did not always arrive at a correct answer
to some of the problems, they were able to write down
heuristics which they felt would be applicable.
Lucas (1972) analyzed protocols of college students
following the teaching of heuristics in calculas. During an eight-week period of instruction one class was taught using an heuristic style based on Polya's writings, and received papers which defined and demonstrated heur istic strategies. They also received problem assignments which encouraged the use of these strategies, while another class learned calculus without Attention to heuristics.
When the problems were scored on approach, plan, and result, subjects from the heuristic treatment were judged superior in their ability to solve calculus problems.
These subjects used the following heuristics more frequently: working backwards, using methods or results of related 23
problems, devising workable plans, and organizing and intro
ducing mnemonic notation. Lucas also concluded that
heuristics can be taught without infringing on normal con
tent.
Putt (1978) studied two instructional techniques for
improving the problem-solving ability of fifth grade child
ren. He compared the effects of the two instructional
approaches with one another and with the effects of no in
struction. One teaching technique incorporated studying the
aspects of the problem-solving process as proposed by Polya,
focusing on writing questions for understanding, selecting
strategies, and using the strategies to solve similar prob
lems. The other instructional approach provided practice in
solving problems and exposure to various methods of solution
of problems. Putt found that both of the instructed groups
scored significantly higher on a problem-solving test than
the non-instrueted group, yet neither of the instructed..
groups proved to be superior to the other. However, he did
% find that the first group appeared to have a different * perception of the problem-solving process, as shown by the nature of the questions that they wrote for understanding the problem, and by the types of strategies that they named.
In addition to research influenced by Polya, there have been other techniques suggested worth examining. Wilson
(1964) contrasted the effect of a "wanted-given" approach which involves instruction in understanding a problem by 24
analyzing the data (the givens), the conditions, and the
goal (the wanted), with an "action sequence" approach.
The latter method requires that the problem solver look
for the operations suggested by the sequence of actions
in the problem. Wilson concluded that the "wanted-given"
approach was superior to the "action-sequence" approach and
to a program providing no direct instruction program con
cerning the problem situation. In fact, the wanted-given
group performed better than the other two groups after
three, six, and nine weeks of instruction on measures of
ability to choose the correct operations, ability to solve
problems, and speed in solving problems.
The Productive Thinking Program uses a comic book
format to lead students to develop heuristics for non- mathematical problems. Crutchfield and Covington, the
developers of the Program (1963), tested it with three pairs
of fifth and sixth-grade classes. They obtained superior
performance by the instructed children over the control
children on measures of divergent thinking, originality, and
perceived value of problem solving (Covington and Crutch
field, 1965).
Treffinger (1969) investigated the effectiveness of the
Productive Thinking Program in.developing verbal creativity and problem-solving ability in children from grades four through seven. He sought, among other things, to determine
the transfer of problem-solving skills from the Program materials to an Arithmetic Puzzles Test. Two (of four) classes at each grade level were randomly assigned to the instructional condition, the other two serving as controls.
The experiment ran for sixteen consecutive school days during which time the control groups continued with normal classroom instruction. At each grade level significant differences favoring the experimental groups were found in pupil's attitudes toward creative thinking and problem solv ing. In comparing pretest and posttest scores, no result was found which indicated that the Program developed skills or abilities that transfer to the arithmetic problem solving tests used in the study.
Jerman (1971) found no significant difference with fifth grade students on an arithmetic word problem test when he compared the Productive Thinking Program with a Modified
Wanted-Given Program (after Wilson, 1964). However, he did find a significant difference in favor of the wanted- given program on a follow-up test of word problems, and he concluded that either of the systematic approaches to problem solving is more effective than not providing any systematic instruction.
Working with 296 sixth grade students.Early (1967) attempted to assess the effects of the presence or absence of word clues in routine verbal problems in word clues in routine verbal problems in mathematics. For the entire sam ple tested, students performed significantly better when word clues were present than when they were absent. Low
performers were found to rely more heavily on word clues
than middle or high level performers. Early also noted that more practice with word problems seemed to lessen the
student's dependency upon the word clues.
Dahmus (1970) suggests a "direct-pure-piecemeal-complete"
(DPPC) approach to solving verbal problems. In this method,
the student is encouraged to translate the data presented in the problem into mathematical sentences by concentrating on a few words at a time. He gradually learns to put together the "piecemeal" mathematical statements into equations and finally, into systems of equations. Dahmus found that this method resulted in better performance than did Polya's method of heuristically solving the whole problem. How ever, as Bassler (1975) points out, the problems used by
Dahmus were very easily translated into mathematical sen tences and therefore were not representative of many word, problems. When Bassler compared Polya's method to the Dahmus method on a different set of problems, the results indicated that a correct equation was generated more often using
Polya's method.
Characteristics Related to Students' Problem Solving Perfor mance
In observing the learner involved in solving a problem, one becomes aware of certain characteristics associated with successful problem solvers. A number of studies have 27 examined this phenomenon.
Dodson (1972) attempted to characterize successful
insightful problem solvers in terms of four categories of
variables: a) mathematics achievement; b) cognitive and
effective variables; c) variables of teacher, background
and attitude; and d) variables of school, community, and
curriculum. Results indicated that good problem solvers
differed from poor ones in several ways. Among the most
prominent differences were that good problem solvers are
superior with respect to a) overall mathematics achievement;
b) verbal and general reasoning ability; c) spatial abil
ity; d) positive attitudes; e) resistance to distraction;
and g) divergent thinking. He found that field independence
was one of the strongest characteristics of successful prob
lem solvers. Blake (1977) also found that good problem
solvers were field independent and that field independent
students used a greater variety of heuristics and were more willing to change their mode of attack on a problem than
field dependent students.
In his classic twelve-year study, Krutetski (1976)
found that a major difference between good and poor problem
solvers lies in their perception of the important elements
of problems. Good problem solvers typically had certain
abilities that poorer problem solvers lacked: a) the ability
to distinguish relevant from irrelevant information; b) the
ability to see quickly and accurately the mathematical 28
structure of a problem; c) the ability to generalize across a wide range of similar problems, and d) the ability to remember the formal structure of a problem for a long time.
Silver (1979) concurred with Krutestski's conclusion that good problem solvers are superior to poor problem solvers in their ability to perceive the mathematical struc ture of a problem having similar structure. More specifi cally, he found that the extent to which individuals sorted problems on the basis of mathematical structure was related to their problem solving ability as well as to their general verbal and mathematical abilities.
After observing some 700 intermediate grade children solving problems for an entire school year, Webb, Moses and Kerr (1977) concluded that willingness, preserverence, and self-confidence were three of the most important influ ences on problem solving performance. They did note, how ever, the difficulty in adequately measuring the extent to which these factors changed over time. Trimmer (1974) found that confidence, lack of anxiety, flexibility, lack of rigidity and an ability to cope with uncertainty were all traits associated with successful problem solving.
The correlation between problem solving success and reading ability has also received considerable attention as an area of research for a number of years. However, the relationship is not clear. Whereas there is considerable evidence to suggest that reading ability is directly 29 associated with performance on standard word problems (e.g.
Chase, 1960; Linville, 1970; Martin, 1964), there is also support for the belief that reading plays only a minor role in determining problem solving success per se (e.g. Balow,
1964; Knifong and Holton, 1976).
In addition, several researchers have considered differences between males and females in problem solving ability. Again, the results are inconclusive; some studies indicate males to be superior to females (e.g., Wilson, 1972), and others conclude that males do not typically perform better than females (e.g., Fennema and Sherman, 1978).
According to Fennema, conclusions reached about male super iority have been gathered from old studies or from studies in which the number of mathematics courses taken was not controlled. Therefore, a better mathematically educated group of males was being compared to a group of females with less mathematics education. In reality, what was being com pared were not females and males but students who had stud ied mathematics 1-3 years in high school with students who had studied mathematics 2-4 years in high school. Fennema and Sherman's (1978) study controlled the number of courses taken; they found that the main differences between sexes in regards to problem solving lay with two affective varia bles, confidence in learning mathematics and belief that mathematics is a male domain. Several observations can be made based on the research reviewed. First, many of the studies involved students who were identified as "above average." There is a need to investigate the behavior of the "average" student or one who is not presently excelling in mathematics. Secondly, there is some agreement as to the effectiveness of instruc tion in problem solving; however, what that instruction should be or how it should be presented is still being de bated. And finally, the results of research aimed at deter mining the differences in problem solving performance based on the sex of the students are inconclusive. CHAPTER III
METHODOLOGY
This study is an exploratory investigation of the
relative effects of problem solving instruction on fifth
grade students' performance in solving mathematical verbal
problems. The following sections are included in this
chapter: 1) a description of the subjects in this study,
2) the schedule followed for the study, 3) classroom pro
cedures, 4) instruments used for data collection, and
5) methods used for analysis of the data.
Subjects
Thirty-two elementary students were selected for this
study from two fifth grade classrooms at a public elemen
tary school in Columbus, Ohio. The school was selected because: a) the student population is representative of
the local community; (due to court-ordered desegregation,
parents of the students in the school represent a broad
spectrum of social and economic categories ranging from welfare recipients to university faculty and business and
professional people); b) the principal of the school recog nized the need for improvement in the area of problem
solving and was eager to assist in a research project attempting to address this problem; c) the fifth grade classroom teachers were both conscientious and willing to 32 adjust their schedules in order to learn more about the problem solving process.
The experimenter identified thirty-two "average" stu dents in mathematics. This rating was based on two factors:
1) the student's score on the California Achievement Test
(mathematics portion), and 2) the teacher's assessment of the student's mathematical ability. The California Achieve ment Test, a standardized test designed to assess a student's achievement in the areas of reading, mathematics, and lan guage, was administered in April of 1979. The experimenter listed the student's names according to their scores on the mathematics portion of this test. The students were then ranked according to the way the classroom teacher perceived their mathematical ability. The teacher and experimenter reached a consensus as to the "average" students in each classroom. In order to get sixteen students from each class the experimenter accepted two to three children who were slightly above average and the same number who were slightly below average in forming two comparable groups, labeled A and B. The names of the students were then listed by room, sex and ability (from high to low). The first boy from Room
8 was assigned to Treatment A, the next to Treatment B, the next to B, then A, and so on. The girls were assigned in the same manner. The experimenter began the assignment pattern for Room 11 with Treatment B for both boys and girls in an attempt to equalize the two.(see Assignment Chart, AppendixA). 33 The experimenter worked with the eight students from one class assigned to Treatment A as a group and then with
the Treatment B group from that same classroom. Since there were also two groups in the second classroom there were four groups, two receiving Treatment A and two receiving B. The size of the groups seems very realistic for those teachers interested in replicating the instructional strategies, since
teachers often work with groups of comparable size within the classroom.
Classroom Procedures
The experimenter met with each group of eight in a small classroom equipped with a chalkboard and eight desks arranged in groups of four. The students in each treatment group were free to share ideas with one another and this seating arrangement was designed to encourage this exchange
(see the figure in Appendix A). The experimenter strived to create a non-threatening, non-competitive atmosphere in both treatment situations.
In each treatment group the subjects received indi vidual copies of each problem. The students silently read their personal copies and then the teacher had one volun teer read the problem aloud to insure that any poor readers were not at a disadvantage. From this point on an observer would recognize behavioral differences, on the part of the teacher and students when comparing Treatments A and B. The treatments differ in two major respects;.namely: the role the teacher plays in the instruction, and the behaviors 34 expected of students when solving problems. These differ ences are outlined in the following descriptions of the two treatments.
Treatment A
The method of instruction for Treatment A is based on
Polya's model. The students were introduced to the model during the first teaching session. The teacher discussed briefly the importance of problem solving in everyday life and the many attempts to assist students in becoming "better problem solvers." Next the students were encouraged to con sider the concept of "problem" and to speculate as to how they would approach one. The teacher intentionally avoided reference to a specific situation in order to allow the stu dents to conceptualize problem solving. The students made numerous suggestions such as: "You would have to really know what the problem is all about," "You would have to know what to do about it," "You would have to try something to solve it," etc. The teacher then shared the four-step model devised by George Polya. The students concurred that the mo del made sense and that the steps that they had suggested were similar but not as organized nor inclusive*. During the subsequent six weeks students became very conversant with all four stages of Polya's model.
Understanding'the Problem
The teacher/experimenter posed questions which focused on the conditions and the objective of the problem as an 35
aid to understanding. (Examples of questions for specific
problems are given in Appendix B). The teacher/experi
menter encouraged the students to ask questions that would
clarify the problem statement and insure a clear under
standing of what was being said. The students circled what
they were asked to find and then stated in their own words
the "unknown". Relevant and irrelevant information was
discussed and problems containing both types of information were shared (examples given in AppendixB). Another teach
ing strategy the experimenter used was purposely ignoring
one or more conditions of the problem when suggesting a
possible solution, hoping that the students would detect
the.omission.
The experimenter also shared several problems with
either extraneous or insufficient information. The students
identified the information that was unnecessary and supplied additional information where it was lacking. The students were then asked to "remodel" textbook problems creating problems with extraneous and those with insufficient infor mation. The students exchanged the "new" problems; they
either wrote down the facts that were not used or listed
the additional facts needed in order to solve the problem.
This exercise was designed to 1) reinforce the idea that one must read the problem carefully rather than perform an operation on the first two numbers in the problem, and
2) prepare the student for problems in the real world which 36 are usually either confounded with extraneous facts or lack information needed to solve the problem.
Devising a Plan
The students in .Treatment A were encouraged to suggest methods for solving the problem that was now understood.
Initially they suggested operations: "I think we ought to multiply." They didn't even specify the factors. It was as if the call for an operation was instinctive. The teacher discussed a variety of problem solving strategies such as making a table, chart or organized list, drawing a diagram, "guess and test" and acting out a problem. The question of when to use each strategy was explored as the teacher shared a number of exemplary problems.
When discussing the solution of each problem, the teach er focused on the planning stage, encouraging the students to share the strategies that they had both considered and elected to use in solving the problem. This phase of in struction was aimed at exposing the students to a variety of cognitive strategies in problem solving and practice in selecting one that is appropriate and will lead to a solu tion of a given problem.
Carrying Out the Plan
Students were free to "carry out the plan" independently, in pairs or in small groups. Initially the majority of 37 students chose to work with one or two students; however,
as time went on a large number of them preferred solving the
problem on their own.
When the teacher perceived that the student or students
had struggled with an inappropriate strategy for a reasonable
length of time, the teacher would ask questions that would
re-focus the student's thinking on the information, the condi
tions and/or the goal of the problem. This redirection
usually led to an appropriate plan.
The importance of computational accuracy was mentioned
on several occasions in this phase and again as a part of
"looking back."
Looking Back
In general, the teacher encouraged the students to
think about the strategies they used, the reasonableness
of their answers in terms of the given conditions, accuracy,
and plans for solving extensions or variations of the prob
lems.
After completing a problem, one student was asked to
describe the method he or she used to the rest of the group.
The other students checked to see if the solution satisfied
the question posed and pointed out any computational errors.
Additional solution strategies offered by the group were
also discussed.
It was apparent that many students were content to end with the "carrying out the plan" phase for very few looked 38
to see if the strategy they chose seemed logical and was
executed properly, the answer they arrived at was reason able, or that it met the conditions of the problem. In order to deal with one part of this situation, the teacher provided a number of problems with "answers" and the stu dents were to determine whether the answers were reasonable or unreasonable (see examples of problems in Appendix B).
A discussion followed this exercise; the purpose was to im press upon the students the importance of Polya's fourth step/'looking back. *'
Often the teacher would alter the information or the conditions of the problem to allow students practice in certain strategies. Students were allowed to suggest some adjustments on their own.
Noting similarities in problem statements and in the solution processes involved is also a part of Polya's
'looking back stage." Although the teacher/experimenter planned to initiate discussions that made comparisons, this was rarely necessary for the students were the first to say, "Hey, this problem is like the cows and chickens problem," or "I'm going to use a table like I did in that other problem." The experimenter did pursue this line of thinking by asking how the problems were alike and/or dif ferent or why they thought the table would be the best strategy again. 39
Thus the instruction in Treatment A was based on Polya's
model; the experimenter discussed the important aspects of
each stage and the heuristic involved in the problem
solving process. The model was referred to often and the
students were very conversant with each stage.
Treatment B
As previously stated, the major distinction between
Treatments A and B is the teacher's role and the behaviors
expected of the students when solving problems. The stu
dents in both treatments worked the same problems; how
ever, in Treatment B the focus was on the answers to the problems, not the problem solving process. The students were asked to solve the problem after it had been read aloud. The students, as in Treatment A, were free to draw on each other for assistance.
Although Polya's model was not discussed in Treatment
B, the following description is organized around Polya's four stages in order to contrast the two treatments more easily.
Understanding the Problem
The teacher provided a comfortable atmosphere in which the students felt free to ask questions concerning the problem statement conditions, etc., but the teacher did not pose any questions to aid understanding. 40
Devising a Plan
Students were free to discuss different strategies
among themselves. No strategies were identified by name;
however, the students eventually employed nearly all the
strategies promoted in Treatment A. Guess and test, mak
ing a list and drawing a diagram were frequently em
ployed particularly during the last few weeks of the exper
ience.
If a student was unable to devise any plan the teacher would model an appropriate strategy but not elaborate on
it.
Carrying Out the Plan
If there seemed to be an impasse the teacher would
suggest that the student re-read the problem statement. If
this technique was ineffective the teacher would suggest a plan for them to carry out.
Looking Back
One of the students shared his or her solution with
the class. The teacher indicated that the solution was correct or modeled the correct solution if no one in the class could do so.
The teacher's role in the two treatment groups was noticeably different. This difference was immediately recognized by the principal on his first visit to both 41
treatment groups. As far as the second major distinc
tion, expected student behavior, in Treatment A the stu
dents were expected to ask questions to clarify the prob
lem statement, to examine the problem and decide if it con
tained sufficient information, to realize that there were often a variety of methods to solve each problem, to check their calculations for accuracy and to determine whether they had addressed all the conditions of the problem. The students in treatment B were expected to solve problems independently or in groups. The underlying assumption was that one way to learn to solve problems is to have experience in solving problems.
A summary of the instructions in both treatments can be found on page 48. A comparison of the teacher's role in Treatments A and B can be found on page 51.
Instruments
Two techniques aimed at analyzing students' performance on each of the subskills outlined in Polya's model were em- ployed in this study. The first was a 30-item written test developed by the team associated with the Iowa Problem Solv ing Project (1PSP) and the second method involved individual interviews.
The difficulty in evaluating students' problem solving ability is well recognized. Traditionally, developers of standardized tests and teachers have measured students' 42
abilities in problem solving as they would simpler learner
outcomes. That is, the student is presented with a number
of word problems.and asked to solve them; the number or
percentage of correct answers is used as an index of the
students' problem-solving ability (Schoen, Oehrake ,1980).
Due to the complexity of the task, many educators feel that
this simplistic approach results in an inadequate evalua
tion of the problem-solving process and offers teachers
little, if any, assistance in pinpointing the student's weakness in this multi-step process. With this deficiency
in mind, educators associated with the Iowa Problem Solving
Project (IPSP) under the direction of George Immerzeel of
the University of Northern Iowa launched a test-development
effort in order to provide an alternative approach to prob lem solving evaluation. The test that evolved over a three year period is based on Polya's four step model of the problem solving process. The major goals of the IPSP were to use the Polya model as a basis for the development of instructional problem solving materials and a paper-and- pencil test appropriate for children in grades 5 through 8.
The test provides three sub-scores and a total score for each student. It measures the student's ability to:
1) understand the problem, 2) carry out the plan, and
3) look back at the solution, three of the four stages in
Polya's model. The IPSP team were not able to find a valid 43
machine-scoreable method to test a student's ability to
'•devise a plan," step two in Polya's model.
Four forms of the test were developed and tested, two
equivalent forms for grades 5 and 6 and two equivalent forms
for grades 7 and 8. Over 8,000 Iowa children in grades five
through eight, were involved in tryout testing from 1976
through 1979.
The test was found to be valid. There was a strong
relationship between the IPSP scores and interview-based judgments of students' performance on the same three areas
tested. These judgments were made as students thought aloud while solving open-ended verbal problems in a one-to-one
interview setting. A complete discussion of the test validation can be found in Oehmke (1979).
Both forms of the IPSP designed for fith/sixth gra ders were used in this study. Test number 561 served as a pretest and number 562 as a posttest (copies of each
test are in Appendix D). The students were informed that
there were thirty items on the test and that they were to
circle the correct answer for each item. They were advised
to take their time and read the questions carefully. The
tests were collected after forty minutes. The same proce
dure was followed in both pretest and posttest situations.
The second method employed in this study involved indi vidual interviews. This method of analysis has been em ployed in a number of recent studies on problem solving 44 such as those by Lucas (1974), Kantowski (1974), Webb
(1975), and Putt (1979). The student is asked to "think aloud" or verbalize all his assumptions, conjectures, strat egies, etc., as he attempts to solve a problem. This tech nique, although time consuming, provides a picture of a student's problem solving ability. Kantowski (1975) however, noted some limitations of this method:
...an individual might remain silent during moments of deepest thought. The presence of an observer could put con straints on a problem solver in that he might not attempt solutions which might be considered foolish to someone else but which he would try if he were not being observed (p.112).
In spite of these concerns, the interview-analysis method has generated information aimed at improving problem solving instruction.
The experimenter in this study interviewed eight of the sixteen students in each treatment group. The students, chosen at random, were asked to work two problems while
"thinking aloud." Each interview was audiotaped and the student's problem solving behavior was carefully observed.
Pre-interviews and postrinterviews were conducted and compared.
(A checklist used in recording student behavior during the interview can be found in Appendix C ).
Since the "thinking aloud" technique does not reveal all of the student's thinking, questions were asked to supplement the student's comments and the interviewer's observations. These questions were asked when it was not clear to the experimenter what the student was doing as he carried out the problem, for example, or when there was silence and the student seemed to be thinking through a strategy, clarifying in his own mind what was being asked, etc. The interviewer hoped to gain information concern ing: 1) the information the student used in the problem statement, 2) the question the student was attempting to answer, 3) the techniques used to help understand the prob lem, 4) the strategy or strategies which were used, 5) the type of checking which occurred, if any. This question ing occasionally resulted in the student wanting to revise his method of solution or the solution. Students were per mitted to change answers or to rework the problem although only the work done prior to the probing was considered in the evaluation of the student's performance. Transcripts of sample interview sessions can be found in Appendix C .
Each problem solved in the interview session was scored by awarding 0, 1, 2 or 3 points in each of the four stages of Polya's model. A detailed description of the scoring procedure is included in Appendix C.
In order to test the reliability of the scoring scheme, copies of two transcripts from Treatment A and two from
Treatment B were pulled at random and scored by a professor and four graduate students in mathematics. Each person received a copy of the scoring scheme as it appears in Appendix C. Results of their scoring, as compared to the
experimenter's, can be found in Appendix C.
Analysis of the Data
The data collected in this study include the scores
on the Iowa Problem Solving Project (IPSP) tests, pre interview and post-interview, and the student's scores on the interview analysis, pretest and posttest.
A total score and three sub-scores were computed using
the IPSP tests. The sub-scores provided an evaluation of
the student's ability to: 1) understand the problem,
2) carry out the plan to solve the problem, and 3) look back
at the problem solution. The means and standard deviations
of the total scores for both pre- and posttests were calcu
lated. An analysis of covariance was computed for the post
test score of each of the three criterion listed above and
total posttest score using sex and treatment as variates and
the pretest score of that particular criterion as covariate.
A total soore and four sub-scores were computed based
on the student's performance in the interview sessions.
This analysis provided information concerning the student's
ability to understand the problem, devise a plan, carry out
the plan and look back. Again, means and standard deviations
of the total scores for both pre- and post-interviews
were calculated. Analysis of covariance was computed for
the post-interview score of each of the four criterion listed and total post-interview score using sex and treatment as variates and the pre-interview score of that particular criterion as covariate.
A complete description of the statistical findings
is included in Chapter Four. 48
Summary of the Two Methods of Instruction
Understanding the Problem Treatment A Treatment B
Each student receives a Each student receives a copy of the problem. Teacher copy of the problem. Teacher choses one student to read chooses one student to read the question aloud after stu the question aloud after stu dents have an opportunity to dents have an opportunity to read the problem silently. redd the problem silently.
Teacher poses questions Teacher answers students' which focus on the condi questions concerning clari
tions and the objective of fication of the problem.
the problem, as an aid to understanding. Teacher also
encourages students to ask
questions about the infor mation given to insure a
clear understanding of
what is being asked.
Devising a Plan
Teacher has previously Students are free to discuss
discussed several strategies different strategies among them
such as "guess and test", selves; the teacher allows the
organized lists, making a students to share their strate
table and drawing a diagram. gies with one another. 49 Treatment A Treatment B
Teacher encourages students
to suggest a strategy(s) for
solving the problem and to
share their views concerning
their choice of strategy.
Carrying out the Plan
Students are free to solve Students are free to work inde
the problem independently, in pendently, in pairs or in small
pairs or in small groups. groups. Teacher suggests that
Teacher redirects students the students reread the problem
by suggesting other strate statement if there is an impasse
gies if there is an impasse.
An impasse occurs when the
students are unable to suggest
any plan of action.
Looking Back
A student shares his or her A student shares his or her solu
solution with the class. This tion with the class. Teacher and other solutions are dis indicates that the solution is
cussed. Teacher confirms correct or models a correct solu
correct answer (s) and/or models tion if no one.in the class can
alternative solutions. do so. 50
Treatment A Treatment B
Teacher alters problem slightly and the class, as a group, solves the
"new" problem. Students are encouraged to construct problems that are similar in structure. 51
Teacher Behavior in Experimental Situations
Understanding the Problem
Treatment A Treatment B
- The students were asked to - The teacher provided a very
circle what they were asked comfortable atmosphere in
to find in each problem. which the students felt free
- They were also to state, in to ask for clarification of
their own words, the "problem" a problem and assistance as
in question. it was needed. - The teacher encouraged the - The students worked text students to ask questions to book problems just as they clarify the problem state were written. ment and often modeled this behavior.
- The teachers discussed "re levant and irrelevant infor mation" and shared problems that contained both, drawing the students' attention to the distinction between the two.
- The teacher purposely ignored one or more condi tions of the problem when 52 Understanding the Problem (continued)
Treatment A suggesting a possible solution, hoping the students would detect the omission.
- The teacher presented sever al problems that contained extraneous or insufficient infor mation. The students, using textbook problems, either deleted or supplemented the information given in each sit uation. Working with each other's remodeled versions, the students listed either the extraneous information or the information needed to solve the problem. Devising a Plan
Treatment A Treatment B
- The teacher discussed a - No strategies were identified variety of problem-solving by name, however the students strategies such as making a eventually employed nearly all table, chart or organized list, the strateties promoted in Treat drawing a diagram, guess and ment A. 53 Devising a Plan (continued)
Treatment A Treatment B test, and acting out a prob - Guess and test, making a lem. The question of when list and drawing a diagram to use each strategy was ex were frequently employed. plored as the teacher shared a number of exemplary problems.
- When discussing the solution of each problem, the teacher focused on the planning stage, encouraging the students to share the strategies that they had both considered and used in solving the problem.
Carrying ' ; the Plan
- After a student (or students) -After the student struggled struggled with an inappro with an inappropriate strat priate strategy for a reason- egy for a reasonable length able length of time, the teacher of time, the teacher would would ask questions that would model an appropriate strategy. redirect the student to an appropriate plan.
-The strategy that seemed to cause the most frustration was the organized list. The 54
Carrying out the Plan (continued)
Treatment A Treatment B teacher on a number of occa sions helped students organize their data.
- The importance of computational accuracy was mentioned on several occasions but not stressed.
Back
- The teacher asked a student to - The teacher asked a student share his solution with the class to share his solution with the
This solution was checked by the class and the teacher cor class and teacher. They checked rected the problem if neces to see if the solution satisfied sary. the question posed and if there - If a student suggested ano were no computational errors. ther strategy (other than
- The teacher discussed other the one shared) the teacher solution strategies offered by allowed the student to the group. share his findings. (How
- The teacher shared problems ever, this was not encouraged). with reasonable answers and - The students responded to some with unreasonable solu the reasonable/unreasonable tions. A lesson in checking problem but little discussion
followed. Looking Back (continued)
Treatment A
* your answer for "reason
ableness" followed.
- The teacher extended a
number of the problems that
the students had previously
solved, pointing out the
need for alterations in
their original strategies.
- The teacher encouraged the
students to 1)check their
computations for errors and
2) reread the problem state ment to see that they had con
sidered the conditions of the problem. 56
Schedule for the Study
Week One Pretests administered to the fifth grade students.
Interview sessions conducted with half of the stu
dents from each treatment group (randomly selected).
First teaching session (A & B)
Second teaching session (B & A)
Week Two
Third teaching session (A & B)
Fourth teaching session (B & A)
Fifth teaching session (A & B)
Week Three
Sixth teaching session (B & A)
Seventh teaching session (A & B)
Eighth teaching session (B & A)
Week Four
Ninth teaching session (A & B)
Tenth teaching session (B & A)
Eleventh teaching session (A & B) 57
Week Five
Twelfth teaching session (B & A)
Thirteenth teaching session (A & B)
Fourteenth teaching session (B & A)
Week Six
Fifteenth teaching session(A & B)
Posttest administered to the fifth grade students.
Interview sessions conducted with the same students
previously interviewed. CHAPTER FOUR
ANALYSIS OF THE DATA
The purpose of this study was to compare and contrast
the changes, if any, in the problem solving behavior of
students who had received instruction based on Polya's
model (Treatment A) with the behavior of students who
had not received the instruction but who had solved prob
lems and shared their results (Treatment B). More speci
fically, answers were sought to questions regarding the
differential effects of instructional treatment on the
student's ability to:
1) Understand the problem
2) Devise a plan for solution
3) Carry out the plan
4) Look back over the problem.
The data collected in this study include the student's
scores on interview analyses, both pre-interview and post-
interview and their scores on the Iowa Problem Solving
Project (I.P.S.P.) tests, also pretest and posttest.
A technique aimed at analyzing students' performance on each of the subskills outlined in Polya's model in volved interviews. The students were asked to work two
problems while "thinking aloud" prior to the treatment and 58 59 to work two different problems following the treatment.
Each problem solved in the interview session was scored using a scoring procedure determined by the investigator.
(A detailed description of the scoring procedure can be found in AppendixC). A total score and four subscores were computed based on the student's performance in the interview sessions. This analysis provided information concerning the student's ability to understand the prob lem, devise a plan, carry out the plan and look back over the problem solution. Treatment group and sex of student were used as independent variables.
The second source of data, the Iowa Problem Solving
Project tests, consisted of three subscores and a total score. These tests were designed by Harold Schoen and
Theresa Oehmke in 1977, to evaluate three of the four criteria previously listed; only criterion two, devising a plan, was omitted. The scores of the students who had received Treatment A were again compared with the scores of those in Treatment B. Sex and treatment group were again used as independent variables.
The results of the scores from the interview sessions can be found in Tables 1 through 21. Sixteen students were interviewed, eight students who had received treat ment A and eight students who had received treatment B.
Table 1 shows the means and standard deviations of the pre-interview and post-interview scores on the four criteria: 1) Understanding the problem
2) Devising a plan
3) Carrying out the plan
4) Looking back
The means and standard deviations of the total scores for
both pre- and post-interviews can also be found in Table
1. An analysis of covariance was computed for the post
interview scores for each criterion .and total score using
sex and treatment as variates and the pre-interview score
.of that particular criterion as covariate. The signifi
cance used was .05. TABLE 1
Descriptive Statistics of Pre-Interview Scores and Post-Interview Scores
Criterion N Pretest Mean Posttest Mean (standard (standard deviation) deviation)
"Understanding 2.63 4.38 the problem" 16 (1 .86) (1.93)
"Devising a 16 3.31 4.88 Plan" (2.06) (1 .20)
"Carrying out 16 2.63 4.06 the plan" (2 .22) (1.80)
"Looking Back" 16 0.25 1.06 (0.77) (1.48)
Total 16 8.81 14.38 (6.07) (4.91) 61
Table 2 Is an analysis of covariance Cable for Che
cocal post-incerview scores. Table 3 shows Che parCial
sum of squares for Che analysis of covariance of Che CoCal posC-inCerview scores. Table 2 (ANCOVA Cable) shows ChaC
Chere may be a significanC effecC on Che CoCal posC-incerview scores. FurCher analysis of Che parCial sums of squares
(Table 3 ) shows a CreaCtnenC effecC ac .0474 level of significance afcer adjusCing for Che covariace buc no sex nor inCeracCion effecCs (P » .2260, P *» .3430 respecCively).
The mean posc-inCerview score for TreaCmenc A was
16.375; TreaCmenC B was 12.375; Cherefore, TreaCmenc A had a significanCly higher mean score Chan TreaCmenc B (see
Table 5 ).
Tables 4 and 5 show Che cell means and cell sizes for Che CoCal pre-inCerview and posC-inCerview scores respecCively. TABLE 2
Analysis of Covariance (AHCOVA) table for total post-interview scores
Source______DF_____Sum of Squares Mean Square F Value PR F
Total Sources* 4 179.80 44.95 2.72 0.09 Error 11 181.95 16.54 Corrected Total 15 361.75
TABLE 3 Partial sums of squares for ANCOVA of total post-interview scores
Source______DF_____ Sum of Squares______F Value______PR F
Sex 1 27.20 1.64 0.2260 Treatment 1 82.33 4.98 0.0474 Sex * Trt 1 16.o9 0.98 0.3438 Pre-Interview Score 1 58.00 3.51 0.0879
* Sex, treatment, interaction of sex and treatment and pre-interview.score 63 TABLE 4 Cell means (cell sizes) of total pre-interview scores
TREATMENT A B
Hales 13.33 (3) 10.00 (4) 11.57 (7) Sex Females 4.60 (5) 9.25 (4) 6.67 (9)
7.88 (8) 9.75 (8) 8.81(16)
TABLE 5 Cell means (cell sizes) of total post-interview scores
TREATMENT A B Males 18.00 (3) 10.00 (4) 13.43 (7) Sex Females 15.40 (5) 14.75 (4) 15.11 (9)
16.38 (8) 12.38 (8) 14.38(16) 64
Table 6 is the analysis of covariance (ANCOVA) table of post-interview scores on criterion one, "Under standing the problem." Table 7 shows the partial sums of squares for ANCOVA of post-interview scores on criter ion one. Table 6 shows that there are no significant effects (P ■ .4481). Table 7 shows that there are no significant sex effects (P * .2598) nor treatment effects
(P = .2084) after adjusting for the covariate.
Table 8 presents the cell means and cell sizes for the pre-interview scores on criterion one, "Under standing the Problem," and 9 presents the same informa tion for that criterion's post-interview scores. TABLE 6 Analysis of covariance (ANCOVA) table of post-interview scores on criterion 1, "Understanding the problem."
Source______DF_____Sum of Squares Mean Square F Value PR F
Total sources* 4 14.87 3.72 1.00 0.4481
Error 11 40.88 3.72
Corrected Total 15 55.75
TABLE 7 ■ Partial sums of squares for ANCOVA of post-interview scores on criterion 1, "Understanding the problem."
Source DF Sum of Squares F Value PR F
Sex 1 5.25 1.41 .2598
Treatment 1 6.64 1.79 .2084
Sex * Trt 1 0.70 0.19 .6735 Ere-int erview score on Criter ion 1 1 4.67 1.26 .2864
* Sex, treatment, interaction of sex and treatment and pre-interview score on criterion 1 . 66
TABLE 8
Cell means (cell sizes) for pre-interview scores on Criterion 1 , "Understanding the problem"
TREATMENT
A B Males 3.67 (3) 3.25 (4) 3.43 (7)
Sex
Females 1.00 (5) 3.25 (4) 2.00 (9)
2.00 (8 ) 3.25 (8) 2.63(16)
TABLE 9 Cell means (ceill sizes) for post-inter view scores on Criterion 1 , "Understanding the problem"
TREATMENT
A B Males 5.00 (3) 3.00 (4) 3.86 (7)
Sex
Females 4.80 (5) 4.75 (4) 4.78 (9)
4.88 (8) 3.88 (8) 4.38(16) 67
Table 10 is the analysis of covariance (ANCOVA) table of post-interview scores on criterion two, "Devis ing a plan." Table 11 lists the partial sums of squares for ANCOVA of post-interview scores on criterion two.
The F-test (Table 10 ) shows that there is at least one significant effect (P « .04). Further inspection of
Table 11 shows that there may be a significant treatment effect (P ■ .0586) after adjusting for the pre-interview score. No evidence of sex or interaction effects are present (P - .1104, p » .2519 respectively).
Table 12. shows the cell means and cell sizes for the pre-interview scores on criterion two, "Devising a plan," and 13 presents the same information for that criterion's post-interview scores. The post-interview score mean for
Treatment A is 5.25 while the mean for Treatment B is
^.50 (see Table 13 ). TABLE 10 Analysis of covariance (ANCOVA) table of post-interview scores on criterion 2 /'Devising a plan."
Source______DF_____Sum of Squares Mean Square F Value PR F
Total Sources* 4 12.37 3.09 3.62 0.04
Error 11 9.38 0.85
Corrected Total 15 21.75
TABLE 11.
Partial sums of squares for ANCOVA of post-interview scores on criterion 2, "Devising a plan."
Source DF Stim of Squares F Value PR F
Sex 1 2.57 3.01 0.1104
Treatment 1 3.80 4.45 0.0586
Sex * Trt 1 1.25 1.46 0.2519
Pre-interview score on criterion #2 1 4.78 5.61 0.0373
* Sex, treatment, interaction of sex and treatment and pre-interview score on criterion #2 69
TABLE 12 Cell means (cell sizes) for pre-interview scores on criterion 2 , "Devising a plan"
TREATMENT A B Males 4.67 (3) 3.75 (4) 4.14 (7)
Sex
Females 1.80 (5) 3.75 (4) 2.67 (9)
2.88 (8) 3.75 (8) 3.31(16)
TABLE 13 Cell means (cell sizes) for post-interview scores on criterion 2 t "Devising a plan"
TREATMENT A B
Males 5.67 (3) 3.75 (4) 4.57 (7)
Sex
Females 5.00 (5) 5.25 (4) 5.11 (9)
5.25 (8) 4.50 (8) 4.88(16) 70
Table 14 is the analysis of covariance (ANCOVA) table of post-interview scores on criterion three, "Carrying out
the plan." Table 15 lists the partial sums of squares
for ANCOVA of post-interview scores on criterion three.
The F-test (Table 14) shows that there may be a significant
effect (P » .0766), however after inspection of the partial
sums of squares (Table 15) no evidence of sex or treat ment effect is apparent (P = .1447, P = .1463 respectively) after adjusting for the covariate..
Table 16 shows the cell means and cell sizes for the pre-interview scores on criterion three, "Carrying Out the plan." Table 17 shows the cell means and cell sizes for the post-interview scores on the same criterion.
\ TABLE 14 Analysis of covariance (ANCOVA) of post-interview scores on criterion 3, "Carrying out the plan."
Source DF Sum of Squares Mean Square F Value PR F
Total sources* 4 24.87 6.22 2.84 0.08
Error 11 24.07 2.19
Corrected total 15 48.93
TABLE 15
Partial sum of squares for ANCOVA of post-interview scores of criterion 3
Source DF Sum of Squares F Value PR F
Sex 1 5.40 2.46 0.1447
Treatment 1 5.35 2.44 0.1463
Sex * Trt 1 1.87 0.86 0.3747
Pre-interview score on criterion 3 1 14.63 6.69 0.0253
* Sex, treatment, interaction of sex and treatment and pre-interview score on criterion #3 72
TABLE 16 Cell means (cell sizes) for pre-Interview scores on criterion 3 , "Carrying out the plan"
TREATMENT A B Males 4.00 (3) 3.25 (4) 3.57 (7) Sex
Females 1.60 (5) 2.25 (4) 1.89 (9)
2.50 (8) 2.75 (8) 2.63(16)
TABLE 17 Cell means (cell sizes) for post-interview scores on criterion 3 , "Carrying out the plan"
TREATMENT A B
Males 5.00 (3) 2.75 <4) 3.71 (7) Sex
Females 4.40 (5) 4.25 (4) 4.33 (9)
4.63 (8) 3.50 (8) 4.06 (16) 73
Table 18 is the analysis of covariance (ANCOVA) table of post-interview scores on criterion four, "Look ing back." Table '-19. shows the partial sums of squares for ANCOVA of post-interview scores on criterion four.
The F-test (Table 18 ) shows no evidence of significant effect (P - .3917). The partial sums of squares table
(Table 19 ) shows no sex effect (P ■ .6659) nor treat ment effect (P ° .2761) after adjusting for the covariate.
Tables 20. and 21 show the cell means and cell sizes for the pre-interview and post-interview scores on criterion four, "Looking back," respectively.
In general one can say that treatment seemed to affect scores on criterion two, "Devising a plan"
(Table 11 ) and the total scores (Table 3 ), where subjects in Treatment A scored on the average signifi cantly higher than those subjects in Treatment B. There is no evidence, however, that sex had any significant effect on any of the criteria. TABLE 18 Analysis of Covariance (ANCOVA) table for the post-interview scores on criterion 4, "Looking back."
Source______DF_____Sum of Squares Mean Square F Value PR F
Total Sources* 4 9.60 2.40 1.13 0.40
Error 11 23.34 2.12
Corrected Total 15 32.94
TABLE 19 Partial sum of squares for ANCOVA of post-interview scores on Criterion 4, "Looking back."
Source______DF_____Scan of Squares F Value______PR F______
Sex 1 0.42 0.20 0.6759
Treatment ‘ 1 2.79 1.31 0.2761
Sex * Trt. 1 0.42 0.20 0.6659
Pre-interview score on criterion 4 1 2.12 1.00 0.3386
* Sex, treatment, interaction of sex and treatment and pre-interview score on criterion 4 75 TABLE 20
Cell means (cell sizes) for pre-interview scores on Criterion 4 "Looking back"
TREATMENT
A B
Males 1.00 (3) 0.00 (4) .43 (7)
Sex
Females 0.20 (5) 0.00 (4) 0.11 (9)
0.50 (8) 0.00 (8) 0.25(16)
TABLE 21
Cell means (cell sizes) for post-interview scores on Criterion 4 -» "Looking back"
TREATMENT A B Males 2.33 (3) 0.50 (4) 1.29 (7)
Sex
Females 1.20 (5) 0.50 (4) 0.89 (9)
1.63 (8) 0.50 (8) 1.06 (16) 76
The statistical findings for the written pretest and
posttests (I.P.S.P.) can be found in Tables 22 through 38
The thirty-item tests were administered to thirty-two
students, sixteen from treatment A and sixteen from Treat ment B. The written test scores were broken down according
to the same criteria used in the interviews, except for
criterion two, "Devising a plan," which was not possible
to measure in this manner. An analysis of covariance for
the posttest scores on the pretest scores (covariate),
sex and treatment (variates), was done for each criterion and overall total score.
Table 22 shows the means and standard deviations of
the pretest and posttest scores on the three criteria,
"Understanding the problem," "Carrying out the plan," and "Looking back." The means and standard deviation of the total scores for both pretests and posttests can also be found in Table 22. TABLE 22
Descriptive statistics of written pretest and posttest scores
Criterion N Pretest mean (standard Posttest mean (standard deviation) deviation)
Understanding 32 6.34 6.97 the problem (2 .20) (1.84)
Carrying out 32 6.94 6.91 the plan (2 .12) (1-28)
Looking back 32 5.28 6.94 (2.15) (1-44)
Total 32 18.56 20.81 (4.85) (3.45) 78 Table 23 Is an analysis of covariance table for the
total posttest scores. Table 24. shows the partial sums
of squares for the analysis of covariance of the total
posttest scores. Table 23 (ANCOVA table) shows that
there is a significant effect on the overall posttest scores
of the written test. After adjusting for the covariate,
the pretest scores, however only sex seems to have an
effect and it is marginal (P = .0674). There is no evi
dence of treatment effect (P ** .7526). Tables 25 and
26 show the cell means and cell sizes for the overall pre
test and posttest scores on the written tests. The over all mean for males was 20.07 and 21.47 for females . TABLE 23
Analysis of Covariance (ANCOVA) of the overall post-test scores on the written test
Source______DF_____ Sum of Squares Mean Square F Value PR F Total Sources* 4 191.73 47.93 7.31 0.00
Error 27 177.14 6.56
Corrected Total 31 368.88
TABLE 24
Partial sum of squares of ANCOVA of the overall total post-test scores
Source______DF_____ Sum of Squares F Value______PR F Sex 1 23.82 3.63 0.0674
Treatment 1 0.67 0.10 0.7526
Sex * Trt 1 2.86 0.44 0.5144
Pre-test score total 1 172.54 26.30 0.0001
* Sex, treatment, interaction of sex and treatment and total pretest score 80
TABLE 25
Cell means (cell sizes) for Che Cotal pretest scores on the written tests
TREATMENT A B [
Males 20.29 (7) 17.75.(8) 18.93 (15)
Sex
Females 17.56 (9) 19.00 (8) 18.24 (17)
18.75(16) 18.38(16) 18.56 (32)
TABLE 26 Cell means (cell sizes) for the total posttest scores on the written tests
TREATMENT A B
Males 20.57 (7) 19.63 (8) 20.07 (15)
Sex
Females 21.56 (9) 21.38 (8 ) 21.47 (17)
21.13(16) 20.50(16) 20.31 (32) 81 Table 27 is the analysis of covariance table
(ANCOVA) of posttest scores on criterion one, Understand ing the problem." Table 28 shows the partial sums of squares for ANCOVA of posttest scores on criterion one.
The F-test (Table 27 ) shows that there is at least one factor effect (P « .0138). The partial sum of squares, however, shows that after adjusting for the pretest scores, there is no evidence of sex or treatment effects (P *»
.3076, P “ .8453 respectively). In other words, the pretest scores and posttest scores for criterion one are very strongly related (P ■ .0018) and that relationship is the same for both sexes and both treatments.
Tables 29. and 30 show the cell means and cell sizes for the first criterion’s pretest and posttest scores respectively. TABLE 27
Analysis of Covariance (ANCOVA) Table of Posttest Scores on Criterion 1, ''Understanding the Problem"
Source______DF_____Sum of Squares Mean Square F Value PR F Total Sources 4 38.00 9.48 3.82 .0138
Error 27 67.04 2.48
Corrected Total 31 104.97
TABLE 28
Partial sum of squares for ANCOVA of Posttest Scores on Criterion 1, "Understanding the Problem"
Source DF Sum of Squares F Value PR F
Sex 1 2.68 1.08 0.3076 Treatment 1 0.10 0.04 0.8453
Sex * Trt 1 0.09 0.04 0.8473
Pretest score on criterion one 1 29.98 12.08 0.0018
* Sex, treatment, interaction of sex and treatment and pretest score on criterion one. 83
TABLE 29
Cell means (cell sizes) for Pretest Scores, on Criterion 1, " Understanding the problem'*
TREATMENT A B
Males 6.71 (7) 5.25 (8) 5.93 (15)
Sex
Females 6.78 (9) 6.63 (8) 6.71 (17)
6.75(16) 5.94(16) 6.34 (32)
TABLE 30 Cell means (cell sizes) for Posttest Scores on Criterion 1 , "Understanding the problem"
TREATMENT A B
Males 6.71 (7) 6.25 (8) 6.47 (15)
Sex
Females 7.44 (9) 7.38 (8) 7.41 (17)
7.13(16) 6.81(16) 6.97 (32) 04
Table 31 is the analysis of covariance (ANCOVA) table of posttest scores on criterion three, "Carrying out the plan." Table 32. lists the partial sums of squares for ANCOVA of posttest scores on criterion three.
The F-test (Table 31 ) shows no significant effect
(F *» .2753) and the partial sum of squares shows no sex nor treatment effect (P ■ .9728, P - .9609 respectively), after adjusting for the pretest scores.
Tables 33 and 34 show the cell means and cell sizes for the third criterion's pretest and posttest scores respectively. TABLE 31 Analysis of Covariance (ANCOVA) of Posttest Scores on Criterion 3, "Carrying out the plan"
Source DF Sum of Squares Mean Square F Value PR F
Total Sources 4 8.48 2.12 1.36 2753
Error 27 42.24 1.56
Corrected Total 31 50.72
TABLE 32
Partial sum of squares for the ANCOVA of Post-test Scores on Criterion 3, "Carrying Out the Plan"
Source DF Sum of Squares F Value PR F
Sex 1 .00 .00 .9728
Treatment 1 .00 .00 .9609
Sex * Trt 1 .31 .20 .6622
Pretest score on criterion 3 1 7.51 4.80 .0372
* Sex. treatment, interaction of sex and treatment and pretest scores on criterion three 86
TABLE 33
Cell means (cell sizes)for Pretest Scores on Criterion*3, "Carrying out the Plan"
TREATMENT A B
Males 7.71 (7) 7.50 (8) 7.60 (15)
Sex
Females 5.56 (9) 7.25 (8) 6.35 (17)
6.50(16) 7.38(16) 6.94 (32)
TABLE 34
Cell means (cell sizes) for Posttest Scores on Criterion* 3, "Carrying Out the Plan"
TREATMENT A B
Males 7.00 (7) 7.13 (8) 7.07 (16)
Sex
Females 6.67 (9) 6.88 (8) 6.76 (16)
6.81(16) 7.00(16) 6. 91 (32) 87
Table 35 shows the analysis of covariance (ANCOVA)
of posttest scores on criterion four, "Looking back."
Table .36 shows the partial sums of squares for ANCOVA
of posttest scores criterion four. The F-test (Table
35 .) shows that there is a factor effect on the post
test scores of the fourth criterion.(P ** .0024). After
studying the sum of squares (Table 36 ), sex seems to
have a marginal effect (P = .0509) after adjusting for
the pretest scoreB. There is no evidence of a treatment
effect (P ® .5075). The mean posttest score for males was 6.53, while the females was 7.29 which indicates
that on the average females had significantly higher
posttest scores than males (Table 36 ).
Table 37 and 38 show the cell means and cell
sizes for the fourth criterion's pretest and posttest
scores respectively.
In general, treatment did not make a significant
difference on the scores of any of the criteria studied.
If the effect of sex is considered significant, the
females did significantly better than the males on the
fourth criterion,"Looking back,"and on the overall
test. TABLE 35
Analysis of Covariance (ANCOVA) Table of Posttest-on Criterion 4, "Looking back"
Source DF Sum of Squares Mean Square F Value PR F
Total Sources* 4 28.50 7.12 5.44 .002
Error 27 35.38 1.31
Corrected Total 31 63.88
TABLE 36
Partial sum of squares for ANCOVA of Posttest Scores on Criterion 4, "Looking Back"
Source DF Sum of Squares F Value PR F
Sex 1 5.47 4.18 0.0509
Treatment 1 0.59 0.45 0.5075
Sex * Trt 1 0.00 0.00 0.9877
Pretest score on criterion four 1 22.07 16.84 0.0003
* Sex, treatment, interaction of sex and treatment and pretest score on criterion four 89
TABLE 37 Cell means (cell sizes) for Pretest Scores on Criterion-4, "Looking Back" .•
TREATMENT A B
Males 5.86 (7) 5.00 (8) 5.40 (15)
Sex
Females 5.22 (9) 5.13 (8) 5.18 (17)
5.50(16) 5.06(16) 5.28 (32)
TABLE 38 Cell means (cetil sizes) for Posttest Scores on Criterion +, "Looking Back"
TREATMENT A B
H a le s 6.86 (7) 6.25 (8) 6.53 (15)
Sex
Females 7.44 (9) 7.13 (8) 7.29 (17)
7.19(16) 6.69(16) 6.94 (32) CHAPTER V
SUMMARY OF FINDINGS, CONCLUSIONS AND RECOMMENDATIONS
Review of the Study
The purpose of this study was to compare and contrast any changes in the problem solving behavior of students who had received instruction based on Polya's model with
the behavior of students who had not received this instruc
tion but who had solved problems and shared their results.
The experimenter hoped to gain some insight into how stu dents solve verbal problems and how one might improve in struction in this area.
The thirty-two students involved in the study were in the fifth grade and had been identified as "average" stu dents. This classification was based on: 1. the results of a standardized test, and 2 . the classroom teacher's assessment of the student's mathematical achievement.
The study was conducted over a six-week period during which time the experimenter met with students three times a week for 35-40 minute sessions. Two techniques aimed at analyzing students1 performance on the subskills outlined in Polya's model were employed. One technique involved individual interviews, conducted prior to and following 90 91
treatment, and the other was a thirty-item written test,
also administered prior to and following the treatment.
Summary of the Findings
Interview Findings
A total score and four subscores were computed based
on the student's performance in the interview sessions.
This analysis provided information concerning the student's
ability to understand the problem, devise a plan, carry
out the plan and look back over the problem solution. An
analysis of covariance was computed for the post-treatment
interview score, using sex and treatment as variates and
the pre-interview score, on that particular criterion, as
a covariate. The statistics for the total post Interview
scores indicated that there was a treatment effect (p=.047)
after adjusting for the pre-interview score, but no sex
nor interaction effects. The treatment based on Polya's model yielded a significantly higher mean score than the
other treatment. After studying the statistical analysis
for each of the four criteria, the experimenter found that
the students who had received the Polya treatment had
significantly (p**.059) higher post-interview scores on
only one criterion, '{Devising a Plan." This finding was
perhaps predictable for the experimenter/teacher had spent
a considerable amount of time on step two, Devising a 92
a Plan," In this treatment. This was due to the fact that
the students viewed problem solving as simply choosing an
arithmetic operation and applying it to the numbers given
in a problem. The experimenter made a concerted effort to
teach the students to: 1) think about the problem, and
2) be aware of the variety of "plans" available to them.
The experimenter had also emphasized Polya's fourth
step, "Looking Back"; however the results were not as
impressive. There was no evidence of a treatment effect
(p = .276). The students were quite content to end with
step three, "Carrying Out the Plan," despite the experi menter/teacher's efforts to show them the value of checking
the solution against the conditions of the problem, or the
importance of "Looking ;Back."
Written Test Findings
The written test scores were broken down according to
the same criteria used in the interview evaluations, except for step two, "Devising A Plan," which the test (IPSP) did not measure. An analysis of covariance for the posttest
scores on the pretest scores, sex and treatment was done for each criterion and overall total score. The analysis
showed that there was a significant factor effect on the overall posttest scores of the written test. After ad justing for the covariate, the pretest score, however, only sex seemed to have an effect and it was marginal 93 (p ■ .067), with females performing slightly better than males. There was no evidence of a treatment effect on the
total posttest scores.
The statistics for each of the three subscales.were carefully examined. The analysis of covariance for step one, "Understanding the Problem", showed that there was at least one factor effect. The partial sum of squares, how ever, showed that after adjusting for the pretest scores, there was no evidence of sex or treatment effects. In other words, the pretest scores and posttest scores for criterion one are very strongly related and that relation ship is the same for both sexes and both treatments.
The analysis of covariance for step three, "Carrying
Out the Plan," showed no significant effect and the par tial sum of squares showed no sex nor treatment effect, after adjusting for the pretest scores. There was a factor effect on the posttest scores of the fourth criterion, "Looking Back" (p “ .002). Studying the sum of squares, the experimenter noted that sex seemed to have a marginal effect. The mean posttest score for males was 6.54 while the females was 7.29, which indicates that on the average the females had significantly higher posttest scores than males.
In general, treatment did not make a significant difference on the written test scores of any of the three criteria studied. Sex, however, did seem to have an effect on the posttest scores; the female students did mar
ginally better than the males on the fourth criterion,
"Looking Backhand on the overall test.
Discussion of the Results
In focusing on the effects of treatment, one finds that
in the interview analysis the treatment based on Polya's model had an effect on both "Devising a Plan" and the total post-interview scores. In the written test, however, treat ment had no significant effect on the posttest scores. Two factors must be considered in interpreting these effects.
Therfc was a strong relation between the treatment based on
Polya's model and what was asked of the students in the
interview session. Also, according to the results of the
post-interview, the students in the Polya treatment improved
significantly on step 2, "Devising a Plan;" this step was not evaluated on the written test. In the Polya treatment group, the students were taught to solve problems by ad dressing each aspect of Polya's model, namely attempting to understand the problem thoroughly, devising a plan, carrying out the plan and looking back. They had followed this
procedure for six weeks while working with many problems.
They were confident that this approach was a viable means of finding a solution. When asked to solve a problem dur
ing the interview session, many of the students confidently circled "the unknown" and then considered a number of 95
techniques that they had become very familiar with during
the treatment, such as making a list, drawing a picture,
constructing a table, etc. In general the students in the
other treatment group did not convey the same confidence
vdien faced with a problem during the post-interview.
Even though these students eventually arrived at some of
the same "plans" during the treatment period, they did so
by chance and only after they had tried to apply several
operations to the numbers in the problem. Even after six weeks of working problems together and discussing the ans wers, when faced with a problem in a testing situation, many students in this group relied on the approach that had often proved successful in the classroom, that is, "to do
something with the numbers in the problem." (Jimmy's interview on page 122 is a good example of a student deter mined to perform some operation on the numbers given.").
The treatment based on Polya's model therefore seemed
to provide the students with a variety of problem solving
techniques which many of the students used with confidence.
Evidently this aspect of the treatment did positively
effect the student's post-interview scores on step two,
"Devising a Plan," and in turn, their overall scores.
Unfortunately, this aspect of Polya's model was not assessed
in the written test. Since step two was the only criterion
that showed that the treatment had had a significant effect
on the post-interview scores, the experimenter finds the 96 results of both the written and interview evaluation to be
consistent.
Conclusions
The students in the treatment based on Polya's model
evidenced a significant increase in their total interview
scores. This improvement was primarily due to an im
provement on criterion two, "Devising a Plan." The stu
dents in both treatments began with a strong conviction
that problem solving was nothing more than the application
of an algorithm to numerical data in a problem statement.
In the Polya treatment group, the experimenter spent a
great deal of time combating this notion, stressing the
variety of techniques available in "Devising a Plan" for
a solution to a problem. The students in the other treat ment group were also casually exposed to a number of alter
native approaches to problem solving, however, they did not
appear to be confident that "guessing and testing," draw
ing a diagram, making a list, etc. would allow them to
arrive at the correct solution and as a result they often
attempted to apply an algorithm.
At the beginning of the study both groups, in general,
had a very negative outlook concerning verbal mathematical
problems. A number of students made comments such as:
"I don't like word problems at all," and "I've never been
able to get word problems." One student remained after
the first session to thank the experimenter for selecting 97 her for she "really needed help with word problems." After several sessions the experimenter noticed an improvement in attitude in both treatment groups. The students enjoyed the work sessions and the challenge that each problem pre sented. In the opinion of the experimenter, the approach used in the less directed treatment did not give the stu dents the security and confidence that was enjoyed by those in the treatment based on Polya's model. The students in the latter treatment often referred to Polya's four steps and "talked through" the entire procedure. On a number of occasions the students in the Polya treatment asked to remain beyond the session to complete a problem and cautioned the other students not to tell him or her the answer. Further more, on several occasions, a student in this group, upon completing a problem, also asked if he or she could take it home to see if a sister, brother, or parent could solve the problem. This enthusiasm for the once dreaded "word problem" was a clear indication of a change in attitude.
The students in the less-directed treatment also evidenced some of the same positive behaviors but not as frequently.
In general, it was most encouraging to see the changes in the student's attitude concerning verbal problems and the student's ability to use a variety of problem solving techniques. The experimenter spent a considerable amount of time attempting to erase several of the misconceptions that seemed entrenched in the student's minds. These 98 misconceptions included the seemingly instinctive desire to
apply an algorithm at random to the numerical data in
the problem, students ' disregard for the need to check both
the computations and the solution against the conditions of the problem and the idea that problems should be solved as quickly as mathematical exercises.
Recommendations
This study should be replicated using a larger number of fifth grade students. Consideration might also be given to the inclusion of students of varied abilities. Although this investigation dealt with the "average" students, a large number of studies pertaining to this topic, as noted in Chapter Two, have involved exceptionally capable students.
Studies that would compare the progress made by children rated as high, average and low would offer the classroom teacher some valuable suggestions for improving instructional strategies when working, as many do, with all three levels simultaneously. A longer time period for the study should also be considered, perhaps ten to twelve weeks. This addi tional time would allow the teacher to deal with the many misconceptions that students have concerning problem solv ing.
Furthermore, there is need for the development of valid and reliable paper and pencil instruments for measuring the processes used by students in solving problems. 99
The experimenter chose the 1PSP tests, for the present
study, believing that it was the most thorough and reliable
instrument available; however, it fails to evaluate how
a student arrives at a plan of action and the basis for that
decision. The interview evaluation is most imformative and
reliable, however, it is very time consuming and therefore
usually not practical for a teacher to use with the entire
class.
There are a number of ramifications of this study for
the classroom teacher. In general, all of these relate to a clear understanding of problem solving and a realization
that the method is as important as the solution. First,
it is fundamental that the students realize that problem solving is more than simply choosing an operation and applying
it to the numbers given. Instruction that reinforces this simplistic approach to problem solving more than likely con
tributes to the student's difficulty in solving problems
that do not bear headings such as "Some Problems to Multi ply," or "Problems to Test Your Division Skills." An over emphasis on one-step problems often reinforces the students ' notion that they do not have to think about what
they are doing or analyze the problem in any detail.
Teachers should encourage students to think about a problem and develop a plan for solving it, based upon the data 100 given and the ''unknown" they are asked to find. In order
to do this teachers should offer the students a variety of problems such as problems that have extraneous or in sufficient information; those that can be solved by several methods; problems without numbers, allowing the students to focus on the "plan"; those which require several steps in order to be solved, etc. Since the majority of textbook
"problems*' do not fall into any of the categories listed, the teachers must search for or create such problems.
In regard to the importance of the method over the solution, several other recommendations can be made. The classroom teacher should encourage students to take suffi cient time to study the conditions in the problem. This
"incubation" time is most important and is often not con sidered, giving the students the Impression that problems are to be solved as quickly a& computationalexercises. Teach ers often sustain, perhaps subconsciously, the misconception that the correct answer is more important than the proce dure employed by focusing on the solution and giving little
t attention, if any, to how or why a certain method was chosen.
The previous suggestion to use problems without numbers would force both teacher and students to focus on an Impor tant aspect of Polya's model i.e. "Devising a Plan."
Teachers should also study student's "paper work" in pre ference to just their answers, in an attempt to determine 101
their strategies, misunderstanding or perhaps computa
tional errors. In addition to evaluating the student's
written Work, the teacher could use the interview tech
nique described in detail in this study. This procedure
is time consuming; however, used selectively with certain
students or in small groups, it can be a very effective
diagnostic tool.
Finally, as a result of the present study, the
experimenter/teacher recommends the Polya model as a very
effective means of providing the students with a structure
or frame of reference, enabling them to approach problems
with confidence. This model, while offering direction, also
gives the students the flexibility to apply its principles
to problems not only in a mathematical realm but in the
realm of daily life.
If "learning to solve problems is the principal reason
for studying mathematics" (NCSM, 1977, p.20) then teacher
educators should assist pre-service and in-service teachers
in developing programs that speak to this concern. How many classroom teachers are familiar with Polya's model and/or
the suggestions that he and others have offered concerning
this topic? How often have they been asked to engage in problem solving activities themselves? Are they aware of
the variety of problems that they can and should offer 102
students? Teachers generally do not feel comfortable with
this topic and as a result, rely solely on the inadequate
textbook. Teacher educators have an obligation to 1) help
teachers see that the most important goal in mathematics
is to develop each child's ability to solve problems, and
2)provide teachers with the curriculum, materials, and
confidence essential in the development of competent prob
lem solvers.
One method of fulfilling this second aspect of the
teacher educator's role would be the establishment of a
"problem bank," or large collection of noteworthy problems
for use in the classroom. The creation of such a resource
would involve categorizing, evaluating, comparing and
creating problems by pre-service and in-service teachers.
The time and effort expended on such a project would cer
tainly reinforce the teacher's personal skill in problem
solving as well as provide an abundant supply of problems.
Armed with diverse problems and with the confidence gained
in solving them, the teacher should be better able to meet
the challenge of developing each child's ability to solve problems. APPENDIX A
PROCEDURE FOR ASSIGNING GROUPS
SEATING CHART
103 104
Procedure for Assigning Groups
The students were chosen from two fifth grade classrooms.
Initially these classrooms were formed by random assignment.
The "average" students from each classroom were ranked sep
arately according to the criteria explained in Chapter Three,
page 32, The "8" and "11" in the student's identification number refers to the student's classroom number (see chart
below). The first boy from room eleven, 1-11, was assigned
to treatment A, the next to treatment B, the next boy to B
and the next to A and so on. The girls were assigned in
the same manner. In room eight, the experimenter began the assignment pattern with treatment B for both girls and boys
in an attempt to equalize the two groups.
Assignment Chart
Identification Treatment Identification >Treatment
1-11 male A 1-11 female A
2-11 male B 2-11 female B
3-11 male B 3-11 female B
4-11 male A 4-11 female A
5-11 male A 5-11 female A 6-11 male B 6-11 female B
7-11 male B 7-11 female B
8-11 male A
9-11 male A 105
Identification Treatment Identification Treatment
1-8 male B 1-8 female B
2-8 male A 2-8 female A
3-8 male A 3-8 female A
4-8 male B 4-8 female B
5-8 male B 5-8 female B
6-8 male A 6-8 female A 7-8 female A
8-8 female B
9-8 female B
10-8 female A
Treatment A Treatment B
1-11 male 2-11 male 4-11 male 3-11 male 5-11 male 6-11 male 8-11 male 7-11 male 9-11 male 2-11 female 1-11 female 3-11 female 4-11 female 6-11 female 5-11 female 7-11 female
2-8 male 1-8 male 3-8 male 4-8 male 6-8 male 5-8 male 2-8 female 1-8 female 3-8 female 4-8 female 6-8 female 5-8 female 7-8 female 8-8 female 10-8 female 9-8 female 8 males, 8 females 7 males, 9 females 106
SEATING CHART
X X
X X
X X
XX
X «* student APPENDIX B
A SAMPLE OF THE QUESTIONS POSED IN TREATMENT A
EXAMPLES OF PROBLEMS USED THROUGHOUT THE STUDY
107 103 . A Sample of the Questions Posed In Treatment A
Problem: Fifteen couples have been invited to a birthday
party. The host has several small tables that can seat
one person on each side. He plans to set the small
tables end to end to make one long table to seat all
of the guests. How many small tables will be needed
to seat the 15 couples with one person on each end?
Understanding the Problem
- How many people were invited to the birthday party?
- Can four people sit at each small table?
- Can you tell in your own words what the problem is
asking?
Devising a Plan
- Name some of the strategies that we have discussed.
(Since the teacher and students have solved several
problems together, the students were familiar with
a number of strategies and referred to them by
name, guess and test, organized list, draw a dia
gram) .
- Which strategy do you think would be the best one to
use in solving this problem?
(The teacher encouraged the students to suggest
strategies. Throughout the devising-a-plan step
the teacher emphasized the type of strategy each
child suggested). 109 Carrying Out the Plan
The students usually did not have any difficulty sol ving the problem once a strategy had been selected. If
the strategy chosen was inappropriate the teacher, after a reasonable length of time, suggested that the student return to step two and devise another plan.
Looking Back
There are two aspects of this step: one is looking back over the steps taken and the other is extending the problem situation to create variations or an entirely new problem.
- What strategy did you use?
- Were you able to solve the problem using that
strategy?
- Did you reread the problem to see if your solution
fits the question in the problem?
- Does the answer make sense?
- Did anyone use a different strategy to help them
solve the problem? To extend the problem, the teacher asked:
- How many small tables would be needed to seat 14
couples, 13, 12, etc.? How do the number of tables
change when the number of couples change?
-Suppose the couples wanted to sit directly across from
one another. How many tables would be needdd for
fifteen couples? 110
The following are examples of problems used throughout: the study:
Don bet Paula that he could score a total of exact ly 34 points on his dart board. What is the least number of darts needed to score 34? EM Jim works in an ice cream store. Ice cream cones cost 25 cents. Show the ways that Jim can be paid exactly 25 cents for an ice cream cone.
A battery on a baseball team consists of a pitcher and a catcher. If a team has 6 pitchers and 2 catchers, in how many different ways can a battery be chosen?
A caterpillar is put on the bottom of a jar that is 8 inches high and 7 inches across. The jar has a lid with holes in it. Every day the caterpillar crawls up 4 inches. Every night it slips back 2 inches. How many days will it take for the caterpillar to touch the lid of the jar.
Steve brought his can collection to show his class. He displayed the cans by stacking them in the shape of a triangle. On the top of the stack was one can. On the next row down were 2 cans. The third row had three cans. If Steve used 6 rows, how many cans did he display altogether?
There were 8 people at a party. If each person shook hands with everyone else, how many handshakes were there? Ill
Betty has $54 is savings, and her brother Tom has $30. They would like to save more money for their vacation. If Betty can save $3 a week, and Tom can save $5 a week, how long will it take until Tom has the same amount of savings as Betty?
Fifteen couples have been invited to a birth day party. The host has several small tables can can seat one person on each side. He plans to set the small tables end to end to make one long table to seat all of the guests. How many small tables will be needed to seat the 15 couples with one person on each end?
Below are examples of problems used when addressing "rele vant and irrel ivant information."
Linda had a piece of wood that was 29 inches long and 2 inches thick. She wanted to cut it into 2 equal pieces. How long would each piece be?
There are 82 pirates on a sinking ship. How many 7-man rowboats are needed to save all the men? Each rowboat is 18 feet long.
The following are examples of problems used when discus sing the importance of checking to see if ones answers are reasonable. Fred earns $.75 an hour. He worked 12 hours one week. He said he earned about $75 that week. Is that reasonable? 112
Patty had 8 boxes of straight pins. Each box contained 525 straight pins. She said she had less than 2000 straight pins. Is that reason able?
Tom was going to do 50 sit-ups on Monday and add ten sit-ups each day of the week. On Saturday Tom would do 100 sit-ups. Is that reasonable? APPENDIX C
SCORING OF INTERVIEW PROBLEMS INTERVIEW CHECKLIST SAMPLE INTERVIEW TRANSCRIPTS INTERSCORER RELIABILITY CHART
.113 Scoring of Interview Problems
Each problem solved in an interview situation was
scored by awarding 0, 1, 2 or 3 points in each of the four areas on which the interviews were based. Similar scoring
schemes have been used by Lucas (1974), Kantowski (1974),
Webb (1975), Putt (1978), and Proudfit (1979).
Understanding the Problem
(The investigator attempted to determine whether the
student had complete, partial or no understanding of the pro
blem presented.) The following questions were considered.
1) Does the student attend to all conditions?
2) Does the student disregard irrelevant information?
3) Does the student understand the question?
4) Does the student make correct decisions based
upon the information given?
Points for this subscore were awarded as follows:
0 Ignores more than one condition in the problem.
1 Ignores one condition in the problem
2 Misinterprets part of the problem
3 Completely understands the problem
Devising a Plan
(Through the interviews, the investigator attempted
to gain an understanding of the method or methods employed 115 by the students while trying to solve the problem.) The points were awarded as follows:
0 No attempt at a solution.
1 Totally inappropriate plan.
2 Method could not lead to a correct solution, yet
a modification would result in an appropriate
method.
3 A plan which could lead to a correct solution,
provided there are no executive (arithmetical)
errors.
Carrying Out the Plan
(The behaviors examined in this phase of the problem solving process dealt with the manner in which the children used the selected strategy in an attempt to reach a solu tion. The investigator analyzed the student's work in terms of the following questions: 1) Is the method system atically employed? 2) Are any errors made while carrying out the plan? 3) Are these errors structural that is caused by inadequate understanding of the problem or executive, caused by incorrect calculations or carelessness? Points were awarded as follows:
0 The plan was carried out incorrectly due to a
structural error
1 An appropriate plan was partially carried out 116 2 An appropriate plan was carried out incorrectly due to an executive error. 3 An appropriate plan was correctly carried out to
reach the correct solution.
Looking Back
(The investigator was interested in recording any checking that occurred. A distinction was made between checking to determine if the plan was appropriate for the problem and checking.for computational errors.) Points were awarded as follows:
0 No checking occurred
1 The student checked for computational errors.
2 The student checked the solution against the
conditions of the problem.
3 The student checked the computation as well as
the solution against the conditions of the
problem.
Total Score The total score was obtained by adding the scores for each category. This gave an approximately equal weight to each of the four aspects of the problem-solving process.
While it might be argued that weights should not be equal, research does not provide a valid alternative. 117
Interview Checklist
Name______Interview______
Understanding the Problem , rereads the problem asks questions the information in the problem asks questions £e. the question to be answered looks for relevant information focuses on numbers, rather than on total situation ignores condition misinterprets question uses a diagram
Devising a Plan draws a diagram writes numbers from problem introduces additional conditions recalls a related problem
Carrying Out the Plan uses trial and error uses irrelevant information selects solution on irrelevant basis makes inaccurate model of problem forgets question forgets condition makes executive error
Looking Back checks calculations rereads problem checks that solution satisfies condition solves problem again using the same strategy solves problem again using a different strategy Interview Transcript 118
Student: Ed
Problem: Diane was practicing free throws. Each day she doubled the number of free throws that she had made the day before. Sunday she made 1 free throw. On Monday, she made 2 free throws. How many free throws did she make on Friday.
E. On Sunday she made 1 free throw and on Monday 2 free throws and that's it - so that's three.
I.* Are you finished?
E. You can't find out because they didn't tell you enough.
I. What didn't they tell you, Ed?
E. What about the other days?
I. Do you think it would help to read the problem again?
E. (Rereads problem) I don't know about the rest.
I. Do you know what doubling means?
E. Yeah, it's like twice as much.
I. If I doubled 3 what would I get?
E. Six.
I. What did it say about doubling in the problem?
E. (Looking at the problem) She doubled the number of free throws that she made the day before.
I. How many did she make on Sunday?
E . One.
I. On Monday?
E. Two (pause) Did she make 4 on Tuesday?
I, Why do you think it might be 4?
E. I doubled it. 119 I. OK
E. On Wednesday 8 , on Thursday 16, Friday 32. Thirty- two - if she kept going she would have to make a lot of baskets.
I. Should we keep going?
E. I don't think so, they want Friday.
1. Did the problem tell you enough, Ed?
E. Yeah, but I didn't think it did.
*Interviewer Interview Transcript
Student: Julie
Problem: Someday Scott is going to be a great basketball player. On Saturday, he made 20 baskets. Each day he makes 10 more baskets than the day before On which day did he make 100 baskets?
J. First I circle what they want. I'm going to make a list. On Monday he made 20, Tuesday 30, Wednesday A0, Thursday 50, Friday 60, Saturday 70, Sunday 80, Monday 90 and Tuesday 100. (See paper work.) So on Tuesday he made 100! (pause)
I.* Julie, you know that there are different ways to solve problems, why did you choose a list?
J. Well, the days just make a list. You know, Monday, Tuesday, Wednesday and it's not one for "guess and test."
^Interviewer 121 Interview Transcript
Student: Sam
Problem: Diane was practicing free throws. Each day she doubled the number of free throws that she had made the day before. Sunday she made 1 free throw* On Monday she made 2 free throws. How many free throws did she make on Friday?
(Pause)
S. On Sunday she had one, Monday 2, Tuesday 4, Wednesday 6 Thursday 8 , and Friday 10. So Friday she had 10 baskets. (He wrote down the numbers as he spoke.)
I.* How did you figure that out, Sam?
S. I just read the problem and went down the days.
I. You said 4 on Tuesday, 6 on Wednesday, 8 on Thursday and so on, how did you know that?
S. (Pause) because she doubled it each day.
I. Sam, what does doubling mean?
S. Two, like twins! You just add two. (There was silence as Sam added up the column of numbers.)
S. She made 31 baskets.
I. Is that your answer?
S . Yeah.
* Interviewer Interview Transcript 122
Student: Jimmy
Problem: Someday Scott is going to be a great basketball player. On Saturday he made 20 baskets. Each day he makes 10 more baskets than the day before. On which day did he make 100 baskets?
J. Let's see (pause)
Jimmy writes 30 /TOO
I.* Jimmy, where did you get the 30?
J. I added 10 and 20.
I. Why did you decide to do that?
(pause)
J. Oh, I should take 30 times 100
Jimmy writes: 30 100 “00 000 3000 3000 J. No, that can't be right, he couldn't make 3000 baskets!
I. How did you know what to do, Jimmy?
J. I want to use the numbers in the problem.
I. How do you know what to do with the numbers?
(pause)
J. Oh, I know.
Jimmy writes: 20 20 20 20 20 TOO
J. In a week he made 100 baskets. I. How did you know it was a week?
J. Well, (he counts the 20*s) in five days.
1. Is that what the problem asks?
J. (Looking at the problem) On which day did he make 100 baskets? J. (Pointing to each 20 and the 100) Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. Saturday.
*Interviewer 124 Interview Transcript
Student: Ed
Problem: David's family has bikes and trikes. There are 10 wheels in all. How many bikes and trikes could there be in David's family?
E. Trikes - they have three wheels, don't they?
I.* Yes. E. My sister has a trike. Well if they had 2 bikes and 3 trikes they would have - (pause) - let’s see, 3 times 3 equals 9 and 10, 11, 12, 13 wheels. That's too many. Did it say anything about unicycles?
I. I don't think so, but how could you find out?
E. (Looking at the problem) No, just bikes and trikes. (Pause)
I. What are you thinking about?
E. I guess I have three too many wheels and that's what is on a trike so he could have 2 bikes and 5, 6 , 7, 8 , 9, 10, and 2 trikes. That will do it. That's 10.
1 . How did you solve that problem?
E. I just guessed it.
^Interviewer 125
Interscorer Reliability Chart
Evaluator # 1 2 3 4 5 Aver- Experi- age menter
Student A Step 1 0 0 0 0 0 0 0 Step 2 2 1 1 1 1 1.2 2 Step 3 0 0 0 0 0 0 0 Step 4 2 2 _* 2 0 1.5 2
Student B Step 1 0 0 0 0 0 0 0 Step 2 3 3 3 3 3 3 3 Step 3 3 3 3 3 3 3 3 Step 4 2 2 2 _* 2 2 3
Student C Step 1 3 3 3 3 3 3 3 Step 2 3 3 3 3 3 3 3 Step 3 3 3 3 3 3 3 3 Step 4 3 3 3 2 -* 2.8 3
Student D Step 1 2 1 1 2 3 1.8 2 Step 2 2 2 2 2 2 2 2 Step 3 0 0 0 0 0 0 0 Step 4 1 0 0 0 0 .2 0
* Omissions were not used in determining the average score 126
Step 1 - Understanding the problem
Step 2 - Devising a plan
Step 3 - Carrying out the plan
Step 4 - Looking back
The experimenter had the advantage of observing the
child as he "looked back" over his calculations and/or
re-read the problem as well as listening to his comments.
Since the evaluators did not have this opportunity,
they were instructed to omit a score for "looking back" when this step was not evident from the transcript.
This omission was designated by a dash. 'APPENDIX D
IOWA PROBLEM SOLVING TESTS AND THE STATISTICAL DATA FOR TESTS
127 PLEASE NOTE: Copyrighted materials In this document have not been filmed at the request of the author. They are available for consultation, however, 1n the author's university library. These consist of pages: 128-140
University Microfilms International 300 N. ZEEB RD.f ANN ARBOR. Ml 48106 <3131 761-4700 141
BIBLIOGRAPHY
Ashlock, R. B. Error patterns In Computation (2nd ed.). Columbus, Ohio: Merrill, 1976.
Ashton, Sister M. R. Heuristic methods in problem Solv ing in ninth grade algebraT Unpublished doctoral dissertation, Stantora University, 1962,
Balow, I. H. Reading and computational ability as determinants of problem solving. The Arithmetic Teacher. 1964, 11, 18-22,
Bartless, F. C. Remembering. Cambridge: Cambridge University Press, 19327
Bassler, 0. C., Beer, M. I. and Richardson, L. I. Comparison of two instructional strategies for teach ing the solution of verbal problems. Journal for Research in Mathematics Education, 1975*] 6 , (3), m r-v m ------Bell, M. S. Applied problem solving as a school empha sis: An assessment and some recommendations. In R. Lesh, D. Mierkiewica and M, G. Kantowskl (eds), Applied Mathematical Problem Solving. Columbus, Ohio : ERIC/5MEAC, 1979.------Blake, R, N. The effect of problem context upon the problem solving processes used by field dependent and independent students: A clinical study. (Doctoral dissertation, University of British Colum bia, 1976), Dissertation Abstracts International, 1977, 37, 4191A-41922T]
Bloom, B. S. and Broder, L. J. Problem solving processes of college students. Supplementary Educational Monographs. 1950, 73.
Bourne, L. F. Ekstrand, P. R. and Dominowski, R. L. The Psychology of Thinking. Englewood Cliffs, New Jersey: Prentice-Hall, 1971.
Brownell, W. A. Psychological considerations in the learn ing and teaching of arithmetic. In W. Reeve (ed.) HCTM The Teaching of Arithmetic, 1935, 10, 1-31, 142
Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Linquist, M. M. and Reys, R, G. Solving verbal problems-: Results and implications from national assessment. Arithmetic Teacher. 1980, 28^, 8-12.
Chase, C. I. The relationship of certain skills and intellectualfactors to problem solving in arith metic . Unpublished doctoral dissertation, Univer sity of California, Berkeley, 1959.
Conference on the K-12 Mathematics Curriculum. Snowmass, Colorado: Published by the Mathematics Education TDfcvelopment Center, Indiana University, Bloomington, Indiana, 1973.
Covington, M. V. and Crutchfield, R. S. Facilitation of creative problem solving. Programmed Instruction, 1965, 4, (10), 3-5.
Crutchfield, R. S. and Covington, M. V. Facilitation of creative thinking and problem solving in school children. Paper presented in a symposium on learn ing research pertinent to educational improvement, American Association for the 'Advancement of Science, Cleveland, Ohio, December, 1963.
Dahmus, M. G. How to teach verbal problems. School Science and Mathematics. 1970, 2, 121-138.
Damarin, S. X. "Are we teaching children to have 'math anxiety1?" The Ohio State University (mimeographed).
Damarin, J3. Y Problem Solving: Polya’s Heuristic Applied to Psychological Research.(ERIC Document Reproduc tion Service Ho. ED. 129 631).
Dewey, J. How We Think. Boston: Heath, 1933.
Dodson, J . W. Characteristics of Successful Insightful Problem Solvers. MtsilA keport No. 31. Stanford, California: School Mathematics Study Group, 1972.
Duncker, K. On problem solving. Pscyhological Mono graphs , 1945, 58^ (1, Whole No. 270).
Early, J. F. A study of children's performance on ver bally stated arithmetic problems with and without word clues (Doctoral dissertation, University of Alabama, 1967), Dissertation Abstracts International. 1968, 2£, 2889A (University Microfilms No. 68-1037). 143
Educational Services Incorporated. Goals for School Mathematics. Report of the Cambridge Uonirerence on School Mathematics. Boston: Houghton Mifflin. 1963.
Fennema, E. and Sherman, J. Sex-relatdd differences in mathematics achievement and related factors: A further study. Journal for Research in Mathematics Education. 1978, 9 <3)7 IB9'-203'.
Hatfield, L. L., and Bradbard, D. A. Mathematica Problem Solving : Papers from a Research Workshop. Columbus, Ohlb; ERIC/5MEa.gr,' T9/H.------
Hatfield, L. L. Heuristical emphases in the instruction of mathematical problem solving : Rationales and research. In L. L. Hatfield and D. A. Bradbard, (eds.), Mathematical Problem Solving : Papers from a Research Workshop. Columbus, Ohio: feRIC/SMEAC, 19/8.
Higgins, J. L. A new look at heuristic teaching. The Mathematical Teacher, 1971, 64_, 487-495.
Jerman, M. E. Problem solving in arithmetic as a trans fer from a productive thinking program (Doctoral dissertation, Stanford University, 1971). Disserta tion Abstracts International. 1972, 32, 5671X1 (University microfilms No. 72-11, 5757)
Joint Commission of the Mathematical Association of Amer ica and the National Council of Teachers of Mathe matics. The Place of Mathematics in Secondary Education, fifteenth year book' of the New York: Bureau of Publications, Teachers College, Columbia, University, 1940.
Kantowski, M. G. Processes involved in mathematical problem solving. (Doctoral dissertation, University of Georgia, 1974). .Dissertation Abstracts Interna- tional, 1975. 36. 2737X1 (University microfilsm No."73-23 , 76477
Kilpatrick, J. Analyzing the solution of word problems in mathematics: An exploratory study. (Doctoral dissertation, Stanford University, 1967). Disserta tion Abstracts International. 1968, 28, 4383X1 (University Microfilms tab. 6b -6 , 442.7"
Kilpatrick, J. Problem solving in mathematics. Review of Educational Research, 1969, 39, 523-534. 144
Kohler, W. The Mentality of Apes. New York; Harcourt Brace, 19^7.
Krulick, S. and Reys, R. E. Problem Solving In School Mathematics. Yearbook ot the National Council ot Teachers of Mathematics, Reston, Virginia; The Council, 1980.
Krulick, S. and Rudnick, S. A. Problem Solving ; Hand book for Teachers. Boston: Allyn and bacon, 1980.
Krutetskii, V. A. The Psychology of triathematical Abilities in school children. CJ.Riipatrick and I. WirB2u p , eds. ; J. Teller, trans.). Chicago: University of Chicago Press, 1976.
Le Blanc, J. F. You can teach problem solving. The Arithmetic Teacher, 1977, 2J5, 16-20.
Lester, F. K., Jr. Mathematical problem solving in the elementary school: Some educational and psychologi cal considerations. Paper presented at tne Research worKstiop on Problem Solving in Mathematics at the Center for the Study of Learning and Teaching Mathe matics, University of Georgia, May 1975.
Lester, F. K. Mathematical problem solving in the Elementary school: Some educational and psychologi cal considerations. In L. L. Hatfield and D. A. Bradbard (eds.), Mathematical Problem Solving : Papers from a Research Workshop. Columbus, Ohio : ERIC/SMEAc , 1978.
Lesh, R . , Mierkiewicz, D. and Kantowski, M. G. Applied Mathematical Problem Solving. Columbus, Ohio: ERIC/ SMEAC, 1979, *
Lucas, J. F. An exploratory study on the diagnotic teaching of heuristic problem solving in Calculus (Doctoral dissertation, University of Wisconsin, Madison, 1972), Dissertation Abstracts International, 1972, 12, 6825A (University microfilms "NoY'72-15’, 3btJ 0
Lucas, J. F. The teaching of heuristic problem solving strategies in elementary calculus. Journal for Research in Mathematics Education. 1974T37T5T" 36-46':------145 Mathematical Problem Solving Project. Problem solving Strategies and applications of mathematics in the e1ementary school. (Pinal keport). Bloomington: Indiana University Mathematics Education Develop ment Center, May 1977.
National Council of Supervisors of Mathematics. "Posi tion Paper on Basic Mathematical Skills." NCSM, 1977.
National Council of Teachers of Mathematics. An Agenda for Action. Recommendations for School Mathematics of the 1980's. Reston, Virginia: The Council, 1980.
Newell, A. and Simon, H. A. Human Problem Solving. Englewood Cliffs, New Jersey: Prentice Ha117 1972.
Nicely, E. J. An exploratory study of the effects of heuristic problem-solving techniques in the sixth grade. Unpublished toaster's Thesis, l/he Pennsyl vania State University, 1976.
Oehmke, T . Development and validation of a testing instru ment for problem solving strategies of children in rades five through eight. Unpublished doctoral gissertation, University of Iowa, 1979. Polya, G. How to Solve It. Nev; Jersey: Princeton University Press, l9A5.
Polya, G. Mathematical Discovery: On Understanding Learning and Te'aching Problem Solving CVol. 1)7 New York: John Wiley, I9b2.
P61ya, G. Mathematical D iscovery: On Understanding Learning and Teaching Problem Solving cVol 2). New York : John W ile y , 19b5.
Polya, G. "On Teaching Problem Solving." In the Confer ence Board of the Mathematical Sciences, The Role of Axiomatics and Problem Solving in MathemaFTcs" Bob ton : Ginn and Company, l9bb. * Proudfit, L. The development of a process evaluation instrument (Tech Report v ) . Mathematical Problem Solving Project. Bloomington, Indiana? Mathematics Education Development Center, 1977.
Proudfit, L. The examination of problem-solving processes by fifth-grade children and its effect on problem- solying performance. Unpublished doctoral disserta tion, Indiana University, 1980. 146
Putt, X. J. An exploratory investigation of two methods of instruction in mathematical problem solving at the fifth grade level (Doctoral dissertation, Indiana University, 1978). Dissertation Abstracts Interna- tional. 1979, 39. 53B2A. (University microfilms ho. 79-OFT 255.)
Riedesel, C. A. Verbal problem solving: Suggestions for Improving instruction. The Arithmetic Teacher, 1964. 11, 321-316.
Riedesel, C. A. Problem solving: Some suggestions from research. The Arithmetic Teacher. 1969, jL6 , 54-58.
Robinson, E. On the uniqueness of problems in mathematics. The Arithmetic Teacher. 1977, 2J5, 22-26.
Scandura, J. M. Mathematical problem solving. The Ameri can Mathematical Monthly. 1974, 81(3). 273-280^
Schoen, H. L. and Oehmke, L. A new approach to the measurement of problem-solving skills. In S. Krulick and R. E. Reys (eds.), Problem Solving in School Mathematics. NCTM Yearbook, Reston, va. : The Council, i m r . ------
Silver, E. A. Student perceptions of relatedness'afoong mathematical verbal problems. Journal for Research in Mathematics Education. 1979, 10(3), 195-210.
Suydam, N. and Weaver, J. F. Using Research: A Key to Elementary School Mathematics? Columbus, Ohio: ERIC'/SHEAc, 1975.------
Treffinger, D. J. The effects of programmed instruction in productive thinking on verbal creativity and problem solving among pupils in grades four, five, six and seven (Doctoral dissertation, Cornell University, 1969). Dissertation Abstracts International. 1969, 30, 1031A.
Trimmer, R. G. A Review of the Research Relating Problem Solving and Mathematics Achievement to Psychological Variables and kelatlng these Variables to Methods in volving or Compatible with Self-Correcting Manipulative Mathematics Materials. 1974. (ERlcY)ocuraent Repro- duction Service Mo. ED 092 402). 147
Webb, N. L., Moses, B. E., and Kerr, D. R. Developmental Activities Relatedtb Summatlve Evaluation (1975-76. Tech7 Report 4). Mathematical problem Solving Pro- ■ ject. Bloomington, Indiana: Mathematics Education Development Center, 1977.
Wertheimer, M. Productive Thinking. New York: Harper and Row, 19<+5 (Enlarged edition, 1959).
Wilson, W. The role of structure in verbal problem solving in arithmetic: An analytical and experimental compar ison of three problem solving programs (Doctoral dissertation, Syracuse University, 1964). Dissertation Abstracts International. 1965, 25, 6442. (University microfilm No. 65-3445.)