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IMPLICATIONS
OF THE DEVELOPMENT OF MATHEMATICAL

PROBLEM SOLVING, 1894-1983

by

John Peter Orehovec

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Administration and Curriculum

1984

@Copyright by
JOHN PETER OREHOVEC

1984

ABSTRACT

IMPLICATIONS OF THE DEVELOPMENT OF
MATHEMATICAL PROBLEM SOLVING, 1894-1983

by

John Peter Orehovec

Since problem solving in mathematics has been
described, questioned, discussed, researched, and criticized
frequently over the past century, one could question why
such organizations as the National Council of Teachers of
Mathematics, as well as individual mathematics educators,
have taken so long to recognize its importance to the extent
of making problem solving the focus of mathematics instruc-
tion over an entire decade, the 19808.

With such importance being placed on problem solving
in mathematics during the 19803, this investigation was
developed to consider the following questions:

1. What ideas, concerns, and approaches helped formulate
problem solving in mathematics?

2. What ideas, concerns, and approaches helped elevate
problem solving to its current level of importance in mathe-
matics education?

3. Did past practices provide any indication for a
successful problem-solving movement for the remainder of the
19805 and the future?

The focus of this historical study was to trace the

development of mathematical problem solving over a ninety-year

John Peter Orehovec

period. Hundreds of ideas, concerns, and approaches to
mathematical problem solving were considered for use in this
investigation. Each--in its own way--contributed to the
development of problem solving.

In summarizing the conclusions, it was found that
concerns for teaching and learning mathematical problem solv-
ing similar to those found sixty years ago continue to exist.
Problem-solving models developed decades ago continue to be
utilized in mathematical problem solving. The following
"model" is a composite of problem—solving models developed
over the past ninety years:

1. Read the problem carefully.
2. Look for the "known" facts contained in the problem.
3. Determine the "unknown" portion of the problem.

4. Use the "known" and “unknown" portions to determine
a procedure for solving the problem.

5. Estimate the final answer.
6. Solve the problem (computation).

7. Compare the answer in the solution with the estimated
answer (look for "reasonableness").

8. Label the final answer.

It was further concluded that mathematical problem
solving can be taught--successfully. For this to happen,
there must be a willingness on the part of classroom teachers,
building principals, curriculum coordinators, and school
superintendents to fully implement a problem-solving approach
to teaching mathematics. Based upon the conclusions, this
investigation provides fourteen recommendations for the future

teaChing of mathematical problem solving.

This work is dedicated to
four special people:

My parents, Martin (1915-1965)
and Wanda (1919-1983)

Monsignor Andrew A. Andrey,
for giving me
the first teaching job of my career

Mrs. Evelvia Baeckler,
for making me
solve problems many years ago

This is for them.

ii

ACKNOWLEDGMENTS

It is a pleasure to acknowledge Dr. Calhoun C.
Collier for his assistance, advice, and constructive criti-
cism throughout this investigation and during my doctoral
program.

A special thank you to the remaining members of my
graduate committee, Dr. William L. Cole, Dr. Walter G.
Hapkiewicz, and Dr. William J. Walsh. Their thoughts and
concerns will always be a part of my professional makeup.

For critiquing the initial draft of this study, I
express my graditude to Dr. Floyd G. Parker and Mr. Michael
A. Nash. To Mike and his wife, Jane, a special thanks for
the many years of unwavering friendship.

Deepest appreciation to my step-father, José, my
mother-in-law, Nancy, and my father-in-law, Joseph, for
their encouragement throughout my doctoral program.

Thank you to my brother, Paul, and my sister-in-law,
Maribeth, for their love and support at a time when I really
needed it.

Finally, I share the joy of completing this study
with my wife, Barbara. For her assistance in organizing
data, typing and proofreading, for her love, support, and
encouragement, and for putting up with'my periodic outbursts
of "weird" behavior, I simply say "Luv!"

iii

Chapter

TABLE OF CONTENTS

I 0 THE PRO BLE M O O O O O O O O O O O O O O 0

II. A

III. A

IV. A

Introduction . . . . . .
Statement of the Proble

Historical Background . .
Methodology . . . . . . .
Theoretical Framework . .
Organization of the Study
Limitations . . . . . . .
Assumption . . . . . . .
Research Questions . . .

REVIEW OF MATHEMATICAL PROBLEM SOLVING
PERIOD 1’ 1894-1928 c o o o o o o o 0

Summary of Period I: 1894-1928 . . . .

REVIEW OF MATHEMATICAL PROBLEM SOLVING
PERIOD II: 1929-1953 0 o o o o o o o 0

Summary of Period II: 1929-1953 . . .

REVIEW OF MATHEMATICAL PROBLEM SOLVING
PERIOD III: 1954-1983 0 o o o o o o 0

Summary of Period III: 1954-1983 . . .

V. SUMMARY, CONCLUSIONS, RECOMMENDATIONS:

APPENDICES

AND IMPLICATIONS . . . . . . . . . . .

Summary . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . .
Recommendations . . . . . .
Implications for Future Research .

A. THORNDIKE'S ORGANIZATION OF LEARNING . .
B. THORNDIKE'S TITLES OF LESSONS . . . . . .

iv

Page

16

41

44

91

95

151

154

154
156
159
165

167
169

Page

C. KNIGHT'S ANALYSIS OF STEPS IN THE

PROCESS OF PROBLEM SOLVING . . . . . . . . 170
D. SKILLS INVOLVED IN PROBLEM SOLVING
BY BRUECKNER AND MELBY . . . . . . . . . . 172
E. POLYA'S MODEL: THE QUESTIONS . . . . . . . . 174
F. MATHEMATICS PROJECTS AND PROGRAMS . . . . . . 176
G. FENNELL'S STAIRCASE OF PROBLEM SOLVING . . . 177
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 178

CHAPTER I
THE PROBLEM

Introduction
In the 1922 opening statement to the teachers in
Michigan, Superintendent of Public Instruction, Thomas
E. Johnson wrote:
The subject of arithmetic is a troublesome one
because of the fact that we have had so many
movements which appear now to have been rather of
the nature of fads (p. i).
This statement prefaced a bulletin produced by the
Michigan Department of Public Instruction which attempted
to improve the course of study in arithmetic. Since that
time there have been numerous attempts to improve
arithmetic learning and instruction. A number of indivi-
duals, organizations, and committees have suggested changes
in mathematics instruction both within Michigan and the
United States.
In 1980, the National Council of Teachers of
Mathematics (NCTM) made eight recommendations for improving
school mathematics within its An Agenda for Action:

Recommendations for School Mathematics of the 1980§. The

Council recommended that:

problem solving be the focus of school
mathematics in the 19805;

basic skills in mathematics be defined to
encompass more than computational facility;

mathematics programs take full advantage of
the power of calculators and computers at all
grade levels;

stringent standards of both effectiveness and
efficiency be applied to the teaching of
mathematics;

the success of mathematics programs and
student learning be evaluated by a wider
range of measures than conventional testing;

more mathematics study be required for all
students and a flexible curriculum with a
greater range of options be designed to
accommodate the diverse needs of the student
population;

mathematics teachers demand of themselves and
their colleagues a high level of professional-
ism;

public support for mathematics instruction be
raised to a level commensurate with the impor-
tance of mathematical understanding to
individuals and society (p. 1).

Furthermore, the Council recommended the following

"actions" to make "problem solving" the focus of school

mathematics in the 19803:

1.

2.

The mathematics curriculum should be organized
around problem solving.

The definition and language of problem solving
in mathematics should be developed and expanded
to include a broad range of strategies,
processes, and modes of presentation that
encompass the full potential of mathematical
applications.

Mathematics teachers should create classroom
environments in which problem solving can
flourish.

Appropriate curricular materials to teach
problem solving should be developed for all
grade levels.

5. Mathematics programs of the 19805 should
involve students in problem solving by pre-
senting applications at all grade levels.

6. Researchers and funding agencies should give
priority in the 19805 to investigations into

the nature of problem solving and to effective

ways to develop problem solvers (pp. 2-5).

By making "problem solving" the focus of the 1980s,
it suggests that the National Council of Teachers of
Mathematics is emphasizing some new concept or idea. In
truth, "problem solving" ideas, Concerns, and approaches
have long been described, discussed, and applied in mathe-
matics instruction. Hence, a review of the history of

problem solving, based upon the questions in the problem

statement, is the focus of this investigation.

Statement of the Problem

Since problem solving in mathematics has been
described, questioned, discussed, researched, and
criticized for nearly a century. a reader could question
why sudh organizations as the National Council of Teachers
of Mathematics, as well as individual mathematics
educators, have taken so long to recognize its importance.
The magnitude of this importance is evidenced by the
NCTM statement that problem solving should be the focus of
instruction over an entire decade.

With sudh importance being given to the emphasis
on problem solving in mathematics during the 1980s, this

study was developed to consider the following questions:

1. What are the historical ideas, concerns, and
approaches that have helped formulate problem
solving in school mathematics?

2. What are the historical ideas, concerns, and
approaches that have helped elevate problem
solving to its current level of importance in
mathematics education?

3. Compared to events and influences of the past,
what practices indicate a successful problem-
solving movement for the remainder of the 19805

and the future?

Historical Background

A number of educational philosophers, mathemati-
cians, psychologists. and teachers have influenced
mathematics instruction. For example, the philosophy of
Johann Heinrich Pestalozzi centering on the child's
perceptions, the social-utility theory of John Dewey,
Edward L. Thorndike's connectionist-theory of psychology,
William A. Brownell's meaningful arithmetic, and George
Polya's questioning techniques included in his problem-
solving model have affected the teaching of mathematics in
some way.

In addition to many individuals affecting the
study of arithmetic, numerous organizations, committees,
and projects have provided further direction and influence.

A list of organizations providing guidance through their

membership would include the National Council of Teachers
of Mathematics (NCTM) and the National Society for the
Study of Education (NSSE).

During the past century, a number of exemplary
projects have provided alternative methods and materials
for use in mathematics education. Enjoying common profes-
sional visibility were the efforts of the Greater Cleveland
Mathematics Program, the School Mathematics Study Group,
the Cambridge Conference on School Mathematics, the
Nuffield Mathematics Project, the Madison Project, and the
Unified Science and Mathematics for Elementary Schools
project.

Finally, committees were established at the
national level to provide direction and to suggest changes
in the teaching of mathematics. These committees include
the Committee of Ten on Secondary School Studies, the
Committee of Fifteen on Elementary Education, and the
Committee of Seven on Grade-Placement in Arithmetic.

Each of the aforementioned educators, organiza-
tions, projects, and committees affected mathematics
instruction. Within the realm of mathematics, numerous
innovations occurred over the past ninety years. As a
result, change also took place in mathematical problem
solving.

In teaching and learning mathematics, the idea of

problem solving is crucial. How did the problem-solving

"movement" get its start? Why has problem solving become
an increasingly popular area of research? How has problem
solving maintained an aura of popularity--and mystique--
in mathematics instruction?

Beginning with John Dewey's (1910) ideas on
"reflective thinking," each decade since has seen a rise
in the number of studies, books, and other sources of
commentary related to mathematical problem solving.

Edward L. Thorndike accelerated the idea of "problem
solving" in The Thogndike Arithmetics (1917), The
Psychology of Arithmetic (1922), The Psychology of Algebra
(1923), and the Mgthematics Teacheg (1922). In The
Psychology of Arithmetic (1922), Thorndike called for a
clearer definition of problem solving. In discussing the
general function of problem solving, he wrote:

The aim of the elementary school is to provide for

correct and economical response to genuine prob-

lems, such as knowing the total due for certain

real quantities at certain real prices, knowing

the correct change to give or get, keeping

household accounts, calculating wages due, com-

puting areas, percentages, and discounts,

estimating quantities needed of certain materials

to make certain household or shop products, and

the like (p. 9).
Thus, Thorndike believed "problems should be solved in
school to the end that pupils may solve the problems which
life offers" (p. 11).

In its Third Yearbook (1928), the National Council

of TeaChers of Mathematics included an entire section

devoted to "Problem—Solving in Arithmetic" (pp. 223-267).

In this yearbook, Lucie L. Dower noted the specific
preparations that must be met by providing boys and girls
the perfect fundamentals of arithmetic, and the abilities
to "solve the quantitative situations that arise in every-
day activities" (p. 223). In Dower's discussion of problem
solving, the following topics were considered: sources of
problems, essentials of good problems, classification of
problems, emphasis on problem-solving (difficulties).
degree of difficulty (of a problem), process of problem-
solving, causes of failures in problem-solving, recent
investigations and experiments in problem-solving, objec-
tives of problem-solving, psychology of problem-solving,
standard tests, and selected topics on experimental work,
remedial instruction, and problem-solving analysis (pp. 224-
263). A prime objective of the section was to "train boys
and girls to solve problems met in actual life situations"
(p. 249).

Harry Grove Wheat, in The Psychology and Teaching
of Arithmetic (1937), referred to computation and problem
solving as "The Persistent Concerns" (p. 126). In compar-
ing the "modern" curriculum with the "old" curriculum, he
wrote:

In content, the modern curriculum differs markedly
from the curriculum of the past. Internally,
however, the influence of the past still persists
in the modern school. Computation and problem-

solving were the interests of earlier days:
computation and problem-solving remain as interests

of the present. The extremes to which earlier
concern in computation and problem-solving were
carried have led to sharp reactions in later days
(pp. 126-127).

For Wheat, the purpose of instruction went beyond
teaching children to solve problems. The following
observation was made in 1937:

...the purpose is to provide them with methods of
thinking, with ideas of procedure, with meanings
inherent in number relations, with general princi-
ples of combination and arrangement, in order that
the quantitative situations of life may be handled
intelligently and without doubt and uncertainty

p. 140 .

In its Forty-first Yearbook (1942), the National
Society for the Study of Education devoted Chapter XII to
the topic of problem solving. In writing this chapter on
"problem solving," William A. Brownell noted:
The reasons for the relative paucity of educational
research on problem solving are not hard to find:
0n the one hand, the problems set in the classroom
are exceedingly complicated, more so, in all prob-
ability, than those set in the psychological
laboratory; on the other hand, children's behavior
in the face of problem situations is so variable
from problem to problem and from child to child
that exact and comprehensive quantitative descrip-
tions are most difficult to attain (pp. 419-420).
George Polya (1945) gave attention to the idea of
problem solving with the publication of How to Solve It:
A New Aspect of Mathematical Method. In this book, Polya
describes the following four-phase procedure for solving

problems:

1. Understanding the problem.
2. Devising a plan.
3. Carrying out the plan.

4. Looking back (pp. inside-front and inside-back
covers).

The National Council of Teachers of Mathematics included
Polya's How to Solve It problem-solving model, with its
extensive questioning, as an insert for the inside-front
and inside-back covers of its 1980 Yearbook, Problem
Solving in School Mathematics.

In 1953, the National Council of Teachers of
Mathematics focused part of its attention on "problem
solving" by including a chapter on "Problem-Solving in
Mathematics" in its Twenty-first Yearbook, The Learning of
Mathematics: Its Theory and Prgctice. In opening the
chapter, Kenneth B. Henderson and Robert E. Pingry (1953)
stressed the importance of teachers' understanding the
theory of problem solving when they wrote:

The present chapter on problem—solving in mathe-
matics is written on the assumption that mathematics
teachers should understand the basic theory of problem-
solving which is derived from research in the subject
and also see clearly the implications of this theory
for methods and procedure in the classroom. Both
are necessary. Theory apart from the implications
and consequences is largely sterile. Methods and
procedures apart from a conceptual framework become
little more than a bag of tricks (p. 228).

In a summary of research on teaching elementary

school mathematics, C. Alan Riedesel and Paul C. Burns

(1973) noted an increase in the number of research studies

completed in mathematical problem solving. Although the

10

quality of research design was quite low, Riedesel and
Burns indicated "the improvement of problem-solving skills
has been the topic for more research studies than any other
single topic" (p. 1160). They cited the "practical
answers" provided by research as having a greater effect
upon the improvement of problem solving "than for any other
area of the elementary-school mathematics curriculum"

(p. 1160).

Further consideration of the development of
mathematical problem solving was highlighted by Arlene
Gilda Luc Dowshen (1980). She found that:

The number of research studies in problem

solving by decade remained fairly consistent until ‘

the 19605 when there was a marked increase in the

number of studies concerning problem solving and

this increase continued into the 19705. In the

first half of the 19705 there were as many studies

on problem solving as there were in the 19505 and

19605 combined (p. 149).
An increase in the number of doctoral dissertations was
given as the major reason for the overall increase in the
number of studies cited in the area of mathematical problem
solving (p. 149).

The findings of Brownell (1942), Riedesel and Burns
(1973), and Dowshen (1980), indicate concerns for mathe-
matical problem solving. These concerns have now become
the focus of an entire decade, not just a section in an

educational encyclopedia or a chapter in a yearbook. This

investigation takes into account these concerns and

ll

considers their influences on mathematical problem solving

over a ninety-year period.

Methodology

This historical investigation utilizes both primary
and secondary sources of reference material. Egimggy
sources are defined as original documents, such as text-
books, manuscripts, and school records (Ary, Jacobs and
Razavieh, 1972; Best, 1981: Borg and Gall, 1979). Textbooks,
such as The Psychology of A;gebrg (1923) and How to Solve It:
A New Aspggt ogyMgthemgticgifiMethod (1945), are examples of
primary sources. Secondary sources are defined as published
bibliographies, referenced works, histories, and encyclo-
pedias (Ary, Jacobs, and Razavieh, 1972; Best, 1981; Borg
and Gall, 1979; Gay, 1976). For example, writing in the
Engyclopedia of Educational Research (1969) is considered to
be a secondary source.

Both primary and secondary sources are considered
to be important to the development of this study. Depend-
ing on the type of reference, journal articles are
considered either primary or secondary sources. However,
for purposes of this investigation, journal articles found
in School Science and Mathematics (1902-1983). the

Mathematics Teadher (1910-1983), and the Arithmetic

 

Teacher (1954-1983) are considered to be primary sources
even though the format and style of the journal was Changed

in a reprinting process.

12

Theoretical Frgmework
The National Council of Teachers of Mathematics

has given considerable emphasis to the importance of
"problem solving" in several of its yearbooks (1928, 1953,
and 1980). The Council, in its An Agendg;§or Action:
Recommenggtions for Sghool Mgthemgtics of the 19805 (1980),
gave top priority to the teaching of problem solving. How
this priority, given by the National Council of Teachers
of Mathematics, evolved, provides the basic framework for

this investigation.

Organization of the Study

This study traces the mathematical problem-solving
movement through three periods over the past ninety years,
1894-1983. Chapter I includes the introduction, statement
of the problem, historical background, methodology,
theoretical framework, limitations, assumption, and
research questions. Chapters II, III, and IV present
reviews of mathematical problem solving for the three
periods: 1894-1928, 1929-1953, and 1954-1983. The data
included in Chapters II, III, and IV are presented in
sequential order based upon the year in which the sources
were published or cited. Hence, the development of these
three chapters is chronological. Chapter V includes the
summary, conclusions, recommendations, and implications for

future research in mathematical problem solving.

13

Limitations

Several limitations were imposed on this study in
order to provide certain parameters from which this
research evolved. The starting date for the data covered
in this investigation was determined by the curricular
Changes influenced by the Report of the Committee of Ten
on Secondary School Studies (1894). The three periods were
selected according to the emphasis given mathematical
problem solving in the National Council of Teachers of
Mathematics yearbooks for the years 1928, 1953, and 1980.
Thus, the ending date for the first period was determined
by the date of the 1928 yearbook. The next year, 1929,
established the starting date for the second period. The
ending date was determined by the date of the 1953 yearbook.
The third period was established by the remaining years
included in the interval, 1954 through 1980. Hence, the
three periods are: Period I (1894-1928), Period II (1929-
1953), and Period III (1954-1980). For purposes of making
this study as current as possible, the years 1981-1983
were added to the third period, changing Period III from
1954-1980 to 1954-1983. Finally, although "general" ideas
on problem solving were not ignored, major attention was
given to the ideas, concerns, and approaches more closely

aligned with mathematical problem solving.

14

Assumption

The assumption of this study is that ideas on
problem solving found for any particular period are based
upon the development of problem solving from the preceding
period. Any attempt to make recommendations for the
future teaching of problem solving in mathematics is
predicated on the previous development. Thus, ideas for
Period II are based upon the findings found from Period I;
and ideas for Period III are based upon the findings
found from Period II. Finally, recommendations for the
future development--and teaching--of mathematical problem

solving are based upon the findings found from Period III.

Resegrch Questions

In reviewing mathematical problem solving, this

 

investigation examines the following research questions:

1. What ideas, concerns, and approaches on
problem solving have had an impact on the
teaching of mathematics?

2. What are the peak periods during the past
ninety years in which the development of prob-
lem solving caused changes to take place in
mathematics instruction?

3. What problem-solving models have been most
popular in mathematics instruction over the

past ninety years?

15

4. What ideas, concerns, and approaches on problem
solving developed in the past continue to affect

the teaching of mathematics?

CHAPTER II

A REVIEW OF MATHEMATICAL PROBLEM SOLVING
PERIOD I: 1894-1928
Development of problem solving in mathematics

during the period 1894 to 1928 was influenced by the cur-
ricular changes of the Committee of Ten and the works of
educators such as David Eugene Smith, John Dewey, and
Edward Lee Thorndike. Prior to this period, much of the
teaching and learning of arithmetic was considered to be
less than "practical," revolving mostly around "principles,
rules and facts." Education, in general, consisted of
learning the three R's: reading, 'riting and 'rithmetic.
As such, work in arithmetic consisted mainly of drill and
practice.

On a fair estimate, not less than one-fourth of

the pupil's time, for the first eight or ten years

of his school life, is given to the study of this

subject; but the results are too often quite in-

adequate to this large expenditure of time, the

most that can generally be claimed being a toler-

able familiarity with the processes of the

fundamental rules, common fractions, and denom-

inate numbers, with a very imperfect knowledge

even of the processes of decimal fractions,

proportion, evolution, and the business rules of

arithmetic (Kiddle and Schem, 1877, p. 40).

However, during the 18905, educators began seriously

questioning the education process in America. In 1892, the

National Education Association (NBA) appointed a committee

16

 

17

to analyze the school curriculum throughout the country.
The appointment of the Committee of Ten, chaired by
Charles W. Eliot, President of Harvard, typified two forces
in mathematics education at this time:

(1) the concern of persons with a major initial

interest in education as a whole for the special—

ized subject-matter field of mathematics and

(2) the influence of national committee reports

as stimulators of reform (NCTM, 1970, p. 5).
In 1893, the Committee of Ten endorsed the Herbart method
because it believed the bottom line to teaching mathematics
was "...to store the mind with clear conceptions of things
and their relations" (Kramer, 1970, p. 23).

The Report of the Committee of Ten on Secondggy
School Stgdies (1894) provided a number of curricular
changes that would have an effect upon education for many
years. Recommendations for mathematics included the need
for shorter and more enriched assignments, the use of con-
crete solutions to problem solving, an earlier introduction
to algebra, an earlier and more intuitive approach to
geometry, and the integration of arithmetic with other
subjects in the curriculum (Sizer, 1964, pp. 232-235).

Even with the Committee Repgrt, change did not come
about immediately. Mathematics teachers were slow to
Change, and drill in computation was still a major focus in
teaching arithmetic. In the "Editor's Introduction" to

The Teaching of Eigmentary Mathematics by David Eugene

Smith (1900), Nicholas Murray Butler wrote:

18

Perhaps no single subject of elementary in-
struction has suffered so much from lack of
scholarship on the part of those who teach it as
mathematics. Arithmetic is universally taught
in schools, but almost invariably as the art of
mechanical computation only. The true signifi-
cance and the symbolism of the processes employed
are concealed from pupil and teacher alike. This
is the inevitable result of the teacher's lack
of mathematical scholarship (p. ix).

With such lack of scholarship, the suggested changes set
forth by the Committee of Ten led to curricular scrutiny by
liberal arts colleges and universities. The aim of the
Committee's analysis was the academic program of the normal
school with its elementary content. "The criticism became
particularly serious when the normal schools began to con-
sider it their legal and professional responsibility to
provide programs for the preparation of teachers for the
public high schools as well as for the elementary sChools"
(NCTM, 1970, p. 308).

During the 18905, many theories of teaching began
making their way to the front in American education.
Though not a new theory, the principle of "apperception,"
developed decades earlier by Johann Fredrich Herbart,
played an important role during this period of time. The
theory, whereby learning takes place by "teaching the new
in terms of the old," began as a four-step process:
preparation, presentation, comparison, and generalization.

Proponents of Herbartian Philosophy later expanded the

process to include the following five steps:

19

1. Preparation--a review of related materials.

2. Presentation—-new material was presented,
analyzed, and clarified.

3. Association or comparison--new material was
compared to old material: an organization of
facts.

4. Generalization or conclusion--generalizations
made from stated facts.

5. Application-~an application of the principles
learned (Kramer, 1970, pp. 22-23; Meyer, 1967,
p. 23 .

It! Mn‘w ‘

John Dewey provided a philosophy quite different
from that of Herbart. He believed that the learning of
arithmetic should be related to daily living experiences.
Furthermore, Dewey advocated that the learning of numbers
should become an active process by which children actually
measure or count (Baur and George, 1976, p. 20: Kramer,
1970, p. 24). In summary, Baur and George (1976) wrote:

The net result of these influences led to a
period in education commonly called the social-
utility period. During this period those aspects
of mathematics which were not directly applicable
to everyday living were dropped from the curri-
culum. The tool (sic) dimension of mathematics
was brought to a peak as children measured to
bake cookies and played "grocery store" at
school. The "how" of the operations became more
important than the "why." Obtaining the correct
answer was of utmost importance. Once again,
children were taught the sequence of steps to a
procedure and then they were drilled on their
learning. This approach to teaching corresponded
to the theories of learning being set forth by
Edward Thorndike and other connectionist psychol-
ogists of the period (p. 20).

Change in the teaching of mathematics in the early
19005 came about as a result of many influences. According

to David Eugene Smith (1909), these changes occurred at

20

both the elementary and secondary levels. There was a
great interest evolving in the psychological development

of children. The influence of business also had an effect
upon curriculum and textbook development (p. 209). Foreign
influences in commerce, practical psychology, and science,
and the "dogma of thoroughness," played important roles in
changing mathematics instruction in the United States prior
to 1909. Smith (1909) provided several suggestions for
improvement including one of calling for a mathematics
curriculum that would link the teaching of mathematics in
grades 8 through 12 by combining algebra, concrete geometry,
and arithmetic (pp. 212-217).

Commerce, business, and life itself played impor-
tant parts in the development of the mathematics curriculum
during the first two decades of the twentieth century. As
Myrtie Collier wrote, in 1914:

Arithmetic should be taught in suCh a way that

the number facts learned may be usable in the life
of the Child, and also usable in the everyday life
of the individual after leaving school. To secure

this result in our school work two factors of arith-
metic need special emphasis, namely:

1. The removal of all mystery in the fundamental
operations and principles of number relations.

2. The application of the number facts through
practical problems to bring the child into vital
touCh with actual business and industry (p. 294).
The focus of Collier's comments was to suggest
ways of developing principles of arithmetic so that the

"mystery in mechanical operations" could be eliminated.

 

21

Collier highlighted the inductive approach by illustrating
the five steps in the Herbartian plan with a goal: To get
children to learn a principle, verbally, and apply it in
number form (p. 296).

Entering the 19205, considerable emphasis on
mathematical problem solving was provided by Edward Lee

Thorndike. In The New Methods in Arithmetig (1921),

A inn-a." V

the author summarized "the Older system" of organizing

is

arithmetic content. In calling the older system a beauti-

ful thing "to look at, but very hard to learn by" (p. 83),

the author included a list of concepts that ”the pupil was

supposed to learn" (p. 83). The entire list of Thorndike's
"organization of learning" is contained in Appendix A.

For Thorndike, to teach arithmetic was to teach
life itself. "Life organizes its arithmetical demands, not
so muCh by the nature of the processes as by the situations
involved" (p. 96). Furthermore, in discussing "the new
methods," he allowed for the organization of arithmetic to
focus on those situations in which the learner frequently
found himself. The list of "organized centers for arith-
metical training" for fourth grade children is included in
Appendix B, "Titles of Lessons."

According to Thorndike (1921). an important limita-
tion in_providing "appropriate" work in arithmetic for

children was found in the classroom itself because

22

mathematics problems were frequently described in words
rather than solved in "actual" situations. As was noted
at the time:

One important limitation due to the conditions
of classroom teaching is that the facts of the
problem can so seldom be presented to sense--must
so often be described in words. The problems of
life are most often questions about situations or
facts actually existing before the pupil's eyes,
less often questions which the person puts to him-
self in connection with his past affairs or future
plans, and least often questions put to him in
words by another. In proportion as we can escape
this limitation and actually present the situations,
we not only are surer of preparation for life, but
also find it easier to teach the pupils how to
attain correct solutions (p. 126).

In order to compensate for the "word limitations," Thorn-
dike provided a three-pronged problem-solving model. The
three main elements in this model were: "(1) to know just
what the question is, (2) to know what facts you are to
use to answer it, (3) to use them in the right relations"
(p. 126).

Finally, Thorndike (1921) included the idea of
problem solving in an effort to provide the proper tools
included in the "new methods." The "new methods" provided
so-called "real" situations to every learning experience.

Consequently the newer methods try (1) to pro-

vide real situations or projects where that is
feasible, and (2) to encourage the pupil to identi-
fy himself with the person whom the problem repre-
sents as acting or planning. If the reality cannot
be supplied, and if the sense of personal partici-
pation cannot be aroused, they try at least (3) to
free the problem from difficulties due (a) to its

vocabulary and structure or (b) to lack of experi-
ence by the pupils of the facts described (p. 127).

 

it-

23

Middlesex A. Bailey (1923) summarized Thorndike's
influences on the philosophy of teaching arithmetic. In
doing so, he reviewed the general principles and instruc-
tional ideas set forth by Thorndike in The ngchology of
Arithmetic (1922), The New Methods of Arithmetic (1921), and
The Three-Book Seriesygf Arithmetic (1917). In the summa-
rization, Bailey (1923) outlined Thorndike's system of
solving problems in the following four-step process:

1. Introduce each process and principle by a prob-
lem illustrating its need.

2. Tell the learner what to do.

3. Require him to verify his answers from known
facts.

4. Expect him to conclude that the procedure is
right "Because doing so always gives the right
answer" (pp. 129-130).

In critiquing Thorndike's system, Bailey analyzed each step,
made several modifications, and proposed the following out-
line:

1. Introduce each process and principle by a prob-

lem illustrating its need.

2. Ask the learner to solve the problem by a
method known to him.

3. Invite him to join with you in finding a shorter
process.

4. Require him to fix the new process in mind by
solving one or more other problems, first by
the old method and then by the new.

5. Require the solution of other problems by the
new method without statement of the rule, or
by the statement of the rule and its application.
In either case, require a proof of the answer
by some check (PP. 134-135).

Bailey was concerned about the way in which children not

only learned arithmetic but how much of the problem-solving

24

process was actually retained. Thus, his proposed outline
was an effort to provide children with a series of steps in
problem solving that could include more "reasoning skills"
in external situations. As Bailey noted in 1923:
At present, pupils on leaving school are almost
helpless to think out by themselves the solution
of problems that depart from type. or even to
solve type problems when stated in unusual forms
(p. 140).
The author, however, was not alone in attempting to pro-
vide more "reasoning" skill in the problem-solving process.
Ability in thinking skill was becoming more promi-
nent in mathematics education during the mid-19205. The
literature in arithmetic began to include topics on
questioning (Gould, 1923), and relationships between think-
ing and memorization (Atkins, 1923). Clarence G. Gould
included the following comment in an article written on the
"Art of Questioning" in 1923:
Reflective thinking is one of the most impor-
tant things in mathematics. In the form of problem-
solving, and the working of examples, reflective
thinking plays a large part (p. 52).
Thus, mathematics educators began looking at the notion of
problem solving as meaning something more than just "finding
an answer."
In getting children to develop independent reasoning
skill, Gould (1923) suggested that teachers include "reflec-
tive thinking" as a part of the process in solving problems.

Children were encouraged to organize their thoughts, formu-

late principles, and to recall the principles that had been

25

learned earlier (p. 53). K. W. Atkins (1923), in stating
that "Thinking is problem solving," provided the following
description:

The individual is confronted with some situation
which he must meet by a series of appropriate acts.
(The need for the solution must be keenly felt
before thinking takes place. Otherwise, the solu-
tion will be dodged or delayed.) When the problems
becomes real to the individual, various solutions
will be recalled from previous experiences with
somewhat similar problems. These are each tried
out to see if they will meet the confronting situ-
ation. Finally, some combination of experiences
is found which will effect (sic) a solution to the
problem. Then, and then only, may the thinking
process be said to be complete (p. 762).

In 1925, John R. Clark and E. Leona Vincent compared
two different methods of analyzing--and solving--mathemat-
ical problems. Included in this analysis was the following
"conventional" problem-solving method:

1. What is asked for in the problem.
2. What facts are given in the problem.

3. How should these facts be used to secure the
answer.

4. What is the answer to the problem (p. 226).
In contrast to this four-step method, the two authors
introduced a "Graphical Method" in which pupils had to
"determine what is to be found in the problem, what it
depends upon, what each of these dependents in turn depends
upon, and so on until he has unravelled the essential facts
and relationships in the problem" (p. 226). In comparing

the two methods, Clark and Vincent (1925) found that

26

children who used the "Graphical Analysis Method" improved
during practice more than those children who used the
"Conventional Analysis Method" (pp. 232-233).

In 1925, Clifford Brewster Upton, in "A Plea for
Professionalized Subject Matter," called for providing
practice in problem solving (p. 415). Along with a "provi-
sion for bringing each normal-school student up to some
standard skill in the fundamental operations," Upton called
the problem—solving provision a "complicating factor of

considerable importance" (p. 415). He suggested connecting

practice in problem solving with work done in socio-economic

arithmetic (business or personal arithmetic).

J. M. Kinney (1925). in writing about "The value
of the verbal Problem," considered the use of such problems
in teaching algebra and cited the following four reasons
for including verbal problems.

1. Tb give a concrete application of an abstract
theory.

2. To develop the ability to translate a quanti-
tative relationship, expressed in words, into
the symbolic language of algebra.

3. To develop the ability to think.
4. To develope (sic) the ability to solve the
problems of a quantitative nature that may
rise in the various fields of human endeavor
(pp. 267-268).
Finally, in discussing the "ability to think," Kinney
wrote, "Of course to solve any problem is to think. And,
no doubt, most of us feel that practice in thinking im-

proves the ability to think" (p. 267).

Ill?

27

In the monograph, Diggngstig Studies in Arithmetic

(1926), G. T. Buswell discussed the failures in elementary
school arithmetic and attributed most of the failure in
the elementary school to arithmetic. In the introduction,
he noted:

The failures in the elementary school are
caused more frequently by arithmetic than by any
other subject in the curriculum. The failures
due to arithmetic are to be traced to three con-
ditioning factors, namely. (1) the materials of
arithmetic, consisting of textbooks, practice
exercises, and special devices: (2) the teacher's
methods of instruction and her manner of pre-
senting arithmetic to the pupils; and (3) the
methods and mental processes of the pupils (p. 1).

Much of the monograph considered work related to the four
operations-—addition, subtraction, multiplication, and
division. The author believed that proper methods of work
should be emphasized before children became involved in
drill. However, in terms of ability in working with the
four operations, Buswell (1926) wrote:

...a mastery of these fundamentals alone does not
guarantee ability to meet all the demands of the
school or of society so far as arithmetic is con-
cerned. After the fundamental operations have been
mastered, there is still the difficulty of apply-
ing them to the solution of problems. Problem-
solving is, in a very real sense, the test of
one's ability in arithmetic. There need be no
conflict, however, between the emphasis on the
fundamentals and the emphasis on problem-solving.
Both must be adequately taught. While a mastery
of the fundamentals does not guarantee ability to
solve problems, a lack of knowledge of the funda-
mentals very seriously interferes with problem-
solving. Pupils frequently fail to solve problems
correctly because their methods of manipulating the
four fundamental processes are so clumsy that their
attention is diverted from the problems to the
details of the number combinations (p. 195).

28

Buswell was not alone in his belief that problem solving
should be an integral part of the mathematics program.
The purpose of arithmetic and the methods by which it was
taught were continually being discussed, updated, and
improved. Problem—solving improvement was important in
this updating process.

In the First Yearbook entitled A General Surveyyof
Progress in the Last Twenty-five Years, the National
Council of Teachers of Mathematics provided a general
review of the mathematics taught during the first quarter
of the 19005. In a survey of advances in arithmetic,
David Eugene Smith (1926) wrote:

Perhaps the most important Change of all is
seen in the purpose of teaching arithmetic. A
quarter of a century ago it was felt that the
subject should be hard in order to be valuable,
and it sometimes looked as if it did not make so
much difference to the school as to what a pupil
studied so long as he hated it. The old idea that
this was good for the mind and soul was not at
that time fully discarded. There was also prevalent
the idea that as many applications of arithmetic
should be introduced as the time allowed, irrespec-
tive of whether they were within the mental horizon
of the pupil or within the probable needs of his
life after leaving school. This view has now been
Changed; the purpose of teaChing arithmetic has
come to be recognized as the acquisition of power
to calculate within the limits of the needs of the
average well-informed citizen. It has also come
to be recognized that the problem is primarily de-
signed to show a need for computation, by giving
applications that add to the interest in calculation
and by introducing the puzzle element of problem-
solving, which may add further interest. A sec-
ondary purpose of the problem is the imparting of
some knowledge of the economic conditions, that
the pupil will find in daily life, this being pre-
sented to him in a simple manner that will make it
seem interesting and worth while (p. 19).

:r‘ "em" ‘-

29

The Second and Third Yearbooks accorded a more in-depth
look at specific aspects of the mathematics curriculum and
of selected topics in teaching the subject of arithmetic.
One of the concepts reviewed and scrutinized was problem
solving in mathematics.

The Second Yearbook of the Council focused upon
the Curriculum Pgoblems in Teaching Mathemgtics. In it,
F. B. Knight (1927) assembled information on nine specific
topics in mathematics instruction. One of those, Discussion
Nine, concentrated on "Remedial Drill on Arithmetic Problem
Solving." In reacting to the question, "What Can Be Done
for Pupils Who Need Remedial WOrk?“ the author included the
work of H. A. Greene. In doing so, a five-step problem-
solving model was displayed (see Appendix C). The five
steps--comprehension, analysis and organization, recogni-
tion, solution, and verification--closely resembled an
analysis of the thinking process attributed to John Dewey
(p. 65). In discussing the plight of "verbal problems,"
Knight (1927) provided the following discussion:

Mest problems now supplied to children are too

difficult for them. For example, the counterpart
of problems known to be useful in determining the
intellectual difference of twelve- and thirteen-
year-old pupils can frequently be found in material
supplied to ten- and eleven-year-old pupils. Our
present practice relative to problem material is
based on a gross over-estimation of the child's
ability to reason. This practice doubtless is
largely responsible for the frequent over-
helpfulness of teachers on problem material,
which is a crutch few would defend. It is also

probably responsible for the nervous and often
ill-formed attempts to use devices to teach

30

Children how to solve problems. It is moreover
probably responsible in part for the criticism of
the public relative to problem solving in the
schools.... we lead the public to think that a
Child can solve problems of a given nature since
we give him these problems to solve. But either
the child is too immature to solve these problems
or the available methods for teaching problem
solving are so inferior that the child cannot
work them. The public notes the persistent gap
between what we ask the child to do and what he
actually can do, and blames the school (p. 20).

Finally, in discussing effective methods for

improving skills, Knight (1927) reported a study by O. S.
Lutes, the purpose of which was to discover effective
techniques for teaching problem solving in the elementary
grades. Children's errors in problem solving were found
to fall under three main headings: "(1) ignorance of
principle, or wrong operation, (2) comprehension diffi-
culties, (3) computation errors" (p. 49). Furthermore, of
the five comments in the conclusions of the study, two
specifically made note of problem solving in mathematics.
These comments stated that "all pupils of normal intelli-
gence can profit from instruction in problem solving" and
"motivation is an important factor in securing improvement
in problem solving ability" (p. 51).

G. T. Buswell also made an extensive contribution
to the Second Yearbook. In observing that the subject of
arithmetic had become "a productive field of researCh,"
Buswell (1927) discussed five general problems found in

arithmetic. They were:

31

l. the difficulties in reading encountered in

arithmetic;

2. the teaching contribution of arithmetic text-
books;

3. the treatment given to the number system as
such;

4. the preparation of drill exercises keyed to
specific needs as revealed by diagnoses of
pupils' work: and

5. the grade location of the different arithmetical
processes (p. 73).

In elaborating on the second problem regarding "the teaching
contribution of arithmetic textbooks," Buswell (1927) in-
cluded the following discussion on problem solving:

Problem-Solving. The mental processes employed in
problem-solving have not been subjected to the
same degree of analysis as the processes employed
in working with the four fundamental operations.
Ultimately, such detailed diagnoses will be made.
It has been assumed quite generally that children
solve problems by the logical methods which are
supposed to be characteristic of adult thinking.
This logical procedure is described ordinarily in
suCh steps as the following: (a) defining the
problem; (b) recalling related facts which bear
upon it: (c) setting up and evaluating hypotheses
as to its solution: (d) selection of one hypothe-
sis; and (e) the final verification of the hypothe-
sis and the sclution of the problem. “If one.will
observe carefully the mental processes employed by
Children in the actual solution of problems, he
will find that there are many deviations from this
so-called logical method. Children make frequent
short cuts in their thinking. They frequently act
upon the first hypothesis which comes to their
minds rather than make a careful evaluation of
several hypotheses. Mere commonly still, they take
their cue from certain words in the problem or from
certain forms of expression and proceed with very
little logical thinking at all. While the logical
steps by which the process of reasoning has been
described may represent the way one should think,
they certainly do not represent the way in whiCh
many children do think. An investigation which is
badly needed is a detailed individual diagnosis

of the actual thinking carried on by children in

32

solving the ordinary problems presented in

arithmetic. Until such a diagnostic survey of
children's reasoning is available, it will be
difficult to supply suitable instructional material
for problem-solving (pp. 85-86).

The inclusion of, and emphasis on, verbal problems

were seen as important to the mathematics curriculum.
discussing the history and significance of problems in

algebra, vera Sanford (1927) discussed the use of the

verbal problem as a means for developing skill in problem

solving. The following comments are noteworthy:

One of the most important reasons for includ-
ing verbal problems in our work in elementary
algebra is to provide practice in analyzing a
given situation to see what mathematical rela-
tionships are present and how they may be
manipulated to answer a quantitative question.
This involves the ability to read intelligently
and to organize given facts.

It is this problem-solving attitude that will,
it is hoped, transfer to other situations in so
far as the student sees them to have like elements
(p. 3 .

In conclusion, Sanford (1927) added:

The object, however, is not convincing the student
that things happen in the world as they do in
algebraic problems, but providing him with work of
such a degree of reasonableness that he will not
be distracted from the purpose of problem solving
by unnatural and irrelevant situations (p. 7).

Furthermore, E. H. Taylor (1927) related problem-

solving techniques within the realm of algebra. The
author provided the following eight-step process in

"teaching pupils how to solve problems":

1.
2.
3.

33

Read the problem carefully.
Decide what is to be found.

Decide what facts needed in the solution are
given in the problem.

Decide what other facts are needed, and find
these facts.

Determine the processes needed in the solution.
Estimate the result.

Perform accurately the necessary computation.
Check the results (pp. 106-107).

The eight-step process described by Taylor is not limited

to algebraic solutions. Similar processes can also be

found in general arithmetic.

In discussing "The Systematic Solution of Arithmetic

Problems," Paul Ligda (1928) summarized the so-called

"directions" given by textbook writers for solving problems.

The following set of "directions" were cited:

1.
2.
3.
4.
5.
6.

Read the problem with understanding.
State what is given.

State what is to be found.

State the processes you will use.
Solve the problem.

Check (p. 24).

In response to the six-step procedure which Ligda (1928)

referred to as an "analysis," the following five-step

process was suggested:

1.
2.

Read the problem.

State the main quantitative thoughts as briefly

as possible.

Or: Make statements of comparison between sets
of quantities.

Or: Write verbal equations.

34

3. Identify the quantities with the terms of (2).

4. Substitute if necessary, then do the arithmetic
work indicated.

5. Interpret the result, check, and prove (p. 25).
Originally, this process was designed for algebra students.
However, Ligda believed that it could be easily modified
to meet the problem-solving needs of younger pupils.

In 1928, G. W. Myers described the three leading
objectives in public school arithmetic. They were: "(a)
intelligent control of number and space procedures, (b)
calculatory skill, and (c) problem-solving power" (p. 281).
Myers believed that although problem solving was the high-
est objective of all, it was too high to be effective for
children. In calling for greater control in the teaChing
of problem solving. the writer made the following observa-
tion:

we must however keep up the unremittent struggle

for some appreciable measure of problem-solving

power, but let us not forget that there can be

high social mastery of arithmetic without any

considerable degree of mastery of such problem-

skill as the customary "word" problems of current

texts represent. Excessive preoccupation with

this type of problem-solving skill to the extent

of making it the ruling objective cannot but con-

tinue to result in futility (p. 282).

As an alternative to "teaching by drill," Myers (1928)
included the following eight steps in teaching an arithme-

tic process or topic:

1. MOtivate the topic.
2. Develop the concepts.
3. Assimilate the ideas.

35

. Practice for skill.
Apply the new skill.

. Problem tactics and strategy.
. Consolidate new with old skills.
. Keep recalling old skills to hold them (p. 283).

034me

Although a certain amount of drill was necessary throughout
the procedure, Myers looked for the establishment of a
routine and an emphasis on recall to develop skill in
problem solving or for what was referred to as assimilation,
skill-building, and skill-holding.

In an address read before the National Council of
Teachers of Mathematics Conference in Boston (February 24,
1928), Walter F. Downey reiterated the progress made by
pupils in mathematics during the preceding twenty-five
years. In doing so, he cited the major changes that had
transpired in mathematics education. At the top of the list
was an effort to focus more on what the author referred
to as "the human phase of the subject." In considering
the human factor in education, Downey made the following
three important points regarding the processes in education:

(1) that the pupils should be given opportunity to
be problem finders as well as problem solvers, be-
cause problem finding and solving are infinitely
more productive in the development of vital minds
than is problem solving alone: (2) that whatever
activity is undertaken, whether it be academic
study, meChanic arts, practical arts, fine arts,
or athletics, the principle should be accepted and
followed that if the thing is worth assigning and
is properly assigned it is worth MASTERING one
hundred percent, not sixty percent or seventy per-

cent only, before passing on to the next bit of
work: and (3) that, before considering any problem

36

as completed, the pupil should feel sure in his
own mind, through the use of checks and other
means, that his work is correct (p. 242).
Downey believed that by accomplishing problem solving
through guidance of the three steps, students would
accomplish "power and habit" in their problem-solving
development.

The National Council of Teachers of Mathematics
analyzed mathematical problem solving in its Third Yearbook
(1928). By devoting an entire chapter to the concept of
problem solving, NCTM accorded mathematics educators with
a comprehensive look at the strengths and weaknesses found
in the teaching of arithmetic. In the chapter on "Problem-
Solving in Arithmetic," Lucie L. Dower (1928) included a
two-fold description of how Children acquired problem-
solving ability. First, teachers provided children
experience "by giving specific preparation for the kinds
of problems the pupils will meet in life," and second, "by
giving general preparation for all kinds of problems"

(p. 223). For Dower, preparation in the latter form,
general problem solving, was considered more important than
teaching for specific types of problems. She believed

that through a general preparation of problem solving,
children would "acquire the ability to judge a given
problem on its own merits, to study the relationships that
exist between the various quantities involved, and to think

out the solution" (p. 223).

37

In considering a general preparation toward the
teaching of mathematical problem solving, Dower (1928)
noted:

This approach aims to develop skill in planning

the solution as well as ability to execute the

plan. This mastery of problem-solving comes only

from meeting many different kinds of problems,

from seeing many relations, and from reasoning

out each problem in terms of the relationships

that are involved. By this method, pupils are

more likely to recognize similar problems in any

new situation and to apply the correct solution

(p. 223).
With the aforementioned comments, she set the tone for
the entire chapter in which the idea that preparation of
general problem-solving ability was an essential part of
the education process.

In analyzing mathematical problem solving, Dower
(1928) divided the chapter into three major components.
The first part investigated the elements and facts that
made up problems and analyzed the facts involved in problem
solving. The second part provided an investigation,
described research completed, noted important psychological
aspects involved, and provided a list of tests that could
be used in testing Children's ability in problem solving.
The third part provided a description of experiments
(activities) that had been carried out in a normal school
setting.

In addition to some of the ideas and theories on

problem solving in mathematics already mentioned in this

study, Dower discussed other studies, ideas, and theories

38

on problem solving that were pertinent during the 19205.

In a summary of the causes of failure in problem solving

described in the Second Yearbook of the National Council

of TeaChers of Mathematics, Dower (1928) provided the

following list:

1.

The language used in the problems is too diffi-
cult. It is beyond the reading standard for
the grades in which it occurs.

The pupils have not had sufficient training in
interpreting thought from silent reading.

The pupils lack understanding of the technical
terms involved.

The situations described by the problems are
not understood by the pupils, because they
are outside the range of the pupils' experi-
ences.

The fundamental combinations, facts, or pro-
cesses called for in the solution of the
problems have not been habituated.

The pupils are unable to see the relations
between the steps called for in the solution
of the problem.

The pupils are so burdened with undue labeling
and elaborate indication of steps that their
minds are diverted from the real process of
solution (p. 239).

In another summary taken from the Fourth Yearbook

of the Department of Superintendence, Dower (1928) noted

the following eight causes of failure in problem solving:

1.
2.

Lack of general ability in silent reading.

Lack of familiarity with technical terms in
arithmetic.

Carelessness in reading.

Lack of experiences necessary to understand the
setting of the problem.

Inadequate skill in computation.

39

6. Lack of knowledge of such essential facts as
tables of weights and measures.

7. Inability to see the relationships in the
problems so as to Choose the proper operation.

8. Inability to do reflective thinking (p. 239).

Dower provided mathematics educators with a number
of summaries of works by individuals who were investigating
mathematical problem solving in the 19205. For example,
she reported P. R. Stevenson's description of failure in
problem solving:

1. Physical defects.
2. Lack of mentality.
3. Lack of skill in fundamentals.

4. Inability to read, which of necessity affects
the ability to read arithmetic problems.

5. Lack of general and technical vocabulary.
6. Lack of proper methods or technique for
attacking problems (p. 239).
Finally, in acknowledging work completed by F. B.
Knight and G. T. Buswell in the Second Yearbook of the
National Council of Teachers of Mathematics, Dower (1928)
cited six suggestions concerning the future of research in
mathematical problem solving. In quoting Knight and

Buswell, she wrote:

1. We should find out what types of problems
Children can do and should be required to do.
We constantly over-estimate the child's ability
to reason. Mest problems given to children are
too difficult for them.

2. we should discover the most effective classroom
technique for teaChing the skill of problem-
solving in elementary school arithmetic.

4O

3. we should attempt a standardization of the
vocabularies used in the different books and
a correlation, grade by grade, between the
vocabularies used in arithmetics and the
vocabularies encountered in general reading
in the elementary schools.

4. There is need of a detailed individual diagno-
sis of the actual thinking carried on by
children in solving the ordinary problems pre-
sented in arithmetic; perhaps, to begin with,
those dealing with one particular unit only.
Until suCh a survey of children's reasoning
is available, it will be difficult to supply
suitable instructional material for problem-
solving.

5. There is need of the preparation of a textbook
which is composed essentially of explanatory
material, with an accompanying manual of prac-
tice exercises. The space which has been
previously used for examples and problems may
well be given to detailed instructions to the
pupils relating to methods of procedure, an
explanation of arithmetical operations, and a
presentation of social situations in which
arithmetic is to be applied.

6. The following are possibilities for the devel-
Opment and application of remedial instruction
units:

a. Exercises stressing vocabulary.
b. Exercises stressing problem comprehension.

c. Exercises stressing what is given in the
problem.

d. Exercises stressing what is called for in
the problem.

e. Exercises stressing the estimation of
answers.

f. Exercises stressing choice of procedure.

9. Exercises stressing relationships in
problems (pp. 248-249).

The Third Yearbook, sponsored by the National Council
of Teachers of Mathematics, summarized the problems facing

.mathematics educators in the 19205. Lucie L. Dower (1928),

41

in examining ideas on problem solving, provided a "new

challenge" for those involved in mathematics education.

§gmm§ry QEgPegiod I: 41894-l9gg

During the period, 1894-1928, a number of ideas,
concerns, and approaches on problem solving came to promi-
nence in mathematics education. The influences of John
Dewey and Edward Lee Thorndike played important roles during
this period in which problem solving became a viable concept
to teach and a powerful ally in the mathematical repertoire
of students in the elementary and secondary schools. Cer-
tainly, the notion of problem solving was not new, but the
resurgence of theories and the argument of proponents helped
popularize the teaching of problem solving.

The development of problem solving during this
period can be seen initially in the extension of the
Herbartian Mede1--preparation, presentation, comparison,
and generalization--into a five-step process with the addi-
tion of "application." The recommendations of the Committee
of Ten played an important role in encouraging teachers to
"teach the new in terms of the old."

Change in arithmetic came slowly in the early 19005.
Teaching arithmetic in terms of "daily living" was popular
during the initial part of this period, and problem-solving
tasks related to business, commerce, and science were pro-
vided. Dewey's approach, albeit different from the

Herbartian four-step process, advocated an “active approach"

42

to learning arithmetic. Along with ideas on "reflective
thinking," the arithmetic-in-life theme was prevalent
throughout the period: and although "drill" was still seen
as an important criteria, practice was made more applicable
by the organization of learning centered on "practical
arithmetic." The lessons established by Thorndike during
the second decade and the early 19205 provided an organized
meChanism for both teachers and students to follow.

During this period, skill in reading became an ini-
tial step for many problem-solving models. For example,
Paul Ligda (1928) called for reading with understanding
while Lucie Dower (1928) observed that the vocabulary used
in problem solving tended to be too difficult for many
children. In an extensive list, F. B. Knight (1927) in-
cluded a set of factors which the author believed played an
important role in encouraging children to read arithmetic
successfully and to comprehend language indigenous to math-
ematics.

The development of the "Conventional Medel" became
an important prototype from which many other models would
evolve. This process, whereby several questions are asked,
is highlighted in the following illustration:

1. What is asked for in the problem?
2. What are the facts?

3. How are the facts to be used in answering the
question (or problem)?

4. What is the answer?

43

While reading is implied in step 1, step 4 includes aspects
of computation. Later models added a step requiring a check
of the final answer. As ideas on mathematical problem sol-
ving became more prominent, the four-step model evolved
into a more elaborate and expansive process. Aspects re-
lated to "reading the question" became important criteria.
Emphasis on looking for main ideas and thoughts, determining
"what was to be found" and "what was given," were other im-
portant additions to the processes of problem solving.
Finally, the attention given to the "human phase"
of problem solving and the inclusion of "everyday activities"
played major roles in the development of problem solving
during the 19205. The concern for problem solving was
highlighted by the amount of consideration given it by the
National Council of Teachers of Mathematics in its 1928
Yearbook. The NCTM Yearbook included a number of ideas on
mathematical problem solving. These ideas provided a frame-
work for experimentation, research, and development on
problem solving in mathematics for years. The importance
of suCh concern for solving problems set the tone for the
development of mathematical problem solving for Period II,

1929-1953.

CHAPTER III

A REVIEW OF MATHEMATICAL PROBLEM SOLVING
PERIOD II: 1929-1953

The problem-solving approach in mathematics was
greatly enhanced by a number of new ideas and models
developed during the period, 1929-1953. Although a number
of "new" theories on problem solving were fostered by the
principles set forth by John Dewey and Edward L. Thorndike,
there were other ideas that came to the front in mathematics
education during the period. A sampling of educators who
influenced theories on problem solving in mathematics
during the period, 1929-1953, includes Harry Grove Wheat,
Robert Lee Morton, William A. Brownell, George Polya,

Leo J. Brueckner, and Foster E. Grossnickle.

In 1929, Paul R. Hanna, in a dissertation entitled
"Arithmetic Problem Solving," provided a comparison of
three methods of problem solving: the Dependencies Method,
which utilized "graphics and diagrams" whereby children
followed a particular thought pattern; the Conventional
JMethod, in which children followed a four-step plan: and
the Individual Method, in which Children used any method

they so desired. Included in the analysis was a summary of

44

45

the problem-solving steps found in several fourth-grade

and seventh-grade textbooks. Hanna (1929) summarized the

following steps found in the Alexander-Sarratt Arithmetics

(grade 4):
1. What does the problem tell me?
2. What does it ask me to find?
3. What operations must I use?
4. What will my estimated answer be?
5. How can I check my work? (p. 46)

Children were encouraged to use the following three-step

plan in the Standard Segvice Agithmetics (grade 4):

l.
2.
3.

What does it ask you to find?
What facts do you use to find the answer?

Do you add, subtract, multiply, or divide?
(p. 46)

In a third series, Strayer-Upton Agithmetics (grade 4),

the writer found the following four-step model:

1.
2.
3.

4.

What does the problem mean?
What is to be found?

How can the numbers given in the problem be
used to get the right answer?

Is the work right? (p. 46)

Hanna (1929) utilized the following four-step model as one

of the three models used in his experiment:

1.
2.
3.
4.

What is asked for?

What facts are given?

What operations are necessary?
What is the answer? (p. 47)

By citing the four-step "Conventional Medel," he suggested

lasing a general procedure for problem-solving in mathematics.

46

In assessing problem-solving steps included in
"courses of study" for teachers, Hanna (1929) found five
of the nine programs studied provided no technique for
problem solving while three suggested the "Conventionall
Method." The author found two alternative methods that
were used in two of the courses. The first, "course of
study in arithmetic," relied on the following six steps:

1. Read the problem.

2. What does the problem ask you to find?

3. What does the problem tell you?

4. What process will you need to use?

5. Find your answer.

6. Is your answer a reasonable one? (p. 48)

In the second course, the "Tentative Course of Study in
Arithmetic," Hanna (1929) found a three-step model for
problem solving. Those steps included:

1. Silent reading of problems by whole class.
2. Clearing up language difficulties if necessary.
3. Giving the answer if within primary facts,

indicating what is to be done if process work

is required (p. 48).

He included an analysis of "professional litera-
ture" and concluded that twelve of the sixteen references
studied gave preference to the four-step "Conventional
jMethod." Overall, Hanna (1929) concluded that the Con-
ventional Method was the technique most widely recommended

in textbooks, courses of study, and professional literature.

.Although the "Conventional Method" was discovered to be

47

the most popular method, he concluded that its utiliza-
tion was not always in the best interest of the learner.
In summary, Hanna (1929) noted:

The conventional-formula method of problem
solving was found to give the least gain in ability
when compared with the individual or the depen-
dencies methods. When the fourth and seventh
grades are viewed as a whole, there seems to be
no difference between the results of the depen-
dencies and individual methods. In the final
analysis of the results of this experiment, one
would not be justified in advocating the use of
the conventional-formula method nor could one say
definitely whether the dependencies or the indi-
vidual method is of greater value in aiding
children to solve arithmetic problems (p. 53).

The models discussed so far were applicable to most
mathematical situations. The emphasis on skill in solving
verbal problems also played an important role in the de-
velopment of problem solving at all levels of mathematics
including algebra.

The inclusion of verbal problems was cause for
concern in mathematics because reading played an important
role in the development of skills found in problem solving.
In discussing "Teaching the verbal Problem in Intermediate
Algebra," Barnet Rudman summarized a guide to solving the.
verbal problem. In doing so, Rudman (1929) described the
following five-step procedure:

1. Read the whole problem carefully.

2. Represent the quantity to be found by x or
some other letter.

3. If more than one quantity is to be found,
represent the smallest quantity by x and
the others in terms of x.

48

4. Translate the remaining statement of the
problem into algebra in the form of an
equation.

5. Solve the equation and check in (sic) the
conditions of the problem (p. 85).

Rudman noted that the problem with the five-step guide was
its inadequacy when data had to be interpreted. Recom-
mendations were made for a more "analytic approach" to

the solving of verbal problems and for a more systematized
arrangement of problem material. For many educators,
specific difficulties were found in the descriptions and
language related to problem solving.

In an effort to clarify the meaning of problem
solving, Harry Grove Wheat (1929) provided insight into its
meaning and inclusion in arithmetic. In discussing "The
Purpose of Problem Solving in Arithmetic," the author made
the following observation:

The language of the problem should be so clear

as to leave no doubt in the child's mind as to what

the situation really is. The language should not

be difficult nor strange. Unusual terms have no

place (p. 13).
Wheat (1929) also provided a definition of problem solving.
In doing so, the difference between "problem" and "problem
solving" was noted. He defined the two in the following
citation:

By "problem" is meant "practice exercise," and by

"problem-solving" is meant the mental process of

recognizing the general ideas of addition, sub-
traction, multiplication, and division (p. 14).

49

By defining terminology, Wheat helped clarify certain
aspects of problem solving. However, difficulties in deal-
ing with processes of problem solving continued to exist.
The idea of "improving problem solving in arithme-
tic" was popularized following its inclusion and emphasis
in the 1928 Yearbook sponsored by the National Council of
Teachers of Mathematics. In 1929, Leo J. Brueckner, writing
in the Elementary English Review, made the following four
recommendations for improving mathematical problem solving:

1. Increased use should be made of problematic
situations that arise naturally in the activi-
ties of the school and community in which
number is needed in its normal setting, in
order that the pupils may learn to apply
processes being taught as they are applied
in life.

2. Stress should be placed on accuracy in compu—
tation and comprehension of the meaning and
function of arithmetic processes. This can be
accomplished by the use of carefully construc-
ted instructional materials in which special
consideration is given to the known diffi-
culties of pupils in computation.

3. A systematic attack should be made on the
teaChing of problem solving. This would in-
clude the use of specially constructed reading
exercises on elements in problem solving, the
teaching of techniques in problem solving, and
the solution of many problems that are within
the experiences and comprehension of the pupils
and in the solution of which they would be
interested. Little is known as to the rela-
tive effectiveness of various types of
exercises. It is therefore recommended that
as wide a variety of exercises in problem
solving be used as is practicable.

4. Obviously, standard tests of problem solving
have an important place in such a program since
they aid the teaCher to determine the needs of
the class, their level of ability, and to
select the pupils in need of special help in

50

problem solving. The results of experimental
work suggest that pupils whose work in problem
solving is below standard profit much from the
training received through the use of exercises
in problem solving (p. 139).
Not only has problem solving played an important role in
the development of elementary mathematics but in the
development of mathematics at all levels.
In 1930, William Betz included "problem solving"
in a list of objectives for an introductory course in
algebra. The following six objectives were considered to

be the "central core" of the teaching of algebra:

1. The language and the ideas of algebra.

2. The formula.

3. The equation.

4. The graph.

5. The fundamental principles and processes.
6. Problem-solving (p. 120).

In his writing, Betz stressed the ideas of "understanding,"
real "application," and actual "thinking."

Orlie M. Clem and Bertha Adams Hendershot (1930)
provided further insight into the inclusion of problem solv-
ing in algebra. In discussing the difficulties involved in
solving verbal problems, the two authors made the following
five suggestions for the inclusion of verbal problems in
algebra.

1. The general opinion that problem solving in

algebra causes more difficulty than meChanical
manipulation, seems justified.

2. Special emphasis should be given to the teach-
ing of problem solving in elementary algebra.

51

3. A significant ability demanded in problem
solving is the ability to think things
through.

4. ...Teacher training courses in mathematics should
give more specific attention to problem solving.

5. The most common causes of failure in solving
verbal problems are: lack of preparation and
knowledge of techniques on the part of the
teacher; on the part of the pupils,--inability
to read, lack of logical reasoning ability,
poor labeling, lack of statements, lack of
knowledge of arithmetic, and lack of proper
Checking (p. 147).

A description of how children solve problems was
provided by Paul Ligda (1930). In an attempt to "help
children out of their bewilderment," he described the

problem-solving process in the following manner:

The solution of a problem may be considered to
consist of four somewhat overlapping parts: the
analysis, the synthesis, the translation into
symbols, and the symbolic solution. The analysis
consists in breaking the verbal statement into
distinct and separate parts and finding the rela-
tionships among these parts. The synthesis con-
sists in rearranging the results of analysis in
suCh a way that equations are obtained. The last
two parts do not need discussion (pp. 514-515).

In discussing "The Skills Involved in Problem
Solving in Elementary School Arithmetic," Guy A. west
summarized some of the frustration found in the teaching
of problem solving. In 1930, west justified the idea of
problem solving with the following comments:

The fundamental skills of arithmetic have been
analyzed by several authorities, but little has been
accomplished toward setting up a reliable and com-
plete analysis of the skills involved in problem
solving. No one would argue that the teaching
of problem solving is similar in principle to
the teaching of the basic facts of arithmetic.

On the other hand, few would doubt that a

52

detailed analysis of the skills involved in the
former would have considerable value for the
teacher who finds that John Jones can not solve
certain problems. His inability to do so re-
mains the unsolved mystery to many teachers

(p. 379).

West (1930) investigated the utilization of "problem

solving" found in a dozen textbooks on the teaching of

arithmetic. The following list was included as an "example"

of the problem-solving process:

1.
2.
3.
4.
5.

Grasp the conditions (Comprehension).
Separate into smaller units.

Plan the solution.

Solve in the best way.

Check the results (pp. 379-380).

Leo J. Brueckner (1930) included the Chapter

"Diagnosis in Problem Solving," in a textbook titled

Diagnostic and Remedial Teaching in Arithmetic. In dis-

cussing the nature of pupil difficulties in problem

solving, Brueckner noted the five—step process of W. S.

Monroe for solving arithmetic problems:

Reading the statement of the problem with
understanding.

Recalling of principles applicable to the
problem.

Formulating of a plan of procedure concerning
the operations to be performed, this being based
upon the elements of meaning and the recalled
principles.

verifying of the procedure which generally
does not constitute an explicit step.

Performing of the operation which is also,
strictly speaking, not a step in the reason-
ing process (p. 266).

53

In describing the causes of difficulty in problem solving,

the author made the following comments on children's inabil-

ity to clearly "see" a problem.

In other words, the problem is not concrete

to the child unless he is able to form a clear
mental picture of the situation described. It
may be that he has not read the problem care-
fully or that he may have read it carefully and
yet, through lack of experience in the situation
described, he may not be able to form a picture
of the situation (p. 271).

In summarizing the chief causes of difficulty found

in problem solving, Brueckner (1930) compiled many of the

ideas found in the major investigations on problem solving

initiated during this period of time. The six major causes

are provided in the following list:

1.

Lack of ability to perform the necessary
computations accurately or to select the
operation needed.

Lack of systematic method of attack in solving
a problem.

Careless reading or lack of vocabulary.

Lack of knowledge of essential facts, data,
or principles involved.

Failure to complete the problem.
Failure to comprehend the problem in whole or
in part (p. 308).

In joint authorship with Ernest O. Melby, Brueckner

further discussed the idea of problem solving in a textbook

titled Diagnostic and Remedial Teaching. In eleborating on

"the elements in problem solving," Brueckner and Melby (1931)

provided the following description of the factors involved

in problem solving:

54

The solution of verbal problems found in textbooks
involves four major factors: (1) the ability to
comprehend the meaning of the statements in the
problem and the situation that is presented; (2)

the knowledge of essential facts and principles
needed to arrive at the solution; (3) the ability
to select the processes to be used in solving the
problem; (4) and the ability of the pupil to per-
form the necessary computations accurately (p. 222).

In suggesting ways to detect weaknesses of pupils involved
in problem solving, Brueckner and Melby (1931) included a
list of twenty-eight skills that were considered important
to problem solving, such as the "ability to tell what
facts are given" and the "ability to tell what question
the problem asks." Appendix D contains the complete list
of twenty-eight skills.

In 1932, the National Council of Teachers of
Mathematics focused its entire yearbook on The Teaching of
Algebra. In discussing the "Recent and Present Tendencies
in the Teaching of Algebra in the High Schools," JOseph
Jablonower (1932) cited Thorndike and made the following
observation in a section on "problem solving" in algebra:

Thorndike has given us, in his Psychology of
Algebra, the most exhaustive study of the
psychology of problem solving. He points out

the need that the problem be genuine and that

it have the tang of reality. But his Chief
contribution is his emphasis on the true nature
of the problem and his consequent catalogue of
problems. A problem is a task in connection with
which the individual has to select his tools and
processes. A problem necessarily involves novel
elements, or a novel situation to which familiar
elements must be applied in a novel way. This
notion of the problem is broader than is the one
which makes the problem synonymous with the verbal
problem. While the verbal problem has its uses,
it is not the whole of the story. Pupils may have

55

difficulty with the verbal problem and yet have
acceptable mastery of algebra in its more impor-
tant aspect as a tool for representing quantitative
and functional relations (p. 14).

During this time, Edward L. Thorndike provided numerous
ideas on problem solving in mathematics. The thoughts of
John Dewey also had their impact.

In 1933, Dewey, in the revised edition of How We
Thigk, included a description of the "Essential Functions
of Reflective Activity." In essence, he claimed that
there were two limits--pre-reflective and post-reflective--
to every unit of thinking. In between these two units were
five states of thinking. They were described in the follow-
ing manner:

1. Suggestions, in which the mind leaps forward
to a possible solution.

2. An intellectualization of the difficulty or
perplexity that has been felt (directly
experienced) into a problem to be solved, a
question for which the answer must be sought.

3. The use of one suggestion after another as a
leading idea, or hypothe§i§, to initiate and
guide observation and other operations in
collection of factual material.

4. The mental elaboration of the idea or supposi-
tion as an idea or supposition (reasoning, in
the sense in which reasoning is a part, not
the whole, of inference).

5. Testing the hypothesis by overt or imaginative
action (p. 107).

In 1934, Paul Klapper described six factors that
determined ability to solve problems. The author, in

defining "intelligence" as the "native ability to recognize

56

quantitative relationships" (p. 440), called it the first
and most important factor in determining ability to solve
problems. The remaining five factors included:

1. Skill in silent reading.

2. Familiarity with the technical language of
arithmetic.

3. Understanding of situations that give rise to
arithmetical problems.

4. A high degree of skill in the fundamental
operations.

5. An attitude towards accuracy that leads to
questioning and checking of answers (p. 440).

Klapper (1934) differentiated "problems" from both
"exercises" and "examples." In discussing the differences,
the author defined an "exercise" as providing "a
direction to perform an operation" and an "example" as
"an exercise clothed in words so that the task seems more
social and becomes, therefore, more interesting" (p. 439).
Accordingly, the author defined a problem as "a chal-
lenging situation that invites solution. It cannot be
resolved without careful analysis and planning" (p. 439).

In discussing problem solving, Klapper (1934)
included a series of five steps for the purpose of solving
a problem. The five steps were identified in the following
manner:

1. Grasping the situation.

2. Ascertaining (a) what is to be sought, the
unknown, and (b) what is given, the known.

3. Planning the solution.
4. Carrying out the plan. .
5. Checking the answer (p. 458).

57

Finally, Klapper (1934) made the following comment
regarding the important role problem solving played in the

arithmetical process:

Problem-solving vitalizes arithmetic by infusing
purpose into its computations. Problems intensify
both the utilitarian and disciplinary values of
arithmetic by providing quantitative interpreta-
tions of social, economic and civic experiences.

In problem-solving the child finds the real
challenge of arithmetic (p. 439).

For better than a decade, since the influence of
Edward L. Thorndike in the early 19205, the teaching of
mathematics had undergone a considerable amount of scrutiny.
Many critics in the mid-19305 found it fashionable to judge
the merits of mathematics. Perhaps Arthur E. Robinson
(1935) summarized the situation with the following comments:
ARITHMETIC was introduced into the elementary
school in 1548. For 400 years it has held a place
of importance in the elementary school curriculum
second only to that of reading. Yet in spite of
its venerable age, the subject has at no time in
its long history been subjected to such severe
criticism as that of the past decade (p. 215).
In discussing the education of teachers, the author
noted:
...few studies or investigations have been
directed at what would seem to be one of the
first and most vital problems in the teaching
of arithmetic, the professional equipment of the
prospective teacher in the field of elementary
school arithmetic (p. 216).
In analyzing pedagogical aspects, Robinson (1935) concluded
that "Mbst of the teaching of arithmetic observed impresses
one of the fact that the theory of teaching and the practice

of teaching are still quite unrelated" (p. 219).

58

In examining "The Curriculum and the Mastery of
Academic Skills" for the previous one hundred years,
Harold Rugg provided a look at various subjects in the
curriculum. As he noted, most of the arithmetic that
had been included in the curriculum focused on the develop-
ment of skill in addition, subtraction, multiplication,
and division of integers and fractions. Rugg (1936) made
specific references to the teaching of problem solving.
In doing so, the following comments provided insight into
the teaching of arithmetic:

On the side of problem-solving there was the
same emphasis upon mechanics rather than upon in-
terpretation and understanding. It was implicitly
assumed that skill in the use of numerical tech-
niques could be abstracted from "life situations"
and developed apart. Arithmetical "problems" were
merely word descriptions of number situations, not
actual social situations in which a child would
naturally use numbers. Furthermore, the problems,
as they appeared in arithmetic books, were outland-
ish and unreal. Trained in the doing of these
isolated, unreal word-problems, by some mysterious
process the pupils were to be able later in life
to transfer their skill to actual situations re-
quiring arithmetical techniques (p. 142).

In writing about "Problem Solving in Algebra,"
D. McLeod and Daniel McIntyre (1937) provided insight into
some of the frustration associated with the teaching of
problem solving during the 19305. The authors put the
idea of problem solving into perspective with the follow-
ing comment:
No apology is needed for re-introducing in
the pages of The Mathemggigs,Tegche£_a topic whiCh
may be time-worn. The subject of problem solving

is always in place and ever calls for solution
(p. 371).

59

McLeod and McIntyre (1937) cited a concern of mathematics
educators when the following question was posed:
Immediately the question arises: Is this
condition universal? What is so difficult about
problem-solving as compared with simplifying
fractions, factoring, deriving formulas, or
finding the root of an equation? (p. 371)
The fact that the teaching of problem solving has
historically created difficulties was made more prevalent

by Harry Grove Wheat (1937) in a textbook titled The

Psycholggy and Teaching of Arithmetic. In discussing "The

Development of Arithmetic," the writer stated:

The early interests in computations and in problems
persist in present-day arithmetic, which divides
into two parts; namely, computation and problem-
solving. Ancient interests dominate the arithmetic
of the modern school (p. 102).

Later, in comparing "The Relation of Present versus Past
Conditions to Problem-Solving," Wheat (1937) added:

In view of the present organization of the
curriculum in arithmetic with its emphasis upon
problems and problem-solving, what has just been
said may seem heretical in the extreme. But we
should remember that the school continues to
emphasize problem-solving in arithmetic, because
problem-solving was, and had to be, a major
activity in the earliest developments of the
subject (p. 141).

What was it that he was calling heretical? It was a

call for teachers to do more than teach children simple
solutions to arithmetic problems. He claimed that the
purpose of education was to get children to think. Wheat
(1937) described a response to the question in the follow-

ing manner:

60

The purpose is so to order and systematize the
child's methods of dealing with combination and
arrangement of objects that he may go through

life freed from the necessity of confronting
problems of an arithmetical nature. The purpose
rests upon the assumption that the individual has
a higher function to perform in life than to ex-
pend his energy in solving what were once problems
in arithmetic but are problems no longer. He must
be set free from the necessity of ever having
problems in arithmetic to solve (pp. 140-141).

Wheat (1937) defined "problem solving in arithmetic" as:
...practice in the recognition (a) of general
ideas in familiar situations, and (b) of new sit-
uations with which are involved the general ideas
that now should be familiar. Thus, the double
purpose of "problem-solving" is distinguished
(p. 211).

Robert Lee Morton (1937), in a mathematics methods
textbook titled Teaching Arithmetic in the Elementagy
School (volume I, Primary Grades), described problem
solving in the following citation:

The processes of addition and subtraction are not

of importance in themselves. They are a means to

an end; the end is problem solving (p. 107).
In discussing "the method of formal analysis" as a pro—
cedure for solving problems, Morton (1938) cited (volume II,
Intermediate Grades) the ideas of educators from the 19205.
In describing the so-called "Conventional Method," he
included the problem-solving models of Edward Lee Thorndike
and Fletcher Durell. The three-step model provided by
Thorndike (1921) differed little in theory from the follow-

ing six-step model presented by Durell in 1928:

61

State what is given.
State what is to be found.
Make a list of the operations to be performed.

. Estimate the answer.
. Make the computations.
. Check the answer (Morton, 1938, p. 468).

O‘tflbNNl-J

A two-fold reason for utilizing the "Conventional Method"
was given: first, children were forced to thoroughly read
through a problem; and second, children were trained to
think critically.

In 1940, the idea of problem solving was given an
even greater dimension. W. S. Schlauch, in an article
titled "The Use of Calculating Machines in Teaching
Arithmetic," suggested that interest in problem solving
could be stimulated by including "calculating machines" in
the mathematics curriculum at the junior and senior high
school levels. Schlauch (1940) provided the following
summary:

we may conclude that the use of calculating

machines in teaching arithmetic is justified by its
results, and that they should be used wherever the
cost of installing such machines can be met. They
are used to best advantage in the upper grades of
the junior high school and in the senior high school.
They lend speed, accuracy, and confidence in compu-
tation, and stimulate an interest in problem solving
calling for their use (p. 38).
This was not the first time that an educator had called for
the use of "calculating machines" in mathematics. In 1937,
Evelyn M. Horton provided a discussion of "Calculating

LMachines and the Mathematics Teacher" in the Mathematics

Teacher.

62

In 1940, Ben A. Sueltz discussed the "Recent Trends
in Arithmetic" at the conference of the Eastern-States
Association of Professional Schools for Teachers. In a
follow-up article Sueltz noted these four trends.

First, subject matter continues to be impor-
tant but it is redirected in terms of functional
living. It is no longer an end in itself.

Second, the modes of learning as well as the
ends of learning tend to dissolve subject-matter
barriers.

Third, textbooks continue in use both as
learning and reference materials, but the text-
book is freely supplemented with excursions,
investigations, and interviews.

Fourth, a more serious attempt is being made
to develop in pupils the intelligent participa-
tion in, and the feeling of, responsibility for
their own affairs (p. 270).

In 1942, the National Society for the Study of
Education (NSSE) provided attention to the concept of
problem solving in its Forty-first Yearbook. In this Year-
book, William A. Brownell discussed the many facets of
problem solving including researCh in problem solving,
processes, growth and the teaching of problem solving, and
suggestions for the development of abilities in problem
solving. Brownell (1942) highlighted the significance of
the "special treatment" given to problem solving with the
following comment:

Since all the other chapters in this Yearbook deal
in one way or another with learning, the present

Chapter might appear repetitious, if not redundant.
Yet, here is a separate chapter on problem solving.
Its presence attests the belief that, for the pur-
poses of education at least, problem solving needs

to be considered separately from other kinds of
learning (p. 415).

63

Brownell (1942), in noting "(1) the rise of field
theories of learning, with consequent changes in the design
of experimentation, and (2) the attempt to get at the nature
of problem-solving behavior without regard to any particular
systematic point of view in psychology" (p. 419), discussed
three Changes that had occurred over a fifteen-year period.
In describing the three changes in psychological research
(with special reference to "problem solving"), the author
wrote:

One of these changes consists in the attempt to
set problems which "mean" something to the subject
(animal or man), or at least envisage the learning
task as it most probably is envisaged by the sub-
ject. A second change is the tendency to concen-
trate research interest, not merely on errors and
successes, but on the way in which the subject
proceeds to attack and solve its problem. The
latter trend has not meant a wholesale abandonment
of objective data: after all, the systematic
analysis of errors, for example, reveals much
concerning the pattern of behavior involved in
problem solving. Rather, it has meant that the
experimenter has been willing to advance explana-
tions and interpretations, anthropomorphic, if
necessary, in the case of animal subjects, but
nevertheless designed to understand what the
problem and its solution mean to the subject.

A third Change, closely associated with the
second, is the greater importance now attached to
qualitative descriptions of significant behavior
to supplement or to replace purely quantitative
descriptions (PP. 418-419).

Children in the early grades were taught a system
of problem solving that asked the following series of
questions summarized by Brownell in 1942:

1. "What is asked?" (or, "What am I to find?")
2. "What is given?" (or, "What do I know?")

64

3. "What process or processes should I use?"

4. "What is the probable answer?" (the last
question being followed by actual computation).
(p. 432)

In response to the development and use of the models that
had been patterned after Dewey's "Analytical Thought"
method of solving problems, Brownell (1942) presented two
criticisms in the following manner:

In the first place, the method of step analysis
represents a logical pattern of thinking which
may or may not characterize expert thinking on
the part of adults, but whiCh certainly has not
yet been shown to characterize good thinking on
the part of children. According to this method
of teaching, a formal abstract pattern, possibly
suitable to adults, is imposed upon children
before they are ready for it. A preferable pro-
cedure is first to ascertain the level of think-
ing whiCh children have attained and then to
lead them on to more mature and economical levels
as rapidly (but only as rapidly) as they can adopt
them.

In the second place, this method puts too much
trust in technique alone and disregards other
essentials in effective problem solving (p. 432).

In a discussion on "Growth in Problem-Solving
Ability," Brownell (1942) noted the work of Jean Piaget and
associates at the Maison des Petits, in Geneva, Switzerland.
In citing four studies completed by Piaget in the later

19205 and early 19305, the writer made the following
summation:

Piaget represents growth in problem solving
as influenced by two sets of factors. The first
set is highly personal and narrowly individual-
istic, the result of the egocentrism of early
childhood. Opposed to the first set of factors is
another set, social factors, which are steadily
imposed upon the child and which have the effect
of leading him to substitute objective reality for

65

his own subjective schemas and to replace his
illogical, if personally satisfying, mental
processes by others which are rational and can
meet the requirements of impersonal appraisal.
The conflict between the two sets of factors is
resolved finally in favor of the social factors,
though the egocentric factors are by no means
easily, quickly, and completely surrendered

(p. 428).

Brownell offered four criticisms of Piaget's work
on problem solving. In discussing the first criticism,
the author made note of "Piaget's failure to consider
sufficiently the prejudicial character of the problem tasks
with which he worked" (p. 430). In the second criticism,
Brownell took exception to Piaget's definition of reason-
ing. In commenting on Piaget's use of "reasoning" in
place of "problem solving," Brownell (1942) provided the
following three objections:

...first, that this kind of thinking is rare;

second, that it overvalues verbal expression as

a measure of thinking; and third, that it tends

to encourage the notion that young children can-

not solve problems of any kind (p. 430).
The third criticism dealt with Piaget's ideas on age levels
and the development of skill in reasoning. According to
Brownell (1942), it could be interpreted "that children at
certain rather definite ages achieve equally definite

levels of thinking" (p. 430). The following rebuttal was

noted:

...it is probably true that the changes in problem
solving which Piaget attributes to age are better
explained as the effects of increases in general
experience and in control over language. In this
case, age makes its contribution chiefly by pro-
viding opportunity (p. 430).

66

The fourth and last criticism had to do with Piaget's
notion that adult reasoning was different than the reason-
ing of children. Brownell (1942) cited the works of those
who found that both children and adults displayed the same
tendencies in reasoning skill (p. 431).

Brownell (1942) included twelve "Practical
Suggestions for Developing Ability in Problem Solving."
In doing so, he elaborated on problem situations, puzzles,
learning situations, meanings and understandings, problems,
mistakes, abilities, and attitudes. Three of the "sugges-
tions" were presented in the following manner:

d. Skill in problem solving is partly a matter of
technique and partly a matter of meanings and
understandings. Highly formal and abstract
techniques should never be imposed upon the
child. Instead, they should be viewed as the
end-products of development. Teaching should
start with whatever technique the child uses
proficiently and should guide him in the adop-
tion and use of steadily more mature types of
problem solving.

9. To be most fruitful, practice in problem solving
should not consist in repeated experiences in
solving the same problems with the same tech-
niques, but should consist in the solution of
different problems by the same techniques and
in the application of different techniques to
the same problems.

k. A problem-solving attitude, an inquiring and
questioning mind, is a desirable educational
outcome, and it is possible of development.

The practice of "learning" by cramming does

not produce this outcome, nor does the practice
of accepting from others truths and conclusions
which ought to be established by the learner
himself. The attitude gg produced by continued
experience in solving real problems, one
consequence of which is that the learner comes
to expect new problems and to look for them
(pp. 439-440).

67

In 1944, Harry C. Johnson provided a review of
literature on "Problem-Solving in Arithmetic." In the
Opening paragraph, he noted a recurring theme that had
concerned mathematics educators for years. This theme was:

Educators interested in the improvement of learning

in the elementary school will readily agree that

the teaching of problem-solving in arithmetic

offers one of the greatest challenges to elementary-

school teachers (p. 396).
It is noted that a majority of the references included in
the Johnson review dated back to the 19205 and early 19305.
In fact, over two-thirds (28 out of 39 references) were
written prior to 1934 and only three were dated in the 19405.
The "average" date of all the references included in the
Johnson review of literature was 1931. Although not very
scientific, the statistics illustrate the wide-spread
emphasis that was given to "problem solving in arithmetic"
during the latter stages of the 19205 and the early 19305.

During the mid-19405, the idea of "meaningful
arithmetic" became important. The "meaning theory," pre-
5ented a decade earlier by the National Council of Teachers
of Mathematics in its Tenth Yearbook (1935), was enhanced
by mathematics educators suCh as William A. Brownell, Leo
J. Brueckner, Foster E. Grossnickle, and Harry Grove Wheat.

The "meaningful arithmetic" movement, based upon
the idea "that children must understand the structure of

the number system and be able to perform number operations

meaningfully" (Kramer, 1978, p. 28), provided the overall

68

theme for the Council's Sixteenth Yearbook in 1941. In
addition to the individuals cited on the previous page,
many of the top mathematics educators in the country--
including Robert Lee Morton, Guy T. Buswell, and B. R. Buck-
ingham--had a hand in putting together the yearbook titled
Arithmetic in General Education. The Sixteenth Yearbook
included a chapter by C. L. Thiele on "Arithmetic in the
Early Grades." After discussing the socialization aspect
of arithmetic, Thiele (1941) described the teaching of mean-
ingful arithmetic. In noting the difference between the
ideas contained in the Sixteenth Yearbook and those ideas
found ten years earlier, the author wrote:

The significant difference between the program of

arithmetic which finds support in this Yearbook

and that of a decade ago is in the extent to which

children see meaning in the numbers which they use

and operate. The keynote of the new arithmetic is

that it should be meaningful rather than mechanical

(p. 45).
Thiele included a brief description of the role "problem
solving" was to play in the teaching of "meaningful arith-
metic." In noting that "experience" was an important
factor and that the use of dramatization, verbal description,
and illustration was vital, the writer provided the follow-
ing discussion:

Problem solving. In the meaningful program of

arithmetic instruction, problem solving instruction

does not assume an independent role. Instead, it

is ultimately bound up with the whole teaching pro-

cess. The experiences with concrete settings
through whiCh abilities are developed provide

69

experiences in using number for purposes of quanti-

tative thinking. Following this, the applications

of these abilities to situations which described

rather than "present-to-sense," contributes to the

development of problem-solving abilities (p. 78).

In 1945, Harry Grove Wheat approached the question

concerning "the sense and utility of teaching meaning."
In discussing the development of the "meanings approach"
for children in the early grades, he wrote:

The meanings in early arithmetic are simple: in

later arithmetic, they still are simple, though

eaCh in succession is a step beyond those that

precede. Why meanings in later arithmetic seem

complex and beyond apprehension is that they are

frequently considered in isolation and without

due regard for the earlier meanings which give

them support (pp. 101-102).
During the same year, Wheat (1945), in a summary of theses
in arithmetic completed at West Virginia University, wrote
about the "types of training" that were related to the
teaching of problem solving. In analyzing six studies on
"problem solving," the author distributed the "types of
training" into two distinct categories. The first category
included the type of training students received in prepara-
tion for problem solving. The second category included the
type of training "that pupils should be expected to get
from problem solving" (p. 31).

G. Polya (1945) produced the often-mentioned book

How to Solye It: A New Aspect of Mathematical Method. In

it, he described a four-phase approach to solving problems.

70

The Polya Model has been popularized and used by teachers
of mathematics at all levels of instruction. The four steps
were presented in the following manner:

First. YOu have to understand the problem.

Second. Find the connection between the data and
the unknown. YOu may be obliged to con-
sider auxiliary problems if an immediate
connection cannot be found. You should
obtain eventually a plan of the solution.

Third. Carry out your plan.

Fourth. Examine the solution obtained (pp. inside-
front and inside-back covers).

The key words underlined above, highlight the steps in the
Polya Medel for problem solving. Each step in the model
contains a series of questions. The complete list of
questions is included in Appendix E of this investigation.

In 1946, J. T. Johnson, in summarizing a study of
four factors affecting problem solving, provided the follow-
ing introduction:

The solution of the problem of problem solving

is a major trend right now in the teaching of
arithmetic. Let us hope that the end of this dec-
ade will see a part of this solution (p. 256).

In addition to Johnson (1946). a panel of four
members--Elizabeth F. Jeffords, G. T. Buswell, w. B. Storm,
and Lucile B. Gates--attempted to answer the following
questions concerning the factors contributing to problem
solving:

1. Is It Reading?

2. Is It Number Relations?

3. Is It Meaning of Numbers and Their Operations?
4. Is It Teaching? (pp. 256-266)

71

A case was built in support of each of the questions cited

on the previous page as having an influence on problem

solving.

In answering the fourth question, Lucile B. Gates

(1946) included the following list of what were believed to

be the chief causes of failure in problem solving (as seen

by teachers of arithmetic):

Failure to read the problem correctly.

Lack of knowledge of arithmetical vocabulary.
Using the wrong computational process.
Guessing in computation.

Inability to make correct judgments concerning
the problem.

Lack of knowledge of place value.
Failure to estimate answer.
Problems of little or no practical value.

Lack of number knowledge such as the number of
feet in a mile, quarts in a gallon and so forth
(p. 265).

In terms of problem solving, Gates (1946) provided the

following six-step process in an attempt to answer the

question, "What does correct solution require?"

1.

Reading with understanding, which means that the
pupil must be able to read the problem so that
he will know what he is given and what he is
required to find.

Deciding what processes he will have to use and
using them correctly.

Estimating his answer.
Carrying out all steps until final completion.

Knowing when to stop, when he has arrived at
the point, when he has answered the question,
"To Find."

Proving the result (p. 265).

72

In 1946, Harry Karstens, in discussing the "Effec-
tive Guidance in Problem Solving," provided some insight
into the inclusion of problem solving in the mathematics
curriculum. Regarding the difficulties of teaching problem
solving, he made the following comment:

The teaChing of problem solving, which makes
greater demands on the ability and resourcefulness
of a teacher than any other phase of arithmetic,
is a particularly delicate and exacting procedure
when handling low-ability groups (p. 172).

The author noted that "the first requirement for teaching
problem solving is a clear concept of the function of
problems" (p. 172). With this in mind, the idea of problem
solving and its role in the computational process were
noted in the following comment:

Computation in arithmetic achieves its objective
in problem solving. Computation, in itself, is
meaningless: problem solving, without computa-
tional ability, is futile. Computation makes
problem solving possible: problems give sense to

computation. Problems are the humanizing element
in arithmetic (p. 172).

Karstens (1946) discussed the use and teaching of
problem solving. On the teaching of problem solving, he
provided the following look at the concerns and steps in-

volved in the process:

The term "teaching problem solving" carries
unfortunate implications. "Teaching problem solving"
implies formal methods and set routines while the
term "guidance in problem solving" connotes a sympa-
thetic molding of the pupil's thinking and an
informal directing of his work habits. The problem
solving material in many texts and courses of study

73

is based primarily on a pattern of formal analysis
in which the pupil asks himself three questions.

1. What am I to find out?
2. What facts (numbers) are given?
3. What shall I do with the numbers? (p. 173)

As a substitute for the three questions listed above,
Karstens (1946) suggested a multi-step method. In noting

that "any method of formal analysis tends to stress tech-

niques instead of understandings" (p. 173), he provided the

following plan for solving problems:

1. What is the problem about?

2. What am I to find out?

3. What are the essential phrases or sentences?
4. What must I do to solve the problem?

5. What facts will I need?

After this, the computation, checking, and
evaluation of the answer should follow in due order
(p. 174).

Karstens (1946) saw the ideas of "meaning" and
"thinking" as important tools in the problem-solving process.
In summarizing his thoughts on "effective guidance programs
in problem solving," the author provided the following cita-
tion which described much of the attitude toward arithmetic

instruction during the 19405:

...meaning is all impprtant in a guidance program
in ppobiem solvin . The teacher's problem is to

take stock of each individual's "mental tools,"

suCh as they are, and guide him into more efficient
and more mature ways of using those tools. Her
efforts should be devoted toward making each child
a thinking individual. Since p11 other phases of
arithmetic culminate in the problem solving program,
guidance in problem Solving should be the most in-
teresting and at the same time the most challenging
phase of arithmetic teaChing (p. 175).

74

It has been mentioned frequently in this chapter

that "meanings" and "understanding" played major roles in

mathematics instruction during the 19405. In discussing

the "Aspects of Problem-Solving Ability," William L. Schaaf

(1946) combined the ideas of meaning and understanding with

problem-solving ability. In writing about the ability to

solve problems, he elaborated in the following manner:

Most of the common methods used in teaching problem-
solving are unsuccessful because they do not get at
the heart of the matter, which is an adequate under-
standing of the meaning of number and of the nature
of relationships between quantities, together with
a ready facility in recognizing such relationships
in a wide variety of settings, verbal or otherwise
p. 495 .

In suggesting a "realistic approach" to teaching problem

solving, SChaaf (1946) recommended the following areas of

emphasis:

1.

mmbw

(13%

Understanding numbers and number relations and
the rationalization of operations and processes.

Understanding the mathematical relationships
involved in a problem situation.

Understanding the essence of a problem situation.
Understanding the vocabulary used in the problem.
Understanding the relevance of data.

Recognizing the arithmetic operations or proce-
dures to be used.

Analyzing two-step problems.
Estimating the answer.
Checking the answer (PP. 496-497).

In 1947, Leo J. Brueckner and Foster E. Grossnickle

provided a look at instruction in meaningful arithmetic

with the writing of How to Make Arithmetic Meaningfu .

75

Included in the textbook was an extensive chapter on "The

Place of Problem Solving in Arithmetic." In the following

list, they summarized the causes of difficulty in solving

problems:

1.

10.

Failure to comprehend the problem in whole or
in part because of lack of experience and ina-
bility to visualize the situation.

Deficiencies in reading, such as inability to
locate information, inability to remember what
is read, inability to organize what is read,
and inability to read for details.

Inability to perform the computations involved,
either because the pupil has forgotten the
procedure or has failed to learn it.

Lack of understanding of the process, resulting
in the random trial of any process that may
come to mind in order to get an answer.

Lack of knowledge of essential facts, rules,
and formulas, such as how many inches there
are in a yard or the rule for finding the
perimeter of a rectangle.

Lack of orderliness in arranging written work.
Ignorance of quantitative relations due to a a a

llimited vocabulary or to lack of understanding

of principles, such as the relations between
selling price, cost, profit, and margin.

Lack of interest, due to inability to solve the
problems because of their difficulty, unattrac-
tiveness, and general low level of merit.

Level of mental ability too low to grasp the
relations implied.

Lack of practice in solving verbal problems
(p. 452).

In reviewing research, Brueckner and Grossnickle

(1947) looked at "good" and "poor" achievers in problem

solving.

Describing the analysis, the two authors included

the list on the next page which categorizes the differences

76

between good and poor achievers in problem solving in terms

of having either "significant" or "no significant" differ-

ence:

A. Diffegences Highly Significant

l. Psychological factors

a.
b.
c.
d.

e.

General reasoning ability
Non-verbal mental ability
Delayed and immediate memory
General language ability

General reading level

2. Computation abilities

a.
b.
c.
d.

Skill in fundamental operations

Ability to estimate answers of examples
Ability to see relations in number series
Ability to think abstractly with numbers

3. Problem solving reading skills

a.
b.
c.
d.

9.

Steps in problem analysis

Finding the key-question of the problem
Estimating answers to problems

Ability to read graphs, charts, tables

Range of information about arithmetic
uses

B. No Significgnt Differences

1. Range of general information

2. Gates Tests in General Reading

a.
b.
c.
d.

Grasp of Central Thought
Prediction of Outcomes
Following Directions

Reading for Details (p. 454).

Throughout the textbook, Brueckner and Grossnickle

(1947) pointed out that experience played a major role in

the way children learned arithmetic. In looking at "Direct

Experience in Problem Solving," they noted that "the most

77

effective way to develop the ability to use quantitative
procedures is through their direct application" (p. 435).
Direct experience was defined as "the actual use of number
in social situations that arise in daily life in and

out of school" (p. 435).

In 1947, G. T. Buswell edited the monograph,
:Agithmeticg947y" which included a series of papers pre-
sented at The University of Chicago. In writing the opening
chapter, L. J. Brueckner (1947) discussed "arithmetic in
elementary and junior high schools." Included in the chapter
was a description of the "Minnesota State Guide for the Im-
provement of Instruction in Arithmetic." Six major activ-
ities--problem-solving activities, construction activities,
appreciation activities, creative activities, excursions,
and practice activities--were described. The list of
"problem-solving activities" included the following items:

a. The formulation of a problem.

b. Consideration of the scope and significance of
the problem.

c. Planning a method of attack.
d. The assignment of tasks to individuals or groups.

e. The location and gathering of necessary infor-
mation from persons and printed sources.

f. Research and experimentation needed to get new
data.

9. The assembling, organizing, and presenting of
findings.

h. Drawing conclusions and making decisions.
1. Taking steps to carry out decisions (p. 6).

78

Numerous ideas pertaining to problem solving
appeared during the 19405. In 1948, H. van Engen commented
on the changes that were taking place in mathematics in-
struction. Writing about the implications of research on
the organization and learning of arithmetic, the author
noted the following citation about problem solving:

There is evidence that the teaching world is
gradually changing its conception of what consti-
tutes problem solving in arithmetic. The so-called
problems found in the textbook are now, at times,
being called examples. There is a growing reali-
zation that at best the book problems are exercises
in using the language of arithmetic and that very
little problem solving activity may accompany the
procurement of answers to a page of textbook prob-
lems (p. 262).

In 1948, Lee J. Cronbach provided a working defini-
tion of problems and exercises. Writing in the monograph,
"Arithmetic 1948," edited by G. T. Buswell, Cronbach
included the following ideas on problem solving:

...the major purpose in teaching problem-solving
is to prepare the pupil to solve problems when he
encounters them in the flesh, far from the class-
room. Moreover, any proper psychology of problem-
solving should encompass problems which are non-
mathematical in nature.

The characteristic of problem-solving that

separates it from all other behavior is that one
cannot solve problems by habit (p. 32).

In summarizing the comments on "The Meanings of
Problems," Cronbach noted, in the following citation, the
important role that "problem solving" and "meanings" play in

arithmetic instruction:

79

The implications of present knowledge of problem-
solving can be summarized concisely. First, meaning-
ful problem-solving will take place if the problem
has meaning for the Child. The selection of sensible
problems is the easiest way to increase the soundness
of the pupil's thinking. Second, problems must not
degenerate into exercises, since exercises make small
demands on higher mental processes. Third, if arith-
metic lessons must make use of puzzles remote from
the pupil's experience, the topic can be made mean-
ingful by giving the child the necessary experience,
directly or vicariously. Finally, the setting of the
problem and the emotional experiences of the child
must be carefully considered, in order to remove
factors which will distract the child from giving his
full attention to the problem. If this is done, the
child should have just one concern in thinking of his
problem, namely, "What is the answer?" (p. 43)

Also writing in :Agithmetic 1948," Maurice L. Hartung
(1948) elaborated on the "Advances in the Teaching of Problem
Solving." In discussing the "behaviors" needed for solving
problems, he included the following "key words:"

1. Recognize and formulate the problem.
2. Collect and organize data.
3. Analyze and iptegpret the data.

In arithmetic, it is often (but by no means
always) useful to replace this group by a more
restrictive set, namely, choosing and carrying
ppp the processes.

4. Egg! and verify conclusions (p. 45).
Hartung included a six-step model for solving problems that
had originally been cited by Raleigh Schorling, John R. Clark,
and Rolland R. Smith in the textbook Apiphmetic fogngung
America in 1944. The steps for problem solving read as
follows:

a. Read the problem carefully to learn what is
given and what you are to find.

b. If you do not know the meaning of any word or
expression, find out its meaning.

80

c. Think through the steps of solution and decide
what process to use in each step.

d. If possible, estimate your answer.
e. Solve the problem and check each process.
f. Check your answer for reasonableness. See that

it is expressed correctly (Hartung, 1948, pp.

In looking at the relationship between reading and
problem solving, Hartung (1948) took a dim view of the types
of problems that were used to teach problem solving. The
author provided the following comment:

Although attention to reading skills may improve
problem-solving scores, improvement in reading,
alone, will not make good problem-solvers. Much
more needs to be done. Perhaps one reason that
many of the efforts to improve problem-solving
abilities have proved to be disappointing is that
the kinds of problems to be used have been too
narrowly conceived (p. 52).

Along the same line, J. T. Johnson (1949), in writ-
ing about the "nature" of problem solving, called for
psychologists to better define "memory" and "reasoning."

In discussing different aspects of problem solving, the
author looked at the existing relationship between reading
and problem solving. Johnson provided the following comment
in an effort to get educators to consider something other
than reading skill as the sole indicator of success in prob-
lem solving:

For many years we have heard it stated that the

reason children cannot solve problems in arithmetic
is because they cannot read. This is only a half
truth. It seems to fall under the mathematical cate-
gory of "necessary and sufficient" reasons. In this

case reading ability is the necessary but not suffi-
cient reason for problem-solving in arithmetic.

81

That there is something besides reading ability

which is required in problem-solving is evidenced

by the fact, well known to every elementary school

teacher, that there are many good readers in our

schools who are poor in problem-solving (p. 110).

Lucy Lynde Rosenquist, in Ygung_Children Learn to

Use Arithmetic, provided a four-step example of problem
solving. In writing a "things-to-do" list, she relied
heavily upon "Visualizing the Problem Situation." As a part
of the process, children were taught to analyze their own
errors and to "become conscious of the various places in the
solution of a problem where mistakes may be made" (p. 86).
Rosenquist (1949) stated that the following abilities were

needed to solve problems:

1. Visualizing the Problem Situation
a. Dramatizing Problems
b. Making Pictures of Problems
c. Clarifying WOrd Meanings
d. Stating the Question for a Problem

e. Stating Original or Personal Experience
Problems

2. Selecting the Process to be Used in Computation
3. Performing the Computation
4. Checking Results (pp. 86-93).

In 1950, Millie Almy discussed the use of problems
and problem-solving skills for young children. After
pointing out that many educators believed problem solving
to be an adult activity, she endorsed the use of problem
experiences with very young children. In prefacing her
comments on providing young children with problem-solving

activities, Almy (1950) wrote:

82

When the relationship of problem-solving to
child behavior is clearly understood there can be
little question as to the importance of problem-
solving in the education of young children (p. 148).

Writing on "The Need for Extending Arithmetical
Learnings," Ben A. Sueltz and John W. Benedick (1950) pro-
vided insight into some of the growing discontent with
competence and achievement in mathematics. The following
observation is particularly noteworthy:

The current trend toward teaching arithmetic
and mathematics for "meaning" and "understanding"
coupled with the aim to achieve functional compe-
tence on the part of the pupil is causing rever-
beration among mathematics teachers. Teachers ask,
"What and how muCh can pupils really understand?"
and "Shall I teach this topic if I cannot find more
than a few poor examples of functional usefulness?"
While these questions do not represent the point of
view of all teachers, they are indicative of a
growing discontent with the achievement in mathe-
matics at both the elementary and secondary school
levels (p. 69).

In answering the question, "What is Arithmetic?," the two
writers provided the following response:

...arithmetic is a study of the significance and

uses of numbers in the social, cultural, economic,

and industrial situations most commonly found in

our society. This does not preclude a study of

"the science of numbers" but rather provides that

both the science of numbers and the algorisms of

computation shall be important elements and stages

in the study of arithmetic (p. 69).

In 1950, Max R. Goodson, in writing about "Problem-

Solving in the Elementary School," referred to the "act" of
problem-solving "as an orderly sequence of steps" (p. 145).
In describing this "formalized" process, the author cited

the following five-step model:

83

1. definition of the problem;
2. formulation of hypotheses or ideas;

3. testing hypotheses by reasoning through their
logic and by experimentation:

4. applying the affirmed hypothesis in changing
the conditions of the problem so as to bring
about a correction or control: and

5. generalizing the affirmed hypothesis to the

limits of its supporting facts for applica-

bility to as wide a range of phenomena as

possible (p. 146).

In looking at the question, "Why Problem—Solving in the
Elementary SChool?," Goodson (1950) concluded that the
"Changing society" and the "maturity required by our times"
provided the greatest possible reason for including problem
solving as "one of the basic functions of the elementary
school" (p. 147).

In 1950, Robert L. Thorndike, writing in The Forty-
ninth Yearbook of the National Society for the Study in
Education (NSSE), expanded the theory of problem solving in
a Chapter titled "How Children Learn the Principles and
Techniques of Problem Solving." In the Yearbook, Learning

and Instruction, Thorndike (1950) looked at three questions:

1. What is a problem?

2. What do we know about successful techniques for
problem-solving?

3. What can teachers do to develop the problem-
solving abilities of their students? (p. 192)

In answering the first question, Thorndike suggested that
Children become more "aware" of the problems when they are
"involved" in the process. Interestingly, in answering the

second question, he divided the section into five distinct--

84

and familiar--steps. For purposes of analysis, Thorndike
utilized five steps described by John Dewey (1933) in How
We Think. The five steps are:

l. Becoming aware of a problem.

2. Clarifying the problem.

3. Proposing hypotheses for solution of the problem.
4. Reasoning out implications of hypotheses.

5. Testing the hypothesis against experience
(Thorndike, 1950, p. 196).

At this point, Thorndike (1950) expanded the think-
ing on problem solving. In doing so, the author suggested
suCh factors as expanding life's experiences, bridging the
gap between "the given" and "the desired," positive use of
facts based upon a person's experience, understanding of
individual maturity, understanding unique situations, and
the ability to appraise problem situations (pp. 196-208).

He called for more awareness of problem situations and im-
proved questioning techniques on the part of teachers. The
author made several noteworthy observations which included
calling the nature of problem solving a "complex and variable
behavior" (p. 215). Thorndike (1950) referred to the process
as an "attack upon problematic situations for which the
individual has no ready-made response patterns" (p. 215).
His comments were summarized in the following manner:

There is no simple pattern or routine of problem-

solving which can be isolated and taught in the

schools as a simple unitary skill. Rather, problem-

solving is an integration of a host of more

particular knowledges, skills, and attitudes with
which the schools can appropriately be concerned.

85

A wide range of interests and experiences, an organ-
ized and functional stock of background information,
efficient skills for locating and organizing needed
information, perseverance yet flexibility in attack-
ing problem situations, a willingness to suspend
judgment until the evidence is in, habits of test-
ing critically any proposed solutions, attitudes of
critical appraisal of the reliability and bias of
sources, skill in "if-then" thinking--these and many
more are the qualities which the school must try to
develop if it is to improve problem-solving ability
in its pupils (p. 215).

In 1951, C. Newton Stokes, in the textbook Teaching
the Meanings of Arithmetic, included a chapter on "Problem-
Solving." The concept of problem solving was defined as
follows:

...problem-solving in arithmetic is the determina-
tion of the nature of the relations involved in a
challenging quantitative situation and consequent
activity that unifies properly chosen relations
into a satisfactory result. Thus problem-solving
is thoughtful action. It is reflective thinking

in general, or it is reflective thinking in such a
specific task as finding the how-many or how-much.
It is the action involved in a purpose-to-end mind
process, executed by analyzing (differentiating and
discriminating) the quantitative relations found in
a problem situation and then unifying (integrating)
relevant elements into a meaningful and u5able out-
come (pp. 187-188).

In discussing the application of problem solving in
arithmetic, Stokes (1951) provided a set of related concepts
that had been identified by The Committee on the Function of
Mathematics in General Education. In describing the follow-
ing ideas, the author noted "that the child's understanding
of these concepts involved in problem-solving in arithmetic
should give him a more mature and a more operative ability

to resolve problem situations" (p. 199). The concepts were:

86

a. Recognize and formulate a problem.
b. Collect needed information and data.
c. Determine the relationships involved.

d. Express the relations in symbolizations--
words or number symbols involving operations.

e. Find the solution or consequence (p. 195).

Like so many educators who discussed the importance
of "experience" in problem solving, Stokes (1951) alluded
to the socialization aspect of solving problems. The
following comment was made regarding the "source of
problems":

Problem-solving ability in arithmetic requires
a background of experience. Then, to develop this
ability so that it parallels growth in other fields,
there must be provision for a wide variety of ex-
periences in problem-solving. Sources must be
found, both in the social and mathematical aspects
of arithmetic (p. 199).

Finally, Stokes (1951) made the following observation re-
garding the "mathematical aspect" of problem solving:

The mathematical aspect of problem-solving is
work with symbols. Once the relations expressed
in words are formulated in symbols, then thinking
takes place through the manipulation of these
symbolic representations of quantitative measures.
The teacher must see that her problems are suffi-
ciently comprehensive in terms of the mathematical
work to be done. Every learning element must re-
ceive attention, and the amount of attention must
be commensurate with the difficulty of the concepts
involved in the element (p. 354).

J. Allen Hickerson provided insight into a more
recent phenomenon found in the mathematics classroom.

Writing in the textbook, GuildingChildren's Arithmetic

87

Experignces: The Expepience-Language Approach to Numbers,

Hickerson (1952) included the following comment on problem

solving:

The term ppobiemgolving in arithmetic par-
lance has come to mean, unfortunately, not the
solving of one's own problems first hand, but the
solving of vicarious problems. The theory of such
problem solving is that if a child can read about
someone else's out-of-school problems and learn to
solve them while sitting in the classroom he pre-
pares himself for solving his own out-of-school
problems when and if he meets them (p. 8).

To the concept of problem solving, Hickerson attached the
significance of the term "word problems." In his language-
approach to teaching arithmetic, Hickerson (1952) provided

the following observation:

The newer arithmetic textbooks and workbooks
include word-problems that describe experiences
and situations many children encounter. They
are for the most part within the range of activity
of many children. In spite of this excellence,
however, the teacher who expects his children to
solve these word-problems should be aware of cer-
tain things.

Since solving a printed word-problem is pri-
marily a matter of being able to read, the teacher
must be sure that the child is ready to read the
particular word-problem. A child may be ready to
read one problem with meaning, but not another (p. 8).

In their revised textbook, Making Arithmetic
Meaningful, Leo J. Brueckner and Foster E. Grossnickle
(1953) expanded the material on "problem solving." In the
chapter on "The Scope of Problem Solving in Arithmetic,"
they attempted to get teachers to "see" problem solving
as an integral part of the mathematics curriculum. In
Opening the chapter on "problem solving," Brueckner and

Grossnickle made the following comments:

88

Instruction in an arithmetic program that
emphasizes meaning and understanding is based
largely on problem solving. Quantitative think-
ing is the basis of problem solving (p. 491).
As for "problem solving" playing a major role in the math-
ematics curriculum, Brueckner and Grossnickle (1953) added:

The teacher should not look upon problem

solving as a separate aspect of the work in
arithmetic but as an integral part of all phases
of the work.... The goal of the teacher should
be to lead him Ehe studenfl gradually to use
increasingly mature procedures that involve the
use of abstract symbols and formulas, and higher
levels of thinking (p. 491).

Regarding the diagnostic aspect of problem solving,
Brueckner and Grossnickle included a section on "Techniques
for Diagnosing Difficulties in Problem Solving." The two
authors suggested using textbooks and workbooks to perform
informal tests on the following items:

1. Telling what is to be found in a problem,

2. Telling what facts are given,

3. Naming the process to use in solving a problem,
4. Estimating answers of problems, and

5. Checking computations (p. 515).

In its Twenty-first Yearbook titled The Learning
of Mathematics: Its Thegpy and Practice, the National
Council of Teachers of Mathematics included a chapter on
"Problem-Solving in Mathematics." In writing the chapter,
Kenneth B. Henderson and Robert E. Pingry (1953) included
an "Analyses of Problem-Solving." As part of their in-
vestigation, the writers presented an "analysis of

reflective thinking" first described by John Dewey in 1933.

89

In observing the inadequacies of the Dewey analysis,
Henderson and Pingry (1953) identified the processes they
believed occurred "regularly" during problem solving.
The two authors noted a three-step process which had
been introduced by Donald Johnson in 1944. The Johnson
model included the following steps: "(a) 'Orientation
to the problem': (b) 'Producing relevant material, an
elaborative function': and (c) 'Judging, a critical
function'" (p. 236). The authors reworked the Johnson
model and provided the following three steps in their
analysis of problem solving:

1. Orientation to the problem.
2. Producing relevant thought material.
3. Testing hypotheses (p. 237).

Henderson and Pingry, in calling problem solving
"a very complex process" (p. 247). made suggestions to
teachers on how they could help students improve skill in
solving problems. The authors suggested a greater under-
standing and inclusion of the psychological aspects of
learning. They wrote:

...psychologists find it difficult to distinguish
between problem-solving and learning generally.
From this point of view there is a sense in which
this entire Yearbook, rather than just this chapter
concerns problem-solving. MOtivation, attitudes,
transfer of training, drill, concept formation,
language, and logic, are all aspects of problem-
solving. A teacher who is seeking to improve
problem-solving ability must necessarily give
proper emphasis to each of these aspects. A
program of instruction, however, that involves
these necessary phases of learning is not suffi-
cient. It is important that specific experiences

90

designed to foster problem-solving abilities
be provided in the program of instruction

In discussing the teacher's responsibility in
teaChing problem solving, Henderson and Pingry (1953)
made the following observation:

The teacher's task is two-fold concerning problem-
solving. One aspect is that of helping the students
with the problems at hand. The second aspect is
that of helping students understand the problem-
solving processes per se (p. 249).
The two authors, in writing about the inclusion of problem
solving in the mathematics curriculum, cited the following
concern:

Of course, before the teacher can teach
problem-solving the teacher must understand
problem-solving. Mathematics teachers need to
be students of problem-solving processes as
well as students of mathematics. There is
considerable evidence that many mathematics
teachers do not understand what problem-solving
is: or if they know, they do not have it as an
objective of instruction (p. 249).

In providing suggestions for improvement in the
teaching of problem solving, Henderson and Pingry (1953)
noted the importance of the organization of the body of
knowledge: the emphasis on reading skill (especially with
verbal problems): the recognition of individual differences:
the encouragement of verbal responses: a better utilization
of diagramming techniques, dramatization and modeling: and
the inclusion of open-ended questions to improve problem-

solving skill in the classroom (pp. 248-255). The authors

suggested the use of questions such as those provided by

91

G. Polya in How to Solve It. In conclusion, they
elaborated on the problem-solving abilities of both
students and teachers. In summary, they presented the
following discussion:

Mathematics teachers believe that the ability
of a student to solve mathematics problems is de-
pendent upon how deep his understanding of mathe-
matics is. The student's ability to solve problems
also depends upon the student's understandings.
attitudes, and skills concerning problem-solving
processes. This implies that the teacher of
mathematics must understand mathematics as well
as the psychological processes of problem-solving
to be of help to the student. To provide suCh an
understanding of the latter, this chapter attempted
to set forth a conceptual framework of problem-
solving and point out some of the implications of
this for classroom procedure. The hope is that
this will afford a teacher fruitful hypotheses
concerning his own efforts to teach students the
set of understandings, attitudes and skills con-
ducive to solving problems (p. 268).

The chapter on problem solving was aimed at the classroom
teacher. Henderson and Pingry provided information on
problem solving in the hopes of changing the problem-
solving behavior of teachers. Hopefully, this change in
attitude would be passed on to children in the mathematics
classroom. The result would be the development of more

proficient problem solvers.

Summary of Pepapd II: 1929-1953
Paul R. Hanna's dissertation on arithmetic problem
solving signaled the advance of problem-solving research,
theories, and ideas during the period, 1929-1953. Hanna

summarized work that had been accomplished on problem

92

solving and provided a comparison of three distinct problem-
solving "models": the Dependencies Model, the Conventional
Model, and the Individual Model. In addition to focusing
on the problem-solving processes described in professional
literature and problem-solving activities used in courses

of study, Hanna studied the problem-solving models found in
children's textbooks. As this period progressed, sugges-
tions were made for more analytical approaches to problem
solving and for a more systematized arrangement of problem
material.

The emphasis on "meaningful arithmetic" during the
late 19305 and early 19405, the emphasis on psychological
aspects of problem solving, the National Society for the
Study of Education emphasis on problem solving in 1942, and
the National Council of Teachers of Mathematics emphasis on
problem solving in the early 19505, provided the peak peri-
ods for this time frame. As a result of some of the work
started during the late 19205, an attempt to emphasize prob-
lem solving in algebra was important during this period.

Harry Grove Wheat, William A. Brownell, and Leo J.
Brueckner provided much of the stimulus for the development
of problem solving in mathematics during the 19305. Re-
search revealed that the three major causal factors involved
in problem-solving difficulties were: failure to accurately

or completely read the problems and problem statements, lack

93

of ability to perform accurate computations, and the inabil-
ity to read technical (mathematical) language.

A number of problem-solving models similar to the
"conventional model" originated during the 19305. Medels
presented by Fletcher Durell and Paul Klapper were espe-
cially important in establishing a foundation for the
development of later models. The model by Klapper (1934)
included the basic components of grasping and ascertaining
the problem, planning, carrying out the plan, and checking
the final answer. These steps compare quite favorably with
the steps found in the model by George Polya (1945)--
understanding the problem, devising the plan, carrying out
the plan, and looking back (or examining). Partially be-
cause of its extensive questioning, the Polya MOdel
continued to be important to problem solving in mathematics.

As had been the case in the late 19205, finding a
solution to the "problem" of problem solving became a domi-
nant pursuit in the mid-19405. Toward the end of the
decade, there was a move to change some of the terminology
involved in problem solving. A number of textbooks and
educators referred to Problems as "examples," and emphasis
was put on "exercises" in textbooks. Toward the end of the
19405 and at the beginning of the 19505, there was a renewed
interest in the "scientific-method" approach to teaching
problem solving. Along with the "meaningful" approach to

teaching arithmetic, the emphasis on "an orderly sequence

94

of steps" and the ideas of "experience" and "critical think-
ing" were seen as important parts of the problem-solving
process.

Entering the 19505, mathematical problem solving was
still viewed as a complex process. Emphasis on the theory
of problem solving, and providing teachers with problem-
solving methods and procedures, were considered necessary
parts of the learning process. Finally, it was believed
that in order for teachers to teach problem solving, they
must be proficient in problem-solving techniques themselves.
It is noted that a new influence began to "appear" during
the period 1929-1953 with "calculating machines" being rec-
ognized as devices that would someday revolutionize the way
mathematics--and mathematical problem solving--would be
taught. The development of problem solving for Period III,
1954-1983, would see extensive use of various "calculating
machines" such as counting frames, calculators, and comput-

ers in mathematics education at all levels of instruction.

CHAPTER IV

A REVIEW OF MATHEMATICAL PROBLEM SOLVING
PERIOD III: 1954-1983

In February, 1954, the teaching of arithmetic in
the elementary school was accorded additional recognition
when the National Council of TeaChers of Mathematics pub-
lished the first volume of the Arithmetic Teacher. In
writing about "The Revolution in Arithmetic," William A.
Brownell reviewed some of the changes that had taken place
in arithmetic during the 19005. In noting the changes
that had occurred over a fifty-year period, he discussed
the "Formal Disciplines" of mathematics in the early 19005,
the "functional" approach that began around 1910, the
"social aim" that prospered during the late teens and early
19205, the move from "product" learning to "process"
learning in the mid-19205, the "meaningful" approach of
the 19305 and 19405, and the "child development" approach
that evolved from the "child study" movement of the early-
and mid-19205 (Brownell, 1954, pp. 1-5). In summary,
Brownell (1954) suggested that the so-called "revolution"
was actually change that had evolved over an extensive

period of time. He made the following observation:

95

96

Actually of course the process of change has been
one of evolution, for each modification has emerged
from a given status and has led to the next modi-
fication. The steadying and stabilizing influence
in this period of evolution has been what I have
called the search for a functional curriculum

(p. 5 .

In finalizing his comments, Brownell (1954) pointed out
that mathematics, in order to be effective, should have a
mathematical aim and a social aim. He concluded his
article with the following comment:

To be intelligent in quantitative situations
Children must see sense in the arithmetic they
learn. Hence, instruction must be meaningful
and must be organized around the ideas and re-
lations inherent in arithmetic as mathematics.
But they must also have experiences in using the
arithmetic they learn in ways that are signifi-
cant to them at the time of learning, and this
requirement makes it necessary to build arith-
metic into the structure of living itself (p. 5).

Although he was discussing mathematics in general, the
same two aims--mathematical and societal--could well be
applied to the teaching of problem solving in arithmetic
during the 19505.

In the same volume of the Agitpmegic Teagpap,
Foster E. Grossnickle (1954) discussed the "Dilemmas
Confronting the Teachers of Arithmetic" and observed:

The teaCher of arithmetic today frequently
is caught between opposing forces in the fields
of the curriculum and in the psyChology of learn-
ing. The dilemma resulting from the curriculum
is caused by the need for meeting standards of
aChievement in arithmetic so as to satisfy the
demands of the business world on one hand and the
operation of the policy of continuous promotion
on the other. Frequently, these two forces work
in opposite directions. The dilemma resulting
from the psychology of learning is caused by the

97

time needed for mastery of a topic in meaningful
learning as compared to the time needed for manipu-
lation of symbols by rote learning (p. 14).

Grossnickle (1954) concluded his comments with the follow-
ing observation:

The acceptance of meaningful arithmetic is in
jeopardy as long as a teacher knows that an eval-
uation of her instruction will be measured in
terms of standards set for rote learning (p. 15).

Such an evaluation, measured in terms of rote learning,
might have an effect upon the development of problem
solving in mathematics.

In the second issue of the Agaphmetic Teacheg,
Charlotte Junge reviewed the mathematics curriculum of
1954 and highlighted two areas of arithmetic. First,
Junge (1954) noted that the development of mental skills
provided "a keen number sense, a healthy self-reliance,
and the power to think with numbers" (p. 5). Second, she
noted the attention given to the "development of abilities
in problem-solving," with the following comment:

verbal problems are included as a part of the work
at all grade levels--even the first, and effective
problem-solving helps are systematically provided.
These problem-solving activities aim at helping

the child understand and see the relationships be-
tween what he wants to find out and the known facts.
They seek to help the Child develop his own way of
solving problems and to leave him with a method of
attack on quantitative situations. Consequently,
modern programs in arithmetic encourage the use of
problems to introduce new concepts, for practice

on concepts which have been developed and for eval-
uation.and testing of concepts learned. Problem-,
solving is assuming a role of major importance in
programs based on the development of meanings (p. 5).

98

In discussing the "link" between computation and

problem solving, Harry Grove Wheat (1954), criticized the

teaching of mathematics during the 19505 with the follow-

ing comment:

Computation and problem solving have always been
widely-separated activities in the arithmetic of
the school. We do very little in present-day
teaching to get our pupils able to recognize them
as a single thinking procedure in different dresses
(p. 5 .

In the textbook, Guilding Arithmetic Learning,

John R. Clark and Laura K. Eads (1954) noted that "think-

ing" and "concept development" were integral parts in the

problem-solving process. In stating that problem-solving

situations could be found "at the experience level, at the

materials level, at the generalizing level, at the compu-

tation level" (pp. 258-259), Clark and Eads (1954) included

the following eight steps for "determining readiness in

problem solving" for individuals and for groups of children:

1.
2.

They can find the solution readily.

They can change the numbers in the problem to
other reasonable numbers.

They are not distracted by extraneous data.

They can change the items in the problem to
other reasonable items.

Often they can solve the problem in more than
one way.

They can devise other problems using the same
situation.

They can talk about the problem, tell things
about it that were not stated, invent circum-
stances that created the problem, etc.

They can explain why they used the method they
used in solving the problem (PP. 264-265).

99

In writing Practical Plans for Teaching Arithmetic
in 1954, Ruth H. Drewes, Ada S. Mermer, and Winifred P.

von Boenigk discussed "training" in problem analysis as a
part of the development of problem-solving ability,
especially in solving verbal problems. In developing
skill in problem analysis, the following series of
questions and directions were suggested:

1. Read the problem silently (Teacher supplies
needed words.).
2. What does the problem ask?

3. What facts must we know in order to answer
the question?

What process shall we use?

Estimate the answer.

WOrk the problem.
Label the problem.
Check the problem.
Compare the answer with the estimate (p. 99).

(oooqmmb

In 1954, Herbert F. Spitzer, in the textbook zap
TpaChing of Arithmetic, made the following observation
regarding the treatment of problem solving in the elementary
classroom:

The treatment of problem-solving in many books
on the teaChing of arithmetic includes extensive
material on the kinds of subject matter with whiCh
verbal problems should deal. It is generally
recommended that problems should deal with things
and conditions that are within the experience of
the elementary school child--that problems should
be real. Where problems are used for illustrative
purposes, as in initial instruction, this matter
of having the subject matter of problems within
the experience of the children is important
(Pp. 183-184).

100

In discussing the inclusion of problem solving
activities and processes in materials used by children in
the classroom, Spitzer (1954) summarized some of the
suggestions on problem solving made by publishers. He

wrote:

The manuals and advertising which accompany
pupils' textbooks make extensive claims regarding
the program of teaChing problem-solving included
in the books. The procedures actually used in the
books are not always as impressive as the state-
ments made about them. Textbooks emphasize most
often the technique of selecting the fundamental
process needed to solve the problem. Other pro-
cedures frequently found in the books are these:
(1) systematic analysis of problems (stating what
is given, what is to be found, etc.)3 (2) finding
hidden questions in problems: (3) estimating the
answer to problems: (4) solving problems contain-
ing superfluous numbers: (5) dealing with problem
situations which lack sufficient data: (6) admoni-
tion to read carefully: and (7) use of sets of
problems based on one social scene, such as a State
Fair, the Grocery Store, or Ranch Life (p. 188).

Spitzer (1954) called for a more elaborate plan
for the development of skill in problem solving. In
addition to the procedures noted in the previous citation,
he presented the following activities and ideas for the
development of problem-solving ability:

1. The non-pencil-and-paper, or oral solving of
problems.

2. The use of diagrams or drawings.

3. Writing the number question.

4. Pupil formulation of problems.

5. Intensive study of number operations.

6. Class solution of difficult quantitative
problems encountered in other school work.

101

The work or study habit whereby the pupil
asks himself questions about the problem.

Solving the same problem by several teChniques.
Other procedures for improving problem-solving
ability (pp. 189-199).

In a texbook titled The Diaggosis and Treatment

pf Learning Difficulties, Leo J. Brueckner and Guy L.

Bond (1955) provided insight into "Diagnosis in Problem-

Solving and Quantitative Thinking." The authors suggested

several types of tests "to measure the ability of pupils

to solve problems and to think quantitatively" (p. 233).

In 1955, Brueckner and Bond noted that in diagnosing

problem solving, the measures found in computational

ability and reading ability should supplement the measures

found in the following five "tests":

1.

the ability to solve verbal problems, one of
the major skills measured by available standard
tests:

the ability to read graphs, charts, tables, and
similar materials and to answer questions based
on them;

knowledge of vocabulary;

knowledge about social applications of arith-
metic fundamental in problem-solving: and

quantitative understandings (p. 233).

In citing research, Brueckner and Bond (1955)

included a list of nine areas in which good problem solvers

were found to be superior to poor problem solvers. The

categories were:

1.
2.

Computational ability.

Ability to apply the sequence of steps involved
in problem-solving.

.b

(DmQOWU'I

102

Ability to estimate answers to verbal problems.

Range of information about social uses of arith-
metic.

Ability to read graphs, charts, tables.
Ability to see relations in number series.
General and nonverbal reasoning ability.
General reading level.

Level of mental ability (p. 290).

The authors recommended the following specific procedures

for treating deficiencies in problem solving:

1.
2.

11.
12.
13.

Making number operations meaningful.

Experience in using operations in social
situations.

Using objects to show the meaning of processes
used in solving problems.

Using manipulative materials to work out solu-
tions of problems.

visualizing solutions of problems.

Improving the quality of verbal problems.
Developing relationships among number processes.
Explaining reasons for using processes.
Identifying processes to use.

Problems without numbers.

Learning to sense relationships in equations.
Making the vocabulary of problems meaningful.
General program for improving problem-solving
(pp. 294-300).

E. H. Taylor and C. N. Mills, in 1955, classified

problem-solving difficulties into five distinct categories.

Beginning with reading, the categories were listed in the

following manner:

103

Difficulties with reading.

a. Difficult vocabulary.

b. Carelessness in reading.

c. Technical terms.

d. Unfamiliar units of measure.

e. Unfamiliar forms of statements or
questions.

Difficulties with numbers.
a. very large numbers.
b. very small numbers.

c. Kinds of numbers--integers, fractions,
decimals.

Difficulties with computation.
a. Errors in computation.

b. Inability to perform the necessary
computation.

Difficulties with facts.

a. Lack of knowledge of facts assumed to
be known.

b. Not enough facts given to make a problem.
c. Confusion caused by extraneous facts.
Difficulties with comprehension and analysis.

a. Inability to comprehend the meaning of
the problem.

b. Inability to analyze the problem into
separate conditions.

c. Inability to image the conditions of the
problem.

d. Inability to decide the necessary process
or processes to be performed (pp. 227-228).

The authors suggested the following "directions" in solving

problems:

1.
2.
3.

Read the problem carefully.
Decide what is to be found.
Decide what facts are given in the problem.

Decide what other facts are needed, and deter-
mine these facts.

104

. Determine the processes needed in the solution.
Estimate the result.

. Perform accurately the necessary computation.
Check the results (pp. 242-243).

(DQOWW

In addressing the question, "How do you teach
problem solving?," Howard F. Fehr (1955) made the following
suggestions:

Perhaps most significant are (1) always look at
the whole problem, the whole situation; (2) seek
the relationship of the parts to the whole, and
the whole to the part: (3) analyze, organize and
reorganize the relationships until what is known
is directly related to what is wanted, then in-
sight will occur. Thus problem solving demands
"ceaselessyattention to the building of clear,
well interrelated arithmetic concepts in all the
apeas of common experience" (pp. 30-31).

Regarding the solving of word problems, Fehr (1955)
noted the reliance upon "active thinking" in the problem-
solving process. "Estimation," "mental solutions," and
"the association of language with an operation" were cited
as three characteristics that needed greater attention
in the problem-solving activities associated with word
problems (p. 31).

In 1956, Shirley Stillinger Brewer utilized an
often-used concept in order to improve the problem-solving
abilities of her college-level students. In working
experiments with electricity, Brewer introduced the

"scientific method." The steps are:

1. Define problem
2. Research
3. Hypothesis

105

4. Experiment
5. Conclusion (P. 117).

Then, in applying the scientific method to solving word
problems, Brewer (1956) reworked the method and provided
the following five—step problem-solving model:

1. What do we want to know?

2. What do we know?

3. Best estimate.

4. WOrking the problem.

5. What did we find out? (p. 117)

At the conclusion of Brewer's comments in the Arithmetic
Teachag, the editor said:

EDITOR'S NOTE. Although the scientific
method or approach to problem solving may be very
old, each new generation should discover it. Mrs.
Brewer's pupils did this by simple transfer from
work in science. There are many opportunities
for "tieing together" arithmetic with other areas
of learning. Each of these ties tends to give
greater understanding and significance to each
of the areas involved. The most important element
in Mrs. Brewer's development is the role of the
pupils in thinking (p. 118).

In an article titled "Developing Facility in
Solving verbal Problems," Herbert F. Spitzer and Frances
Flournoy (1956) elaborated on the use of problem-solving
techniques utilized in the mathematics classroom with the
following comments:

The improvement of pupil achievement in verbal
problem solving is an important objective of most
upper grade arithmetic teachers. That this objec-
tive is not often reached with any degree of satis-
faction is evident to all students of arithmetic
teaching. It is also quite evident to students of
arithmetic teaching that, although there are many
problem-solving improvement procedures in use, the

106

most widely used procedure is that of just having
pupils work problems without specific directions

or suggestions. The other problem-solving improve-
ment procedures practically all suggest specific
steps for pupils to engage in (p. 177).

Spitzer and Flournoy (1956) criticized instruction
in problem solving. On a pessimistic note, they suggested,
in the following comment, a simple modification of existing

practices:

In view of the rather long time that instructors
have been concerned with problem solving, it is

very doubtful whether any one entirely new pro-

cedure of merit will turn up. Improvement will,
then, most likely be the result of modification

and refinement of plans now in use (p. 177).

In their conclusion, Spitzer and Flournoy provided two
recommendations for improving abilities in problem solving.
In citing "the sketchy problem-solving improvement program
offered by any one textbook, the lack of agreement on
procedures among textbooks, and the omission from text-
books of what appear to be promising procedures" (p. 182).
they recommended the following:
1. The typical textbook program for improving or
developing problem-solving ability has to be
supplemented by providing more experience with

the techniques recommended and by using prom-
ising teChniques not included in the textbook.

2. Students of arithmetic teaching need to make
studies to determine whether or not proposed
problem-solving improvement procedures actually
contribute to this ability (p. 182).

Harry Peeler (1956), in an article on "Teaching
verbal Problems in Arithmetic," noted that "a meaningful
approach to problem solving calls for problem solving

directed toward an outgrowth of generalizations,

107

understandings, and number facts--using these tools to

lead into other phases of problem solving" (p. 244).

Peeler elaborated on how the phases of problem solving

are considered to be an integral part of the arithmetic

program.

The following comments were made:

In a program of teaching arithmetic meaningfully,
problem solving is not one separate part of the
whole but is an element running through and inter-
twined with the whole. As concepts, skills, gen-
eralizations and understandings progress and grow
so does ability in problem solving grow, and as
problem solving grows so grow the skills, concepts
and understandings (p. 245).

In 1957, Leo J. Brueckner, in the textbook,

Improving the Agithmetic Program, cited the following

procedures for improving problem-solving ability:

Have the children solve many easy, interesting,
well-graded problems.

Seize opportunities that arise to have the
children apply numbers in social settings.

Give the slower learners direct guidance in
the reading and analysis of problems, tables,
graphs, maps, diagrams, and so on. The more
able learners can devise their own procedures.

Try to increase the child's understanding of
the meaning of the four number processes and
the situations under which they are operative.

Insist on the checking of computations per—
formed in problem-solving.

Exercises in the carefully directed reading of
problems will be of real value.

VOcabulary exercises to broaden and enrich
meanlngs of words and expreSSlon are desirable.

Have the children demonstrate problem situations
with objects, drawings, illustrations, and
diagrams to make them meaningful.

Be sure that the children are familiar with
systems and instruments of measurement.

10.

11.

108

Show the child the progress he is making by
the results of standard and informal tests
of reading.

Have the children try to suggest several ways
of solving a problem so as to give extra prac-
tice in problem-solving and to improve quanti-
tative thinking (pp. 78-80).

Leo J. Brueckner, Foster E. Grossnickle, and John

Reckzeh (1957) included a section on the "Logical Pattern

of Problem Solving" in the textbook titled Developing

Mathematical Understandings in the Upper Grades. A five-

step plan, called a "logical approach," was outlined in

the following manner:

1.
2.
3.

Find what the problem question is.
Then find what facts the problem gives.

Try to think of ways to find the answer to the
question asked, or seardh for a familiar pattern
or model in the problem.

Do the necessary computation.
Check the answer to see if it is sensible
(p. 319).

In contrast to the above steps, the three authors

suggested the utilization of the following steps for

children who lack sufficient background ability in problem

solving and for children who are slow learners:

1.
2.

3.

Kinney,

Find what the problem question is.

Identify the information and numbers given in
the situation the problem presents.

Show how the answer to the question asked de-
pends upon what is given (p. 320).

John L. Marks, C. Richard Purdy, and Lucien B.

in the textbook TeachingArithmetic for Understand-

ing (1958), noted that children who follow a systematic

109

plan were more liable to succeed in solving word problems

than children who did not follow a plan. They observed

that "no one systematic plan is superior" (p. 329) and

included the following problem-solving procedure:

4.
5.
6.

Read the problem and decide: What does the
problem ask me to find? What does the problem
give me to use?

Make a drawing, if needed.

Select the process or processes. Do I add?
Subtract? Multiply? Divide? In what order?
With what numbers?

Estimate the answer.
Compute and check the computations.
Compare the answer to the estimate (pp. 328-329).

In 1958, Robert H. Koenker authored an article

titled "Twenty Methods for Improving Problem Solving."

He included the following suggestions which had been

"proven to be of value by research and/or clasroom prac-

tice" (p. 74):

1.
2.
3.
4.
5.
6

Use the whole method of attacking problems.
Estimating answers to problems before solving.
Diagramming problems.

Dramatizing problems.

Orally solving problems.

Encourage children to work problems using
different methods.

Differentiating problems for the various
ability levels.

Making up problems.

An understanding of arithmetic is prerequisite
to problem solving.

Using concrete objects and devices in solving
problems.

110

11. Going over assigned problems with children.
12. Checking or proving answers.

13. working problems without numbers.

14. Solving problems with irrelevant facts.

15. Finding missing facts in problems.

16. Finishing incomplete problems.

17. Stressing careful reading of problems.

18. Developing an understanding of arithmetic
vocabulary.

l9. Stressing neatness of work.
20. Mixing problems of different types (pp. 74-77).

In a critical look at "Twentieth Century Mathe-
matics for the Elementary School," H. van Engen highlighted
some of the difficulties found in the teaching of arith-
metic. Buoyed by the launching of the Russian satellites,
Sputnik I and Sputnik II, the comments by Van Engen (1959)
were part of a trend criticizing the teaching and learning
practices in mathematics and science. Regarding problem
solving, he made the following notation:

The schools have not been successful in de-

vising a sensible approach to problem-solving.
This audience is all too familiar with the various
proposals for improving the problem-solving ability
of the elementary school pupil. In spite of all
the proposals and the research, it is probably not
too far amiss to summarize the results of present-
day research by the single statement: The best
way to teach children how to solve problems is to
give them lots of problems to solve. Certainly
a fresh approach to problem solving is needed
(p. 74).
van Engen (1959) recommended a more mathematical approach
to problem solving. In calling the procedure "a mathema-
tician's approach to problems in miniature" (p. 75), he

provided the following observation:

111

One first searches for the fundamental structure
of a problem situation; then he finds the appro-
priate symbols to express this structure. Once
the problem has been structured, a knowledge of
previous problems and problem—solving techniques
can be applied. Certainly, no "cue" method or
mere admonitions to THINK hold the mathematical
power that the search for the structure of the
physical situation can command. The failure of
the older methods over the past years should be
reason enough to banish them from the classroom
and search for methods with more mathematical
power (p. 75).

The Twenty-fifth Yearbook sponsored by the National
Council of Teachers of Mathematics included a short section
on "The Psychology of Problem Solving:' Embedded in a
Chapter on "Reading in Arithmetic," David H. Russell
(1960) discussed and characterized "problem solving" as
"a systematic and logical process containing from three to
nine steps" (p. 216). The author noted that most of the

problem-solving procedures included suCh activities as

" ettin to understand the roblem, search, suggesting
solutions and eliminating sources of error" (p. 216). In

summarizing the research findings into what problem

solving entails, he provided the following commentary:

The researCh suggests that problem solving is not
a unitary factor, best described by one term suCh
as reasoning, but rather a complex of different
abilities. While the specifics are not always
clear, the essential parts of problem solving seem
to be an orienting function, an elaborative and
analytical function, and a critical function. The
problem-solving process varies with the nature of
the form of stating the problem, the methods of
attack known by the solver, the personal charac-
teristics of the solver, and the total situation
in which the problem is presented (p. 216).

112

In 1961, Wilbur H. Dutton and L. J. Adams included
a six-step problem-solving model in Arithmetic for Teacherg.
The steps in the model had a dual purpose-~a set of rules
to be used in the solving of arithmetic problems in both
textbook and non-textbook situations. The six "rules" in
the problem-solving technique were stated in the following
manner:

1. Read the problem.
2. Decide what is given.
3. Decide what is to be found.

4. Decide which operations are necessary to take
what is given and use it to find what is to be
found.

5. Solve the problem.
6. Check the result (p. 178).

In commenting on the use of the above-mentioned problem-
solving model, the two writers solicited the inclusion of
"application" as a substitute for the so-called "realistic,
practical" problems that had permeated mathematics instruc-
tion. Dutton and Adams (1961) made the following observa-
tion:

Problems do not have to be realistic to be use-
ful. So-called fantasy problems may generate in-
terest more readily than realistic, practical
problems, and lead to genuine desire on the part
of the student to learn more about arithmetic and
mathematical principles. Besides, the use of the
word "practical" can lead to argument: what is
practical for one may not be practical for another.
It is not necessary for a problem to be of immediate
value to be justifiable. Instead of describing
problems as practical it may be better to describe
them as applications; that is, many problems are
applications in the sense that they illustrate how
arithmetic may be applied in situations that are

113

practical for some people and not necessarily
practical for the learner at the time that he
studies them. In this sense many problems
illustrate how arithmetic can be used (p. 180).

"Trends and issues" in elementary school arithmetic
provided the theme for the book of readings titled
Improving Mathematics Programs: Trends and Issues in the
Elementary School edited by M. VEre Devault (1961). In
addition to a number of citations on solving problems
appearing throughout the book, a Chapter written by Lowry
W. Harding (1961) was devoted to "Productive Approaches to
Problem Solving." He noted in the introduction that "the
essence of problem solving is search and discovery"

(p. 194). Harding then followed the above comment with:
And discovery is creative-~it requires originality.
Thus, the teacher faces both temptation and oppor-
tunity. If he succumbs to the temptation merely to
drill his pupils on routine operations he deadens
their interest, inhibits their creativity and hin—
ders their intellectual development. But, if he
stimulates the interest of pupils by considering
what are problems to them and challenges curiosity
and effort by suggesting other questions appropriate
to their ability and knowledge he may give them a

desire for, and some competence in independent
thinking (p. 194).

In a textbook titled The Teaching of Arithmetig,
James Robert Overman (1961) described problem solving as a
"thought-process." In a chapter on "Developing Patterns
and Habits of Thinking Useful in Problem-Solving," he
included a series of "useful thought patterns in solving
problems." The following four-phase problem—solving

process was cited:

114

1. Getting a clear understanding of the conditions
of the problem.

2. Planning the solution.
3. Carrying out the plan.
4. Checking the result obtained (pp. 381-382).

Overman noted that the four phases of thinking do not
always "occur separately" nor do they always occur "in a
fixed order" (p. 382).
Writing about "those problem-solving perlexities,"
Cleata B. Thorpe (1961) made the following introductory
comment about teachers and their dealing with the concept
of problem solving:
We teachers are in large measure responsible

for problem-solving being the obstacle that it is

to many a pupil in elementary schools. In the

first place, we toss the terms "problem" and

"problem-solving" about quite indiscriminately.

we seem to have no clear and definite concepts

for those terms in our own minds (p. 152).
After discussing the teachers' vague concepts of problem
solving, she included recommendations for improved instruc-
tion in solving problems. In noting that "the use of
reflective teaching in problem-solving lessons in arith-
metic is most effective when certain conditions prevail"
(p. 155) the author included the following provisions for

the betterment of problem solving:

1. The atmosphere of the classroom must be con-
ducive to pupils feeling accepted and at ease.

2. The attitude of the class must be cooperative
and receptive to the ideas and opinions of
others.

3. Suitable problem material must be provided.

115

4. There must be time for unhurried thought and
relaxed activity.

5. Teachers and pupils remember that concepts do
not come as single flashes of insight; they
unfold slowly, step by step, even with guidance.

6. There is recognition of the fact that there is
usually more than one correct and acceptable
solution to every problem, though some may be
too "round-about" for practicality.

7. Any correct solution is commended (pp. 155-156).
Thorpe also called upon teachers to do their part in im-
proving the teaChing of problem solving in the elementary
school. She wrote the following summary in 1961:

The values of such learning processes in the
teaching of problem-solving in arithmetic have
come to be more widely recognized than in the past,
and there is every indication that they will play
an increasingly important part in elementary arith-
metic-learning in the immediate future. By way of
summary, we may conclude that there are some very
promising possibilities for vanquishing the problem-
solving perplexities--if teachers will do their
part (p. 156).

In 1962, George Polya produced volume I of

Mathematical Discovery on Understanding, Learning, and
Teaching Problem Solving. In the Preface, he commented
upon the importance and applicability of teaChing problem
solving in mathematics as follows:

What is know-how in mathematics? The ability
to solve problems--not merely routine problems but
problems requiring some degree of independence,
judgment, originality, creativity. Therefore, the
first and foremost duty of the high school in teaCh-
ing mathematics is to emphasize methodical work in

problem solving (p. viii).
Charles F. Howard and Enoch Dumas (1963) included

a chapter titled "Developing Problem-Solving Ability" in

116

the textbook Basic_§£ocedure§4in Teaching_Arithmetic.

In the discussion, they noted that "traditionally, the
arithmetic program in the elementary school has relied
heavily upon verbal problems, which are referred to often
as 'word problems' or 'story problems,‘ to develop the
pupils' problem-solving abilities" (p. 339). The authors
suggested an understanding of the setting of a verbal
problem (including reading and vocabulary skills), the
identification of mathematical relationships in problem
situations, and improved computational procedures in an
effort to strengthen the problem—solving abilities of pupils
in arithmetic (PP. 343-356).

In the fourth edition of Qigcovering Meanings in
Elementggy School Mathemapigg, Foster E. Grossnickle and
Leo J. Brueckner (1963) referred to "problem solving" as
"the highest level of quantitative thinking" (p. 301). In
making the above comment, they noted that "it is a primary
function of the arithmetic program to arrange experiences
that will develop in children the ability to 'think
through' problematic situations that they encounter and
to deal with them intelligently and skillfully" (p. 301).
In discussing the "scope of quantitative thinking," the two
authors connected aspects of quantitative thinking and
reflective thinking with problem solving in the comments

that follow:

117

Problem solving is the highest form of re-

flective thinking. A mathematics program...that
emphasizes meaning and understanding is based
largely on problem solving. Quantitative think-
ing is the basis of effective problem solving

p. 302 .

Grossnickle and Brueckner (1963) provided insight into the

development of problem-solving ability with the inclusion

of five "elements of a problematic situation" which were

described in the following manner:

1.
2.

A desired goal is to be attained.

There is a blocking of the path to be taken to
attain the goal.

Available habitual responses are not suitable
or adequate to attain the goal.

various possible solutions (hypotheses) are
proposed and tested.

A tentative conclusion is reached (p. 303).

In commenting on "Guidance in Problem Solving,"

Grossnickle and Brueckner (1963) included the following

list of factors that contributed to the successful solving

of problems:

1.
2.
3.

The problem was of concern to the children.
The problem was solvable.

The problem was clearly defined in terms that
each pupil understood.

The children gave suggestions as to possible
solutions and evaluated them.

The children worked out a tentative plan for
solving the problem and gathered, evaluated,
and organized the necessary data.

A variety of operations was used to obtain the
data.

The children were guided by the teacher in
planning the solution and in reaching conclu-
sions as the occasion demanded.

118

8. The children used the solution that they had

arrived at in dealing with the situation the

original problem presented and realized that

they had been successful (p. 305).

Foster E. Grossnickle (1964), in an article titled
"verbal Problem Solving," noted that the concept of "problem
solving" is taught to help the pupil discover a pattern
used in solving problems. The following six-step process

' was included to help develop a pattern for solving problems:

1. identification of the problem question,
2. recognition of the operation to use,

3. writing the mathematical sentence to express
the relationship between the numbers given,

4. finding the number which will make the
sentence true,

5. Checking the solution by evaluating the
equation, and

6. labeling the answer (p. 14).

In "A Look at Problem Solving in Elementary School
Mathematics," Kathryn V. Herlihy (1964) noted that "the
development of problem-solving ability may well be the
main objective of education" (p. 308). In the discussion
on problem solving, she suggested a plan that was closely
aligned with the problem-solving model first proposed by
G. Polya in the 1940s. The steps included devising a plan,
which Herlihy called "the main achievement in problem
solving" (p. 310). After the plan was devised, she sug—
gested knowledge of computational skills, a process for
Checking the steps, and the final solution to better develop

skill in problem solving (p. 310).

119

J. Houston Banks, in 1964, discussed the "objec-
tive" of problem solving in the textbook Learning and

Teaching Arithmetic with the following comment:

Skill in problem solving is not the sole ob-
jective of arithmetic instruction. Nor is problem
solving a separate topic of arithmetic. It should
be a vital part of all phases of the subject.
Meaningful problems should serve to introduce each
new arithmetical topic. The problem should estab-
lish a need for the new skill. For example,
problems most efficiently solved by division should
be used as a preliminary to learning the division
algorithm. They enable the pupil to appreciate the
usefulness of the process (pp. 405-406).

In elaborating on the "Improvement of Problem-Solving
Ability," Banks (1964) included the "Analysis Mbdel" which
he noted was "held in high favor by many arithmetic text-
book writers" (p. 414). The "Analysis MOdel" consisted of

the following formal steps:

Read the problem to determine (a) what is given,
and (b) what is required. (c) Determine from the
relationships between the quantities, given and
required, what operations are necessary. (d) Esti-
mate the answer. (e) Solve by performing the
operations in (c). (f) Check the answer (p. 414).

Wilbur H. Dutton (1964), in Evaluating Pupils'

Understanding of Arithmetic, provided the following concern
regarding the change that had been taking place in mathe-

matics instruction during the 1960s:

No other educational problem is receiving as
much attention as the new mathematics curricula.
we are, in fact, in the midst of a revolution whiCh
began after WOrld War II and which is assuming
amazing proportions today. Persistent problems
relating to the kind and amount of mathematics to
be taught in the elementary school have been
acCentuated. Problems pertaining to the teaching

120

of arithmetic have been created by the introduction
of new mathematical content and recent interpreta-
tions of the meaning theory. Because of rapid
developments within the field of mathematics and
the rapidity of change within our culture, it is
not clear what the content or methods will be a
few years from now. Thus the need for pupils and
teaChers to learn mathematical concepts thoroughly
and to be able to adjust to additional changes in
application and methods in the future seems para-
mount (p. 1).

In discussing the "new math," he made the following
comments regarding Change in elementary school arithmetic
instruction:

In methodology, emphasis is placed upon dis-
covery and intuitive thinking. The primary aim
of all teaChing of the new mathematics is to help
Children understand the fundamental structure of
mathematics and methods of mathematical reasoning

p. 13 .

Enoch Dumas, in writing a section on "Solving the
Problem of Problem Solving" in Updating Mathematics (edited
by Francis J. Mueller), described a multi-step problem-
solving process aimed at the improvement of solving word
problems. Dumas (1964) provided the following procedure:
1. Recognize each word for what it is and attach

the meaning intended by the author.

2. See the relationships among parts of a sentence
and among a group of sentences.

3. Understand the social setting.

4. Identify the mathematical relationships and the
needed computations.

5. Carry out the necessary computation.
6. Label the answer.
7. verify the correctness of his answer (pp. 179-180).

121

In the same book, Gerald W. Brown (1964) described
a seven-step problem-solving sequence in a discussion on
"Mental Arithmetic in the Problem-Solving Process." In
citing the sequence of steps taken from a number of text-
book series, he noted the following procedure:

1. Be certain you understand the problem.

2. Think what you need to know to solve the
problem.

Look for a hidden question.

Decide what computations to make.
Estimate or decide on a reasonable answer.
Perform the necessary computations.

40301.5(»

Check your final answer by comparing it to
your estimate (p. 195).

Brown advocated plenty of practice in getting children to
understand the above process of problem solving.

In 1966, Leslie A. Dwight included a chapter on
problem solving in the textbook, Modern Mathematics for
the Elementary Teacher. The author included the "con-
ventional" use of problem solving which was described in
the following manner:

...problem solving refers to a set of statements,
oral or written, which gives information, usually
related to everyday life situations, involving
quantitative data and which implies finding a
quantitative answer without indicating how the
quantitative answer is to be obtained (p. 470).

The conventional use of problem solving implies
work with word problems, story problems, or verbal prob-

lems. Dwight (1966) provided the following seven-step

procedure for solving verbal problems:

122

1. Read the problem to get an over-all view of the
whole problem.

2. Reread the problem and identify:
a. The problem question--what is to be found
b. The data relevant to the problem question.

3. Formulate, in words, a sentence (or sentences)
expressing the relation (or relations) of the
known and unknown numbers implied by the pat-
tern or action described in the verbal problem.

4. Write the word sentence (or sentences) in sym-
bolic (horizontal) form in which a symbol (or
symbols) is used to represent the unknown
number (or numbers).

5. Find the set of numbers that makes the mathe-
matical sentence (or sentences) of (4) a true
statement (or true statements).

6. Check the solution obtained in (5).
7. State a word sentence, using the solution, to
answer the problem question (p. 471).

In writing Building Mathematical Compgtence in the
Elementary School, Peter Lincoln Spencer and Marguerite
Brydegaard (1966) introduced the chapter on problem solving
with the following comments:

In a very real sense, the process of mathe-
maticking is one of problem sensing and problem
solving. It seems axiomatic that problem-solving
behavior occurs only when one is confronted with a
problem that he wills to solve. Hence, the first
concern with the development of problem-solving
abilities is that of leading the student to sense
a problem and to desire to find a way to cope with
it (p. 349).

In a discussion on "Developing Competency with
Problem Solving," they noted the separate procedures of
Leslie Beatty and E. H. C. Hildebrandt. In describing the

Beatty Model, Spencer and Brydegaard (1966) included the

123

"phases" incorporating procedures and materials that had

been utilized in a 1959 project involving 2000 children.

The model was described in the following manner:

\lO‘U’lwal-J

8.

Identifying "What is the question?",

Analysis of the problems independent of number,
Children writing their own problems,
Estimating and judging problems,

Graphic structuring,

Labeling the answer to a problem,

Identifying "On what does the answer depend?",
and

Mental arithmetic (p. 353).

The Hildebrandt Model was described as follows:

CDQOWUl-waH

Presentation,

Attention,

Observation and Exploration,
Classification,

Further Exploration,

Formulation,

Generalization, and

verification and Application (p. 354).

In 1966, Robert M. Gagne discussed the idea of

"Discovery in Problem Solving" in Learning by Discovery:

A Critical Appraisal, edited by Lee S. Shulman and Evan R.

Keislar.

He made the following comment:

A still more complex kind of learning situation

within which it is appropriate to suppose that dis-
covery may occur is called problem solving. In
order to define this kind of learning, it is neces-
sary to distinguish two things. First is the fact
that problem solving typically requires the learner
to acquire what may be called a higher-order prin-
ciple, formed by putting together two or more
simpler principles.

124

The second characteristic of the problem-
solving situation is a matter of method, rather
than content. Problem-solving situations are
usually designed to re uire discovery on the
part of the learner. This is an inevitable part
of their makeup; otherwise, they would probably
not be called problem-solving (p. 147).

Louis S. Cohen and David C. Johnson (1967) sum-
marized some of the behaviors found in "problem solving"
in the A£;thmetig Teacher. They provided a list of
"behaviors that might be exhibited by the problem solver."
The list included: "observing, exploring, decision making,
organizing, recognizing, remembering, supplementing, re-
grouping, isolating, combining, diagramming, guessing,
classifying, formulating, generalizing, verifying, and
applying" (p. 261). Furthermore, the authors noted "the
ability to translate a given situation into mathematical
symbolism is considered to be the mogt uggful tool in
problem solving" (p. 262).

In the textbook Multiple Methods of Teaching
Mgthematics in the Elementggy.Sgpggl (1968), Charles H.
D'Augustine defined problem solving as "the process of
reorganizing concepts and skills into a new pattern of
application that opens a path to a goal. This is in con-
trast to the application of a habitual pattern to reaCh a
previously attained goal" (p. 25). In the chapter on

"problem solving," he cited two factors which the teacher

had the responsibility for "nurturing." The two factors

125

were "the concepts and skills that a child brings to a
problem-solving situation and his repertoire of previously
solved problems" (p. 26).

The October, 1969, issue of Review of Educationgl
Research (RER) provided a review of research in science
and mathematics. In summarizing the research on "Problem
Solving in Mathematics," Jeremy Kilpatrick (1969) intro-
duced the section by noting: "The preeminence of increased
problem-solving ability as a goal of mathematics instruc-
tion has long been admitted; but like the weather, problem
solving has been more talked about than predicted, con-
trolled, or understood" (p. 523). In discussing the re-
search on problem solving completed during the 1960s, he
noted that most of the research in mathematics education
had been completed by doctoral students working on disser-
tations. The following comment was made:

Problem solving is not now being investigated

systematically by mathematics educators. Few
studies build on previous research: few studies
have an explicit theoretical rationale. There are
signs, however, that some mathematics educators

are beginning to borrow ideas and techniques from
recent psychological studies of higher cognitive
processes. As more educational researchers appear
who are trained both in mathematics and in psychol-
ogy, research on problem solving in mathematics
may attain a direction and cohesiveness it current-
ly lacks (p. 523).

In 1969, Ruth S. Jacobson provided "A Structured
Method for Arithmetic Problem Solving in Special Education."

The following "Guide for Problem Solving" was included to

help "solve the unknown":

126

1. Read the problem.

2. What is the action? Is it combining or
separating equal or unequal groups?
(In a comparison problem, omit Step 2.)

3. What is the operation?
4. Show the equation using frames

OXA=CJ

5. Fill in the knowns.
6. Compute (p. 25).

The focus of the lesson on problem solving included in

the discussion was twofold: "One, to give the child who

has had difficulty in solving word problems a method of

attack, and two, to give the classroom teacher some ideas

so that he may develop new lessons for the future" (p. 27).
In 1969, J. D. Williams considered three "deter-

minants" in the success of solving problems. These were:

1. Factors in the problem-solving situation.

2. Factors in the problem-solver's previous
training.

3. Characteristics of the problem-solver (p. 261).
In developing abilities to solve problems, Williams pro-
vided the following list of "general advice" in suggesting
an eight-step process that had originally been cited in
1957 by A. N. Frandsen:

1. Determine what is wanted.
2. Find which of the given facts are relevant.

3. State in a single sentence what is wanted as
a function of the data given.

4. Restate this in arithmetical language.

5. Try to recognise (sic) this statement as one
of the standard operations used in arithmetic,
and plan the solution.

127

6. Estimate an answer.
7. Make necessary computations.
8. Check the solution. (a) Does the answer

approximate to the estimated answer?

(b) Perform any possible arithmetical check

(pp. 274-275).
The author noted the importance of "readiness" in the
ability to provide "transfer" in the problem-solving pro-
cess (p. 274).

Shortly after the launching of the Russian
satellites Sputnik I and II in 1957, and continuing
through the 19608 and into the 1970s, a number of projects
were developed in an effort to improve the overall quality
of learning and instruction in science and mathematics.
These mathematics projects looked at changes in curricular
materials and content, ideologies, and methodologies. Most
of the projects incorporated some aspect of problem solving
into the teaching/learning process, either through the
structure of learning or through the use of materials in
the project. For example, the Greater Cleveland Mathematics
Program (GCMP) emphasized problem solving throughout the
K-12 program (Weiss, 1978, p. 9). "The project made ex-
tensive use of the discovery approach to learning and drew
heavily upon the principles of mathematics to help children
learn the underlying structure of the material presented"

(Mahaffey and Perrodin, 1973, p. 247). See Appendix F for

a list of projects.

128

A number of ideas were fostered by the mathematics
programs and projects started in the late 1950s and 19603.
Anne W. Schaefer and Albert H. Mauthe (1970) cited the
work of the Nuffield Teaching Project. In noting its
theme created by an ancient Chinese Proverb--I hear, and
I forget; I see, and I remember; I do, and I understand--
the authors made the following comment:

A promising approach to the teaching of problem

solving is the use of the mathematics laboratory.
The purpose of the laboratory is to provide Chil-
dren with opportunities to discover mathematical
concepts through their active involvement in solv-
ing problems. The emphasis is on learning by
doing (p. 7).

In 1970, Wilbur H. Dutton, Colin C. Petrie, and
L. J. Adams included an extensive chapter on "Problem
Solving" in the textbook Agithmetic for Tegchggp. The
Chapter included: (1) types of problems; (2) children's
thinking; (3) concept formation; and (4) strategies for
solving problems. In noting the "types of problems," the
authors described various levels of cognition. Part of
the discussion involved "The Taxonomy of Educational
Objectives for the cognitive domain" cited earlier in the
Taxonomy of Educational Objectives by D. R. Krathwohl,

B. S. Bloom, and B. B. Masia (1964). In discussing the
levels of cognition, Dutton, Petrie, and Adams (1970)

state that the "classification enables the curriculum

developer and the teacher to select the level of cognition

129

intended for a particular unit or lesson" (p. 141). The

six levels of cognition were described in the following

manner 8

1.

Knowledge, requiring recall of specific facts,
trends and sequences, classifications, criteria,
theories, and structures.

Comprehension, requiring understanding, trans-
lation, interpretation, and extrapolation.

Application, requiring the use of abstractions
in particular and concrete situations.

Analysis, requiring a breakdown of communica-
tion into its constituent elements such that
the relative hierarchy of ideas is made clear
or the relations between the ideas made expli-
cit.

Synthesis, requiring the putting together of
elements so as to form a whole.

Evaluation, requiring judgments about the value
of material and methods for given purposes
(p. 141).

In terms of problem-solving techniques, Dutton,

Petrie, and Adams (1970) emphasized various aspects of

"discovery learning." In discussing the procedures used

in the "discovery" process, the following problem-solving

techniques included in a non-referenced textbook series

were cited:

Using objects and pictures to begin instruction
on a topic or problem,

Relating the new work to meaningful, everyday
experiences,

Presenting and developing mathematical ideas
and concepts before introducing abstract sym-
bols or other formalizations,

Developing effective study procedures,

varying teaching procedures to meet individual
differences (p. 151).

130

Dutton, Petrie, and Adams (1970) provided the following
set of rules to assist children in the problem-solving
process:

1. Read the problem.
2. Decide what was given.
3. Decide what is to be found.

4. Decide which operations are necessary to take
what is given and use it to find what is to be
found.

5. Solve the problem.
6. Check the result (p. 151).

The authors recommended that "problem-solving work
should be organized around the abilities and experiences
of children" (p. 151). Thus, it is not surprising that
they cited Piaget's three stages of mental structuring--
(l) sensori-motor group structures, (2) concrete-operations
group structures, and (3) formal mental structures--and the
four principles of conceptual learning formulated by
Zoltan P. Dienes--The Dynamic Principle, The Perceptual
variability Principle, The Mathematical variability
Principle, and The Constructivity Principle (pp. 143-148).

In the textbook Guiding Chilggen to Mgthematiggl
Discovery (1970), Leonard M. Kennedy provided a broad
description of problem solving. The author stated that
"a problem should be considered as any situation an
individual faces for which no immediate solution is
apparent, but for whiCh the possibility of solution exists"

(p. 358). It was noted that word problems were "designed

131

to supplement the real problems children encounter during
their in-school and out-of-school activities" (p. 358).

In a sense, word problems were "practice exercises in
vicarious problem-solving" (p. 358). Kennedy (1970) in-
cluded the following "schematic representation of problem-

solving procedure" (p. 359).

 

 

Recognition of the Examination of the Selection of
nature of the situation to note -€#procedures
problem situation similarities and for solution
a dissimilarities to of problem
past situations

Checking and judging
results of computation

 

 

 

 

 

 

 

 

 
 
     

 

  

Performance
of the
operations

 

 

 

As a part of the process, he suggested use of the following
activities to assist children in the problem-solving pro-
cess: mathematical sentences; concrete materials, pictures,
and diagrams; tables and graphs; reading word problems;
dramatizations; child-composed word problems; simplifying
problems; estimating answers; using alternate methods of
solution; word problems with too much or too little
information, without questions, or without numbers; and
mental arithmetic (pp. 359-368).

Maribeth Henney (1971) discussed a multi-step
"problem analysis" in an attempt to improve verbal problem-

solving ability through reading instruction. The eight

132

steps included the reading of the "verbal problem," a

look at the "main idea," asking a pertinent "question,"
deciding what the "important facts" are, writing a "relation
sentence" followed by a "mathematical sentence," performing
the "computation," and stating an "answer sentence"

(pp. 226-227).

In 1973, Charles H. D'Augustine included a chapter
on "problem solving" in Multiple Methods_ngeagpipg
Mgthemgticsin the Elementary SChool. He provided a six-
step set of "rules" that closely resembled those "rules"
found in many elementary mathematics textbook series. The
author referred to the use of such "rules" as the "trans-
lation approach" whereby the learner reads a story problem
word-by-word, phrase-by-phrase and "translates" the words
and phrases into expressions that are combined to form
mathematical equations. The following "rules" were cited:

1. Find out what question you are supposed to
answer.

2. Find the essential information.

3. Decide on the appropriate operation.

4. write the equation.

5. Solve the equation.

6. Put the solution into an English sentence
whiCh answers the question (p. 50).
D'Augustine (1973) also included the following

"Guidelines for Developing Problem-Solving Skills":

1. NOt only should the child be given the skills
necessary to solve problems, but he should also
be taught how to identify and delimit problems.

133

NOt only should the child be taught how to
translate a problem into a mathematical sen-
tence, but he should also be taught how to
translate the problem into a simpler model
of the problem.

Not only should the child be taught how to
find alternative paths to his goal, but he
should also be taught how to decide which
of these is the most efficient path.

Not only should the child be taught how to
derive a numerical answer, but he should also
be taught how to interpret and use the in-
formation practically.

Not only should a Child be taught to check his
results, but he should also be taught to modi-
fy his solution as new data become available

to him. In other words, he should be made aware
of the fact that the answer for today may not
be the answer for tomorrow.

Not only should a Child be taught to solve
problems, but he should also be taught to
create problems (p. 47).

D'Augustine (1973) noted that "problem solving is the

process of striving to reach a goal by a previously un-

traveled path. Problem solving takes many guises" (p. 52).

In the process of solving problems, he stated that all or

some of the following activities will occur:

1.
2.

A problem is recognized.

The basic structure of the problem is reduced
to a simplified model.

Data are gathered or avenues for solution are
selected.

A value judgment is rendered in selecting the
"best" avenue for a solution.

The problem is solved and the choice of solu-
tion is evaluated.

Alternate solutions are tested (p. 52).

134

In 1974, Wayne A. Wickelgren wrote How to Solve
Problems: Element§_gf a Theopy;of ngblems and Problem
Solving. In discussing "General versus Special Methods"
of problem solving, the author stated:

The relation between specific knowledge and
methods, on the one hand, and general problem-
solving methods, on the other hand, appears to be
as follows. When you understand the relevant
material and specific methods quite well and al-
ready have considerable experience in applying this
knowledge to similar problems, then in solving a
new problem you use the same specific methods you
used before. Considering the methods used in simi-
lar problems is a general problem-solving technique.
However, in cases where it is obvious that a par-
ticular problem is a member of a class of problems
you have solved before, you do not need to make
explicit, conscious use of the method: simply go
ahead and solve the problem, using methods that
you have learned to apply to this class. Once you
have this level of understanding of the relevant
material, general problem-solving methods are of
little value in solving the vast majority of home-
work and examination problems for mathematics,
science, and engineering courses (p. 3-4).

In the first Chapter, he discussed the methodology involved
in the problem-solving process. The following summary on
the development of skill in problem solving was provided:

There is no particular reason to engage in
this careful, conscious analysis of a problem when
you can immediately get some good ideas on how to
solve it. Just go ahead and solve the problem
"naturally." However, after you solve it or, even
better, while you are solving it, analyze what you
are doing. It will greatly deepen your understand-
ing of problem-solving methods, and you might dis-
cover new methods or a new application of an old
method.

As you get extensive practice in using these
problem-solving methods you should become so skilled
in their use that the process becomes less conscious
and more automatic or natural. This is the way of
all skill learning, whether driving a car, playing
tennis, or solving mathematical problems (p. 6).

135

Stephen Willoughby (1975), in writing about
"Lessons Learned and Pitfalls to be Avoided in the
Continuing Curricular Reform Efforts in Mathematics
Education," included a brief discussion on problem solving.
In reference to the practical value of problems in the
"new math" programs, he provided the following remarks:

The most common valid criticism of "the new
mathematics" is that it really doesn't teach
Children to solve real problems. But this has
also been the easiest valid criticism to make of
any mathematics education program in history.
When a program does not have enough good work
with problem solving it is not because the authors
of the program are evil or even stupid. It's be-
cause having enough good problem-solving work is
undoubtedly one of the.hardest possible things to
do in mathematics (p. 6).

In summarizing the section on the practicality of solving
problems, Willoughby (1975) wrote:

As we develop problem-solving material, I
suggest that it is desirable, where possible, to
allow the student to learn about the problem with-
out reading it in a book. The reason for this
suggestion is that many children find their great-
est difficulty with mathematics to be the reading
of "word problems" and this is not what we really
want them to be able to solve anyway. It often
seems to be the case that the words used to simu-
late the real problem are what keep the pupil from
solving the problem (p. 6).

Jack C. Gill and Morris I. Beers (1976) provided a
"look" at the relationship between the "real world" and
the "mathematical world." In describing the complexity of
the problem-solving process, they included the following

diagram (p. 217):

136

 

 

 

 

 

 

real world mathematical world
real or
physical abstraction mathematical model
problem
application
Of
mathematical
V principles
/
physical translation mathematical
solution r: solution

 

 

 

 

 

For purposes of illustration, Gill and Beers (1976) focused
on "verbal" problem solving and suggested the following

four strategies to assist the problem solver in the learning
process: estimation, graphs or tables, pattern recognition,
and equations or number sentences (pp. 217-218).

In 1977, Kenneth J. Travers, Len Pikaart, Marilyn
N..Suydam, and Garth E. Runion approached the teaching of
problem solving through "heuristics." The authors described
the teaching of heuristics as "general student activities to
help in solving problems" (p. 140). In separating the
"heuristics" into two parts--Initiating Heuristics and Look-
ing Back Heuristics--the four authors included the following
list:

Initiating Heuristics

1. Select appropriate notation.

2. Make a drawing, figure, or graph.

3. Identify wanted, given, and needed information.
4. Restate the problem.

5. Write an open sentence.
6. Draw from a cognitive background.
7. Construct a table.

8.
9.
10.
11.
12.

137

Guess and check.
Systematize.

Make a simpler problem.
Construct a physical model.
WOrk backwards.

Looking Back Heuristics

13.
14.
15.
16.
17.

Generalize.

Check the solution.

Find another way to solve it.

Find another result. _

Study the solution process (pp. 150-152).

Frank K. Lester, Jr. discussed "Research on Mathe-

matical Problem Solving" in the Professional Research Series

presented by the National Council of Teachers of Mathematics

in 1980.

The plight of problem solving was provided in the

following comments:

The fact that problem solving has been the ob-

ject of so much research, a focal point for several
curriculum-development efforts, and the subject of
innumerable books, articles, and conference reports
attests to its importance in the study of mathe-
matics. Indeed, there is substantial support for
the notion that the ultimate aim of learning mathe-
matics at every level is to be able to solve
problems. Despite this well-recognized importance,
the role problem solving should play in the mathe-
matics curriculum is less clear. A cursory look
at the most popular mathematics textbooks gives
ample evidence of the lack of generally accepted
tenets about the role of problem solving (p. 287).

Lester (1980), in noting "that problem solving is a very

personal type of activity" (p. 287), provided the following

citation on solving problems:

...the way a problem is presented and the type of
information provided may significantly influence
success in solving it. A few of the many factors
that play a role in mathematical problem solving

138

include previous experience, mathematical back-
ground, level of interest, motivation, and problem
structure. In short, problem solving is an
extremely complex area of human behavior (p. 287).

In making the observation that "mathematical problem
solving appears, to a certain extent, to be so complex and
subtle as to defy description and analysis" (p. 288), Lester
(1980) listed the following four types of factors associated
with problem solving:

The Problem Itself: Task Variables
Characteristics of the Individual: Subject Variables
Problem-Solving Behavior: Process variables

Environmental Features: Instruction variables
(pp. 288-289)

The author also provided seven key issues in his
discussion on problem solving. The issues are highlighted
in the following list:

Theory-related Issues

Issue 1 Past problem-solving research in mathematics
has suffered from the absence or neglect of
theory.

Issue 2 Problem solving is a chaotic area of inquiry
largely because of the widely diverse types
of tasks used. The tasks used in problem-
solving research significantly affect the
generalizability of results.

Issue 3 Characteristics of problem solvers greatly
affect behavior and consequently severely
limit the generalizability of results.
What kinds of subject to use in problem-
solving research is a topic of muCh '
discussion.

Instruction-related Issues

Issue 4 There is little agreement regarding how
best to improve problem-solving performance
beyond the obvious fact that attempting to
solve problems is a necessary ingredient.

139

Issue 5 In addition to a lack of consensus
regarding the best ways to enhance prob-
lem solving, there is no accord about the
nature of improvement in problem solving.
Some researchers interested in problem-
solving instruction have focused on the
improvement of students' abilities to use
particular strategies or skills, and
others have considered improvement only
in terms of an increase in the number of
correct solutions.

Issue 6 The extent of instructional treatments in
recent research varies from about one week
to several months, with relatively short
treatments being the most common. Treat-
ments should be extensive enough not only
to allow for full explication of ideas and
procedures but also to provide ample
opportunity for students to practice the
procedures they are being taught.

Research Methodology Issue

Issue 7 No generally accepted methods or instru-
ments for measuring performance or
observing behavior during problem solving
are clearly reliable and valid. Thus,
the kind of instrumentation that is appro-
priate for a particular purpose remains
an issue (pp. 315-316).

Thomas P. Carpenter, Henry Kepner, Mary Kay Corbitt,
Mary MOntgomery Lindquist, and Robert E. Reys summarized the
"Results and Implications of the Second NAEP Mathematics
Assessment: Elementary School" in 1980. The authors
offered a pessimistic review of the results in problem-
solving skill in the elementary school:

If it were necessary to single out one area that
demands immediate attention, it would clearly be
problem-solving. At all age levels, and in virtually
every content area, performance was extremely low
on exercises requiring problem-solving or applica-
tions of mathematical skills. In general,
respondents demonstggted a lgck of even the most
basicppgblem—solving skills. Rather than attempt-
ing to reasOn a problem through and figure but what

140

needs to be done to solve the problem, most
respondents simply tried to apply a single arith-
metic operation to the numbers in the problem.

The results indicate that students are not familiar
with such basic problem-solving strategies as
drawing a picture of a figure described in a prob-
lem, substituting smaller numbers in a problem to
attempt to find a solution method, or checking the
reasonableness of a result (p. 47).

Writing in the book, Problem Solving: A Handbook

for Teachers, Stephen Krulik and Jesse A. Rudnick (1980)

included the following set of "workable heuristics":

1.

Read

la. Note key words.

lb. Get to know the problem setting.

1c. What is being asked for?

1d. Restate the problem in your own words.
Explore

2a. Draw a diagram, or construct a model.
2b. Make a chart. Record the data.

2c. Look for patterns.

Select a strategy

3a. Experiment.

3b. Look for a simpler problem.

3c. Conjecture/guess.

3d. Form a tentative hypothesis.

3e. Assume a solution.

Solve

4a. Carry through your strategy.

Review and extend

5a. verify your answer.

5b. Look for interesting variations on the
original problem (pp. 20-29).

The two authors also provided the following list of

"activities" in "The Pedagogy of Problem Solving":

141

1. Create an atmosphere of success.

2. Encourage your students to solve problems.
3. Teach students how to read the problem.

4. Involve your students in the problem.

5. Require your students to create their own
problems.

6. Have your students work together in pairs or
small groups.

7. Encourage the use of freehand drawings.

8. Suggest alternatives when the present approach
has apparently yielded all possible information.

9. Raise creative, constructive questions.

10. Emphasize creativity of thought and imagination.
ll. Encourage your students to use a calculator.

12. Use strategy games in class.

13. Have your students flow-chart their own
problem-solving process.

14. Don't teach new mathematics while teaChing
problem solving (pp. 37-63).
In 1980, the National Council of Teachers of Mathe-

matics (NCTM) devoted its entire yearbook to the teaChing of

problem solving. The yearbook, titled Problem Solving in
School Mathematics, was written for those individuals who
have the greatest impact upon the teaching of problem
solving--the classroom teacher. This importance was noted
by Editor Stephen Krulik in the following passage extracted

from the Preface of the 1980 Yearbook:

Although Problem Solving in School Mathematics
was never intended to be a definitive work on prob-

lem solving (in keeping with the concept of this
series of yearbooks), the papers were all written
with an eye to the classroom teaCher. Here are
ideas to be used in the classroom at all levels of
instruction. Here are problems, examples, and il-
lustrations. And if these ideas are used, then the

142

authors will have done their job well! Problem

solving should be the major focus of all mathemat-

ics instruction; Epgblem Sglving is an attempt to

help the classroom teacher in this important effort

(p. xiv).
With the emphasis on problem solving, the National Council
of Teachers of Mathematics embarked upon a journey it pre-
scribed for all individuals and organizations involved in
the teaching of mathematics: Problem solving was to be the
focus of mathematics instruction during the 1980s.

In Chapter 6 of the yearbook, Alan Osborne and
Margaret B. Kasten (1980) discussed the "Opinions about
Problem Solving in the Curriculum for the 1980s: A Report."
The authors described the results of a survey of nine popu-
lations: subscribers to Agithmetic Teggpgg and Mathematics
TeaCher, members of the American Mathematical Association
of Two-Year Colleges, members of the Mathematical Associa-
tion of America, teacher educators, mathematics supervisors,
principals, school board presidents, and presidents of
parent-teacher associations. Osborne and Kasten (1980)
reported the following conclusions:

CONCLUSION 1. Problem solving should receive more

emphasis in the school mathematics program
during the coming decade.

CONCLUSION 2. There is broadly based unanimity
for the aims or purposes of instruction on
problem solving.

CONCLUSION 3. The strategies of (l) translating a
problem to an equation, (2) constructing a table
and searching for patterns, (3) drawing pictures
or diagrams to represent a problem, and (4)
solving a simple problem first and extending the
solution to the original problem were identified
as appropriate content for both the elementary
and secondary school levels.

143

CONCLUSION 4. No extreme positions concerning
teaching methodology for problem solving
were preferred by any group.

CONCLUSION 5. Using a problem as the means or
the vehicle to develop and introduce mathe-
matical topics is a preferred methodology.

CONCLUSION 6. Problem solving is important for
all students and should begin early in their
mathematical experience.

CONCLUSION 7. A modification of the mathematics
curriculum to provide unique problem-solving
experiences for special groups, such as women,
the college bound, or ethnic minorities, is
perceived as inappropriate (pp. 51-60).

Nicholas A. Branca wrote in Chapter 2 of the
1980 Yearbook, "Problem Solving as a Goal, Process, and
Basic Skill." Both the difficulty and the importance
of understanding the multiple interpretations of problem
solving were discussed. The following comments summarize
the ideas of the chapter:

We as teachers must realize the importance of
problem solving with respect to each of the three
interpretations. We must be aware that the stu-
dents who enter school during this decade will
spend the majority of their productive lives solv-
ing the problems of the twenty-first century.
Although we have no way of knowing what these
problems will be like, considering the different
interpretations of problem solving can help us
prepare for them.

Considering problem solving as a basic skill
can help us organize the specifics of our daily
teaChing of skills, concepts, and problem solving.
Considering problem solving as a process can help
us examine what we do with the skills and concepts,
how they relate to each other, and what role they
play in the solution of various problems. Finally,
considering problem solving as a goal can influence
all that we do in teaching mathematics by showing
us another purpose for our teaching.

144

Each of these interpretations is important,
but they are different. When we encounter the
term problem solving, we should consider which
interpretation (or interpretations) is intended.
The multiple meanings for the term can easily
lead a writer to ambiguity and a reader to mis-
understanding. Problem solving has too many
facets for us always to look at it from the same
angle (p. 7).

In discussing the "general nature of problem

solving," Marilyn N. Suydam provided a composite list of

steps in the problem-solving process in Chapter 5 of the

1980 Yearbook. These steps are:

1.

Understanding the problem--an awareness of the
problem situation that stimulates the person
to generate a statement of the problem in
writing, orally, or merely in thought.

Planning how to solve the problem.

a.

b.

C.

Break down the components; enumerate data;
isolate the unknown.

Recall information from memory; associate
salient features with promising solution
procedures.

Formulate hypotheses or a general idea of
how to proceed.

Solving the problem.

a.

b.

C.

Transform the problem statement into a
mathematical form, or construct repre-
sentations of the problem situation.

Analyze the statement into subproblems for
which the solution is more immediate.

Find a provisional solution.

Reviewing the problem and the solution.

a.
b.

Check the solution against the problem.

verify whether the solution is correct; if
not, reject the hypotheses, the method of
solution, or the provisional solution.

Ascertain an alternative method of solution

145

In 1981, Francis M. Fennell, in the Elementary
Mathematics Diagnosis and Correction Kit, cited the im-
portance of teaChing problem solving in the following
manner:

Problem solving transcends all other strands

or topic areas within the elementary mathematics

curriculum. Developing the ability to solve rele-

vant problems is a major objective in all areas

(p. 361).
He described fifteen approaches to better meet the problem-
solving abilities of children. They were: oral problems,
open-ended problems, illustrated problems, interdisciplin-
ary examples, problems without numbers, mini-problems,
pupil problems, challengers, experiencing charts, deletion
of extra information, card file, calculator, bulletin
boards, interpretation interviews, and the problem of the
day (pp. 362-364).

In terms of diagnosis, Fennell (1981) included a
"staircase" of problem solving. The diagram lends itself
to a "Problem-Solving Process Checklist" in which the
teaCher keeps track--via yes or no responses--of the
problem-solving abilities of children (pp. 365-366).

The "staircase" of steps--What's the problem?, Make a Plan,
Do It!, and Checkup-~is illustrated in Appendix G.

Frank K. Lester, Jr. and Joe Garofalo edited

Mathematical Problem Solving: Issues in Research in 1982.

In the introduction, the following comment regarding inter-

est in and research of problem solving was noted:

146

The high level of interest in problem solving
and thinking among educators and psychologists
has been accompanied by a corresponding level of
concern for developing a stable and useful body
of knowledge about the phenomena associated with
these highly complex areas of human behavior.
There is no question that our understanding of
problem-solving behavior has increased greatly
in recent years. However, for various reasons...
the research on the whole has been rather unsys-
tematic and has lacked clarity of purpose and
focus (p. ix).

In a section titled "Building Bridges Between
Psychological and Mathematics Education Research on Problem
Solving," Lester (1982) cited three reasons for what was
called "the rather chaotic state" of problem-solving
researCh. The three factors were:

1. a neglect of theory to guide systematic inquiry;

2. the exteme complexity of the nature of problem
solving; and

3. the rudimentary state of the research methodol-
ogies employed (p. 55).
In addition to the three factors, the author noted the

following issues:

1. The role of theory in problem-solving research;
2. The types of researCh tasks to use;

3. The relative emphasis to place on developing
competency models or performance models of
problem-solving behavior;

4. TeaChing problem solving (if in fact it can be
taught) and what the teacher's role should be;

5. The nature of problem-solving performance
changes;

6. The types of research methodologies to employ
(p. 55).

147

Lester (1982) also considered a series of seven questions,

each of which was related to one or more of the issues list-

ed on the previous page. The seven questions were:

Can problem solving be taught?

What is the role of understanding in problem
solving?

To what extent does transfer of learning occur
in problem solving?

What are the primary task variables that affect
problem solving?

What is the role of metacognitive behavior in
problem solving?

How do successful and unsuccessful (good ver-
sus poor, expert versus novice) problem solvers
differ with respect to their problem-solving
behavior?

What are the most appropriate research meth-
odologies? (p. 56)

Eugene D. Nichols and Merlyn J. Behr (1982) wrote

Elementary Schogl Mathematics and How to Tegch It. In a

chapter on "problem solving," they provided the following

description:

When we speak of problem solving, we are refer-

ring to what are commonly called by teachers and
students story problems or verbal problems. A
substantial part of elementary school mathematics
instruction is devoted to teaching students how to
solve suCh problems.

The central emphasis given to the matter of

problem solving is correctly placed, for this is
important in everyday life as well. Many problems
encountered in everyday activities lend themselves
to solution by mathematical methods. It is there-
fore necessary that these methods be developed
systematically in a mathematics curriculum (p. 277).

The following activities were suggested in teaching skill

in problem solving: writing equations (translation), writing

and solving equations, solving number puzzles, writing

148

miniproblems, pictures in problem solving (diagrams), using
logical reasoning, asking "What additional information is
needed?," looking for "extra information," estimating
answers, solving "multi-step problems," making up problems,
reading a chart, and using flow charts (pp. 277-296). In
terms of "heuristics," the authors recommended using the
Polya Model (pp. 296-298). Finally, they suggested using
the calculator in problem-solving activities because that
"accomplishes at least two things: it frees the child to
think about the problem-solving methods rather than concen-
trating on the computations, and it forces the child to
think about the accuracy of the solution and the reasonable-
ness of the answer" (pp. 323-324).

In 1983, Daniel T. Dolan and James Williamson, in
TeaChing Problem—Solving Strategies, offered the following
caution:

One must be extremely careful not to fall into

the trap of defining problem solving as another
basic skill which can be treated as an algorithm.
The National Assessment of Educational Progress
findings on mathematics achievement made in 1979,
reported that "evidence shows that students proceed
mechanically and thoughtlessly through problems--
if they forget the rule, then they are unable to do
the problem on their own." This analysis of re-
sults of achievement in problem solving is
consistent with the usual methodology of teaching
it. Problem solving in mathematics has become a
set of words which are wrapped around some compu-
tational exercises (p. ix).

In recommending the teaChing of strategies, the authors

included the following activities: guessing and checking,

149

making a table, patterning, making a model, eliminating,
and simplifying (pp. 3-105).

The 1983 Yearbook of the National Council of
Teachers of Mathematics was written in response to the
recommendations included in An Agendg for Action: Recom-
mendations for School Mathematics of the 19805. Titled
The Agenda in Action, the 1983 Yearbook provided a "frame-
work for exploring current developments leading to the
improved teaching of mathematics at all levels" (p. ix).
More than a third of the yearbook highlighted the first
recommendation made by the Council in 1980, that "problem
solving must be the focus of school mathematics in the
19805." Several citations taken from the 1983 Yearbook are
particularly noteworthy.

writing a section on "Problem Solving as a Focus:
How? When? Whose Responsibility?," Peggy House, Martha L.
Wallace, and Mary A. Johnson (1983) stated the following:

Problem solving is a process, not a step-by-step

procedure or an answer to be found; it is a journey,
not a destination. Successful problem solvers can
be identified by the processes or the attitudes of
mind they display. Four characteristics that help
identify good problem solvers are desire, enthusiasm,
facility, and ability (p. 10).
The writers discussed each of the four characteristics and
concluded:

The scenario of a future in which problem solving

is truly the focus of mathematics instruction is ex-

citing and challenging. However, the degree to whiCh
the Agenda actually becomes action depends on the

150

degree to which each mathematics teacher, teacher
educator, and supervisor accepts the responsibility
to create that future. There is no question that
we have the facility and the ability to do this and
to make it stick. The real question is, do we have
the desire and the enthusiasm? (p. 19)

Richard Brannan and Oscar Schaaf, writing in the
1983 Yearbook, considered the topic of "An Instructional
Approach to Problem Solving." In their opening remarks, the
two authors summarized the concern for teaching mathematical
problem solving with the following comment:

Problem solving has been a subject of research
by mathematics educators, educational psychologists,
and philosophers since the 1930s. Yet tests, in-
cluding the recent mathematics tests of the National
Assessment of Educational Progress, indicate that
our sChool students and even graduates have serious
deficiencies in problem-solving ability. An exam-
ination of current mathematics texts and the results
of classroom visitations reveal that problem solving
is only a minor part of the mathematics instruction
in both elementary and secondary schools (p. 41).

Brannan and Schaaf (1983) made several recommendations for
making "problem solving the focus of mathematics education."
In the summary, they provided the following:

1. The mathematics curriculum should be organized
around problem solving.

2. Appropriate curricular materials to teach prob-
lem solving should be developed for all grade
levels.

3. Mathematics teachers should create classroom
environments in which problem solving can
flourish (p. 59).
The criteria listed above provide a foundation for
making problem solving the focus of mathematics instruction

for the remainder of the 19805. Over the next seven years,

151

teachers must recognize suCh propositions--and those that
preceded them--so that the concerns for teaChing mathemati-
cal problem solving so prevalent in the 19805 will not be

repeated during the 1990s.

Summgpy of Period III: 1954-1983

Changes in curriculum and the psychology of learning
played important roles in the development of mathematical
problem solving during the period, 1954-1983. "Thinking"
and "conceptual development" became integral parts of the
problem-solving process. Noteworthy was the fact that in
teaching problem solving there were two aims, societal and
mathematical. Although a number of problem-solving models
continued to reflect the "conventional method" of the 1920s,
most models retained the initial stages of "reading the
problem thoroughly or carefully" and "comprehending the
problem." verbal problem solving continued to be an issue.

During this period, problem solving was incorporated
throughout the field of mathematics. Research played an
important function in the teaching of problem solving, and
mathematics educators were encouraged to include theoretical
research in the instructional process.

A number of mathematics educators contributed to the
development of mathematical problem solving during Period
III. However, the problem-solving model presented by George
POlya in the course of Period II continued to gain in popu-

larity. Toward the end of Period III, the Polya Model was

152

cited widely in journal articles, mathematics methods' text-
books, and in the 1980 Yearbook of the National Council of
Teachers of Mathematics. The importance of this fact is
recognized because the decade of the 19805 was--and still
is--viewed by the Council as the decade of problem solving,
and the Polya Model, with its extensive questioning has been
frequently highlighted.

In addition to many individuals, numerous mathemat-
ics programs and projects, developed shortly after the
launching of Sputnik I and II, affected the teaching of
problem solving in mathematics. This period spawned a
series of mathematics programs initiated by private and
governmental agencies. During the 19603, there was renewed
interest in providing for extensive training in the area of
problem solving with a variety of techniques suggested. In
numerous attempts to translate word problems to number prob-
lems as a part of the problem-solving process, children were
encouraged to solve problems "naturally." Discovery became
an important ingredient in the process and students were
invited to discover new problem-solving strategies and to
apply them in settings other than the problem itself.

Further development in the teaching of mathematical
problem solving during this period can be found in the em-
phasis accorded problem analysis, critical thinking skill,
and "mental" arithmetic. Ideas prevalent during Period III

include the use of diagnosis in problem solving and continued

153

emphasis on a meaningful arithmetic program (Brueckner and
Bond, 1955), a more mathematical approach to problem solving
(van Engen, 1959), emphasis on the methodical work found in
mathematical problem solving (Polya, 1962), emphasis placed
on discovery and intuitive thinking in the "new math" pro-
gram (Dutton, 1964), emphasis on Piaget's stages of mental
structuring and Dienes' Principles of Conceptual Learning
(Dutton, Petrie, and Adams, 1970), the relationships--and
differences--between general and specific methods of solving
problems (Wickelgren, 1974), and the recommendation that
problem solving be the focus of mathematics instruction for
an entire decade, the 19803 (NCTM, 1980).

Finally, with the pessimistic outlook cited during
the late 19703, and the NCTM focus on problem solving for
the 19803, it is clear that many problems associated with
the teaching of mathematical problem solving continue to
exist. In the final analysis, the gap between the theory of
problem solving and the inclusion of problem solving in the
mathematics classroom might be wider than most educators

realize.

CHAPTER V

SUMMARY, CONCLUSIONS,
RECOMMENDATIONS, AND IMPLICATIONS
This chapter (1) summarizes the problem; (2) tenders
the conclusions; (3) provides recommendations for the future
teaching of problem solving in mathematics; and (4) states
the implications for future research in mathematical problem

solving.

Summary
This investigation considered the following ques-
tions:

1. What are the historical ideas, concerns, and
approaches that have helped formulate problem
solving in school mathematics?

2. What are the historical ideas, concerns, and
approaches that have helped elevate problem
solving to its current level of importance in
mathematics education?

3. Compared to events and influences of the past,
what practices indicate a successful problem-
solving movement for the remainder of the 19803

and the future?

154

155

Teaching problem solving in mathematics has been a
matter of interest since the 19203. Yet, with continuing
emphasis through the intervening decades some of the prob-
lems found in the 19203 continue to exist in 1983.

The focus of this study is to trace the development
of mathematical problem solving over a ninety-year period
and make recommendations for the future teaching and learn-
ing of problem solving. Hundreds of ideas, concerns, and
approaches on mathematical problem solving have been dis-
cussed in this investigation. Each--in its own way--has
contributed to the development of problem solving in
mathematics.

Improved research techniques, greater interest in
methodology, and the incorporation of various ideas and
theories of learning into the instructional process have had
an impact on current practices found in the teaching of
mathematical problem solving. But with the ideas, thoughts,
theories, models, approaChes, and research included during
the past ninety years, the development, improvement, and
implementation of problem solving in mathematics continues
to be a source of concern. The following section provides a

list of conclusions gathered from this investigation.

156

Conclusions

Based upon the data included in this study, the

following conclusions are cited:

1.

Concern for the teaching of mathematical problem
solving has been a source of major interest in the
mathematics community for over ninety years. Seri-
ous consideration for teaching problem solving in
mathematics began in the 19203. Concerns similar
to those found sixty years ago continue to exist.
The National Council of Teachers of Mathematics
provided extensive material on mathematical problem
solving in its 1928, 1953, and 1980 Yearbooks. The
goal was to encourage classroom teachers to better
understand the basic ideas and theories of mathe-
matical problem solving.

During the past nine decades, numerous ideas, con-
cerns, and approaches have provided teaChers with a
number of descriptions and suggestions on how'pggp
to implement problem solving into the mathematics
curriculum. Some descriptions can be confusing to
classroom teachers. For example, in certain cases,
problem-solving descriptions are limited strictly
to word or story problems. In other cases, descrip-
tions of problem solving are used in conjunction

with mathematical problems of all types, for example,

157

2 + 7 = n, 45.6 - 7.8 + 8.99 = z, and 5% x 3% = m,
and are not limited to story or word problems.
Models developed decades ago continue to be utilized
in mathematical problem solving. For example, the
Polya Model and the "conventional" models that pre-
ceded it, continue to appear in textbooks, journal
articles, and other professional literature. Thou-
sands of steps have been included in hundreds of
problem-solving models. M03t of the steps "fit"
into eight basic categories. The following "model"
is a composite representation of the problem-solving
models developed over the past ninety years:

a. Read the problem carefully.

b. Look for the "known" facts contained in the
problem.

c. Determine the "unknown" portion of the problem.

d. Use the "known" and "unknown" portions to
determine a procedure for solving the problem.

e. Estimate the final answer.
f. Solve the problem (computation).

g. Compare the answer in the solution with the
estimated answer (look for "reasonableness").

h. Label the final answer.

The "problem" of problem solving has been a source
of concern for classroom teachers throughout the
ninety-year period. The "problem" includes diffi-

culties with:

158

a. teachers' inability to implement problem-solving
approaches and techniques in mathematics instruc-
tion. The recent data indicate that problem
solving occupies only a minor part of the mathe-
matics programs found in elementary and secondary
schools; and

b. children's inability to read and understand the
technical language of mathematics and to compre-
hend various problem-solving strategies. Recent
literature indicates that children lack an under-
standing of the basic problem-solving skills.

Problem solving can be taught! The recent literature

indicates that a successful problem-solving movement

can be realized. For this to happen, there must be

a greater awareness on the part of classroom teachers

in providing for alternative approaches, strategies,

and materials used in mathematical problem solving.

Furthermore, there must be a willingness on the part

of all classroom teachers, building principals, cur-

riculum coordinators, and school superintendents to
fully implement a problem-solving approach to teach-
ing mathematics. This includes daily problem-solving
activities in mathematics, in-service programs that
center on mathematical problem solving, and a mathe-
matics curriculum that follows the National Council

of Teachers of Mathematics recommendation that

159

problem solving be the focus of mathematics instruction

during the 19803. The next section provides specific rec-

ommendations for implementing particular procedures in

teaching problem solving in mathematics.

Recommendations

Based upon the conclusions of this investigation,

the following recommendations for the future teaching of

mathematical problem solving are suggested:

1.

There should be greater emphasis placed on mathemat-
ical problem-solving activities--such as the use of
drawings, tables, and charts, guessing, trial-and-
error, and estimation--to solve problems in the early
grades. The strategies learned in the early grades
should be expanded as children progress through the
curriculum. At eaCh level, teachers should recog-
nize children's readiness in problem-solving skill.
Pupils should be strongly encouraged to use a variety
of problem-solving strategies and to look for alter-
native solutions to problems. They should be
provided with specific problem-solving experiences
such as working with brainteasers, product-purchasing
problems, and other mathematical situations that
reflect not only mathematical growth but also growth
in their ability to solve problems in the real world

as well.

160

Teachers should stress and integrate mathematical
problem-solving practice with other subject-matter
areas such as science, social studies, physical
education, industrial arts, and other parts of the
curriculum. Such problem-solving activities include:
longitudinal and latitudinal problems in geography;
relating statistical concepts to predicting outcomes
for local, state, and national elections; applying
mathematical principles of distance, speed, and time
to physical education activities; using newspapers
and magazines to solve comparison-buying problems;
preparing alternative classroom designs for seating
arrangements and learning centers; and developing
large-scale problem-solving activities involving the
entire class (for example, determining the expenses
involved in purchasing a computer for the classroom
including software, hardware, and miscellaneous
materials).

Teachers should provide for the development of crit-
ical thinking skills such as analysis, synthesis,
and evaluation which are important to understanding
mathematical problem-solving processes at all levels
of mathematics. Skill in critical thinking is an
important factor in recognizing a problem, compre-
hending the question(s) in a problem, determining

computational procedures, estimating the final

161

answer, and determining the "reasonableness" of the
final solution. These skills must be developed if
students are to comprehend the scope of mathematical
problem solving.

Children should be challenged to ask.questions when
solving problems in mathematics. The teacher should
encourage children to question various strategies,
provide time for follow-up questions, and recognize
unusual solutions that are both mathematically
correct and important to the problem solver.

Various problem-solving strategies and techniques
should be included in preparing pre-service teachers
in mathematics. A problem-solving approach to teach-
ing mathematics is recommended. Mathematics programs
should reflect the theme of the Nuffield Mathematics
Teaching Project: I hear, and I forget; I see, and

I remember; I do, and I understand.

In order to teach mathematical problem solving,
teachers must understand processes of problem solving,
exhibit confidence in their own problem-solving abil-
ities, and utilize alternative methods in presenting
problem-solving material. These practices should
begin in the preschool and continue through all levels

of the mathematics curriculum.

162

All classroom teachers should become familiar with
the suggestions and recommendations made by organi-
zations such as the National Council of Teachers of
Mathematics. The recommendation that during the
19803 mathematics instruction focus on problem solv-
ing must be known and understood by each teacher,
building principal, curriculum coordinator, and
school superintendent. The ideas contained in var-
ious journals, yearbooks, and bulletins provide the
classroom teacher with numerous problem-solving
strategies which should be incorporated into mathe-
matics instruction at every opportunity.

Local school districts should adopt a problem-solving
approach to teaching mathematics. In-service pro-
grams on mathematical problem solving should be
provided several times during the school year. In
these in-service programs, teachers should experience
and understand various problem-solving strategies and
approaches. They should become familiar with current
literature and research on problem solving in mathe-
matics.

Textbooks should reflect, emphasize, and utilize
various problem-solving strategies and approaches
beginning in the early grades and continuing through

college-level courses in mathematics. They should

10.

11.

12.

163

make use of the various problem-solving models, and
provide for extensive questioning techniques similar
to those found in the Polya MOdel.

Reading continues to be a source of concern in
mathematical problem solving. Teachers should be
aware of the reading abilities of both good and poor
readers in mathematics. Children should be helped
to recognize the type of reading found in mathe-
matical symbolism and relate this type of language
to mathematical story or word problems. Teachers
must stress the reading aspects of all types of
mathematical problems. Students must realize that
the initial step in any problem-solving model or
procedure is to read the problem carefully.

In working with a variety of problem-solving strate-
gies, children should use concrete aids and other
manipulative devices such as nuts, bolts, sticks,
straws, geometric shapes, abaci, counting frames,
and calculators. Each student should be aware of
not only manipulative aids but also the various types
of strategies used in solving problems. This list
includes the use of drawings, diagrams, charts,
tables, mathematical sentences, and pictures in the
problem-solving process.

The development of mathematical problem solving must

include the use of computers. A variety of computer

13.

14.

164

experiences should be provided for students at all
levels of instruction. Children should become famil-
iar with computer literacy in the primary grades.

The use of computer material should not be limited
strictly to drill and practice. Exercises must
include elements of strategy and provide for alter-
native solutions. Teachers and curriculum
coordinators should select "software" materials that
reflect a variety of approaches to problem solving
and incorporate unique activities that will provide
students with various experiences in solving prob-
lems.

The Polya Model continues to be used extensively in
mathematics education. Because of this, teaChers
should become familiar with and utilize the question-
ing techniques included in the model. Students
should be able to apply the questions in the Polya
Model to all types of problems in mathematics and not
just to story or word problems.

The Polya Mbdel should not be the only model used in
the mathematics classroom. Teachers should become
familiar with general problem-solving steps similar
to those cited earlier in this chapter. Those steps

are:

a.
b.

e.
f.

g.

h.

165

Read the problem carefully.

Look for the "known" facts contained in the
problem.

Determine the "unknown" portion of the problem.

Use the "known" and "unknown" portions to
determine a procedure for solving the problem.

Estimate the final answer.
Solve the problem (computation).

Compare the answer in the solution with the
estimated answer (look for "reasonableness").

Label the final answer.

It is recommended that the teacher add at least two steps

to the above procedure. They are:

i.
j.

Double check the answer to guarantee correctness.

Look for alternative methods of solution to
verify the final answer.

Implications for Future Research

Based upon the findings of this investigation, the

following implications for future research in mathematical

problem solving are noted:

1.

Research on the historical development of mathemat-
ical problem solving for the 19703 and 19803 should
be expanded and include an in-depth look at the im-
pact of various "calculating machines" on problem
solving in mathematics.

Research on the degree of impact that various indi-
viduals, projects, committees, and organizations had
on mathematical problem solving should be more close-

ly investigated.

166

Research on the historical development of problem
solving in mathematics should compare and contrast
the development of problem solving found interna-

tionally, for example, Great Britain, Japan, and

the Soviet Union.

APPENDICES

11.
12.
13.

14.

15.

l6.

17.

18.

19.

20.

167

APPENDIX A

THORNDIKE'S ORGANIZATION OF LEARNING
To read, write, and understand integers.
To add with integers.
To subtract with integers.
To multiply with integers.
To divide with integers.
To read, write, and understand United States money.
To add with United States money.
To subtract with United States money.
Tb multiply with United States money.
To divide with United States money.
To read, write, and understand fractions.
To reduce them to higher and lower terms.
To find the least common multiple.

Tb add with common fractions, then with mixed
numbers.

To subtract with common fractions, then with
mixed numbers.

To multiply with common fractions, then with
mixed numbers.

To divide with common fractions, then with
mixed numbers.

To read, write, and understand decimal numbers.

To reduce common fractions to decimals and vice
versa.

To add with decimal fractions and decimal mixed
numbers.

(continued)

168

APPENDIX A - continued

21.

22.

23.

24.
25.
26.
27.
28.
29.
30.

31.

32.

33.

34.

To subtract with decimal fractions and decimal
mixed numbers.

To multiply with decimal fractions and decimal
mixed numbers.

To divide with decimal fractions and decimal
mixed numbers.

To understand denominate numbers.
To reduce them, "ascending" and "descending."
To add with denominate numbers.
To subtract with denominate numbers.
To multiply with denominate numbers.
To divide with denominate numbers.
To read, write, and understand percents.
To manipulate the "three cases" of percentage:
I. Multiplying by a percent.
II. Dividing one number by another and
expressing the result as a percent.
III. Dividing a number by a percent to find what
number it is that percent of.
To understand the uses of percents in computing
interest, discounts, insurance premiums, taxes,

dividends, yields of bonds, etc.

To understand and compute square root and cube
root.

To compute the areas of certain surfaces and the
volumes of certain solids or the contents of
certain receptacles (Thorndike, 1921, pp. 83-84).

1.

9.
14.
15.
18.
19.
20.
21.
24.
45.
46.
51.
53.
54.
58.
60.
61.
67.
72.
74.
79.
87.
98.
99.

169

APPENDIX B

THORNDIKE'S TITLES OF LESSONS

vacation Activities
School Supplies

Playing "How Far"
Playing "Saving"
Telegrams, Express, and Freight
Playing "Cashier"

House Plans

Drawing to Scale

The School Program
weighing

Buying Candy

School Marks and Averages
Keeping Accounts

Buying Fruit

Henry's Orchard

How Lewis Earns Money
How Elsie Earns Money
At the Fish Market

A Christmas Party
Earning and Saving

At the Butcher Shop
Buying in Quantity
Report Cards

Earning and Saving
(Thorndike, 1921, p. 96)

170

APPENDIX C

KNIGHT'S ANALYSIS OF STEPS
IN THE PROCESS OF PROBLEM SOLVING

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171

APPENDIX C - continued

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10.
11.
12.
13.
14.
15.

16.

17.

18.

19.

172
APPENDIX D
SKILLS INVOLVED IN PROBLEM
SOLVING BY BRUECKNER AND MELBY

Ability to tell what facts are given.
Ability to tell what question the problem asks.
Ability to select essential facts.
Ability to estimate answers.

Ability to tell how to check answers.

Ability to name the process to use in solving
one-step problems.

Ability to name process in order used to solve
two-step problems.

Knowledge of vocabulary.

Knowledge of essential denominate numbers and
units of measure.

Knowledge of essential principles and concepts.
Ability to judge absurdities.

Ability to check true and false statements.
Ability to assemble essential data.

Ability to read accurately and exactly.
Ability to follow directions.

Ability to attack the solution of a problem
in a systematic manner.

Ability to apply processes in local situations.

Ability to interpret tables found in reference
books.

Ability to use the index, table of contents,
etc., as aids in studying.

(continued)

173

APPENDIX D - continued

20.

21.

22.

23.

24.
25.
26.
27.

28.

Ability to understand quantitative concepts
in map reading.

Range of information in application of
arithmetic.

Ability to make analogies.

Ability to answer specific questions about
problems.

Ability to formulate problems from given data.
Ability to illustrate uses of processes.
Ability to detect cues in solving problems.
Knowledge of historical background.

Ability to restate a problem in the words

of the pupil (Brueckner and Melby, 1931,
p. 227).

174
APPENDIX E

POLYA'S MODEL: THE QUESTIONS
First. UNDERSTANDING THE PROBLEM

What is the unknown?

What are the data?

What is the conditigp?

Is it possible to satisfy the condition?

Is the condition sufficient to determine the unknown?
Or is it insufficient?

Or redundant?

Or contradictory?

Draw a figure.

Introduce suitable notation.

Separate the various parts of the condition.
Can you write them down?

Second. DEVISING A PLAN

Have you seen it before?

Or have you seen the same problem in a slightly
different form?

Do you know a related problem?
Do you know a theorem that could be useful?
Look at the ppknown!

And try to think of a familiar problem having the
same or a similar unknown.

Here is a problem reigted togyou£§_gnd gplved before.
Could you use it?

Could you use its result?

Could you use its method?

Should you introduce some auxiliary element in order
to make its use possible?

Could you restate the problem?
Could you restate it still differently?
Go back to definitions.

(continued)

175

APPENDIX E - continued
If you cannot solve the proposed problem try to
solve first some related problem.
Could you imagine a more accessible related problem?
A more general problem?
A more special problem?
An analogous problem?
Could you solve a part of the problem?

Keep only a part of the condition, drop the other
part; how far is the unknown then determined, how
can it vary?

Could you derive something useful from the data?

Could you think of other data appropriate to determine
the unknown? “

Could you change the unknown or the data, or both if
necessary, so that the new unknown and the new data
are nearer to each other?

Did you use all the data?
Did you use the whole condition?
Have you taken into account all essential notions
involved in the problem?
Third. CARRYING OUT THE PLAN

Carrying out your plan of the solution, check each
step.
Can you see clearly that the step is correct?

Can you prove that it is correct?
Fourth. LOOKING BACK

Can you check th§_£esulp?

Can you check the argument?

Can you derive the result differently?
Can you see it at a glance?

Can you use the result, or the method, for some other
problem? (Polya, 1945, pp. inside-front and inside-
back covers)

11.

12.

13.

14.

176

APPENDIX E

MATHEMATICS PROJECTS AND PROGRAMS
The Boston College Mathematics Institute
The Cambridge Conference on School Mathematics

The DevelOpmental Project in Secondary
Mathematics at Southern Illinois University

The Greater Cleveland Mathematics Program (GCMP)

The University of Illinois Committee on School
Mathematics (UICSM)

The University of Maryland Mathematics
Project (UMMaP)

The Minnesota Mathematics and Science Teaching
Project (MINNEMAST)

The Nuffield Mathematics Project
The Ontario Mathematics Commission
The School Mathematics Study Group (SMSG)

The Stanford Program in Computer-Assisted
Instruction (CAI)

The Southern Illinois Project-Comprehensive
School Mathematics Project (CSMP)

The Syracuse University-webster College Madison
Project

Unified Science and Mathematics for Elementary
Schools (USMES)

(National Council of Teachers of Mathematics,
1970, p. 284; weiss, 1978. PP. 6-9)

177

APPENDIX G

FENNELL'S STAIRCASE OF PROBLEM SOLVING

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BI BLIOGRAPHY

 

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