atical
Sponsored by the Georgia Center
for the Study of Learning and
Teaching Mathematics
and the
Department of Mathematics Education
University of Georgia
Athens, Georgia
Larry L, Hatfield, Editor
David A. Bradbard, Technical Editor
ERIC/SMEAC CENTER FOR SCIENCE, MATHEMATICS,
AND ENVIRONMENTAL EDUCATION.
an information center to organize and disseminate
information and materials on science, mathematics, and
environmental education to teachers, administrators,
supervisors, researchers, and the public. A joint project of
the College of Education, The Ohio State University and
the Educational Resources Information Center of NIE.
-------
MATHEMATICAL
PROBLEM
SOLVING
Papers from a Research Workshop
Sponsored by the Georgia Center
for the Study of Learning and
Teaching Mathematics
and the
Department of Mathematics Education
University of Georgia
Athens, Georgia
Larry L. Hatfield, Editor
David A. Bradbard, Technical Editor
ERIC Clearinghouse for Science, Mathematics,
and Environmental Education
College of Education
The Ohio State University
1200 Chambers Road, Third Floor
Columbus, Ohio 43212
January 1978
-------
These papers were prepared as part of the activities of the Georgia
Center for the Study of Learning and Teaching Mathematics, under
Grant No. PES 7418491, National Science Foundation. The opinions
expressed herein do not necessarily reflect the position or policy
of the National Science Foundation.
This publication was prepared pursuant to a contract with the National
Institute of Education, U.S. Department of Health, Education and Welfare.
Contractors undertaking such projects under Government sponsorship
are encouraged to express freely their judgment in professional
and technical matters. Points of view or opinions do not, therefore,
necessarily represent official National Institute of Education position
or policy.
ii
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MATHEMATICS EDUCATION REPORTS
The Mathematics Education Reports series makes available recent
analyses and syntheses of research and development efforts in mathe-
matics education. We are pleased to make available as part of this
series the papers from the Workshop on Mathematical Problem Solving
sponsored by the Georgia Center for the Study of Learning and Teach-
ing Mathematics.
Other Mathematics Education Reports make available information
concerning mathematics education documents analyzed at the ERIC
Information Analysis Center for Science, Mathematics, and Environ-
mental Education. These reports fall into three broad categories-
Research reviews summarize and analyze recent research in specific
areas of mathematics education. Resource guides identify and analyze
materials and references for use by mathematics teachers at all
levels. Special bibliographies announce the availability of docu-
ments and review the literature in selected interest areas of
mathematics education.
Priorities for the development of future Mathematics Education
Reports are established by the advisory board of the Center, in
cooperation with the National Council of Teachers of Mathematics,
the Special Interest Group for Research in Mathematics Education,
and other professional groups in mathematics education. Individual
comments on past Reports and suggestions for future Reports are
always welcomed by our Clearinghouse.
Jon L. Higgins
Associate Director for
Mathematics Education
iii
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Contents
Acknowledgements and Overview
Leslie P. Steffe, Thomas J. Cooney, and Larry L. Hatfield vii
Research on Mathematical Problem Solving: An Overview
Larry L. Hatfield 1
Variables and Methodologies in Research on Problem Solving
Jeremy Kilpatrick ?
Hnuristical Emphases in the Instruction of Mathematical Problem
Solving: Rationales and Research
Larry L. Hatfield 21
The Teaching Experiment and Soviet Studies of Problem Solving
Mary Grace Kantowski ^3
Mathematical Problem Solving in the Elementary School: Some
Educational and Psychological Considerations
Frank K. Lester, Jr 53
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Acknowledgements and Overview
The Georgia Center for the Study of Learning and Teaching Mathematics
(GCSLTM) was started July 1, 1975, through a founding grant from the
National Science Foundation. Various activities preceded the founding
of the GCSLTM. The most significant was a conference held at Columbia
University in October of 1970 on Piagetian Cognitive-Development and
Mathematical Education. This conference was directed by the late Myron
F. Rosskopf and jointly sponsored by the National Council of Teachers of
Mathematics and the Department of Mathematical Education, Teachers
College, Columbia University with a grant from the National Science
Foundation. Following the October 1970 Conference, Professor Rosskopf
spent the winter and spring quarters of 1971 as a visiting professor of
Mathematics Education at the University of Georgia. During these two
quarters, the editorial work was accomplished on the proceedings of the
October conference and a Letter of Intent was filed in February of 1971
with the National Science Foundation to create a Center for Mathematical
Education Research and Innovation. Professor Rosskopf's illness and
untimely death made it impossible for him to develop the ideas contained
in that Letter.
After much discussion among faculty in the Department of Mathematics
Education at the University of Georgia, it was clear that a center devoted
to•the study of mathematics education ought to attack a broader range of
problems than was stated in the Letter of Intent. As a result of these
discussions, three areas of study were identified as being of primary
interest in the initial year of the Georgia Center for the Study of
Learning and Teaching Mathematics—Teaching Strategies, Concept Develop-
ment, and Problem Solving. Thomas J. Cooney assumed directorship of the
Teaching Strategies Project, Leslie P. Steffe the Concept Development
Project, and Larry L. Hatfield the Problem Solving Project.
The GCSLTM is intended to be a long-term operation with the broad
goal of improving mathematics education in elementary and secondary schools.
To be effective, it was felt that the Center would have to include
mathematics educators with interests commensurate with those of the
project areas. Alternative organizational patterns were available—
resident scholars, institutional consortia, or individual consortia.
The latter organizational pattern was chosen because it was felt maximum
participation would be then possible. In order to operational!ze a
concept of a consortia of individuals, five research workshops were held
during the spring of 1975 at the University of Georgia. These workshops
were (ordered by dates held) Teaching Strategies, Number and Measurement
Concepts, Space and Geometry Concepts, Models for Learning Mathematics,
vii
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and Problem Solving. Papers were commissioned for each workshop. It
was necessary to commission papers for two reasons. First, current
analyses and syntheses of the knowledge in the particular areas chosen
for investigation were needed. Second, catalysts for further research
and development activities were needed—major problems had to be
identified in the project areas on which work was needed.
Twelve working groups have emerged from these workshops, three in
Teaching Strategies, five in Concept Development, and four in Problem
Solving. The three working groups in Teaching Strategies are: Differential
Effects of Varying Teaching Strategies, John Dossey, Coordinator;
Development of Protocol Materials to Depict Moves and Strategies, Kenneth
Retzer, Coordinator; and Investigation of Certain Teacher Behavior That
May Be Associated with Effective Teaching, Thomas J. Cooney, Coordinator.
The five working groups in Concept Development are: Measurement Concepts,
Thomas Romberg, Coordinator; Rational Number Concepts, Thomas Kieren,
Coordinator; Cardinal and Ordinal Number Concepts, Leslie P. Steffe,
Coordinator; Space and Geometry Concepts, Richard Lesh, Coordinator; and
Models for Learning Mathematics, William Geeslin, Coordinator. The
four working groups in Problem Solving are: Instruction in the Use of
Key Organizers (Single Heuristics), Frank Lester, Coordinator; Instruction
Organized to use Heuristics in Combinations, Phillip Smith} Coordinator;
Instruction in Problem Solving Strategies, Douglas Grouws, Coordinator;
and Task Variables for Problem Solving Research, Gerald Kulm, Coordinator.
The twelve working groups are working as units somewhat independently
of one another. As research and development emerges from working groups,
it is envisioned that some working groups will merge naturally.
" The publication program of the Center is of central importance to
Center activities. Research and development monographs and school mono-
graphs will be issued, when appropriate, by each working group. The
school monographs will be written in nontechnical language and are to be
aimed at teacher educators and school personnel. Reports of single
studies may be also published as technical reports.
All of the above plans and aspirations would not be possible if it
were not for the existence of professional mathematics educators with
the expertise in and commitment to research and development in mathematics
education. The professional commitment of mathematics educators to the
betterment of mathematics education in the schools has been vastly under-
estimated. In fact, the basic premise on which the GCSLTM is predicated
is that there are a significant number of professional mathematics
educators with a great deal of individual commitment to creative scholar-
ship. There is no attempt on the part of the Center to buy this scholar-
ship—only to stimulate it and provide a setting in which it can flourish.
viii
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The Center administration wishes to thank the individuals who wrote
the excellent papers for the workshops, the participants who made the work-
shops possible, and the National Science Foundation for supporting
financially the first year of Center operation. Various individuals have
provided valuable assistance in preparing the papers given at the workshops
for publication. Mr. David Bradbard provided technical editorship; Mrs.
Julie Wetherbee, Mrs. Elizabeth Platt, Mrs. Kay Abney, and Mrs. Cheryl
Hirstein, proved to be able typists; and Mr. Robert Petty drafted the
figures. Mrs. Julie Wetherbee also provided expertise in the daily
operation of the Center during its first year. One can only feel grateful
for the existence of such capable and hardworking people.
Thomas J. Cooney Leslie P. Steffe Larry L. Hatfield
Director Director Director
Teaching Strategies Concept Development Problem Solving
and
Director, GCSLTM
Ix
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Research on Mathematical Problem Solving:
An Overview
Larry L. Hatfield
University of Georgia
As human endeavors go, it is a complex task to help someone else to
become a better problem solver. Yet teachers, parents, and even children
routinely engage in this task. And their efforts yield some degree of
success with most learners—witness the increasing complexity of society
and the solving skills necessary to cope with and advance civilization.
But it is still largely a mystery why certain efforts with certain learners
seem to produce either lesser or greater results. Attempts to describe
why or how a person solves a mathematical problem have resulted in rather
shallow, primitive and incomplete pictures. Explanations of how mathe-
matical problem-solving competence builds across a person's experiences
are similarly thin. Predictive theories of human problem solving are
non-existent.
Most mathematics educators consider any learning goals relative to
problem solving to be of major importance. The contributions which the
researcher might make to these goals require careful deliberation and
planning. While this may have been also necessary in the past, the
research enterprise in mathematical education has not always built upon
a thoughtful analysis of the researchable problems and scholarly methods
of solution for studying the daily functioning of classroom instruction
and learning of mathematics.
It appears that tremors of change may be assuming quake proportions
within certain sectors of educational research. Reconstructionists, such
as Cronbach (1966,1975), Shulman (1970), Snow (1974), and Magoon (1977),
are calling for educational researchers to adopt philosophies and metho-
dologies that require a break from contrived, laboratory-oriented settings.
They are advocating that the dominant context of formal education—the
classroom with groups of students studying standard schooling subjects—
must again become the experimental environment. Current psychological
theories of learning are incapable of explaining or directing activities
in such classrooms. Yet to the leading American psychologists at the
turn of this century the study of the educational process in classrooms
was the vital focus of their discipline. The subsequent rejection of all
unobservable mental processes, which characterized the transformation of
psychology during the antimentalistic revolution, led to the psycholgists1
retreat to ivory-tower laboratories and to non-human subjects.
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To a large extent the infant field of mathematics education research
continues to emulate the tactics and academic standards imposed by this
Fisherian tradition. Prototypic studies are usually short, narrowly
circumscribed, focussed on behavioral outcomes, and quickly analyzed.
Though perhaps conducted in a classroom, the research methodology usually
ignores most of the complexity of that environment by intentionally ignor-
ing the constructive processes or the situational (environmental) variables.
Shulman (1970) discussed several aspects in the study of educational envi-
ronments. He observed that if research is conducted in a setting with
characteristics similar to school situations, then one may be able to
make reasonable extrapolations to the classroom milieu. He urged a renewed
concern for external validity so that "the experimental conditions can
serve as a sample from which to make inferences to a population of external
conditions of interest" (1970, p.377). In order to study the characteris-
tics and effects of educational environments, he recommended a "distinctive
features" analysis:
To deal with the discontinuity between the settings of research
and of educational application, a common language or set of
terms for characterizing both experimental educational settings
and curricula is needed. Researchers must seriously strive to
develop a means of analyzing the characteristics of both exper-
imental and school settings into a complex of distinctive features
so the task validity of any particular experiment can be estimated
in terms of the particular criterion setting to which inferences
are being made...
I envisage ultimately a situation in which use of such a
distinctive features approach would allow one to characterize
the instructional settings to which a particular body of exper-
imental research would most effectively be applicable. Conversely,
one could begin with a curriculum of interest and use such an
approach to identify critical experiments that might be conducted
to examine particular features of the complex curricular Gestalt.
(Shulman, 1970, p. 379)
Cronbach (1966),discussing the logic of experiments on "discovery
learning," observed that a particular educational tactic is part of a total
instructional system. Placement in a context always in combination with
other tactics prohibits conclusions to be drawn about the tactic considered
simply fay itself. Educational researchers are called upon to study an
educational tactic in its proper context. The approaches used by Soviet
psychology through the conduct of "teaching experiments" offers an
important framework for such studies.
In addition to returning the educational research spotlight onto
the classroom, Magoon (1977) advocated refocusing our research on another
overlooked aspect. Based upon acceptance of cognitive views in psychology
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and sociology, he speculated that a construetivist perspective will
likely gain credibility among educational researchers. Constructivism
assumes that knowledge is a phenomenon built-up within and by the human
subject. The mechanisms and processes for such constructions are crucial
to an understanding of their knowledge and consequently for interpreting
the behaviors and actions of a subject. He proposed that approaches to
the study of such phenomena must be primarily ethnographic, involving
extensive descriptive and interpretive efforts at explaining the complex-
ity. Cronbach (1975), offering explanations for the absence of strong
experimental evidence of aptitude-treatment interactions in school settings,
recommended a similar methodological shift: researchers reverse the
priority from building generalizations about variables to attending to
each particular situation and the localized effects along with any factors
unique to that locale. Scriven (1972) also suggested a general relaxation
of the constrained traditional experimentalism. In examining the tradi-
tional concept of "reliability" he noted that if the usual objections to
self-reports were overcome, educational researchers would naturally pay
more attention to people's reasons for action in contrast to their present
attempts to determine causal accounts of it.
What this suggests for research on mathematical problem solving must
be decided by the interpretations of scholars in our field. We do not
lack for ideas and numerous successes in the teaching of a variety of
problematic emphases to differing children. Perhaps we do need to open
up to new perspectives in the study of problem-solving instruction and
learning with alternative methodologies, such as ethnographic, anthropo-
logic, and even artistic. We need to shift attention to longitudinal case
studies. New emphases on situational variables and analysis of environ-
ments should be assumed. Many scholarly, artistic mathematics educators,
essentially "turned-off" by the Fisherian tradition of experimentalism,
would find new acceptance as researchers.
The papers of this monograph present differing but compatible
perspectives for investigating mathematical problem solving. Kilpatrick
offers a careful analysis of variables and methodologies for research on
problem solving. Adopting the traditional separation of dependent and
independent variables, he presents a thorough typology for studies of
learning and for studies of teaching problem solving. In his suggestions
for methodologies he also promotes "intensive study of the same classrooms
over an extended period of time."
Following the theme for the Research Workshop of "instruction in
heuristical methods," Hatfield reviews pedagogical rationales and recent
studies in this area. Several general qualities of needed research are
offered toward the planning of future investigations.
Kantowski's paper describes the Soviet "teaching experiment" and its
origin in the U,S.S.R. Specific suggestions for its potential use in
investigations on mathematical problem solving are presented. The paper
by Lester offers an extensive review of research efforts directed partic-
ularly at the elementary school levels. The activities of the tri-site
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Mathematical Problem Solving Project, based at Indiana University, are
described as a context for several interesting research results and
questions. Since the writing of Lester's paper the project has been
discontinued, but several of the project's participants have continued
with their research efforts in this area.
The Problem Solving Research Workshop served to provide an initial
impetus to the formation of an intellectual consortium of researchers of
mathematical problem solving. The productive efforts of the participants
during the intervening years has led to solidifying the first stages of
collaborative research. Optimistically, new conceptions and results will
be forthcoming.
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References
Cronbach, L. J. The logic of experiments on discovery. In L, S.
Shulman and E. R. Keisler (eds.), Learning by discovery: A
critical appraisal. Chicago: Rand McNally, 1966.
Cronbach, L. J. Beyond the two disciplines of scientific psychology.
American Psychologist, 1975, _30, 116-127.
Magoon, A. J. Constructivist approaches in educational research.
Review of Educational Research, 1977, 47(4), 651-693.
Scriven, M. Objectivity and subjectivity in educational research.
In L. G. Thomas (ed.), Philosophical redirection of educational
research. Chicago: National Society for the Study of Education,
1972.
Shulman, L. S. Reconstruction of educational research. Review of
Educational Research, 1970, 4CI(3), 371-396.
Snow, R. E. Representative and quasi-representative designs for
research on teaching. Review of Educational Research, 1974,
44, 265-292.
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Variables and Methodologies in Research
on Problem Solving*
Jeremy Kilpatrick
Teachers College, Columbia University**
The primary consideration a researcher ought to keep in mind as
he plans and conducts a study is: What am I trying to find out by doing
this study? Anyone conducting research on problem solving in mathe-
matics needs to be especially clear about the purpose of a study since
there is so much unexplored territory in which to get lost.
Only when one has the purpose clearly formulated is it appropriate
to ask what variables are involved in the study, whether additional
variables should be considered, and what methodology or methodologies
should be used to gather data on these variables. Although it makes
little sense to choose a variable or methdology before one has settled
on the research question to be investigated, there may be some value
in discussing the kinds of variables and methodologies that are avail-
able—as an aid to the researcher who has framed a question but who has
not yet decided how to investigate it. This paper is an attempt to
survey some variables and methodologies that one might use in research
on problem solving in mathematics, with particular attention to those
that appear most promising. Research on problem solving per se is con-
sidered separately from research on the teaching of heuristics.
Variables in Research on Problem Solving in Mathematics
Variables can be classified in a variety of ways, depending on
one's purpose. Classifications include stimulus variables, response
variables, and intervening variables (Travers, 1964), and active
variables versus assigned variables (Ary, Jacobs & Razavieh, 1972).
The most common scheme, borrowed from mathematics and science, is to
classify a variable in a research study as independent or dependent.
In the narrow sense, "independent variable" refers to the condition
manipulated in an experiment. (One should recall the admonition of
David Hawkins, 1966, that "to call something an independent variable
is not to use a name but to claim an achievement [p. 6].") In the
broad sense, however, a variable is classed as independent if it is
used in making predictions. Variables referring to the behavior
being predicted are called dependent variables. A given variable can
be independent or dependent, depending on its role in the study, but
most variables tend to be used in only one way.
*Thanks are due to Nick Branca, Sandi Clarkson, Dorothy Goldberg,
Howard Kellogg, Ed Silver, and Phil Smith, who discussed a preliminary
outline of the paper and made many helpful comments and suggestions.
**Now at the University of Georgia.
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Independent Variables
Any study of problem solving in mathematics involves a person
(subject) solving a mathematical problem (task) under some condition
(situation). Each of these components can be used to define a class
of variables.
Subject Variables
Subject variables can be categorized according to whether or not
they are based on a sample of the subject's behavior. Some variables
describe the subject as a person: his sex, height, educational status,
etc. Such variables can be measured by direct observation, examination
of records, or the subject's own report. Other variables are based on
inferences derived from a sample of the subject's behavior, say, in
response to a test or questionnaire or as observed by a clinician. Such
variables include various kinds of aptitudes, abilities, attitudes, and
achievement.
A related, and perhaps more useful, classification of subject
variables is based on the extent to which they can be modified experi-
mentally. Variables not open to such modification are sometimes termed
"organismic" or "assigned" variables. Variables open to some modifica-
tion (and requiring a sample of behavior) may be termed "trait" varia-
bles. (The last two terms are not standard and perhaps not even
satisfactory since such traits as general mental ability are considered
all but unmodifiable in most research contexts, and not all modes of
modification can be termed "instruction." For the purpose of this
paper, However, the terms will suffice.)
Organismie variables. In research on problem solving in mathema-
tics, information on organismic variables such as age, sex, race, and
social class may be gathered to assist in describing the sample, but
with the exception of age and sex, such variables are seldom used as
dependent variables in the design (and then typically in subsidiary
hypotheses only). Samples of different ages are sometimes drawn in
studies of developmental change, but inferences about the development
of problem-solving proficiency must be especially tentative in view of
the large role instruction apparently plays. Researchers often must
weigh the advantages of a sample of young, inexperienced subjects (so
that one can study problem-solving processes in a relatively pristine
form, with greater opportunity for detecting "developmental" changes)
against the advantages of a sample of older, more experienced subjects
(so that one can study a greater variety of more sophisticated pro-
cesses).
Trait variables. Traits include abilities (such as spatial visual-
ization ability or memory for problems); attitudes, interests, and
values (such as attitude toward mathematics or interest in proving
theorems); and other personality variables relating to perceptual style,
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cognitive style, self-concept, persistence, anxiety, need for achieve-
ment, sociability, etc. Except for the variables dealing with style,
which by definition refers to a consistency in behavior across a wide
class of situations, each kind of trait variable ranges from the
general to the specific (as the examples in parentheses above are meant
to suggest). A general trait such as persistence, for example, not
only is likely to be less manipulatable than its more specific cousin
persistence in solving problems of type X, but is also likely to be
less strongly related to ability in the processes used in solving pro-
blems of type X. Although ultimately one wants—for purposes of a more
powerful theory—to link problem-solving process abilities to variables
that are as general as possible, one is probably best advised to begin
with variables of some specificity.
As an example, consider the ability to estimate. Psychologists
tend to think of this ability—when they think of it at all—as
general and presumably unitary. A study by Paull (1971), however,
suggested not only that there are distinct estimation abilities in
mathematics but also that they may relate in different ways to
problem-solving performance.
Abilities relating to memory, classification, generalization, esti-
mation, judgment, verification, and the like are required in the solution
of a complex mathematical problem (see Krutetskii, 1976, for a delinea-
tion of these and other abilities). Researchers should consider
including measures of such abilities in their studies of problem solving,
but again the measures should be specific to the phenomena being studied
(memory for problems, say, as opposed to associative memory) if a choice
between general and specific measures must be made.
Trait variables that seem to have particular promise of being asso-
ciated with problem-solving performance in mathematics include the
ability to generalize a relationship from a small number of instances,
the ability to classify problems according to their mathematical struc-
ture, the ability to recall structural features of a problem, the
ability to estimate the magnitude of a numerical solution, the ability
to detect extraneous and insufficient data, a resistance to fatigue in
performing mathematical tasks, a sensitivity to problem situations, a
preference for elegance in problem solutions, a reflective cognitive
style, and a field independent cognitive style. For most of these
variables, measuring instruments need to be developed and refined much
further. Two categories of trait variables that might be explored in
relation to mathematical problem solving are individual differences in
brain hemisphere functions (Wittrock, 1974) and in ability to handle
semantic versus syntactic processes (Simon, 1975).
instructional history variables. The instructional history of the
subjects—the topics they have studied, the problems they have attempted
previously, the techniques of problem solving they have been taught,
the types of instruction they have received—generates a set of varia-
bles that can be used in describing the sample of subjects, used in
selecting the sample, or (as treatment variables—see below) manipulated
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10
as part of the study. Such variables are clearly relevant to the
problem-solving process, but they have seldom been considered expli-
citly except in studies comparing the relative effectiveness of two
or more instructional methods. Failure to take account of variation
in prior instruction may account for some of the failure to find
differences between methods. Even in studies that do not involve
treatment comparisons, specification of instructional history varia-
bles is likely to assist in the interpretation of results.
Task Variables
A simple-minded classification of problem tasks (Kilpatrick, 1969)
is according to content and structure. Both of these categories bear
further examination and elaboration.
Context variables. Suppose two mathematical problems involve the
same numbers in the same relation but one deals with rabbits and chickens
in a barnyard, the other with two boats on a river. Most people
would agree that the two problems are the same in (mathematical) struc-
ture, but what word expresses their difference? "Content" might seem
suitable at first, but on reflection, "context" appears marginally
better since it avoids the connotation that the mathematical content
is different.
Iftiatever the term, the semantic variables characterizing the
differences between the physical situation modeled in the problem, as
well as the syntactic variables characterizing the language in which
the problem is expressed, need to be explored both analytically and
empirically.
Structure variables. The issue of problem structure is also more
complicated than first thought might suggest. Consider the following
problem:
Find the volume F of the frustum of a right pyramid with
square base, given the altitude h of the frustum, the
length a of a side of its upper base, and the length b of
a side of its lower base.
One way to characterize the structure of this problem is to say that
the formula
a2 + ab + b2
expresses the relation among problem elements and that any other pro-
blem in which elements , , , and are in this relation, with ,
, and given, has the same structure as this problem. The two
problems might be said to have the same syntactic structure.
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11
Another way to characterize the structure of the problem is in
terms of the network of all possible steps from one state of the pro-
blem to another: the state-space (Goldin & Luger, 1975). For the
problem above, a sketch of the state-space is given on the inside
front cover of Volume 2 of Mathematical Discovery (Polya, 1965).
Another problem having the same state-space might be said to have the
same semantic structure. Or perhaps the distinction is better
expressed as the structure of the problem (in terms of its mathe-
matical formulation) versus the structure of the problem space (in
terms of the set of all possible steps in solving the problem).
Again, regardless of the terminology adopted, the underlying ideas
deserve consideration. A host of research problems revolve about the
issue of structure: Can subjects classify problems according to struc-
ture? Is there an advantage to training subjects to make such classi-
fications? What problem features facilitate transfer across problems
differing in context but not structure? Until some dimensions of
structure are identified more clearly, the effects of similarities
and differences in problem structure cannot be studied systematically.
Format variables. A problem may be presented to a subject orally
or in written form. It may or may not involve the manipulation of some
apparatus. The instructions may involve the presentation of rules or
boundary conditions to be observed in solving the problem, or the sub-
ject may be expected to induce these rules or conditions as part of the
problem. The problem may be given all at once or one part at a time.
The subject may or may not be given hints or encouragement as he solves
the problem. He may be asked to think aloud as he works or to retro-
spect over the course of his solution. He may or may not be permitted or
be encouraged to record scratch work. All these variables can be
classed as format variables. They are seldom manipulated systemati-
cally in a study since they are not ordinarily of interest to the
mathematics educator. By ignoring them, the researcher tacitly assumes
they do not affect the relationships he observes. Since all generali-
zations from research on problem solving need to be validated across
situations and data gathering methods, however, variation in problem
format can be included as part of the validation.
Situation Variables
The dividing line between format variables and situation varia-
bles is not entirely clear. If a subject were given insufficient or
misleading information about a problem, the issue would appear to be
one of format. If he were told that the experimenter was interested
in how fast he solved the problem, when in fact the experimenter was
recording how many errors he made, the issue would appear to be one
of situation. Both cases involve instructions, but it seems useful
to distinguish between variation in the content of the instructions
(format) and variation in the subject's perception of the purpose of
the task (situation).
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12
A situation variable concerns the conditions, physical and psycho-
logical, under which the subject solves, or attempts to solve, the
problem. Situation variables include whether or not the subject volun-
teers for the study, whether or not he is given extrinsic rewards such
as money or grades, whether he works alone or in a group, the time of
day at which data are gathered, the presence or absence of distrac-
tions, the characteristics and behavior of the interviewer, the nature
of the problems given previously and the subject's success with them,
and whether or not the subject was told how he did on the previous pro-
blem. Like format variables, situation variables themselves are of
relatively little interest to the mathematics educator, although they
may interest the social or educational psychologist. They are nuisance
variables—since they will not go away, one can only hope they will not
make much difference. Unfortunately, they may.
Dependent Variables
Some dependent variables are derived from the subject's responses
to a problem task; others require additional samples of behavior. Let
us consider the latter type first. (Recall that the studies under
discussion concern problem solving per se; studies involving instruc-
tion in heuristics are yet to be considered.)
Concomitant Variables
Any of the trait variables mentioned previously may be used as a
dependent variable in a problem-solving study. For example, one might
investigate how a subject's classification of problems changed after
he had solved a set of problem tasks. Or one might ask whether his
attitude toward problem solving had changed.
While working on the problem tasks, the subject may have acquired
new knowledge of or skill in mathematics beyond simply learning how to
solve the problems. Measures of this knowledge or skill could also
serve as dependent variables.
As before, one would expect that the more specific the trait,
knowledge, or skill, the greater the likelihood it will be influenced
by the problem tasks. Experience suggests that one cannot expect much
change in a concomitant variable, however specific, if the number of
tasks is small.
Product Variables
Product variables are based on dimensions of the subject's solu-
tion to the problem: its correctness, its completeness, its elegance
and economy, the speed with which it was attained, and the number and
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13
diversity of the alternative solutions the subject finds. Speed and
correctness are the most commonly used product variables, but others
ought to be considered if possible.
Process Variables
Process variables are based on the solution path the subject
takes; they are derived from either the subject's verbal report of
his thinking or the manipulations he makes with an apparatus. Pro-
cess variables relate to such things as the subject's strategy (as
inferred from the sequence of steps he takes), the heuristics he uses,
the algorithms he uses, the efficiency of his solution path, the
extent of his perseveration in blind alleys, the nature and number
of the errors he makes, and his response to hints.
Any respectable study of problem solving in mathematics should
include measures of process variables. Almost nothing is known about
the generalizability of these variables across problems and occasions
although the words "strategy" and "style" connote such generalizabil-
ity (Branca & Kilpatrick, 1972). Research on subjects' consistency
in their use of process variables would be a valuable contribution.
Evaluation Variables
It would be nice to have a map of a subject's cognitions after he
has solved a problem. How does he view the problem? How does he
relate it to other problems he has solved, other information he
possesses? Is he aware of the processes he used and the errors he
made? How confident is he of his solution? Such questions may be
difficult for subjects to answer directly. Considerable ingenuity
may be needed in devising instruments to get at the subject's cogni-
tive (and affective) map of the problems he has solved. A subject's
report of what he was trying to do as he solved a problem and how he
perceives the problem after having worked on it is no less valuable
for being subjective. His report provides data that can be obtained
in no other way. There is no particular virtue in labeling such data
as unscientific and ignoring them.
Variables in Research on the Teaching
of Heuristics in Mathematics
Up to this point the only "treatment" involved in the research
studies discussed has been the administration of problems to be
solved. Consequently, the distinction between independent and depen-
dent variables has necessarily been made in the broad sense of
predictor versus behavior predicted rather than the narrow sense of
condition manipulated versus outcome observed. In studies of the
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teaching of heuristics, however, one has full-fledged treatment varia-
bles; namely, the methods, materials, and other conditions of instruc-
tion. Whether or not these treatment variables are manipulated
experimentally, they must be considered in designing such studies.
Independent Variables
The same independent variables considered earlier can appear in
research on heuristics. The subject variables remain as before. The
task variables can be used to characterize problems used in instruc-
tion. The situation variables are essentially the same, but the
researcher may need to give more attention to questions of school
climate and organization. (The category of "setting variables" used
by Richard Turner, 1976, in discussing research on teaching strate-
gies appears to be roughly equivalent to "situation variables.")
Categories of variables that need to be added are instructional treat-
ment variables, classroom activity variables, and teacher variables.
Instructional Treatment Variables
Some variables characterize an instructional treatment in general:
The extent to which the treatment is integrated into ongoing school
instruction, whether the treatment is the same for all subjects or
individualized, whether the treatment is determined in advance or
modified according to the subject's response, the extent to which the
treatment tasks resemble the outcome tasks, etc. Other variables
refer to one facet or another of the treatment.
Method variables. What heuristics are taught as part of the treat-
ment? In what sequence are they taught? Is the instruction itself
heuristic? (open-ended? Socratic? inductive?) Does the teacher
illustrate how the heuristics are used? Are students given names for
the heuristics? What problems are used in instruction? Are problems
grouped by type? Are model problems taught for each type? What pro-
blems and solutions are discussed with students? What is the nature
of this discussion? Are students given a chance to discuss problems
and solutions with other students? Each question implies one or more
variables that might be used to characterize instructional methods, and
the list of such variables is limited only by one's ingenuity in asking
such questions.
As noted elsewhere (Kilpatrick, 1973), methods-comparison studies
are typically flawed in both conception and execution. The common fail-
ure to define methods operationally in such studies appears even more
serious when one considers the manifold ways in which methods can vary.
Rather than comparing methods, researchers interested in Instruction in
heuristics should put their energies into devising the best instruc-
tional program they can and then demonstrating in detail how the program
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15
functions and how effective it is in the classroom. The creation,
tryout, and revision of program components and instruments for measur-
ing effectiveness are research activities of far greater potential
than the comparison of methods.
Materials variables. The line between methods and materials is
difficult to draw, but it is probably worthwhile to separate the two
—at least conceptually. Materials variables in research on heuris-
tics include the nature of the instructional media used, the devices
used to represent problem situations, and the nature of the accompany-
ing prose. Materials variables in themselves are likely to be of
little interest to mathematics educators.
Classroom Activity Variables
Instructional treatments ordinarily involve classroom activity.
Some of this activity is in accordance with the researcher's plan,
and as such is part of the intended treatment. Much of the activity,
however, is not under the researcher's control. Although the activity
can ultimately be considered part of the instructional treatment
package—as it works out in practice—a separate category of class-
room activity variables is useful, if only to permit the researcher
to check the variation in activity within a treatment group and the
congruence between the actual and the intended treatments. (In the
latter case, the classroom activity is functioning as a dependent
variable.)
The Teaching Strategies Project of the Georgia Center for the
Study of Learning and Teaching Mathematics has been examining one
part—albeit an important part—of classroom discourse (see Cooney,
1976). Some idea of the broader field of research in classroom acti-
vity is given by Dunkin and Biddle (1974). Despite the rapid expansion
of the field, apparently only one researcher (Stilwell, 1967) has done
a descriptive study of the teaching of problem solving in mathematics.
Further work along this line is needed. Although teachers may not pay
much attention to heuristics during instruction, someone should be pre-
pared to describe their activity when they do.
Teacher Variables
Nothing is more distressing than to hear researchers talk of "the
teacher variable"—as though there were only one. If only there were.
But teachers differ in age, sex, teaching experience, self-confidence,
enthusiasm, philosophy of education, attitude toward mathematics, pre-
ference for unstructured classroom activity, and love of children.
They also differ in problem-solving ability, problem-solving exper-
ience, knowledge of heuristics, interest in problem solving, value
placed on instruction in heuristics, willingness to delay in supplying
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16
a solution, and ability to accept and to transcend erroneous solutions.
One has only to glance at a list of "teacher competencies" to see the
variety of ways toachers can differ. Just as no one knows which
teacher competencies, if any, are prerequisite to effective teaching,
so no one knows which teacher variables, if any, might predict the
learning of heuristics.
The analogous issue of teacher competencies is raised here by
design. Nothing would be more fruitless than to attempt a catalog
of teacher variables on the chance that some might prove to be good
predictors of learning. A much better approach would be to identify
teachers who seemed to have had some success in teaching heuristics,
to see whether this success held up over time and across situations,
and then to explore dimensions of similarity among these teachers and
dimensions of contrast between these teachers and others deemed less
successful. Only then might one be ready to conjecture some relevant
teacher variables.
In most research on instruction in heuristics the role of teacher
variables will be either to aid in describing the sample of teachers in
the study or to suggest plausible reasons for the differences likely to
occur in the performance of students taught by different teachers.
Since the sample of teachers is likely to be small, the researcher
should be able to gather considerable information about each teacher,
which would presumably improve either the description or the conjecture.
Dependent Variables
The dependent variables in research on heuristics are the same
dependent variables discussed earlier, plus some new ones. Classroom
activity variables can be taken as dependent variables (more precisely,
as process variables) if one wishes to learn how the instructional
treatment influenced classroom activity. Additional dependent varia-
bles that are product variables include all the various measures one
could make of what was learned during instruction. One of the models
for mathematics achievement (see Wilson, 1971, for examples) might
help in organizing these product variables.
Methodologies in Research on Problem Solving
and the Teaching of Heuristics in Mathematics
The preceding discussion was intended to suggest some variables a
researcher designing a study ought to consider and then either ignore,
eliminate as variables in the design, control through randomization or
matching, or build into the design as independent or dependent varia-
bles. The effect of the discussion, however, may have been to over-
whelm any reader not already paralyzed by the complexity of empirical
research on problem solving. Such research is complex, certainly, but
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17
as long as one keeps coming back to one's research question and asking
what variables and methodologies bear on it, the complexity ought to be
manageable.
In this paper, methodologies relating to historical research, sur-
vey research, and reviews of the literature are not considered.
Although such studies can be valuable, they demand special methodolo-
gies. Most of the numerous books on research methods in education
(such as Ary, Jacobs, & Razavieh, 1972; Isaac & Michael, 1971; Travers,
1964) treat these topics.
Methodologies in research on problem solving and the teaching of
heuristics in mathematics are so multi-faceted as to defy classifica-
tion. Consider two of the facets:
1. Type of comparison or contrast. The researcher may be looking
for similarities or differences regarding the same or different sub-
jects' responses to the same or different tasks or treatments on the
same or different occasions or under the same or different conditions.
Each combination of alternatives implies a somewhat different approach.
2. Method of gathering data. The researcher may administer tests
or questionnaires; use an apparatus that presents a problem and either
record the subject's response himself or have it recorded mechanically;
interview subjects given a problem and asked to think aloud or retro-
spect; use personality inventories, protective tests, or such techniques
as word association, the Q-sort, the semantic differential, or the
repertory grid; observe subjects solving problems in the classroom or
elsewhere; make video- or audio-tape recordings of classroom activity;
rely on teachers or students as observers and possibly confederates; act
as a participant observer in a group problem-solving situation; act as
the teacher in an instructional situation, keeping a log and using
recordings to prompt introspection; use the computer to simulate problem-
solving processes from protocols gathered by other means; or use instru-
ments to monitor subjects' physiological processes during problem solving
or instruction.
The Cartesian product of all possible types of comparison or contrast
with all combinations of data-gathering methods only begins to suggest
the variety of methodologies one could employ.
The most promising methodologies for research on problem solving
in mathematics are those involving intensive study of the same set of
subjects over an extended period of time. The subjects must solve a
large number of problems of diverse types in order to permit confident
generalizations about the processes they use. Numerous measures of
trait variables should be obtained, and control should be exercised
over instructional history variables. Case studies of subjects selected
because of notable giftedness in mathematics or notable difficulty with
mathematics may be particularly useful. Cross-age studies of develop-
mental trends in problem solving may help to suggest process variables
that should be studied further, but longitudinal studies are obviously
to be preferred.
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The most promising methodologies for research on the teaching of
heuristics in mathematics are those involving intensive study of the
same classrooms over an extended period of time. Of special interest
are designs in which the experimenter works with the teacher during
the course of an academic term or so, observing the effects of various
modifications in instruction and using interviews with students to
supplement test and observational data. Such studies should lead to
the development of materials for instruction in heuristics. The class-
room activities of teachers identified as especially effective in
teaching problem-solving techniques should be analyzed and contrasted
with the classroom activities of teachers having more ordinary attain-
ments. Studies in which the instruction is programmed to control
sources of teacher variation may help to suggest which heuristics are
most teachable, but studies involving at least some instruction by
teachers should predominate.
Experimental studies in which all variables are under tight
control are not likely to be of much value in the present state of
our ignorance as to how people solve complex mathematical problems
and how they might be led to use heuristic methods. Too much develop-
mental work is needed before experimentation could be effective. For
example, instruments and techniques must be developed and validated
for assessing most of the variables discussed in this paper.
No one is suggesting that researchers abandon the designs and
techniques that have served so well in empirical research. But a
broader conception of research is needed, and an openness to new
techniques, if studies of problem-solving processes and the teaching
of heuristics are to have an impact.
Some years ago a group of researchers gave a battery of psycho-
logical tests each summer to mathematically talented senior high school
students attending special summer institutes at Florida State Univer-
sity. The scores on the tests were intercorrelated, and some
correlation coefficients were significant, some not. Several research
reports were published (Kennedy, 1962; Kennedy, Cottrell & Smith, 1963,
1964; Kennedy and the Human Development Staff, 1960; Kennedy, Nelson,
Lindner, Turner & Moon, 1960). As Krutetskii (1976) notes, the process
of solution did not appear to interest the researchers—yet what rich
material could have been obtained from these gifted students if one
were to study their thinking processes in dealing with mathematical
problems. Why were the students simply given a battery of tests to
take instead of being asked to solve mathematical problems? It's a
good question.
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19
References
Ary, D.; Jacobs, L. C.; and Razavieh, A. Introduction to research in
_gducat:ign. New York: Holt, Rinehart & Winston, 1972.
Branca, N. A. and Kilpatrick, J. The consistency of strategies in the
learning of mathematical structures. Journal for Research in
Mathematics Education, 1972, 3^, 132-140.
Cooney, T. J. (Ed.). Teaching strategies: Papers from a jresearch work-
shop. Columbus, Ohio: ERIC/SMEAC, 1976.
Dunkin, M. J. and Biddle, B. J. The study of teaching. New York: Holt,
Rinehart & Winston, 1974.
Goldin, G. A. and Luger, G. F. Implications of problem structure for
teaching problem-solving heuristics. Paper presented at the
meeting of the National Council of Teachers of Mathematics,
Denver, April 1975.
Hawkins, D. Learning the unteachable. In L. S. Shulman and E. R.
Keislar (Eds.), Learning by discovery^ A critical appraisal.
Chicago: Rand McNally, 1966.
Isaac, S. and Michael, W. B. Handbook in research and evaluation. San
Diego: Knapp, 1971.
Kennedy, W. A. MMPI profiles of gifted adolescents. Journal of Clini-
cal Psychology. 1962, 18, 148-149.
Kennedy, W. A.; Cottrell, T.; and Smith, A. Norms of gifted adolescents
on the Rotter Incomplete Sentence Blank, Journal of Clinical
Psychology. 1963, 19, 314-315.
Kennedy, W. A.; Cottrell, T.; and Smith, A. EPPS norms for mathemati-
cally gifted adolescents. Psychological Reports, 1964, _14_, 342.
Kennedy, W. A. and the Human Development Clinic Staff, A multidimen-
sional study of mathematically gifted adolescents. Child
Development, 1960, 31, 655-666.
Kennedy, W. A.; Nelson, W.; Lindner, R.; Turner, J. ; and Moon, H.
Psychological measurements of future scientists Psychological
Reports, 1960, 7_, 515-517.
Kilpatrick, J. Problem solving in mathematics. Review of Educational
Research, 1969, _3J9, 523-534.
Kilpatrick, J, Research in the teaching and learning of mathematics.
Paper presented at the meeting of the Mathematical Association
of America, Dallas, January 1973,
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Krutetskii, V. A. The psychology of mathematical abilities in school
children. Chicago : University of Chicago Press, 1976.
Paull, D. R. The ability to estimate in mathematics. Unpublished
doctoral dissertation, Teachers College, Columbia University,
1971.
Polya, G. Mathematical discovery; On understanding, learning, and
teaching problem solving, Vol. 2. New York: Wiley, 1965.
Simon, H. A. Knowledge and skill in problem solving. Invited address
presented at the meeting of the Special Interest Group for
Research in Mathematics Education, American Educational Research
Association, Washington, March 1975.
Stilwell, M. E. The development ^gncL analysis of a category system for
systematic observation of teacher-pupil interaction during geo-
metry problem-solving activity. (Doctoral dissertation, Cornell
University) Ann Arbor, Michigan: University Microfilms, 1968,
No. 68-893.
Travers, R. M. W. An introduction to educational research. (2nd ed.)
New York: Macmillan, 1964.
Turner, R. L. Design problems in research on teaching strategies in
mathematics. In T. J. Cooney (Ed.), Teaching strategies; Papers
from a research workshop. Columbus, Ohio: ERIC/SMEAC, 1976.
Wilson, J. W. Evaluation of learning in secondary school mathematics.
In B. S. Bloom, J. T. Hastings, and G. F. Madaus (Eds.), Hand-
book on formative and summative evaluation of student learning.
New York: McGraw-Hill, 1971.
Wittrock, M. C. A generative model of mathematics learning. Journal
for Research in Mathematics Education, 1974, _5» 181-196.
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21
Heuristical Emphases in the Instruction of Mathematical
Problem Solving: Rationales and Research
Larry L. HatfieId
University of Georgia
The significance of the goal of improving the learner's prohlem-
solving competence within school mathematics is well established.
Recommendations for emphasizing problem-solving techniques in teaching
mathematics can be found throughout the history of mathematics educa-
tion (Jones, 1970). However, in spite of an implored need that
educators must know much more about using problems to stimulate inde-
pendent and creative thinking, the teaching and learning of problem
solving has only occasionally been investigated by mathematics educa-
tion researchers.
The general goal of the Problem-Solving Project of the Georgia
Center for the Study of Learning and Teaching Mathematics (GCSLTM) is
to organize and conduct coordinated studies of the conditions for and
the effects of learning and instruction of mathematics which emphasizes
problem solving. Several distinct areas of study encompassing this
goal can be identified. These areas include:
(a) studies devoted to identification of strategies and pro-
cesses used in solving various mathematical problems,
including a search for aptitudes related to these strate-
gies and processes;
(b) studies devoted to development of clinical procedures for
observing and analyzing mathematical problem-solving
behaviors;
(c) studies devoted to development of instructional procedures
aimed at improving a student's problem-solving capabili-
ties;
(d) studies devoted to development of teacher training proce-
dures to result in deliberate employment of instructional
methods aimed at enhancing the growth of problem-solving
capabilities of students;
(e) studies of an expository nature, including analytical
developments as well as technical reports and interpre-
tative reports of activities and findings of this problem-
solving research and development.
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The specific project goals during the first year are: (a) to
conduct various investigations in mathematical problem solving, (b)
to provide an organized synthesis of the literature in mathematical
problem solving appropriate to the investigations underway and to
the interests of the investigators, (c) to identify detailed speci-
fications for studies to be undertaken in the second round of the
GCSLTM's activities, and (d) to establish a working consortia model
for conducting coordinated series of investigations. The conduct of
the Problem-Solving Research Workshop represents a major step toward
obtaining these first-year goals.
The research on problem-solving behavior to be found in the
literature of psychology and education (including mathematical educa-
tion) is considerably varied. As a matter of project strategy it was
decided that the theme of the Problem-Solving Workshop would be
"instruction in heuristical methods." While intending to provide a
focus and a direction for our initial research efforts particularly
into area c (development of instructional procedures), such an empha-
sis should eventually span the five areas for studies noted earlier.
The purpose of this paper is to contribute a common perspective
for conceptualizing investigations reflecting the theme of teaching
mathematical problem solving by, and for, heuristical methods. The
following sections include a discussion of rationales for emphasiz-
ing heuristical precepts in teaching and researching mathematical
problem solving, a review of selected recent mathematical education
research involving heuristical methods, and an identification of
possible directions and dimensions of studies to be undertaken in
the Problem-Solving Project.
Instruction in Heuristical Methods: What, Why and How
The most widely known contemporary and practical treatment of
heuristic is due to the eminent mathematician Polya (1957, 1962,
1965). Recently Wickelgren (1974) has sought to provide an extension
of such treatments, blending significant developments from the field
of artificial intelligence and information processing models with
Polya's maxims. In addition, Higgins (1971) offers an interpretation
of heuristic as it applies to a methodology for mathematics instruc-
tion termed "heuristic teaching." It must be assumed that Workshop
participant- are reasonably familiar with these elaborate writings.
At the same time a brief review of particular aspects will serve to
highlight and clarify certain points of view.
What are "heuristical methods" for problem solving? A synopsis
from Polya's discussions is presented in search of clarification.
Heuristic, or heuretic, or "ars inveniendi" was the name of
a certain branch of study, not very clearly circumscribed,
belonging to logic, or to philosophy, or to psychology, often
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23
outlined, seldom presented in detail, and as good as for-
gotten today. The aim of heuristic is to study the methods
and rules of discovery and invention. (Polya, 1957, p. 112)
Heuristic, as an adjective, means "serving to discover,"
(Polya, 1957, p. 113)
Modern heuristic endeavors to understand the process of
solving problems especially the mental operations typically
useful in this process, It has various sources of informa-
tion none of which should be neglected, A serious study of
heuristic should take into account both the logical and
psychological background, it should not neglect what such
older writers as Pappus, Descartes, Leibnitz and Bolzano
have to say about the subject, but it should least neglect
unbiased experience. Experience in solving problems and
experience in watching other people solving problems must
be the basis on which heuristic is built. (Polya, 1957,
pp. 129-30)
Heuristic reasoning is reasoning not regarded as final and
strict but as provisional and plausible only, whose purpose
is to discover the solution of the present problem. We are
often obliged to use heuristic reasoning. We shall attain
complete certainty when we shall have obtained the complete
solution, but before obtaining certainty we must often be
satisfied with a more or less plausible guess, CPolya, 1957,
p. 113)
Polya approaches heuristic from a practical teacher~oriented
aspect: "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" CPolya, 1362, p, vi). His detailed "case
histories" of problem solutions feature questions and suggestions
which, organized into four phases of work on the solution (under-
standing, planning, carrying out, looking back), have come to be
known as his "planning heuristic," According to Polya, these ques-
tions and suggestions have two common characteristics, generality
(in that they indicate a general direction of action and thus may
help unobtrusively) and common sense (in order that they can occur
naturally or easily to the solver himself). But the significant
assumption by Polya is the following:
If the reader is sufficiently acquainted with the list and
can see, behind the suggestion, the action suggested, he
may realize that the list enumerates, indirectly, mental
operations typically useful for the solution of problems.
(Polya, 1957, p. 2)
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24
Thus, "heuristical methods" for solving problems include plaus-
ible but uncertain actions of a general yet natural character. To
know and to use "heuristical methods" at some level of effective-
ness suggests that one also knows in some fashion the cognitive
operations bearing on one's own problem-solving behavior.
This notion hints at potentially important, yet typically
subtle, distinctions: curriculum or instruction aimed at teaching
for problem solving versus teaching about problem solving versus
teaching via problem solving. Most contemporary school mathematics
textbooks claim to teach for problem-solving outcomes; in the sense
that organized, usable knowledge in mathematics (concepts, princi-
ples, skills) is essential to being an effective problem solver in
mathematics, these claims are partially justified. Few texts, how-
ever, seek to teach rather explicitly about problem solving in the
sense of heuristic. And correspondingly few textbooks approach
content as heuristicnlly oriented problem solving as Polya or
Higgins recommend.
What should constitute "instruction in heuristical methods,"
particularly as it relates to mathematical education, is still an
ill-defined matter. Again, one can find various characterizations
and exemplifications offered by Polya (1962, 1965), Wickelgren
(1974), Covington and Crutchfield (1965), Wilson (1967), Butts
(1973), and others. An essential feature seems to be the explicit
identification and use of heuristical ploys within the act of
solving mathematical problems. Sometimes these are modeled by the
teacher or instructional materials to be observed and to be imitated
by the learner, while at other times they are to be initiated and to
be practiced in the learner's own problem-solving actions. Further
discussion on this question will be offered in the final section of
this paper relating particular strategies for our research.
Why should mathematics educators choose to give "instruction
in heuristical methods" of problem solving? Here again, the cogent
arguments offered by Polya (see especially 1957, pp. 1-32; 1965,
pp. 99-142) encompass learner motivation, educational relevancy,
general culture, enhancing common sense reasoning, and active
learning. These arguments must certainly be accompanied now by
acceptable research findings regarding the efficacy of learning
and using heuristical percepts for improved problem-solving capa-
city. Herein lies a central point of departure for the investiga-
tions to be stimulated in this Project.
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1
Selected Investigatjjms Emphasizing Heuristical Methods
Recently several investigations of mathematical problem solving
have attempted to teach explicitly heuristical precepts or with heur-
istical methods or to analyze problem-solving protocols using systems
based on Polya's ideas. A number of studies have used various treat-
ments of task-specific and general "heuristics" in attempting to
improve subjects' problem-solving competence through instruction,
Ashton (1962) gave ten weeks of heuristic-oriented instruction based
on Polya's work to ninth grade algebra students. These students,
when compared with a control group receiving conventional instruction,
were better able to solve word problems,
Covington and Crutchfield (1965) also obtained superiority of
the instructed children in measures of divergent thinking, original-
ity, and perceived value of problem solving. These subjects used
comic book format to engage students in developing "heuristics" for
non-mathematical problems. Olton (1967) conducted an extensive test
of the revised version of this program and confirmed the positive
effects on a diverse set of performance indicators. His students
achieved up to 50 percent higher scores on post-test measures where
the teacher discussed each lesson, a finding which seems to support
the value Polya has assigned to a "looking back" phase in his
"planning heuristic." Jerman (1971) used The Productive Thinking
Program and a Modified Wanted-Given Program (after Wilson, 1967)
with fifth grade students and concluded that teaching problem solving
in mathematics to students of this age can best be done in a mathe-
matical context using a wanted-given approach, whereas either system-
atic approach to problem solving was more effective than not providing
any systematic instruction,
Wilson (1967) and Smith (1973) compared subjects taught mathema-
tical problem solving using either "task-specific heuristics" or
"general heuristics" via self-instructional booklets. In each study
the dependent variables were time measures and number of correct steps
for each section of the criterion test. Both investigators hypothe-
sized that task-specific heuristical instruction would lead to
superior performances on the training tasks but poorer performances
on the transfer tasks than would instruction in the use of general
heuristics. Analysis of performances of Wilson's subjects on the
training tasks and five transfer tasks revealed that only the general
(planning) heuristic was superior to the others (a task-specific heur-
istic applicable only to the topic under study and a means-end
heuristic): it is suggested that general heuristics learned in the
first training task were practiced on the second task, thereby facil-
itating transfer. Thus, Wilson failed to confirm his central hypo-
thesis and, in fact, found that one of his general heuristics (the
The reader is urged to examine other more comprehensive reviews
of problem-solving research, including Kilpatrick (1969), Riedesel
(1969), Simon and Newell (1971), and Suydatn (1972).
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26
planning heuristic) led to better performance on a training task than
did the heuristic specific to that task. One possible explanation
lies in the potential power of a heuristical maxim: the availability
of the maxim to the problem solver and his efficiency in using the
advice. In part, the problem solver must recognize that general
heuristics are indeed general and therefore possibly applicable to
solving an unfamiliar problem wherein known task-specific strategies
are not useful. The limited learning time (two 3-hour sessions) and
number of problem-solving episodes (about 20 problems) may have
resulted in less practicing of general heuristics than may be neces-
sary for subjects to become operable.
Smith's (1973) study, patterned after Wilson's, attempted to
emphasize the generality of its general heuristics to a greater
extent than Wilson did with his. Subjects were also given more
opportunity to practice similar heuristic procedures in a variety
of settings. The data for Smith's subjects failed to support the
hypothesis that instruction in heuristics differing in level of
generality leads to differences in performance on transfer tasks.
Questionnaires and interviews suggested that, while subjects claimed
to have used the heuristics when completing a given learning task,
almost no subjects attempted to use the general heuristic on the
transfer tasks. Smith conjectured the possible reasons for the
apparent abandonment of the planning heuristic on these unfamiliar
tasks. Often the more general heuristic does not always suggest
itself; the problem solver must often make a conscious effort to
reach into his "bag of heuristics" when he is stymied. Besides know-
ing strategies which he believes will be of use, the subject must
think of trying them when he gets stuck. Adequate practice and
encouragement, couched in successful applications of heuristical
advice, seem absolutely essential for true operationality of such
strategies. In positing suggestions for further study, Smith notes:
Investigators of human problem solving are probably well
advised to incorporate some means of studying a subject's
behavior in addition to examining test or time scores.
Selected interviews or problem-solving questionnaires are
one possibility. Some form of protocol analysis might
provide valuable insights into the apparent lack of trans-
fer power of general heuristic advice. In fact, exploratory
studies might be more valuable than experimental ones, given
the present state of our knowledge about human problem solv-
ing. A researcher who devotes his full energy to studying
the problem-solving processes of his subjects rather than
the products they produce may discover revealing behavior
patterns. (pp. 100-1),
Kilpatrick (1967), using a system based on heuristical processes
identified by Polya, analyzed problem-solving protocols of 56 junior
high school students in relation to their performances on a battery
of aptitude, achievement, and attitude scales. The processes used in
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27
solving word problems seemed unrelated to observed systematic styles
of approach to spatial and numerical tasks. Subjects who attempted
to set up equations Cpre-algebra students) were significantly super-
ior to others on measures of quantitative ability, mathematics
achievement, word fluency, general reasoning, logical reasoning, and
a reflective conceptual tempo. Those subjects who used the most
trial and error were higher than the others in quantitative ability
and mathematics achievement. Those subjects who used the least trial
and error had the most trouble with the word problems, spent the least
time on them, and got the fewest number correct.
Lucas (1972) and Goldberg (1973) employed a modified version of
Kilpatrick's (1967) system for coding and analyzing the problem-
solving protocols of subjects following the application of instruc-
tional treatments designed to exhibit and use a general planning
heuristic. Lucas made an exploratory study of the effects of teaching
heuristic in calculus. Following an eight-week instructional period
during which one class was taught using the style of teaching suggested
by Polya along with prepared materials that defined and demonstrated
the use of heuristical advice while another class learned calculus
without explicit attention to heuristics, volunteers from both classes
were interviewed individually and asked to solve problems while think-
ing aloud. The protocols of these thirty subjects were analyzed to
identify strategies.
Lucas was able to isolate thirty-eight variables that represented
heuristical ideas, indicators of difficulty, types of errors, and per-
formance measures of time and product score. He found evidence of
differences in performance between subjects in the heuristical and
non-heuristical treatments, but his conclusions were stated tentatively,
reflecting the purpose of identifying behavioral variables rather than
hypothesis testing. Subjects from the heuristical treatment were judged
superior in their ability to solve problems when the criterion was score
on approach, on plan, or on result; they read the problems more easily
and hesitated less as they worked. Some heuristics were used more fre-
quently by subjects taught heuristically: working backward, using the
result or method of a related problem, and introducing suitable nota-
tion. Heuristical training did not appear to affect the use of
diagrams, the use of trial and error, the number of errors, the number
of aborted solutions, or the ability to write proofs. Lucas concluded
that Kilpatrick's system can be used to characterize college students'
problem-solving behavior, and that heuristical maxims can be taught in
calculus without infringing on course content. That heuristical
instruction was no more time consuming than the kind of instruction
given to the control class may be explained because the written
instruction in heuristics was reserved for extra-class time. Larsen
(1960) had found that calculus students instructed in heuristical ideas
learned them, but apparently at the expense of normal course content.
Goldberg (1973) examined the effects of instruction in heuristical
advice for writing proofs, suggested by Polya, on the ability of college
students not majoring in mathematics to construct proofs in number
theory. Two sets of seven self-instructional booklets written by the
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investigator were used with 238 subjects enrolled in nine classes.
Classes were randomly assigned to one of three different treatments:
reinforced heuristic, unreinforced heuristic, and non-heuristic.
Following the six-week experimental period, measures of understanding
of number theory concepts, ability to construct proofs, attitude
toward the self-instructional booklets, and attitude toward problem
solving were obtained. The analyses indicated that heuristic instruc-
tion with reinforcement in class is relatively more beneficial than
unreinforced or non-heuristic instruction with respect to understand-
ing number theory concepts and writing proofs for high ability
students. This result was also found favoring unreinforced heuristic
instruction over instruction by imitating examples (non-heuristic).
The non-heuristic self-instructional booklets were found more helpful,
easier, and generally more appealing than the heuristic self-
instructional booklets. Goldberg suggests that the non-heuristic
materials were less threatening and more fun due to the inclusion
of puzzles and number tricks whose purpose was to help to make the
time to complete the booklets more comparable to the heuristical
treatment.
A more important effect was observed in the more positive atti-
tudes toward the problem-solving process of the subjects in the
reinforced heuristic treatment than students whose instruction in
heuristics was not reinforced in class. Applying a coding system to
the written proofs of students scoring in the top third of the proofs
posttest, she found that students given reinforced heuristic instruc-
tion used heuristics more than students who had studied number theory
by imitating examples. Among the precepts noted, these students wrote
"given" or "prove," rephrased the conclusion of the problem, used
theorems more frequently than definitions, introduced suitable nota-
tion, and worked backwards in their proofs more frequently than
proficient proof writers who were not given reinforced heuristic
instruction.
Vos (1976) compared the effects of three instructional strategies
on problem-solving behaviors in secondary school mathematics. Five
particular behaviors (drawing a diagram, approximating and verifying,
constructing an algebraic equation, classifying data, and constructing
a chart) were identified as problem-solving behaviors or "heuristics."
Each of three experimental treatments, classified as Repetition (R),
List (L), and Behavior Instruction (B), involved an exposure to twenty
mathematical problems but with variations in the placement of, and
emphasis on, one of the five implied problem-solving behaviors. Simply
stated, the treatments were: R presented the problem task only; L
presented the problem task which was momentarily interrupted with a
checklist of desirable problem-solving behaviors and individual written
instruction in a specific useful solving behavior followed by a return
to the problem task; and B initially presented individual written
instruction in a specific problem-solving behavior followed by the
same training problem task used in L and R. Each problem task adminis-
tered through self-instructional materials took about twenty minutes to
complete. Subjects in six mathematics classes (grades 9, 10, 11) at a
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private Iowa school were randomly assigned within classes to one of
the three treatments which occurred over about fifteen weeks. Post-
treatment data included scores from a Problem-Solving Approach Test
(PSAT) and a Problem-Solving Test (PST). PSAT consisted of two parts
each having problem statements with choices indicating an approach
that could best be used to solve the problem. Part I offered choices
directly related to the five instructed problem-solving behaviors
whereas Part II sought to measure transfer in using other problem-
solving behaviors. The PST consisted of seven problems seeking a free
written response with encouragement to write all their thoughts about
the situation. In summarizing the results found from the various data
analyses, Vos asserts that specific instruction in utilizing problem-
solving behaviors increased the effective use of the behaviors. The
evidence supports the idea that classroom mathematics instruction
should involve specific instruction in a set of problem-solving
behaviors.
Webb (1975) studied the problem-solving processes used by forty
second year high school algebra students while solving eight problems.
The aims of his study were (a) to consider how cognitive and affec-
tive variables and the use of heuristical strategies are related to
each other and to the ability of high school students to solve pro-
blems, (b) to identify problem-specific strategies from those used in
solving problems in general, and (c) to identify problem-solving modes
of groups of students. Protocols from individual interviews were
analyzed using a coding system adapted from Kilpatrick (1967) and Lucas
(1972). Data from sixteen cognitive and affective pretests, frequen-
cies of problem-solving processes, and total problem inventory scores
were analyzed using principal component, regression and cluster analy-
ses.
Webb observed that mathematics achievement was the variable with
the highest relation to mathematical problem-solving ability. The use
of heuristical strategies had some relation to mathematical problem-
solving ability not accounted for by mathematics achievement. In
particular, the component Pictorial Representation accounted for a
significant amount of the variance. Thus, the processes used by
students in this study added to their ability to solve problems beyond
their mathematical conceptual knowledge (mathematics achievement).
Furthermore, Webb noted that students who used a wide range of
heuristical strategies, on the average, were better problem solvers.
Most of these heuristical strategies were found to be problem-specific.
This implies that in order to solve several different problems, a
range of problem-specific strategies needs to be employed. Strategies
such as "specialization" and "successive approximations" were used at
least once on six of the eight problems, but were used by more students
on one or two of the problems. One possible reason for the restricted
use of these strategies is that students only used the strategies in
obvious ways and did not employ the strategies where they could be
strategically used. Webb suggests that one direction for research
would be to examine whether students can be taught to use such strate-
gies to solve a wide range of problems.
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Particular combinations of strategies appeared to relate to per-
formance. Students who used a moderate amount of trial-and-error and
a moderate amount of equations or who use equations often and trial-
and-error seldom performed about the same on the Problem-Solving
Inventory. Students who used a high frequency of trial-and-error and
had a low use of equations did not do as well. These results are
somewhat counter to Kilpatrick's (1967) observations of the relative
effectiveness of trial-and-error methods. However, for these high
school students and for the problems in the inventory, it appeared
that trial-and-error as an approach to problem solving had a value
as a supplementary process to the use of equations, but not as a
replacement for the use of equations.
Kantowski (1974) conducted a "teaching experiment" (quite in the
Soviet sense) as a clinical exploratory study of processes used by
eight grade nine subjects in solving non-routine "to show" problems
in geometry. She noted Talyzina's (1970) observation that subjects
who were successful problem solvers in geometry introduced an "opera-
tive proposition" ("heuristic") into the solution. Kantowski observed
that if the "heuristics" used by her students were goal-orj^ented (that
is, specifically related to the conclusions of the problem) the solu-
tion tended to be more efficient. She observed a dramatic increase
in the use of goal-oriented "heuristics" across her instructional
treatments on "heuristics" and geometry problem solving.
But she made an even more penetrating observation. Of course,
valid deductions are often essential to solving mathematics problems.
Such deductions are commonly made by analysis and synthesis. She took
analysis to be what Polya refers to as "decomposing" or making infer-
ences from what is known. Synthesis, on the other hand, is a "recom-
bining" of facts to form a new whole. She examined where these
analytic and synthetic deductions occurred in the sequence of processes
during solution and their relationship to the use of "heuristics." The
manifestation of regular patterns of analysis and synthesis among suc-
cessful problem solvers is striking. In a high percentage of cases
these regular patterns of analysis and synthesis were immediately
preceded by a goal-oriented "heuristic." In cases where non-goal-
oriented "heuristics" were introduced, the patterns of analysis and
synthesis were generally irregular and superfluous.
McClintock (1975) reported the relative effects of verbalization
of "heuristics" on transfer of learning. In three one-hour periods
of instruction (methods were discovery, expository or control) general-
izations for the sums, sums of squares, and sums of cubes of the first
n natural numbers were taught (modes were teacher verbal instruction,
self-instructional reading, or combined teacher verbal and self-
instructional reading). Following initial learning students attempted
to two-item practice test whereupon a "heuristics verbalization" group
responded verbally to the request to state what they recalled having
done in solving the problems with encouragement to use Polya's four
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31
phases for their description. Thereafter, all groups were given trans-
fer tasks (12 problems to solve) related to the taught generalizations.
Significant effects for method of instruction, "heuristics verbaliza-
tion," and interactions were found. It appeared that a combined
instructional mode followed by the "looking back" of the verbalization
of "heuristics" tended to produce greater transfer to problem-solving
tasks.
"Looking Back" at These Studies
What do these few premier investigations offer as results or
directions for future research? What features seem common among these
studies? Are there research methodologies, variables or designs appear-
ing to encompass more salient aspects of the behaviors we should or
might be studying?
Defining constructs. The plague of ill-defined constructs which
permeates much of educational research is manifested in most of these
investigations. The broad implications of the notion heuristic result
in its usage being at best varied but more often unclear. In one
sense it involves the "science" (or art) of studying and describing
the mental processes or operations of solving problems. This would
seem to connote an applied epistemology: The study of, and use of,
knowledge itself. Despite its focus on mathematical problem solving
serving to provide considerable clarification, Polya's more practical
approach does not relay a cognitive psychological theory of the nature
and usage of heuristic.
The studies described earlier generally use the term "heuristics"
(plural) in the sense suggested by Kilpatrick: ". . .as any device
technique, rule of thumb, etc., that improves problem-solving perfor-
mance" (1967, p. 19). The maxims, questions, precepts, ploys, and
suggestions offered by Polya and others are most often referred to as
"heuristics." Usually no more than implicitly are the actual behaviors
of the problem-solving act considered. Particularly missing are dis-
cussions of the mental operations and cognitive structures which are
(a) necessary to assimilate the heuristical precept as a potentially
powerful item of knowledge for future problem solving, (b) required
to recall and then apply the heuristical advice, (c) "triggered" by
the recall or suggestion of an heuristical statement.
Finally there is further confusion resulting from the use of
heuristical notions in conceptualizing instructional methods for
teaching about the maxims and their use in problem solving. It would
seem that one could teach heuristical advice as generalizations via
expository instruction. Yet Polya implores the teacher to use a
pedagogy of problem setting, teacher questioning, active learner
participation and choice, and post-solution discussions that reflect
the techniques of an heuristical problem-solving approach.
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Another notion requiring further clarification is problem-solving
strategy. Various references can be found to a "trial and error
strategy," an "inductive strategy," "heuristic strategy," "information
processing strategy," an "algorithmic strategy," or a "strategy of
indirect proof." The considerable range of interpretation occurring in
these examples illustrates the confusion. Yet the intuitive idea of
strategy has appeal as a useful idea for describing or contrasting
approaches to a problem solution,
Studying processes. Surprisingly, few of these studies used as
dependent measures the actual process sequence trace of the subject's
problem-solving act. Rather time to solution, number of correct steps,
and correct problem answers are used. Kilpatrick, Lucas, Kantowski,
and Webb interviewed subjects to obtain protocol analyses of the pro-
cess sequences. Yet even here the primary dependent measures became
type and frequency of occurrence of various heuristical precepts. Only
Kantowski reports identified patterns in the sequences noted for her
subjects.
The particular manifestation of student behaviors, including
explicit as well as apparent use of heuristical ploys, may be a way
to characterize solution strategies. Similarities across problems
(both in content and sequence) of processes used by a subject could
be described as a solution strategy known to that problem solver.
Or consistent displays of process sequences across subjects for a
particular problem or type of problem could empirically describe the
solution strategies commonly associated with that problem.
The protocol analyses undertaken in some of these studies engaged
the seminal coding system, or variations thereof, provided by Kil-
patrick (1967). Obviously the nature and quality of process analyses
will be contingent upon the sensitivity and comprehensiveness of the
interview and the template of the coding scheme subsequently used on
the protocols.
Generality of heuristics. Several of these investigations dealt
with a "generality of heuristics" dimension. Recall Polya's recogni-
tion of the generality characteristic of his questions and suggestions.
As a teacher, he seeks to offer advice as unobtrusively as possible so
that steps in the emerging solution path do not become too obvious to
the solver. At the same time he wants the advice to be widely appli-
cable to many different problems. Any research findings about "general
vs. task-specific" heuristics are very tentative. Most investigations
were able to observe positive effects from subjects being taught or
using more general heuristical precepts. Interestingly, few of these
studies offer direct evidence regarding whether the subjects indeed
learned the taught "heuristics." Most studies employed transfer tasks
to observe knowledge of heuristical advice by observing its use in
problem solving.
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Instructional intervention. Following the artistic lead offered
by Polya, many of these investigators have sought to Improve the pro-
blem solving performances of their subjects by directly providing
special instruction in "heuristics." Programmed instruction booklets
were used in several studies. Some investigators incorporated especial
concern for accompanying such booklets with teacher reinforcement (dis-
cussion) of the "heuristics" employed. Yet the results of the effects
of explicitly developing a "looking back" phase in the instruction are
mixed.
The characteristics of the instruction actually offered are not
clearly detailed in reports of some of these investigations. Among
the variants one finds (1) the extent to which certain heuristical
maxims are specifically taught, (2) the manner in which maxims are
isolated and illustrated as the single or primary tactic in a problem
solution, and (3) the degree to which an "heuristic" question or
advice is explicitly stated as opposed to being modeled but never
"pointed at."
Notable results. Both conclusion-oriented and exploratory inves-
tigations are represented in the studies reveiwed here. Obviously
caution must be exercised in accepting the results. Yet certain find-
ings appear to be evident across several of these studies:
1. A student's background knowledge of mathematics appears to
be a dominant factor in successful mathematical problem-
solving performances. This observation supports the
importance of carefully building-up the problem solver's
knowledge of mathematical ideas. However, the relation-
ships among instructional variables and problem-solving
competence are not clear from this research. In particular,
it is unclear what effects explicit "instruction in heuris-
tical methods" may have upon knowledge structures,
especially within problem-solving tasks.
2. Students given special treatments which feature problematic
tasks and solving processes are often rated as relatively
better or improved problem solvers. Thus, solving problems
and attending to solving methods do appear to obtain posi-
tive results. Yet, beyond this global maxim, "solve
problems and reflect upon your solutions," considerably
more detailed information is needed to guide mathematics
teachers and students. Some potential directions are
suggested by the instructional and task variables included
in these investigations.
3. Certain heuristical ploys or maxims appear from these
studies to be more commonly taught or used. These included
trial and error, successive approximation, working back-
wards, drawing a pattern or representation, and inductive
pattern searching. Perhaps these are more immediately used
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because they are among the solving tactics used by
humans in coping with all manner of task situations.
They are deeply habituated and, therefore, more
naturally and automatically called upon. As Polya
noted, the qualities of generality and common sense
may be significant in obtaining heuristical competence.
4. The influence of idiosyncratic traits which a student
brings to a problem-solving episode is unclear. Few of
these investigations attempted to study subject varia-
bles. Yet it seems crucial to effecting improved
mathematical problem solving to be clear about the roles
played by a student's aptitudes, preferences, cognitive
structures, memory, learning styles, or personality.
Furthermore, it may be important to discern individual
trait (i.e., relatively stable and long-term) and state
(or situational) factors in problem solving.
'Looking Ahead": Toward Coordinated Research of
Mathematical Problem Solving
One of the unique features of a research consortium ought to be
the manifestation of a more concerted thrust on the problems under
consideration than the individual investigators, working separately,
might produce. Put another way, the "whole should somehow become
greater than the simple sum of all parts." Coordinated team research
in an area as complex as the learning and teaching of mathematical
problem solving will not be easy. Yet perhaps the conditions for
creating focussed research efforts and results are now only becoming
existent in mathematical education in this country.
This section examines certain points of view regarding prospec-
tive research in the Problem-Solving Project. These ideas are
expressed to engender discussion both during and following the
Research Workshop. It must be clear that all committed participants
in the intellectual consortia connoted by the GCSLTM must shape its
eventual contributions. And this will certainly be a long-term
effort. To execute change, real change, in the teaching and learn-
ing of mathematical problem solving in our schools will require no
less.
The choice of emphasis in "instruction in heuristical methods"
will cast a certain direction to the Problem-Solving Project. Inher-
ent in this choice are several hypotheses. Heuristical methods of
problem solving:
a. can be learned,
b. can be taught,
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c. if effectively used, do Improve problem-solving perfor-
mances , and
d. along with its pedagogical counterpart, heuristical
methods of teaching, can become a viable part of mathe-
matical curricula.
In a global sense it could become the overriding mission of the Problem-
Solving Project to study and/or test these general hypotheses or conjec-
tures. At the same time the focus on "instruction in heuristical
methods" must not inordinately constrain our research. While having
a definite "applied" (i.e., classroom-oriented) characteristic, we must
also recognize and encourage more basic, theory-oriented efforts. The
pattern of the recent Soviet research, incorporating both aspects, may
be a useful paradigm to follow.
Directions for Future Research
Several general features of investigations to be encouraged in our
consortium are next proposed. First, the processes, per se, of mathe-
matical problem solving must be studied. Most past mathematics education
research which considered problem-solving outcomes has examined various
treatments and dependent measures that ignore the actual processes used
by the subjects during their problem-solving acts. The solution (i.e.,
final "answer" or proof) of a mathematics problem, however lucidly set
down, is typically an inadequate trace of the processes used to arrive
at that solution object. Ample direction for studying the cognitive
processes used by students in problem solving can be found in the works
of Brownell (1942), Duncker (1945), Buswell (1956), Wertheimer (1959),
Polya (1962), Kilpatrick (1969), and Kantowski (1974). Of particular
interest is the emphasis taken by a number of Soviet researchers in
studying the dynamics of mental activity during mathematical problem
solving (Kilpatrick and Wirszup, 1969),
Secondly, the problems used with subjects should be non-trivial
mathematical problems of the sort they might meet in the classroom.
By a mathematical problem is meant a challenge encountered in a task
environment, which is itself perhaps only partially known to the sub-
ject, wherein the concepts, relations, operations, transformational
procedures, and models of mathematics provide the major elements or
vehicles for solving the challenge. Laboratory studies of the psycho-
logist have rarely dealt directly with the complex behavior appropriate
for solving a challenging mathematics problem. Many reasons can be
noted for selecting simple tasks, such as card sorting or level pull-
ing. At the same time mathematics educators have generally hesitated
to apply any conclusions stemming from such research because the tasks
have been unrelated to the type of mathematical problems posed by the
mathematics teacher. To assure relevance to mathematical education,
we should emphasize commonly used as well as nonroutine settings which
utilize appropriate mathematical concepts, principles, and skills
either known or readily learned by the subjects.
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The research methodology should make major use of qualitative
methods, small groups of subjects, and long-term genetic approaches
to study the learning and^ development of problem-solving competence^
There may be a growing sense of the need for the careful conduct of
clinical investigations in mathematics education research. Brownell
noted promising changes in psychological research on problem solving,
including the setting of problems which "mean" something to the sub-
ject, concentrating attention on not merely the errors and successes
but on the way the subject proceeds to attack and solve the problem,
and attaching greater importance "to qualitative descriptions of sig-
nificant behavior to supplement or to replace purely quantitative
descriptions" (Brownell, 1942, p. 419), Certainly the influence of
Piaget and his followers in demonstrating the efficacy of such method-
ologies has been great. The recent appearance of the series, Soviet
Studies in the Psychology of Learning and Teaching Mathematics (Kil-
patrick and Wirszup, 1969) has generated further interest in approaches
aimed at penetrating into the child's thoughts to analyze his mental
processes. Menchinskaya (1969) described various methodologies used
within the genetic approach, including the "teaching experiment" in
which study is combined with pedagogical influence and entire classes
of children are involved over a number of years in order that more
valid judgments about the changes that occur in mental activity as a
result of instruction might be made. Kilpatrick advocates this
methodology by suggesting that the researcher "who chooses to investi-
gate problem solving in mathematics is probably best advised to under-
take clinical studies of individual subjects. . .because our ignorance
in this area demands clinical studies as precursors to larger efforts"
(Kilpatrick, 1969, p. 532).
Wilson (1973) discussed the role, features, and credibility of
clinical intervention research. Three major purposes research must
fulfill were identified: generating hypotheses with antecedent prob-
abilities confirming hypotheses, and constructing guiding models
and explanatory theories. Theory construction was assigned to a
class referred to as Analytic-Synthetic Research while the confirming
of hypotheses was assigned to Experimental Research. Normative
Research deals with activities designed to generate hypotheses con-
cerning facts and those connections between facts which exist in
nature. Clinical Intervention Research, also conducted from the
generative purpose, is aimed at producing hypotheses about the connec-
tions among facts which might be brought into natural existence by
some intervention. Among the distinguishing features of Clinical
Intervention Research, Wilson noted the methods of data collection
(emphasizing interviews), the type of data collected (primarily qual-
itative), the analyses performed on these data (often categorical),
the selection of subjects (typically non-random) the number of
subjects (usually only a small group), the length of time for subject
involvement (extended periods), the nature of the treatments (loosely
pre-planned but dynamically modified for individual subjects), and
the contents of reports (extensive, systematic descriptions of treat-
ments and apparent effects coupled with a conscious, guided search for
patterns among the idiosyncratic performances that can be translated
into testable hypotheses).
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A research methodology which seeks to elicit information about
the subjects' cognitive processes must utilize carefully conceptual-
ized task environments wherein the subject can operate naturally,
openly, and productively. Our tasks must embody the potential for
stimulating cognition either directly or isomorphically characterized
as mathematical thought; Within such task environments our subjects
would operate with other participants or, at times, with only the
experimenter. The productive problem solving of children working
together toward a common goal will have direct importance for class-
room applications of this research. Although we can recognize the
inadequacies and potential distortions, we should explore probing
techniques to encourage the subject to give a verbal self-report of
what he is or has been thinking, thereby prompting a phenomenological,
steam-of-consciousness record of mental processing. Thus, to summar-
ize, in the current state of knowledge with respect to mathematical
problem-solving behavior, primary emphasis must now be given to
designing appropriate task environments, carefully observing over a
long period the individual child's spontaneous and learned problem-
solving behavior, reporting detailed case studies directed at portraying
the child's learning and development of problem-solving competence, and
generating hypotheses, procedures, measuring instruments, and designs
for future experimental investigations,
The primary emphasis in our immediate research should be on con-
ducting "teaching experiments" (in the _Soviet_sense_) to obtain Jfurther
detailed information on mathematical problem-solving heuristics as
teachable-learnable-transferable knowledges. These investigations
would likely have school-based, yet clinical, treatments which: (a)
explicitly teach heuristical precepts within mathematical problem
solving, (b) use "heuristic teaching" methods, and (c) study the
cognitive prerequisites for, and mental processes of, acquiring and
using such general precepts in mathematical problem solving. Results
of such investigations should lead to empirically derived "maps" of
the domains to be considered in learning and teaching problem solving.
Careful delineations of taxonomies or typologies of problems, "heuris-
tics" or "strategies," and mathematical knowledge structures need to
be formulated.
At the same time, it seems of paramount importance to sponsor and
encourage analytical research. Little or no substantive theory-
building has been done relative to the teaching, learning, or using
of heuristical methods in mathematical problem solving. In recent
years there has been a notable increase in the interest of mathematics
education researchers for studying the use of heuristical advice in
problem solving. Yet these scattered investigations have not been
predicated upon, nor led to, any but the narrowest of theoretical
bases. The assumptions upon which an instructional emphasis in heur-
istical methods is based must be better explicated. Those variables
that may be accounting for productive problem-solving performances,
learner difficulties in problem solving, and effective teaching and
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modeling of problem solving must also be identified and defined. In
short, fundamental questions need to be systematically generated and
studied. Which heuristical maxims are "teachable objects"? What
teaching "moves" or strategies might foster the acquisition and use
of heuristic? What is the nature of a child's learning with respect
to heuristic? To what extent might a learner's stage of cognitive
development account for the ease or difficulty in acquiring or using
heuristical methods during problem solving? Are there useful taxono-
mies or typologies of mathematical problems for illustrating various
heuristical maxims? These types of questions demand extensive analyses
as well as sensitive empirical treatment.
The interface of available theoretical formulations in cognitive
deve^lggment psychology and in teaching strategies with our heuristical
emphases must be considered. The proposed emphasis in "instruction in
heuristic methods" necessitates concerns for teaching factors. The
model upon which the Teaching Strategies Project of the GCSLTM is based
does not include an analysis of "moves" in teaching problems, in teach-
ing about problem solving, or more specifically in teaching heuristical
maxims (though the latter might well be construed to be principles or
generalizations within the present model). Yet the power and general-
izability of this teaching strategies model suggests that its fundamental
features may be useful in building a similar model for teaching heuristi-
cal methods and problem solving. Such model building and subsequent
model testing ought to be a central feature of our future work. The
production of a potentially huge bank of recorded lessons or episodes
from our "teaching experiments" would allow an easy access to a variety
of excellent teaching exhibits to be analyzed in such model building.
The relationships between cognitive developmental psychology and
the development of mathematical problem-solving competence must be
studied. In particular, aspects of Piaget's concrete and formal oper-
ational thought should be examined for possible connection with the
learning and use of heuristical methods. Conscious use of heuristics
in problem solving would seem to involve thought which encompasses
multiple operations, combinatorial processes, isolation of task varia-
bles, logical operations, flexibility (e.g., reversibility-reciprocity)
or other aspects of Piaget's theory. We ought to be able to teach the
heuristical questions as items of knowledge. But can we teach all sub-
jects to selectively use such heuristical advice? Polya noted the
importance of the mental operations implied by an heuristic precept.
We need to map out in finer detail how mental operations and structures
may describe contingencies for successful problem solving. That is, a
more penetrating analysis is needed of the cognitive operations essential
to allow the mind to consciously employ executive control over, and
choice among, the problem-solving strategies known to the student.
In particular, caution must be exercised that the operativity we wish
to see exhibited in the application of heuristical methods of problem
solving does not become mechanical rule-use by students who have been
unable to assimilate such knowledge.
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Perhaps an equally viable question would pursue the apparent
effects of heuristical knowledge on cognitive operativity. To be
sure, many of the task environments used by Piagetians are problem-
solving ventures. A subject's spontaneity in approaching and dealing
with the task is often crucial to ascertaining the stage of opera-
tivity. Yet heuristical knowledge would conceivably influence the
fashion in which a subject encoded and operated on a task. For
example, would subjects who knew how to approach problems with quite
well-organized, inductive pattern searching strategies and who did
in fact solve some of the classical formal operations tasks (e.g.,
pendulum, balance beam, hidden magnet) as an apparent result, have
essentially been accelerated in their development?
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40
References
Ashton, Sister M. R. Heuristic methods in problem solving in ninth
grade algebra. Unpublished doctoral dissertation, Stanford Univer-
sity, 1962.
Brownell, W. A. Problem solving. In The psychology of learning,
Forty-first Yearbook of the National Society for the Study of
Education, part II, 1942.
Buswell, G. T., & Kersh, B. Y. Patterns of thinking in solving problems.
University of California Publications in Education, 1956, 12, 63-148.
Butts, T. Problem solving in mathematics. Glenview, Illinois: Scott,
Foresman, 1973.
Covington, M. V. & Crutchfield, R. S. Facilitation of creative problem
solving. Programmed Instruction, 1965, 4_(4) , 3-5, 10.
Duncker, K. On problem-solving. Psychological Monographs, 1945, 58,
(5, Whole No. 270).
Goldberg, D. J. The effects of training in heuristic methods in the
ability to write proofs in number theory. Unpublished doctoral
dissertation, Columbia University, 1973.
Higgins, J. L. A new look at heuristic teaching. The Mathematics
Teacher, 1971, 64(6), 487-495.
Jennan, M. Instruction in problem solving and an analysis oj[ structural
variables that contribute to problem-solving difficulty (Tech. Rep.
No. 180). Stanford: Institute for Mathematical Studies in the
Social Sciences, Stanford University, 1971-
Jones, P. S. (Ed.). A history of mathematics education in the United
States and Canada. Washington: National Council of Teachers of
Mathematics, 1970.
Kantowski, E. L. Processes involved in mathematical problem solving.
Unpublished doctoral dissertation, University of Georgia, 1974.
Kilpatrick, J. Analyzing the solution of word problems in mathematics:
An exploratory study. Unpublished doctoral dissertation, Stanford
University, 1967.
Kilpatrick, J. Problem solving and creative behavior in mathematics.
In J. W. Wilson & L. R. Carry (Eds.)» Reviews of recent research
in mathematics education. Studies in mathematics (Vol. 19).
Stanford: School Mathematics Study Group, 1969.
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41
Kilpatrick, J. , & Wirszup, I. (Eds.)- Soviet studies In the psychology
of learning and teaching mathematics (6 vols.). Stanford, California:
School Mathematics Study Group, 1969-1972.
Larsen, C. M. The he_uris_tic_ standpoint in the teaching of elementary
calculus. Unpublished doctoral dissertation, Stanford University,
1960.
Lucas, J. F. An exploratory study on the diagnostic Egjgchin_g_o_f heuristic
problem-solving strategies in calculus. Unpublished doctoral
dissertation, University of Wisconsin, 1972.
McClintock, C. E. The effect of verbalization of he_uristics_ on transfer
of learning. Paper presented at the National Council of Teachers
of Mathematics 53rd Annual Meeting, Denver, Colorado, April, 1975.
Menchinskaya, N. A. Fifty years of Soviet instructional psychology.
In J. Kilpatrick & I. Wirszup (Eds,), Soviet studies in the psychology
of learning and teaching mathematics, (Vol. 1). Stanford, Cali-
fornia: School Mathematics Study Group, 1969.
Olton, R., Wardrop, P., Covington, M., Goodwin, W., Crutchfield, R.,
Klausmeir, H., & Ronda, T. The development of productive thinking
skills in fifth-grade children (Tech. Rep. No. 34). Madison,
Wisconsin: Wisconsin Research and Development Center for Cognitive
Learning, 1967.
Polya, G. How_ to solve it (2nd ed.). Garden City, N. Y.: Doubleday,
1957.
Polya, G. Mathematical discovery: On understanding, learning, and
teaching problem solving. (Vol. 1). New York: Wiley, 1962.
Polya, G. Mathematical, discovery: On understanding, learning, and
teaching problem solving. (Vol. 2). New York: Wiley, 1965.
Riedesel, C. A. Problem solving: Some suggestions from research.
The Arithmetic Teacher, 1969, 16, 54-58.
Simon, H. A., & Newell, A. Human problem solving: The state of the
theory in 1970. American Psychologist, 1971, 26, 145-159.
Smith, J. P. The effect of general versus specific heuristics in mathe-
matical jjroblem-splving tasks. Unpublished doctoral dissertation,
Columbia University, 1973.
Suydam, M. N. Annotated Compilations of Research on Secondary School
Mathematics, 1930-1970. Washington, B.C.: U. S. Department of
Health, Education and Welfare, Project No. l-C-004,1972.
Talyzina, N. F. Properties of deductions in solving geometry problems.
In J. Kilpatrick & I. Wirszup (Eds.), Soviet Studies in the psycho-
logy of learning and teaching mathematics, (Vol. 4). Stanford,
California: School Mathematics Study Group, 1970.
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42
Vos, K. The effects of three instructional strategies on problem-
solving behaviors in secondary school mathematics. Journal for
Research in Mathematics Education, 1976, 1(5), 264-275.
Webb, N. An exploration of mathematical problem-solving processes.
Paper presented at the meeting of the American Educational Research
Association, Washington, C.D., April, 1975.
Wertheimer, M. Productive thinking. New York: Harper, 1959.
Wickelgren, W. A. How to solve problems: Elements of a theory of
problems and pjroblem solving. San Francisco: W. H. Freeman
and Company, 1974.
Wilson, J. W. Generality of heuristics as an instructional variable.
Unpublished doctoral dissertation, Stanford University, 1967.
Wilson, J. An epistemologically based system for classifying research
and the role of clinical intervention research in that system.
Paper presented at the Annual Meeting of the American Educational
Research Association, New Orleans, February, 1973.
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43
The Teaching Experiment and
Soviet Studies of Problem Solving
Mary Grace Kantowski
University of Florida
The purpose of this paper is threefold; first, to examine the
typically Soviet research methodology known as the teaching experiment;
next, to review several of the Soviet Studies related to mathematical
problem solving in which some form of the teaching experiment was used;
and, finally, to reflect on ways in which aspects of this methodology
could be applied in this country in research dealing with the processes
involved in mathematical problem solving.
Rationale for a New Methodology
A perspicacious grasp of the Soviet concept of the "teaching experi-
ment" requires a thorough understanding of the forces that led to its
conception and some reflection on the rationale for its development.
Among the primary forces that necessitated the evolution of a new
research methodology in the U.S.S.R. was the influence of the philosophy
of the collective in the post-revolutionary society. The Soviet attitude
toward learning and instruction was a strong reaction to the concept of
the class system of pre-revolutionary Russia. In the spirit of the
theory of dialectical materialism the Soviets assumed instruction, not
native ability, to be the major factor in intellectual achievement.
They believed that except for cases involving organic damage or severe
retardation, all children have the same potential for academic accom-
plishment. 1
Whereas learning theories and pedagogical principles had previously
been based on experimental and theoretical psychology, the Marxist per-
cention of the dynamic interrelationship between pedagogy and psychology
emphasized the influence of instruction and content on psychological
lOnly a minority of pedagogical researchers led by V. A. Krutetskii
are somewhat dissonant with this main stream of thought. Krutetskii,
whose research will be discussed in a later section of this paper, is
investigating variation in ability.
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44
growth. This point of view compelled a search for a research methodology,
different from the "cross-sectional" type of investigation, that would
permit researchers to observe qualitative effects of various forms of
instruction. Such a direction obviously mandated the organization of
research as well as techniques that would allow researchers not merely
to observe complex processes involved in learning such content as
reading, grammar, and mathematics, but that would, in fact, influence
the development of these processes. The new research methods would have
to include longitudinal observation and evaluation; they would have to
permit a researcher to study changes in mental activity as well as the
effects of planned instruction on such activity.
To this end, pedagogical and psychological research were tied
together in their organization under the Academy of Pedagogical Sciences
(Reitman, 1962). Since the primary value of psychological research under
the Soviet regime was seen to be the improvement of instruction, Academy
studies of thinking necessarily involved the teaching methods most
effective in producing learning and independent thinking, and, conversely,
studies related to instruction dealt with complex mental processes.
The Influence of Vygotsky
The "teaching experiment" grew out of the "individual psychological
experiment" introduced in the twenties by Lev Serayonovich Vygotsky, the
psychologist-educator who left an indelible mark on Soviet pedagogical
research before his early death in 1934.
In the Marxist tradition, Vygotsky asserted that specifically human
mental processes are not inborn but formed, and that their development
is totally dependent on how they are taught. He characterized intellec-
tual development as evolutionary or shaped by adaptation to external
environment and not embryonic which he interpreted as development flowing
more or less smoothly according to a stereotype. According to Vygotsky,
all mental processes occur only by acquisition as a result of internali-
zations after some external activity (El'konin, 1967; Gal'perin, 1967).
Moreover, clinical data collected in early studies convinced him that
the development of certain mental processes was accompanied by changes
in cognitive structure at various levels of sophistication of function
of these processes (Kostyuk, 1968). He found, for example, that the
processes of analysis, synthesis, comparison, and generalization exhibi-
ted definite levels, and noted positive effects of various pedagogical
practices on these levels.
These factors in the revolutionary concept of the formation of men-
tal processes, the indication of changes in cognitive structure with
mental growth in certain operations and this conviction of the primacy
of instruction in mental development led to Vygotsky's conception of a
genetic instructional research methodology that would focus in the
qualitative aspects of thinking and learning. He conceived of an
"instructional experiment" that would be a systematic reproduction of
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45
processes as they develop under various instructional procedures. In
his "laboratory," often the school setting, he tried to follow the
course of development by "experimentally evoking the genesis of volun-
tary attention" (El'konin, 1967). Because the processes were observed
only periodically, Vygotsky attached great importance to his concept
of the "zone of proximal development" (Gal'perin, 1967; Kostyuk, 1968),
where the passage from lack of knowledge or lack of ability to operate
to possession of knowledge of operational ability and the corresponding
change in cognitive structure could be observed.
Perhaps artificially, Vygotsky distinguished between "simple" and
"scientific" concepts, a difference not in the content of concepts
but in the way concepts are mastered (El'konin, 1967; Menchinskaya,
1969b; Talyzina, 1962). He saw the former as learned spontaneously
and inefficiently from "object to definition" through daily experience
while the latter were learned from "definition to object" through
carefully planned instruction. This distinction, which further empha-
sizes the Soviet view of the primacy of instruction was seen as
necessary to Vygotsky since he felt that in the acquisition of "scien-
tific" concepts and relationship between instruction and development
was most clear and capable of most complete investigation. It was with
the development of "scientific" concepts that Vygotsky's studies
were concerned.^
The method introduced by Vygotsky was, in a sense, modeling rather
than empirically studying the processes as they developed and, studying
the results of the learned behavior in a clinical setting.
Vygotsky began using these genetic experimental techniques in
studying the relationship between language and thought. His influence
soon spread to other disciplines, and although his basic concept of the
pedagogical experiment remained the same, it began to take on different
forms to correspond to varying research needs.
Characteristics of the^ "Teaching Experiment"
If one word had to be chosen to characterize the "teaching experi-
ment," it would most likely be the term "dynamic" since it is movement
that interests the Soviet researchers—movement from ignorance to knowl-
edge, from one level of operation to another, from a problem to a
solution. The aim of this research is to "catch" processes in their
development and to determine how instruction can optimally influence
these processes. Unlike most experimental designs used in this country,
2Recently, Menchinskaya (1969b) and Talyzina (1962) have taken
issue with Vygotsky's assumption that concepts are best learned
"scientifically." Studies related to activity learning (Zykova, 1969;
Kalmykova, 1962) support their objections.
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46
the "teaching experiment" cannot be completely characterized by
describing sampling procedures, experimental groups and test statistics.
The label is actually a generic term for a variety of pedagogical
research forms in which the strictly statistical analysis of quantita-
tive data is of less concern than the daily subjective analysis of
qualitative data. Most studies deal with some aspect of the formal
school situation although the data are often gathered only from a
sampling of "strong" "average" or "weak" students who are generally
categorized and selected with the aid of the classroom teacher. The
data collected are often qualitative, obtained in a clinical setting
by recording verbal protocols for future analysis. Underlying this
procedure is one of the salient features of the "teaching experiment,"
its compensatory nature. The quantity of macroscopic data (such as
objective test scores) generally acquired in an experimental study is
exchanged for microscopic detail of processes observed using a small
sample. Probing interviews and exchanges with individual students add
to any group data collected to support generalizations resulting in
decisions for future instructional sequences.
Other general characteristics of the "teaching experiment" include
its longitudinal nature (instructional treatment is applied and data
are gathered over an extended period), the planning of instruction in
the light of observations made during the previous session, and extensive
co-operation among classroom teachers and researchers. It is proced-
urally acceptable to give hints to the subjects during testing, so that
any learning in the testing situation may also be observed. In most
cases results are reported in the form of a narrative that includes an
analysis of observed behaviors and conclusions drawn from the analysis.
Any quantitative data are generally reported using descriptive statis-
tics. Inferential test statistics are seldom used.
Some "Teaching Experiments"
Menchinskaya broadly defines the "teaching experiment" as "study
combined with pedagogical influence" (1969a). She describes two
forms of this research widely in use (Menchinskaya, 1969b). The first
is the "experiencing" form in which only one mode of instruction is
employed and observations are made in a clinical setting to determine
its influence on mental processes. No explicit comparison is made to
any other instructional procedure.
The Gal'perin and Georgiev study (1969) and the follow up study
by El'konin (1961) are examples of research using the "experiencing"
form. Both involved the introduction of mathematical concepts using
the unit of measurement instead of the concept of number. In the
El'konin study, although no explicit comparisions are made, the sub-
jects chosen were judged to be very low in mathematical concepts
initially. Thus, the fact that this group learned all mathematical
concepts and skills required at their grade level implied a judgment
regarding the value of the method. The study employed the latest form
in the evolution of the "teaching experiment" and one that should be of
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47
interest to American researchers who are attempting to bridge the gap
between research and what actually occurs in the classroom. The experi-
mental classroom used was equipped with one-way glass and with a TV
camera lens mounted in the classroom and connected to a screen in the
adjoining laboratory, which was also equipped with booths for individual
experiments. All group and individual sessions were recorded on tape
for future analysis. Although the general course outline and content
to be covered were determined in advance, the experimenters, teachers
and aides observed the class sessions held during the day, discussed
the lessons and planned activities and instruction for the following
session on the basis of what occurred each day. The individual experi-
ments highlight another feature of the "teaching experiment," that of
probing for hunches on which to base new instructional strategies.
Again, the dynamic nature of the "teaching experiment" is evident here
as the experimenters attempt to capture processes as they are being
formed and to determine optimal strategies of instruction. This proce-
dure may be useful as a preliminary to pilot testing to determine
plausible hypothesis to be tested in future experimental studies.
The "experiencing" mode of the teaching experiment was also used
by Krutetskii in his studies of mathematical abilities (1965, 1969,
1973). The investigations were conducted by Krutetskii and his stu-
dents between the years 1959 and 1965 using students from the second
through the tenth grades. By analyzing solutions of carefully organized
sets of mathematical problems generated over periods of about two years
with the same students, Krutetskii was able to delineate characteristics
of studants with high ability in mathematics. The organization of the
problem presentation was instructional; as the problems were solved,
mental processes were observed in their development. The problems
included those requiring generalizations and algebraic proofs, and those
with visual-graphic and oral-logical components, among others. Krutetskii
emphasized that although some quantitative data were gathered (e.g., the
number of problems solved and the time to solution), the dynamic indices,
such as progress in qualitative aspects of problem solving, were more
valuable than the static, quantitative ones. These are reflected in the
components Krutetskii enumerated in the structure of mathematical abilities,
namely (1) the formalized perception of mathematical material (2) quick
and sweeping generalization of mathematical material (3) curtailment of
thought (4) flexibility of thought (5) striving for economy and (6) a
mathematical memory. -* In a recent publication, Krutetiskii took a
definite stand on the existence of levels of ability (1973), a position
opposed to the basic Soviet philosophy.
3 In his earlier writings Krutetskii included spatial skills in the
structure, but later removed it from the "obligatory" structure.
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The second form of the "teaching experiment" is the "testing" mode,
one more closely related to our own experimental studies. This pro-
cedure was used by Kalmykova (1962) in a study related to mathematical
applications in physics. Aspects of the research that clearly distin-
guishes it from our experimental studies include the type of data
collected and the form of analysis of these data. With the help of the
classroom teacher the subjects in the study were divided into "weak"
and "strong." One half of each group was assigned to each of two methods
of instruction—method "A" which was essentially expository and in which
the teacher outlined the procedures to follow in completing the exercises,
and method "B" which was essentially a heuristic teaching technique. A
variable called "rate of learning" was determined by the number of problems
needed in the instruction and the time necessary for a subject to com-
plete exercises independently. Kalmykova states that there was no
significant difference in the effects of two methods for the "strong"
students. In spite of the fact that the average number of problems and
the time required for mastery were not significantly different even in
the case of the "weak" pupils she concludes, one the basis of analysis
of clinically obtained data, that method "B" was superior for the "weak"
students since they exhibited higher levels of analytic-synthetic activity
in the solution of the control problems.
In another study Kalmykova (1975) uses a form that does not clearly
fall into either the "experiencing" or the "testing" category but con-
tains elements of both. In her initial research study on analysis and
synthesis in problem solving, Kalmykova observed various teachers as they
taught problem solving in elementary school classrooms and then examined
the problem solving behaviors of their students. She analyzed in detail
the instructional strategies of one particularly successful teacher,
V.D. Petrova, and suggested elements that should be included in all
problem solving instruction based on her analysis of Petrova's techniques.
Kalmykova herself then applied these techniques with some success in
instructing a group of "weak" students. Petrova's method of instruction,
which emphasized both analysis and synthesis in problem solving, was
compared to the more structured "classical analysis" methods used by the
other teachers. These comparisons used observations of problem solving
behaviors of the students from all participating classes. Thus, both
the "experiencing" form and the "testing" form were used to some extent
by Kalmykova in the same study.
The "Teaching Experiment" and Our Research
Any attempt to compare the Soviet methodology to research designs
used in this country would be analagous to an attempt to answer the
question: "Who is the better athlete Chris Evert or Olga Korbut?"
Just as an athlete is judged on standards related to her event, each
research methodology must be examined in the light of its purposes
and the philosophy of education encompassing it. The Soviets are
concerned with the qualitative aspects of mathematics learning and
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49
problem solving. Thus far we in this country, with few exceptions,
have focused on the quantitative. Perhaps the answer to effective
problem solving research lies in a compromise—a merger of the two
methodologies through studies that involve both aspects.
Paradoxically, the essential differences between the "teaching
experiment" and research in this country account for what could here
be considered the most severe limitations as well as the desirable
strengths of the Soviet methodology. Because the instruction is often
determined by what is found in preliminary analysis and the duration
of the experiments is often a year or longer, this approach introduces
variables that would be considered invalidating to most American
researchers. The time between the initial testing and the intro-
duction of the instruction could result in learning on the part of
subjects who were identified as not having the desired skills initially.
Because analysis is logical, the introduction of hints during testing is
not considered undesirable in Soviet research. Experimental control in
our sense gives way to the opportunity to "catch" the learning of a
concept or strategy and to suggest instructional techniques to insure
mastery. On the other hand, the instructional experiment allows the
researcher to observe how a subject is operating and to determine levels
of sophistication (for example, elegance in problem solving) instead of
mere numbers of correct solutions. Such diagnostic techniques permit
the discovery of erroneous concepts as well as "strokes of genious."
Few researchers in mathematics education in this country have con-
cerned themselves with the detailed study of the development of pro-
cesses in mathematics. Kantowski (1974) investigated processes used in
the development of skills in solving geometry problems; Lucas (1972)
and Goldberg (1973) studied processes related to problem solving in the
Calculus and number theory, respectively.
Although process research in this country has been sparse, several
recent studies related to problem solving could be modified to include
a process component. For example, in the Wilson (1967) study involving
the level of generality of heuristics used in instruction, a matrix
sampling technique could be used and qualitative data collected from
representative subjects in each of the cells. Analysis of such data
could serve as a valuable supplement to the statistical analysis of the
quantitative data by providing information on how the development of
processes is affected by the level of generality of the heuristics used
in instruction in each of the content areas, and perhaps by suggesting
other instructional techniques.
Another possibility would be to randomly select individuals from
intact classes such as those used in the Goldberg (1973) and Lucas (1972)
studies and to follow identical instructional techniques with these
individuals while gathering qualitative data from their verbal and
written protocols along with the quantitative data from the remainder
of the classes. Such observations made using subjects on various ability
levels could suggest Aptitude-Treatment-Interaction studies for teaching
problem solving. Other suggestions for studies related to process
research may be found in Kantowski (1974).
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The gathering of qualitative data is but one step in progress toward
understanding the processes of mathematical thinking. Methods of
analyzing the data are needed, and more importantly, methods for com-
municating the results to other researchers, and to classroom teachers
must be explored. Finally, the ultimate goal is to find ways to use the
data to improve classroom instruction and to positively affect mathe-
matics learning and problem solving in the classroom.
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51
References
El'konin, D. B. A psychological study in an experimental class. Soviet^
Education, 1961, 3_ (7), 3-10.
El'konin, D. B. The problem of instruction and development in the works
of L. S. Vygotsky. Soviet Psychology, 1967, 5_ (3), 34-41.
Gal'perin, P. Ya. On the notion of internalization. Soviet^ Psychology,
1967, 5_ (3), 28-33.
Gal'perin, P. Ya., and Georgiev, L. S. The formation of elementary
mathematical notions. In J. Kilpatrick & I. Wirszup (Eds.),
Soviet studies in the psychology^ of learning and teaching
mathematics (Vol. 1). Stanford, California: School
Mathematics Study Group, 1969.
Goldberg, D. J. The effects of training in heuristic methods in the
ability to write proofs in number theory. Unpublished doctoral
dissertation, Teacher's College, Columbia University, 1973.
Kalmykova, Z. I. Dependence of knowledge assimilation level on pupils'
activity in learning. Soviet^ EducatjLpn, 1958, 1^ (11), 63-68.
Kalmykova, Z. I. Analysis and synthesis as problem-solving methods. In
J. Kilpatrick, E. G. Begle, I. Wirszup, & J. A. Wilson, (Eds.),
Soviet studies^ in the psychology of learning and teaching mathe-
matics (Vol.11). Stanford, California: School Mathematics Study
Group, 1975.
Kantowski, M. G. Processes involved in mathematical problem solving,
Unpublished doctoral dissertation, University of Georgia, 1974.
Kostyuk, G. S. The problem of child development in Soviet psychology.
Soviet Psychology, 1968, 6 (3-4), 91-111.
Krutetskii, V. A. Age peculiarities in the development in abilities
in students. Soviet Education, 1965, 8^ (5), 15-27.
Krutetskii, V. A. An investigation of mathematical abilities in school
children. In J. Kilpatrick & I. Wirszup (Eds.), Soviet studies in
the psychology of learning and teaching mathematics^ (Vol. 2).
Stanford, California: School Mathematics Study Group, 1969b.
Krutetskii, V. A. The problem of the formation and development of
abilities. Soviet Education, 1973, 16, 127-145.
Lucas, J. F. An exploratory study in the diagnostic teaching of elemen-
tary calculus. Unpublished doctoral dissertation, University of
Wisconsin, 1972.
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52
Menchinskaya, N. A. Fifty years of Soviet instructional psychology. In
J. Kilpatrick & I. Wirszup (Eds.), Soviet studies^ in the psychology
of JLearning and teaching mathematics (Vol. 1). Stanford, California:
School Mathematics Study Group, 1969a.
Menchiaskaya, N. A. The psychology of mastering concepts; fundamental
problems and methods of research. In J. Kilpatrick & I. Wirszup
(Eds.), Soviet studies in the psychology of learning and teaching
mathematics (Vol. 1). Stanford, California: School Mathematics
Study Group, 1969b.
Reitman, W. Some Soviet investigations of thinking, problem solving, and
related areas. In R. A. Bauer (Ed.), Some views on Soviet psychology.
Washington: American Psychological Association, 1962, 29-61.
Talyzina, N. F. Manner of acquisition of initial scientific concepts.
Soviet Psychology and Psychiatry, 1962, 1 (1), 39-42.
Vygotsky, L. S. Thought and language. Cambridge, Mass.: MIT press, 1962.
Wilson, J. W. Generality of heuristics as an instructional variable.
Unpublished doctoral dissertation, Stanford University, 1967.
Zykova, V. I. Operating with concepts when solving geometry problems.
In J. Kilpatrick & I. Wirszup (Eds.), Soviet studies in learning
and teaching mathematics (Vol. 1). Stanford, California: School
Mathematics Study Group, 1969.
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Mathematical Problem Solving In The Elementary School:
Some Educational And Psychological Considerations*
Frank K. Lester, Jr.
Indiana University
Introduction
One of the most important goals of elementary school mathematics is
to develop in each child an ability to solve problems. In recent years
more and more emphasis has been placed on problem solving in the
elementary mathematics curriculum. A cursory look at the scope and
sequence charts of the most popular textbook series points out this
trend. In each of these series problem solving is identified as one
of the key strands around which the mathematics program is built. At
the same time there is concern among teachers, mathematicians, and
mathematics educators that these programs are doing a poor job of
developing problem solving ability in children. Points of view which
are representative of the dissatisfaction with current programs are
found in the reports of the Snowmass Conference on the K-12 Curriculum
and the Orono Conference on the National Middle School Mathematics
Curriculum held during the summer of 1973. These reports called for
extensive modification of current mathematics programs to include a
more systematic approach to providing instruction in problem solving.
The current concern should raise a number of questions in the mind
of anyone interested in the mathematics education of children. Exactly
what is problem solving? Can students really be taught to be better
solvers? If problem solving is so important and good problem solvers
are not being developed, what steps should be taken to change present
instructional practices? Certainly an answer to the first question must
be obtained. So, before proceeding any further a definition of problem
solving should be provided.
*The author is indebted to Dr. Norman L. Webb and other members of
the Mathematical Problem Solving Staff at Indiana University for their
valuable suggestions. The views expressed in this paper do not constitute
an official statement of policy regarding the goals of the Mathematical
Problem Solving Project. Th author accepts sole responsibility for all
of the positions and views stated in this paper.
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Definition of a Problem
A problem is a situation in which an individual or group is called
upon to perform a task for which there is no readily accessible algorithm
which determines completely the method of solution.
Any one of a number of other definitions of a problem would be
satisfactory for the purposes of this paper (e.g., Bourne, Ekstrand &
Dominowski, 1971; Davic, 1966; Henderson & Pingry, 1953; Newell & Simon,
1972). Let it suffice to say that any reference to a problem or problem
solving refers to a situation in which previous experiences, knowledge,
and intuition must be coordinated in an effort to determine an outcome
of that situation for which a procedure for determining the outcome is
not known. Thus, finding the length of the hypothenuse of a right triangle
given the lengths of the two legs probably does not involve problem
solving for the student who understands the Pythagorean Theorem, but
may be problem solving of a complex nature for the student who has not
been exposed to the Pythagorean Theorem.
Since problem solving is viewed as such an important part of learning
mathematics, it seems natural to analyze carefully what is involved in
the process so that effective instructional techniques can be developed.
There is little or no argument on this point. Everyone agrees that serious
attention must be given to instructional issues related to problem
solving. However, beyond this point there is little, if any, unanimity
of opinion concerning the process of problem solving.
Even the most successful problem solvers have difficulty in identify-
ing why they are successful, and even the best mathematics teachers are
hard pressed to pinpoint what it is that causes their students to become
good problem solvers. Unfortunately, in spite of the volumes that have
been devoted to problem solving what is now universally accepted know-
ledge about problem solving can be boiled down to George Polya's words
of advice to mathematics students: "Use your head." (Professor Polya's
final statement in a presentation at the 1974 annual meeting of the
American Mathematical Society.)
Out of frustration over an inability to deal successfully with the
problem solving dilemma, mathematics educators have turned to psychology
for guidance. The nature of problem solving and the measurement of problem
solving ability have been the objects of considerable attention by psycho-
logists (representative reviews of psychological research in problem solving
have been written by Bourne £. Dominowski, 1972; Davis, 1966; Green, 1966).
Typically, psychological reports of problem solving research begin with
a statement like: "Research in human problem solving has a well-earned
reputation for being the most chaotic of all identifiable categories of
human learning" (Davis, 1966, p. 36). Indeed, it has only been during
the last twenty-five years that a major point of view or technique has
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55
developed which attempts to isolate the important variables which influ-
ence problem solving behavior.1
There appear to be a number of reasons for this condition. First,
a variety of tasks has been used in problem solving research. The tasks
found in the literature include such diverse problems as matchstick,
Tower of Hanoi, jigsaw puzzles, anagram problems, concept identification
problems, arithmetic computation problems, and standard mathematics text-
book word problems. Also, problem solving research has been conducted
by experimenters with quite different positions on the nature of problem
solving. The traditional cognitive-Gestalt approach of such psychologists
as Wertheimer (1959), Maier (1970), and Duncker (1945) is quite different
from the associative learning theory approach characterized by the work
of Maltzman (1955) and the Kendlers (Kendler & Kendler, 1962). More
recently, especially within the past fifteen years, considerable effort
has been devoted to the development of an information processing approach
to the study of problem solving. The well-known work of Newell and Simon
(1972) is representative of the information processing view of the pro-
blem solving process. Thus, although much exciting and potentially
fruitful work is being conducted by psychologists, very few definitive
answers to the questions concerning the nature of learning and instruction
in mathematical problem solving are available at the present time. It
is likely that these answers will result only from several years of
intensive study that reflects a cooperative effort by mathematics
educators, psychologists, and classroom teachers.
Overview of This Paper
The intent of this paper is to describe the philosophy and activities
of the Mathematical Problem Solving Project (MPSP) at Indiana University.
The paper will contain four main sections:
1. the critical issues and questions related to
mathematical problem solving,
2. nature of the MPSP,
3. thrust of the work of MPSP at Indiana University, and
4. plans for future research.
The main focus of this paper is on the research and development efforts
underway at Indiana University. Included in this effort is a serious
•"•Kilpatrick (1969) suggests that serious attention to problem solving
by mathematics educators has developed primarily within the last ten or
so years.
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attempt to develop a conceptual framework for mathematical problem
solving. The development of such a framework will center on the creation
of a model for mathematical problem solving. Since the creation of such
a model is considered to be of utmost importance in developing a frame-
work for future research and development efforts, an extensive discussion
of models of problem solving is included. It is hoped that the positions
posed and the efforts described will stimulate valuable discussion con-
cerning the key issues related to mathematical problem solving in the
elementary schools.
Critical Issues and Questions Related
to Mathematical Problem Solving
The opening sentence of this paper stated that the development of
children's problem solving abilities is a major goal of elementary school
mathematics. It is interesting that while few educators would disagree
with this claim there is little evidence that a serious attempt is being
made to attain this goal. No single factor can be identified as causing
this state of affairs to exist. Instead the problem can be attributed
to a number of causes. The following are among the most prominent:
1. Problem solving is the most complex of all intellectual
activities; consequently, it is the most difficult intel-
lectual ability to develop.
2. Elementary school mathematics textbooks typically are
deleterious rather than facilitative in developing
problem solving skills and processes in children.
3. Elementary school teachers do not view problem solving
as a key feature of their mathematics programs.
Before suggestions are presented for remedying the present situation it
is appropriate to elaborate on causes 2 and 3.
It is the author's opinion that the overwhelming-majority of the
activities presented in elementary mathematics texts as problems are
actually little more than exercises designed for practicing the use of
a formula of algorithm. A second criticism is that textbooks do not
include enough situations which involve real-world2 applications of
mathematics.
The term "real-world" is difficult to define since a real-world or
real-life problem for one person may not be a real-life problem for
another. Although interest rate and grocery shopping problems are very
real in the sense that such problems are encountered daily by adults, they
are often not even problems for children because children are not inter-
ested in them.
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The third cause is the result of several factors. It is a fact
that 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. Of
course elementary teachers cannot be blamed for their perception of
mathematics since it is based primarily on educational experiences which
stressed memorization of rules, formulas, and facts. However, the view
of mathematics which is held by elementary teachers is a part of a
vicious cycle which has developed. Children are not learning to become
good problem solvers because their mathematics textbooks do not provide
appropriate opportunities for them to solve problems and because their
teachers do not view problem solving as important. At the same time,
teachers do not view problem solving as important because it was not
given priority status when they studied mathematics. This condition
cannot be rectified by attempting to convince preservice teachers of the
importance of problem solving. At Indiana University preservice elemen-
tary school teachers are required to take nine semester hours of mathe-
matics and three semester hours of methods of teaching mathematics.
Even this uncommonly good situation does not allow sufficient time to over-
come ten or more years of "bad" experiences with mathematics. Also,
young teachers are prone to model their teaching behavior after the
behavior of their supervising teachers. Consequently, if little or no
provision is made for developing children's problem solving skills by a
student teacher's supervising teacher, it is unlikely that the student
teacher will consider problem solving as an important part of the mathe-
matics program.
Remedies for the existing conditions cannot ignore the need to
improve current teacher training programs, but improved teacher training
is only a small part of the solution. Even if teachers can be trained
to view mathematics as an area accessible through experimentation and
independent thought, they will probably resort to using whatever written
materials are available in the classroom and these materials are, for
the most part, not conducive to enhancing the development of problem
solving abilities. Thus, serious and extensive efforts must begin to
develop exemplary instructional materials in mathematics which have
problem solving as their main focus. The Mathematical Problem Solving
Project (MPSP), which will be described later, is attempting to satisfy
the need for such problem solving materials by producing a series of
modules devoted to the development of certain problem solving techniques
and by collecting and categorizing problems suitable for use in the
intermediate grades.
Attempts to develop instructional materials of any type must involve
considerable reflection about the most important aspects of the topic
being considered. In the course of developing modules which will teach
children fundamental skills and processes of problem solving the following
questions are among those which should be considered.
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1. What kind and how much direction should be given in a
module?
2. What instructional format is best suited to teaching
children how to solve problems?
Of course, these are important questions, but they are not specifi-
cally related to mathematical problem solving. Instead, they are ques-
tions which are raised by writers of any sort in instructional materials.
It is premature to attempt to answer these questions until answers to
several more basic questions are found. Unfortunately, the knowledge that
exists about how children solve problems and how problem solving should
be taught is very limited. For example, no confident answers have been
found for the most basic questions such as:
1. What prerequisite skills,abilities, etc. must children
have to solve particular kinds of problems?
2. What aspects of the problem solving process can be
taught to intermediate grade children?
a. Can children use various problem solving strategies
effectively?
b. Can children learn to coordinate the cognitive processes
which are needed in solving complex problems?
Clearly the answers to these questions to a certain extent must be
based upon the intuition and experience of the persons involved in
writing the materials. However, it is equally as important that these
questions be attacked by considering the theoretical and research base
underlying the various views toward teaching problem solving. It would
be most unfortunate to have another curriculum project which devotes
all its energies to the development of materials to the exclusion of
attempting to further the scientific knowledge regarding learning and
instruction in mathematical problem solving.-'
The issues raised thus far have been concerned primarily with the
role of problem solving in the existing mathematics curriculum and the
development of instructional materials. Before these issues can be dealt
with in an appropriate way it is essential that several more fundamental
issues and questions be considered. These issues include the four pre-
viously mentioned and are listed with some discussion following.
1. Can problem solving be taught?
-*This view is also held by Richard Shumway and is presented in a
position paper prepared by him for the MPSP (1974).
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2. If problem solving can be "taught,"^ what type of experiences
most enhance the development of this ability?
3. What are the specific characteristics of successful problem
solvers?
4. What prerequisite skills, abilities, etc. and what level of
cognitive development must a student have in order to solve
a particular class of problems?
5. Educators and psychologists generally agree that there are
several factors which influence problem difficulty. What are
the primary determinants of mathematical problem difficulty
for children in grades 4-6?
6. There are several motivation factors which influence children's
ability and willingness to solve mathematical problems. For
example:
a. What types of problems are interesting to children in
grades 4-6?
b. To what extent does a child's cognitive and emotional
style influence her/his willingness to solve problems?
7. What problem solving strategies can children (grades 4-6) learn
to use effectively? More fundamentally, can problem solving
strategies be taught which are generalizable to a class of
problems?
8. Since problem solving is also important in nonmathematical
areas, the question arises concerning the extent to which
learning to solve various types of mathematical problems
transfers^ to solving nonmathematical problems (the issue is
just as important if modified to read ". . . transfers to
solving other types of mathematical problems").
9. There are a number of issues related to the method of instruction.
Among the most important are:
a. Is the small group mode of instruction a better mode than
either the large group mode or individual instruction in
terms of teaching problem solving?
^"Taught" is being used here in the sense that teaching can be
viewed as facilitating the understanding of or knowledge about something.
It does not imply necessarily direct intervention in the student's learn-
ing process.
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b. What aspects of the problem solving process should
influence the choice of method of instruction? For
example, should the type of problem solving strategy
appropriate for a problem affect the instructional
mode used?
c. The specific role of the teacher in problem solving
instruction is an open issue. Are there certain aspects
of the problem solving process which suggest a more
directive role by the teacher than others?
d. How should problem solving instruction be organized and
sequenced? For example, should specific skills (e.g.,
making tables) be developed before attention is directed
toward teaching a particular strategy? To what extent
should a hierarchy (in the sense of Gagne 1970) be
followed in planning instruction in problem solving?
10. How do such characteristics of problems as difficulty, interest,
setting, strategy, and mathematical content relate to one
another?
11. Several models of the problem solving process have been suggested.
Do any of these models adequately describe mathematical problem
solving? Is there a need for developing a model for instruction
in problem solving? An instructional model might be fundamentally
different from a model of the solution process.
Specific Questions Under Study by the MPSP
The MPSP at Indiana University has selected several of these issues
and questions for study: namely, NOS. 1, 5, 6(a), 7, and 11. Since
these questions and issues have been given some careful thought, it is
appropriate to discuss them briefly.
Question 1. Can problem^ solving be taught? Clearly, this is the
most important question of all. Kilpatrick's (1969) review of mathe-
matical problem solving indicated that very little research has been
done regarding the influence of instruction on problem solving ability.
The answer to this question probably will not be determined until more is
known about the nature of solving problems and the relationships among
the many factors which influence mathematical problem solving.
Question 5. What are the primary determinants of mathematical
problem difficulty for children in grades 4-6? Psychologists generally
focus on four main areas for investigating problem difficulty: (a)
type of problem task; (b) method of presentation of the problem; (c)
familiarity of the problem solver with acceptable solution procedures
(strategies, skills, etc.); (d) problem size (e.g., a problem with several
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dimensions, both relevant or irrelevant, is more difficult than a problem
having fewer dimensions). Each of these areas has direct relevance for
elementary school mathematical problem solving. Clearly, not all types
of problems are appropriate for children of this age. What is less clear
is the best method of presenting particular classes of problems to chil-
dren. Language factors, complexity of the problem statement, role of
concrete and visual materials, child's prior experiences, and type of
problem are among the several factors determining the most appropriate
method of presentation. Much valuable information could be gained by
posing problems to students in different forms and versions and under
varying conditions.
That the student's familiarity with acceptable solution procedures
is an important determinant of problem difficulty raises a number of
questions which must be considered.
1. Which skills and strategies are most important for aiding
problem solving in mathematics in grades 4—6?
2. Which skills and strategies should be taught first?
3. Which, if any, strategies do students use naturally?
4. Which skills and strategies can be taught efficiently and
effectively? Can any be taught?
5. Should the skills (e.g., making a table) be developed before
concentrating on teaching a strategy (e.g., pattern finding),
or should they be developed as the strategy is taught?
6. Does teaching a particular strategy really improve problem
solving ability in the sense that for any problem a student
will be able to choose the most appropriate strategy to use?
More questions are being raised than answers in this paper. This
reflects the author's earlier statement that there are few definitive
answers to the questions about learning and instruction in mathematical
problem solving. The questions posed in the preceding paragraph are no
exceptions. However, despite the lack of answers based on firm research
evidence, there is considerable agreement that strategies can and should
be taught. This claim will be discussed when Question 7 is considered.
Issues related to problem size and problem complexity are a major
focus of the research efforts of the MPSP. Since these efforts will be
discussed in the last section of this paper no more will be said about
problem size in this section.
The four determinants of problem difficulty that have been discussed
are certainly not the only ones. Rather, they are the ones to which
psychologists have devoted the most attention. Maier (1970) stated that
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there are several other important factors which make a problem difficult.
In determining a list of causes of difficulty, he began with the assump-
tion that there is no lack of knowledge on the student's part. Based upon
this assumption he listed five potential causes of difficulty in addition
to the four that have already been mentioned: (a) misleading incorrect
solutions, (b) type of demands made upon idea-getting processes versus
idea-evaluation processes, (c) difficulty in locating subgoals that can
be reached, (d) lack of motivation, and (e) high degree of stress.
The factors which have been listed in the previous paragraphs
illustrate the extreme complexity of problem solving. In addition
psychologists have determined these factors primarily through highly
controlled experimentation. In many of the "laboratory" studies there
was no need to consider factors such as mathematical content, level
of understanding of concepts, processes, and skills, and environmental
influences since ability to perform the tasks used is not contingent upon
these factors. Unfortunately, these factors are present in normal class-
room instruction. Consequently, in addition to the determinants of pro-
problem difficulty which have already been mentioned, the teacher is
confronted with the task of dealing with even more confounding factors in
planning appropriate mathematical problem solving activities.
Question 6 (a). What types of problems are interesting to children
in grades 4-6? This question cannot be answered without considerable
knowledge of a student's background, experiences, cognitive ability,
and psychological makeup. There is substantial evidence that learning
is-enhanced when instruction is meaningful and relevant to the student.
It is reasonable to expect that this is also the case in learning to
solve problems. There are no hard-and-fast rules for determining if
a particular problem is interesting, but there are some general rules-
of-thumb which can guide problem selection.
1. Be sure the problem statement (if written) is easy for the
student to read.
2. Use personal words and terms in the statement of the problem.
Try to make the student feel like he is a part of the problem.
3. Although "real-world" problems are often difficult to find,
such problems have a high motivational value. (Most of the
"interesting" real-world problems are too sophisticated for
the level of mathematical understanding which intermediate
grade children have).
4. Encourage students to make up their own problems.
5. Do not place the student in a stressful situation. For example,
insistence on getting a correct answer in a short period of
time is a good way to kill enthusiasm for working a problem.
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The MPSP is developing a problem bank for grades 4-6. One of the
criteria for selecting a problem for inclusion in the bank is that is
must be interesting to children. Interest will be determined through
extensive interviewing and observing children as they solve problems.
Question 7. What problem solving strategies can children learn
to use effectively.? In papers prepared for the MPSP, Greenes (1974)
and Seymour (1974) offer specific recommendations regarding skills and
strategies which should be taught. Greenes not only listed several
strategies which can be taught to children in grades 4-6 but also made
suggestions for sequencing problem solving activities. The skills and
strategies Greenes identified include: estimate or guess, simplify, con-
duct an experiment, make a diagram, make a table, construct a graph,
write an equation, search for a pattern, construct a flowchart, partition
the decision space, and deductive logic.
Seymour considers such skills as "making a table" and "constructing
a graph" as valuable aids to mathematical problem solving but would prob-
ably classify such skills as substrategies because they are really tools
for applying a strategy. The strategies he considers appropriate for the
intermediate grades include: analogy, pattern recognition, deduction,
trial and error, organized listing, working backwards, combined strategies,
and usual strategies which are unique to a problem.
The belief of mathematics educators like Greenes, Seymour, and Polya
(1957) that strategies can be taught should be given serious consider-
ation. Most of our knowledge about learning and instruction is based on
the experiences of teachers who have thought long and hard about ways
to help children learn. Although little research has been done on the
effectiveness of teaching problem solving strategies, the fact that
several master teachers are convinced of the feasibility of teaching
children the use of certain strategies should encourage teachers who are
planning to include problem solving as a part of their mathematics pro^
gram.
Question 11. Do any of the models of the problem solving process
adequately describe mathematical problem solving? The primary purpose
of a model is to describe the salient and essential characteristics of
the process or phenomenon which is being modeled. Any model of the
problem solving process should be evaluated on the basis on the extent
to which it not only identifies the essential aspects of the process
but also the extent to which stages and relationships among those stages
are ident i f ied.
An investigation of this question has evoked considerable inquiry
within the MPSP, and it is a major theme of this paper. A discussion
of models of mathematical problem solving is included in a later section
of this paper.
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The Nature of the Mathematical Problem Solving Project
The Mathematical Problem Solving Project (MPSP), which is cosponsored
by the National Council of Teachers of Mathematics and the Mathematics
Education Development Center at Indiana University and funded by the
National Science Foundation, is working toward the development of mathe-
matical problem solving modules which can be inserted into existing
curriculum of grades 4-6. Many types of problem situations will be
included in these modules: real-world applications of mathematics (i.e.,
"real world" as the student sees it), problems related to the mathematics
studies in the standard curriculum, mathematical recreations, and problems
involving various strategies such as guess and test and pattern finding.
While the MPSP is primarily a development project the materials being
developed will be based upon research into the teaching and learning of
problem solving and will be pilot tested in a number of elementary
schools.
The project is in operation at three different centers: the
University of Northern Iowa, the Oakland Schools (Pontiac, Michigan),
and Indiana University. While the project has identified the central
goal as being the development of problem solving modules for use in
grades 4-6, each center plays a distinct role.
The Role of_the University of Northern Iowa (U.N.I.) Center5
The MPSP site at the University of Northern Iowa is directed by
George Immerzeel. The primary role of the site is to develop a series
of "skillsfl6 booklets and associated problem solving experiences.
Specifically, the center at U.N.I, is identifying the spectrum of required
skills that are not part of the present curriculum and writing materials
that build this spectrum for particular problem solving strategies.
After considering an extensive list of required problem solving
skills and classifying these skills into those that are simple (require
a limited set of tactics) and complex (requiring a variety of tactics),
seven were identified as appropriate for students in grades 4 through 6:
description summarizes the role of U.N.I, as reported by
George Immerzeel and his associates.
Tliere is a semantics problem in trying to communicate ideas about
problem solving. Terms like "skill," "strategy," "heuristics," and
"techniques" connote different things to different people. The word
"skill," as used by the University of Northern Iowa staff, refers to
generic problem solving techniques which are needed in order to use a
particular strategy. Thus, "making a table" is a skill, whereas "pattern
finding" is a strategy.
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1. using an equation,
2. using a table,
3. using resources (reading, formulas, dictionaries, encyclopedias),
4. using a model (physical model, graph, picture, diagram),
5. make a simpler problem,
6. guess and test, and
7. compute to solve.
Each of these skills is simple in that they involve a single principle
tactic. They do not depend upon an interrelation among tactics as is
the case in strategies such as pattern finding and goal stacking 7
A "skills booklet" will be written for each of the seven skills.
These booklets will be designed to teach the subskills needed to use a
particular skill. For example, for the Guess and Test Skills Booklet,
approximately 100 problems were written and the skills necessary to
solve the problems were identified. These skills were then incorpo-
rated into the booklet.
The skills booklet is written so that a student can use the booklet
independent of teacher input and also so the teacher can use the booklet
in a regular classroom setting. After completing each booklet, the
student is given an evaluation that not only determines the student's
success in the skills but is a guide to group placement for the problem
solving experiences designed for the skills.
The problem solving experiences consist of a set of cards for each
skill. These cards represent five levels of difficulty and a variety of
interests. Although a majority of the problems are supposed to have a
"real world" setting, there are also examples from all aspects of the
curriculum. From this set of problems each student should be able to
find problems that not only fit her/his interests but also are at a level
of difficulty where the student will be challenged but have a reasonable
chance for success. Also included in the problem set are problems in
which the use of the mini-calculator is appropriate. These problems are
identified so the student knows the calculator is suggested for the prob*-
lem. A separate skills booklet for the mini-calculator will be developed
which can be used with any type of problem solving strategy.
As the skills booklets and problem sets are developed, they will be
field tested with students in grades 4-6 in the Malcolm Price Laboratory
School of the University of Northern Iowa.
See Newell and Simon (1972) and Wickelgren (1974) for a description
of goal stacking.
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The Role of the Oakland Schools Center8
David Wells is the director of the Oakland Schools Center. This
center is responsible for preparing teachers to field test and help
develop materials. The teachers will use their classrooms to field
test the materials developed at Oakland, U.N.I., and Indiana University.
Thus the Oakland Schools center operates the major field testing com-
ponents of the project. Currently, there are twelve teachers partici-
pating actively in solving problems, discussing problem difficulty,
identifying problem solving strategies, developing problems for use in
modules, and contributing to the development of modules.
The participation of classroom teachers is an essential part of
the project. It is also essential that these teachers teach
in a school system which offers diverse socio-economic groupings of
children. The Oakland Schools Center is ideally suited in this respect
since it has approximately 260,000 students and 14,000 teachers and
contains industrialized centers, surburban communities, and rural areas.
The Role of Indiana University (I.U.) Center
The Mathematics Education Development Center, under the direction
of John LeBlanc, is the third site involved in MPSP. The role of the
I.U. center is twofold. First, it is involved in the development of
one or more modules based on information gathered through work with
-individual and small groups of students. Second, the center has major
responsibility for evaluating the materials developed at the other
centers and for making suggestions for revision. At the same time, the
staff of the Mathematics Education Development Center is best qualified
among the three centers to conduct developmental research into the
questions which will arise inevitably as the modules and problems are
being created. To date, research problems have been identified relate
to problem difficulty and complexity and techniques for observing and
interviewing children as they attempt to solve problems. The thrust of
the work of the I.U. center will be discussed in more detail in a later
section.
Interrelationships of the MPSP Centers
The roles of the three centers have been described briefly but the
interrelationships among the centers has not been specified. Interaction
description summarizes the role of the Oakland Schools as
reported by Stuart Choate, Assistant Director of the Oakland Schools
Center.
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among the centers is determined on the basis of need for reaction to
ideas being investigated and materials being developed. For example, it
is expected that materials devised by one center will be reacted to
by the other centers. In this respect, there is a cyclic pattern of
continual development, testing, and evaluation of materials which are
produced (see Figure 1). Also, all three centers will be involved in
identifying research-able issues for close scrutiny by the l.TJ. center.
Develop
problem solving
materials
Evaluate
materials
University of
Northern Iowa
Oakland
Schools
Indiana
University
Figure 1. Interrelationship of primary roles of the MPSP centers.
A final word should be said regarding the feasibility of a tri-
site project. Such an organizational structure necessitates some
confusion, inefficiency, and duplication of efforts that must be taken
into account in assessing the project. However, despite these short-
comings the tri-site aspect is viewed as a strength rather than a weak-
ness of the project. The collaboration of educators with different
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68
interests, experience, and expertise has been proposed by several leading
curriculum developers. Having three centers offers a broader base for
disseminating the materials which will be developed and provides a
wider range of expertise in the areas of teaching, materials development,
evaluation, and research.'
Focus of Efforts in the MPSP at Indiana University
This section is devoted to a description of the research and develop-
ment work at Indiana University during 1974-75. Also, the current status
of the model of mathematical problem solving which is evolving will be
discussed. Although the development of a model has been given little
direct attention during the past year, it seems appropriate to present
it in this paper in order to elicit the reader's reactions.
The work of the I.U. center during the past year focused primarily
on intensive observation of students' problem solving behavior, the
development of a problem bank, and the creation of a problem solving
module. The details of each of these three aspects of this work are
discussed in the sections which follow.
Observation of Fifth-Grade Students
In order to get a better feeling for what types of problems students
find interesting and to investigate if students employ any discernible
strategies as they solve problems, the decision was made to spend some
time (approximately 6 weeks) observing fifth-grade children as they
attempted to solve problems without having any prior instruction. Fifth-
graders were used because it seemed reasonable to fix the age level of
the children so that developmental factors related to age would not have
to be dealt with.
Approximately eighty problems were found that were suitable for
most fifth-graders. The problems were selected on the basis of: rele-
vance to fifth-grade mathematics, potential interest for fifth-graders,
and "nonroutineness" (i.e., problems that are not standard textbook
"story problem"). Consideration also was given to selecting problems
which could be solved in more than one way. Ten of the problems were
selected for use in interviewing students.
9This view was articulated by James Gray who is the N.C.T.M.
representative on the MPSP Advisory Board.
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Two classes comprising approximately sixty fifth-grade students were
interviewed as they attempted to solve some of the ten problems. The
first class of students was interviewed individually and in groups of
two, three, and four as they worked a set of four problems. When groups
of students were interviewed, it proved too difficult to identify from
audio recordings the processes used by individuals. Thus, all students
in the second class were interviewed individually. The findings from
the interviews were:
1. Very few of the students wrote anything down. Sorae drew
a figure, but only after it was suggested by the interviewer.
2. Most students had difficulty retaining multiple conditions
and considering two or more conditions at the same time.
3. Students often solved a problem that was not the stated
problem. They misread the problem or misinterpreted the
problem.
4. Students in general did not use strategies, although a few
attempted to identify patterns for some problems.
The observation that many students were unable to coordinate multiple
conditions in a problem (finding 2) deserves elaboration. One of the
problems presented to students was the following:
There are 5 cups on the table. John has 9 marbles, and he
wants to put a different number of marbles under each cup.
Can he do this? Explain.
There are three different conditions to coordinate: five cups,
nine marbles, and a different number of marbles under each cup. (Of
course, "John" cannot perform this task.) Some students ignored the
third requirement and came up with 2, 2, 2, 2, 1 as their answer. Other
students ignored the condition of having nine marbles and arrived at 4,
3, 2, 1, 0 for an answer. It should be pointed out that although many
students did not initially coordinate all of the conditions, they were
able to do so after rereading the problem or being given a. simple clue
by the interviewer. It should be added that it is possible that
students did not use all of the conditions because they would not have
found a way to put the marbles under the cups otherwise. It is likely
that they have been conditioned to find an "acceptable" answer at all
costs. To them, getting an answer is the most important thing; getting
an answer that makes sense is something else. This situation is probably
not the fault of the students but the fault of a society which stresses
immediate results and values quanitity more than quality.
Mini-Instruction of fifth-grade. The results of the interviews
suggested that although the students were unsuccessful for a variety of
reasons, they did benefit from the questions asked and the hints given
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by the interviewer. Thus, it seemed feasible to devise short sequences
of instructional activities which would focus on helping children in the
areas that appeared to cause them the most trouble.
A fifth-grade class, different from those interviewed, was divided
into four groups (3 groups of 8 children and 1 group of 7 children). The
groups were approximately equal in ability based on the scores from a
pretest on mathematical reasoning. Each group was given forty-five minute
of instruction on each of four consecutive days. The instruction varied
among groups by what was stressed. The four different instructional
stresses were based on the findings from the interviews. They were:
1. Using Strategies: This group worked on using
"pattern finding" and "simplifi-
cation" in solving problems.
2. Coordinating Conditions: This group considered the condi-
tions of the problems and checked
that the solution satisfied all of
the conditions.
3. Understanding the Problem: This group was given ways to help
understand what a problem is asking
such as drawing a figure or dis-
tinguishing between relevant
and irrelevant information.
4. Working Problems: This group was given no particular
instruction. The students were
given the problems and asked
to work them. They were told if
they had the solutions right or
wrong and given hints when necessary.
Each group was given nearly the same set of problems over the four-
day period. These problems were selected because they were appropriate
for instruction in each group. At the end of the four-day instructional
period, a posttest of four problems was given to all the students to see
if any change in their problem solving behavior had occurred. In addition,
two students from each group were individually interviewed as they worked
the posttest.
There was no attempt to compare the groups statistically in terms of
problem solving performance. This was not an experimental study to deter-
mine which of four instructional techniques was the best, but rather an
exploratory investigation of the feasibility of providing instruction in
very specific aspects of the problem solving process. As this point the
primary interest was to try out ideas in order to gain a narrower focus,
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not to conduct careful planned and controlled experiments to test well-
formed hypotheses.
The results of the mini-instruction were inconclusive. Although
the group which received instruction on using strategies seemed to bene-
fit the most from the instruction, the teacher variable may well have
been the factor that caused this to happen since each group had a
different teacher. In general, the extent of the influence of the small
group instructional sessions is unclear. However, the insight gained
into the behavior of fifth-graders in small group problem solving situations
was invaluable. Interviewing and observing students as they work on
mathematical problems has continued to be a primary activity at the I.U.
center.
Development of a Problem Bank and Problem Categorization Scheme
The second major thrust of the I.U. center has been toward the
development of a large bank of problems of a wide variety of types.
As the size of this bank has grown, it has become necessary to determine
a scheme for categorizing the problems so that retrieval of problems
will be efficient. A substantial effort has been undertaken to devise
a suitable categorization scheme. In pursuit of this scheme the purposes
of having a problem bank had to be clarified. The purposes of the prob-
lem bank are:
1. to provide classroom teachers with a source of problems
of various types, and
2. to have available a wide range of problems with respect
to structure and mathematical complexity, mathematical con-
tent, problem setting, strategies used in solving the
problems, interest, etc. for use in development of problem
solving materials.
One important use of the problem bank is as a source of problems
exemplifying a particular strategy. For example, if a teacher wishes
to illustrate the use of the "pattern finding" strategy, he/she can go
to the problem bank and choose problems designated as "pattern finding"
problems.
In order to categorize the problems in the bank four dimensions
were identified: the setting of the problem, the complexity of the
problem, strategies applicable for a problem, and the mathematical
content of the problem. Initial attempts to sort out the components
of each category resulted in the following outline for a categorization
scheme.
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I. The setting of problems
A. Verbal setting
1. Simple statement
2. Statement In story form
3. Statement in game form
4. Statement in project form
B. Auxiliary nonverbal setting (a verbal setting accom-
panied by nonverbal information or materials which are
not essential to solving the problem)
1. Diagram/picture/graph
2. Concrete objects
3. Acting out the problem
4. Hand calculators and other "facilitative" devices
C. Essential nonverbal setting (nonverbal information or
materials essential to solving the problem)
1. Diagram/picture/graph
2. Concrete objects
3. Acting out the problem
4. Hand calculators and other "facilitative" devices
II. Complexity of problems
A. Complexity of the problem setting
1. Number of words
2. Number of conditions (numerical and nonnumerical)
3. Type of connectives among conditions
4. Familiarity of setting
5. Amount of superfluous information
6. Number of clues provided (verbal and nonverbal)
B. Complexity of the solution process
1. Familiarity with the type of solution
2. Number of questions posed
3. Type of connectives among questions
4. Number of variables
5. Type of connectives among variables
6. Number of different operations required
7. Type of operations required
8. Number of steps required to reach solution
III. Problem solving strategies
A. Pattern finding
B. Systeraatization
C. Visual perception
D. Inference
E. Trial-and-Error
F. Use and/or development of visual aids
G. Use and/or development of simpler problems
H. Recall and use of previous experiences
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IV. Mathematical content
Since the problem bank will be used within the structure of
the existing mathematics curriculum, the components of this
category should be determined on the basis of topics
included in various grade-five mathematics textbooks.
Problems which exemplify the use of various strategies have not
been difficult to find. Carole Greenes and Dale Seymour have provided
the MPSP with large collections of excellent problems which illustrate
particular strategies and which are appropriate for use in the inter-
mediate grades. Complexity has proven to be the most challenging
category to consider. Several weeks of intensive study resulted in a
revision of the outline related to the complexity of problems. The
revised outline is presented here without discussion. Work is now underway
to determine if factors included in this outline are critical in the
determination of problem complexity.
I. Complexity of problem statement
A. Vocabulary
1. Word frequency
2. Specialized use
B. Sentence factors (conceptualization of phrases)
1. Number of simple sentences
2. Average number of words per sentence
3. Decodability of phrases
C. Amount of information
1. Numerals and symbols
2. Necessary numerical and nonnumerical data
3. Questions asked
D. Interest factor
1. Number of personal words
2. Number of concrete nonmathematical words
II. Complexity of the focusing process
A. Interrelationships of conditions
1. Number of bits of irrelevant data
2. Types of connectives between conditions (and, or,
if . . . , then)
3. Order of presentation of the givens and/or operations
4. Logical structure of the problem
B. Interrelationships of goals
1. Leading questions
2. Corollary questions
3. Completely disjoint questions
4. Related questions
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III. Complexity of the solution process
A. Unique vs. non-unique vs. no solution
B. Mathematical content involved
C. Types of strategies that could be used effectively
D. Minimum number of subgoals
E. Types of goals
IV. Complexity of evaluation
A. Ease of checking solution
B. Ease of generalizing solution
Module Development
The development of instructional materials on pattern finding was
begun. Pattern finding was chosen as the focus of the module because
the students had an accurate understanding of the word "pattern" and used
it in conversation. Also, there is a wealth of problems which involve
pattern finding in their solutions. Preliminary versions of parts of the
module have been tested in fifth grade classrooms. No formal evaluation
of the extent to which students learn to use a pattern finding strategy
has been conducted. Instead, the testing has concentrated on readability
of the materials, clarity of presentation, format used, and interest
level.
The attempt to develop a problem solving module on pattern finding
and determine a scheme for categorizing mathematical problems necessitated
a careful examination of the behaviors, both affective and cognitive,
which are demonstrated as a student tries to solve a problem. This
analysis involved an attempt to determine a model of the problem solving
process which emphasizes the most important components of the process and
provides an accurate description of how successful problem solvers think.
Toward a Model of the Problem Solving Process
A search of the literature of problem solving revealed that several
attempts have been made to devise a model which describes problem solving.
It was appropriate to study some of these models in order to create a
model which approximates the process for solving mathematical problems.
Dewey's model of reflective thinking. In his classic book, How we
Think, Dewey proposes five phases of reflective thought (Dewey, 1933).
While reflective thought is not synonomous with problem solving, it is
clear that reflective thought is an essential part of problem solving.
The five phases are: 1. suggestion,, direct action upon a situation is
inhibited thereby casuing conscious awareness of being "in a hole"
(p. 107); 2. intellectualization, an intellectualization of the felt
difficulty leading to a definition of the problem; 3. hypothesizing,
various hypotheses are identified to begin and guide observations in the
collection of factual material; 4. reasoning, each hypothesis is mentally
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elaborated upon through reasoning; and 5. testing the hypothesis by action,
"some kind of testing by overt action to give experimental corroboration,
or verification, of the conjectural idea" (pp. 113-4).
Dewey is careful to point out that these phases do not necessarily
follow one another in any set order. This analysis is valuable in identi-
fying stages in reflective thinking and thus, in problem solving. How-
ever, it considers only the logical aspects of reflective thought but does
not consider nonlogical "playfulness" or intuition. It has been suggested
(Getzels, 1964) that Dewey's formal steps are more a statement of one
type of scientific method than an accurate description of how people
think. As a result, this model of the process of solving problems may
describe how students ought to think, but it does not describe how stu-
dents usually do think when they are solving problems.
Johnson's model of problem solving. Whereas Dewey's model reflects
a logical analysis of problem solving, Johnson (1955) has provided an
analysis which is oriented to the psychological processes related to
problem solving. Johnson's model is of particular interest because it
provides a framework in which "to interpret measures of problem difficulty
such as solution time" (cited in Bourne et al., 1971, p. 56). Three
stages are included in his model:
1. preparation and orientation—the student gets an idea of
what the problem involves;
2. production—the consideration of alternative approaches
to a solution and the subsequent generation of possible
solutions; and
3. judgment—the determination of the adequacy of a solution
and the validity of the approach used to arrive at the
solution.
In addition to providing information about problem difficulty this
model offers a dimension that is not present in Dewey's model—it leads
to speculation about the effects of instruction. In Johnson's model
preproduction activity by the problem solver is just as important as the
production stage. Unfortunately, little is known about the preparation
state because researchers have preferred to investigate problem situations
which ate well-defined for the student. Thus, the preparation stage plays
a less important role. Future research efforts should include studies
which focus on the preparation stage of problem solving by examining
problems for which the student is not fully prepared.
Polya's model of problem solving. George Polya's extensive writings
have been a source of much valuable information regarding the problem of
teaching problem solving in mathematics (Polya, 1957, 1962). Unlike Dewey
and Johnson, Polya's concern lies primarily with mathematical problem
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solving. To him, teaching problem solving involves considerable exper-
ience in solving problems and serious study of the solution process. The
teacher who wants to enhance her/his student's ability to solve problems
must direct the student's attention to certain key questions and sugges-
tions which correspond to the mental operations used to solve problems.
In order to group these questions in a convenient manner Polya suggests
four phases in the solution process:
1. understanding the problem,
2. devising a plan,
3. carrying out the plan, and
4. looking back.
Since Polya's four phases are familiar to most mathematics educators
interested in mathematical problem solving, no discussion of his model
will be presented here. It should be pointed out that instead of being
a description of how successful problem solvers think, his model is a
proposal for teaching students how to solve problems. While this model
may be valuable as a guide in organizing instruction in problem solving,
it is too gross to be of much help in identifying potential areas of
difficulty for students or clearly specifying the mental processes
involved in successful problem solving.
Webb's model of problem solving. After reviewing the existing liter-
ature on mathematical problem solving, Webb (1974) devised a model which
is purported to be a synthesis of the various models described in the
literature. This model contains three main stages in solving a problem:
1. preparation—includes defining and understanding the problem;
understanding what is unknown, what is given, and what the
goals are;
2. production—includes the search for a path to attain the
goals; recall of principles, facts, and rules from memory;
generation of new concepts and rules to be used in solving
the problem; and development of hypotheses and alternative
plans that may lead to one or more goals; and
3. Evaluation—includes checking subgoals and the final solu-
tion; and checking the validity of procedures used during
preparation and production.
Webb stated that his model "is not a hierarchial model in that preparation
always comes before production which always must precede evaluation. This
is more a cyclic model" (Webb, 1974, p. 4). The models of Polya and Webb
have proven to be useful to the staff at the Indiana University center of
the MPSP as rudimentary models from which a more detailed and refined
model can be developed.
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Some other models of problem solving. In addition to the models
proposed by Dewey, Johnson, Polya and Webb, at least two other thought-
ful models have been developed. The first is the model of Klausmeir
and Goodwin (1966). The major aspects of their model are highlighted
below without discussion:
1. setting a goal,
2. appraising the situation,
3. trying to attain the goal,
4. confirming or rejecting a solution, and
5. reaching the goal.
The major points of the second model by Wallas (1929) are also high-
lighted below without discussion:
1. preparation,
2. incubation (a mulling-over period),
3, illumination (the conception of a solution), and
4. verification,
A Working Model of Problem Solving for the MPSP at Indiana University
The primary limitation of each of the models that have been discussed
is that they are either prescriptive (viz., Dewey and Polya) or
only grossly descriptive (viz., Johnson, Klausmeir and Goodwin, Wallas,
and Webb). The prescriptive models suggest techniques to help the student
to be a better problem solver. The descriptive models may be more valu-
able in the sense that they identify phases the student goes through
during problem solving. A goal of the MPSP is to devise a more detailed
and refined descriptive model.
The search for such a model has led to an investigation of informa-
tion processing approaches to problem solving research. With the possible
exception of gestalt psychology, information processing theory seems to
be the only psychological theory which has problem solving as a central
focus. A primary thrust of information processing theory is to develop
a description of specific types of problems that is precise enough to
enable an explanation of problem solving behavior in terms of basic cog-
nitive processes. The most complete description of information process-
ing theory has been presented by Newell and Simon (1972). Wickelgren
(1974) has attempted to develop an operationalized theory of problem
solving by combining elements of information processing theory and the
ideas of master teachers like George Polya.
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The work of'Newell and Simon, and Wickelgren has led the author to
the model for solving mathematics problems which is described in the
following paragraphs. This model is, of course, not as refined as it
should be nor does it necessarily generalize to all types of successful
mathematical problem solving behavior. However, it does pinpoint some
critical components of problem solving behavior which are missing in
the other models. Six distinct, but not necessarily disjoint, stages
are included in this model:
1. problem awareness,
2. problem comprehension,
3. goal analysis,
4. plan development,
5. plan implementation, and
6. procedure and solution evaluation.
It should be emphasized that these stages are not necessarily sequential.
In fact it only rarely happens that these stages do occur sequentially
and distinctly from each other.
In keeping with an information processing approach to building a
model, it would be desirable to devise a flow chart that would describe
the student's cognitive processes as progress is made from Problem Aware-
ness through Procedure and Solution Evaluation. However, since the
stages are not hierarchically ordered or even distinct, for most problems
it is not possible to devise a completely accurate diagram of the flow of
progress during problem solving. Figure 3 (page 82) is a rough descrip-
tion of the way in which the stages of the model are related.
Stage 1: Problem awareness. A situation is posed for the student.
Before this situation becomes a problem for the student, he/she must
realize that a difficulty exists. A difficulty must exist in the sense
that the student must recognize that the situation cannot be resolved
readily. This recognition often follows from initial failure to attain
a goal. This view of what constitutes a problem is consistent with
Bourne's description of a problem situation as one in which initial
attempts fail to accomplish some goal (Bourne et al., 1971). A second
component of the awareness stage is the student's willingness to try to
solve the problem. If the student either does not recognize a diffi-
culty or is not willing to proceed in trying to solve the problem, it is
meaningless to proceed (see Figure 2),
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Problem
posed
Stop
or
repose
Is
aware of
difficulty?
Can
difficulty
be resolved?
willin
to attempt a
solution?
S
Decides
to work
Stop
or
repose the
problem
Figure 2: Schematic representation of problem awareness.
Stage 2; Problem comprehension. Once the student is aware of the
problem situation and declares a willingness to eliminate it as a problem,
the task of making sense out of the problem begins. This stage involves
at least two sub-stages: translation and internalization. Translation
involves interpretation of the information the problem provides into
terms which have meaning for the student. Internalization requires that
the problem solver sort out the relevant information and determine how
this information interrelates. Most importantly, this stage results in
the formation of some sort of internal representation of the problem
within the problem solver. This representation may not be accurate at
first (or it may never be accurate, hence the student fails to solve the
problem), but it furnishes the student with a means of establishing goals
or priorities for working on the problem. It is here that the nonsequen-
tial nature of the model shows up for the first time. The accuracy of
the problem solver's internal representation may increase as progress is
made toward a solution. Thus, the degree of problem comprehension will
be a factor in several stages of the solution process.
Stage 3: Goal analysis. It seems that the problem solver may jump
back and forth from this stage to another. For some problems it is
appropriate to establish subgoals, for others subgoals are not needed.
It is often true that the identification and subsequent attainment of
a subgoal aids both problem comprehension and procedure development.
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Goal analysis can be viewed as an attempt to reformulate the problem
so that familiar strategies and techniques can be used. It may also
involve an identification of the component parts of a problem. It is
a process which moves from the goal itself backwards in order to separate
the different components of the problem. Thus, goal analysis actually
includes more than a simple specification of given information, specifi-
cation of the relationships among the information, and specification of
the operations which may be needed (see Resnick & Glaser, 1976, for a
more detailed discussion of goal analysis).
Stage 4: Plan development. It is during this stage that the pro-
blem solver gives conscious attention to devising a plan of attack.
Developing a plan involves much more than identifying potential strate-
gies (e.g., pattern finding and solving a simpler related problem). It
also includes ordering subgoals and specifying the operations which may
be used. It is perhaps this stage more than any other that causes diffi-
culty for students. It is common to hear mathematics students proclaim
after watching their teacher work a problem: "How did he ever think of
that? I never would have thought of that trick." The main sources of
difficulty in learning how to formulate a plan of attack emanate from
the fact that students are prone to give up if a task cannot be done
easily. Of course, if problems can be done too easily, they are not
really problems. A good problem causes initial failure which too often
results in a refusal to continue. This state of affairs is not the fault
of students, but rather the fault of teachers who do not recognize that
initial failure is a necessary condition for problem solving (Shumway,
1974). It may also be true that students are unable to devise good plans
because they have few plans at their disposal. There is preliminary evi-
dence from work done at the Indiana University center of MPSP that many
children in grades 4-6 proceed primarily in a trial-and-error fashion
until they either find a "solution" that satisfies them or give up.
Equipping students of this age with a few well-chosen strategies may
facilitate their ability to plan.
Another source of difficulty for students at this stage is in order-
ing subgoals and specifying the operations to be used. For many students
the hardest part of problem solving lies with knowing what to do first
and organizing their ideas. Consequently, in addition to teaching stu-
dents strategies, attention must be given to helping them organize their
thinking and planning.
Stage 5: Plan implementation. At this stage, the problem solver
tries out a plan which has been devised. The possibility that executive
errors may arise confounds the situation at this stage. The student who
correctly decides to make a table and look for a pattern may fail to see
the pattern due to a simple computation error. Errors of this type prob-
ably cannot be eliminated, but they can be reduced if instruction on
implementing a plan also considers the importance of evaluating the plan
while it is being tried. Thus, while stages 5 and 6 are distinct, they
are not disjoint. The main dangers of stage 5 are that the problem
solvers may forget the plan, become confused as the plan is carried out,
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or be unable to fit together the various parts of the plan. Fitting
together the parts of a plan can be a very difficult task in itself.
This difficulty may arise from the fact that the best sequencing of
steps in the plan or the best ordering of subgoals may not be clear
to the problem solver. For some problems the sequencing of subgoals
does not matter, while for others it is essential that subgoals be
achieved in a particular order. The reader is referred to Chapter 6
of Wickelgren's book How To Solve Problems for an in-depth analysis
of techniques for defining subgoals and using them to solve problems
(Wickelgren, 1974).
Stage 6: Procedures and solution evaluation^ Successful problem
solving usually is the result of systematic evaluation of the appro-
priateness of the decisions made during the problem solving and thought-
ful examination of the results obtained. The role of evaluation in
problem solving goes far beyond simply checking the answer to be sure
that it makes sense. Instead, it is an ongoing process that starts as
soon as the problem solver begins goal analysis and continues long after
a solution has been found. Procedure and solution evaluation may be
viewed as a process of seeking answers to certain questions as the
problem solver works on a problem. Representative of the questions
which should be asked by the problem solver at each stage are the
following:
1. problem comprehension stage—What are the relevant and irrel-
evant data involved in the problem? Do I understand the
relationships among the information given? Do I understand
the meaning of all the terms that are involved?
2. goal analysis stage—Are there any subgoals which may help
me achieve the goal? Can these subgoals be ordered? Is my
ordering of subgoals correct? Have I correctly identified
the conditions operating in the problem?
3. plan development stage—Is there more than one way to do
this problem? Is there a best way? Have I ever solved a
problem like this one before? Will the plan lead to the
goal or a subgoal?
4. plan implementation stage—Am I using this strategy correctly?
Is the ordering of the steps in my plan appropriate,
or could I have used a different ordering?
5. solution evaluation stage—Is my solution generalizable?
Does my solution satisfy all the conditions of the prob~
lem? What have I learned that will help me solve other
problems?
Figure 3 attempts to illustrate the interrelationships that exist
among the stages in the model. It also suggests how a student might
proceed in solving a problem.
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Problem
Posed
Awareness
Attained
Problem
Comprehension
Procedure
Evaluation
Solution
Evaluation
Figure 3: Schematic representation of a model of mathematical problem
solving.
How the model may be used. The most valuable aspect of this model
is that it provides a conceptual framework for identifying the factors
which most influence success in problem solving. This framework can be
useful to the teacher who is trying to organize appropriate problem
solving experiences for students by highlighting various potential
sources of difficulty for problem solvers. It also emphasizes that
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teachers cannot be content to teach students how to solve problems by
simply showing a few "tricks of the trade." Of course, the model does
not describe problem solving for all types of problems and, in this
sense at least, it is incomplete. But, it does supply a partial
explication of a theory of problem solving which, although not fully
conceptualized, is being created. The development of a theory of pro-
blem solving will give direction and add focus to any research efforts.
Such a theory is needed critically within mathematics education at the
present time. Many of the research efforts in mathematical problem
solving which have been conducted were well-conceived and carefully
done, but the results of these efforts have had little impact on
instructional practice. This is partially due to the diversity of
types of research and the conflicting results which have been obtained.
It is also due to the fact that none of the results seem to be general-
ized to all types of mathematical problems. It may be that no single
theory, and hence no single model, can accurately depict problem solving
for all types of problems and all types of problem solvers. Even with
the possibility of such a state of affairs, it is worthwhile to continue
the search for a suitable model since such a search will provide valuable
information about the nature of the problem solving process.
Plans For Future Research
Although the MPSP is primarily a development project, an investiga-
tion of a few research questions will be included as a part of the efforts
during 1975-76. Much of the work done at I.U. during the past year can be
classified as exploratory. Emphasis was placed on intensive observation
of students, the collection of problems, the creation of a problem solving
module, and the design of a suitable model for mathematical problem solv-
ing. While none of these endeavors can be considered research in the
usual sense, all of the work at I.U. was conducted with a research spirit.
That is, every effort was made to approach each issue in an open-minded
and objective manner and to apply the scientific method of inquiry.
Perhaps the most valuable result of the work at the I.U. center was
the identification of three areas within the problem solving process which
cause difficulty for fifth-graders. Two of these difficulties are related
to problem comprehension, while the third is related to plan development
and implementation.
1. Students often misread or misinterpreted problems.
2. Students had difficulty retaining and coordinating
multiple conditions in a problem.
3. Students do not appear to use any strategies during
problem solving.
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Further investigation of the first difficulty suggested that students
often perceive a simplified version of a stated problem. The students
then proceed to solve the problem as they perceive it. In a few cases,
the students were not even aware that a problem existed. In other
cases students had trouble understanding phrases in problems (e.g., "a
checker in every row and in every column" and "every sixth night").
Clearly, students cannot solve problems they don't fully understand.
It is important, then, to pay special attention to the factors which
influence problem comprehension. More specifically, it is important
to determine the primary determinants of reading difficulty since most
mathematical problems are presented in a written form.
Several measures of comprehension of written passages have been
developed by reading specialists. However, there is reason to believe
that these measures may not be appropriate for written mathematical
passages since mathematical English appears to be much different from
ordinary English. Kane (1968) has suggested that there are at least
four differences between mathematical English and ordinary English:
(a) redundancies of letters, word, and syntax are different, (b) names
of mathematical objects usually have a single denotation; (c) adjec-
tives are more important in mathematical English than in ordinary
English; and (d) the grammar and syntax of mathematical English are
less flexible than in ordinary English.
If mathematical English is significantly different from ordinary
English, it is essential that the nature of these differences be deter-
mined. Two members of the MPSP staff at I.U., Norman Webb and Barbara
Moses, have designed a study which aims at identifying a reliable and
accurate measure of comprehension of written mathematics problems.
Their study will investigate the following questions:
1. Is the Cloze procedure a reliable measure of comprehen-
sion for individual mathematical problems?
2. What is the relationship of certain stimulus measures of
mathematical problem statements to the mean Cloze score
percentage?
3. What stimulus measures are the best predictors of mean
Cloze score percentage?
Stimulus measures will include such variables as the number of one-
syllable words, nouns, personal words, symbols connectives, sentences,
and clauses per 100 words as well as the number of words with special-
ized mathematical meanings and the average sentence length.
The Cloze procedure is a popular technique for measuring reada-
bility of long passages. The procedure Involves deleting every nth
word or symbol of a passage and replacing them with blanks. The student
must fill in the blanks. The score is determined by the number of
responses matching the deleted material. A high score indicates high
readability.
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85
Webb and Moses expect that one or two stimulus measures will be
found that can be used to predict the difficulty of comprehending a
mathematical problem. They also expect the Cloze procedure to prove
to be an adequate measure of readability for mathematical problems.
If such expectations are supported, the task of classifying problems
according to complexity will be greatly reduced.
The fact that many of the fifth-graders were unable to coordinate
and retain the conditions given in a problem has led to the design of
a study to investigate particular issues related to this fact. Another
MPSP staff member, Fadia Harik, has decided to explore the influence
the number of conditions in a problem has on success in solving prob-
lems. In addition, she will investigate the effect certain types of
teacher clues has on problem solving success. This aspect of her study
arose from the observation that although fifth-graders do not initially
coordinate multiple conditions simultaneously, they are able to do so
in some problems if the teacher provides clues or asks the students to
reread the problem.H
Research studies like those of Webb and Moses, and Harik have been
carefully conceived, organized, and planned. Their questions have
risen from a concern for developing a sensible theory of mathematical
problem solving. It is only by conducting research based on a sound
conceptual framework that any significant progress will be made toward
developing instructional materials which will enhance children's
ability to solve mathematical problems.
Both the study by Harik and the one by Webb and Moses have been
completed since this paper was written. The interested reader can
obtain information about the results of these studies by contacting
the author of this paper.
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86
References
Bourne, L. E., & Dominowskl, R. L. Thinking. Annual Review of Psychology,
1972, 23, 105-130.
Bourne, L. E., Ekstrand, B. R., & Dominowski, R. L. The psychology of
thinking. Englewood Cliffs, New Jersey: Prentice-Hall, 1971.
Davis, G. A. Current status of research and theory in human problem
solving. Psychological Bulletin. 1966, £6(1), 36-54.
Dewey, J. How we think. Boston: D. C. Heath & Co., 1933.
Duncker, K. On problem-solving. Psychological Monographs. 1945, 58(5,
Whole No. 270).
Gagne, R. M. The conditions of learning (2nd ed.). New York: Holt,
Rinehart & Winston, 1970.
Getzels, J. W. Creative thinking, problem-solving, and instruction. In
H. G. Richey (Ed.), Theories of learning and instruction, Part I.
Chicago: The National Society for the Study of Education, 1964.
Green, B. F. Introduction: Current trends in problem-solving. In B.
Kleinmuntz (Ed.), Problem solving: Research, methods and theory.
New York: John Wiley, 1966.
Greenes, C. Untitled paper prepared for the Mathematical Problem Solving
Project, 1974 (available in mineograph form only).
Henderson, K. B., & Pingry, R. E. Problem-solving in mathematics. In
H. F. Fehr (Ed.), The learning of mathematics its theory and prac-
tice. Twenty-first Yearbook. Washington, D.C.: National Council
of Teachers of Mathematics, 1953, 228-270.
Johnson, D. M. The psychology of thought and judgment. New York: Harper
& Row, 1955.
Kane, R. B. The readability of mathematics books. Journal of Research
in Science Teaching. 1968, _5, 196-298.
Kendler, H. H., & Kendler, T. S. Vertical and horizontal processes in
problem solving. Psychological Review. 1962, 69, 1-16.
Kilpatrick, J. Problem solving and creative behavior in mathematics. In
J. W. Wilson and L. R. Carry (Eds.), Reviews of Recent Research in
Mathematics Education. Studies in mathematics (Vol. 19). Stanford,
California: School Mathematics Study Group, 1969, 153-187.
-------
87
Klausmier, H. J., & Goodwin, W. Learning and human abilities: Educa-
tional psychology (2nd ed,). New York: Harper & Row, 1966.
Maier, N. R. F. Problem solving and creativity in individuals and
groups. Belmont, California: Brooks-Cole, 1970.
Maltzman, I. Thinking: From a behavioristic point of view. Psycho-
logical Review, 1955, j>6, 367-386.
Newell, A., & Simon, H. A. Human problem solving. Englewood Cliffs,
New Jersey: Prentice-Hall, 1972.
Polya, G. Mathematical discovery (Vol. 1). New York: John Wiley,
1962.
Polya, G. Presentation as a part of a panel discussion on mathematical
problem solving. Annual Meeting of the American Mathematical
Society, San Francisco, January 15-18, 1974.
Resnick, L. B., & Glaser, R. Problem solving and intelligence. In
L. B. Resnick (Ed.), The nature of intelligence. New York:
Halsted Press, 1976.
Seymour, D. Considerations for a mathematics problem-solving curricu-
lum for the intermediate grades. Paper prepared for the Mathe-
matical Problem Solving Project, 1974 (available in mimeograph
form only).
Shumway, R. J. Untitled paper prepared for the Mathematical Problem
Solving Project, 1974 (available in mimeograph form only).
Wallas, G. The art of thought. New York: Harcourt, Brace, 1926.
Webb, N. A review of the literature related to problem-solving tasks
and problem-solving strategies used by students in grades 4, 5
and 6. Progress report prepared for the Advisory Board of the
Mathematical Problem Solving Project, 1974 (available in mimeo-
graph form only).
Wertheimer, M. Productive thinking. New York: Harper & Row, 1959.
Wickelgren, W. A. How to solve problems. San Francisco: W. H. Freeman,
1974.
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Participants
1. Jeffrey Charles Barnett
2. Morris I. Beers
3. David Bradbard
4. Nicholas A. Branca
5. Sandi Clarkson
6. Mary Kay Corbitt
7. Thomas J. Cooney
8. Carolyn Ehr
9. Beverly Gimmes tad
10. Dorothy Goldberg
11. Gerald A. Goldin
12. Douglas A. Grouws
13. Fadia Harik
14. Larry L. Hatfield
15. Ruth E. Heintz
16. Mary Grace Kantowski
17. Howard M. Kellogg
18. Jeremy Kilpatrick
19. Gerald Kulm
20. Kil S. Lee
21. Richard A. Lesh
22. Frank K. Lester, Jr,
23. John F. Lucas
24. C. Edwin McClintock
25. Len Pikaart
26. Sid Rachlin
27. H. Radatz
28. Danielle Scheier
29. John Schleup
30. Ed Silver
31. J. Phillip Smith
32. Larry Sowder
33. Leslie P. Steffe
34. Kenneth E. Vos
35. Norman L. Webb
36. Arvum Weinzweig
37. James W. Wilson
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