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A CLINICAL INVESTIGATION OF THE UNDERSTANDING OF EXPONENTS

BY REMEDIAL ALGEBRA STUDENTS AT A FOUR YEAR COLLEGE

BY

Fred J. Wilson

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Educational Administration

1985

ABSTRACT

A CLINICAL INVESTIGATION OF THE
UNDERSTANDING OF EXPONENTS BY
REMEDIAL ALGEBRA STUDENTS AT

A FOUR YEAR COLLEGE

by

Fred J. Wilson

The purpose of the study was to gain information about
the manner in which successful and unsuccessful college students
in beginning remedial algebra and intermediate remedial algebra
view and apply the algebraic processes relevant to the concept
and related principles of exponent.

The theoretical aspect of the investigation took the form
of a synthesis of the theories of Brunner, Skemp, Gagné, and
Krutetskii, combined with the cognitive modes suggested by
Erlwanger to develop a model for use in the determination of
each student's "understanding" in terms of Skemp's "relational"
and ”instrumental", and Bruner's "iconic" and "symbolic" modes
of knowledge representation. A "thinking-aloud" interviewing
technique was used in an effort to rate each of fourteen re-
medial algebra student's thought processes as they solved a
pre-determined set of problems pertaining to exponents. The
students were designated as using one or more of the following

levels of understanding: Instrumental-Iconic, Instrumental-

 

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Symbolic, Relational-Iconic, or Relational-Symbolic.

It was found that both successful and unsuccessful stu-
dents operate primarily at the Instrumental-Symbolic level,
with the successful intermediate algebra students occasionally
using a Relational-Symbolic mode, and only rarely a Relational-
Iconic mode. Neither successful nor unsuccessful students used
numerical imagery to any extent even when suggested by the
interviewer. The major source of difficulty appeared to be in
the transition from a purely arithmetical problem to an algebraic
one involving a variable exponent and a variable base. It was
determined that the beginning algebra students, both successful
and unsuccessful, had an "understanding" only of positive integer
exponents with integer bases.h They had virtually no understand-
ing of the exponential properties. The intermediate algebra
students, additionally, had a limited instrumental understand-
ing of negative integer, zero, and rational exponents as well
as the exponential properties.

Suggestions for future research are indicated. Also, an
extensive bibliography of learning theory as well as imagery

research is provided.

I.
)

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H.

ACKNOWLEDGEMENTS

I wish to express appreciation to the following people
for their encouragement and assistance in this project:

Dr. William Sweetland (deceased) for giving me faith
to begin doctoral studies;

Dr. William Fitzgerald for his patient guidance in this
study;

Dr. Max Raines for his constant encouragement and aid;

Dr. Perry Lanier for his constant probing in an attempt
to keep me on a theoretical path;

My family for their patience, understanding, and
encouragement through several years of study; and

My wife, Mavis, for the many long and late hours spent

typing this paper.

ii

Chapter

I.

II.

III.

TABLE OF CONTENTS

Page
THE PROBLEM AND ITS BACKGROUND ...' ............. 1
Introduction .......... ........................ 1
Purpose of Study ......... ...... ....... ........ 7
Theoretical Background ........................ 9
Skemp's Theory of Understanding ..... ....... 10

Bruner's Theory of Knowledge Representation. 10
Krutetskii's Theory of Mathematical

Generalization ......... ...... ...... ..... . 11
Gagne's Theory of Hierarchical
Prerequisites ...................... ...... 13

A Theoretical Synthesis-Skemp, Bruner,
Krutetskii, and Gagné .................... 15
Definition of Terms ....... ................... . 16
Overview ...................................... 19

REVIEW OF THE LITERATURE ......... .... ........ . 22

Learning Theory . .......... ............ ....... . 23
Associationism ............................. 24
Gestalt Psychology of Learning ............. 31
Meaningful Learning Theory ................. 4S

Piaget's Theory of Cognitive Development . 51
Educational Implications ....... ........ 72
Comparison with Stimulus Response ...... 77

Bruner's Theory of Mathematic Learning ... 78

Skemp's Theory of Mathematical
Understanding ............... ......... .. 87

Imagery Research .......................... 101
Related Mathematical Studies ................. 113

Exponents ................... ......... ..... 114
Concept Learning ...... .................. .. 118
Summary: Review of Literature ...... ......... 132

RESEARCH PROCEDURES........................... 134

The Interviews ... ...... .. ................... . 134
Selection of the Sample ..... .......... ..... .. 136
Selection of the Questions ................... 138
A Model for Analysis of the Protocols ........ 143
Summary ............ ...... ....... ....... ...... 149

iii

Chapter Page

IV. ANALYSIS OF THE INTERVIEWS .................... 150
The Use of the Problem Sets ................... 150
The Interviews ................................ 155

Math 111 - High Examination Scores .......... 157
Math 111 - Low Examination Scores ....... .... 173
Math 121 - High Examination Scores .......... 182
Math 121 - Low Examination Scores ........... 192
Observations Across Students ................. . 198
Category 1: Prerequisite Knowledge ......... 198
Category 2: Variable and Equality ...... .... 201
Category 3: Definition of Exponent ......... 203
Category 4: am.an = am+n; am/an = am-n .... 206

Category 5: (ab)n anbn; (a/b)n = an/bn ... 210

Category 6: (am)n = amn ..... ........... .... 212

Category 7: False Generalizations........... 215

Category 8: (a+b)2 = a2 + 2ab + b2 ....... .. 218

Summary - Analysis of Interviews .............. 219

V. SUMMARY AND DISCUSSION ........................ 224

Procedure ..................................... 224

Results 0.0.0.0... ........ .... ......... 00...... 226

Discussion ................................. ... 237

Limitations of the Study ................... .... 241

Suggestions for Future Research ............... 243
APPENDIX A: Hierarchy of Prerequisite Knowledge

for Exponents ..- .............. . ....... 246

APPENDIX B: Exponent Pre-Test ........................ 247

APPENDIX C: Consent Form ............................ 248

BIBLIOGRAPHY ....................................... .. 249

iv

 

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CHAPTER I

THE PROBLEM AND ITS BACKGROUND

Introduction

 

Why do some students, after completing one, two, or
more years of high school mathematics, including at least
one algebra course, enroll in college, only to find that due
to a low score on an entrance examination he/she has been
assigned to a "remedial" algebra course? Once in the course,
why do they continue to make errors on elementary material

such as stating (x2)3 = x5; x2 ° x3 = x6; and x2 + x3, at

various times to equal either x5 or x6?

It would be tempt-
ing to classify such errors as "careless" due to lack of
attention, or lack of concentration and effort on assigned
problems. However, as the literature on this topic indicates,
this answer is far too simplistic.

In addition to the above concerns, why do some remedial
algebra students, when confronted with the dual tasks of
x2 . x3, and (x2)3, adopt the Strategy of falling back to a
"numerical imagery" mode and substitute a number such as 2
for x, and use this answer to aid in the determination of
the correct result for the initial problem? Why do other
students never adopt this procedure and depend entirely on

their memory of the appropriate "law of exponents"? Do

some students lack practice in the use of such imagery?
l

How does the understanding of the concept of exponent differ
among students that use numerical imagery and those that do
not? Do some students lack an understanding of the use of
variable? Do some students lack an understanding of the
concept of exponent?

- This study was undertaken with the assumption that
difficulty with the concept of exponent is pervasive among
students in remedial algebra courses at the college level.
Due to the spiral nature of mathematical learning, this study
was undertaken with the further assumption that learner dif-
ficulties which occur in the more advanced topics of radicals
and logarithms are due to a lack of understanding of the
concept of exponent. Thus, the remedial algebra students'
understanding of the concept of exponent is an area that is
indeed worthy of investigation. Furthermore, this study was
undertaken with the belief that learner difficulties with
exponents for the most part, are not due to carelessness or
lack of effort, but are attributable to more deep-rooted cog-
nitive confusion, that can be amended by appropriate teaching
and learning strategies.

It should be noted that this study was intended to in-
vestigate the thought processes of students with respect to
the conceptual difficulties manifested in the learning of
remedial algebra. In this regard, the topic of exponents
was chosen due to the fact that inherent in this topic are
the ideas of variable, equality, definitions, and generali-

zations. However, exponent is merely an exemplar of many

3

equally appropriate t0pics that could have been chosen (for
example, factoring).
Robert Davis (1967) has stated:

The entire traditional ninth-grade algebra
course is based on a sequence of tricks to get
children to write down on paper what appear to
be correct answers, although the student more
often than not does not know what it all means,
if anything (p. 16).

Evidence that college and university faculties must
deal with students that lack an understanding of fundamental
algebraic concepts that were previously assumed to be assim-
ilated in a ninth-grade algebra course is strikingly presented
by Keimig (1983) in a report on academic standards. She notes:

The demographic depression and the prevailing
mood of decline, diminished resources, and
threatened retrenchment are new, at least to
this generation of faculty. So are the kinds
of students new to many institutions that have
altered their admissions practices and curric-
ula, as most institutions have done (p. 1)..

Citing a report by Roueche (1981), Keimig (1983) elaborates
on the characteristics of students entering college today:

The average high school graduate today has a "B"
average over four years of high school, yet reads
at the eight-grade level, a loss of two grade
levels in the last 10 years (p. 51).

Akst (1981), explaining that arithmetic and algebra
have traditionally been considered primary or secondary
school topics, states:

Yet in 1980 there were more than 600,000 students
enrolled in college remedial or developmental math
courses covering precisely this content. To aid
these students, twice as numerous as they had been
ten years, earlier, virtually all campuses have
established basic math programs... (p. l).

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Killian's (1980) research (cited in Keimig) indicates
that fifty percent of freshmen are concrete thinkers with
respect to Piaget's developmental stages (p. 46). Clearly,
the changing nature of college enrollments will challenge
the college mathematics professionals.

One of the underlying assumptions pertinent to today's
higher educational system is that there is great diversity
of intelligence within the student population. This intelli-
gence factor is generally seen as coinciding with an achieve-
ment factor. That is, when great differences in performance
are noted between individual students, it is expected that
there is a like difference in intelligence. Fifty years ago
only the most "capable" students were admitted to college in
the United States. Today's college students appear to be
more variable in terms of measurable attributes of background
and ability. However, while admissions policies have become
increasingly liberal, traditional expectations and attitudes
toward student achievement persist. With regard to the study
of mathematics, mathematics faculty frequently and implicitly
divide the students into two groups: those who can "under-
stand” theory, and those who must limit themselves to step-
by-step procedures that depend primarily on rote-learning
processes. It is difficult to determine if traditional
assumptions are in fact valid or merely a self-fulfilling
prophecy (Hackworth, 1981, pp. 51-53).

Research does exist to indicate factors other than

intelligence play a role in achievement. It has been

 

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5
conjectured that "cognitive entry skills" are accountable
for approximately 50 percent of the students variance from
the norm with respect to achievement using conventional
instruction techniques (Bloom, 1976) (reported in Hackworth,
1981, pp. 49-50). In addition, Bloom's research indicates
that approximately 20 percent of variance in end of course
achievement is based on "affective entry skills". Thus,
according to Bloom, nearly 70 percent of end of course
achievement can be attributed to factors which, at least
partly, are deficient for the remedial mathematics students.)
However, there is some hope for both students and faculty.
Bloom (1976) asserts that 95 percent of students can achieve
mastery (90 percent) of the instructional objectives when
provided with appropriate instruction (Hackworth, 1981 p. 51).
In addition, Carroll (1963), cited in Hackworth (1981),
suggests that based on his research, that if speed of learn-
ing is disregarded, there is a relatively small variation in
achievement level within the population (p. 50).

One of the key difficulties with most remedial instruc-
tional programs is that they are based on knowledge of the ‘
students "understanding" of assumed prerequisite concepts.
However, several researchers have pointed out with dismay
that instruments for measuring understanding are not available
(see for example, Underhill, 1976). Thus, teachers are faced
with with the twin delemmas of being asked not only to teach
for understanding, but also to determine the mathematical

understanding that remedial students bring to the learning

6

situation. Additionally, there is little agreement as to
the definition of mathematical "understanding".

David, Jockusch, and McKnight (1978) propose the
following as a definition of understanding:

Comparing input data with many existing things
you already know; looking for apparent incon-
sistencies or contradictions; making careful

note of the areas which can be used in the future
to guide future selection of solution methods;
trying to identify and retrieve an appropriate
”assimilation paradigm" or schema, and to
synthesize a new one if no appropriate old one
can be found in memory; making a careful critical
appraisal of how well the present situation
matches the retrieved schema that has been
selected; and trying to develop apprOpriate
"meta-language" in order to be able to analyze
the mathematical situation effectively (pp. 283-
284).

Skemp (1976) has described two levels of understanding:
"instrumental" understanding, and "relational" understanding.
Skemp described “instrumental understanding" as "rules with-
out reason" and "relational understanding" as "knowing what
to do and why". Skemp states:

In instrumental understanding I would until

recently not have regarded as understanding at

all. It is what I have in the past described

as "rules without reasons", without realising

that for many pupils and their teachers the

possession of such a rule, and ability to use

it, was what they meant by "understanding" (p. 20).
Skemp (p. 21) cautions that instrumental understanding
”usually involves a multiplicity of rules" as students en-
counter new problems in mathematics. Eventually the massive

number of such rules becomes unmanageable. In contrast,

relational understanding has more general applications:

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To understand something means to assimilate it
into an appropriate schema (Skemp, 1971, p. 46).

He further notes:

An appropriate schema is one which takes into

account the long term learning task, and not

just the immediate one (Skemp, 1971, p. 51).

This then brings us to the heart of the problem for
educators who are involved in remedial mathematics instruc-
tion at the college level: When a particular concept is to
be presented by an instructor to the student in a learning
situation, how does the instructor know if the "appropriate"
schema is available? Secondly, if the determination is made
that the schema is not available, what instructional tech-
niques are appropriate? 'Due to the wide diversity of student
backgrounds in a beginning algebra class (remedial), are

there any commonalities with respect to relational under-

standing, in terms of prerequisites, that can be assumed?

Purpose of Study

 

This study focused on the preceding questions for the
concept, and the related principles, of exponent. It was
the purpose of this study to investigate the following
questions with respect to both successful and unsuccessful
(as indicated by test scores) remedial students in both
beginning algebra and intermediate algebra at the college
level:

1. Do remedial algebra students have a relational, instru-
mental, or no understanding of the prerequisites con-

jectured as necessary (as advocated by Gagné) for success

8

in dealing with the concept of exponent.

2. Do remedial algebra students have a relational, instru-

mental, or no understanding of the concept of exponent?

a.

How does the understanding of positive, negative,
(both integral and fractional) and zero exponents
differ in the same student? Between students?
How does the understanding of explicit number
exponents and literal exponents differ in the

same student? Between students?

3. Do remedial algebra students have the ability to general-

ize (as defined by Krutetskii) the various properties of

exponents?

a.

Can the source of "false generalizations" be
determined?

Have students that appear to have generalized
the properties of exponents (relational under-
standing), merely generalized the symbolic

notation (instrumental understanding)?

4. What types of imagery (Bruner's enactive, iconic, and

symbolic) do students use when working with the concept

of exponent?

a.

Does the imagery used differ, and in what respect,
for students at the relational and instrumental
levels of understanding?

Can a student who is operating at the instrumental
level be "pushed" by way of hints and guided

questioning to use numerical imagery as an aid

9
to relational understanding?

5. Do successful students (as determined by a letter grade
on a test) differ from unsuccessful students with respect
to the four questions above?

Despite its inherent limitations, a clinical interview-
ing methodology has been deemed by many reSearchers as the
appropriate procedure for investigating internalized thought
processes (see, for example, Suydam and Dessart, 1980;
Lester, 1980; Fennema and Behr, 1980; Kantowski, 1977;
Confrey and Lanier, 1980). .

Seven students were selected from each of a remedial
beginning algebra class and a remedial intermediate algebra
class. One-half of these students were determined by their
instructor as obtaining the highest scores, and one-half as
obtaining the lowest scores, on a unit examination pertaining
to exponents. A semi-structured interview procedure, using
the "thinking-aloud" technique was followed as students solved
problems relating to the concept and principles of exponent.
The interviews were recorded on audio-tapes, and significant
body actions of the interviewees were recorded on paper by
the interviewer. ,The tapes were later qualitatively analyzed
for any commonalities of behavior with respect to the re-

search questions previously mentioned.

Theoretical Background
A theoretical framework for the investigation of the

preceding research questions was developed by combining

10

critical features from each of the theories of Skemp, Bruner,
Krutetskii, and Gagnél
Skemp's Theory of Understanding

Skemp (1979A) has proposed three types of "underStanding".
Two types: "instrumental" understanding, and "relational"
understanding were deemed as pertinent to this study. Skemp
gives the following definitions:

”Instrumental understanding" is the ability to

apply an appropriate remembered rule to the

solution of a problem without knowing why the

rule works. '

"Relational understanding" is the ability to

deduce specific rules or procedures from more

general mathematical relationships (p. 45).
Skemp's general definition of "understanding" is given in
one of his earlier writings: "To understand something means
to assimilate it into an appropriate schema" (Skemp, 1971,
p. 45).
Bruner's Theory of Knowledge Representation

Bruner has conjectured that human beings process and
represent information by way of three parallel systems #-
one through manipulation and action, one through perceptual
organization and imagery, and one through symbolic apparatus.
Bruner has called these modes of representation, respectively
"enactive", "iconic", and "symbolic" (Bruner, 1970, p. 291).
He notes, "Their appearance in the life of the child is in
that order, each depending upon the previous one for its
development, yet all of them remaining more or less intact

throughout life" (p. 291).

Bruner (19663) has specified that learning of

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mathematics follows precisely this order.

we would suggest that learning mathematics
reflects a good deal about intellectual
development. It begins with instrumental
activity, a kind of definition of things

by doing them. Such operations become
represented and summarized in the form of
particular images. Finally, and with the
help of a symbolic notation that remains
invariant across transformations in imagery,»
the learner comes to grasp the formal or
abstract properties of the things he is
dealing with. But while, once abstraction

is achieved, the learner becomes free in a
certain measure of the surface appearance of
things, he nonetheless continues to rely upon
the stock of imagery he has built en route to
abstract mastery. It is this stock of imagery
that permits him to work at the level of heur—
istic, through convenient and‘nonrigorous means
of exploring problems and relating them to
problems already mastered (p. 68).

In a different statement relating to the extreme im-
portance of adequate imagery background he states:

We reached the tentative conclusion that it
was probably necessary for a child, learning
mathematics, to have not only a firm sense of
the abstraction underlying what he was working
on, but also a good stock of visual images for
embodying them. For without the latter it is
difficult to track correspondences and to check
what one is doing symbolically (Bruner, 1966B,
p. 66).

Krutetskii's Theory of Mathematical Generalization
Krutetskii (1976) asserted:

...there has been no fixed definition of
mathematical ability that would satisfy everyone.
Perhaps the only thing about which all investi-
gators agree is that one should distinguish
between ordinary, "school" ability for mastering
mathematical information, reproducing it, and
using it independently and creative mathematical
ability, related to the independent creation of
the original product that has a social value
(p. 21).

12

Krutetskii (1976) investigated the mathematical
"ability" of "capable", "average", and "relatively in-
capable" students (p. 1976). Placed among the relatively
incapable students were those students "...who could not
work problems that went beyond the limits of the standard
they had mastered", and students whose "...mathematical
habits were formed with difficulty, required a large number
of exercises, and were shaky -— disintegrating easily in
the absence of practice" (p. 177).

Krutetskii has subdivided mathematical ability into
the categories: "generalization", "reversibility", "flexi-
bility”, and "curtailment" (Krutetskii, 1976, pp. 195-198).

Of particular interest to this study is Krutetskii's
definition of the ability to generalize as the

...ability to see something general and known to

him in what is particular and concrete (subsuming

a particular case under a known general concept),

and (2) the ability to see something general and

still unknown to him in what is isolated and

particular (to deduce the general from particular

cases, to form a concept) (Krutetskii, 1976,

'p. 237).

Rachlin (1981) notes that Krutetskii's definition of
generalization ability as the ability to generalize algebraic
operations and mental processes, to a large extent is equi-
valent to the ability that is traditionally, in America,
called ”transfer".

Kruetskii (1976) reports that mathematically capable

students readily found the generality behind particular, and

externally different, details (frequently, "on the spot").

13
Incapable students on the other hand had to be extensively
tutored on material which covered all the various cases and
combination of irrevelant features, to attain even a very
elementary degree of generalization (pp. 240—241).
Gagne's Theory of Hierarchical Prerequisites

Gagne's approach to the teaching of mathematical con-
cepts comes primarily from a combination of the neobehavior-
ist psychological position and the task analysis model that
historically has developed from the industrial and military
training models. Gagné suggests that instruction should
always begin with a task analysis of the instructional ob-
jectives. The primary question to Gagné is, "What should
the learner be able to do when instruction is completed?"
This "capability" must then be stated specifically and be-
haviorally (Crosswhite, et a1, 1973, p. 7).

The capability can be conceived of as a terminal beha-
vior which eventually is placed at the top of an often complex
pyramid. This complex pyramid is developed by asking at each
level, "What would the learner have to know in order to do
this?" Gagné would continue this determination of pre-
requisite knowledge until the fundamental units of learning
are reached - For Gagné this would be classical or condi-
tioned responses (Crosswhite, et al, 1973, p. 7).

Gagné (1970) has stated:

Thus it becomes possible to "work backward" from

any given objective of learning to determine what

the prerequisite learnings must be-—if necessary,

all the way back to chains and simple discrimin-
ations. When such an analysis is made, the result

14

is a kind of map of what must be learned. Within
this map, alternate "routes" are available for
learning, some of which may be best for one learner,
some for another. But the map itself must represent
all of the essential landmarks; it cannot afford to
omit some essential intervening capabilities (p. 242).

He further notes:

Representations of learning hierarchies are limited
to the description of and interrelations of intel-
lectual skills. Thus, a hierarchy does not rep-
resent external conditions of learning, as they
have been described in previous chapters. Accord-
ingly, the learning hierarchy does not picture

the procedures of instruction. The intention is
not to depict how an individual may come to learn
a particular intellectual skill -— what kind of
instruction to give, how much guidance of learn-
ing to introduce, what sequence of communications
to follow, and so on. What is shown is only the
internal conditions for learning, the prerequisite
capabilities that will provide the positive trans-
fer to a new learning event. Identifying these
capabilities and assuring their availability are
matters of critical importance for instructio

(p. 242). -

In this study the concern was not with Gagne's behavior-
istic approach to learning, but with his admonition that an
analysis of prerequisite knowledge and skills is vital to
the design of any instructional unit. Gagne has indicated
the omission of any essential step in the hierarchy of pre-
requisite knowledge may lead to long-term "blocking" of
future understanding (Gagné, 1970, p. 243). In particular,
this investigation was interested in the question of whether
a student's understanding of the concept and principles of
exponents was ”blocked" by the lack of understanding of pre-

requisite concepts.

 

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15

A Theoretical Synthesis —- Skemp, Bruner, Krutetskii, and
629.13

That portion of Skemp's theory which pertains to "in-
strumental" understanding and "rleational" understanding was
combined with Bruner's modes of internal representation of
information, ”iconic" and "symbolic" to form a two-by-two
matrix for the investigation of the understanding of exponents.
The understanding of the concept of exponent by each student
interviewed was qualitatively determined to belong to one
cell of the following matrix, which was developed from sug-

gestions by Erlwanger (1975C) and Alexander (1977):

UNDERSTAND ING
Instrumental Relational

 

Iconic

Symbolic

MODE

 

 

 

 

 

 

Problems and questions presented to the interviewees
were designed to elicit information that enabled a diagnosis
to be made with respect to a pre-determined set of conjectured
prerequisite concepts as advocated by Gagné. In addition,
problem sets and interviewer questions were arranged in an
order that anticipated the determination of generalizability
(as defined by Krutetskii) on various exponential concepts

and principles.

16

Definition of Terms

Canceling The dividing of a common factor from numerator

 

and denominator of a fraction, based on the identity and
multiplicative properties. For example:

.4_=.Z_L_2=2.Z—1
6 2-3 2 3‘

2:3
3 3°

Explicit Number A non-literal number such as 0, 2, k, and

\/'7'.

Exponential Definitions
a. Positive Integer Exponent

If n is a positive integer (called the "exponent"),
and a any real number, (called the "base"), then an
is a short notation for the number of factors of a.
For example, 23 means 2 ° 2 ' 2.

b. Negative Integer Exponent
a-n, where a is a non-zero real number and n is a

l I I n
pOSitive real number, means the rec1proca1 of a .

For example, 2.3 is equivalent to 1/23.
c. Zero Exponent
a0 is defined to be 1, where a is any non-zero real
number. For example, 20 is equivalent to l.
d. Rational Exponent
an/m’ where a, m, and n are real numbers, is the mth
root of an or the nth power of the mth root of a.
For example, 82/3 is the cube root of 8 squared;
3\/g3 = 4 , or equivalently the square of the cube
root of 8; (3\/8')2 = 4.

Exponential Properties For any of the variations of exponents

l7

and bases previously mentioned, these are generalizations of

manipulative techniques summarized in the following:

a. am . an = am+n
As an example, 22~ 23 would be (2-2) - (2-2-2) and
by arithmetical properties equivalent to 22+3 or 25.

b. (am)n = amn

As an example, (22)3 by arithmetical properties would

2 2+2+2 = 23(2) 6

be (22) (2 ) (22) and thus 2 = 2 .

m -
c. a /an = am n

As an example, 24/23 would be (2-2-2-2)/(2-2-2) and
by arithmetical properties three of the four factors

could thus be "canceled", Zl-Zl-Zl-Z, leaving one

1 1 1
factor of 2, that is 24"3 = 2.
d. (a-b)n = an/bn
As an example, (2'3)2 means (2'3) ° (2'3) and by
arithmetical properties is equivalent to (2:2) - (3-3)
and thus 22°32.
e. (a/b)n = an/bn
As an example, 62/32 is equivalent to 6-6 and by

3:3
”canceling" the factors of 3, 62-62 equivalent to

1 1

2, that is, (6/3)2.

2-2 or 2
False Generalization This term is used in this present study
to represent two possible kinds of erros. First, a student
may change a problem to make it "fit" a known generalization,

and secondly, the student may change a generalization to

make it fit the problem. The second sense may, in fact, occur

 

 

aw.“ Q

18

when a student is not aware of the generalization, but forms
an incorrect generalization on his/her own.

Literal Number In this study the term is equivalent to the
term "variable".

Numerical Imagery In this study the term is used for the
situation in which a student working with an expression con-
taining variables, uses the substitution of explicit numbers
for the variable in order to gain more understanding of the
variable expression.

Remedial Algebra Algebra that is taught at the college level,
but is a normal part of a standard high school algebra sequence.
In this study, there are two such courses. Beginning Algebra,
Math 111, is basically equivalent to the first year of high
school algebra and Intermediate Algebra, Math 121, which is
basically equivalent to the second year of high school algebra.
They are remedial in the sense that a student having taken
them in high school and understanding enough to successfully
reach the minimum score on the ACT examination, is not re-
quired to take them at Ferris State College. Additionally,

no credit toward a mathematics major is granted for these
courses. I

Variable In this study, the assumption was made that students
would visualize "variable" as a placeholder for a number, or
simply a letter which stands for a number. Variable is used
in the present study in two senses. First, as a "generalized
number" in such expressions as 2a + 3a = 5a. That is, the

expression is true when any number is substituted for a.

19
Secondly, as a "specific number", as in the solution to
x + 4 = 7, where only the number 3 can be substituted for x

to make an equality.

Overview

In Chapter II, Review of the Literature, the concept
of understanding as used in this study, is placed in an
American historical perspective of understanding. Connection-
ism, as defined through the theories of major writers, such
as Thorndike, is tied in historically with the concept of
”meaningful learning", as developed from the theories of
Brownell, Piaget, Bruner, and Skemp. In addition, what re-
searchers had to say about "algebraic thinking" is reviewed.
In this regard, studies are reviewed which have emphasized
exponents, remedial algebra for college students, algebraic
errors, and algebraic concepts. Also, to place Bruner's
concept of imagery into the major present-day psychological
theories of imagery, the works of the major researchers in
the field of imagery are reviewed.

In Chapter III, Research Procedures, the method of
selection of the interviewees is detailed. Also, the pre-
determined questions and problems for the interviews are in-
dicated along with the theoretical basis for their selection.
In addition, the interview techniques and procedures are
elaborated.

A model for analysis of the interview data is presented
in Chapter IV, Analysis Of Interviews. The theories of Skemp

and Bruner are integrated to form a matrix, as suggested by

20
Erlwanger (1975C) and Alexander (1977), for the purpose of
classifying the understanding of exponents exhibited by the
individual during the interviews. Excerpts from the audio-
taped interviews are presented verbatum so that the reader
may see the model in action as it is applied to the research
questions previously indicated.

This study was undertaken with the idea that while
"understanding" is a somewhat elusive concept with respect
to determination, its importance, however, cannot be over-
looked, due to the massive number of "remedial" students
entering the higher educational system in America.

It has been stated by Brown (1942):

In general, I've come to the conclusion (there
are exceptions, of course) that the ease and
accuracy with which any educational outcome is
measured is in direct proportion to its unimpor-
tance. That is, the easy items to measure
accurately are the ones which make least dif-
ference whether they are measured or not (p. 354).

In a similar vein Craig (1966) notes:

Finally, there is admittedly insufficient
evidence of how well the concepts derived in
psychological theory or the laboratory apply
in the classroom. The risks of overgeneral-
ization are great; but they may be reduced
somewhat if extensions of psychological con-
cepts of teaching are proposed, not as fact,
but as hypotheses that merit further tryout
in classrooms. This is what we have advised.
With this proviso, we suggest that any re-
maining risk in the use of learning theories
is preferable to the alternative hazards of
dependence on hunches, uncritical imitation,
or habit (p. 82).

Perhaps Bayles (1960) best articulated the hope for

teaching the remedial mathematics students enrolling in

21
American colleges:

In reality, preference for understanding-level
over memory-level teaching means belief in the
proposition that the only way to make teaching
genuinely practical is to make it basically
theoretical (p. 194).

It is with this attitude that this study was undertaken.

CHAPTER II

REVIEW OF THE LITERATURE

A thorough seanfl10f the literature has revealed no
in-depth studies using interviews to examine college students'
concepts of exponent. Several studies have classified the
various types of errors that students make in algebraic study.
Some researchers have studied and classified the factors
which contribute to algebraic success. In a similar vein
other studies have been carried out to classify concepts
which are fundamental to algebraic learning.

Since this present study is about understanding, the
review includes a historical perspective of "meaningful learn-
ing" therory. Additionally, because of the importance of
imagery to this study, a review of imagery literature is
included.

Elementary algebra is traditionally a course which is
taught at either the eighth or ninth grade level of school,
consequently, this review will cite not only literature per-
tinent to college students, but also will include pre-college
studies. Since to some researchers algebra is "generalized
arithmetic", it follows that there is not a fine line that
distinguishes between studies which relate to arithmetic.

This review then, does include literature which pertains

22

23

particularly to arithmetic, but is deemed general enough

to be pertinent to this investigation of algebra learning.

Learning Theory

Heidbreder in 1924 (cited in Bourne, 1966, pp. 24-25)
anticipated the two major lines of development which grew
to characterize the theories of concept development. One
line views the human organism as a passive recipient of environ-
mental information. A "composite photograph" is gradually
built up from examples in which the commonfeatures stand
out and the irrelevant features are washed out. The learner
is viewed as an organism which contributes nothing except
memory of previous examples. Internal activity which operates
on an incoming event is seen as non-existent as far as contri-
buting to concept formation. That is, the learner is passive.
This line of reasoning has been used in various forms to
define the arguments of psychologists who adhere to the
"associationistic" and "behavioristic" therories of learning.

A second theoretical line conjectures that humans are
not passive learners, but actually are active participants
in the concept formation process. The learner always develops
some hypothesis about the unknown concept. Over a sequence
of examples and nonexamples, the learner, after possibly
forming and rejecting several hypotheses, will settle on
a correct one and consequently the concept is formed. In
this therory the learner is always active, registering informa-
tion modifying any hypothesis which is incompatible with

incoming data (Bourne, 1966, p. 25). In this way concepts

24

are "constructed" by the learner.

In this section the major features of researchers in
both the stimulus-response associational theory and the con-
structivist hypothesis-testing theories will be detailed.
Additionally, the Gestaltist-field theory, which is in some
sense a middle ground between the associationistic and the

constructivist theories, will be reviewed.

Associationism

The primary psychological figure in America in the early
part of the 20th century was Edward L. Thorndike. Although
Thorndike's work built upon the works of his teacher, William
James, who in turn was influenced by Alexander Bain, who in
turn was influenced by associationist theorists all the way
back to Aristotle, it is Thorndike's name that has become pre-
dominant (Sandiford, 1942, p. 107). Thorndike is perhaps
remembered best for his statement of the "law of effect",
which in modern-day language is thought of as "reinforcement"
(Resnick and Ford, 1981, pp. 12—13). In Thorndike's theory,
learning is seen as a problem of establishing proper con-
nections. This blended well with the doctrine of "social
utility" prevalent in his day. The social usefulness could
be used as a measure to decide which connections should be
encouraged in the learner. With his tradition of laboratory
experimentalism, he was thoroughly committed to the task of
transforming laboratory findings into a theory of classroom
instruction (McDonald, 1964, p. 8).

Thorndike formulated his ”law of effect" based on animal

25
experimentation. The experiment most frequently associated
with Thorndike in this regard consisted of placing a cat
in a wooden box that could be Opened by tripping a latch.
Eventually the trapped cat in frustration would accidentally
trip the latch and escape. The cat would then be placed
in the box again and the process repeated. Each time the
experiment was repeated, less time would be required for
the cat to escape. In Thorndike's view, the cat was not
"figuring out” how to escape from the box, rather the reward
of escape served to strengthen the bonds between the experi-
mental situation and the response that permitted escape
(Resnick and Ford, 1981, p. 12). Thus,

When a modifiable connection between a situation

and a response is made, and is accompanied or

followed by a satisfying state of affairs, that

connection's strength is increased: When made

and accompanied, or followed by, an annoying state

of affairs, its strength is decreased (Thorndike,

1913, p. 4).

Although he experimented mostly with animals, Thorndike
believed his principles applied equally well to humans. He,
along with other psychologists, called "connectionists" or
"associationists" believed all human behavior could be analyzed
in terms of two simple constructs. When broken down to irredu-

cible units, behavior was found to consist of stimuli (exter-

nal events) and responses (the subjecth reaction to the stimuli).

 

If a certain response given in reaction to a particular stimuli
brought a reward of some nature, then a bond, or association
(or "connection") was formed between the stimulus and response.
The more frequently a particular stimulus - response pair

was rewarded, the stronger the connection. Thus, the law of

26
effect suggested that practice followed by reward was the
principle manner in which human learning took place (Resnick
and Ford, 1981, p. 13)..

Thorndike started a tradition of bringing laboratory
results to teachers and educators. He had a particular interest
in the teaching of arithmetic and addressed this topic in
his 1922 book: The Psychology of Arithmetic. What teachers
needed to do was to find and make explicit the particular
domain of bonds that constituted arithmetic. After this,

"drill and practice" involved presenting bonds in a carefully
fashioned manner so that important bonds were practiced fre—
quently, and lesser bonds, less often. It should be noted

that bonds had an effect on each other, and that any particular
bond should be formed with the consideration for every other
bond that has been or will be formed. The rewards that served
to strengthen the particular bonds were obtained when arithmetic
problems were made interesting, practical, and fun (Resnick

and Ford, 1981, p. 15). This appears to be a part of Thorndike's
theory of learning mathematics that was ignored by many of

his disciples.

Thorndike's (cited in Alexander, 1977, p. 26) perception
of the learning of algebra, in terms of topics covered, was
indicated when he surveyed high school teachers as to which
topics should be taught. His summary included:

1. Involved manipulation of polynomial ex-

pressions is not a justifiable way of using the

high-school student's time.

2. Since the application of equations in
other high-school subjects is chiefly in the pro-

portion form, mastery of that form and other easy
equation forms should be secured.

27

3. It would be profitable to extend the field
of application of the construction of formulas
as well as their evaluation.

4. There is need for the careful development
of the art of criticism as applied to graphs.

5. The presentation of laws of means of mathe-
matical graphs should be encouraged.

6. The function concept should be used when
advantageous, but with economy (Thorndike, 1923,
pp. 82-83).

That algebra for Thorndike was mainly rote, is indicated

by his statement that learning algebra requires learning
.about one hundred and fifty rules" (p. 228).

Thorndike, however, also seemed to indicate that more
than rote learning was necessary. He felt the formulations
of ”habits" and "principles" was an integral part of algebra.
He notes (p. 239):

If we leave to habit everything that can be done

as well by habit, we gain an added dignity for

the matters that really are matters of principle...
It is not because he does not value rules and prin-
ciples in algebra that the psychologist often prefers
to use habits instead: it is rather because he

does value principles and does not wish them to

be misused and cheapened.

Thorndike's meaning of the words "habit" and "principle"
can be gleaned from his statement with respect to literal
numbers:

The principle We can represent numbers by letter,
develops into at least ten distinct habits of thought,
namely:

1. A letter may mean a particular number
of things, like men, boys, or eggs.
2. A letter may mean a particular number
of units, like cents, quarts, feet.
3. A letter may mean any one of a number
of numbers, like the number of dollars in the cost
of any number of suits of clothes of a certain
sort, or the number of square feet in any rectangle.
4. A plettefi may mean any number, as in (p + q)
(P ‘ q) = q .
5. If you call a certain number p. you may

~‘U

.u.‘

Q

28

call 3 times that number q or r or s or any letter
except p that you please, but it is commonly useful
to call 3 times that number 3p.

6. If you call a certain number p, you may
call 3 more than that number any letter except
p that you please, but it is commonly useful to
call it p + 3.

7,8,9. The same principle of consistency
and utility with p - 3, p/3 and 3/p.

10. If we call a certain number (say, the
profit Mr. A. made in January, 1922) p we don't
call it anything else and don't call p something.
different so long as we are thinking about the
problem to answer which we called that number p
(Thorndike, 1923, pp. 228-229).

Although, specifically referring to arithmetic, Thorndike
seems to indicate that his idea of mathematical learning
is more than the simple stimulus - response episode that
many of his followers advocated: He notes:

...I hasten to add that the psychologists of today
do not wish to make the learning of arithmetic

a mere matter of requiring thousands of disconnected
habits, nor to decrease by one jot the pupil's
genuine comprehension of its general truths. They
wish him to reason not less than he has in the

past, but more. They find, however, that you do

not secure reasoning in a pupil by demanding it,

and that his learning of a general truth without

the proper development of organized habits back

of it is likely to be, not a rational learning

of that general truth, but only a mechanical memorizing
of a verbal statement of it. (Thorndike, 1922,

p. 174-175).

Thorndike's view of algebra learning can be summarized
by the following statements:

If the 'set' or attitude of the mind toward the
first hundred or so operations with literal numbers
is permitted to become that of learning a queer
game, where you pretend to add, subtract, multiply
and divide letters, there is certainty that these
bonds themselves will be weak, and probability

that all later practice will be much less effective
than it should be (Thorndike, 1922, p. 329).

and

Algebra to most learners...is in large measure
forming more or less particular bonds or connections,

29

such as a x ab = azb, a(a + b) = a2 + ab, a means

la, -a x -b = +ab, learning to operate several

of these together as needed, organizing them further

into more inclusive habits and insight, summing

up what one has learned to do in rules, and thus

gradually attaining a sense of what it is right

to do with literal numbers and why (p. 246).

In summary, Thorndike believed the successful algebra
student is one who, upon the presentation of a problem, perceives
the structure of the problem, chooses the appropriate sequence
of connections, and from memory is able to produce the necessary
series of connections to solve the problem.

Thorndike's rules for generating specific sequences
of drill and practice were largely intuitive. Which bonds
were the easiest? How much practice was enough? How should
practice be organized? Perhaps Thorndike's greatest legacy
in mathematics education was the step he took in the direction
of bringing psychological theory to bear on instructional
problems. He focused attention on the content of learning
mathematics. Some of the questions he posed are still imr
pacting both research and education even today (Resnick and
Ford, 1981, p. 16).

Simultanemmfly with Thorndike's development of connection-
ism was Watson's promotion of "behaviorism". By "behavior”
Watson referred merely to muscular activity. Watson believed
humans learned the same as Pavlov's dogs, in terms of classical
conditioning. watson formulated the theory that learning
can be explained without Thorndike's law of effect. Learning
is dependent upon "frequency” and "recency". Watson's prin-

ciple of frequency states the more frequently one has responded

to a particular stimulus, the more likely that same response

30

will be given should the stimulus occur again. His principle
of frequency said the same held true with respect to elapsed
time since the response to the stimulus.

Watson and his disciple, Guthrie, and their followers,
became known as "contiquity" theorists due to their belief
that the stimulus - response bonds are strengthened by the
response occurring in the presence of the stimulus (Sandigard,
1942, p. 107; Harrison, 1967, p. 28). In contrast, in "operant",
or "instrumental" conditioning, the response must be made
before either a positive or negative reinforcer. The "effect"
of the stimulus response bond is the strengthening agent.

The followers of this theory are known as "reinforcement"
theorists (Klausmeier and Goodwin, 1971, p. 25).

Connectionism, as advocated by Thorndike, and Watson's
behaviorism, fell into disfavor in America during the later
1920's and 1930's. This was forthe most part based upon
the growing mood in America of “education for democracy".
Education became the hope of all classes, an instrument of
social reform. John Dewey with his psychology of pragmatism
that interrelated "aim", ”interest", and "intelligent action"
was chosen by educators as the prime spokesman against the
anti-equalitarian ideas of Thorndike. Dewey attached the
reflex arc of the connectionists, arguing that the stimulus
and response were not sharply distinguishable, but must be
viewed as organically related. For Dewey, "mediated experiences"
were the main psychological event. What the learner is involved
in when a stimulus is given will determine how the stimulus

is perceived, and consequently the response (McDonald, 1964,

'(J
o
H

an H
:' A.

.
._., . '-
ovo.‘

I
, A.
1'. Nu
.h”

o;
I'D-
(I)
(I!

31

p. 11). Dewey (1933) noted, "to grasp the meaning of a thing, an
event, or a situation, is to see it in relation to other things;
to note how it operates or functions, what consequences follow
it, what uses it can be put to" (p. 137). In his attack on con-
nectionism, and his view that learners' interests and aims are
paramount in education, Dewey's philosophy was easily reconciled
with the growing Gestaltist view of learning (Harrison, 1967, p. 30).
Gestalt Psychology of Learning

In the late 1920's and early 1930's the domination of the
American psychological scene by behaviorism and connectionism was
challenged by the importation of Gestalt psychology from Germany.
A Cornell psychologist RuM. Ogden translated Koffka's German writ-
ings into English. Hohler's works were also translated. Although
the Gestalt theory had been developing in Germany since first
announced by Max Wertheimer in 1912, it was Kohler and Koffkals
writings, plus their extended visits to America, that brought the
new theory to the attention of the American psychologists.
(Hilgard, 1964, p. 54).

Koffka's 1924 book, Growth in Mind (cited in Hilgard and
Bower, 1966), detailed the criticism of trial-and-error learning
as conceived by Thorndike (and upon behaviorism). Koffka's view
was supported by Kohler in his 1925, book Mentality of Apes.
Kohler showed how apes could by-pass the laborious processes of
stamping out incorrect responses and stamping in correct ones, as
indicated in Thorndike's theories developed from experiments with
cats (Hilgard and Bower, 1966, p. 231). A typical problem

as described by Kohler, was the "detour" in which a barrier was placed

32
between an ape subject and the goal object. The animal had
to detour around the barrier to obtain the goal object, such
as a banana. Other settings required the animal to join
two sticks to form a rake in order to obtain food. The be-
haviorial descriptions of the animals solving problems indi-
cated that the apes observed the goal object for some period
of time, then very rapidly solved the problem. According
to the Gestalt point of view, the animals were able to re-
organize their perception of the world, thus achieving "in-
sight” into the problem, and consequently perceive most aspects
of the problem in its entirety or "whole", a "gestalt" (Ellis,
1972, pp. 185-186).

Gestalt principles were particularly attractive to the
educational progressives, who under the leadership of people
such as Dewey, had already taken the position against rote
learning and in favor of insight as an educational goal.

The principles made it unnecessary to rely on the conditioning
stimulus-response process to explain all human learning
(Woodring, 1971, p. 91). There were voices raised in protest
against the acceptance of insight as a competitor to associa-
tive learning. Guthrie (1935) (cited in Hilgard, 1964) felt
that sudden learning would have to fall into the category

of luck, and hence lay outside the influence of science.
Others thought of insight as an extreme case of transfer

of training. Even others saw insight in the descriptive
sense: learning could be wigh insight but not p1 insight
(Hilgard, 1964, p. 61). Hilgard notes that some advocates

of associationism taught practical skills by such methods

33

as carefully constructed motion pictures, accompanying narra-
tives, and practice in actual manipulations. The emphasis
was not on habits, but upon inherent organization, meaningful-
ness of parts, and perceptual patterns. This is exactly
the emphasis that one alerted to insight would propose (Hilgard,
1964, p. 62). Hilgard maintains that criticism of association
theory was not based on the importance assigned to past exper-
ience, but rather on the notion that past experience guarantees
the solution to a problem, no matter how the problem is pre-
sented.. In contrast, the insight point of view is that,
with sufficient past experience, more difficult problems
can sometimes be solved by a learner due to the structural
display of the problem, but other learners will solve the
problem because they are better able to generalize and not
be misled by the structural features (Hilgard, 1964, pp.
61—62).

Gestalt psychology had its greatest success in the field
of perception. It demonstrated the role of organization
and background upon phenomenally perceived processes in its
aeanflt.on the associationist hypothesis that a percept is
made up of sensation-like elements bound together. When
the gestalt psychologists turned to learning theory they
used the same arguments against the associationist's reflex
arc of learning. Koffka's starting point for learning was
the assumption that the laws of organization in perception
are applicable to learning. Since the difficulty of a problem
depends upon the initial structuring of the field, as is

open to observation of the learner, the correct solution is

34

largely a matter of perception (Hilgard and Bower, 1966,
pp. 232—233). The guiding principle of gestalt learning
theory is that psychological organization tends to move in
one general direction, toward the "good" gestalt. A "good
gestalt has such properties as regularity, simplicity and
stability. Psychological organization tends to follow laws
of "similiarity", "proximity", "closure", and "good continu-
ation". As an example of the law of "closure", in a problematic
situation the whole is seen as incomplete and a tension is
set up toward completion. The strain for the learner to
complete acts as an aid to learning, and the achievement
of closure is satisfying. The gestalt preference is for
conceiving psychological processes as a product of the present
field, and the role of past experience is explained in a
theory of "memory traces" (Harrison, 1967, p. 32). The essen-
tial features are:

A trace is assumed from prior experience and repre-

sents the past in the present, a present process

is assumed which can select and communicate with

the trace, and there is a resulting process of

recall or recognition. Against the theory that

memory leads to decay and fuzziness, the Gestaltists

theorize that it leads to changes in the direction

of increased clarity. In skill learning the trace,

as part of the field process, exerts an influence

in the direction of making it similar to the process

which initially produced the trace. While the

skill is being learned, the trace is somewhat un-

stable. The trace and process interact and greater

stability is gradually reached. Since the trace

system obeys dynamic laws, it also undergoes stabil-

izing changes over periods of no practice, greater

improvement of skill learning will occur with distri-

buted practice as proposed to mass practice (Hilgard

and Bower, 1966, p. 239).

Wertheimer (1959), summed very succintly the Gestaltist

position on problem solving:

35

(S is the situation in which thought process start,
S is when after a number of the process ends, the
pgoblem is solved).

When one grasps a problem situation, its struc-
tural features and requirements set up certain
strains, stresses, tensions in the thinker. What
happens in real thinking is that these Strains
and stresses are followed up, yield vectors in
the direction of improvement of the situation,
and change it accordingly. S is a state of affairs
that is held together by inner forces as a good
structure in which there is hanmony in the mutual
requirements, and in which the parts are determined
by the structure of the whole, as the whole is
by the parts (p. 239).

The process does not involve merely the given parts
and their transformations. It works in conjunction with
material that is structurally relevant but is selected from
past experience, from previous knowledge and orientation.

In all this, those movements and steps are strongly
preferred which change the state of affairs in S1 along a
structurally consistent line into 82 (pp. 239-240).

He added advice for teachers:

In the light of my teaching experiences it would
seem best -- especially at the start to show, to
"teach" as little as possible. It would seem desi-
rable to avoid as much as possible the giving of
ready-made steps. The child should be confronted
with tasks with which he tries to deal himself.

Let the child face problems, let him receive co-
operative help as he needs it, but let him not
simply copy or repeat given procedures. I would
avoid as much as possible anything that might intro-
duce a mechanical state of mind, an attitude of
drill (p. 276).

Wertheimer pointed out the inadequacies of the associa-
tionistic theory of learning in comparison to the Gestalt
conception of learning.

Association may mean chaining items together
in an and-sum of connections that has by nature

no structure, as in the usual theory of learning
rote syllables. Or at the opposite extreme, it

 

36

may mean realizing of structural belonging in which
items require each other as parts in a context--
including the enduring effects of that realization.

Repetition may mean that the same piecemeal,
blind connection occurs over and over again; or
it may mean the change from an un-understood and
sheer additive pairing to the realization of a
structure in which the meaning of the items becomes
that of parts in a characteristic whole.

Trail and error may mean a heedless succession
of blind proceeding with random order of directions;
or, again, it may mean that some sensible hypothesis
is structurally tested. In the latter case, the
very failure may elucidate the situation and suggest
another hypothesis which fits the given structure
better.

Learning on the basis of success may mean
that an action is singled out because of the success
that follows the action only factually, but is
not understood; or it may mean that, in learning,

a subject grasps why just this kind of action leads
to just this effect for intrinsic structural reasons.
It is the latter form of "learning by success"

that enables the subject to vary his action in

a structurally sensible way when the situation

is no longer the same (pp. 250-251).

Wertheimer's statements show the progression of Gestalt
psychology into more of a "field" theory of psychology. Field
therories of learning give unusual prominence to the organized
"whole"; whose properties and structure both explain the
localized occurrence that it embraces and at the same time
permits increased control over it (Hartmann 1942, p. 166).
Phrased another way:

Our motor performances have an identifiable beginning

and end; our perceptions are clearly in terms of

spread-out wholes or 'figures', such as entire

persons, objects, and events; and our 'inner life'

is almost always structured, being rarely in an

unstructured or nonstructured phase (p. 171).

Hartmann (1942) notes that the "field“ in theory is derived
from the scientific use of the word, and quotes Kantor to

indicate that it "consists of a definite frame of reference

:marking the limits of interactions of phenomena during the

37

occurrence of events" (p. 172).

There are many versions of field theory, Gestalt theory
being only one. Lewin's "field theory" was more explicit
in his advise to educators. Lewin notes, "A teacher will
never succeed in giving proper guidance to a child_if he
doesn't learn to understand the psychological world in which
that individual child lives” (Lewin, 1942, p. 217). Lewin
indicated psychologists and educators must not pick out iso-
lated elements within a situation, but must characterize
the situation as a whole. According to Lewin the "atmosphere"
surrounding a learning situation is just as important as
the bearing physical entities have on the field of gravity
and field of electricity (Lewin, 1942, p. 218).

Lewin specified four types of learning: 1) as a change
in "cognitive structure", 2) as a change in motivation, 3)
as a change in ideology, and 4) as a voluntary control of
body musculature (p. 220). With respect to learning as a
change in cognitive structure, Lewin notes:

It is correct that a change in cognitive structure

may occur on the occasion of repeated experience.

However, it is important to realize that it is

not the repetition itself, but the change in cognitive

structure which is essential for learning (p. 229).
He uses the scenario (later adopted by Skemp) of a person
attempting to find his/her way about a new city without a
map. Even after many trips the person is still basically
lost. In contrast, once given a map, the person is able
to view the situation as a whole, and consequently make the

appropriate changes to his/her cognitive structure (p. 229).

To Lewin, learning results from the two entities, the structure

38

of the cognitive field, and from certain need or motivations.
Entities pertaining to motivation are those such as basic
needs, goal structures, level of aspiration, and group belong-
ingness (pp. 238-239).

One other significant theory in the continuium of histor-
ical development of learning theory from pure stimulus-response
theory to the cognitive learning theories prevelant in the
1970's and 1980's, is Tolman's "sign-gestalt" theory. Initially
Tolman's theory was called "purposive behaviorism; and in
addition to "sign-gestalt", has been. called a.Psign-signifi-
cance" theory, and an "expectancy" theory (Hilgard and Bower,
1966, p. 191). Tolman's theory was a genuine "behaviorism"
and as such rejected introspection and conscious experience.

It was concerned only with objective behavior, but did, how-
ever, emphasize the cogpitive nature of learning. Tolman's
system was “molar" rather than the "molecular" behaviorism

of Watson. An act of behavior was considered to have
properties of its own, which could be described and identified
regardless of whatever muscular, glandular, or neural processes
underlie it. The molecular facts of physics and physiology,
which are the basis of behavior, have identifying properties

of their own, but these are not the properties of behavior

as molar. Thus, Tolman's theory was independent of physiology
(pp. 191-192). Although, the complete act of behavior is
initiated by environmental stimuli and physiological states,
certain processes intervene. These "intervening variables"
include such processes as cognitions and purposes. For Tolman,

the problem of psychological analysis at the molar level was

39
to infer these processes which intervene between the initiation
of action in the world of physics and physiology and the
resulting observable consequences. Tolman believed that
in learning an individual formed a cognitive "map" by learning,
not movements, but meanings (PP. 193-195). Hilgard and Bower
(1966) have noted the importance of Tolman's contribution
to the American psychological movement:
It was Tolman's contribution then to show

that a sophisticated behaviorism can be cognizant

of all the richness and variety of psychological

events, and need not be constrained by an effort

to build an engineer's model of the learning machine.

With the diversification of behaviorism under

the influence of Tolman and others, the old brittle-

ness of watsonian behaviorism has largely disappeared,

and what virtues there are in the behavioristic

position have now become part of the underlying

assumptions of most American psychologists--without

most of them thinking of themselves as behaviorists

at all (p. 219).

The contemporary status of Tolman's sign-gestalt theory,
as well as gestalt and field theories is rather a matter
of opinion. Woodring (1971) feels that Gestalt psychology
has already passed its peak and has now been absorbed into
what is psychology. However, Kohler (1958) (cited in Hilgard,
1964) complained behaviorism was much in ascendance in American,
and that problems such as perceptual contours were being
reduced to atomistic terms. He took the position that the
major teachings of Gestalt psychology had been neglected
(p. 58). Hilgard feels, however, that the upsurge of interest
in problem-solving and in cognition generally, have hidden
beneath them the basic notion of "insight" as advocated by

the Gestaltists (Hilgard, 1964, p. 581; Harrison, 1967, p.

35).

4O

Behaviorism, which had suffered in the late 1920's and
the early 1930's at the hands of the gestaltists, and at
the hands of their own theorists such as Tolman, was given
a new lease on life by Skinner's (1938) publication of The
Behavior of Organisms. Skinner rejected intervening variables,
and broke with conventional stimulus-response psychology
by making a distinction between "respondent" and "Operant"
behavior. Respondent behavior is a direct response to a
stimulus, as in classical conditioning. An example of respon-
dent behavior would be the flow of saliva in response to
food in the mouth. On the other hand, operant behavior simply
happens, apparently spontaneously, rather than in response
to a specific stimulus. As an example of operant behavior,

a baby alone in a crib may twist and "coo" spontaneously,

in response to nothing in particular (Hilgard, Atkinson,

and Atkinson, 1979, p. 198). One aspect of Skinner's theory
may be compared to Thorndike's law of effect: when the occur-
rence of an operant is followed by the presentation of a
reinforcing stimulus, the strength is increased. What is
important is that the response of the organism produces the
reinforcing agent. This type of conditioning became know

as "instrumental" conditioning to distinguish it from classical
conditioning (Hilgard and Bower, 1966, p. 110).

Skinner is more interested in individual behavior as
Opposed to group behavior. He believes that given a strong
independent variable, a reliable dependent variable, and
adequate experimental controls, that behavior is lawful enough

to permit accurate prediction of what an individual organism

41

will do under a given set of conditions (Hill, 1964, p. 37).
Skinner and his followers have classified reinforcement schedules,
interpreted avoidance learning, analyzed verbal behavior,
and attempted to interpret social phenomena in terms of learning
principles (pp. 39-40).

In 1954 Skinner embarked upon a series of investigations
which led to inventions designed to increase the efficiency
of teaching in the areas of arithmetic, reading, spelling,
and other school subjects. Skinner's inventions, and others
modeled after his, came to be known as "teaching machines?
(Hilgard and Bower, 1966, p. 132). Skinner (1971) has stated,
”Fortunately for us all, the human organism is reinforced
by many things. Success is one of them" (p. 39). Skinner
held that since the teacher could not be with every student
on all tasks, the machines, with "programs" that provided
immediate feedback with respect to success, could have a
significant impact by acting as reinforcers. This method
of "programmed instruction" eventually became a major commercial
and educational enterprise in the 1960's (Hilgard and Bower,
1966). The interest has been rekindled in the 1980's due
to the advent of the micro-computer and the consequent prolifer-
ation of educational "software".

According to Hill (1964) in addition to Skinner, two
other major approaches to stimulus-response theories of academic
psychology have continued into modern times. The names of
Guthrie and Hull are generally associated with these traditions
(p. 28). Guthrie (1942), citing animal studies, reaffirmed

his conviction that association by contiguity is the basic

42
law of learning (pp. 46-47). Later, Guthrie did expand his
theory to put more emphasis on the modification of stimulus
reception by changes in receptor orientation. His expanded
concept included scanning, a systematic variation in receptor
orientation service to discover a stimulus. That is, what
is being noticed becomes a signal for what is being done
(Guthrie, 1959) (cited in Hilgard and Bower, 1966, p. 92).
The importance of Guthrie's expanded theory was in the trend
toward cognitive interest (Hill, 1964, p. 41). £311, on
the other hand postulated that a number of intervening variables
serves to link the independent and dependent variables. The
most basic of these intervening variables is "excitatory
potential", which refers to the strength of the tendency
to give a certain response to a particular stimulus. The
excitatory potential for any given response depends upon
a number of other intervening variables such as interest,
habit, strength, drive, and incentive motivation. The importance
of Hull's modification is primarily due to the cognitive
slant given to his stimulus-response theory by the distinction
between learning and performance (Hill, 1964, pp. 34-35).

W.F. Hill (1964, p. 27) has indicated that most learning
theories within academic psychology have a stimulus-response
orientation. He notes that this is true whether one considers
conditioning, verbal learning or thinking. Hill notes that
although in many cases the paradigm may be greatly modified
and elaborated the stimulus and response theory is basic
to most theoretical work in learning. Nowhere is this more

evident than in the theories of Robert Gagne'. Gagne's

43

approach to learning comes essentially from a combination

of the neobehaviorist psychological position in combination
with the task analysis model that dominates the fields of
industrial and military training. Crosswhite, et al., 1973,
p. 8). Gagne' (1970) has postulated that there are eight
different types of "learning":

Type 1: Signal Learning. The individual
learns to make a general , diffuse response to
a signal. This is the classical conditioned ree
sponse of Pavlov (1927).

Type 2: Stimulus-Response Learning. The
learner acquires a precise response to a discrim-
inated stimulus. What is learned is.a connection
(Thorndike, 1898) or a discriminated operant
(skinner, 1938), sometimes called an instrumental
response (Kimble, 1961).

Type 3: Chaining. What is acquired is a
chain of two or more stimulus-response connections.
The conditions for such learning have been described
by Skinner (1938) and others, notably Gilbert (1962).

Type 4: Verbal Association. Verbal association
is the learning of chains that are verbal. Basically,
the conditions resemble those for other (motor)
chains. However, the presence of language in the
human being makes this a special type because internal
links may be selected from the individual's previously
learned repertoire of language (see Underwood 1964b).

Type S: Discrimination Learning. The individual
learns to make different identifying responses
to as many different stimuli, which may resemble
each other in physical appearance to great or lesser
degree. Although the learning of each stimulus-
response connection is a simple type 2 occurrence,
the connections tend to interfere with each other's
retention (Postman, 1961).

Type 6: Concept Learning. The learner acquires
a capability of making a common response to a class
of stimuli that may differ from each other widely
in physical appearance. He is able to make a response
that identifies an entire class of concepts or
events (see Kendler, 1964). Other concepts are
acquired by definition, and consequently have the
formal characteristics of rules.

Type 7: Rule Learning. In simplest terms,
a rule is a chain of two or more concepts. It
functions to control behavior in the manner suggested
by a verbalized rule of the form, "If A, then B".
where A and B are previously learned concepts.
However, it must be carefully distinguished from

44

the mere verbal sequence, "If A, then B", which,
ofcourse, may also be learned as type 4.
Type 8: Problem Solving. Problem solving

is a kind of learning that requires the internal

events usually called thinking. Two or more pre-

viously acquired rules are somehow combined to

produce a new capability that can be shown to depend

on a "higher-order" rule (pp. 63-64).

Gagne' rejects the notion that learning is the same
for all eight types; their differences are said to be more
important than their similarities. In Gagne's system the
eight types are hierarchically related in that each type
namfires the next lower type as a prerequisite--except possibly
for Type 1 and Type 2 (Hilgard and Bower, 1966, p. 570).
For Gagne', or the programmed-instruction position which
is patterned after his theories, the objectives of instruction
are "capabilities". These are behavioral products that can
be specified operationally. He insists that objectives must
be stated clearly in behavioral terms (Crosswhite, et al,
1973, p. 8). Additionally, Gagne' has extended the concept
of hierarchical prerequisites to school subjects such as
mathematics. The terminal behavior (capability) is placed
at the top of what will eventually be a complex pyramid.
Subsequent to analyzing the task Gagne' would ask, "What
would you need to know in order to do that?" Once these
prerequisites are determined, the question would be asked
again, and continued until the pyramid is completed (Crosswhite,
et al, p. 7). Gagne' has stated, "If one wants to insure
that a student can learn some specific new activity, the

very best guarantee is to be sure he has previously learned

the prerequisite capabilities. When this in fact has been

45
accomplished, it seems to me quite likely that he will learn
the new skill without repetition" (Gagne', 1973, p. 111).

Although conditioning is at the base of his eight learning
types, the sufficiency of conditioning is rejected:

Despite the widespread occurrence of conditioned
responses in our lives, they remain unrepresentative

of most of the events we mean by the word "learning".

Voluntary acts can be conditioned only with difficulty,

if at all. If a child wants to learn to ride a

bicycle, he will get no help in this activity by

arranging the pairing of a conditioned and unConditioned

stimulus, because voluntary control of his actions

is not acquired in this way. The same is true,

needless to say, for most other kinds of things

he must learn, beginning with reading, writing,

and arithmetic. There can be little doubt that

watson's idea that most forms of human learning

could be accounted for as chains of conditioned

responses is wildly incorrect; and this has been

pretty generally conceded for many years (Gagne',

1970, pp. 12-13).

The message Gagne' transmits is, that faced with the
problem of improving training, one should look for much less
help from well known learning principles than from the impli-
cations of the techniques of task analysis, and component
task sequencing. Gagne' has been especially active in apply-
ing the principles to general education. In particular student
achievement in mathematics has been analyzed in terms of
hierarchies of knowledge and component task achievement in
the course of the acquisition of knowledge (Glaser, 1964,

p. 173).
Meaningful Learning Theory of Mathematics

P.D. Woodring (1971) notes that by 1940 many of the
authors of educational psychology text books had become con-

vinced that they must present a variety of psychological

theories to teachers -- behavioristic, association theories,

46

gestalt, and psychoanalytic interpretation (p. 92). McDonald
(1964) notes that such theories as "functionalism", as ex-
pounded by Judd, absorbed the major ideas of other systems--
heredity and the nervous system are there, social influences
are prominent, measurement and individual differences receive
their due, the psychological aspects of school subjects is
included, and the cognitive processes are emphasized (p. 23).

Against this historical background of general learning
theory, those psychologists and educators with a particular
interest in mathematics learning were rebelling against the
rote memorizations and drill and practice routines of the

stimulus-response advocates. William Brownell (1935) in criti-

 

cizing connectionist theory described the "meaning“ theory
of arithmetic:
This theory makes meaning, the fact that chil-
dren see sense in what they learn, the central

issue in their arithmetic instruction. Drill is

recommended when ideas and processes, already under-

stood, are to be practiced to increase proficiency,
to be fixed for retention, or to be rehabilitated
after disuse. The "meaning" theory conceives of
arithmetic as a closely knit system of understandable

ideas, principles and processes. (p. 19).

Brownell (1973), identified four major faults of con-
nectionist learning theory: 1) The magnitude of the task
of attempting to memorize the required multitudes of bonds,
2) the use of product measures given to evaluate process
measures, 3) the invalidity of the analysis of arithmetic
performance by adults, and 4) the probability of negative
effect on later mathematics learning. In a similar vein,

he identified four instructional weaknesses that he attri-

buted to connectionist learning theory. First, the teacher's

47
attention is directed away from the processes by which children
learn, while they are overly concerned about the product
of learning. According to Brownell, since to connectionists
all learning is the formation of connections, the process
of learning is making of connections. He notes:
...identification of the learning process

with the formation of connections, however valid

for ultimate psychological and neurological theory,

is not useful to teachers. Teaching is the guidance

of learning. We can guide learning most effectively

when we know what the learners assigned.to'us‘

really do in the face of their learning tasks.

In a word, we as teachers can be helpful in guidance

to the degree to whiCh we know our pupifs processes.

I do not mean that the product of those processes

is no concern of ours; but I do mean that processes

are of at least equal importance with products.

The teacher wholomws the product which is to be

finally achieved, but who also knows how to discover,

evaluate, and direct the processes of her pupils

as they approach this goal - that teacher is probably

a good teacher (PP. 63-64).

A second instructional weakness of connectionist theory,
according to Brownell, is over-rapid instruction. He indi-
cates the connectionistic view of learning leads us to give
the pupil at the outset the form of response which we want
him to ultimately have, with the consequent result of memori-
zation and superficial, empty verbalization. Brownell indicates
that a more valid picture of learning should be plotted in
terms of process and would look something like a series of
steps, each successive one somewhat higher on the maturity
scale than the preceeding one. Each stage serves its purpose
for a time, then is abandoned, but not forgotten. The older
procedure is overlaid by another, but the old pattern remains

for use if for any reason the new procedure does not function

smoothly (p. 64). Brownell indicates that educators must

48
take the time to use whatever sensory aids that are available.

He states:

When the goal of understanding is accepted,
the function of temporary aids is seen in its proper
perspective. Such aids contribute meanings when
meanings are needed; and the more meanings, the
deeper the understanding, and the greater the chances
of successful transfer to new and unfamiliar situations

(p. 66).
He continues:

Learning is progressive in character. The
abstractions of mathematics are not to be attained
all at once, by some coordinated effort of mind
and will. Instead, we must start with the child
wherever he is, at the foot of the ladder, or at
some point higher up. Well chosen sensory aids
reveal the nature of the final abstractions in
a way which makes sense to the child. If he can
work out the new relationships in a concrete way
and can himself test their validity in an objective
setting, he has faith and confidence at the start;
and he is the readier to learn with understanding
the more abstract representations of mathematics.
Sensory aids, like many so-called crutches, are
then not only admissable under the conception of
learning which I am outlining: they are obligatory
(p. 66).

The third instructional weakness of connectionist learning
theory indicated by Brownell, is faulty practice. Brownell
feels the connectionist exhortation of "make the proper con-
nection under satisfying conditions; exercise the connection
until it is firmly established”, often leads to superficial
learning. He feels that repetitive practice often "freezes"
the learner at an undesirably low level of maturity. Brownell
advocates that in the early stages of mathematics learning,
activities should be instituted which will enable the learner
to explore the new material. He advocates "varied" practice
to aid the student in discovering the right combination of

ideas. Repetitive practice should only be used to promote

49
efficiency once the learning has occured (PP- 67-68).

The fourth instructional weakness attributed to the
connectionistic school by Brownell was the use of inappropriate
remedial measures. Brownell indicates that remediation for
the connectionist would be merely to show the pupil the correct
connection and have the student practice until this proper
connection is formed. He believes that errors in mathematics
are the result not of imperfectly learned symbols, but of
imcomplete understandings, of inappropriate thought processes,
and of faulty procedures. In Brownell's opinion mathematical
errors comes from failure to traverse the stages and levels
of thinking in an orderly fashion. Called upon to perform
- at a higher level than any he has yet attained, and without
guidance to reach the higher level, the child will choose
to either refuse to learn, try to learn by blindly following
rules, or to fool the teacher by being proficient at a lower
level procedure. Brownell notes:

The child whose attitude toward mathematics has
been ruined needs to have that attitude corrected.

The working of masses of unenlightened and unenlight-

ening examples and problems will not reach the '

source of difficulty. If the undesirable attitude

arose because of inability to understand and of

a consequent series of failures, the child's attitude

will improve only when he understands and when

he has had ample experiences of a successful kind.

(p. 70).

The appropriate remedial instruction according to Brownell
is a guided questioning approach.

He should go as far as he can on his own;

when he can go no further, he should be questioned

and guided through questions to locate his difficulty

and to analyze its nature. Through continued question-

ing he should be led to suggest possible next steps
and then to evaluate these steps himself. But at all

50

stages the student should be required to make sure

of his own knowledge (to the extent that he has

any) and he should be allowed to identify his de-

ficiencies himself and to feel that he is making

progress by his own efforts. Remedial instruction

of this kind is worthy of the name, and the results

justify the time and energy that must be expended

to secure them (p. 71).

Brownell, citing Katona's research, indicates that problem
solving which is based upon understanding is superior to
"problem solving" (quotes by Brownell) based upon memorization.
He notes also thattmderstanding is a matter of degree, and
that varying degrees of understanding react differently with
problem solving. He further believed that the degree of
understanding engendered is a function of instruction given,
and that the form of instruction which enables the learner
best to organize his previous experience of learning is favored
to other kinds (Brownell, 1942, pp. 436-437).

Brownell (1948), writing with educational research in
mind, indicated measures to be used in addition to speed
and accuracy of problem solving. He suggests that more basic
are changes in the process level, time of retention, and
the ability to transfer the learned ideas to new situations.
Brownell (1946) also identified the processes of understanding
analysis, synthesis, discrimination, generalization, and
comparison. He notes:

...Understanding of a principle means that

one can identify its appropriateness and usefulness

in situations where it has not been seen or used

previously. Understanding of a process implies

that one knows when and how to use it effectively

(Po 41)-

Brownell (1942), sounding somewhat like Piaget,

states:

51

...one needs only to point out that all learning
starts with some inadequacy of adjustment, some
disturbance of equilibrium -- and so, with a
"problem" -- and that in the process of achieving
adjustment and returning to a state of equilibrium
one, "solves" the problem (p. 415).

That Brownell was in fact an admirer of Piaget's work,
and in a sense an "American voice" for Piaget, is indicated
by the following statement from an aritcle by Brownell (1942)
on problem solving:

Much of this chapter, an amount of space which

may appear to be disproportionate, is given to

Piaget's account of the development of ability

in problem solving. The.writer had as an alter-

native the possibility of citing numerous other

valuable researches, but these would have had to

be dismissed with a.word. The decision in favor

of Piaget's research was made in full recognition

of the loss of prestige which it has suffered during

the past few years. Some of Piaget's investigation

have been repeated in this country and elsewhere,

but with different results; and some of his inter-

pretation have been challenged. Nevertheless,

with all their limitations, Piaget's studies seem

to provide the most illuminating single description

of the way in which children attain power in problem

solving (p. 428).

However, Brownell does indicate several criticisms of
Piaget's theory. They include: the prejudicial character
of the problem tasks, the definition of reasoning as a highly
formalistic type of thinking, the impression that at definite
ages children reach definite levels of thinking, and the
faflzthat Piaget makes adult reasoning unlike children's problem
solving (pp. 430-431). Perhaps as Piaget's theories were
further explained and developed in America over the next
twenty years, Brownell would have found himself primarily
in agreement with Piaget.
lflgget's Theory of Cognitive Development

Piaget is often referred to as a child psychologist,

52

but he has characterized himself as a "genetic epistemologist".
The central question guiding Piaget's research was not "what
are children like?", but "How does the relationship between
KNOWER and the KNOWN change with the passage of time?" Piaget's
methodology consiSts of first observing children's reaction

to their surroundings. Then, based on these observations,

he forms hypotheses about the sort of biological and mental
structures that underlie their reactions. Next, he recomposes
the hypothesis in the form of questions or problems that

he then poses to children in order to reveal their thinking
processes and so test the hypotheses. Piaget was extremely
productive, and over the years has produced large numbers

of problem situations that subsequently have been used by
researchers world wide (Thomas, 1979, pp. 289-292). In this
regard Skemp (1979) reports that Flavell ntending to write

a one chapter summary of Piaget's theory completed the task
seven years later, after having completed a book of about

500 pages. Piaget himself published five books during the
years 1923 to 1932. The books were referred to by Piaget

as his "adolescent" works. These early books gave his pre-
liminary and tentative conclusions about children's intel-
lectual development. In the United States the books were
initally well received, and during the 1920's and 1930's
Piaget's work was highly regarded. However, his views, as
expressed in his early books, came under extreme criticism

and America's interest in his research faded. But with the
translation and publication of several of Piaget's later

books in the 1950's, interest.was revived (Ginsburg and

53

Opper, 1969, pp. 1-7).

Smock (1976) feels that Piaget was greatly influenced
by advances in theoretical physics during the 1920's and
1930's. The fundamental aspects of relativity that are in-
herent in Piaget's theory are that an conceptual judgements
are always relative to the postion of the observer, and that
b) analysis of knowledge acquisition requires a description
of its operational basis; that is, the mental operations
of the individual is associated with the construction and
maintenance of consistent patterns (structure) of his con-
stantly changing relations with his physical and social en-
vironments. Thus, Piaget is unique among psychologists in
that his emphasis on a "constructivist“ theory of knowledge
is indissoluble from his interpretation of "operationism”..
According to Piaget, "reality" is constructed by the child,
not imminent in mind or stimulus (Smock, 1976, p. 10).

Smock (1976) notes that Piaget's view of a child as
one who is seen facing phenomenal disorder from which he
must construct a coherent.view of reality is a very different
child than the one confronted by a stable reality as viewed
by the typical "naive realist" psychologist of America (p. 10).

Thomas (1979) has indicated that Piaget's theory of
knowledge and its acquisition.«differs from the "common sense",
or popular beliefs, in four fundamental aspects. First common
sense holds that knowledge is a body of information of beliefs
a person has acquired through instruction, or through direct
experience with the world. Second, common sense holds that

a person's knowledge is a fairly faithful and accurate

54

representation of what the person has been taught or witnessed.
Third, common sense indicates that an individual's vast store-
house of knowledge is increased if he/she adds to it from
his/her daily experiences, item upon item. Fourth, common
sense holds that whenever an item of knowledge is recalled
from the storehouse of memory, the item can be recovered
in essentially the same condition as it was when first ac-
quired (p. 292).

Piaget in Opposition to his "common sense" approach
Idoes not agree that knowledge is a body of acquired information,
but instead he conceives of knowledge as a process. To know
something means to as; on that thing, with the action being
either physical or mental or both. Young children have only
physical knowledge of an item, but as they grow older, they
gain more experience with such direct, physical knowing,
and mature internally so that they are increasingly freed
from direct physical behavior in order to know something.
They become capable of producing mental images and symbols
that represent objects and relationships, and consequently,
the older<flfild4s knowledge increasingly becomes mental activity.
He thinks about things by carrying out interiorized actions
on symbolic objects. Thus to Piaget, knowledge is a process
or repertoire of actions, rather than an inventory of stored
information (Thomas, 1979, pp. 292-293).

Piaget diagrees with the "common sense" idea of the
way objects or events are recorded in a child's mind (per-
ception). To Piaget the child does not take in a picture

of objective reality, but the picture he obtains is biased

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55

by both the child's past experiences and his current stage
of internal maturation. Thus, the way two children know
(action) the same object will not be identical (Thomas, 1979,
p. 293). .

Piaget does agree with the "common sense" idea that
the result of a person's past actions can be stored as memo-
ries to be retrieved when wanted. He agrees that the quantity
of memories increase with age and experience, but he does
not believe that remembering is simply a matter of summoning
images of past events from memory and placing them in conscious-
ness whereby they can be viewed in their passive, original
condition. Remembering is a reenacting of the original process
of knowing, but it is not simply a repetition of the original
knowing. .The child's mind has been altered by subsequent
experiences and internal maturation, thus the interiorized
recitation or rehearsal of the stored event is, as it were,
"now performed on somewhat altered mental state" (Thomas,
1979, pp. 293-294).

To summarize, Piaget's system views knowledge as a pro-
cess of acting, either physically and/or mentally, on objects,
images, and symbols. The objects are from the world of direct
experience, while the images and symbols can be derived not
only from the "real world, but from memory as well. As Piaget
(cited in Thomas, 1979, p. 294) has stated, "All knowledge
is continually in a course of development and of passing
from a state of lesser knowledge to one which is more complete
andeffective".

The preceding paragraphs have indicated an overall

A*‘

a

 

56

direction of intellectual development that Piaget's theory
is designed to explain. The subsequent paragraphs in this
section will detail the more technical aspects of the mech-
anisms that bring about these kinds of developments.

Piaget has used the term schemes of an action to refer
to the general structure of this action which conserves itself
during repetitions, consolidates itself by exercise, and
applies itself to situations with varying surroundings. Piaget
(1968) states:

A "scheme" is that part of an action or opera-

tion which is repeatable and generalized in another

action or operation; it is that part which is essen-

tially characteristic of action or operation (p. 15).
According to Mick and Brazier (1979), Piaget uses the word
”schema" to refer to simplified images (p. 52). Piaget's
"schemes” parallels Bartlett's (1932) notion of a “schema”
as ”...an active organization of past reactions, or of past
experiences, which must always be supposed to be operating
in any well-adapted organic responses" (p. 201). Thomas
(1979) indicates that the only difference between Piaget's
"schemes" and "schemata" is the English translation. His
later writings were translated as "scheme", while the initial
translation of his earlier writings from the original French
contained the word "schema". Consequently, many reviewers
of Piaget's work use the word "scheme" (cf. Thomas 1979)
and others the word "schema" (of. Harrison, 1967) in a similar
fashion.

According to Piaget, the purpose of all thought and

behavior is to enable the individual to adapt to his/her

57

environment in ever more satisfactory ways. A child's develop-
ment must be conceptualized in terms of schemes. The child
acquires ever greater quantities of schemes that become inter-
linked in ever more SOphisticated patterns. Crucial to the
understanding of the process of evolution of these schemes

is Piaget's notion of assimilation and accomodation. Piaget

 

 

used the word "assimilation" to refer to that process of
taking in or understanding events of the environment by match-
ing the perceived features of those events to the individual's
schemes. Piaget (1958) has states:
To assimilate an object to a schema means
conferring to that object one of several meanings

and it is that attribution of meaning which thus

requires, even when it takes place by observation,

a system of more or less complex inferences. In

brief, one could say that assimilation is an asso-

ciation accompanied by inferences (p. 59).

However, when the perceived structures of the environment
fail to fit the child's available schema, even with some
perceptual shaping of that structure, one of two consequences
will result. First the event is ignored or passed by, and
thus not assimilated. The second possible outcome of a poor
match between the perceived environment and the individuals
available schemes is not of total rejection, but of dissatis-
faction, along with continued attempts to achieve a match.

In this case schemes themselves may be altered in form or
multiplied to match the perceived environmental realities.
Piaget used the word "accomodation" to identify this process
of altering existing schemes in order to permit the assimi-

lation of otherwise incomprehensible events (Thomas, 1979,

pp. 294-299). The importance of the roles played by assimilation

58
and accomodation in Piaget's theory can be noted in the follow-
ing statement:

Thus it may be seen that intellectual activity
begins with confusion of experience and of awareness
of the self, by virture of the chaotic undifferentia—
tion of accommodation and assimilation. In other
words, knowledge of the external world begins with
an immediate utilization of things, whereas knowledge
of self is stopped by this purely practical and
utilitarian contract. Hence there is simply inter-
action between the most superficial zone of external
reality and the wholly corporal periphery of the
self. On the contrary, gradually as the differentiation
and coordination of assimilation and accommodation
occur, experimental and accomodative activity
penetrates to the interior of things, while assimila-
tory activity becomes enriched and organized.....Intel-
ligence thus begins neither with knowledge of the
self nor of things as such but with knowledge of
their interaction, and it is by orienting itself
simulaneously toward the two poles of that interaction
that intelligence organized the.world by organizing
itself (Piaget, 1954, pp. 354-355).

Piaget (1954) has used the term equilibrium to specity
the ideal relationship between assimilation and accomodation:

In their initial directions, assimilation
and accomodation are obviously opposed to one another,
since assimilation is conservative and tends to
subordinate the environment to the organism to
the successive constraints of the environment...
Assimilation and accomodation are therefore the
two poles of an interaction between the organism
and the environment, which is the condition for
all biological and intellectual operation, and
such an interaction presupposes from the point
of departure an equilibrium between the two tenden-
cies of opposite poles (pp. 352-353).

Piaget views knowledge as "invariance under transformation:
(Smock, 1976, p. 11). Piaget distinguishes between two aspects

of an act of knowing: the figurative and the operative.

 

The figurative aspect refers to the actions by which the
individual produces a "copy“ of reality, and in this case

cognition is concentrated on the state of reality rather than

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me:

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O—On

:Iaw

Ni~.‘

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:n.‘
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59

transformations. Figurative knowledge is acquired through
the activity of accommodation to the properties of the object.
Piaget makes three sub-divisions among. the figurative compo-
nents of knowing. First is perception, which is a system
that refers to the senses. The copy of a child's environ-
ment based on perception is often inexact, as in the case

of visual illusions. A second component of cognition is
imitation. It is by imitation that the child reproduces
actions of persons or things. Although the child is involved
in action, imitation produces only a copy of reality and
isthus considered figurative. A third aspect of figurative
knowledge is mental imagery. Mental imagery is used to "picture"
a copy of absent object or events (Ginsburg and Opper, 1969,
pp. 152-153).

Operative knowledge is the dynamic aspect of cognition
and refers to actions that are used to change reality. Opera-
tions are the "action schemata" that construct "logical"
transformations, or coordinated actions, on "states" of reality.
Such logical systems of transformations operate either upon
representation of events, or upon the cognitive system's
own logical operations (Smock, 1976, pp. 16-17). Piaget

makes a distinction between concrete operations and formal

 

Qperations. Operations are ways of manipulating objects-

in relation to each other. If the objects are physically
present, and the individual actually moves the objects, or
by observing the objects determines how they might be moved,
then these operations are said to be concrete operations.

However, when concrete operations are transposed into

60
propositions about the relationships that exist, or might
exist among objects, and then these propositions are mentally
manipulated, these intellectual actions are called formal
operations.

It should be noted that just not any manipulation of
an object is an operation. To be classified as operations,
these actions must be internalizable, reversible, and coordi-
nated into systems that have laws that apply to the total
system, and not just the single manipulation (Thomas, 1979,
p. 308). As Piaget (1972) has noted,

operations...are actions, since they are carried

out on objects before being performed on symbols.

They are internalizable, since they can also be

carried out in thought without losing their ori-

ginal character of actions. They are reversible...

Finally, since operations do not exist in isolation,

they are connected in the form of structured wholes

(p. 8).

Piaget recognized that from day to day intellectual
growth is continuous with no major jumps ahead from one day
to the next. However, when he viewed the entire spectrum
of growth years, he was able to distinguish what breaks in—
dicated that the child had completed one phase of development
and was now engaged in a further one. Consequently, Piaget
identified a number of different series of developmental
stages. For example, one series concerned understanding
of physical causality, another the steps in imitation and
play, another in the conception of moral principles and jus-
tice, and others in understanding number, space, and movement.

However, underlying these specific series of stages is a

basic set that provides for overall sensorimotor-intellectual

61

development. Most interpreters of Piaget have viewed his
writings as advocating four major periods of intellectual
development: sensorimotor (birth to 2 years); preoperational
(2 years to 7 years); concrete operational ( 7 years to 11
years); and formal operational (11 years and above) (Thomas,
1979, pp. 3039318; Ginsburg and Opper, 1969, p. 26). It
should be noted that there is some measure of confusion among
Piaget's reviewers as to the number of major periods, and
corresponding ages that he advocates. Some interpreters
see him as advocating three major periods, others four, and
some five (cf. Mick and Brazier, 1979, p. 46 and Phillips,
1969, p. 11). This disagreement seems to hinge on the fact
that sometimes Piaget will call a period of growth a "stage"
and in another instance a “substage”. Piaget (1958, pp.
4-15) indicates three stages (with various substages) while
in another article he indicates four stages (1963, p. 3).
Thelclose interplay between motor activity and percep-
tion in infants led Piaget to label the first two years as
a sensorimotor period. During this period infants discover
the relationships between sensations and motor behavior.
They learn, for example, how far to reach when grasping,
what happens when they push their food to the edge of the
table, and that their hand is part of their body while the
chair is not. Through many "experiments" infants deve10p
a conceptualization of themselves as separate from the exter-
nal world. A major discovery for them is the concept of
"object permanence" - an awareness that an object continues

to exist even when it is not present to the senses. For

62

example, if a cloth is used to cover a toy within reach of
an eight-month old, the infant will not attempt to search
for the toy, and will act as if it failed to exist. On the
other hand a ten-month old will continue to search for the
hidden object. The older child appears to realize the object
still exists although it is out of sight. It should be noted.
that even during this sensorimotor period, Piaget views the
infants activities, such as grasping and letting go, and
other repetitive activities as indicating the acquiring know-
ledge is not accomplished by the environment's imposing reality
on the child's mind. Rather, the repetitive activity is
purposive, designed either to preserve or rediscover an act
or skill. That is, it functions as practice (Hilgard, Atkinson,
and Atkinson, 1979, pp. 70-71; Thomas, 1979, pp.304-305).
It should.be noted that Piaget has subdivided the sensorimotor
level of development into.six stages: reflexive, primary
circular reactions, secondary circular reaction, coordination
of secondary schemes, tertiary circular reaction, and beginning
of thought (Ginsburg and Opper, 1969, pp. 29-66). Ginsburg
and Opper have noted Piaget's notion of "curiosity" or "novelty"
has impact even at this young age. According to Piaget's
idea, the child actively seeks out new stimulation. Also,
this motivations aspect is a relativistic concept. That
which catches a child's curiosity is not the object per se,
but rather the relation between the new object and the child's
previous experience (p. 39).

The second period of intellectual development in Piaget's

theory has been labeled the preoperational thought period.

63

The preoperational stage extends from the start of organized
symbolic behavior, especially language, until about seven
years of age. During this period the child will reconstruct
his experiences from the sensorimotor level into representa-
tional thought.- Piaget indicates that during this period
when the child cOnsiders static states of a situation he

will explain them in terms of the static perceptual configu-
rations at the given moment, rather than in terms of the
changes leading from one situation to another. When the
child does consider transformations, he assimilates them

to his own actions and not as reversible operations. The
child lacks any notion of conservation at this level. For
example, when having transferred a given quantity of liquid
into a beaker which is more elongated than the initial con—
tainer, the child believes the quantity has increased because
the form of the container is different (Inhelder and Piaget,
1958, pp. 245-248). However, as the child reaches the age

of five or six he enters into a stage Piaget calls "intuitive
thought“. It is a stage of transition between depending
solely On perception, and depending on logical thinking.

A child at the age of four or five, presented with a row

of six red beads on a table, when asked to put an equivalent
number of blue beads, will put a row of blue beads the same
length as the red beads, without bothering to count. A child
at the age of six or seven will line up the blue beads opposite
the red ones, thus showing progress toward recognizing equiva-

lent quantities. However, if the experimenter spreads out

the r
old c
lcnge
perce
(Piag

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bu“.

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64

the red beads to form a longer line, the six or seven year
old child thinks the number of blue and red beads are no
longer equivalent. The child is unduly influenced by the
perception of line length rather than the logic of quantity,
(Piaget, 1950, p. 132) (cited in Thomas, 1979, p. 311).

The next level of development in Piaget's scheme is
called the concrete operational stage, and covers the approx-
imate ages from seven to twelve years. Inhelder and Piaget
(1958) state:

In a general sense, by concrete operations
we mean actions which are not only internalized,
but are also integrated with other actions to form
reversible Systems. Secondly, as a result of their
internalizedland integrated nature, concrete opera-
tions are actions accompanied.by an awareness on
the part of the subject of the techniques and coordi-
nations of his own behavior. These characteristics
distinguish operations from simple goal-directed
behavior, and they are precisely those character-
istics not found at this first stage: the subject
acts only with a view toward achieving the goal;
he does not ask himself why he succeeds (p. 6).

 

They note also:

From the standpoint of form, concrete operations
consist of nothing more than a direct organization

of immediately given data. The operations of classi-

fication, serial ordering, equalization, correspon-

dence, etc., are means of inserting a set of class
inclusions of relations into a particular content

(for example: lengths, weights, etc.), means which

it is presented to the subject ( p. 249).

By "internalizable" actions Piaget means that actions
can be carried out in thought and not lose the "original
character of actions” (Piaget, 1953, p. 8). By "reversible
operations" he means "...the capacity to execute the same
action in both directions, but being conscious that we are
dealing with the same action" (Piaget, 1957, p. 44) (cited

in Battro, 1973, p. 152). Or in another place, "reversibility"

. V
n‘fl

...“

.
n,‘ ‘
...

's

V.‘
‘1

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.~

65

is defined as the permanent possibility of returning to the
starting point of the operation in question" (Inhelder and
Piaget, 1958, p. 272). Piaget has designated two distinct

and complementary forms of reversibility: negation or inver-
sion, in which one can return to the starting point by canceling
an operation which has already been performed: and reciprocity,
in which one can return to the starting point by logical
compensating a difference. At the concrete operational level,
the two forms of reversibility are employed independently

of each other, but are not integrated into a structured whole
(pp. 272-278).

The primary differences between the preoperational level
and the concrete operational level of thinking relate not
only to the greater command of the notions of conservation
and reversibility, but also to the fact that the concrete
operations child is capable of decentering his attention,
of recognizing the way two or more dimensions of an event
interact to produce a given result. This is illustrated
by the previous example of the two beakers, one wide and
one narrow. Where as the preoperational child believed the
elongated beaker contained more water, the concrete operations
child considers both~ dimensions simultaneously and recognizes
their interaction (Thomas, 1979, p. 315).

Inhelder and Piaget (1958) have indicated that the most
general property characterizing formal thought is a combina-
tional system.whereby the individual is able to determine
all possible outcomes of an experiment. Operations are no

longer restricted to dealing only with physical objects or

66

reality. The individual at the formal thought stage can
logically derive all possibilities in terms of hypothetical
statements. Piaget has referred to this as "reflective think-
ing". These possibilities can be considered as a system
of second order operations due to the fact that they describe
relations between hypotheses rather than referring to objects
directly (pp. 341-342). Piaget has indicated that the adoles-
cent's thought structures come to approximate a mathematical
lattice structure when the individual is using methods involv-
ing proportionality (Inhelder and Piaget, 1958, pp xxii-
xiii). Piaget did not start with the ideathat cognitive
development conforms to the laws which govern mathematical
structures. However, Piaget (1971) notes that many years
of research into cognitiVe development led him to conclude
that thought structures do come to approximate mathematical
structures. He states, "yet mathematics constitutes a direct
extention of logic itself, so much so that it is actually
.impossible to draw a firm line of demarcation between these
two fields (p. 44).

The fourth developmental stage designated by Piaget

is the formal operational Stage. This period extends from

 

approximately eleven years of age to about fifteen years
of age. This stage is characterized by the development of
formal, abstract thought operations, with which the child
can reason with hypotheses rather than being restricted to
reasoning only in terms of objects. Inhelder and Piaget
(1958) have stated:

Formal thinking is both thinking about thought

67

(propositional logic is a second-order operational
system which operates on propositions whose truth,

in turn, depends on class, relations, and numerical
operations) and a reversal of relations between
what is real and what is possible (the empirically
given comes to be inserted as a particular sector
of the total set of possible combinations (pp.
342-343).

Inhelder and Piaget (1958) note in the concrete operation-
al stage children can use both of the complementary forms
of reversibility (inversion for classes and numbers and recipro-
city for relations), but they never integrate them into the
single total system found in formal logic. In comparison,
the adolescent child:

...superimposes propositional logic on the logic

of classes and relations. Thus, he gradually struc-

tures a formal mechanism (reaching an equilibrium

point at about 14-15 years) which is based on both

the lattice structure and the group of four trans-

formations. This new integration allows him to

bring inversion and reciprocity together in a single

whole. As a result, he comes to control not only

hypothetico-deductive reasoning and experimental

proof based on the variation of a single factor

with the others held constant (all other things

being equal), but also a number of operational

schemata which he will use repeatedly in experimental

and logico-mathematical thinking (p. 335).

Piaget notes the attaining of the formal operations
stage is not the end of intellectual growth. The framework
of thought is complete, but the framework is not entirely
filled in. The most obvious distinction between adult and
adolescent thought is the greater lingering egocentrism dis-
played by adolescents. The adolescent, with the newly acquired
skills of logical thought, is an idealist who expects the
World to be logical. Such an individual fails to recognize

or accept the reality that people do not operate solely on

the basis of logic, and thus the adolescent becomes a reformer

68

and critic of the older generation, envisioning a glorious
future in which the younger generation will right today's
wrongs. According to Piaget this idealism and egocentrism

is tempered and becomes more realistic when the youth enters
to occupational world or else enters a field of professional
training. Consequently, over the years of youth and adulthood
the framework of thought is filled in with more complex
schemes or greater knowledge. This is, even though the adol--
escent is capable of all forms of adult logic, the adult

in a sense doeslmxnrmore (Thomas, 1979, p. 317; Inhelder

and Piaget, 1958, pp. 340-346).

Piaget (1976) has indicated that there are four factors
which explain mental development: maturation, experience,
social transmission, and equilibration. He states that each
of these factors although vital to the sort of knowledge
the child acquires in terms of mental schemes, and the time
at which they are acquired, none is by itself sufficient
to account for mental development. However, when taken together
these factors regulate the four stages of cognitive development
previously described (p. 74).

Piaget (1976) acknowledges that internal maturation

 

plays a role throughout mental growth. Heredity not only
provides the newborn child with the equipment to cope with
problems, but also establishes a time schedule for new develop-
mental POSSibilitieS throughout the child's growing years.
Each act of maturation creates possibilities for new schemes

to be created which would have been impossible previously.

However, Piaget is explicit when noting that intelligence

69

is not programmed. Thus, even though an individual is organ-
ically mature enough for a particular scheme to be created,

the extent to which such potentialities are realized is deter-
mined by the types of experiences a child has with his environ-
ment. (pp. 72-74; Thomas, 1979, pp. 300-301).

Piaget (1976) has separated interaction with the environment
into two varieties direct, unguided physical experience,
and the guided transmission of knowledge, that is, education
in the broad sense (social transmission) (pp. 72-74). In
the case of physical experience the child manipulates, observes,
listens to, and smells objects to note what occurs when they
are acted upon. It is not the observation of the passive
objects that develops intelligence, but the set of conclusions
that child draws from those actions that bring about events
and influence objects. Piaget (1976) has called this develop-
ing of conclusions, based on learning the result of the coordi-
nation, "logico-mathematical" experience. Piaget notes that
physical experience is not a simple recording of phenomena,
but."constiun£s an active structuration", since it always
involves an assimilation to logico-mathematical structures.

He offers the example that comparing two weights presupposes
the establishment of a relation, and therefore the "construction”
of a logical form (p. 73).

Social transmission is the educative factor (broad sense)
in that it refers to the transmission of knowledge to the
individual from without. However, for a child to receive
societal information, the child must have cognitive structures

available which enable him to assimilate the information.

I

70

In the case of schooling the linguistic structure would be
vital. Piaget (1976) indicates that even though social trans-
mission is important, it by itself is insufficient. He states:

Although necessary and essential, it also
is insufficient by itself. Socialization is a
structuration to which the individual contributes
as much as he receives from it, whence the inter-
dependence and isomorphism of "operation" and
"cooperation". Even in the case of transmissions
in which the subject appears most passive, such
as school-teaching, social action is ineffective
without an active assimilation by the child, which
, presupposes adequate operatory structures (PP.
73-74).

An example of the above could be cited when a parent
(or school teacher) attempts to teach advanced mathmetics
to a child that does not have the mental structures available
to understand.

Equilibration is the factor.which maintains a.ba1ance

 

among the previous three factors. Piaget (1976) has defined
"equilibrium" (later referred to as "equilibration”) as a
series of active compensations on the part of the individual
in response to external disturbance, and an adjustment "that
is both retroactive (loop systems or feedbacks) and anticipatory,
constituting a permanent System of compensations" (p. 74).
Equilibration is a process of "self-regulation" (p. 74).
Piaget maintains that growth in knowledge is due to a conflict
between events and structures, an imbalance between new exper-
iences and what the individual knows. When new experiences
are neither too novel nor too familar, they may be assimilated
and may influence and change those existent structures. That
is, these structures have accommodated to the new information.

This cognitive conflict creates the transition from one stage

71

to another (Sigel and Cocking, 1977, p. 21).

It should be noted that Piaget believes that affectivity
and motivation cannot be isolated from the cognitive and
intellectual evolutionary factors. He states (1959):

It may even seem that affective dynamic

factors provide the key to all mental developw

ment... There can be no affective states without

the intervention of perceptions or comprehensions

which constitute their cognitive structure... The

two aspects, affective and cognitive, are at the

same time inseparable and irreducible. (p. 75).

Piaget explains stage changes and motivation in terms
of "cognitive conflict" and "logical necessity" (Sigel and
Cocking, 1977, p. 22). Certain environmental demands and
events require responses that have an inherent logic. The
child thus learns to respond to these demands in logical
fashion out of necessity. The situation of logical necessity
frequently creates for the child a cognitive conflict. As
an example, the child must learn that two objects cannot
be in the exact same space at the same time. The conflicts
that children have in their own activities as well as with
other people, creat a "disequilibrium", or tension, and it
is in the solution to these problems that propels the child
from one competence level to another (p. 22).

Piaget.also stresses the role of "self-regulations"
in development. He notes (1976):

...it is impossible to interpret the development

of affective life and of motivations without stress-

ing the all-important role of self-regulations,

whose_importance, moreover, all the schools have

emphasized, albeit under various names (p. 75).

The importance that Piaget attaches to these self-regula-

tions (partial compensations) and the equilibration factor

72
cannot be over-emphasized. Such importance is indicated
by Piaget's (1976) statement:
The regulations are directly dependent on

the equilibration factor, and all later develop-

ment (whether of thought, or moral reciprocity,

or of cooperation) is a continuous process lead-

ing from the regulations to reversibility and to

the extension of reversibility. Reversibility

is a complete--that is totallylxflanced system of

compensations in which each transformation is

balanced by the possibility of an inverse or

reciprocal.

Thus equilibration by self-regulation consti-

tutes the formative process of the structures we

have described (Pp. 75-76).
Educational Implications

Piaget's theory suggests a general proposition that
should have important consequences for education. This proposi-
tion is that the young child differs in several ways from
an adult: in methods of approaching reality, in the views
of the world, and in the uses of language. Piaget's investi-
gations concerning concepts of number and verbal communication
have enabled him to contribute to a basic change in the way
educators view learners. As a result of Piaget's work, educa-
tors have become convinced that the child is not just a minature
and less wise adult, but a unique being with a mental structure
which is distinctive and qualitatively different from that
of adults. For instance, the child below seven years of
age sincerely believes that water may gain or lose quantity
when poured back and forth from different shaped containers
(Ginsburg and Opper, 1969, p. 219).

It should be noted that Piaget considers himself a student

0f human knowledge and development, not of educational design.

However, by implication, a surprising variety of educational

73

recommendations and practices have been derived from his
theory (ResniCk and Ford, 1981, p. 186). Perhaps the single
most important proposition that can be inferred from Piaget's
work for use in the classroom, is thatrfluldnralemnlbest from
concrete activities. For this reason a school should encourage
the pupils' activity, and his manipulation and exploration
of objects. Sigel and Cocking (1977) paraphrase Piaget:
“Each time one teaches a child something he could have dis-
covered for himself, the child is kept from inventing it
and consequently from understanding it completely" (p. 20).
Since Piagetian theory indicates that social interaction
is essential to developing a multiperspective view which
is essential for objectivity, the teacher should encourage
group activities and projects (p. 20).

The teacher does, however, have an indispensable role
in the education process. The teacher's role should be that
of a mentor, stimulating initiation and research. He should
determine the child's current stage of development in the
various areas the curriculum is designed to promote, and
then construct the initial devices and create the situations
which present useful problems to the child. The teacher
is needed to provide counter examples to encourage reflection
and reconsideration of hastily drawn conclusions. The teacher
should not be a go-between that transmits ready-made solutions
and facts to the student (Piaget, 1974, pp. 15-16).

The teacher must be acutely aware of Piaget's notions
<16 assimilation, accommodation, and equilibration. According

tC> Piaget, these factors are primary in motivating students

:01

Na“
5 (A

Q

‘trl

 

 

74
to learn. Piaget (1959) states: "...to say that a subject
is interested in a certain result or object thus mgggg that
he assimilates it or anticipates an assimilation and to say

that he needs it means that heppossesses schemas requiring

 

its utilization" (p. 86). Disequilibrium occurs when the

 

student assimilates data from his immediate experience into
existing mental structures. As cognitive structures change
to accommodate the new data, equilibrium is restored. Cognitive
conflict generates a need to establish equilibrium between
the new information and existing schemas. This condition
is the motivation for cognitive activities. Equilibration
theory holds that what is learned depends on what the learner
can take from the given situation by means of the cognitive
structures available to him. The child will take interest
in those situations that generates cognitive conflict. If
the task demands are so unusual as to be unassimilable or
so obvious as to require no mental work, the student will
not be motivated (Smock, 1976, pp. 14-15). Thus, the teacher
must choose materials and topics with extreme care. Lovell
(1971) notes:
The job of the teacher is to use his profes-

sional skill and provide learning situations for

the child which demand thinking skills just ahead

of those which are available to him. It is a ques-

tion of keeping the carrot just ahead of the donkey's

nose. When a child is almost ready for an idea,

the learning situation provided by the teacher

may well "precipitate" the child's understanding

of that idea (p. 17).

Smock (1976) indicates that a large amount of confusion

Chancerning Piagetian theory for instructional practice and

ediucationalzxsearch derives from a failure to consider the

75

"figurative" and "operative" aspects of intellectual functioning.
Figurations and associated acts are based on physical and
social environment. On the other hand, operational knowledge

is not based on abstractions from physical objects and specific
events, but rather is derived by abstractions from coordinated
actions relevant to those events. That is, operations are

those action schemata that construct "logical" transformation

of "states“. Although, admitting that figurations and opera-
tions have implications for, but not necessarily causal effect
on each other, Smock feels that relationships between figurative
and operational structures has not been completely researched.
He believes that the failure to consider both aspects has

lead to many contradictory research findings with respect

to Piagetian theory. In particular, he indicates that many
studies have generated little or no "cognitive conflict"

and consequent negative findings, due to the fact that any
disparity belongs to the experimenter's reality and is external
to the child's logical operational system (pp. 16-20). Support-
ing Piaget's view that intellectual development brings a

gradual transformation of overt actions into mental operations,
Harrison (1967) proposes that the teacher should assist the
internalization and schematization process by having students
perform actions with less and less direct support from external
activities. For example, a student might be led to operate
directly on physical objects, then on pictorial representations,
and finally on cognitive anticipations of operations not
actually performed until the original external operations

tak:e place internally independent of the environment (pP- 73-74).

".1

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76

Specifically referring to the educational difficulties
in the learning and teaching of mathematics, Piaget has in-
dicated that the problem is not one of aptitude, but rather
from affective blocking or inadequate preparation. He feels
the failure of formal education can be traced to the fact
that it begins with language and illustrations rather than
real practical action. Piaget emphasizes that verbalization
does not guarantee understanding, nor does understanding
depend on verbalization. When the teacher tries to impart
knowledge in a verbal manner, the result is frequently learning
of a very superficial nature (Ginsburg and Opper, 1969, pp.
221-222). Preparation for mathematics learning should begin
in the home with concrete manipulations that foster awareness
of basic logical, numerical, and mensurational relationships.
This type of activity should be systematically developed
and expanded throughout the primary grades until it takes
the form of elementary physical and mechanical experiments
by the time secondary education begins (Piaget, 1951, pp.
95-98) (cited in Harrison, 1967, p. 71). Duckworth (1964)
quotes Piaget as taking exception to those who would Egagh
the "structure" of mathematics to children. Piaget stated:

Teaching means creating situations where struc-

tures can be discovered; it does not mean transmit-

ting structures which may be assimilated at nothing

other than a verbal level (p. 498).

Finally, Resnick and Ford (1981) suggest that in terms
0f contributions to mathematics education, Piaget's greatest

Contribution is perhaps the clinical interview technique.

1"“3

kn-

 

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(I)

 

 

J
in

77
They state:
It provides a means by which teachers can

understand what children understand. This is a

crucial step in an educational strategy that seeks

to match instruction to children's development.

Teachers can cultivate their own skills of observ-

ing and questioning, as well as of setting interest-

ing problems. As they become better at this, they

begin to note details of children's thinking that

had not been apparent before and find themselves

able to follow children's lines of reasoning more

clearly. Under these conditions, mistakes are

not seen as poor thinking, but as information about

each child's current understanding. On this basis,

tasks and questions can be posed that represent

the best match in terms of intellectual "stretching"

0.. (p. 192).
Comparison With Stimulus-Response and Gestalt Theories

Piaget indicates that the fundamental relation involved
in all developement and all learning is not the relation
of association. In the stimulus response schema, the relation
between stimulus and response is understood to be association.p
Piaget believes the fundamental relation is assimilation
rather than association. He feels that a stimulus is important
only when it can be assimilated to a cognitive structure.
Piaget indicates that the activity on the part of the learner
is underplayed in the stimulus-response schema. Existing
between the stimulus and response is an organism with its
cognitive structures. The response exists in the cognitive
structure before the stimulus is enacted. Piaget cites a
study by Berlyne, who spent a year working in Geneva, in
Which an attempt was made to translate Piaget's theories
on the development of operations into Hullian stimulus-response

language. Berlyne found it necessary to introduce the concepts

Of "transformation responses" and "internal reinforcements".

 

78
Piaget noted that these are nothing more than his "operations"
and "equilibrations". Piaget, referring to his theory, states:
So you see that it is indeed a stimulus-response
theory, if you will, but first you add operations

then you add equilibration. That's all we want!

(Piaget, 1976, pp. 78-79).

Piaget (1976) indicates less disagreement with Gestalt
theory. He notes that he had been aware of the works of
Wertheimer and Kohler when he began his own research, that
he would have become a gestaltist (p. 121). The two theories
parallel each other in that cognitive activities and the
reality on which they act are structured totalities. Both
consider intelligence and perception as part of the system
,of equilibrations. The primary difference between the two
theories is that the gestalt is more static than Piaget's
dynamic and modifiable schema. In addition, Piaget's schema
is always the result of differentiation, generalization,
and integration of earlier schemata, while the gestalt is
an outgrowth of a certain level of neural maturation, given

a particular perceptual field, and not of past environmental

interactions (p. 121; Harrison, 1967, p. 67).

Bruner's Theory of Mathematics Learnipg

The noteworthy flaws in Piaget's extensive theory of
cognitive development, as far as many American psychologists
were concerned in the 1950's, was Piaget's lack of definite
statements as to how or why a child passes from one stage
of operations to another, and Piaget's belief that logical
Structures are independent of language (Kagan, 1966, pp.

98-112). In the 1950's American experimental psychologists,

’H

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79
under the leadership of Jerome Bruner, were beginning extensive
research into the analysis of the variables affecting and
processes underlying concept learning. These psychologists
not only found points of disagreement with Piaget, but also
pushed the associationistic view of learning into disfavor
in the United States (Bourne, et al, 1979, p. 169).

Bruner and his colleagues conducted experiments with
adults in which they examined the strategies people employed
in the complex process of sorting and classifying that consti-
tutes concept develOpment. Those pioneering efforts of Bruner
and his associates resulted in the publication of A study
of Thinking (Bruner, Goodnow, and Austin, 1956). A major
finding from these concept attainment studies was the realization
that researchers can describe and evaluate strategies in
a systematic way in terms of their objectives. It was this
laboratory experimentation with adults that led Bruner to
begin to examine the cognitive processes of children. He
became particularly interested in how children mentally repre-
sented the concepts and ideas they were learning.

That Bruner is essentially Piagetian, is well illustrated
by the fact that Bruner's 1966 publication Studies in Cognitive
Growth is dedicated to Piaget (Bruner, 1966A, p. xv). He
notes, "manypoints of disagreement are nevertheless minor
by comparison with the points of fundamental agreement we
share with Professor Jean Piaget" (p. xv). Bruner, like
Piaget, was a constructivist, that is, reality is constructed
by the individual. Anglin (1973) gives a succinct description

of Bruner's theory, which shows the many parallels to Piaget's

Al)

(I!

80
theory:

Though obviously different in basic ways,
several modes of acquiring knowledge - including
the perception of an event, the attainment of a
concept, the solution of a problem, the discovery
of a scientific theory, and the mastery of a skill -
can be described at a formal level in ways that
are strikingly similar. In short, each can be
viewed as a kind of problem whose solution is
actively, though not necessarily consciously,
constructed. Construction usually involves a
recursive process, in which the first step is an
inferential leap from sense data to a tentative
hypothesis achieved by relating incoming information
to an internally stored model of the world based
upon past experience. The second step is essen-
tially a confirmation check in which the tentative
hypotehsis is tested against further sense data.

In the face of a match the hypothesis is main-
tained, in the face of a mismatch the hypothesis

is altered in a way that acknowledges the discre-
pant evidence. That the sense data might be called
cues, clues, instances, or experimental results;
that the hypothesis might be called a category,

a rule, a principle, or a theory; that the internal
model might be called a generic coding syStem,

a focus, a system of representation, a cognitive
structure, a schema, or a paradigm; that the recur-
sive process might be called inference and confir-
mation check, strategy, analysis by synthesis,
induction and deduction, conjecture and refutation,
or the hypothetico-deductive method all these
should not be allowed to obscure the underlying
formal similarity of diverse kinds of mental activi-
ties (p. xxii).

For both Bruner and Piaget, cognitive development involves
qualitative rather than quantitative changes in the cognitive
structures which are available at different age levels. Both
believe in the importance of action in infancy. Both view
growth in childhood as moving the individual from a state
of domination by immediacy and appearance to a state whereby
the individual is able to realize connectedness over time
as well as invariance in face of surface change (Anglin,

1973, xviii).

81

However, there are some major differences between the
theories of intellectual development by Bruner and Piaget.
Perhaps, the most critical difference concerns the significance
attributed to cultural agents in the course of development.
Piaget's theory, while not being a wholly hereditary account,
basically gives little weight to the environment and views
the course of development as relatively fixed. Piaget seemed
little concerned with "the American question" of accelerating
intellectual development. (Anglin, 1973, p. xx). In contrast
Bruner, who believes that a theory of development should
go hand in hand with a theory of instruction, argued that
mental growth to a very considerable extent is dependent
upon mastering techniques that are embodied in the culture°
and thus passed down to the child by cultural agents (Bruner,
19663. p. 21). In line with this belief, Bruner and his
colleagues consistently attempted to design pedagogical means
for accelerating intellectual achievements. Another real
difference between Bruner and Piaget concerns the question
of whether the changes that occur in a child's problem solving
skills at around six or seven years of age are mediated by
language. Bruner believes that symbolic, especially linguis-
tic transformations play an important role in guiding thought.
Piaget argues that the linguistic correlates of success in
Bruner's Studies were symptoms of concrete operation, but
in no way could be causally linked to the increased competence.
It should be pointed out that Piaget's theory is more of
a ”descriptive" nature, while Bruner's theory is more funda-

Inentally psychological, and in comparison to Piaget's theory,

82

of a "prescriptive" nature. Also, in contrast to Piaget,
Bruner had a persistent concern with the relevance of his
theories to practical concerns, particularly education (Anglin,
1973, pp. xviii-xxi).

Piaget's theory of equilibration, as the mechanism responsi-
ble for the child advancing from one stage to another, was
criticized by Bruner for its lack of "specificity" and for
its "circularity" of prediction about growth. However, Bruner
did admit that "cognitive conflict" notion of growth is valid,
but noted, "The rub is that there are many cognitive conflicts.
of this kind that do 22; lead a child to grow" (Bruner,
1966A, p. 4). In fact Bruner proposed a three "stage" model
of intellectual development, whereby growth was given an
"impulsion" by disequilibrium.

Bruner (1966B) stated:

What comes out of this picture, rough though

I have sketched it, is a view of human beings who

have developed three parallel systems of processing

information and for representing it - one through

manipulation and action, one through perceptual

organization and imagery, and one through symbolic
apparatus (p. 28).

Bruner (1966A) calls the mode based on manipulation and action
enactive (pp. 16-21). This is a mode for representing past
events through an appropriate motor response. This mode

is thought to be the only way infants can remember things
during what Piaget has called the sensorimotor stage. Adults
would use this mode for example, when riding a bicycle for
the first time in years. Resnick and Ford (1981) suggest
that children who add numbers by perhaps tapping their fingers

are using the enactive mode of representation (p. 112) .

 

83

Bruner (1966A) called the mode of representation based on
perceptual organization and imagery the iconic mode (p. 21).
Images stand for perceptual events in the same manner that
a picture stands for an object pictured. The mode of represen-
tation based on symbolic apparatus, Bruner (1966A) called
the sympolic mode. Bruner hypothesized such properties as
"categoriality", "hierarchy", "predication", "causation",
and "modification" for the symbolic mode, and noted that
any symbolic activity is not logical without these properties.
According to Bruner, the three modes of representation occur
in the life of the child in the order of enactive, iconic,
and symbolic. Each mode depends on the preceeding one for
its development, yet all remain more or less intact throughout
life. At the enactive level, actions cannot be transformed.
At the iconic level, which is based on images, they can be
transformed but lack a generalization quality. It is the
attainment of the symbolic level and the successful internaliza-
tion of language, that enables the individual to represent
and transform environment regularities with power and strategy.
At the symbolic level, the child can use actions and images
arbitrarily, and perhaps will use all modes of representation
simultaneously. Bruner indicated that without special training
in the symbolic representation of experience, a child will
grow into adulthood depending in large measure on the iconic
and enactive modes of representing and organizing the world
(pp. 21-48) (Harrison, 1967, p. 82).

Bruner does not seem to distinguish between the words

"thinking" and "understanding" (Herscovics, 1979, p. 98).

84

However, he has postulated two modes of thinking: "intuitive"
thinking and "analytic" thinking. According to Bruner (1960)
thinking does not advance in well-defined steps, but seems
to involve an implicit perception of the total problem with
only very little awareness of the processes used. Typically
intuitive thinking rests on familiarity with the domain of
knowledge involved and with its structure. On the otherhand,
analytic thinking proceeds characteristically one step at
a time. A specific plan of attack as well as full awareness
of the operations employed is normal (PP. 55-68).

Bruner (1966B) indicated that good teaching involves
instruction which facilitates student comprehension of a

structure of a discipline, and in instruction which develops

 

learner understanding of process. He favored a "discovery"
format of teaching. "optimal structure" for Bruner referred
to a set of propositions for which a larger body of knowledge
could be generated. Since ideal structure depends upon its
power for simplifying information, generating new propositions,
and increasing the manipulative ability of a body of knowledge,
structure must always be related to present ability and
knowledge of the learner (pp. 44-48).

He states:

...a theory of instruction seeks to take account
of the fact a curriculum reflects not only the
nature of knowledge itself but also the nature
of the knower and of the knowledge getting process...
To instruct someone in these disciplines is not
a matter of getting him to commit results to mind.
Rather, it is to teach him to participate in the
process that makes possible the establishment of
knowledge. We teach a subject not to produce little
living libraries on the subject, but rather to
get a student to think mathematically for himself

85

to consider matters as a historian does, to take
part in the process of knowledge getting. Knowing
is a process, not a product (p. 72).

Reporting on his collaboration with Dienes, Bruner (1966B)
had speCific ideas with respect to mathematics learning:

We reached the tentative conclusion that it
was probably necessary for a child, learning mathe-
matics to have not only a firm sense of the abstrac-
tion underlying what he was working on, but also
a good stock of visual images for embodying them.
For without the latter it is difficult to track
correspondences and to check what one is doing
symbolically (p. 66).

Again, with respect to mathematics learning, he made the follow-
ing observations about children using a wooden model in attempt-

ing to square quadratic expressions:

The children always began by constructing an embodi-
ment of some concept, building a concrete model for
purposes of Operational definition. The fruit of the
construction was an image that "stood for" the concept.
From there on, the task was to provide means of represen-
tation that were free of particular manipulations and
specific images. Only symbolic operations provide the
means of representing an idea in this way. ...what
struck us about the children as we Observed them is
that they not only understood the abstractions they
had learned but also had a store of concrete images
that served to exemplify the abstractions. When they
searched for a way to deal with new problems, the task
was usually carried out not simply by abstract means
but also by "matching up" images (p. 65).

And in another statement:

We would suggest that learning mathematics reflects
a good deal about intellectual development. It begins
with instrumental activity, a kind of definition of
things by doing them. Such operations become represent-
ed and summarized in the form of particular images.
Finally and with the help of symbolic notation the remains
invariant across transformations in imagery, the learner
comes to grasp the formal or abstract prOperties of
the things he is dealing with. But while, once abstrac-
tion is achieved, the learner becomes free in a certain
measure of the surface appearance of things he nonethe-
less continues to rely upon the stock of imagery he
has built en route to abstract mastery. It is this
stock of imagery that permits him to work at the level

86

of heuristic, through convenient and nonrigorous

means of exploring problems and relating them to

problems already mastered (p. 68).

Bruner (1966B) disagreed with Piaget's stage development
of "readiness". He believed that complex notions could be
presented in an understandable form at any age. He states:

Any idea or problem or body of knowledge can

be presented in a form simple enough so that any

particular learner can understand it in a recogniza-

ble form (p. 44).

This Often misunderstood statement, is not advocating, for
example, symbolic, mathematics to very young children, but
indicates that, if a teacher is clever enough, any idea can
be presented in a somewhat recognizable form in one of the
three modes: enactive, iconic, or symbolic, to any child.
One consequence of the previous statement by Bruner is that
he advocates a "spiral" curriculum (Bruner, 1973A, pp. 423-
425). What he is not advocating is the premature presentation
of symbols:

What is most important for teaching basic

concepts is that the child be helped to pass pro-

gressively from concrete thinking to the utiliza-

tion of more conceptually adequate modes of thought.

But it is futile to attempt this by presenting

formal explanations based on a logic that is dis-

tant from the child's manner of thinking and sterile

in its implications for him. Much teaching in

mathematics is of this sort. The child learns

not to understand mathematical order but rather

to apply certain devices of recipes without under-

standing their significance and connectedness (p.

416).

He further states: "For it is in the development of symbolic
representation that one finds, perhaps, the greatest thicket

of psychological problems (1973B, p. 327)." And, finally,

87
with respect to the teaching of algebra:

We provide training in addition, then we move
to abstract symbols like a + a + a and see whether
3a emerges as the answer. Then we test further
to see whether the child has grasped the idea of
repeated addition--(or) multiplication. We devise
techniques of instruction along the way to aid
the child in building a generic code to use for
all sorts of quantities. If we fail to do this,
we say that the child has learned in rote fashion
or that--welwwe given the child 'mechanical' rather
then insightful' ways of solving the problem. The
distinction is not between mechanical and insightful,
really, but whether or not the child has grasped
and can use the generic code we have set out to
teach him (1973C, p. 222).

Thus, on the basis of Bruner's theory, a researcher
investigating a student's understanding of algebra should note
the actions that are performed, the imagery used, as well

as any symbolic system used.

Skemp's Theory Of MathematiCal Understanding

Richard Skemp (1979B) of Great Britain has developed
a comprehensive model of intelligence which he offers as
a replacement for the current models that are based on "I.Q.".
Skemp (1979A) indicates that such I.Q. models can only rank
order individuals with respect to intelligence. They do
not deal with what intelligence is for, how it works, or
how learners can be aided to make the best use of whatever
"intelligence" they possess (p. 44). Skemp's model purports
to do all of these.

The starting point for Skemp is his observation that
most human behavior is "goal directed"; together with the
conjecture that cumulatively, success in achieving goals

is a major factor of survival. For goal directed activity

88

operating on the physical environment, there exists a "director
system", which receives information about the present stage
of the "operand" (what is being acted on), compares this
with a selected "goal state", and with theaid of a "plan"
which it constructs from available "schemas“, takes the operand
from its present state to its goal state and keeps it there.
This director system, Skemp called "delta-one" (Skemp, 1979A,
p. 44). Skemp has also designated a "delta-two" director
system. However, its operands are not in the "outside" environ-
ment, but in delta-one. That is, they are "mental objects".
The purpose of delta-two is to optimize the functioning of
delta-one. The job of delta-two is goal-directed mental
activity of many kinds which includes "learning". Learning
includes the construction and testing by delta-two within
delta-one of the schemas which delta-one must have to function
properly.

Skemp (1979B) makes a distinction between the environment
in which phySical actions and activities take place and a
"mental realm" (p. 21). Individulas build "mental models"
of the physical environment. For a director system to function
properly, the mental models must match "actuality". Important,
also, is the accuracy with which the goal state as represented
in an individual's mental model "matches the consequences
of achieving that state", and the accuracy and completeness
of the paths by which the Goal state may be reached (p. 23).
Skemp notes that mental models differ in significant ways

from physical ones. One of the chief characteristics of

these mental models "is that their elements nearly always

89
represent not just one actual object or event, but what is
common to a number of these." He calls a mental entity of
this type a ”concept". By representing what is common to
a variety of actual states, concepts enable an individual
to act successfully in a wide range of actual systems by means
of the same director system. Skemp calls models made of
a number of interconnected concepts a "conceptual structure".
The process by which certain qualities of objects and events
are internalized as concepts, he call "abstraction". Skemp
emphasizes that the development of conceptual structures
and the ability to use the process of abstraction is indicative
of the director system's openness to change - to "learn".
A distinctive quality of intelligence can be inferred by
an organism's ability to construct new director systems,
and to improve the Ones it has (Pp. 24-25). He states:
1 Intelligent learning is now conceptualized
as a process involving two director systems.
These are: _
(a) a director system delta-one, whose state

can be changed in this way.

Such a director system is described as

teachable.

(b) a director system delta-two, who operand

is the director system described in (a),

and whose function is to take it towards

states which make possible optimal function-

ing (pp. 89-90).

Skemp notes that when an individuals director system
does not have the ability to change as in "intelligent learning"
and change toward a goal-state gives rise to frustration
and the inability to change away from an anti-goal state

gives rise to anxiety (p. 16).

Skemp (1979B) calls the formation of a concept based

90
on incoming sensory-data "perceptual learning", and such a
concept a "primary concept". Concepts which the individual
derives from other concepts are called "secondary concepts"
(pp. 118-120). For Skemp a "schema" is "a structure of con-
nected concepts" (p. 190). He terms the ability to makes one's
mental processes and schemas the objects of conscious view
"reflective intelligence" (p. 175). Skemp writes of "assimila-
tion" - "This emphasis in our experience of actuality towards.
what is like our existing schemas", "expansion" of schemas,
and "reconstruction" of schemas. It is interesting to note that
Skemp in this 1979 formulation of his theory does not use the
word “accomodation", but earlier (Skemp, 1972, p. 190), did
use this word.

It is not the purpose of this review to completely explore
Skemp's model of intelligence in detail (Skemp's explanation
required 324 pages), but to give a synopsis that will aid in
the description of his theories with respect to the teaching
and learning of mathematics. His model of intelligence is pri-
marily Piagetian, though he denies it emphatically. That Skemp's
”assimilation", "expansion", and "reconstruction" equals Piaget's
”assimilation" and "accommodation" is obvious. However, Skemp
feels that Piaget fails to distinguish between "psychological"
and "physiological“ assimilation. Skemp‘indicates that his use
of "schema" as a mental structure existing in its own right
contrasts with Piaget's "schema" as being linked to actions.
Importantly, Skemp believes that "Piaget's view that any act
or mental state, can in itself be described as intelligent“

is inferior to Skemp's notion that intelligence cannot be

91

inferred from behavior in itself, but only from "changes
in behavior" (Skemp, 1979B, pp. 212-221).

That Skemp's theoretical differences with Piaget are
not major has been seen by the following reviewers. Bolton
(1977) states, “...the main features of Skemp's schematic
learning are essentially Piagetian" (p. 137). Fehr (1966)
commenting on Skemp's early work on concepts,. comments,
"This is quite in agreement with Piaget's phiIOSOphy, developed
through his study of children's thinking, namely, that to
'know a concept is to act on it'“ (p. 224). Harrison (1967)
writes, "Skemp's assertion that he has added to Piaget's
description in his discussion of reflective intelligence
seems somewhat more illusory than real in the light of an
examination of Piaget's whOle theory" (p. 125).

The primary feature of Skemp's theory is the fact that
he has in particular addressed the teaching and learning
of mathematics for not only young children, but also those
at the secondary leVel. In this regard, he has extended
the work of Piaget and Bruner, whose research in mathematics
learning dealt mainly with young children that had not attained
the formal level of thought. Skemp has written extensively
about the teaching and learning of mathematics, particularly
algebra, as it pertains to students that are on the verge
of the formal stage, or have in fact reached it.

Skemp views algebra as a "generalized arithmetic". This
is indicated by his statement, “...the number schema, combined
with the idea of a variable, together lead straight into

algebra" (Skemp, 1971, p. 237). In another place he wrote:

92
(cited in Harrison, 1967, p. 106).
Just as the number concepts and arithmetical
processes are abstractions from and generalization

of experiences with material objects, so are the

algebraic concepts generalizations of experiences

with numbers and arithmetical processes.

Skemp (1971) as previously noted, indicates that concepts
were either "primary”, which are based on sensory experiences,
or "secondary", which are derived from other concepts (PP.
118-119). He believes that all mathematical concepts are
secondary concepts. Skemp states that, "a concept is a (learnt)
realization of some-regularity in actuality" (Skemp, 1979B,
p. 29). He also ordered concepts: *If concept A is an example
of concept B, then we shall say that B is of a higher order
then A" (Skemp, 1971, p. 25). He defines relationships between
concepts and speaks of a "conceptual hierachy" for interrelated
concepts (p. 25). Of extreme importance to the field of
mathematics, and for this study as well, is Skemp's assertion
(that:

In general, concepts of a higher order than
those which a person already has cannot be communi-

cated to him by a definition, but only by collecting
together, for him to experience, suitable examples

(p. 26).
Definitions are useful for adding precision to the boundaries
of a known concept, or of indicating its relation to other
concepts. They can also be used to communicate new concepts
of a 1232; order (p. 26).

Skemp, like Bruner, gives a great amount of credit to
language in the intellectual developmental process. He defines
a “language" to be a "set of symbols, with their associated

concepts, such that for certain relations between the symbols

93
there correspond certain relations between the concepts"
(Skemp, 1979B, p. 159). A symbol is a primary concept, and
reflective thought is closely connected with associating
the involved concepts with a symbol. He suggests that verbal
symbols concentrate our attention on one part of a schema
at a time, and is analytic; whereas visual symbols is better
at showing how parts relate to each other, and is mainly
synthetic (p. 158). If concepts are sufficiently well formed
to be activated by encountering of thinking Of the associated
symbol, then symbols aid in helping to show structure, making
routine thinking automatic, understanding information, and
retrieving information. Symbols are essential for reflective
activity; they help acquire the raw data, and then enable
the individual to manipulate the ideas, first in one way
and then in the other (p. 160).

Particularly in mathematics, symbols make a major contri-
bution. It is essential in mathematics that elementary processes
become automatic, thus freeing the individual's attention
to concentrate on new ideas. In mathematics, this is done
by detaching the symbols from their concepts, and manipulating
them accoring to well-formed habits without attention to
their meaning. This automatic performance of routine tasks
must be clearly distinguished from the mechanical manipulation
of meaningless symbols, which is not mathematics. A machine
does not know what it is doing. A mathematician, working
automatically, can at any time he wishes pause and re-attach
their meanings to the symbols; and he must be able to pass

easily from one form of activity to the other, according to

the requirements of the task (Skemp, 1971, p. 89).

94

Skemp

emphasizes that without the ability of the individual to

invest symbols with the appropriate meanings, they are useless

(p. 89). It follows that in
and learning of mathematics,
student and teacher have the

the symbols (Skemp, 1979, p.

using symbols in the teaching
it is imperative that both the
same concepts associated with

153).

Skemp (19793, p. 158) notes that reflective activity
which involves making an idea conscious seems to involve

associating the idea with a symbol. Concepts are elusive
objects and it may bethat symbols are the most abstract

kind of concept of which an individual can be aware. It

is largely by the use of symbols that humans achieve voluntary
contrOl Over their thoughts (Skemp, 1971, p. 83). A far
reaching type of reflective activity is that which leads
to what Skemp call "mathematical generalization". An example
is offered that deals with exponential notation. After defining

the notation by examples such as:

2
a3 = a-a
a4 = a-aoa
a. = a-a-apa, etc.

it can be seen rather easily that

2 3

a ' a = a-a ' a-a-a

Having developed these schema the individual can "formulate"

m n m+n

it symbolically as a a a = a , where a is any real number

and m and n are natural numbers. If the learner reflects

on the "form“ of the method, while ignoring the content

and original restructions, and considers such "illegal"

operations as 20 - 23 = 20+3 = 23(as being valid operations),

95

and 2.3 . 23 = 2-3+3 = 20 and determines that logically 20 = 1

3 3

and 2- = 1/2 , then consequently Obtains an extension of

the concept of exponent and the method involved, and writes
aman = am+n for all integers m and n; Skemp would say the
individual has used mathematical generalization (PF- 59-
61; Mick and Brazier, 1979, pp. 60-61).

Perhaps Skemp's major contribution to the teaching and
learning of mathematics is his conjectures pertaining to
"understanding". He states:

To understand a concept, a group of concepts,

or symbols, is to connect it with an appropriate

schema. To understand an experience is to realize

it within an appropriate schema (Skemp, 1979B,

p. 148) I
and,

We may mis-understand, meaning we think we understand

something when we do not...because we connect it

wrongly to an appropriate schema...or we may connect

it to an inappropriate schema (p. 148).,
and also,

In the complete absence of an appropriate

schema, not only can there not be any understanding

at all, but there may be complete unawareness of

its absence (p. 147).

Skemp (1979A) indicates that an "appropriate schema"
is "appropriate to the subject matter" and "appropriate to
the gOal to be achieved" (p. 45). He notes, without details,
that there could possibly be other considerations.

Skemp does not elaborate on what an "appropriate" schema
is, but does indicate the "qualities" of a schema. In short
he notes "the difference between a schema and a set of isolated

concepts is, that in the schema the concepts are connected..."

($«mp, 1979B, p. 187). Skemp describes "connections" as either

96

"associative" or "conceptual", but cautions that a connection
which is conceptual for one person may be associative for
another. The higher the proportion of conceptual links to
associative links the higher the quality of the schema. A
critical feature for connected concepts is that the "activation"
of one should lead to the activation of the other (PP. 187-189).
Skemp uses the analogy of a "cognitive map" (like Tolman)
to indicate that a schema is "a structure of connected concepts"
(p. 190). Among the features important for the effective
use of a schema are, relevance to the task at hand, extent
of its domain, the quality of organization which makes it
possible to use concepts for a lower or higher order, the
quality of connections (associative or conceptual), the degree
to which it can function when confronted with irrelevant
information, the degree to which it can assimilate new facts
and assimilatory power relative to other schemas (PP. 190-
191). Apparently for Skemp, the learning of a new concept
in algebra would require a well-conceptualized bank of prerequi-
site concepts.

Skemp (1979B) has put forth two types of understanding:
“instrumental" and "relational":

Instrumental understanding, in a mathematical situ-

ation, consists of recognizing a task as one of

a particular class for which one already knows

a rule. To find the area of a rectangle, multiply

the length by the breadth. For a triangle, calcu-

late half times the base times the perpendicular

height.... The kind of learning which makes this

possible, which I call instrumental learning, is

.the memorizing of such rules.

Relational understanding, in contrast, consists
primarily in relating a task to an appropriate
schema... If he has an appropriate parent schema,

someone who does not know the rule can still calculate
the area of a given trapezium by dividing it into

97

a triangle and a parallelogram... (p. 259).

In an earlier article Skemp (1976) indicates that the
terms instrumental understanding and relational understanding
were suggested to him by Steig Mellin-Olsen of Bergen University.
Mellin-Olsen and Skemp justify their terminology by their
belief that students and teachers say they understand something
if they can use a rule or procedure to obtain the correct
answer, thus using mathematics in the same way an individual
learns to use an instrument. In relational understanding
the individual is aware of the underlying mathematical relation-
ships - which includes inStrumental understanding (p. 20).

Skemp (1976) notes that it is not a self-evident truth
that mathematics teachers should strive for relational under-
standing. Instrumental understanding has several (apparent)
advantages: instrumental mathematics is usually easier to
"understand" ("minus times minus equal plus"), the rewards
are more immediate (success breeds confidence and effort),
often less content coverage is required, and the right answer
can usually be obtained faster (even theoretical mathematicians
use instrumental thinking). On the other hand relational
understanding is more adaptable to new tasks (one does not
need a new rule for each new situation), it is easier to
remember (but harder to learn), it Often can become a goal
in itself, and relational schemas are organic in quality--
once an individual sees relational understanding as a goal,
he will seek out new material to understand (PP. 23-24).

(Skemp believes the long-term educational advantages

of relational understanding are obvious, but offers several

98
reasons why instrumental mathematics is taught in many class-
rooms: (1) relational understanding takes too long when
only one technique or problems needs to be learned, (2) the
topic is too difficult, but the students need to "know" it
for an examination, (3) a skill is needed in another area,
such as science, before all the prerequisite schemas are
available for relational understanding, and (4) a new teacher
may choose to teach instrumentally because all of the other
faculty teach this way. Situational factors which contribute
to the adopting of an instrumental technique by a given teacher
include; good examination scores are needed to impress super-
visors, over-burdened syllabi, the difficulty of assessing
relational understanding, and the psychological difficulty
of adapting to relational goals after many years of teaching
instrumentally (p. 24).

Skemp cautions that aside from which type of understanding
is better, there are some very serious mathematical mis-
matches that can occur. Students whose goal is to understand
instrumentally, may be taught by a teacher that wants them
to understand relationally, or the other way about. Also,
the text may aim at either instrumental or relational understand-
ing (perhaps changing according to topic), and thus create
more chances for mis-matches. Skemp believes that much of
the "modern mathematics" used texts whose aim may have been
relational understanding, but were taught by teachers with
an instrumental philosophy. Consequently, students would
have been better off to have a "traditimrflfl text, for at least

they would have become proficient at many mathematical techniques

99
that would have been of use in other subjects (p. 22).

Backhouse (1978) as well as Byers and Herscovics (1977)
_have disagreed with Skemp's distinguishing between two types
of understanding. Byers and Herscovics believe there should
instead be four categories. Byers and Herscovics would add
"intuitive" and "formal" understanding. Intuitive understanding
(like Bruner) "is the ability to solve a problem without
prior analysis of the problem", and formal understanding
"is the ability to connect mathematical symbolism with relevant
mathematical ideas and to combine these ideas into chains
of logical reasoning” (Byers and Herscovics, pp. 24-27).
Herscovics suggests that the introduction of a formal approach
to the teaching of algebra without the formal notation being
given meaning is what leads to instrumental understanding.

The significance of research into "understanding" is expressed
by Herscovics (1976):
Because it has been so difficult to define
understanding, it has often been expressed in the

form of vague generalities. This inability to

define understanding clearly has made it difficult

to counter the criticism of those evaluators using

exclusively as their criteria the so-called products

of learning, namely, the right answers (p. 104).

Skemp (1979A) did amend his theory to add formal under-
standing, which he calls "logical" understanding. He also
conjectures two "modes of mental activity", "intuitive" and
"reflective” (p. 45). His logical understanding, like Byers
and Herscovics, is the ability to connect mathematical symbolism
and notation with relevant mathematical ideas and to combine

these into chains of logical reasoning (p. 45). Skemp compares

this with Bruner's ”analytic apparatus of one's craft", and

100
notes that the objective is not the development of new schemas
or concepts, but to ascertain that the existing schemas which
have been constructed and the solutions which have been devised
are accurate (p. 47). Skemp's logical understanding may be
thought of as the techniques which are involved in formal proof.

It should be pointed out that there is not general agree-
ment as to Skemp's logical understanding. Choat (1981) has
argued strenously that relational understanding, "...provides
for the formation of concepts in two categories - physical
experience and logico-mathematical experience" (p. 19). He
states:

Logic is implicit throughout relational under-
standing and does not apply only to inductions

to regulate form so agreement cannot be reached

to accept a separate category of logical understand-

ing as defined by Skemp (p. 19).

In fact Choat indicates that at about the "junior school"
(England) age, children are deemed to no longer need play
and physcial experience in learning, and that subsequent
teaching is based on the medium of language, and a consequent
formal approach, which leads inevitably to instrumental under-
standing (p. 21).

This summary of Skemp's theory will end with two statements
by Skemp (1971) which emphasize in no uncertain terms his
Stance against what he considers the prevalence of the teaching
of mathematics based on meaningless rules.

Instruction of this kind may fairly be described

as a series of insults to the intelligence; for

they purport to be based on reason, but (usually)

are not. To the extent what is.being communicated

is not intelligible, the receiver is trying to

accomodate his schemas to assimilate meaninglessness.
To do this would be equivalent to destruction of

101

these schemas; the mental equivalent of bodily
injury (p. 117).

Imagery Research

 

That a review of the research on "imagery" was essential
for any study which relates to the uses of Bruner's "iconic"
mode or Piaget's "concrete" stage is succinctly4illustrated by
Clements (1981) in an article in which he discusses the some-
what confused state of imagery research. He notes that the
theoretical formulation of imagery might seem irrelevant to
mathematic educators, but that the recent use of visual aids
and the growing realization and research indicating that
imagery can be of fundamental importance in the teaching and
learning of mathematics, is likely to generate in the mecha-
nisms of how images are evoked. He notes:

A second, and equally important, reason why
mathematics educators should concern themselves

with theoretical formulations of imagery is that

unless they do they will not be able to take full

advantage of the substantial body of research on

imagery which has been carried out by psychologists,

following the remarkable return to favour of imagery

research in the 1970's. Mathematics educators will
continue to give the impression of naivete' in mat-

ters related to research concerning imagery and

spatial ability unless they become better acquainted

with the relevant psychological literatures (p. 7).

Psychologists, in addition, to studying the "what" and
"how” of mental events have generally been interested also in
the "why" of mental events. Due to the fact that mental acti-
vities occur completely "inside the head" no other person can
be aware of the actual occurrence of a mental event. Since
scientists can only study observable forms of behavior, the

study of mental imagery has a long and controversial history

(Holt, 1970, p. 254).

102

The fOllowing is a short synOpsis of the early history of
mental imagery in America as reported in Kosslyn and Jolicoeur
(1980, pp. 139-144).

Self-reporting methods are the most ancient techniques for
studying imagery. The belief that people differed in their abili-
ty to recall sensory experiences was recorded by Fechner in 1860.
In 1883, Galton developed this idea more thoroughly by way of a
"breakfast table" questionnaire, whereby people were asked to
imagine their breakfast table that particular morning with respect
to color, brightness, and detail. Slightly over 10 percent of
his subjects appeared not to have images. The nonimagers tended
to be successful scholars and scientists, while the higher imagers
were women and children. Kosslyn and Jolicoeur note that McKellar
(1965) found that virtually all members of Mensa (who score very
high on IQ tests) report some imagery in contrast to Galton's
findings. Thus, as the authors note, either Galton's methods were
flawed or times change (p. 141). Betts (1909) refined question-
naires by Galton by having people assign a numerical value to
vividness with respect to sensory modes. He found that auditory
and visual imagery were slightly more vivid than other modes.

In the early 1900's Wilhelm Wundt, often identified as
the founder of "modern psychology", and his followers believed
that all thought processes were accompanied by images. When
they could not substantiate their beliefs by research, a
hotly contested psychological debate ensued. Somewhat as a

result, the behavioristic philosophy of John Watson gained

 

wide acceptance. Watson rejected the ideas of mental images

and studied behavior in its own rights, disregarding

103
any attempts to draw conclusions about internal events from
observation. Consequently, imagery fell out of favor during
the reign of the behavioristic brand of psychology in the
United States. It was not until the 1960's that the limitations
of behaviorism were perceived and "cognition" came to the
forefront, that imagery research once again flourished.

Contemporary research in imagery is in the following
areas (Kosslyn, 1980, pp. 2-3):

1. Attempts to determine thereffects of the use of
imagery on a person's ability to perform various tasks.

2. Attempts to demonstrate how imagery is involved
with modality-specific perception - e.g. auditory image vs.
visual image.

3. Attempts to determine situations in which people
use imagery spontaneously.

4. Attempts to ascertain the structure of imagery.

During the 1970's several researchers have put forth
theoretical models of the imagery construct. The models
fall primarily into either of two classes; "picture-in-the-
mind" theories, and "propositional" theories (Clements, 1981,
p. 3).

One of the most influential picture-in-the-mind theories
was developed by Paivio. Paivio's "dual coding" theory suggests
that a particular stimuli may bring into play both an imagery
system and a linguistic system. To Paivio, memory is dual--
both pictorial and verbal. Images are thought to be concrete
and parallel while verbal representations are abstract and

sequential. In a finding that has definite implications

104
for mathematics, Paivio indicates that concrete words are
remembered better than abstract words, and that pictures
are remembered better than either. Paivio's model has been
criticized for failing to address the questions of generation,
inspection, and transformation of images (Pinker & Kosslyn,
1983, p. 48-49).

Another picture-in-the-mind theory was developed by
Kosslyn. In this model the "picture" is a two-dimensional
rapidly decaying icon of how the object really looked at
some particular time. The images have no long term memory
role. Long-term information is stored as abstractions, and
images are "generated" out of these representations. In
Kosslyn's view the same storage format does not apply for
the storing and generating of mental images and verbal informa-
tion. Images have two components, a pictorial "surface represen-
tation", and a "deep representation" contained in long term
memory. Long term memory stores not just quantitative facts
but also lists of facts in "symbolic" format which allows
an individual to generate images in novel combinations. Kosslyn
has used the analogy of a cathode ray tube (CRT) to describe
his model, and has used a computer simulation program to
produce a "picture" of a "car" on the CRT (Kosslyn, 1980,
jp. 154; Clements 1981, p. 4-6).

One researcher who feels he has supplied empirical evidence
for the picture-in-the-mind model was Shepard (1978) (cited
by Clements, 1981, p. 4, and Yuille and Marschark, 1983,

P- 143). Shepard designed experiments which purported to

Show that mental images could be "rotated" in a manner analogous

105

to the way the physical object would be rotated. His conclusions
are based on the "time lag" for the subject to mentally rotate
an object.

The preceding theories are models that use the image
as an essential element of cognition. That these models
are not universally accepted is well illustrated by Pylyshyn.
In a debate that has some of the same features as "cognition"
vs. "stimulus-response", Pylyshyn (1973, 1980) (cited in
Yuille and Marschark, 1983, p. 147) argued that images do
not exist as "pictures-in-the-mindfi,but that knowledge is
stored as a set of propositions. He feels that there exists
basic similarities between cognitive operations and computational
procedures. That is, cognitive processes can be modeled
by formal operations operating on symbolic structures, much
like computer data structures. The computer is the basic
tool for evaluation by these "computation" theorists. These
theorists reject the notion that it is important to postulate
imagery and believe that both verbal information and mental
images can be generated from a set of propositions which
represent a persons knowledge and are stored in memory. Pylyshyn
has said that KoSslyn's "picture" of a car on a CRT is immaterial,
that what really COUDtS is the computation and symbolic procedures
that were used to develop the "image”, and is therefore redundant

(Clements, 1981, p. 6).

Anderson (1978) believes the problem of defining imagery
may be insoluble. He claims there is no way to determine
Which group is correct, barring strong physiological data.

He feels both models can be made to fit the data. He advocates

106
as a solution a "hybrid" or dual-code model such as Piaget's.
That this debate between the two groups is viewed as exceedingly
serious for the future of psychological study is indicated by
Yuille(l983).
However, there is a stronger, more important
conclusion being offered here: that a continuation

of the current type of theorizing and research

related to imagery will improve the likelihood

of success of the computational approach, and that

approach will be disasterous, not only for imagery

models, but for psychology as a whole. The current
crisis in imagery research stems from the weak
theoretical concepts which characterize the field.

For the most part, definitions of imagery seem

to be based upon intuitive notions of mental pro-

cesses, and the interpretation of research results

appear to originate from the same intuitive source.

This is not an argument that intuition is not a

useful heuristic, but rather that it is not a suffi-

cient basis for an empirical science. What is

required from aspiring theorists is a solid defini-

tion of imagery, and its relationship to other

cognitive processes (pp. 280-281).

There are several other theories which vary somewhat
from the "computational" and "picture-in-the-mind" tenets.
Clements (1981) cites an example of one such theory: the
theoretic account of imagery by the French psychologist M.
Denis. He insists imagery is an active constructive process.
He distinguishes between life-time figural schemata developed

. . . t . .
thnxghout an indiViduals life and stored in long-term.memory,
and the mobilization of figural schema in a particular situation,
which gives rise to mental images. Denis conjectures that
"representational units” are stored in memory. These units
are hierarchically organized components organized according
to their probability Of contributing to image formation.
A key feature in an "activational process" which can be

applied selectively to the task at hand. The word "pie"

107
could evoke the image of a person eating an apple pie or
a circular pie diagram for the solution of fraction problems.
Denis feels the contention that imagery may result in faster
and better recall is not necessarily true. He believes that
on some tasks the extensive use of visual imagery may be
a hindrance (p. 6).

Clements (1981) notes that several other imagery theories
are implicit in what is known as the "representational-develop-
(ment hypothesis" (p. 6). The three main features of this
hypothesis are: (1) internal representations used changes
with age, (2) the representational forms of adults are stronger
than representational forms of children, and (3) the earlier
preferred forms are not erased.but are supplemented, and
overshadowed, by later representational forms. From this.
comes the claim that older people think more "abstractly",
while young children rely more on images.

Of particular interest to this study is the fact that
the leading proponents (implicitly) of the representational-
develOpment theory are Piaget and Bruner. They did not denote
a specific theory of imagery, but it is embedded in their
total theories.' '

Piaget and Inhelder (1971) (cited in Clements, 1981)
denote a difference between "reproductive" images which are
analogous to known objects or events and "anticipatory" images
(at about seven or eight years of age) in which objects or
events not previously perceived are represented. Also, Piaget
and Inhelder refer to images as "static", "kinetic" or "trans-

formational”. Inhelder and Piaget indicate that images are

108
not derived from perception, but are "an interiorization of
imitation". To Piaget it is evident that children are in-
capable of forming mental images until they have proceeded
through a stage whereby objects can only be represented by
actions performed on the objects (sensorimotor period). Only
after about 18 months of age does imagery representation come,
and children can think about things that are not directly per-
ceived. Kosslyn (1980, p. 460) notes that it is extremely dif-
ficult to consider Piaget's theory of imagery in isolation since
it is embedded in the framework of his total theory. Kosslyn
takes issue with Piaget and Inhelder in their insistence that
images are "interiorized imitations". Kosslyn states, "It
never is clear exactly what is imitated, or how such imitation
occurs" (p. 410). Kosslyn describes Piaget's theory as "de-
scribing“ a phenomenon rather than explaining it. In a rather
strongly worded criticism he notes:

Piaget and Inhelder's account is more on the
level of intentionality..., and hence is open to
multiple interpretations on the level of the func-
tion of the brain. They do not specify how interior-
ized imitation operates, nor have they specified the
format or content of the image. This level of dis-
course will never process adequacy, and hence seems
of limited value" (p. 411).

Kosslyn also writes:

Given the long-standing popularity of the idea,
there is surprisingly little evidence that young chil-
dren really do utilize mental imagery more than do
adults (p. 415).

However, that Piaget has strong support for his imagery

theory, comes from Yuille and Marschark (1983, p. 150). They

note that Piaget's model of cognition contains both an amodal

109

and imagery code. Piaget's memory consists of mental structures
and schemata. Images are not elements as such in these struc-
tures, but are tools that the system can use in problem solving.
,When Piaget indicates that young children can imitate aspects
of the environment, or anticipate consequences of their actions
by using images, he implies that such actions could not take
place without the aid of images. Yuille and Marschark state:
The Piagetian approach may be a possible resolu-

tion to the computational-imagery debate. However,

before such an alternative becomes viable, the meaning

of the concept "scheme" and the nature of the mental

structure must be elaborated. Although this constitutes

a major task, it might be a more worthwhile expenditure

of effort than the continuation of the current debate

(p. 150).

'Bruner (1970) built somewhat upon Piaget's notions of
the child's interactive relationship with the environment to
devise a model of how such interactive episodes are represented
in the mind. Bruner felt that the most important thing about
"memory" was not storage of past experience, but the retrieval
of relevant information. The information needs to have been
coded, and have some method of being processed in order to be
adequate. This system of coding and processing, Bruner called
"representation". He called the three modes of representation
(discussed elsewhere) "enactive", "iconic", and "symbolic".
He states:

Iconic representation summaries events by

the selective organization of percepts and of images,

by the spatial, temporal, and qualitative structures

of the perceptual field and their transtrmed images.

Image ”stands for" perceptual events in the close

but conventionally selective way that a picture

stands for the object pictured (p. 291).

That Bruner did not have a complete theory of imagery is evident

110
in his statement that:
We know little about the conditions necessary

for the growth of imagery and iconic representation,

or to what extent parental or environmental inter-

.vention effects it during the earliest years. In
ordinary adult learning a certain amount of motoric
skill and practice seems to be necessary pre-conditions
for the development of a simultaneous image to
represent the sequence of acts involved.
That Bruner (1973) held imagery to be a key ingredient of
his model is indicated by his observations when reporting
on younger children learning to use wooden models to square
binomial expressions:
The children always began by constructing

an embodiment of some concept, building a concrete

form of operational definition. The fruit of the

consruction was an image...(p. 432).

And also,

. ...they had not only understood the abstractions

they had learned but also had a store of concrete

images that served to exemplify the abstractions

(p. 433).

Bruner (1973) has offered the advice that mathematics
should begin by "instrumental activity" which become "summar-
ized in the form of a particular images", and finally using
symbolic notation "that remains invariant across transformations
in imagery" the abstract properties are grasped (p. 436).
He indicates that imagery is not the desired end result:

Translation of experience into symbolic form...
opens up realms of intellectual possibility that

are orders of magnitude beyond the most powerful

image-forming system (Bruner, 1970, p. 295).

But he does indicate that once abstraction has occured. It
is, "...this stock of imagery that permits him to work at

the level of heuristic..." (Bruner, 1973A, p. 436).

Kosslyn (1980) attacks Bruner et al for their lack of

111
specificity. He states:

Simply positing that equilibration (resulting
in reduction in conflict) is accomplished is not
enough; we need to know how this operation is supposed
to occur. Without such specification, the notion
becomes very difficult to disprove. As the theory
now stands, Bruner, Olver, and Greenfield probably
can account for any finding or its converse with
nearly equal ease' (p. 410).

Clements (1981) comments on the fact that "spatial ability"

 

and "mathematical ability" do not necessarily go hand-in-

land. He discusses a study that he and a colleague did with
first-year engineering student in Papua, New Guinea. In

this study they concluded that there was a tendency for students
who preferred to process mathematical information by verbal-
logical means to out-perform more visual students on both
spatial and mathematical tests. Clements notes that after
reviewing the literature pertaining to spatial ability, visual
imagery, and mathematical learning, that in some circumstances
imagery use can have a detrimental effect on abstraction

of concepts. He acknowledges that many studies, for example,
Kent and Hedger (1980), have proven the worth of the visual-
imagery mode in mathematical problem solving. Clements details
his belief that teachers of mathematics generally end up
forcing the verbal mode of learning on students (p. 3).

Wirszup (1974) reported on the significant amount of
research the Russians have done with respect to the learning
and teaching of geometry. As part of their program, they
have researched the relationship between spatial notions
and imagination, the relationship between geometry and other

branches of mathematics and using visual principles in intuition.

112

Wirszup indicates that the Russians agree with Piaget that
geometry in the western world is started much too late --
the teaching of geometry should start early in elementary
school (p. 3).

Krutetskii (1976) studied the extent to which students
rely on visual images in problem-solving. He examined the
role of the "verbal-logical” and "visual-pictorial" components
of mental activity with respect to mathematics learning.
He had two significant conclusions. One, the ability to
visualize abstract mathematical concepts and the ability
for spatial geometric concepts are not necessary components
in mathematics ability. Secondly, the ability to visualize
abstract mathematical relationships and the ability for spatial
geometric concepts showed a very high correspondence (p.
315).

In the United States such investigators as Jencks and
2225 (1972) have followed up the Soviet ideas and have attempted
todevelop more "concrete" procedures in the teaching of
arithmetic. They note:

Fundamental to the idea of mental imagery

is the use of something -- frequently concrete

objects -- from which the learners can find an-

swers for himself (p. 643).
They advocate the use of such objects as graph paper, yardsticks,
diagrams, floor tiles and such. .They state their investigations
(like Bruner) show that a mistake is made, "when symbolic
rules procede mental imagery necessary to give an arithmetic
process a common sense foundation" (p. 644).

Clements (1981) reports research that deals also with

113

older students. Threadgil-Sowder and Juilf's (1980) research
indicated that low mathematical ability students showed signi-
ficant improvement using manipulative materials, but that
high ability students found a symbolic treatment more beneficial.
Horwitz (1981) in a study of college students concluded that
visual properties of a problem affects the solvability by
low performance students but not by high performance students
(Clements, 1982, p. 35).

This review of the literature on imagery is drawn to
a conclusion by a quotation from Clements (1982):

...it is obvious that, considering the large
expenditure of time and money on the research efforts
which have been described, we know precious little
about how and when use of imagery is likely to
facilitate mathematics learning. This state of
affairs should not be regarded as a signal for
a lessening of research activity aimed at increasing
our understanding of the role of visual imagery
in mathematics learning (p. 36).

He notes that one only has to review the literature, for
example, on concept images in geometry, "...to be convinced
that the different images which one associates with certain

mathematical tasks can substantially affect both performance

on, and understanding of, those tasks" (p. 36).

Related Mathematical Studies
Most research studies dealing with the acquisition and/or
formation of mathematical concepts, cognitive processes,
and "remedial" students have been done at the elementary
school age level. There have been some few studies that
dealt specifically with algebra learning and teaching at

the intermediate school and high school levels. Fewer studies

114
still have been conducted with respect to algebra teaching
and learning with college-age students. Since many remedial
courses in algebra at the college level cover essentially
the same material as beginning and intermediate algebra courses,
at the intermediate and high school level, this review included
studies which pertain to algebra learning and teaching regard-
less of the age level of the students. Additionally, studies
which did not directly relate to algebra, but were deemed
to have significant implications for algebra learning were
also reviewed. It should be noted that the number of studies
which used an interviewing technique in an effort to determine
the cognitive processes used by college students in the learning
Of algebra, is rather insignificant in comparison to the

number of "statistically oriented" studies.

Exponents

 

A thorough data base search shows virtually no significant
research has been done with exponent concepts as a dominating
interest. Nearly all research into algebra, and errors in
algebra, list exponents in a "minor" role. Typically, exponents
are noted under "variable errors" or "generalization errors".

This is pointed out dramatically in a study by Alexander‘

 

(1977). In a thorough review of the literature dealing with
both algebraic concepts and error diagnosis at the high school
level, he reviewed 17 studies dating back to 1910. Of these
studies 7 reported that exponents were among the many factors
causing difficulty in algebra. Alexander (p. 10) cites a
study by Fossler (1924, p. 15), which lists 2 x k ‘ 2x15 =

5x as a misconception students had with exponents. Farha

115

(1935, p. 21) (cited in Alexander, p. 18) reports, "94 percent
of all students showed some confusion in dealing with exponents".
A Another study, based on the National Assessment of Education
Progress (NAEP) mathematics assessment testing, during the
1977-1978 school year, reported results that have implication
for the study of exponents. A National Council of Mathematics
Teachers booklet (edited by Corbitt, 1981) compiled, and
commented on the results of this assessment. The following
results are pertinent to this proposed study:

Given the problem: "a4/a20 = ", 30% of the students
that had completed one year of algebra, and 25% of those
students that had completed two years of algebra, gave the
incorrect response of 1/a5 (p. 65).

Given the problem: " b36 = ", 35% of the students with
one year of algebra, and 47% of the students with two years
of algebra gave the incorrect response of b6 (p. 65). The
authors of the report write:

Students generally could simplify expressions

involving positive exponents, but had difficulty

with negative exponents and radicals (p. 63).
and

...both older groups agreed that "there is

always a rule to follow in solving mathematics

problems". The students may be concentrating on

mastering rules to the extent of ignoring concomi-

tant understanding... (p. 146).

The above would indicate that Skemp's warning of a "multi-
plicity of meaningless" rules in algebra should be viewed

with grave concern.

Shlomo Vinner (1977) researched the concept of exponent

 

as related to the "definitional approach" -- some types

116
of numbers are defined by means of a "lower" type. His
sample was one hundred ninety-five college freshmen, all
of whom had attended a calculus course, and fifty-six students,
at the upper level of undergraduate mathematics studies.
He attempted to elicit information concerning the three formula
an = a.a.a...a;a-m = 1/am (a a o); an/m = vafi} where a is
a real number, with m and n whole numbers. The students
were to determine if these formulae represented theorems,
laws, facts about numbers, a definition, or axioms. Only
about one-fourth of the freshmen, and about one-half of the
upper-classmen, correctly identified all the defining formulae.
Two conclusions of Vinner's are pertinent to this study.
First, he advocates that the definitional approach should
not be used until the graduate level. He states, "To teach
the definitional approach before the student is at the suitable
intellectual stage is just useless (although he might pass
the exam)" (pp.24-25). Secondly, he describes a "naive Platonic"
attitude that immature (mathematically) students have toward
mathematics. These students view the existence of mathematical
objects as analogous to the existence of concrete objects
(p. 19). These students believe that, somehow, all arithmetic
operations are discovered. The notion that mathematicians
define operations, contradicts the naive Platonic philosophy.
Vinner's research does seem to have implications for both
teachers and textbook authors in the defining of symbols and
other entities in devising instructional materials for a
remedial algebra class.

Another study, by S. Rachlin (1981), although not limited

117

explicitly to exponents, did deal with them as part of several
other problems. This research stands out due to the fact
that college level students' knowledge'of basic algebra
was studied by clinical means. Rachlin used a Soviet "ascer-
taining" experiement as advocated by Krutetskii. Using both
interview procedures, and paper and pencil tests, he developed
a case study fOr each of four students that were very successful,
grade-wise, in a remedial basic algebra class at the college
level. All four students had been very successful in two
years of high school algebra. Rachlin investigated Krutetskii's
"generalization", "reversibility" and "transfer" in terms
of Skemp's "relational" and "instrumental" understanding.
He concluded that success in.a basic algebra course could
be used to imply generalization ability (defined by Krutetskii)
but could not be used to imply reversibility, or flexibility.
Also, he concluded that the Students differed greatly in
their relational understanding (Skemp's) of particular topics,
even though their test scores were nearly identical in the
basic algebra class.

Of particular significance to this study, are the following
statements by Rachlin:

Rules such as whether to add or multiply exponents

in a particular situation, although finely tuned

for a test, were applied unCertainly a few weeks

later (p. 261).

For example, while working with multiplication of poly-
nomials with variable exponents, all of the subjects indicated
some confusion over whether to add or multiply exponents.

This confusion arose again when multiplying radicals with

118

different indices (p. 248).

Recall Skemp's admonition presented in Chapter I, "An

appropriate schema is one which takes into account the long term

learning task" in light of the preceding paragraph.

more

>4) (a3 + b)2 = a

Rachlin states:

The common behavior observed in the subjects' solution

to the Generalization Test wasmtgeir apparent ease in
generalizing the first rule:(a ) = a , to all its
variants (p. 243).

However, during the application of the second rule: (A+B)2=

+ 2AB + Bz, some confusion did arise as the variants became
complicated:

5 + 2a3b + b2

5) (2x4n + ty)2 = 4x6n + 4x4ny + y2 (p. 86).

Rachlin calls these "false generalizations". Perhaps only

the operational symbols in the first rule have been generalized,

and not the concept. This study will attempt to distinguish

between concept generalization (relational understanding) and

symbolic generalization (instrumental understanding).

what

Also of interest to the present study is Rachlin's comment:

...only three subjects initiated the use of arithmetic
variants of the algebraic tasks as a heuristic for'
solving the tasks. The fourth subject used arithmetic
variants only if she was directed to by the interviewer.
Two subjects experienced difficulty in transferring
processes from arithmetic to algebra (p. 247).

This study will attempt to determine if, when, and under

conditions, remedial algebra students use numerical

imagery as an aid for symbolic tasks.

Concept Learnipg

Davis, Jockusch, and McKnight (1978) over a several year

119
period investigated the "cognitive" or "information handling"
processes that are involved when students in grades seven,
eight or nine begin the study of algebra or "advanced" arithme-
tic. They formulated a rather thorough definition of "under-
standing": (noted earlier)
What is "understanding"? We presume it is (at
least in large part) a composition of many of the
items discussed early in these notes: comparing input
data with many existing things you already know; look-
ing for patterns, contrasts, comparisons; looking for
apparent inconsistencies or contraditions; making care-
ful note of the cues which can be used in the future
to guide future selections of solution methods; trying
to identify and retrieve an appropriate "assimilation
paradigm" or schema, and to synthesize a new one if
no appropriate old one can be found in memory; making
a careful critical appraisal of how well the present
situation matches the retrieved schema that has been
selected: and trying to develOp appropriate "meta-
language" in order to be able to analyze the mathemati-
cal situation effectively (p. 283).
This perhaps is Skemp's "appropriate schema". They point out
that many students do not realize that mathematics can be
understood, but imagine that all people deal with mathematics
by reliance on memorized rote procedures. There are other
students who do recognize that there is such a thing as under-
standing,kmOW when they d9 understand, and recognize the
value of understanding in terms of retaining knowledge as
well as the fact that it is the kind of knowledge that can
be combined with other knowledge to allow greater power in
thinking about novel problems (p. 283).
Davis, et al, state that they have become convinced
that the "purpose" or "intent" cannot be avoided -- students

accomplish exactly what they set out to accomplish. If they

merely desire an answer to get the problem "right", or to please

120
the teacher, this is what they will get. On the other hand,
if a student is really trying to "learn mathematics" by the
definition of understanding (above), this is what they will
do (p. 276).

In an observation that has relevance to this study dealing
with exponents, Davis, et al, note that all teachers they
observed emphasized the fact that algebraic statements are
in.fixn:statements about numbers, and urged students to use
numerical substitutions to check relationships involving
variables. In their words, "Students use this strategy all
too infrequently...” (p. 127). Also relevant to this present
study is their confirmation of Vinner's caution about the
"definitional approach" to algebra. They point out that

many students are confused by the definition of b0 = 1 based

on hxfical arguments, such as b0 - b0 + 2 = b2. Many students
believe that mathematics was created in one "mega-creation"
of the entire system, and are astounded that we are trying
to decide what 20 "ought" to be (p. 118-119). Davis, et
al, note that the, "process of recognizing the general and
separating it from the specific, by imagining variation where
no actual variation has been presented...(p. 95) is fundamental
to mathematical thinking. This agrees with Skemp's "mathema-
tical generalization".

Another study which is deemed pertinent to the one at
hand, is a study by Harrison (1967). Harrison used Skemp's
early work with "reflective intelligence" tests in combination

with a battery of aptitude tests in an attempt to predict

performance scores. The sample was six classes of grade eight

121

students. The significant finding was that Skemp's notion
of "reflecting thinking" in relationship to mathematics learning
was validated. '

.Erlwanger's (1974) case studies of six elementary school
children that were being taught mathematics in an "Individualized
Instruction" curriculum has implications not only for elementary
teaching, but all levels of educational endeavor. All six
students were of average or above intelligence (I.Q.).. Four
of the students were considered by their teachers to be superior
in mathematics ability. The case studies indicated that
emflichild appeared to have "a stable, cohesive system of
interrelated ideas, beliefs, emotions, views, and so on about
mathematics and mathematics learning". The children developed
a view of mathematics as a set of rules for putting symbolic
patterns on paper. They also held strange (in terms of "adult“
thinking) views of the purpose of mathematics and the relation
between rules and answers. The children were dependent upon
a formal type of thinking about rules in which explanations
involved patterns of symbols (Erlwanger, 1975A, p. 157).

The case studies of "Benny" and "Mat” in particular
raise grave doubts about using written test results as the
sole indicator of an individual's mathematical knowledge,
particularly where the tests are totally of a symbolic nature.
Mat, for example, when asked to add "3/4 + 1/4"; used the
algorithm a/b + c/d = a+clb+d to get 4/8, and then used the
rule for reducing to get 1/2. When the interviewer had him
obtain 4/4 using wooden blocks, Mat said he would choose

the 1/2, "...that's what method they've taught me to do in

122
my booklet..." (p. 251). Benny had a unique method of operating
on decimals and fractions: 5/10 = 1.5, because the one stands
for 10; the decimal; then there's 5...shows how many ones".
Benny also thought 4/11 = 1.5, and also that 11.4 = 1.5.
His consistent pattern was: ab/c + a . (b + c), or ab/c =
b/ac = a . (b + c) (Erlwanger, 1975B, pp. 5-6). If either
Mat orBenny got a wrong answer, when they checked the answer
sheet, they merely looked for symbolic patterns to generalize.
This procedure has implications for Davis' "generalization
by instance", and Skemp's "mathematical generalization",
as well as Rachlin's false generalizations. Teachers must
distinguish between "concept" generalization and "symbolic"
generalization.

Benny's case study has serious implications for a rule-
governed content-area such as algebra. When Benny was asked
if he had ever discovered other ways of doing mathematics
than he had been taught, he responded, "No! I stick with
the rules." When the interviewer asked Benny if this was
because it was easier, he said “Yes. When you hardly know
how to do it." (p. 17).

Erlwanger (1975A) postulated "conceptual" thinking and
"procedural" thinking. He has described these as "algorithm"
thought and "non-algorithmic" thought. Conceptual thinking
describes the situation when a child is thinking about "mathemat-
ical quantities" or "mathematical relations". In procedural
thinking, the child is primarily interested with the applica-
tion of some rule (often self-developed on the basis of an

incorrect generalization). Both modes of thinking could possibly

123

involve algorithmic thinking. However, in conceptual thinking
the emphasis would be on the development of an appropriate
algorithm, while in procedural thinking the emphasis would
be on trying to follow an algorithm step-by-step (pp. 192-
193). Elsewhere he has categorized these as "formal" and
"intuitive" thinking (1974, p. 31). It is interesting to
note that Erlwanger's modes of thinking parallel somewhat
Skemp's "relational" and "instrumental" understanding.

Erlwanger's work is of major importance because, like
Piaget, he shows educators that an interview technique is
a necessary, and viable technique, for ascertaining the mathema-
tical knowledge and ideas an individual student possesses.
In conclusion of the presentation of Erlwanger's studies
a final statement is offered as to the impOrtance and implica-
tions of his work for college remedial algebra teachers and
learners. He (1974, p. 285) writes, ”The case studies suggests
that unless a child's "wrong" ideas, beliefs, and views are
detected and corrected, they may develop and become more
complex.“

Another group of studies conducted by Krutetskii Over
a number of years also has major implications for this present
study.

Krutetskii (1976) rejected the product measurement of
mathematics “ability level“ as being ”the study of the level
of knowledge and skills that is attained“ (p. 13), and indicates
that the attempt to find purely statistical methods for describ-
ing the qualitative distinctions among abilities will fail.

He indicates that one must study and analyze the "process" of

124
attaining results in problem-solving to discover the "psycho-
logical essence" of the results (p. 13). For the Soviets,
testing followed by factor analysis without incorporating
the psychological analysis of process, "has not proved its
value" (p. 36).

The Russians somewhat rejected Piaget's stage theory
and believed that cognitive development and schooling are
closely linked. They believed that the curriculum can alter
the development stages of Piaget (Carpenter, 1980, p. 128).

Krutetskii (1976) has postulated three stages in mathema-

tical problem solving: (1) gathering information, (2) processing
information, and (3) retaining information. He relates the
stages to one or more of the following abilities (defined
by.Krutetskii):

1. The ability to "generalize" both to place a "particular
case under a general rule", and to go from "the
particular to what is unknown and general (p. 335).

2. 'The ability of "flexibility": the ability to switch
easily "to a new method of operation, from one mental
operation to another" (p. 227).

3. The ability to reverse ("reversibility") from a
direct train of thought -- a “reconstruction of its
direction" (p. 287).

4. The ability to curtail ("curtailment") the mental
process (unconsciously) in solving problems of a
”familiar type. However, when necessary the student
can return to full detailed reasoning (p. 264).

Krutetskii's (1976) studies are deemed particularly

125
important because of the breaking down the concept of "mathe-

matical ability" into components which can be "ascertained"

qualitatively during problem solving. Krutetskii's studies
are known for the remarkable ability, as related to the above
factors, of some of the students -- (for example, Sonya,
p 251). However, of concern to this study is Krutetskii's
rather pessimistic view of “incapable students". As opposed
to capable students that generalized "correctly and immediately",
incapable students "cannot generalize according to essential
features, even with help from the experimenter and after
a number of intermediate, single-type practice examples"
(p. 155). On the curtailment aspect, incapable pupils were
always marked by superfluous comprehensiveness, detail, and
unnecessary activity...At the same time their reasoning was
not distinguished by accuracy..." (p. 267).

On flexibility, it was, "...as if the solution that
had been found out off any possibility of their switching
to a new method of operations" (p. 278). Discussing an incapable
student Krutetskii notes that after a lengthly time the experi-
menter taught her to use a faster, easier method, "but afterward
she could not reproduce the habitual method right away. And
only after a half-hour did she recall the old one, but...forgot
the new one" (p. 278). Likewise for reversibility, "...incapable
pupils saw the second problem given them as a reverse problem
only in elementary cases..." (p. 289).

Confrgyand Lanier (1980) used Krutetskii's conceptualiza-
tion of abilities in an experiment vdth.general mathematics

students at the intermediate school level with approximately

126

the same results as Krutetskii for "incapable students".
However, they offered some suggestions for educators. They
suggest: teachers should be aware of the psychological process-
es of teaching and learning mathematics (as well as logical),
consideration of how abilities (both student's and teacher's)
influence the process of teaching mathematics, careful attention
to the assumptions that are made with respect to ability
present in the students, and consider developing teaching
units designed to teach specific abilities (as indicated
by Krutetskii) (PP. 551-554).

Serious implication for this present study is indicated
by Confrey and Lanier's report that the students viewed mathema-
tics as a collection of symbols to be manipulated according
to certain rules. The students had little, if any, concrete
or mental representation to fall back on. The reason for
doing mathematics was to get answers, which are either‘right or
wrong if the teacher or answer key so indicates, understanding
is not involved. (p. 554-555).

Wagner (1981) reporting on a study that was designed
to investigate students' "conservation of equation and function
under transfer of variable", documented two common misconceptions
about variables. One misconception is that changing a variable
symbol changes its referent. A second misconception is that
the linear ordering of the alphabet corresponds to the linear
ordering of the counting numbers (p. 116).

Wagner used the equation(s) 7w + 22 = 109 and 7n + 22 =
109 and asked the students "which would be larger, w or n"?

In a sample of size 29, she found that less than one-half

127

of the students did not conserve equation. These were not
all young children in her study. The fact that one-third
of the students who had studied algebra did not recognize
the w and n would be the same, would seem to have very signi-
ficant implication for the teaching of the concept of variable
(pp. 109-116).

M353 (1979) reported similar data. She found variable
errors such as, "concluding that 4x = 46 given the x = 6,
or that xy = -8 given that x = -3 and y = -5" (p. 133).

The first error is a place-value error while the second
is an implicit addition inferred error. Matz feels that
the critical step in the transition from arithmetic to algebra
is the concept of a "symbolic value". For students who initially
fail to realize that a letter represents a number, "operating
with a letter seems totally underconstrained" (p. 131). She
notes that letters are not a very intuitive notation for
symbolic values since they do not obviously refer to a number.
She believed the usual procedure of using boxes for missing
numbers in arithmetic does not naturally generate to the
concept of.a symbolic value. Boxes are not manipulated in
sentences, nor are they divided by two. Boxes have an inherent
connotation of being filled that letters do not. Students
when faced with a letter will not be able to realize that
variables can be "instantiated" by replacing them with a
number (p. 131).

Other researchers have found student difficulty caused
by the multiple-meanings of a variable. Adda (1982) notes

that x + 3, 2x + 3 = 0, and ax2 + bx + c = 0, where both

128
"parameters" and "unknowns" are referred to by letters has

great potential as a source of confusion (p. 210-211). Kuchemann

 

(1978) has found similary difficulties with the concept of
variable. Buxton's (1982) research has lead her to believe
that mathematical symbols not only have a cognitive aspect,
but an emotional side as well. She has compared the "panic"
that mathematical symbols create for a student pushing them
abom:instrumentally to the fear and embarrassment that an
illiterate adult faces when they see printed words (p. 215-
220).

Behry Erlwanger! and Nichols (1980) designed an experiment
to determine how students view the equals sign. It turned
out that students see the equal sign as a "do something"
symbol -- an operator symbol such as in "2 + 3 = 5". In
an interview setting it was found that students would try
to turn statements such as "3 = 3" or "2 + 3 = 3 + 2" into
an addition or subtraction problem; they were not able to
view it as a relational symbol (pp. 13-15). Map; (1979)
noticed a similar situation with students in algebra with
respect to the difference between "tautologies" and "constraint
egutations". As an example she notes the syntatic similarity
between the two semantically different statements: 4(x + 3) =
4x + 12 and 3x + 3 = 2x + 7.

That these difficulties with variables and the equal sign

are not limited to young children was shown dramatically

by Rosnick & Clement (1980) who asked first-year engineering
students to write an equation to symbolize the statement "There

are six times as many students as professors at this university" (p. 4).

129
In a group of 150 only 63% were able to answer correctly.
They conducted clinical interviews and tutoring sessions
with students that had incorrect answers. The interviews
gave evidence that students changed their behavior and gave
answers to similar problems, but Rosnick & Clement believe
that "conceptual understanding of equation and variable re-
mained, for the most part, unchan " (p. 23). They feel it
is essential that students be able to view variables as stand-
ing for "number", and their studies show the "shortcomings
of an educational system that focuses primarily on manipulative
skills" (p. 23).

{M335 (1979) sums up her extensive investigations of
algebra errors by indicating that most errors are not the
result of carelessness, but rather are systematic and rule-
based. The two most common errors are (1) use of a known
rule (as is) to a new inappropriate situation, and (2) incor-
rectly adapting a known rule so that it can be used to solve
a new problem (p. 95). VanLehn (1982) also found that younger
students were very systematic in their development of "buggy"
algorithms in subtraction.

Another study which has some implications for the teaching
and learning of algebra.at the college level was performed
by Gags (1976). Gage used a self-designed concept attaiment
test in conjuction with selected concepts from beginning
algebra to determine the effect of positive and negative
instances of concept on its acquisition. Her determination
was that when a positive instance was followed by a negative

instance, the concept is more easily acquired. Most algebra

130
texts when dealing with a topic such as exponents, will usually
show each "law" as being separate, with no problems presented
where none of the "laws" apply.

Eggis (1968) conducted an investigation which he claimed
shows that the use of "enactive" and "iconic" modes of teaching
were superior in teaching elementary algebraic principles
to "low achieving" college students. He made provisions
for the use of concrete referents for symbols by the use
of diagrams and a calculator. His results showed that this
was better than the "traditional" lecture method. However,
the study seems to place an undue amount of emphasis on the
calculator as the "concrete" referent.

A study which relates significantly to the learning
of algebra atthe college level was conducted by Parete (1978).
Parete using tasks that were designed to determine Piagetian
stages, determined that the 231 college students tested could
be divided as follows: 21% early-concrete, 30% late-concrete,
10% transitional, 29% early-formal, and 10% in the formal
stages.

Of particular significance to any study pertaining to
remedial mathematics is research into the recently named
malady of Mathematics Anxiety. It is defined literally as
"uneasiness or apprehension regarding mathematics" (Widmer
and Chavez, 1982, p. 272).

Fennema and Behr (1980) argue that confidence and anxiety
are similar in relation to mathematical learning. They also
believe that there are sex-related differences in the confidence/

anxiety dimension that helps explain the fewer number of

131
.females.than males entering mathematics related fields of
study. They also indicate that their research shows that
high-anxiety students appear to perform better in a highly
structured learning situation. (pp. 334-335).

Shodahl and Diers (1984) list several possible reasons
for math anxiety: unintelligible texts, lack of concrete learn-
ing experiences in the Piagetian sense, lack of teacher en-
couragement to develop an overall picture with a consequent
trust in intuition, and teaching which leaves out an explana-
tion of process. They have developed a "Math Without Fear”
course which they indicate reduces anxiety. The course is
taught jointly by a mathematician and a psychologist. The
course involves guest speakers to acquaint the student with
the importance of mathematics, journal-keeping, rapport-
building, relaxation techniques, and the dispelling of the
mystique and myths which surrounds mathematics (pp 32-35).

In a similar vein, Sequin (1984) has found success in
reducing math anxiety by a combination exercise and relaxation
program held prior to mathematics classes (pP. 33-35).

Although, not explicity referring to math anxiety, but
more generally to test anxiety, Russo (1984) indicates that
anxiety is due to a preoccupation with failure, negative
comparisons, thoughts and feelings of inadequacy, and self-
blame and criticism. He suggests relaxation and self-monitoring
in an attempt to see the task at hand in a more positive
light (p. 164).

I However, Greenwood (1984) insists that sex-related differ-

ences in mathematical employment is generally a societal problem

132
and has little to do with math anxiety. In fact, he argues that
the principle cause of math anxiety lies in teaching methodolo-
gies based on the "explain-practice-memorize" paradigm with
a consequent lack of the understanding of process. He believes
that until we, "apply the problem-solving process to the teach-
ing and learning of arithmetic and basic mathematics concepts
and skills, we will continue to produce young adults who suffer
from math anxiety" (p. 663).

This review of the literature is brought to a close by a
thought-provoking statement that was made by Doyle (1983) after
researching the literature for those factors which contribute
to success in the academic setting:

The central point is that the type of tasks which

cognitive psychology suggests will have the great-

est long-term consequences for improving the quality

of academic work are precisely those which are the
most difficult to install in classrooms (p. 196).

Summary: Review of Literature

The review of the literature explored the historical move-
ment of "meaningful learning" by tracing its development through
the connectionistic and behavioristic eras, through the period
of opposition to the above theories by Brownell and the Gestal-
tists, and finally into the modern day era of Piagetian cogni-
tive theory.

Modern-day imagery theory was described in an effort to
place Bruner's, Skemp's, and Piaget's developmental concept of
imagery into a more theoretical framework.

Finally, in this review a compilation of studies detailing
the learning of algebraic concepts was discussed. The work of

Krutetskii in terms of his descriptions of various abilities

133

was given prominence. Several studies were cited showing
the apparent inadequacy of "rule-based" understanding to
provide a suitable background for student success in algebra.
The work of Erlwanger and Matz were given special emphasis
in this regard.

The review of the literature served as a theoretical
background for the research procedures and questions detailed

in Chapter III; RESEARCH PROCEDURES.

CHAPTER III
RESEARCH PROCEDURES

Since this study was basically a qualitative investi-
gation of the thought processes of students, a clinical
interview method of gathering data was deemed to be the most
appropriate procedure to follow.

That the clinical interview is in fact a viable method-
ology for research in the cognitive domain was illustrated
well by Piaget and Krutetskii (reported in CHAPTER II). The
procedure has been well documented by several American re-
searchers as well (Erlwanger, 1974; Easley, 1977; Ginsburg,

1981; and Confrey and Lanier, 1980).

The Interviews

 

The interviews were held in a quiet comfortable air-
conditioned office with side-by-side desks for the interviewer
and the interviewee. The desk of the interviewer was large
enough, and situated so that a tape-recorder could be placed
close enough for easy recording, but yet not be the dominant
article on the desk.

The interviewer started the session by an explanation
on the order of the following, which was designed to put the
student at ease as much as possible:

What I am attempting to do is determine how
students think about algebra - how they look at

134

135

it; what goes through their minds as they work

a certain problem. I will be talking with several

students, so in order to be able to determine if

there are common ways that students think, or

common errors that students make, I would like to

tape-record our talk. I will give you several

problems to work out. While you are working these
problems, I want you to tell me what's going

through your mind - that is, think out loud so

that I can better determine how you are working

the problem. I want you to do your best, but keep

in mind this is not a test. I am not after right

or wrong answers. I'm interested in what you are

thinking as you solve the problems.
The tape-recorder was then turned on.

The problems presented had been typed on sheets of
paper with ample work space provided for each problem. No
more than five problems were presented on one piece of paper;
that is, when a student and interviewer were through with a
set of five problems, then an additional set of five was
presented. It was felt that presenting 49 problems initially
would somewhat awe and perhaps discourage the students. In
order not to fatigue either the student or interviewer, no
session lasted more than one hour. If the problem sets had
not been completed, an additional appointment was made. The
range of times for the interviews were from one hour to one
hour and 45 minutes.

To facilitate the analysis of the taped interviews,
the students were asked to state the number of the problem
before starting to work the problem. The student was en-
couraged to not only "think-aloud", but write steps on the

paper (in order to coordinate the written work and oral state-

ments).

136

Although interviews were built around the problem sets,
the interviewer was free to challenge answers, offer hints
of encouragement, challenge and contradict, or to present
related problems in an effort to "draw out" the thought pro-
cesses of the students. The interviewer recorded any
significant non-verbal or significant body actions during
the interview.

Due to the somewhat ambiguous privacy laws with respect
to student records and information, at the conclusion of the
interview each student was asked to sign a "consent form"
(Appendix C), which allowed the obtaining of academic infor-
mation and the publishing of the results. This form was
based on a similar form by Rachlin (1981). Additionally,
the students were asked to fill out a form detailing their

mathematics background.

Selection of the Sample

Eight students were selected from a section of begin-
ning algebra (Math 111), and eight students from a section
of intermediate algebra (Math 121) at Ferris State College,
Big Rpaids, Michigan during the summer of 1984 for the inter-
views. The selections were from those students determined
by their instructors to have scored very low (four students
from beginning algebra, four from intermediate algebra), or
very high (four students from beginning algebra, and four
from intermediate algebra) on a unit test pertaining to

exponents that is a normal part of each course. Two of these

137

students failed to show up for the interview appointment,
even after repeated appointments, thus fourteen students
were actually interviewed.

The beginning algebra course, Math 111, is a course
comparable to a ninth-grade algebra course. It is a terminal
course for many of the "job oriented" programs at Ferris
State College, as well as a prerequisite for the mathemati-
cally unprepared student whose program (e.g. business and
health programs) calls for an intermediate algebra course.
Students are usually placed in Math 111 by; not having taken,
or by not having successfully completed a second-year algebra
course; and/or having received less than a "C" grade in a
first-year high school algebra course; and/or scoring between
seventeen and nineteen on the mathematics portion of the ACT
examination.

Intermediate algebra, Math 121, is a terminal course
for many four-year programs in the business, health, and
technical programs at Ferris State College, as well as a pre-
requisite for any unprepared student whose program requires
further mathematics courses, such as trigonometry or college
algebra. Students are placed in this course by having having
taken no second-year algebra course in high school, or having
received a grade of "C" or less in such a course, and/or
having scored between twenty and twenty-two on the ACT test.
In any event, the guidelines are very flexible, and are ig-
nored frequently, based on the wishes of the student.

The courses are taught in the traditional lecture method.

138

The claSses meet four days a week, with class size varying
from twenty-five to thirty-five. The ages of the students

in this sample varied from 18 years to approximately 55 years.
The textbooks used in Math 111 (Lial and Miller, 1980) and
Math 121 (WOoton and Drooyan, 1980) both develop properties
of exponents from the use of numerical examples involving
positive exponents, then extend the properties to zero, neg-
ative and rational exponents by use of logical arguments.

It is worth noting that their problem sets consist of positive
examples of the particular property involved. That is the
bases are usually the same or the exponents the same. Both
texts define "variable" to be a letter which represents a
number (Lial and Miller, p. 10; Wooton and Drooyan, p. 2).

In a personal conversation the instructor of the students
indicated that due to the shortness of time (10 weeks) that
the course was taught as a "tool" type course, or program
prerequisite.

Approximately thirty-five percent of the students en-
rolled in Math 111 either withdraw from class or receive a
grade of less than "C" in the class. Very rarely does a
student that received less than a "C" in Math 111 manage to
sucessfully ("D" or better) complete a Math 121 class (Totten,

1983).

Selection of theguestions
The hierarchy of prerequisite knowledge for the under-
standing of the concept of exponent was derived from an

analysis procedure based on Gagne's learning hierarchy model

139

(Gagné, 1970). The hierarchy chart (Appendix A) was then
developed and presented to two professors in the Mathematics
Department at Ferris State College for validation.

In an attempt to put the students "at ease" and to
verify that portion of the hierarchy of prerequisite know-
ledge leading up to the concept of exponent, the students were
asked the following questions: (Simplify)

I. 1. 2-3 + 4 - 12
2. 15 - 3 - (2 + 1)

3. 2-4-6

 

 

8
4. 2-5 + 6
8

5. a + a + a
6. 2a + 3a

7. 2a + 3a - b

 

8. 4xyz
2x

9. a + b
a

10. 3+4(x+5)

Due to the difficulties with respect to "variable" and
the "equal sign", that were pointed out by Wagner (1981),
Matz (1979), and Erlwanger, et a1 (1980), the following ques-
tions were developed to probe the student's understanding of
variable and equation:
II. 1. What does it mean to you when we say 2x +3x = 5x?

2. What does it mean to you when we say solve for x in

x + 4 = 7?

140

3. Which of the indicated equations would be the correct
answer for the following:
Write an equation using the variables S and P to
represent the following statement: "There are six
times as many students as professors at this uni-
versity". Use S for the number of students and
P for the number of professors. (Rosnick and
Clement, 1983, p. 4)
a. P = GS
b. S = 6P
4. Is the following true or false?
4x = 46 given that x = 6 (Matz, 1979, p. 134)
5.Without solving, could you compare the solutions for
W and N in the following:
7W + 22 = 109; 7N + 22 e 109 (Wagner, 1981, p. 109)
In the spring of 1984, a diagnostic examination (Appen-
dix B) was given to selected Math 111 and Math 121 classes.
The test was given to Math 111 classes after the completion
of the unit on exponents, and to the Math 121 classes prior
to covering the unit on exponents and radicals. The most
commonly missed problems were: (Simplify)
x0; 20; 82/3; 43/22; 23 + 22; x3 + x2; (xy)2; and (x3)2.
In addition, for each class a record of the most fre-
quently missed problems on the exponential unit examinations
were kept. In Math 111 twenty-seven students out of one

hundred fifty-nine missed 34-32, and sixty-six students missed

42 + 43. In Math 121, out of thirty-two students, twenty-

5, twenty-eight missed 144/75,and thirty-

1

three missed 25-5

two missed x-3/(x- t x2).

The development of the next set of questions was based

141

on the examination results cited above, the National Assess-
ment of Education Progress results (reported in CHAPTER I),
reports of problem difficulties in Alexander's (1977) and
Rachlin's (1981) research, as well as the theories of Bruner,
Skemp, and Krutetskii. The following problems were presented
to all students. Again, no more than five problems were
presented at one time.

III. Find the value and/or simplify the following:

1. 23

2. (-2)3

3 -2-32
4. 22-32
5. 22.23
6. 22 + 23
7. 32 + 23
8. (22)3
9. 44/22
10. (2-3)2

11. (2 + 3)°

12. (2 + 3)2

13. 44/42
14. s"2
15. (2 + 3)’1
16. 142/72
17. 2‘2.23
18. 23/4"1

1 -2

19. (2‘ )

142

20. (xy)3
21. (x2)3
22. xa-xb
23. mn/m2
24. a“b

25. x2 + x3
26. xayb
27. (x6)8

In addition the Math 121 students were given the following:

28. ti- J3-
29. V14 / 5(7-

30. (x + 3)2

31. ((2 + (3)2

32. ¢Z'- (37

33. (x3n + 2)2
1/n 1/m

34. a - a
(Problems 32 and 33 from Rachlin, 1981)

The problems were grouped on the basis of the following system:
The problems with like bases and multiplication or di-
vision were used to determine if students use exponential
properties, or use the operations of arithmetic to
arrive at the result.

The problems with addition or different bases were
designed to check for false generalizations and/ or
instrumental understanding or relational understanding.
The problems with variable bases and/or variable expo-

nents were designed to see if students had generalized

143

the pertinent exponential property, if any type of imagery
was used, and fOr relational or instrumental understanding.

Those problems with zero or negative exponents were
for the purpose of determining if students have "mathemat-
ically generalized" the system of positive exponents, and
if so, whether this was understood instrumentally or rela-
tionally.

The problems that include radicals in addition to
meeting the design criteria as indicated above, were to see
if students viewed the exponential notation and radical no-

tation as parallel notation for the same quantities.

A Model For Analysis of the Protocols

Erlwanger (1975C, pp. 13-16) theorized that students
in arithmetic operate congitively in one of three systems:

"Basic System" - the student operating in this cog-
nitive mode show a spontaneity and natural insight about
quantities and any relations between the quantities. The
behavioral characteristics of students Operating in the Basic
System show little body movement, and an appearance of deep
thought independent of the written work or diagrams on paper.
The student shows a great deal of confidence in any judgement
made or answer obtained.

"Perceptual Manipulation System" - the student is con-
tinuously involved with an attempt to convert all written or
oral quantities given into a diagram. The student is not

concerned with quantitative judgements or about any relations

144

between quantities, but only about procedures and actions

for transforming quantities. The behavior associated with
this mode shows frequent eye movements between the given
problem and his construction. The student has rapid hand and
body movement in dealing with his real or imagined construct-
iOn. Any oral description given by the student is accompanied
by actions indicated above, along with frequent touching of
his constructions.

"Notational-System Manipulation System" - the student
operating in this system is concerned tOtally with arrangement
and order of written terms and their manipulation. He is
concerned with the recognition of the type of problem by the
position of terms and symbols. From the relative position of
the terms and symbols the student hopes to select the apprOp-
riate algorithm and apply it. The student is only concerned
with manipulating symbols and not about any qualitative judge-
ments about the quantities involved. The behavior associated
with a student Operating in this system shows hand, eye, and
body movements all related to and determined by the written
form of the problem...Also, there is frequent hand and eye
movements between the written terms and symbols. This student
also shows inflexibility in oral descriptions of the problem,
and also shows a rigidity when pointing or touching terms
and symbols that are being used. The student shows a lack of
confidence in answers obtained.

Alexander (1977) believed since Erlwanger's model was

designed for elementary children that there was one significant

145

obstacle in attempting to apply the model to adolescent think-
ing. He conjectured that some students at the formal level
of thought could apply algorithms and perform manipulation

of symbols with "spontaneity and confidence" (p. 66). Con-
sequently, Alexander used Bruner's notions of enactive,
iconic, and symbolic representations of Erlwanger's Basic,
Perceptual Manipulation and Notational-System Manipulation
modes of congitive activity (pp. 84-85). Alexander believed
that Erlwanger's one dimensional model was not adequate in
that an individual at the formal operational level could
operate on symbols in either a "mechanical" or "insigtful"
manner. Thus, he developed a two by three matrix whereby

. the three schemas; enactive, iconic, and symbolic were paired
with the two levels of thought; mechanical and insightful.
That perhaps Alexander's model could be somewhat ambiguous

in terms of the word "insightful" is suggested by Bruner
(from whom Alexander borrowed the word):

The distinction is not between mechanical and

insightful really, but whether or not the child

has grasped and can use the generic code we have

set to teach him (Bruner, 1973, p. 223).

The key is the words "generic code". This would appear
to be Skemp's "conceptual structure" or "appropriate schema"
(p. 46).

However, the designations of enactive, iconic, and
symbolic seem to be extremely relevant to a study of the

mathematical thinking of college students. Bruner (1966),

when discussing the roles of the enactive, iconic, and

146

symbolic representations, noted:

When the learner has a well-developed symbolic

system, it may be possible to bypass the first

two stages. But one does so with the risk that

the learner may not possess the imagery to fall

back on when his symbolic transformations fail

to achieve a goal in problem solving (1966, p. 49).
It was one of the purposes of this study to see if in fact
college students do have imagery "to fall back on", or are
they operating symbolically without any real understanding
of the situation. This study (as did Alexander) eliminated
the "enactive" mode of representation, based upon the fact
that physical materials are not available in the algebra
classes at Ferris State College, nor were they available
during the interview. Therefore, the model used to analyze
the interviews in this study combined Skemp's levels of
understanding (instrumental and relational) with Bruner's
iconic and symbolic modes of representation of knowledge to
form a two-by-two matrix as previously indicated in Chapter
(p. 15).

The following characteristics of student behavior with
respect to cell assignment of the matrix was suggested by
Alexander (1977, pp. 85-88).

A. The Instrumental - Iconic Mode

It is concerned with some diagramatic
or imagined representation of the problem.
It is not concerned with the relation be-
tween the problem and the representation or
any general principles and relationships

which may be abstracted from the imagery
employed.

147

The assOciated behavior is marked by:

1. Eye and hand movements related to
imagined or real transformations
on the representation of the problem.
These movements are done in either
a random order or in a rigid pattern
indicative of a habitual rather than
thoughtful response and there is
little or no eye or hand movement to
the original problem.

2. Little oral description of the actions
or transformations or oral description
which lacks spontaneity.

3. Lack of confidence in any answer
obtained.

B. Relational - Iconic Mode

It is concerned with some diagramatic or
imagined representation of the problem, the
relation between the problem and the represent-
ation, and general principles and relationships
,which may be abstracted from the imagery
employed.

The associated behavior is marked by:

1. Eye and hand movements related to
imagined real transformations on
the representation of the problem
with, at least initially, eye or
hand movement relating the image
to the original problem.

2. The actions or transformations are
not random, but follow a flexible
pattern related to the nature of
the problem.

3. The actions or transformation are
accompanied by spontaneous oral
descriptions.

4. Confidence in any answer obtained.

C. Instrumental - Symbolic Mode
It is concerned with symbolic aspects of

the problem : the recognition of the type of
problem from symbolic cues, the selection of

148

an appropriate algorithm, the manipulation
of terms, expressions, and symbols, but not
with general principles and relationships
inherent in logical thought.

The associated behavior is marked by:

1. Eye and hand movements determined by
the written form of the problem and
performed either in a random order or
in a rigid pattern dictated by the
algorithm selected.

2. Little oral description of the actions,
or oral description which lacks spon-
taneity.

3. Lack of confidence in any answer
Obtained.

D. Relational - Symbolic Mode

 

It is concerned with the symbolic aspects
of the problem: the comprehension of the prob-
lem, and pertinent relationships which exist
among the involved quantities, from symbolic
cues, the selection of appropriate principles
and algorithms for the solution of the problem.
The associated behavior is marked by:
1. Little movement of eye or hand.
2. Spontaneous oral descriptions or
relations observed or algorithms
selected.
3. Confidence in any answer obtained.
Each student will be assigned to at least one cell for each
problem in the interview sequence based upon his/her written
work of the problem, his/her spoken words, as well as body
actions.
The use of the model in assigning students' mathematical

thinking to various cells as they work problems will be seen

in Operation in Chapter IV, Analysis of the Interviews. One

149

short example is offered here in order that the reader may
see the model in action.

Sue, a Math 121 (intermediate) student, has been given
problem #17: 2'2 - 23.
Sue: (long pause) umm...(then writes 21)

Interviewer: Now, tell me what you did there.

Sue: Well, what I'm trying to do, I've got the same
bases, so what I just did is added the exponents.

It is clear that Sue is in the Instrumental-Symbolic
mode from her hesitancy, and the fact that she fell back on
a rule. However, she does switch to the Relational-Iconic
mode when pushed by the interviewer.

Interviewer: (As Sue stares at the paper apparently in deep
thought) You're trying to decide if that is
really right?

Sue: Right.

Interviewer: And how do you decide?

Sue: Well, you'd have to go one-half times one-half
...times 8. One-fourth of 8 is 2.

Summagy
Using the models of Erlwanger (1975C) and Alexander

(1977) as guides, a model was developed that related Skemp's
instrumental and relation understanding to Bruner's iconic
and symbolic mode of knowledge representation for use with
an interviewing procedure to analyze the thought processes
of 14 remedial algebra students at the college level as they
worked designated problems. A transcription to paper was
made from the audio-tapes of all material deemed relevant to

the study and will be detailed in the next chapter.

CHAPTER IV
ANALYSIS OF THE INTERVIEWS

The Use of the Problem Sets
This study was intended to investigate the understand-~
ing of college remedial algebra students with respect to the
concept and principles of exponent, variable, and equation.
Additionally, the intention was to investigate the students'
prerequisite knowledge for the concept of exponent, as well
as the role played by imagery in the understanding of all
the above. The problem sets (listed in Chapter III,
pp. 138-142) were broken into eight categories for the deter-
mination of the students understanding with respect to the
above concerns.
1. The following problems were used for the purpose
of the determination of prerequisite knowledge.
a. Order of operations, use of parentheses,
distributive property.
I. 1. 2-3 + 4 - 12
2. 15 - 3 - (2 + l)
10. 3 + 4(x + 5)

b. Reducing fractions

 

8.

9.

151
4xyz
2x

a + b
a

 

2. The following problems were used for the investi-

gation of the students' understanding of variable,

and/or equation.

I.

v II.

5.

6.

a + a + a

2a + 3a

2a + 3a - b

What does it mean to you when we
say 2x + 3x = 5x?

What does it mean to you when we
say solve for x in x + 4 = 7?
Which of the indicated equations
would be the correct answer for
the following:

Write an equation using the var-
iables S and P to represent the
following statement: "There are
six times as many students as
professors at this university".
Use S for the number of students

and P for the number of professors.
(Rosnick, Clement, 1983, p. 4)

a. P = 68

b. S = 6P
Is the following true or false?
4x = 46 given that x = 6 (Matz,

1979, p. 134)

152

5. Without solving, could you compare
the solutions for W and N in the
following:
7W + 22 = 109; 7N + 22 = 109
(Wagner 1981, p. 109)

3. The following problems were used for the investi-
gation of the students' understanding of the
definition of "exponent". (All of the remaining
problems are from problem set III.)

1. 2

2. (-2)

11. (2 + 3)0

l4. 5"2
24. a-13
27.- (x6)8

4. The following problems were used to investigate

the students' understanding of the exponential

properties am-an = am+n and am/an = am-n.
5. 22.23
13. 44/42
17. 2‘2-23
22. xa-xb
23. mn/mz
34. al/n-allm

5. The following problems were used to investigate
the students' understanding of the exponential

properties (ab)n = anbn and (a/b)n = an/bn.

153

4. 22.32
10. (2.3)2
15. 142/72
20. (xy)3
23. v2‘- vfi‘

The following problems were for the investigation

of the students' understanding of the exponential

property (am)n = amn.
a. (22)3
19. (2'1)'2
21. (x2)3

The following problems were used to determine if
the students used "false generalizations" of the
various exponential properties. Also, these were
used to see if the students' could use the con-
cept of exponent with the required arithmetic

operation to complete the problem.

-3. -2-32
6. 22 + 23
7. 32 + 23
9. 44/22

11. (2 + 3)0
2
12. (2 + 3)
15. (2 + 3)-
18. 23/4‘

25. x + x

154

26. xayb
29. m / 3W
32. \/—- [3‘

8. The following problems were used to investigate
the students' ability to square a binominal as
well as their generalization of the exponential
properties.

30. (x + 3)2
31. (\/'2'+ V3)2

33. (x3n 2

+ 2)
It should be noted that the categories were not intended

to be unique. For example, problem III. 9. 44/22 was listed

under "false generalizations", but it could easily be solved

by a relational understanding of the property am/an = am-n.

That is, since 22 = 4, the problem could well be seen as

44/41 (or 28/22). Another good example is problem III. 27.

(x6)%, which is listed under both the category for definition

and the property (am)n = amn.

The problems were deliberately kept on the "simple"
side to allow for an examination of the students' thought
processes as they relate to exponents without involving any
other concepts or "rules". Only category 7 violates this
somewhat, by requiring that the students be acquainted with
the algorithm for the squaring of a binomial. In this re-
gard, as indicated in Chapter III, the Math 111 (beginning

algebra) students were not asked to do problem set III.

(28-34). Also, the Math 111 students were not asked any

155

problems involving radicals since the Math 111 courses do

not cover problems of this type. It was anticipated that the

Math 121 (intermediate algebra) students would at least recall

having Seen this algorithm, and thus, would not be "scared

off" by the form of the problem, and could concentrate on the

questions by the interviewer.

The Interviews

This study was designed to investigate the following

questions (as indicated in Chapter I):

1.

Do remedial algebra students have a relational, instru-
mental, or no understanding of the prerequisites con-
jectured as necessary (as advocated by Gagné) for success
in dealing with the concept of exponent.
Do remedial algebra students have a relational, instru-
mental, or no understanding of the concept of exponent?
a. How does the understanding of positive, negative,
(both integral and fractional) and zero exponents
differ in the same student? Between students?
b. How does the understanding of explicit number
exponents and literal exponents differ in the
same student? Between students?
Do remedial algebra students have the ability to general-
ize (as defined by Krutetskii) the various properties of
exponents? '
a. Can the source of "false generalizations" be de-

termined?

156

\

b. Have students that appear to have generalized the
properties of exponents (relational understanding),
merely generalized the symbolic notation (instru-
mental understanding)?

4. What types of imagery (Bruner's enactive, iconic, and
symbolic) do students use when working with the concept
of exponent?

a. Does the imagery used differ, and in what respect,
for students at the relational and instrumental
levels of understanding?

b. Can a student who is operating at the instrumental
level be "pushed" by way of hints and guided
questioning to use numerical imagery as an aid
to relational understanding?

5. 'Do successful students (as determined by a letter grade
on a test) differ from unsuccessful students with respect
to the four questions above?

In an attempt to gain insight into the research questions,
the interview audio-tapes were analyzed in terms of Skemp's
relational and instrumental understanding paired with Bruner's
iconic and symbolic modes of representation (as discussed in
Chapter III). The student interviews will be detailed one
at a time, by tracing through the various categories. In the
interest of space and redundancy, excerpts from each category
will not be presented for every student. Sufficient excerpts
are chosen to present the reader with enough background to

see the type of thinking that the individual uses. Excerpts

157

were selected that best seemed to indicate the type of think-
ing the students used in solving the problem sets. Those
chosen indicate not only Relational-Symbolic, Instrumental-
Symbolic, Relational-Iconic, and Instrumental-Iconic, but
also show the wavering of the thought process and the conse-
quent transfer from one mode of understanding to another.1
Each of the interview excerpts are numbered for the pur-
pose of referencing. The problem under discussion will be
indicated after the excerpt number, or else will be indicated
in parentheses when the excerpt references another problem
during the discussion. Items of information that are not
part of the dialogue will also be presented to the reader by
use of parentheses. All excerpts are indented, and each in-
dividual's dialogue is typed single spaced. In the following
fictitious example, the excerpt is number 1000, the problem
is xy-x4, and the reader is told there is a long pause, and
that problem number 6 is 24-23. Jed is the student being
interviewed.
1000: xY-x4
Jed: x to the 4y power, no wait,...that's not right.
When you multiply, you (long pause)...yeah, you
multiply exponents.
Interviewer: Is this problem similar to problem number 6?
(24.26)
Math 111 High Examination Scores
The first group of student interviews are from that

group of Math 111 (beginning algebra) students that were

1. More information regarding transcripts and protocols
may be obtained by contacting the author.

158

designated by their instructor as having ranked at the top of

the class on a unit test dealing with exponents.

Art had no high school algebra, but has taken Math 111
two times previously and has failed. This time Art is doing
well by memorizing rules. He eventually received a B+ for a
final grade. Art's instrumental understanding is indicated
in the following excerpt. .Art has successfully performed the
operations on problem I. l.

1. 2-3 + 4 - 12

Interviewer: It looks as if you grouped the two and three
together, and the four and twelve together,
could you have grouped the three and four to-
gether?

Art: I work according to rules, which is multipli-
cation first. You never deviate from the rules
unless you are told to do so. ‘

However, the fact that Art does have some appreciation
for "rules“ as conventions is indicated by his next comment.
Interviewer: Why do you suppose we have that rule?

Art: It seems probably for uniformity.

That Art possesses some relational understanding of
arithmetic is shown in the following. Art has multiplied out

the numerator and divided to correctly get 6.

2. 2-4-6
8

 

Interviewer: Could you have taken a shortut there - reduced
or canceled?

Art: Uh...I assume I could factor out an eight and a
four, and two.

Interviewer: Would you show me?

Art: Factor out 2-2-2-2-3/2-2-2 and cancel out the
twos (which he then does correctly).

159

Art reverts to the Instrumental-Symbolic mode on the

next problem.

3. 2'5 + 6
8

Art: 2 times 5 is 10 + 6 is 16 over 8,...again I
suppose I could...

Interviewer: What answer do you get?

Art: 2.

Interviewer: The other way - could you reduce this problem?

Art: No, not unless I factored it out...you've got
the factor of ten which is 2-5 and the factor
of six, which is two times 3 (writes 2-2-5-3),

2-2-2

which is fifteen over two, which doesn't cancel
out right.

Interviewer: In problem number 3. (2-4-6) you were able to

8

cancel the two into the eight, could I do that
here?

Art: I would just become more confused. It's not
to complex to grasp, it's just not logical.

Art's last statement should not be taken necessarily
as relational understanding. The "logical" means the vio-
lation of his rule to perform the operations in the numerator
then the denominator fist, then divide. Art also has diffi-
culty with the concept of variable.
4. a + a + a
Art: a to the third power...no, that's not right,
three a, because a to the third power would
be a times a times a.
Interviewer: So, a plus a plus a is three a?

Art: Yes.

Interviewer: How do you see that, what if...

160

Art: There are three sets of a

Interviewer: Suppose I asked you to explain to me why it's
three a's, could you use a number or something?

Art: In more complex equations I could.
Interviewer: Suppose a were five, how would that work out?

Art: Then again, I would have to use more complex
equations.

Interviewer: Suppose a were five there, what would you get?

Art: You mean right here was five a? If a were five
a that would be seven a.

Art does eventually see that idea of a = 5, and gets 15.
Art's Instrumental-Symbolic understanding of algebra
fractions is indicated in the next sequence. Art has success-

fully completed problem I. 8. (4xyz) by correctly dividing
2x

out the factors. His statement in excerpt 3. that it was
"not logical" to reduce on problem 4. was merely because it
conflicted with his other answer or rule, is brought out in
this sequence. .

5. a + b
a

 

Art: a + b would simply be a + b over a. We could
get rid of the a's and it would be b over 1
or simply b.

Interviewer: Suppose, I doubt that you have problem 8. or
problem 9. right, let's say problem 9. (a + b).

a
Art: Well, you could turn it around into a multipli-
cation problem.
Interviewer: O.k.
Art: Simply stated would be...huh...b...uh... times,

uh let me see (long pause, sigh)...uh, o.k...
o.k, that would be b times a equals ab not a + b.

Art has switched into the Relational-Symbolic mode due

161

to the fact that even though struggling he was able to use
the relationship between multiplication and division to con-
tradict an answer that he had obtained while in the Instru-

mental-Symbolic mode of understanding.

6. a + b
b
Interviewer: So what does that tell you?
Art: Which just proves my theory.
Interviewer: Which tells you, you made a mistake on that
or...?
Art: (Shakes head yes, agrees with the answer b)

If I disagree with the b.
Interviewer: Could you justify for me it is b?
Art: Seems logically that I could.

Problem number 4. 2.5 + 6, is pointed out to Art. Even-
8

though Art briefly was in the Relational-Symbolic mode, he
now reverts to Instrumental-Symbolic. Apparently his work
with variables has now caused difficulty with the method
that was "not logical" before.

7. 2-5 + 6
8

Interviewer: Could you do some canceling here?

Art: Well let's see, we had 2-2-3-5 and we had 2:2
(crosses out the twos)

It is then pointed out to Art that he had decided before that
the answer was 2, and he then changed his mind and decided 2
was correct. Art's lack of confidence in his answer is one
symptom of the Instrumental mode of understanding.

Art's understanding of variable seems to waver between

instrumental and relational. In problem I. 10. 3 + 4 (x + 5),

162

Art correctly obtains 4x + 23. When the interviewer pushes
him to show that the answer is not 27x, the best Art can do
is "you could prove it if you integrate it into a more complex
set of conditions". Even when pushed, Art never considers a
counter-example by substituting a number, but the following
sequence shows that Art does at times move out of his rule-
based mode of understanding into a Relational-Iconic mode
(even though his computation is bad).

8. What does it mean to you when we say 2x + 3x = 5x?

Art: It means we have 2 times the variable x plus
3 times the variable x and combine the 2 and
3 because we have the plus sign, which means
5x, knowing that we cannot combine x. When
we have plus, we are simply left with the
variable x.

Interviewer: Suppose I looked at that left side and I said
x is 100. What would the simplification of
the left side be?

Art: 5x is 500, or you could say 200 plus 300.

Art's idea of exponent is for the most part an instru-

mental one. Even though he does have rules, he is not quite

sure of these.

Art: O.k, that would be simply 2 times 2 equals 4,
and 3 times 3 equals 9. And then we multiply
9 times 4, which equals...uh 36.

Interviewer: Is there another way I could have done that?
Suppose I could combine the exponents somehow
or the base?

Art: NO.

Interviewer: Suppose I had 2 times 3 is 6, squared is 36.
Is that an accident?

Art: I don't know, I've never really checked into it.
I suppose we could check into it right now.

163

Interviewer: O.k. Let's do.

3~32, gets 6 for the base

Art then uses an example of 2
and can't decide what to do with the exponents. He does not
want to change the problem even though the interviewer points
out that the bases are different.

That the interview has challenged Art's idea of there
being one way to do all types of problems is brought out by
the next excerpt.

10. 22 + 23

Art: Now, I'm starting to think about this. Could
we go back?

Interviewer: Sure.

Art goes back to problem 4. 22°32, and gets 64, which he

calls 24. He is corrected by the interviewer and together

they decide 64 is a "large number", and thus incorrect be-
cause it doesn't equal thirty-six.

Interviewer: This one (problem 4) gives us 36. Looks like
we multiplied the 2 and the 3 and kept the
exponent.

Art: It's starting to boggle my mind.

Interviewer: Sometimes they do.

Art: Sometimes I go back to my book, and often times
look at the rules. You know Knute Rockne said,
"Practice, practice, practice."

Art did decide "any number to the zero power equals one"

on problem 11. (2 + 3)°, but gave (-5) (-5) = 25 for 5'2.

Although, with help from the interviewer, he did finally

remember that 5.2 was 1/25.

However, that Art has not generalized the properties of

exponents to include exponents other than positive whole

164

numbers is indicated by the following excerpt. When con-
1,-2

1/2-1/2 = 1/4. The interviewer and Art look back at problem

fronted with the problem (2- , Art decides that it is

8. (22)3, which with help from the interviewer, Art had
6

finally decided was equivalent to 2 .

11. (2’1"2

Interviewer: Let's transfer this over to problem 19. We've
decided that (22)3 is 26. Could I use the same

property or do the negative exponents lead to
something different? How do you view the nega-

tive exponent? For example could that (2-1)-2

be 22?

Art: That's what I was trying to show right here, I
bad 1/2 times 1/2; but that's not...

Interviewer: Which is 1/4, but if I multiplied the exponents,
I'd get 2 to the second,...

Art: I'm confused about this rule. I'll tell you
I'm really confused about it.

That Art is working continuously in the Instrumental-
Symbolic mode is verified by the next excerpt.
12. (xy)3

Art: O.k., that would be just xy to the third.

(writes (xy)3) Some people I'm sure would
view that as x to the third y to the third.

Interviewer: Are those the same? x to the third y to the
third is that the same as...

Art: NO.

Interviewer: What if I'm one of those who think it's x to
the third y to the third, and I ask you to
show me I'm wrong.

Art: I'd say go back to the text and prove it.

Interviewer: The text would prove it?

Art: The text would give the rule.

165

That Art sometimes does remember the correct rule, is

demonstrated by the fact that on problem 23. (mn/mz), he

immediately wrote mn-2.

The fact that the Instrumental-Iconic mode is often as

misleading for a student as the Instrumental-Symbolic mode is

indicated by Art's response to problem 27. (x6)%. Art drew

the following diagram: xxx/xxx and immediately gave the

correct answer of x3. It was only upon further questioning

that it could be determined that Art had no understanding of
rational exponents.

Art's philosophy of learning algebra is summed up by
a comment made while working on problem III. 9. 44/22.

Art works the problem in a Relational-Iconic fashion by writ-

ing 2-2-2-2-2-2-2-2 and canceling out the twos to get 64.
2.2

Interviewer: Can't divide the 2 into the four or subtract
exponents or anything like that?

Art: Then again I'm just in an elemental class right
now and that's the way I keep from being con-
fused.

Interviewer: Some people do subtract exponents on this type
of problem.

Art: If you could show me a trick, I'd be happy to
use it.

Art works reasonably well as long as he can use numer-
ical imagery and properties of arithmetic, but as problems
become symbolic, Art shifts to an instrumental mode based
strictly on symbolic manipulation. He then is open to any
"trick" that reminds him of a property that he vaguely

remembers.

166

Dea is another Math 111 student that did well on the
unit examination on exponents that was taken approximately
two weeks prior to the interview. She eventually received
an "A" for the class. Dea is much more confident of her
answers than Art. This is apparently due to the fact that
she has more of a relational understanding than Art. Dea
makes an observation about how she views numbers. Dea has
correctly evaluated 15 - 3 - (2 + 1) by first combining the

2 and the l in parentheses.
13. 15 - 3 - (2 +1)

Dea: I don't know why I do it that way. I guess
it's...I always group. See I see numbers in
groups. Maybe that's what helps me in algebra.
Instead of seeing the 2 + l, I see a 3, now
this one here, (starts doing problem 3. 2-4-6),

8
I would do this as 24 times 2. That would be
easier for me than 8 times 6, which is 24 times
2 is 48, divided by 8, is 6.

Interviewer: O.k. Any shortcuts on that maybe? Any cancel-
ing perhaps?

Dea: Yeah, you could cancel this out. 2 times 4
would be 8 and cancel the 8 out and get your 6.

Interviewer: O.k.
Dea: But, I don't see that as a canceling problem.
I would not recognize that as a short step.
I would just go ahead and figure that out.
Dea then works problem I. 4. 2-5 + 6 properly and
8
gets 2. When asked if she could "cancel" here, she indicates
that this is addition and therefore cannot be reduced.
However, Dea's relational understanding does not hold up

when the problem involves variables.

14. a + b
a

Dea:
Interviewer:

Dea:

Interviewer:

Dea:
Interviewer:

Dea:

Interviewer:

Dea:

Interviewer:

Dea:

Interviewer:

Dea:

Interviewer:

Dea:

167

a goes into a, 1 time plus b.

You can always cancel?

You can just...Yes, that's the way I do it.

You know if you have the same variables, or

letters or whatever, you just divide. I see
it as division, but I just cancel when I see
it.

Could I ask you to go back to problem 4.
(2- -5 + 6)?
8

2 times 5 is 10 + 6 is 16 divided by 8 is 2.
What I am asking is if you could cancel there.

(sighs) (long pause) Let's see...yeah, you
could...No, I don't see where you can cancel
before you perform the operation.

O.k.

I'm thinking 2 goes into this (2-5) five times
and 2 goes into 6 three times. So, let's see
that would be 5 plus 6, which is 11 fourths.

I guess it has to be multiplication on top
before you can.

So what abOut number 9. (a + b)?

a
Hum...(pause - no response)

How do you check these out? I see I've created
a little puzzlement in your mind.

I know it.

You look at that and you're asking yourself
can I cancel or can't I? Is there any way,
in your own mind, to check this out?

The way I see it anything over itself is 1.
That's the way I see it is 1 + b. I don't
know how I could check it out.

The interviewer once more has Dea compare problem 3.

2-4-6 and problem 4.
8

problem 9.

a + b,

2:5 + 6 and problem 8.
8

and then pushes agian to see if Dea can

2x2. and
2x

a

168

 

"check it out". Finally, on her own, Dea uses numerical
imagery.
15. a + b
a
Dea: Um, let me think here. Yeah, if you had the
variables. If you just punched in a = 2 and
b = 3.

Interviewer: O.k. Try that.

Dea: You'd have 2 + 3 on top, which is 5 divided
by 2, which equals five-halfs.

Dea then correctly decides that the "a" cannot be can-
celed. However, Dea's numerical imagery technique is short-
lived. On the very next problem she not only does not use
numerical imagery, but makes an error by confusing what Matz
(1979, p. 137) called "tautologies" and "constraint" equa-
tions.

16. 3 + 4 (x + 5)
Dea: You've got 4x plus 20 plus 3 would be 4x + 23.

Interviewer: You can't combine the 4x and 23?

Dea: No, they are unlike terms. Now if you were to
add the 3 and 4 and go 7 times...that would be
7x + 35. (long pause) See they don't work out

the same. There again you should do multi-
plication before addition.

Interviewer: Some people mistakenly add the 4x and the 23
and get 27x. Suppose you had to verify which
answer is correct. How could you check it out?

Dea: Uh...Yea...let's see could I put a number in
there...(mutters to herself)...couldn't do that.
I don't really know how to go about checking
that particular problem. I think what you'd
have to do is put 0 on the other side and solve
for x. Then just check it out and see if it
equals 0.

Dea then solves 4x + 23 + 0, get x = 23/4 and tries to put

169

this value of x in 3 + 4(x + 5) = 0, becomes confused and
drops it. (That this method would actually work out is an
"accident" as far as Dea's thinking is concerned.)

Dea also had difficulty with the "students and pro-
fessors” problem II. 3.
l7.
Dea: (reads problem again) It would be 68 = P.

Interviewer: Could you give me your logic on that?

Dea: When I read that, I see 6 times the students.
So if you have 6 times students equals pro-
fessors.

Interviewer: What if there were 8 professors, how many
students?

Dea: Uh..., 8 times 6 is 48.
Interviewer: 48 students?

Dea: There are 6 times as many students as professors.
(Does not see the inconsistency with her answer.)

Dea's notions of exponent are in general of a relational
type, but she has not generalized the properties in all cases.
18. 22-32
Dea: You've got 4 times 9 which equals 36.
Interviewer: Could you have done that another way?

Dea: Um...No not that I see.

Interviewer: How about the 2 times 3 first...

Dea: Yeah, you would combine them that way.

Interviewer: Is that just an accident?

Dea: I don't know. I've never done it that way. I
was told to do the exponents first. So, I
don't even visualize that problem. That might

be an accident.

However, on problem 10. (2-3)2 which is the reverse of

 

 

 

 

pr
ti
re

WE

I:

D(

 

170

2~32. Dea has no difficulty using the exponen-

2~32. That Dea has a

problem 4. 2
tial property and immediately wrote 2
relational understanding of this prOperty, (ab)n = anbn,
was also indicated on this problem.

19. (2-3)2

(Dea has correctly done this computation in the two ways in-
dicated above and is being questioned by the interviewer.)

Interviewer: How do you know those are the same? ((22-32)
and (2-3)2).

Dea: This is saying ((2-3)2) 2 times 3 times 2 times
3, and just rearrange it. (writes (2:3) (2:3) =
2-2-3-3)

Dea does have only an Instrumental-Symbolic understand-
ing of zero and negative exponent.
20. (2 + 3)°
Dea: 5 to the 0 equals 1
Interviewer: Why?

Dea: Because its a rule. Anything to the 0 power
equals 1.

Interviewer: Why not 0 rather than 1?

Dea: Oh, I don't know why I do that. It's a rule
that stuck in my head. The way I remember it
is, since it's in the problem, it has to equal
something to keep its place.

Interviewer: Well, in that case, could we have called 50 five?
And let that hold its place?
Dea: Well, you probably could, but I don't think it

would work out. I don't know why they come up
with that 1. Well, yeah, I do. One times any-
thing is one - generally in multiplication. It
doeSn't in addition.

Dea: Let's see...I know how to get the answer, but I

171

don't know how to explain it to you.
Interviewer: O.k. What's the answer.

Dea: It's 1 over, bring the 5 down and make a
positive 2. One over S squared. I don't
know how or why, but I know that's what is
supposed to be done.

Dea is able to correctly work out other problems dealing

with negative exponents by using the definition. On problem

-2.23, when pushed by the interviewer, she does use the

I

appropriate exponential property. However, on problem 19.
-2

l7. 2
(2‘1) , she comes up with two different answers using the
definition and never does see the involved property.

That Dea wavers between a Relational-Symbolic and In-
'strumental-Symbolic mode of understanding with respect to
exponents, is indicated in the following excerpts:

2 3
x )

Dea: That another exponent times x squared times 3

equal x6.

22. (

Interviewer: Could you verify that?

Dea: Yes. You could take x~x three times (writes
x-x-x-x-x-x).

Dea is obviously in the Relational-Symbolic mode in the
excerpt above, but when only variables are involved, the
situation changes.

23. xa xb

Dea: Uh...let me see now...unknown exponents would
be x times x. I don't think that can be
changed in any way. Those are all unknowns...
(long pause). We could put x squared ab.

(writes xzab). I would think. Yeah,...No,

Yeah.

Interviewer: Could you verify that?

172

(Dea now resorts to numerical imagery in her attempt to ver-
ify her answer and transfers into a Relational-Iconic mode.)

Dea: Well yeah, you could put a number in. Take
x = 2, a = 2, b = 3. I don't think they'll

get too big that way. (writes 22-23). Which

would be...25. So the bases stay the same.
That goes right back to adding exponents.

That would be xa+b.

b b

However, on problem 26. xay , Dea answers (xy)a+ , and

proving Skemp's admonition that a "multiplicity of rules"

can get out of hand, has'difficulty on problem 27. (x6)%.

Dea answers this as (x6)-'2 = x"12 = l/xlz, because "the one-
half power is the same as minus 2".

In general, Dea seems to operate according to rules with
some relational understanding with the non-variable type
problems. However, she fluctuates between relational and
instrumental when variables are involved. In comparison to
Art, Dea does frequently resort to numerical imagery.

Dea and Art are representative of the type of understand-
ing exemplified by the "better" Math 111 (beginning) algebra
students during the interviews. Dea remembered the properties
better than Art, although neither appeared to have generalized
all properties. Both used many false generalizations. Dea
did appeal to numerical imagery to aid in the understanding
of symbolic relationships. However, she was not consistent
in this regard. The interview for the other student in this
category will not be detailed here due to the fact that for

the most part, it would merely be repetitious of the types

of thinking shown by Art and Dea.

173

Math 111 Low Examination Scores

 

The four Math 111 students that scored the lowest on
the exponent unit examination in general exhibited a lower
level of understanding of the concepts involved in the problem
sets than Art and Dea. Their understanding was primarily at
the instrumental level on arithmetic problems, while the In-
strumental-Symbolic mode was the mainstay for the variable
type problem. However, one student, Tim, did not differ sig-
nificantly from Art and Dea in his thinking. (Although he did
eventually receive an F for the class). The other three
students were very consistent in their thinking. Again, due
to the repetitive nature of their interviews, only one will
be detailed in this section.

Max is a student that has returned to school after
several years. This is the second time Max has enrolled in
Math 111, after having received an F previously. He received
a D- for his final grade this time. Max had no high school
algebra. He needs credit in Math 111 and Math 121 to meet
the requirements of the specific program in which he is en-
rolled. Max does reasonably well on the arithmetic type pre-
requisite category problems. He correctly does problem 1.

2-3 + 4 - 12, problem 2. 15 - 3 - (2 + l), and problem 3.
2;4;§. However, on problem 3., he is asked by the interviewer
ithe could have "reduced" before doing the multiplication.

Max writes: 21°42'53 to get a different answer of 3/2. He is

4
not able to see what went wrong, but does believe 6 is correct.

Max would not attempt to reduce in the next problem

174

(2-5 + 6) because "the 5 was odd and could not be reduced by
8
2". That Max does not really understand the situation re-

lationally is brought out when he gets to problem 9. a + b
a

24. a + b
a

 

Max: That's b. The a's cancel out.

Interviewer: The a's cancel?

Max: Yes, they are common factors.

In an attempt to get Max to use numerical imagery to detect
his error, the interviewer pushes.

Interviewer: Suppose I doubted that the-answer is b. Is

there any way that you could justify that it
really is b?

Max: Well, you can't divide the a into the b, but
the two a's cancel. I know exactly what you
want.

Interviewer: Some people cancel the a into the a, get 1, and
give an answer of l + b.

Max: Well, you would have a 1. It goes in there one
time. I see what you are talking about now.

Interviewer: So you think it's 1 + b?

Max: Probably.

The interviewer asks Max to look back at problem 8. 4xyz,
2x

which Max had done correctly, using the reducing idea.

Interviewer: Up in number 8., you got l's when you canceled
out the x's.

Max: Yes.

Interviewer: When you multiplied by the one, the product
just stayed the same?

Max: Yes.

Interviewer: Down here since the l is added (emphasizes this
word), you have to leave the one?

175

That Max is thinking in the Instrumental-Symbolic mode

is indicated by the fact that he recognizes the difference

(somewhat) between addition and multiplication with the arith-

metic problems, but even with thestrong hint by the inter-

viewer, he finally concludes the answer is "b".

On problem I.

10. 3 + 4(x + 5) Max exhibits a common

tendency of the Instrumental-Symbolic mode of thinking, to

dwell on the physical arrangement of the symbols.
cessfully completes problem 1.

that "you always do multiplication first".

Max suc-
2-3 + 4 - 12 by indicating

Even when the

interviewer suggests adding the 3 and the 4 first Max will

not change.
25. 3 + 4(x
Max:

Interviewer:

Max:

Interviewer:

Max:

Interviewer:

Max:

Interviewer:

Max:

Interviewer:

Max then makes the correct choice.

+ 5)

That would be 7x plus...uh 35.
How did you get the 7?

I add the 3 and the 4 then times the x and
times the 5.

Let's go back and look at problem 1. (2-3+4-12).
Now you didn't get a 7 here because you said you
prefer to...

Do multiplication first. (long pause) Now why
didn't I do multiplication first?

Yes, is there a times sign between the 4 and
parentheses?
I would think so. Just times what's inside.

Why don't you try it this second way, and see
if you get the same answer.

You get 3 + 4x + 20, which is 4x + 23.
Which answer would you go with?

He has somewhat of

176

a relational understanding of variable. However, the various
uses of variable along with the idea of equation does give
him difficulty. Max correctly answers problem 5. a + a + a.
However, when pushed to explain why it was not a3, he never
considers substituting a number. On problem 6. 2a + 3a and
problem 7. 2a + 3a - b, he also does well. Max uses the
only "concrete imagery" of the interviews. When challenged
as to whether the a's and b's could be added in problem 7., he
responds, "You can't add oil and water". The following
excerpt starts with II. 1. It is here that Max shows some
relational understanding of variable.

26. What does it mean to you when we say 2x + 3x = 5x?

Max: That means you combine like terms. 3x plus 2x
equals 5x.

Interviewer: How do you view the x there?

Max: x is an unknown.

Interviewer: If x were 2, what would this statement mean?
(Max substitutes in both sides and gets 2.2 + 3-2 = 5-2).
Interviewer: Are those equal?

(Max adds in the following manner - and answers "yes").

2 2
2.2.

5-2

Interviewer: Suppose x is 10, what would I get on the left -
without adding them up?

Max: 20 plus...
Interviewer: No, No, don't add them up.
Max: Oh, you get 50, just 5 times 10.
Even though he is doing poorly in the class, Max has

just exhibited more of an understanding of variable than some

177

of the "better" students. However, he does have difficulty

in comparing the use of variable in the problem just discussed
and problem 2. What does it mean to you when we say solve

for x in'x + 4 = 7? .

27.

Max: It means you get the difference between plus 4
and 7. (He gets 3).

Interviewer: Do you view the x in problem 2 as being different
than the x in problem 1? Are these problems two
different types?

Max: No, it's the same thing, x is still the unknown
except in problem 1, I've got 2 times the un-
known.

Until this stage of the interview, Max seemed to be on
par with Art and Dea. It was with the start of the questions
on exponents that his Instrumental-Symbolic mode of thinking
became completely dominant. Max had a rule for every problem
and would tend to exhibit superficial confidence by sticking
with his results in the face of contrary evidence.

28. -2-32

Max: I've got to multiply both sides of this. If I
multiply 2 times 3, I have to multiply two

times the power. (Max gets -64).
Interviewer: Could you possibly do that another way?
(Max writes -6-6)
Interviewer: Are there other possibilities?
Max: I could get this. (Writes -2-9 = -18)

Interviewer: Now, we've got three different answers here.
Which do we go with?

Max: I think this one (-64), because we had to
multiply both sides of the problem by 2.

178

Obviously, Max is confusing the distributive property
with exponential properties. Max also has difficulty with
exponential properties.

29. 22-32

Max: I'll get 4 for the power and 6 for the base.
You add the exponents. (Writes 64).

3) (Max writes 45).

Interviewer: Try problem number 5. (22-2

Interviewer: Now how did you get that?

Max: I'm multiplying these numbers (2 and 2) and
adding the exponents. If I multiplied them,
it wouldn't come out right.

Interviewer: I don't understand what do you mean?

Max: Multiply the exponent (writes 2-3 = 6). That's
not right. -

Interviewer: Suppose you didn't know any of these properties.
Suppose you didn't know that property of adding
exponents that you are using. Could you do it
another way?

Max: I'd have to go 2 times 2 and 2 times 2 times 2.

Interviewer: And what would that be equal to? (Max multiplies
and gets 32).

Interviewer: I'll tell you, 45 is equal to 1024. That's not
equal to 32. Which answer would you go with?

Max: I'd put my faith in this one. (points to 45).
I've been taught to add the exponents.

.Max has shown that he does not have relational understanding
of positive exponents.

Max continues throughout the interview to "solve" all
problems dealing with exponents by the "application" of some
rules. He usually is somewhat right, but misses the relevant
features of the problem that should enable him to distinguish

the appropriate property. Without belaboring the point by

m-4

179

giving the transcriptions verbatum, it should be indicated
that Max made the following "errors" with respect to positive

exponents:

2 4 2 2

44/22 = 2 ; 4 /4 = 1 ; 142/72 = 2; (x

mn/m2 = n/2; and xayb = xyab. He did get (22)3 = 2

but changed to 25 when challenged.

6.

Max does have some instrumental understanding of zero
and negative exponents. He attempts to think of negative
exponents in the Instrumental-Iconic mode, but the imagery
used is not really appropriate.

30. (2 + 3)0

Max: One

Interviewer: Why is that?

Max: Zero exponent.
Interviewer: Always?

Max: That's what I understand.

Interviewer: Why do we have zero exponents? When will we
ever use those?

Max: I believe it's an absolute. I'm not very sure.
But you've got to have zero exponents in order
to have negative exponents.

Interviewer: Why do we have negative exponents?

Max: If you take six apples away from four apples,
you're in the hole.
.Max is able to write 5.2 = 1/25 and a.b = l/ab. However,

Max again resorts to false generalizations in the appli-

cation of properties. He makes the following errors:

1)-2 3 -1 3 1 1

(2' = 1/2 ; 2 -2 = 4 ; and 23/4‘ = -2.

In summary, Max appears to use some imagery while using

180

an instrumental level of thinking. He does have some re-
lational understanding, but his tendency to place total
authority in algorithms robs him of the benefits of this
understanding.

It was stated previously that Max's interview was
typical of the students, that did poorly on the exponential
unit examination. It should be noted that this is true with
respect to the tendency to rely on rules to solve problems.
Max was atypical in the sense that he had more of an under-
standing (both relational and instrumental) of variable than
the others. Two students responded "a3" for the answer
to a + a + a. When asked what "a represented", they both
answered "1". The other person was correct on a + a + a = 3a,
but on the next problem 2a + 3a gave 5a2 as the answer. Ap-
parently, the concept of variable, as it relates to exponents,
is a trOublesome one for some students.

As reported earlier in this section, the interviews
for the other three students in the "lower" category of the
Math 111 students will not be extensively detailed. However,
the following excerpts are enlightening as to the type of
"understanding" that some students have with respect to the
combination of variables and exponents. Len, and Nan are the
students in the following excerpts:

31. a + a + a
Len: a to the third. (writes a3).
Interviewer: What procedure did you use to get that?

Len; a + a + a...exponents, I'm just adding the a's.

181

Interviewer: Is there any way to check this out? Suppose
you were not sure whether it was a to the third
or something else? (long pause) Is "a" a
number, or a letter...what does it represent?
Len: One,...right? (laughs) (long pause)

Interviewer: Let me ask you this; suppose a were 2, what
would you have?

Len: If a stood for 2, then 6.
Interviewer: How did you get that?

Len: Well a, a, a. If a were 1, you'd get 3, so if
it's 2, you have two, four, six.

It is interesting to note that Len would have the correct
answer on a paper and pencil test. His confusing of the no-
tation as he passed back and forth from explicit numbers to
variables was such that the errors ”self-corrected".

32. a + a + a
(Nan has just correctly responded "3a")
Interviewer: How do you view the "a" there? What is it?

Nan: Everybody hates those letters. How can you add
a "z" to an "a"?

Interviewer: What does a represent?
Nan: I guess a could represent 1, couldn't it?

Interviewer: Does that check out over here? (points to
a + a + a = 3a)

Nan: 1 plus 1 plus 1 = 3. There would be no a.

Interviewer: What does the 3a represent? Is that 3 + a, or
3 times a or...?

Nan: Just generally means there are three a's...
Would be 3 + a.

(After several conversational exchanges, the interviewer has

Nan substitute in the number 2 in a + a + a = 3a and get 6 = 6.)

182

Nan: I never thought of it that way. Nobody
ever taught it that way.

If Nan's memory is true, apparently she has never been
out of the Instrumental-Symbolic mode since starting algebra.

The total lack of confidence in her own understanding,
as well as her view of algebra is expressed by her statements

2~32. Nan indicated the

2 2

while working on problem III. 10. 2

= (2.3)2.
2

-3
2

numbers were "rigged", and it was luck that 2

The interviewer then gave her another problem 5 -3 and to-

2-32 were

gether they checked out the fact that (5-3)2 and 5
equal by using 25-9 and 152 to get 225. Nan had said that
these two would not be the same, and indicated her surprise
by, “They are not supposed to do that are they?...There is

supposed to be only one given way to get an answer."

Math 121 High Examinations Scores

 

Interview excerpts will be presented for only two stu-
dents in this section. Judy's excerpts are, in general, rep-
resentative of her relational understanding, while Alan's show
his inclination toward an instrumental mode of understanding.
These excerpts express quite well the range of student re-
sponses of those students that did well on the exponential
unit examination.

Judy has a good overall grade point average for all
courses. She completed one year of high school algebra, as
well as Math 111. She received an A grade in both of these
courses, and eventually received an A in Math 121.

Judy will not be pushed into "canceling" out the 2 in

problem 4. 2-5 + 6, and sticks with her result of 2.
8

183

33.
Interviewer: I thought when you had times, you could cancel.
Judy: It has to be all times.

Interviewer: I wonder why that is. What have I really done
wrong if I cancel the 2?

Judy: I don't know.

(Judy is "pushed" by the interviewer and finally exhibits her

appreciation for the structure of arithmetic.)

Judy: (writes 2(5 + 3))...and now divide out the 2.

That Judy can consistentlyause the distributive property is

shown in problem 6. 2a + 3a, when she once more justifies

her answer by writing (2 + 3)a = 5a.

Judy's responses in the next excerpt show that she does

use the Relational-Iconic mode of thinking of "variable".

34. What does it mean to yOu when we say 2x + 3x = 5x?

Judy:. I'm not sure what you want.

Interviewer: How do you view the x?

Judy: As a variable. There is no value for x yet.

Interviewer: Well, let's suppose x is 15, could you tell me
the value of the left side without substituting
15 on the left side? '

Judy: 2 times 15 plus 3 times 15...

Interviewer: No, that's not what I want you to do.

Judy: x is 15? (Interviewer agrees). Then x is 15
on the right and you get 75.

Judy then does problem 2. What does it mean to you
when we say solve for x in x + 4 = 7?, by giving the result
of 3. When asked the difference in problem 1 and problem 2,

she states, "In problem two x has only one value, but in

184

problem one it can have many values".

Judy comes as near as any student to expressing re-
lational understanding of the "students and professors"
problem.

35.

Judy: (long pause, while looking at the problem) b.
(S = 6P).

Interviewer: What makes you choose b?

Judy: Because there are six times as many students
as professors. So for every professor there
are six students. (writes S = 6-6, 36 = 36)

Judy's lack of confidence betrays her wavering into the In-
strumental-Symbolic mode in this next exchange.

Interviewer: But it seems if there are six times as many
students, you would multiply the number of
students by 6.

Judy: (long pause) Well, one student equals six
professors if you want to equal them out,
right? (laughs) I'm terrible at story
problems.

Interviewer: So, which are you going to go with? Are you
going to stick with your answer or did I
change your mind?

Judy: I'm going to switch to a. (P 68)
Judy uses a Relational-Iconic mode when dealing with expon-
ential properties.

35. 22-32

(Judy writes 4-9 = 36)

Judy: I could have gone 2-3 is 6.
Interviewer: O.k. Do that.

Judy: 2-3 to the second power is also 36.

Interviewer: Now you've got me confused. On problem 3.

185

(-2-32), you told me you had to do the
exponentaiation first.

Judy: They are not to the same power. (points to
problem 3.)

Interviewer: How do you know that works?

Judy: That's what we learned in Math 121.

Interviewer: Suppose I doubted that (2-3)2 is the same as
22-32. Could you show me it has to be?

Judy: (writes (2-3) (2-3))

Interviewer: Is that the same as the other?

Judy then writes 2-3-2.3 = 2.2-3-3 = 22.32.

On problem 9. 44/22

Judy is one of the few students

who can use an exponential property. However, she does not
do this until pushed by the interviewer. Judy has written
44/22 = 256/4 = 64 when the next excerpt begins.

4/22

Interviewer: Is there any way I could shorten that down?

Divide the 2 into the 4 or subtract exponents
or whatever?

37. 4

Judy: The bases have to be the same.
Interviewer: Well, could you make the bases the same?

Judy: Probably, I'd have to figure out what the
exponent would have to be up here if I

make 4 into 2 times 2. (writes 28/22 = 26

)
Interviewer: Could you have made the bases 4?

Judy: Yeah. I could have done that too by putting
a 4 in the denominator.

However, Judy has not generalized all the exponential
properties. She has no difficulty in explaining relationally
n m+n m-n

all the problems that pertain to am-a = a and am/an = a ,

but does have difficulty looking at the "reverse" of

(ab)n =
38. (2-3)2
Judy:

Interviewer:

Judy:
Interviewer:
Judy:

Interviewer:

Judy:

186

anbn and (a/b)n = an/bn.

This would be 6 to the second power equals 36.

Suppose I don't work this the way you do, but

go 22-32. Is that alright?

...Um huh (yes)...that's 4 times 9 which is 36
also.

Was I just lucky there because of the 2 and 3's?
Given a problem of multiplication to a power,
could I always do that?

As long as it's multiplication.

Is there a property dealing with this?

Not that I'm aware of.

This could be taken as just momentary forgetfulness on Judy's

part, but the next excerpt adds evidence of lack of knowledge

of the equivalent division property.

39. 142/72

Judy:
Interviewer:

Judy:

Interviewer:

Judy:
Interviewer:

Judy:

Interviewer:

(long pause, starts to cross out the 7 and 14

but changes her mind.)

You were going to divide the 7 into the 14, but
decided it didn't work?

I don't know why I did that?
142/72 = 196/49 s 4)

(Judy then gets

Could we have done that another way? It may be

accidental, but...
2 to the second would have been 4.
Is that just luck?

I don't know what would have happened to the 2

down here. (writes 72 = 14, but corrects it).
Is there any way we can check that out? If we

had 14 to the 7th power, we would be all day.

187

Judy: Not that I know of.
With the help of the interviewer, Judy writes 142/72 =

14-14 = 2.2 = 4, but still does not recognize the general-
7-7 ‘

ization.

Judy has no difficulty with the exponential problems
that contain either variable bases or exponents. When pushed,
she can fall back on numerical imagery to justify her con-
clusions.

40. xa-xb

Judy: That's xa+b

Interviewer: Suppose someone doubted that and thought it was

xa+b. Is there any way you could check this...
to help decide between...

Judy: Well, if you took 1 for x.
Interviewer: What about a and b?
Judy: Use a = 2 and b = 3.

(Judy then gets 12-13 = l5 = l, and concludes that l was a

poor choice for x, when the interviewer indicated that 16
also equals 1.

Judy handled the problems well which involved zero and
negative exponents. It was apparent that her thinking was
primarily instrumental due to the fact that she could offer
no reason why "any number to the zero power is one". How-
ever, she did recognize that negative exponents were used
for "making fractions". Judy did exhibit a relational
understanding of rational exponent on problem 27.

41. (x6);5

Judy: x

188

Interviewer: How did you get that?

Judy: 8 of 6 is 3. You times the,outer exponent by
the inner exponent.

Interviewer: Why do we have fractional exponents? (pause)
Why do we have one-half powers?

Judy: We make square roots out of them. (Judy writes

\/;E = x3).

Interviewer: How do we know the square root is x3? (Judy

writes x3-x3 = x6)

Judy was the only student to show at least some instru-
mental understanding of the problem. VIZ/ 1/7.
42. VIZ/ 3\/7'

Judy: I can't do that unless I change it to fractions.

Interviewer: Do that.

Judy: (writes 141/2/71/3)

Judy is stymied, but with the interviewers help is able to

finally simplify a similar problem, \f2/ 3J2, by switching to

fractional exponents: 21/2/21/3 = 21/2"l/3 = 23/6'2/5 =

21,6 = 5/2. However, she still could not complete the initial

al/m.al/n

problem. When she tries problem 27. , she easily

added the exponents as fractions, but after the interviewer
had her consider the problem as “J3 - 5J3, she could go no
further and did not appear to see the equivalency between the
Problems.

Judy had little trouble with (x + 3)2 and ( J2'+(f3)2,

3n

but needed encouragement on (x + 2)2. She initially wrote

2
6
X n + 4, but immediately scribbled this out and then completed

the problem as the product two binomials - (x3n + 2) (X3n + 2)-

It was only with the interviewers suggestion that she saw the

189

problem as the squareof a binomial.

In general, Judy appears to operate at either a Re-
lational-Iconic or Relational-Symbolic level, and only oc-
casionally dropping into the Instrumental-Symbolic mode of
thinking.

Alan had no high school algebra, but did receive an A
in a beginning algebra course at the college level. He even-
tually received an A- in the Math 121 class.

Alan's interview will not be detailed to the extent of
Judy's. However, there are several problems which point out
their different levels of understanding. Some of these will
be presented in order that the reader may obtain a more pre-
cise picture of the thought processes of these more "capable"
Math 121 students.

That Alan is not as proficient (as Judy) at the instru-
mental level is shown in the following excerpt.

42. 15 - 3 - (2 + 1)

Alan: O.k. Again, I would just work from left to
right, just going from 15 - 3, you'd get 12.
Then I'd do what's in the parentheses first,

and then just subtract that (answers 9).

Interviewer: Could you have removed the parentheses first
before you did all the addition and so on?

Alan: Well...

Interviewer: There's four numbers there, could you somehow
write the four numbers without the parentheses
and still come up with a 9?

Alan: You couldn't take the parentheses away and do it.

Interviewer: Well, what if I did take the parentheses away?
WOuld that change anything?

Alan:

Interviewer:

Alan:
Interviewer:

Alan:

190

It would change the answer.

Well, but I want it equivalent. If I drop the
parentheses, what about the signs in here?

You'd have to change the plus to a minus.
O.k. Would you change the 2 to a minus?

No.

Alan does have a well developed sense of numerical

imagery which helps him make decisions.

43. a + a + a

Alan:

Interviewer:

Alan:.

Interviewer:

Alan:

Oh...(long pause) Here again, just add straight
across, three a's = 3a.

What if I thought that were a3? (pause) Well,
what did you write down underneath there?

Just checking something out.

Oh, good, now that's what I want to know. Now
what did you do on that?

What I did is try to figure out exactly what
3a would be. I put a value on a of 5, which
I got 5 + 5 + S = 15. Then 3 times a would

be 15, since 3 times 5 = 15.

Alan showed that he was extremely capable when operating

in the Relational-Iconic mode. The prime example is the "stu-

dents and professors" problem.

44.
Alan:
Interviewer:

Alan:

Interviewer:

Alan:

This b is correct and the a is incorrect.

O.k. Why did you pick the second one? (S 6P)

Well, because there's 6 times as many students
as professors, so...

So, let me ask you on that, then, how come you
multiplied the professors by 6 if the students...

Because there are less professors than students.
There's 6 times less professors than students.

Interviewer:

Alan:
Interviewer:

Alan:

Interviewer:

Alan:

191

Suppose that I thought it was a, could you
justify that it was really b? Other than

what you're saying. Could you show me I'm
wrong if I chose a?

Well, just by putting in some type of number.
O.k., do that.
Six times as many students, so I'd give a value
of l to profeSsors and 6 to students. 1 = 6'6,
which is not right, 1 does not equal 36.

Then yours would be what?

Mine would be 6 = 6-1, which is 6 = 6.

That Alan has not generalized the exponential properties,

and seems to have difficulty distinguishing among them is

indicated by the following excerpt.

2.3

45. 2 2

Alan:

Interviewer:

Alan:

Interviewer:

Alan:

O.k., here you can't multiply the two bases
because your exponents are different, so I'd
go back to my simple form. (writes 4-8 = 32)

Now could we have manipulated with the exponents
there somehow? Is there a property involved?

No, because they're different exponents.

O.k., but you've got the same base. Does that
matter?
No. To be able to combine the two bases,

you've got to have the same exponents.

That Alan has difficulty with false generalizations is

shown in his solutions to the following two problems.

46.

Alan:

Interviewer:

Alan:

(2 + 3)2

Um...here I would add what's in the middle,
what's in the brackets, then square it,

52 = 25.

Could I square first?

2

Well, you could have, 22 = 4 and 3 = 9, no,

you couldn't.

192

Interviewer: Could you have told me that before you got 13?
Alan: Not without working it out, no.

Alan is in fact prophetic as indicated by his answer to
problem 25. x2 + x3. Alan listed the answer as x5.

Alan's thinking on zero, negative, and rational exponents
paralleled Judy's in the sense that they both used the Instru-
mental-Symbolic mode. Judy was more efficient at this than
Alan.

Alan obtained the correct answers to (x + 3)2 and
(vfif-t \/3)2 by writing each factor and using the "FOIL"
method. However, the fact that Alan has not really generalized
the concept of exponent, nor the properties, is exemplified by
his answer to (x3n + 2)2. Alan again uses the FOIL method,

2

9n
but gave an answer of x2 + 4x3n + 4. Apparently, the con-

cept of exponent combined with variable stretched his exponent
schema too far.
Math 121 Low Examination Scores

Todd's interview excerpts are very much representative
of the students who scored poorly on the exponential unit ex-
amination. In general, they are very rule oriented without
any relational understanding or imagery to fall back on in
case the rule is not an exact fit.

Todd eventually received a D grade for the class. He
did not take an algebra course in high school, but did take
the Math 111 class and obtained a C for the final grade.

Todd seems to have parallel systems of thinking. When

dealing with numbers from arithmetic, he does reasonably well,

193

but his tendency to use the Instrumental-Symbolic mode of
thinking whenever variables appear in a problem overcome any
remembrance or understanding of arithmetic procedures. The
following excerpts are used to illustrate this.

47. 2-3 + 4 - 12

Todd: That would be 2 times 3 is 6 plus 4 is 10,
minus 12 is negative 2.

Interviewer: Could we have added the 3 and the 4 first?
Todd: No, that's the way I learned it.

Todd is not really off-base yet, but in the next excerpt he
applies a different system fOr a similar problem.

48. 3 + 4(x + 5)

Todd: Uh...3 plus 4 would be 7, umm...would give
you 7x + 35.

Interviewer: Would it have been possible to multiply by
the 4 first?

Todd: Yes, it would be 4x + 5...no, you couldn't do
that, it would goof you up.

Interviewer: How about 4x + 20?
Todd: Oh yeah, I see. That would be 4x + 20 plus 3.
That would be 7x + 23. No, you couldn't do

that. There are no parentheses around it...I'd
have to go with the 7x + 35.

49. 2-5 + 6

8
Todd: O.k. Um...I see now that I've worked problem 3.
(2-4o6) that I can cancel. (Todd "cancels" by
8

2 and gets ll/4 for the answer.)
Todd then, on his own, decides to perform the Operations
in the numerator and obtain 16/8 = 2.

Interviewer: Now obviously both of these, 11/4 and 2 aren't
correct. In problem 3. (2-4-6) you did it

194

two ways and get 6 both ways. What went wrong

 

here?
Todd: I'm not sure. (He then decides that 2 is the
correct answer.)
50. a + b
a
Todd: That would just be a + b. This is an add prob-

a
lem. You can't cancel unless it's multiplication.

Interviewer: What about problem 4. (2-5 + 6)? Does that
adding matter, or does the fact that I've got
2-5 enable me to cancel?

Todd: 1 Yes. The 2-5 lets me cancel.

Todd does illustrate some relational thinking in a continu-

ation of the above excerpt.

Interviewer: What if I really thought I could cancel and get
1 + b? Is there any way I could check my answer?

Todd: Times it by a.
Interviewer: Would you do that for me?

Todd: (writes a(l + b) ='a + ab). It (1 + b) doesn't
work.

Todd has only an Instrumental-Symbolic notion of variable.
He does state that, "x is just replacing a number" in problem
II. 1. 2x + 3x = 5x (meaning?). However, he is not able to
follow this up until the interviewer requests that he sub-
stitute 10 for x. When asked to compare the "tautology",
2x + 3x = 5x in problem 1. and the "constraint equation",
x + 4 = 7, Todd dwells on the 2 in 2x, and finally concludes
the two problems would be essentially the same if the second
were 2x + 4 = 7.

Todd does use numerical imagery in the "student and

195

professor" problem. He uses 10 for the nubmer of professors
to get 60 for the number of students. He chose b. (S = 6P)
and sticks with it. However, after this, the interviewer can
not get Todd to use numerical imagery for justification. Todd
always uses a rule instead.

Todd is aware of the exponential properties, but only in
an instrumental way. In the following excerpt, the interview-
er tries several times to get Todd to use the meaning of ex-
ponent along with arithmetic operations. Todd keeps using as
his authority the property (a-b)n = anbn.

2 2

51. 2 '3

Todd: That would be 4-9 is 37. (He corrects to 36).

Interviewer: Maybe it's an accident, but it looks as if I
multiplied the 2 and the 3 to get 6 and squared
that, I would get the 36 answer. Could I multi-
ply first and then square?

Todd: (pause) Yes, they are both the same. So as far
as that goes, you could multiply and square.

Interviewer: Do you mean if the powers are the same, I can...?

Todd: Yes. It's like if you had an x and a y. (writes

lxzyz) uh...(long pause), no...uh...

Interviewer: Why don't you do this. Write (2-3)2. (Todd
writes this.) Is this the same as 22-32?

Todd: Yes, you'd square the 2 and then square the 3.

Interviewer: Yes, but that's what I'm wondering, can I really
do that? Could you show me other than the fact

that both are 36? Can you take the (2-3)2 and

change it into 22-32?

Todd: 2 times 3 is 6, squared is 36.
The:interviewer tries again, but Todd keeps using the property

in order to verify the particular case.

196

Todd makes the following false generalizations:

44/22 = 24-2 = 4; 142/72 = 20 = 1 (does not recognize that
l is not a reasonable answer); (xy)3 = x3y3 = xy6;
\f‘-\/'3=\/3.

Todd does correctly identify (2 + 3)0 by noting "any-
thing to the 0 power is l", but then he generalizes that
(2 + 3)0 = 20 + 30. He doesn't see the difference until
questioned by the interviewer. He then recognized that the
addition "makes it wrong".

Todd, however, has no relational understanding of neg-

ative exponents.

52. 5‘2

Todd: That's an exponent?

Interviewer: Yes, that's a negative 2 power.

Todd: I'm lost.

The interviewer suggests "possibilities" of -52 or 1/52, and
Todd immediately remembers the definition and does reasonably
well on the other problems that have negative exponents by
using this definition. He does not use the exponential prop-

erties for the negative exponents.

Todd does have some instrumental understanding of ra-

tional exponents.

53. (x6)k

Todd: That would be...(very long pause) 6 times 8 is

3
x .

Interviewer: Why do we need fractional exponents?

Todd: I couldn't tell you.

197
Interviewer: What does it mean to have a one-half power?
Does that have any meaning at all?
Todd: It's a root -— a square root.
Todd's instrumental understanding of "roots" along with his
misuse of exponential properties cause him to retract his
correct applications of the rules.

Interviewer: Is x3 really the square root of x6?

Todd: (pause) No, x3, that would be the square root

of 9.
Interviewer: What's the square root of 25?
Todd: 5.
Interviewer: How do you know?
Todd: 5 times 5.
Interviewer: Now, is x3 the square root of x6?
Todd: No, that's the square root of x9.

Interviewer: Check it out for me. (Todd multiplies x3ox3

and gets x6.)

Interviewer: So, is x3 the square root of x6?
Todd: NO.

On the problems dealing with the form (a + b)2, Todd
gave x2 + 9 as the answer to (x + 3)2. He was reminded of
problem 12. (2 + 3)2 where the distributing of the exponent
did not work out. He then changes his mind. He tries to use
the FOIL method, but still gets x2 + 9. On (\/2 + \f3)2. he
reverts again and gets 2 + 3 = 5. However, on the problem

3n + 2)2, he did exhibit an instrumental understanding of

the property (am)n by getting x6n + 4.

(x

Other students in the same category as Todd (test scores)

198

used very nearly the same thinking mode of viewing the prob-
lem sets - Instrumental-Symbolic. They differed on the
specific problems missed, but did share the characteristic of
generalizing the prOperties of exponent to situations, where
in fact they did not apply. It appeared that anytime the
problem situation became a little "fuzzy", these students

lost confidence in their ability and consequently placed their

trust in the most likely algorithm that they remembered.

Observations Across Students

 

The problem sets were subdivided into eight categories
(pp. 150-154) for the purpose of tracing the students' thought
processes through the entire interview. The categories were
designed to examine the students' understanding, both instru-
mental and relational (or lack of either) with respect to pre-
requisite arithmetic knowledge, variable, equality, the various
properties of exponent, and additionally "false generalizations"
of any of these. Another interest was the type and quantity
of imagery used. The format followed here will be to list
each category along with pertinent references from the inter-
views to lend support as to either the existence or non-exist-
ence of understanding in the designated category.
Category 1: Prereguisite Knowledge

This category was intended to investigate the students'
knowledge of arithmetic operations, arithmetic properties, and
simple operational type problems involving variables. The

following problems are in this category: (from problem set I)

199

1. 2-3 + 4 - 12

2. 15-3-(2+l)

 

 

8
8. 4xyz
2x
9. a + b
a

10. 3 + 4(x + 5)

All students interviewed obtained the correct result
for problem 1. The important feature of this problem is the
order in which the operations are performed. Due to the fact
that it is a "convention" that multiplication is performed be-
fore addition, those students who said, "It's a rule”, were
considered to have attained the appropriate prerequisite know-
ledge. It is only when this problem is viewed in conjunction
with problem 10. that the individual student's consistency
can be Observed. All students interviewed correctly answered
problem 1., but one student from the low scoring Math 111,
and one from the low scoring Math 121 obtained an answer of
7x + 35 on problem 10.

The remaining problems 3, 4, 8, and 9 were used to in-
vestigate the students' understanding of the "reducing" process
involving fractions. As indicated earlier, the categories are
overlaping in some respects. Problems 4 and 9 were for the
purpose of determining the thinking of students when they gen-

eralized an algorithm inappropriately. All of the Math 111

200

students "canceled" out the a in §_:_p, but none initially
"canceled" the 2 in 2-5 + 6. The fgct that many of the stu-
dents were in an instrumental mode was shown by the fact that
the interviewer "talked" them into dividing by 2 in the second
problem.

Interestingly, the situation was reversed with the Math
121 students. Only one student incorrectly divided by a, and
he eventually reversed himself by the use of numerical imagery
(Todd). However, five of the seven students incorrectly re-
duced by 2 in the second problem. Most, however, did obtain
the correct answer by performing the indicated operation.

Perhaps the above indicates that the beginning algebra
students are more attuned to arithmetic procedures and just
"grind out" their results, whereas the Math 121 students are
more concerned with finding an algorithm from their more ample
supply. Apparently the Math 111 students saw the more obvious
”a“ in the numerator and denominator, while the Math 121 stu-
dents seemed to notice the multiplication symbol and divided
out the common "factors".

None of the students obtained an incorrect answer for
the other two problems. It is important to note that many
did not use a reducing procedure in g;g;§ until prodded by
the interviewer. One could conjectureathat many were led
"down the garden path" to divide inappropriately on the other
two problems. One of the characteristics of relational under-

standing is faith and confidence in an answer. Only Judy

exhibited this understanding by correctly factoring 2(5 + 3),
8

201

then dividing (excerpt 33).

It would appear that both the Math 121 and Math 111
groups in general have a rather limited instrumental type
understanding of the fraction and rational expression problems.

Category 2: Variable and Egpality

 

Category 2 problems were designed to investigate the
students' understanding of the different uses of variables
and the equals sign. The following problems were used for
this purpose.

I. 5. a + a + a

6. 2a + 3a

7. 2a + 3a - b

II. 1. What does it mean to you when we say 2x + 3x = 5x?

2. What does it mean to you when we say solve for
x in x + 4 = 7?

3. Which of the indicated equations would be the
correct answer for the following:
Write an equation using the variables
S and P to represent the following
statement: "There are six times as
many students as professors at this
university". Use S for the number of

students and P for the number of pro-
fessors. (Rosnick, Clement, 1983, p. 4)

a. P = 65
b. S = 6P

4. Is the following true or false? 4x = 46
given that x = 6 (Matz, 1979, p. 134)

5. Without solving, could you compare the solutions
for W and N in the following: 7W + 22 = 109;

7N + 22 = 109 (Wagner 1981, p. 109)

202

The three problems from problem set I. could have been
placed in the category of prerequisite knowledge, but were
placed in this category because of the interviewer's emphasis
on numerical imagery in the questioning process. Three Math
111 students from the "lower" level did answer a3 on the
problem a + a + a, and two followed up with 5d2 on problem 6.
None of the Math 121 students did this. The Math 121 students
were also able to "justify" their answers by choosing a par-
ticular value for a, to a much greater extent than the Math
111 students. Most of the Math 111 students eventually did
use numerical imagery, but most, only after being helped by
the interviewer. However, only one student, Judy (Math 121),
used the distributive property to justify her answer to 2d + 3a.

The problem taken from Matz's (1979) research; if x = 6
does 4x = 46?, and the problem originated by Wagner (1981);
Compare W and N given that 7W + 22 = 109 and 7N + 22 = 109,
were not missed by any student. One student did note, "I
used to think W and N were equal", on Wagner's problem. Per-
haps Matz and Wagner were working with less symbolically ex-
perienced students.

Problem II. 3., the "students and professors" problem
was primarily an exercise in viewing the surface feature "six
times as many students' for the Math 111 students in both
levels. Just as reported by Rosnick and Clements (1983, p. 4),
they could not see their answers were incorrect even after
substituting a number, which created an inequality. Four of

the Math 121 students also viewed the problem in this fashion.

203

They seemed somehow to view one professor and a group of six
students as being "equal". Perhaps in this situation mental
imagery acted as a detriment to understanding. Three of the
(Math 121 students operated in the Relational-Iconic mode on
this problem and easily showed their understanding of the
particular problem as well as the concept of equality (excerpts
35 and 44).

One ominous implication of this category was the number
of students that did not see a difference in the use of the
variable x in the tautology 2x + 3x = 5x, problem II. 1. and
the conditional equation x + 4 = 7 in problem II. 2. Several
students at both the Math 111 and Math 121 level viewed both
problems as a "solve for x" type, or "add like terms type",
and only with pushing by the ihterviewer were able to sub-
stitute numbers in 2x + 3x = 5x and then recognize that it
held for all numbers. One A student in Math 121 (no excerpt),
after being asked to substitute 10 for x and getting an equal-
ity, noted "I never thought of it like that before."

Category 3: Definition of Exponent

The purpose of this category was to investigate the
students' understanding of positive, negative, and zero in-
tegral exponents. Additionally, a positive rational expon-
ent was used in an attempt to determine not only the under-
standing of this form of exponent, but the understanding of
the relationship between exponents and radicals as well. The
problems were all from problem set III, and are as follows:

1. 23

204

ll. (2 + 3)°

14. 573
24. a'b
27. (x6)Li

All Math 111 students evaluated 23 correctly. One lower
level student gave l/23 for the result to (-2)3. Apparently
her negative exponent schema was activated by the appearance
of both a negative number and an exponent. One other student
needed three attempts before arriving at the correct result.
Most of the students wrote down the three factors of negative
two and used a step-by-step procedure. However, all students
except one were able to give the signs of (-2)50 and (-2)51
immediately upon being asked to do so by the interviewer.
Again, it looks as if the Math 111 group will use arithmetic
procedures in preference to other methods.

All of the Math 121 students handled the two problems
dealing with positive integral powers with ease, and also were
able to give the signs of (-2)50 and (-2)51. However, approx-
imately half did use a step-by-step procedure to evaluate (-2)3.

For the problem (2 + 3)0, all students except one gave
the correct result. Eventually the one student also determined
the correct result. Most of the Math 111 students appeared
fearful of both zero and negative exponents, and no student
showed confidence in the answer given (for example Max, ex-
cerpt 30). A small minority did change their answer to two

0

(20 + 3 = l + 1) when questioned by the interviewer. The

205

understanding was strictly at the Instrumental-Symbolic level
for this problem.

The Math 121 responses paralleled the Math 111 group.
One student answered 2, and one other student wavered between
1 and 2 (excerpt 51). No student in either group presented
a logical argument when the interviewer asked, "Why do we
have 0 exponents?". One student did respond, "one times any-
thing is one...", (Dea, excerpt 20), but could not follow up
on this. Perhaps one Math 121 student summed up the general
feeling toward any exponent other than positive integers.

When asked, "Why do we have negative exponents," he responded,
"Just to screw us".

On the problem 5'2 four of the Math 111 students design-
ated the result as "-25". There were two students from each
level that missed this problem. Two of the Math 121 students
in the lower level missed this problem. All students were
able to recall the correct result when aided by the interviewer.
That most students learned from this problem is evidenced by
the fact that all Math 121 Students, and all but one Math 111
student, gave the correct response to a-b. However, no stu-
dent in either group demonstrated any relational understanding
when quizzed about "Why do we have negative exponents?"
(excerpt 16)

Problem 27., (x6)k, gave the Math 111 student a chance
to generalize from whole number exponents to rational number
exponents due to the fact that these exponents are not covered

as a part of the Math 111 course. It is possible that some

206

of these students did cover this topic in high school. Three
students answered correctly, but only one actually "extended"

the appropriate exponential property without aid from the

. . -2 6 9
lnterv1ewer. Incorrect responses were, (x6) , x 7, x , and

3x. Of those that gave x3, one indicated 1/2 3 x6 = x3 and

one used the diagram x-x-x/x-x-x (Art, excerpt 12).

The Math 121 students had covered this tOpic, there-
fore the interviewer was primarily interested in checking if
these students could see the relationships with the \/;g. All

of the Math 121 students multiplied the exponents to get x3

except one student who answered 1/x3. This student was the
only one who did not recognize the one-half power as equivalent
to the square root. Judy was the prime example of the Rela-
tional-Symbolic understanding as she immediately wrote the
answer and then verified that it was the same as the square
root by multiplying x3-x3 to get x6 (excerpt 41).

The interviews indicated that most students in Math 111
had an instrumental understanding of zero, negative integer,
and fractional exponents. None of the students exhibited a
relational understanding of rational exponents.

m n m+n m n m-n
-a = a ; a /a = a

Category 4: a
These problems were designed to investigate if the stu-
dents had generalized these properties to several variations
of the variables. The "generalization" ability was in Krutet-
skii's (1976) first sense - "subsuming a particular case

under a known general concept" (p. 237). The problems in this

category are taken from problem set III.

207

5. 22.23

13. 44/42

17. 2’2-23

22. xa-xb

23. mn/m2

34. al/n- al/m (Math 121 students only)

With respect to am-an = am+n in the group of Math 111

students, six of the seven students gave the answer 32 for

2 3

2 -2 . However, all six used the arithmetic version and gave

4-8 = 32. When pressed by the interviewer, four of these
students were able to determine it was 25. Not all were con-
fident when they concluded, "You add the exponents". That
the students were perhaps wise to use the obvious arithmetic
procedure rather than the property, is indicated by the 45
response given by the one person to get the wrong answer
(excerpt 29). The.fact that none of the Math 111 students had
generalized this prOperty to variants other than positive
integer exponents is indicated by the results from problems

2~23, five people determined the

17 and 22. On the problem 2-
correct procedure, but all worked with the fractional equival-
ency of 2-2. Some few, at the prodding of the interviewer,
did apply the property, but had no confidence other than the
fact the answer was the same. The incorrect responses were
1/4 and -32. On the problem xa-xb only one Math 111 student
gave the correct response. He was able to use numerical im-
agery in his justification. The other answers given were

xab and x2ab.

208

For the Math 121 students the interviews showed four

students recognized the generalization on the problem 22°23.

However, the fact that they had not generalized the property

2 3

to other variations is indicated in the results to 2- -2 .

Not a single student added the exponents initially. That
these students are more capable when dealing with negative
exponents is illustrated by the fact that six of the seven

students changed 2—2 to 1/4 and completed the problem correctly.

6

One student listed 4- as the result. That for many students

in algebra working with “letters" is easier than working with

"numbers" is manifested by the fact that four of the seven

a+b for xa-xb.

Math 121 students gave an answer of x Two

students were able to use numerical imagery to aid in the de-

termination of their result (excerpt 40). The only incorrect

response from the Math 121 group was xab. One student that

eventually received an A in the class obtained an answer of

xa+b, but stated she obtained the answer by using the rule,

a+b doesn't make sense." The Math 121 students were

l/m.al/n

"but x
also asked to do the problem a . Again showing the
symbolic generalization used, all students added the exponents.
However, the results have implications for the prerequisite

category. In addition to the correct answer a(n+m)/mn’ the

1/n+m was indicated three times. Most, however, did

result a
eventually add correctly when the error was pointed out by

the interviewer. None of the students could convert to rad-
ical form and show any relational understanding (excerpt 42).

For the division algorithm, success at the

209

Instrumental-Symbolic level was approximately the same. Three
of the four low level Math 111 students answered the problem
44/42 correctly using arithmetic manipulation. One student

2 as the

did attempt to use the property, but designated 1
answer. One student in the high level Math 111 group did use
the property correctly. However, one student also answered
12. Most students were able to "recall" the algorithm with

a "reminder" from the interviewer. Only two of the Math 111
students were able to give the correct answer for mn/mz. The
most popular incorrect answer was n/2. (the "m's cancel")

On the problem 44/42, all Math 121 students answered
correctly with four students (high level) applying the algori-
thms immediately. All other students used the arithmetic
method, but used the algorithm with a slight hint from the
interviewer. When challenged on "subtracting the exponents”,
three of the students were able to write the factors in the
numerator and denominator and divide out two of the fours,
thus illustrating relational understanding of the property.

On the problem with a literal base, mn/mz, three people used
the property correctly. Of the three, one had scored low on
the exponential test. Two other students answered "can't be
done", and one student (low scorer) gave an answer of ln-Z.

Again, it appears that the Math 111 students prefer to
use the arithmetic method, because both the multiplication
and division schemas with respect to exponents are incomplete.

The Math 121 students prefer the algorithms, but in fact show

very little more relational understanding of these prOperties

210

than the Math 111 students.

Category 5: (ab)n = anbn; (a/b)n = an/bn
The problems in this category were as follows from prob-

lem set III.

4. 22-32
10. (2-3)2
16. 142/72
20. (xy)3

28. V”. \/'9' (Math 121 students only)
This category caused much more difficulty than the preceding
category as far as relational understanding was concerned.
For the purpose of discussion (2-3)2 and (xy)3 will be

2 2

paired, and 2 -3 and 142/72 will be paired.

The Math 111 students took the obvious arithmetic path

on (2-3)2 and got 62 = 36. It was only when the interviewer

wondered if it was an "accident" that 22-32 also gave 36,

that the real interest in the problem began. Only one student
immediately quoted a property. However, several students
showed a relational understanding of the arithmetic prOperties
of exponent to show that (2-3)-(2-3) is equal to (2-2)-(3-3).
(Dea, excerpt 19). On the similar problem (xy)3, two high
level students and one low level student used the property to
immediately get the result. One high level student wrote
3x3y, and one low level answered 3xy. The students were able
to view this as three factors of xy much more readily than

they could view three factors of (2-3). Perhaps the fixatiOn

with arithmetic operations made 2-3 too enticing to ignore.

211

On the "reverse" direction problems 22-32 and 142/72,

success was not nearly so good. On 22-32

all students wrote
4'9 = 36. When pushed as to whether the 2-3 could be done
first, the lack of understanding of this prOperty was shown.

4 ("You add exponents

One student answered that it would by 6
when multiplying"), however, that conflicted with the known
result of 36, thus he did change to62 without any justifi-
cation. Another student, Dea (excerpt 18) can only answer

that it "might be an accident". Nan (p. 182) thought the
numbers were "rigged". The lack of "reversibility" was
universal on this property for the Math 111 students. When
challenged to use a property, a typical answer (excerpt 29)

was 64. That success was less on 142/72 for the Math 111
students, was due only to the fact that it was arithmetically
more difficult to compute. Four students used long division

to complete the problem (after squaring numerator and denom-
inator). One high level and one low level student did use a
"property". Both gave an answer of 2 ("Divide 14 by 7, sub-
tract the exponents"). Apparently the students ignored the
zero exponent. Art did write 22/12, but his lack of confidence,
and oral responses indicated he was primarily guessing.

The Math 121 students, to a great extent, paralleled
the Math 111 students in their lack of relational understand-
ing of the properties in this category. Judy (excerpt 38)
who, in fact, seemed to have the widest range of relational
understanding was not aware of a property that dictated that

2

(2-3)2 = 2 ~32. However, in excerpt 36, Judy had concluded

212

that there was a property 22-32 = (2-3)2. Likewise (excerpt

2

39) Judy claims ignorance of any "shorcut" on 142/7 , although

2 was the same answer that she had

she very quickly saw that 2
obtained using arithmetic. Some students did try to use a
"property". One student wrote 22, one answered 20, and
another one decided 21. All students, when aided by the in-
structor, were able to use the definition of exponent to write
‘14;14 and consequently concluded that 22 was correct. All of
the7Math 121 students gave "2:3 = 6 as their answer for
\fZ-\f9. All agreed that it was the same aS\/36. However,
the numbers were so obvious, it could not be ascertained if
the students were using a property or just focusing on the
arithmetic equivalency.

In general, the students in both Math 111 and Math 121
exhibited little evidence of acquaintance with these prOperties
from this category. Consequently, what understanding they
possessed was either at the arithmetic level or the instru-
mental level.

mn mn
)=a

Category 6: (a
The problems used to investigate the understanding of
this exponential property are listed below. All are from

problem set III.

8. (22)3
19. (2"1)‘2
21. (x2)3

On (22)3 all of the Math 111 students except one did

"what is in parentheses first", wrote 43, then multiplied to

213

get 64. When pushed by the interviewer, all of these stu-
dents lacked confidence as to whether the exponents could be

5 and 25 or

added or multiplied. Only after calculating 2
writing (2-2)~(2-2)-(2°2) were they convinced to multiply

the exponents. One of the students discussed previously in

the interview portion of this document, Nan, insisted that it
was "just luck" that 64 was obtained when, at the interviewer's
instigation, she used 26. Interestingly, Dea (excerpt 18)

had insisted that exponentiation must be done when working
with 22. 2, but insisted that the 2 should not be squared

first on (22)3. She did use the property and multiplied the
exponents to achieve the correct result. On (x2)3 two low

5

level students and one high level student gave x and 6x as

the answer respectively. All other students gave x6 as the

answer. However, only one (Dea) used the property. All
others wrote something on the order of (x2)-(x2)-(x2) to
arrive at the result. One student (high level) responded,
when asked about the possibility of adding or multiplying
exponents, "that's what I hate about algebra, when you multi-
1)-2

plied the algorithm initially. Their lack of confidence in

ply, you add." On (2- only two students (low level) ap-
their answers indicated that they were not really sure if in
fact there was such a property. Four students eventually
arrived at the result by using the definition of exponents
and the consequent complex fraction. That the Math 111 stu-
dents do not see the exponential properties as applying for

negative exponents is indicated by Dea (excerpts 21 and 22).

214

Without hesitation, she gave x6 as the result to (x2)3, but
even with suggestions by the interviewer was never able to
successfully accommodate her schema to allow for the negative
exponents.

Only one Math 121 student used the arithmetic procedure
on (22)3, all others immediately multiplied the exponents
and were able to use the definition of exponent to verify
their answer. They were equally successful with (x2)3. With

1)-2, two students used the defini-

the negative exponents (2-
tion and worked the problem in the fraction mode. All others
used the property to immediately get their answer, and when
pushed, illustrated a relational understanding by going back
to the fractional mode to verify that the property really
applies.

The Math 111 students generally showed little under-
standing, either instrumentally or relationally of the prop-
erty (am)n = amn. Their primary technique of solving the
problems was to use the definition of exponent, then count
the factors. When this technique failed to apply directly,
as.in the problem with negative exponents, they had much more
difficulty.

The Math 121 student showed a relational understanding
of this property. However, it should be pointed out that
this problem dealing with negative exponents was the fourth
Problem pertaining to negative exponents presented during the

interview. Perhaps any uneasiness these students felt with

reSpect to negative exponent had dissipated by this time.

215

Category 7: False Generalizations

The problems in this category were developed to be used
in conjunction with problems from other categories. It can-
not be said that a student has formed a concept, or generalized
a concept or principle if the student applies the principle
to negative instances as well as positive instances. For
example, if a student multiplies the exponents when given
(x2)3, but also multiplies them when given xz-x3, then he is
generalizing falsely. Skemp (1979B) has stated "...the pos-
session of a concept can be evidenced in this way, namely by
distinguishing between examples and non-examples..." (p. 120).
Confrey and Lanier (1980) noted, "...generalization is not a
unitary process, but requires differentiation among the rel-
evant variables, the constants and the irrelevant variables"
(p. 551). The following problems were designed to investigate
"false generalization" (Rachlin, 1981, p. 49). The previous
categories did offer Opportunities for the students to falsely
generalize exponential properties. Many such false general-
izations were pointed out in the discussions of these categories.

All problems are from problem set III.

3. -2.32
6. 22 + 23
7. 32 + 23
9. 44/22

11. (2 + 3)°
12. (2 + 3)2

15. (2 + 3)‘

216

18. 23/4'1
25. x2 + x3
26. xayb

29. V'l‘4 / 37'7— (Math 121 students only)

32. \f6- - V3 (Math 121 students only)

Of the seven Math 111 students, the number of students
falsely generalizing on the non-variable problems were as

follows: -2-32, one answer of -64 (excerpt 28); 22 + 33,

3, none; 44/22, three answers of 22; (2 + 3)2,

none; 32 + 2
one answer of 4 + 9 = 13; (2 + 3)-1, one answer of -2 - 3 = -5;
and 23/4-1, none. Several students did get the wrong answer
for 23/4-1 due to arithmetic errors - one student designated
4.1 = -4. It is worth noting that the students took the‘
obvious arithmetic procedure to arrive at the correct solution,
but all showed a lack of confidence to the question, "Is there
a property that would apply here?" (excerpts 10 and 20).
Several students did then generalize falsely, but immediately

retracted when the result conflicted with their previous answer.

Three people (two low level and one high level) gave x5 as the
2 3

result for x + x . This is in contrast to no false general-
izations for 22 + 23. Several students did show relational
understanding by using such procedures as x2 + x3 = X°x + x-x-x

to indicate that x5 was incorrect. Two students used numerical

b

imagery in their justification. For the problem xay , four

Math 111 students (three high level, one low) falsely general-
ized by indicating the following: xyab, xya+b, (xy)a+b.

In the Math 121 group, none of the students gave an

217

incorrect response to -2-32, 22 + 32 or 32 + 23. Only one

student was somewhat tempted to apply a property when it was
suggested by the interviewer. The problem 44/22 did cause
three people to give an answer of 22 (excerpt 51). All three
did change their mind after using the arithmetic operations.
Two people showed relational understanding by converting to
either a common base of 2 or 4, then using the appropriate
algorithm (excerpt 35). None of the students answered (2 + 3)0
or (2 + 3)2 incorrectly, although two students did‘write

0 0

2' + 3 and then change their mind. Two Math 121 students

(one high level, one low level) had difficulty with (2 + 3)'1,

both said it was -5. When asked about l/5 as a result, both
decided that either answer was correct "depending on how you
look at it". All students in the Math 121 group answered
23/4-1 correctly, although some did have difficulty with the
fractions. None considered converting to a base of 2 and
using the appropriate property. Very nearly the same results

appeared for the Math 121 students as for the Math 111 stu-

dents on x2 + x3. None of these students falsely generalized

on 22

+ 23, but two students (one high level, one low level)
gave x5, two other students initially wrote the x5, but re-
flected and changed. The two students that wrote x5 corrected

themselves with a slight "push" by the interviewer. None of

the students gave an incorrect answer on xayb, although one

student had difficulty convincing himself not to use a property.
Apparently, x2 + x3 caused more difficulty than xayb because

"the bases are the same". On the problem J14 / 3J7-the false

218

generalizations were in the direction of "can't". Only two
students were able to convert to exponential form (excerpt
42), but neither could complete the problem. Two students
(low level) gave an answer of \/3 for VF- V3.

It appears that both Math 111 and Math 121 students
falsely generalize the exponential properties. This is evi-
denced by the fact that practically no students gave in-
correct responses with the arithmetic type problems, but in
every case with the variable type problems, one or more stu-
dents would incorrectly apply a property. Additionally, the
interviews showed many of the students lacked confidence with
respect to their answer, and would frequently change their
result and then change it back.

Category 8: (a + b)2 = a2 + 2ab + b2

This category was designed not so much to see if the
Math 121 students could square a binomial, but more for the
purpose of determining if they correctly used the exponential
properties when exponents were not the most obvious feature
of the problem.

Two students (low level) of the seven were not able
initially to give the correct response to (x + 3)2, but in-
stead gave x2 + 9. When reminded by the interviewer of their
answer to (2 + 3)2, one relented and used the FOIL method to
get the correct result. The other student tried, but still
did not get the answer (excerpt 53). The same two students

immediately made the similar false generalization on the next

problem and answered 2 + 3 = S for ( V? +- J3)2.

219

However, two other students gave answers of 8 +‘2- V6 (be-
cause (\f3)2 = 6), and 11 (because \J6 +\/6 = 6). The cur-

n + 2)2 was the power the students would

cial feature of (x3
obtain for the first term of the trinomial answer. Responses

2
2 2 9n
were 2x3n, x6n , x9n (twice), x2 , and the correct result

6n (2 times). Most students were able to correct their an-

swer with some hints by the interviewer. One student, showing

consistency left out the middle term on all three problems.
This category showed rather well that the students had

not, for the most part, generalized the properties am-an = am+n

or (am)n = amn. They had shown by their work on previous
problems that they were aware of the properties and could
also combine "like terms", but in fact only two of the seven

students could put it all together on a more "complex" example.

SUMMARY - ANALYSIS OF INTERVIEWS

In this section excerpts from the tape-recorded dialogue
between an interviewer and students were presented from both
a beginning and an intermediate algebra class as they worked
selected problems while "thinking aloud". The interviewer
questioned, gave hints, contradicted, and posed other problems
in an effort to determine if the students were using either
the Instrumental-Symbolic, Instrumental-Iconic, Relational-
Symbolic, or the Relational-Iconic, mode of understanding.
Selected excerpts from the interviews were presented as rep-
resentive of the thinking processes that were used by the

students while working the problem sets.

220

The problem sets were subdivided into eight categories

in order to analyze the totality of the student responses

for the purpose of investigating the research questions posed

in Chapter I. The categories, along with some overall eval-

uation of the student responses are listed below:

1.

Prerequisite knowledge The students interviewed

 

in both beginning and intermediate algebra appear
to have a limited instrumental understanding of
the conjectured prerequisite knowledge of numerical
fractions and rational expressions.

Variable and Equality The students interviewed
in both algebra sections appear to have some
difficulty with variable. They, in general,
believe a variable, "stands for a number".

They also can use Instrumental-Symbolic pro-
cedures for combining like terms. However,

their difficulty with the concepts of variable
and equality is evidenced by the fact that.very
few realize that numerical substitution is one
appropriate means of testing a conjecture about
variables. Additionally, some students did not
notice a contradiction when asked to substitute

a number for a variable, doing so and getting an
inequality, thus indicating a lack of relational
understanding of the equality concept. It was
also found that students did not recognize the

use of a variable for a tautology as Opposed to

221

the use of a variable for a constraint equation.
Definition of Exponent - Various Forms In
general, both the beginning and intermediate
algebra students had a relational understanding
of positive integer exponents. For the zero
exponent both groups appear to have an instru-
mental understanding. However, the beginning
algebra students exhibited very little under-
standing of negative integer or fractional ex-
ponents, while the intermediate students inter-
viewed were judged to be at the Instrumental-
Symbolic level of understanding with respect

to both the negative integer and fractional
exponents. Three of the intermediate stu-
dents showed at least some relational under-
standing Of fractional exponents.

. . m n m+ m n
Exponential Properties a -a = a n and a la =a

m-n
Both groups of students were judged to lack re-
lational understanding of these properties. The
beginning algebra students tended to use arith-
metic procedures to avoid having to use the prop-
erties, while the intermediate group could use

the algorithms better, but showed very little
relational understanding. A few of the students
did exhibit relational understanding by using

the exponential definitions (when working with

positive integer exponents) to justify their use

222

of these properties.

Exponential Properties (ab)n = anbn and

(a/b)n = an/bn

The students interviewed were judged to have

at most some instrumental understanding of

this category. The beginning, algebra students
avoided the use of these whenever possible,
while the intermediate students had some in-
strumental understanding. Both groups appeared
to have great difficulty with the reverse of
the properties - i.e. anbn = (ab)n.
Exponential Property (am)n = amn
Interviews of the beginning algebra students
seemed to indicate that they had very little
understanding of any type with respect to this
property. Some few did show instances of
instrumental understanding. The intermediate
students on the other hand were judged to
possess a relational understanding of this
property.

False Generalizations

 

The beginning algebra students in particular
showed evidence of false generalizations.

Once they have committed a rule to memory,
they use it, whether appropriate or not. The
lower level students in the intermediate group

tended to do likewise. The better intermediate

1.

223

students did use false generalizations, but
appeared to be more cautious about applying

an algorithm. Some students did use numerical
imagery as a basis for their decisions.

Squaring A Binomial (a + b)2 = a2 + 2ab + b2

 

This category was used only for the intermediate
algebra students. The primary purpose was to
look for the generalization of the properties

n m m+ m n
-a = a n and (a ) = amn.

a The majority of
the students were able to square the binomials
presented, but most obtained a wrong answer for
the exponent of the first term of the answer when

working with literal exponents.

 

CHAPTER V

SUMMARY AND DISCUSSION

Procedure

 

During the summer of 1984, fourteen students from Ferris
State College, Big Rapids, Michigan were slected to partici-
pate in this study of the investigation of college remedial
algebra students' understanding of the concept and principles
of exponent. Seven students were selected from a beginning
remedial algebra class, and seven students were selected from
an intermediate remedial algebra class. Approximately one-
half of the students from each class scored the highest grades
of all students on an exponential unit examination given pre-
vious to the beginning of this study. The other students
scored the lowest grades relative to all students on the ex-
ponential unit examination.

Forty-nine questions were developed based on errors
from exponent pre-tests, prerequisite knowledge for the con-
cept of exponent, and the various definitions and properties
of exponent.

The primary instrument for investigating the students'
understanding of exponent was an interview with each student
of approximately ninety minutes. The students were asked to
"think aloud" as they worked through the set of problems.
Although the problems were pre-determined, the interviewer

224

225

was free to develop new problems during the interview, offer
hints of encouragement, question solutions, question procedures,
and contradict. The item of interest was the student's thought
process, therefore the interviewer was free to move in any
direction that was necessary to "draw out" the student's under-
standing.

The interviews were audio-recorded for later analysis.
The tape of each interview was analyzed by use of a two-by-
two matrix developed by combining Skemp's (1979B) theory of
relational and instrumental understanding with two of Bruner's
(1973) modes of knowledge representation - iconic and sym-
bolic. The students were rated as they worked variouSTprOblems
as helonging to either of the four cells: Instrumental-Iconic,
Instrumental-Symbolic, Relational-Iconic, and Relational-Sym-
bolic. Rating techniques were adapted from Erlwanger (1975C)
and Alexander (1977) in order to determine the mode of think-
ing.

Selected excerpts of the interviews for the various
levels of student "capabilities" were given verbatum in order
for the reader to judge the validity of the analysis model
Additionally, the problem set was subdivided into eight
categories for the purpose of tracing an individual's thought
process through the entire interview with respect to each of
eight items of interest; prerequisite knowledge, the expo-
nential properties, various definitions of exponent, and false
generalizations. The problem set results from all interviews

were then tabulated, and the various answers listed. These

226

were coordinated with the verbatum interview excerpts to
specify as far as possible the various levels of understand-

ing at which the groups as entities appeared to be functioning.

RESULTS

This study was designed to be an investigation into the

 

thought processes of how students deal with the concept and
principles of exponents. Several researchers have detailed
both the dangers of attempting to draw firm conclusions from
an interview setting, and the benefits of the proper util-
ization of the procedure (for example, Erlwanger, 1974). The
results here, then, are of a somewhat tentative nature. At
best they can serve as guides to future studies.

This study began with a goal of investigating the fol-

lowing questions:'

1. Do remedial algebra students have a relational, in-
strumental, or no understanding of the prerequisites
conjectured as necessary (as advocated by Gagne')
for success in dealing with the concept of exponent.

2. Do remedial algebra students have a relational,
instrumental, or no understanding of the concept
of exponent?

a. How does the understanding of positive,
negative, (both integral and fractional)
and zero exponents differ in the same
student? Between students?

b. How does the understanding of explicit

227

number exponents and literal exponents
differ in the same student? Between
students?
3. Do remedial algebra students have the ability
to generalize (as defined by Krutetskii) the
various properties of exponents?

a. Can the source of "false generalizations"
be determined?

b. Have students that appear to have general-
ized the properties of exponents (relational
understanding), merely generalized the sym-
bolic notation (instrumental understanding)?

4. What types of imagery (Bruner's enactive, iconic,
and symbolic) do students use when working with
the concept of exponent?

a. Does the imagery used differ, and in what
respect, for students at the relational
and instrumental levels of understanding?

b. Can a student who is operating at the in-
strumental level be “pushed" by way of
hints and guided questioning to use nu-
merical imagery as an aid to relational
understanding?

5. Do successful students (as determined by a letter
grade on a test) differ from unsuccessful students
with respect to the four questions above?

The results of the investigation into these questions

228

will now be addressed, one at a time in numerical order.

1. Do remedial algebra students have a relational, in-
strumental, or no understanding of the prerequisites
conjectured as necessary (as advocated by Gagne')’
for success in dealing with the concept of exponent.

The prerequisites, as developed in terms of Gagne's
chart of hierarchical prerequisite knowledge (Appendix A),
primarily were those of arithmetic operations and knowledge
of variable and equality.

As indicated by the interview excerpts and interview
tabulation, the students in both the beginning and inter-
mediate levels appear to have relational understanding of the
operations involving integers. The interviews showed that
both beginning and intermediate algebra students had a very
limited instrumental understanding of fractions and rational
expressions. A particularly troublesome area was the false
generalization of "canceling" inappropriately. For example,
the problems a_:_p and 2'5 + 6 were missed frequently due to
the "crossing oat" of a common number which actually was not
a common factor of both numerator and denominator. Also,
many of the intermediate algebra students had difficulty

l/m.al/n due to the lack

adding the exponents in the problem a
of understanding of the process used in adding fractions
which contain literal numbers.

2. Do remedial algebra students have a relational,

instrumental, or no understanding of the concept

of exponent?

229

a. How does the understanding of positive,
negative, (both integral and fractional)
and zero exponents differ in the same
student? Between students?

b. How does the understanding of explicit
number exponents and literal exponents;
differ in the same student? Between
students?

The results of the interviews lend weight to the con-
jecture that both beginning and intermediate algebra students,
both successful and unsuccessful, have a relational under-
standing of the concept of positive integer exponent. It is
with the use of the various exponential properties that in-
strumental understanding replaces relational understanding.
The beginning algebra students tend to avoid the exponential
properties to the greatest extent possible. The intermediate
algebra students are much more attuned to algorithms, but
tend to use the exponential properties in inappropriate sit-
uations (false generalizations) (excerpt 46).

In particular, the use of a variable makes the erro-
neous application of an exponential property much more likely.
The interview results indicate that those problems with
variable exponents are more difficult for students at all
levels than those with variable bases. When both the base
and exponent were literal numbers, all levels of students
interviewed were as likely to use an incorrect application

of a property as the correct one. For example, the two

230

problems xa°xb and xayb, lead to many different answers. Of
those students who obtained the correct results, many seemed
to have instrumental understanding to such a low degree that
their answer was little better than a guess. Only a very few
of thestudents could use numerical imagery as a crutch for
decision-making on these problems.

The understanding of the zero exponent for all levels
of students was shown to be at the instrumental level
(excerpt 30). All students did recognize "anything to the
zero power is 1". No student was able to offer an adequate
response to the question, "Why do we need a zero exponent?",
or "Why do we have a zero exponent." Most of the students'
difficulties appeared at the false generalization level on
the problem (2 + 3)a. A few students distributed the zero
and obtained two for an answer.

After variable exponents, the major difficulty appeared
to be with negative exponents. For both the beginning and
4 intermediate algebra students, it was as if the exponential
properties were not intended to be used with negative expo-
nents. At both levels the students tended to use the defi-
nition to convert to fractions, and then work with positive
exponents. This showed a relational understanding of the
definition, but not even instrumental understanding for the
exponential properties with respect to negative exponents
(excerpt 17).

In all the variations of exponent, the intermediate

algebra students exhibited both more relational and

231

instrumental understanding than the beginning algebra stu-
dents. The unsuccessful intermediate algebra students
appeared to have less understanding, of both types, than the
successful beginning algebra students. However, apparently
due to their longer "exposure", they were more apt to recall
a procedure when prodded by the interviewer.

The understanding of rational exponents was virtually
non-existent at either the relational or instrumental level
for the beginning algebra students (excerpt 12). Rational
exponents had not been covered as a topic in their class,
therefore, it is not an unexpected result. Several students
did give the correct result for (x6)8, but the interviews
showed that since they were expected to do something, some
students added and some multiplied. None mentioned the idea
of a square root. On the other hand, the intermediate
algebra students did, in general, show a relational under-
standing on (x6)l5 by using not only the correct property,
but recognizing the equivalency with the notion of square
root (excerpt 41). However, most manifested, at best, an
Instrumental-Symbolic understanding of the problem al/m-al/n.

3. Do remedial algebra students have the ability

to generalize (as defined by Krutetskii) the
various properties of exponents?
a. Can the source of "false generalizations"
be determined?

b. Have students that appear to have general-

ized the properties of exponents (relational

232

understanding), merely generalized
the symbolic notation (instrumental
understanding)?

It cannot be stated with any degree of confidence that

remedial algebra students have the ability to generalize

 

relationally the properties of exponent. The interviews
indicated only one student out of fourteen who was at least
"close" tohaving generalized the properties (Judy). Several
students would do quite well on one or two properties, or on
all properties in one direction, but then fail on another
property, or the same property in the reverse direction
(excerpt 38).

The source(s) of false generalizations appear to come
from a total reliance on instrumental understanding when-
using exponents (excerpt 23).

Those students operating strictly at the instrumental
level have no other way except total memorization of symbols
to serve as a check to false generalization, therefore, most
beginning algebra students and a majority of the intermediate
algebra students engaged in false generalization. Those
students who generalize falsely do not appear to notice all
the relevant features of a problem, and frequently home in
on one surface feature that matches a feature of an algorithm
available to them. For example, several students saw only
the multiplication and exponent for the problem xayb, and

+b

obtained xya , obviously ignoring the essential part of the

algorithm pertaining to the base.

233

Although variable exponents and bases are primarily a
source of false generalizations, several students did engage
in such thinking with explicit numbers (excerpt 28). In
most situations where errors were made on the arithmetic
problems, the interviewee could get the answer arithmetically
with appropriate suggestions from the interviewer. However,
when asked about a property for the problem, it was as if
the arithmetic qualities of the numbers vanished, and in a
sense the problems were of a variable nature (excerpt 29).
The numbers were symbols to be manipulated.

4. What types of imagery (Bruner's enactive, iconic,
and symbolic) do students use when working with
the concept of exponent?

a. Does the imagery used differ, and in what
respect, for students at the relational
and instrumental levels of understanding?

b. Can a student who is Operating at the
instrumental level be "pushed" by way
of hints and guided questioning to use
numerical imagery as an aid to relational
understanding?

As expected (due to lack of concrete materials), no
student used the enactive mode of imagery. Most students
interviewed worked primarily in the symbolic mode. This
included both successful and unsuccessful students from both
beginning and intermediate algebra. None of the students

used the iconic mode spontaneously. The primary difference

234

in imagery usage between the relational thinkers and instru-
mental thinkers appeared to be the fact that the students
with relational understanding recognized that they gguid use
numerical imagery. Judy was a good example of this (excerpt
40). This was not always possible for the people with in-
strumental understanding. Another A student in the inter-
mediate algebra section indicated that "she had never thought
of it that way." when pushed to use numerical substitution.

This investigation idicated that a student could be

pushed to use numerical imagery as an aid to understanding,
at least in the short term. Some students did not use numer-
ical imagery initially, but after being pressed by the inter-
viewer, used it frequently thereafter in an apparent rela-
tional manner. Dea (excerpts 14, 15 and 16) is a good example.
Initially, she was not inclined to use numerical imagery, but
after finally using it to determine that she cannot divide
out the a in a_:_p, she then readily uses it in the next ex-
cerpt. Only a long-term evaluation could determine if Dea,
in fact, will continue to use numerical imagery as an aid to
understanding.

5. Do successful students (as determined by a letter
grade on a test) differ from unsuccessful students
with respect to the four questions above?

a. Prerequisite Understanding
The beginning algebra students interviewed
did not differ to any great extent in their

understanding of prerequisite knowledge.

235

both successful and unsuccessful students
had at most an instrumental understanding.
In the intermediate algebra group, the
unsuccessful student had only an instru-
mental understanding of the prerequisites.
However, the successful student group con-
tained students that understood instrumen-
tally and students that understood re-
lationally. Judy (excerpts 33-41) and

Alan (excerpts 42-46) differed in their
understanding, but both were very success-
ful.grade-wise in the class.

Concept of Exponent

This question was dealt with thoroughly

in the main question number 2. Generally,
the interview results could be capsulized
to the findings that both successful and
unsuccessful students had relational
understanding of positive integer exponents,
instrumental understanding of zero exponents
and negative exponents, and limited instru-
mental understanding of variable exponents.
Generalization and False Generalization
Both the unsuccessful and successful be-
ginning algebra students, according to the
interview results, had not generalized the

concept of exponent to all of the variants

236

and properties. Both groups appeared
equally likely to generalize falsely on
variable related problems. The successful
students knew the "rules" better on the
arithmetic problems. (Max, excerpts 24-30,
and Dea, excerpts 13-23).

The unsuccessful intermediate algebra
students were as likely to falsely general-
ize as the beginning algebra students.
However, they were more likely to falsely
generalize than the successful intermediate
algebra students. Todd is an example of
such a student (excerpts 47-53). The
successful intermediate algebra students also
used false generalization, but particularly
on variable problems.

Imagery

For the beginning algebra students inter-
viewed, imagery was for the most part non-
existent. It was only at the instigation
of the interviewer that numerically imagery
was initially used.

As indicated in the main question number 4,
the successful and unsuccessful student at
the intermediate level used very little
imagery. Some of the successful students

used it when pushed by the interviewer.

237

DISCUSSION

 

This investigation would lead one to believe that re-
lational understanding as defined by Skemp (1979B) has little
bearing on the success or lack of success in either a begin-
ning or intermediate algebra class. All of the successful
students at the beginning level and at least half of the
successful students at the intermediate level lacked rela-
tional understanding with respect to exponential concepts and
principles.

Bruner (1966) conjectured that mathematics learning
should proceed in the order enactive mode, iconic mode, and
finally symbolic mode of representation. The omitting of
the first two modes was cause for concern for the long-term
mathematical education of the individual. The interview data
showed that few students view algebra in any mode other than
symbolic. The data also has implication for the findings of
other researchers.

Krutetskii (1976) indicated that "incapable students":

Cannot generalize mathematical material
according to essential features even with
help from the experimenter and after a
number of intermediate, single-type practice
exercises (p. 254).
This investigation shows that rarely had students, at any
level generalized the concepts and principles of exponent.

Gagne' (1970) interated the importance of prerequisite

knowledge in mathematics learning. Skemp (1979) also in-

dicated that conceptual structures can only be built from

available lower order concepts. The students in this study,

 

238

as indicated by the interviews, had minimal relational under-
standing of the conjectured prerequisite concepts.

If one assumes any "integrity" for the course offerings
and resultant grades attained by the students in this study,
the dramatic implication can only be that these students have
accomplished a goal that the above theorists indicate they
should not achieve. Perhaps a plausible explanation is
Offered by Robert Davis (1967).

Davis, after observing a teacher write on the board
such problems as xz-x3 = x5, plO-p7 = p17, and so on for 45
minutes a day for two consecutive days, had the following to
say:

Whenever I have described this lesson
to mathematicians, I have usually found it
neither necessary nor prudent to add any
further remarks. To professional mathema-l
ticians, the lesson speaks for itself. To
mathematics educators, however, it may be
appropriate to claim that what is at work
here is clearly a teacher belief system of
a particularly Thorndikean sort. These
stimuli were being connected to the re-
sponses with such tenacity as to explain

how it can happen that a college freshman

can write x2 + x3 = x5 (p. 19).

The successful students in this study had received ex-
cellent grades on an exponential examination approximately
two weeks previous to the interviews. The teacher indicated
that for examinations he gave problems from the text.. one
can conjecture that the students earnestly studied the text
and sample board problems, took the examination, and promptly
forgot the rules (the "bonds" were broken). As Skemp (1979B)

would say there was a match among the students, the text, and

239

the teacher.

This is £95 an indictment of the instructor. He in-
dicated that given the short span of ten weeks, along with
a required syllabus, and students that were "unprepared",
that he treated the courses as "tool" courses. He tried to
get the students to learn enough to function in their par-
ticular program, or to meet a prerequisite for the next course.

If, in fact, the interview findings are representative
of remedial students' understanding of algebra and concept-
ualization of the learning process in algebra as they exist
at various two-year colleges, four-year colleges, and uni-
versities; then the findings are perhaps an indictment of
the higher-educational system. Perhaps educational institu-
tions are writing off any hope for these students to "under-
stand" in favor of a stimulus-response situation which will
at least enable the student to "pass" the math requirements
and remain a bona-fide tuition paying student. Is it reason-
able to enroll students who have a history of virtually no
success in mathematics, give them a quick ten-week course
consisting of lectures at the chalk board, place them in
groups from thirty-five up to one-hundred-fifty, and expect
them to understand? To do so violates virtually all that we
have learned from cognitive research in the preceding fifty
years by such men as Piaget, Bruner and Skemp.

Surely, the purpose of a mathematics course at the
college level can not be justified by the concerned academic

department wanting the course(s) to be "on the student's

240

record". This would appear to be one of the real possibili-
ties based on the consistent lack of any relational under-
standing by the majority of interviewees in this study.
Apparently, students are "successfully" completing these two
courses, and thus, meeting the required prerequisites for
their particular programs.

The above would indicate there are two possible program-
matic scenarios for these students. First, perhaps the under-
standing of the algebraic course material is not really
needed in the involved programs, at least in any major role.
In this case, one must question the expenditure of resources
for all parties involved for the "window-dressing" of par-
ticular courses. Secondly, perhaps the algebra topics covered
in the'two remedial courses are indeed required for the pro-
grams, but are "taught" again by other instructors as the
topics are needed in the programs. That is, the algebra
courses could be justified by the conjecture that even though
the students have little relational understanding, it is
easier to memorize the involved algebraic concepts and prin-
ciples the second time because "the students have seen the
material before."

One must question the educational priorities of a system,
which enables a student to methodically move up the educa-
tional ladder by continually memorizing material and con-
sequently passing examinations. If such a student had under-
stood relationally at the elementary school level, perhaps

he/she would have understood the middle school mathematics,

241

and eventually the high school mathematics. That is, re-
lational understanding at any educational level equips the
student to have a fair chance for success of such under-
standing at the next level. In this event, the student would
have been able to enroll in a "college algebra" course when
coming to college (if needed). It seems that at each educa-
tional level the system can not adequately respond due to the
"failure" of the educational system to provide relational
understanding at the previous level.

In summary, the only real answer to the massive numbers
of remedial students enrolling at the college level would
appear to be a real dedication by the educational system at
all levels to expend the time and financial resources neces-
sary to aid students in obtaining a relational, as opposed

to an instrumental, understanding of mathematics.

LIMITATIONS OF THE STUDY

1. This study dealt with remedial algebra students at
Ferris State College who were enrolled during a summer term.
Are the students who attend college during the summer quarter
and enroll in remedial algebra representative of all such
students who enroll during the academic year? Perhaps only
those who failed the class previously, or those "bright"
enough to move ahead, enroll in summer. All of these un-
answered questions limit the scope of this study.

2. The small sample size from each of the two levels of

algebra students limits any conclusion only to very tentative

242

conclusions, and only with respect to the fourteen students
actually interviewed.

3. The model used in this study was derived from models
developed by Erlwanger and Alexander (previously cited) for
the purposes of investigating elementary school children, and
middle school-age children respectively. There were two dif-
ficulties noticed during the application of this model to
college-age students. First, students at this age operating
at an instrumental level of understanding are often supremely
confident of their answers. They believe they truly "under-
stand". Thus, they have all the characteristics the model
predicts for students at the relational level. Many times
during the interviews, it was only after several problems
that the instrumental level of understanding of an individual
could be perceived. In this sense, the study is limited by
the author's ability to actually perceive the correct level
of understanding.

4. Additionally, college-age students, particularly the
"bright" ones, have the ability to "learn" from the prodding
and hints from the interviewer, as well as previously worked
problems. As an example of this, frequently students who had
no inclination to use numerical imagery, used it consistently
after a suggestion by the interviewer. Due to the fact that
the forty-nine problems were for the most part closely re-
lated conceptually, this study is limited by the students'

ability to learn during the interview.

243

SUGGESTIONS FOR FUTURE RESEARCH

l. The interview data indicates that successful stu-
dents in general lack any real understanding (relational) of
the algebraic processes related to the concept and principles
of exponent. The implications of the interview results are
'so ominous with respect to any future mathematical endeavors
by these students, that the study should be replicated at
other institutions, not only with exponential concepts, but
other algebraic concepts, and sampling more mathematically
advanced students as well.

2. Doyle (1983) has indicated that "low ability"
students in mathematics do not do well in an unstructured
setting. They do not develop the strategies and "higher
order executive routines" that enable them to be successful
(pp. 175-177). Confrey and Lanier (1980) have suggested that
Krutetskii's abilities such as generalization reversibility,
flexibility, and curtailment should be addressed directly in
a teaching situation rather than embedded in lessons on other
concepts (p. 553).

The suggestion here is for a researcher to attempt to
design such a course that is relevant to college-age algebra
students.

3. Nigel Ford (1981) after reviewing the literature on
the assessment of learning in higher education, subscribed
to the belief that the quickest way to change student learning
is to change the assessment system (p. 372).

Many students in the interviewing sessions obviously

244

believed that answers were the most important aspect of al-
gebra. All one had to do was memorize how to do enough
problems and algebra was easy. One student accused the inter-
viewer of "messing her up for the next test", by having her
think of some aspects of exponents that she had never con-
sidered.

Is it too late for relational understanding to become a
personal goal of remedial algebra students? By by-passing
the first two levels of Bruner's knowledge representation,
have many of the students become mired in the Instrumental-
Symbolic mode due to affective reasons?

It is suggested here that it would be worthwhile to
teach a remedial algebra class modeled after the "mathematics
laboratory" idea of Fitzgerald (1972) (cited in Fey, 1980,

p. 414):
Primarily, a mathematics laboratory is a

state of mind. It is characterized by a ques-

tioning atmosphere and a continuous involvement

with problem solving situations. Emphasis is

placed upon discovery resulting from student

experimentation. A teacher acts as a catalyst
in the activity between students and knowledge.

Secondarily, a math lab is a physical plant
equipped with material objects...Since a student
learns by doing, the lab is designed to give him
the objects with which he can do and learn.
Not only teach for understanding, but test for under-
standing as well. Perhaps it would take two terms to "cover"
the material usually covered in one term. However, if the

experiment could shed light on the question of whether it is

actually possible to teach remedial algebra college students

 

from a relational standpoint, then it would contribute

245

greatly to what is known about such students.

This study began initially by detailing the massive
numbers of remedial algebra students who are presently en-
rolled in higher educational institutions in the United States.
According to Akst (1981), there were 600,000 such students in
1980. Stanley Erlwanger (1974) observed at the conclusion of
his case studies with Benny et al:

The case studies suggest that unless a child's

wrong ideas, beliefs and views are detected

and corrected, they may develop and become more

complex (p. 285).

This study is concluded with the observation that the

findings herein indicate that Erlwanger was correct. Benny

has grown up. Benny is now in college.

APPENDICES

2116

APPENDIX A

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247

.APPEmHILX B
Exponent Pre-Test

Write 2.2-2.2.2 in exponential form

 

Simplify: 21 22
. . 2
Simplify: 2'3
Simplify: 22+32
Simplify: 234-22
. . 5
Simplify: 2
22
Simplify: (23)2
. 3
Simplify: 4
22
. . 0
Simplify: 2
2/3

Simplify: 8
Simplify: (2'3)2

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Simplify: x- x
Simplify: xy
Simplify: x2+y

Simplify: x +x

 

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Simplify: y
2
x

Simplify: x

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248

APPENDIX C

CONSENT FORM

I confirm that my participation in Mr. Wilson's research
into the learning of algebra is voluntary.

I recognize also, that my participation will in no way
affect my grade in my mahtematics course.

I consent to allow Mr. Wilson to discuss my work in my
algebra class with my instructor.

I realize that my responses may be published in a
research report, but understand that this will be

done in an anonymous fashion.

Name (Printed)

 

Signature

 

Date

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