INSTRUCTION 'IN PROBLEM SOLVING AND PIAGET’S
THEORY OF COGNITIVE DEVELOPMENT ' ‘

Dissertation for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY .
JERRY K. STONEWATER
1977

This is to certify that the

thesis entitled

INSTRUCTION IN PROBLEM SOLVING AND PIAGET'S
THEORY OF COGNITIVE DEVELOPMENT

presented by

Jerry K. Stonewater

has been accepted towards fulfillment

of the requirements for

 

Ph.D. degree in Department of Admin—

 

 

1stration and Higher

Date March 14, 1977

 

0-7639

 

 

 

 

F4

 

PC
are critic
Without t}
large Perc
ever. are
research .'
the devel.
instructi.

ABSTRACT

INSTRUCTION IN PROBLEM SOLVING AND PIAGET'S
THEORY OF COGNITIVE DEVELOPMENT

BY

Jerry K. Stonewater

The Problem

 

For engineering students, formal reasoning skills
are critical for mastery of an engineering curriculum;
without them success in engineering is improbable. A
large percentage of minority engineering students, how-
ever, are not formal-operational thinkers. In addition,
research identifying instructional strategies that promote
the development of formal thought is limited. Hence,
instructional strategies that facilitate development of

formal thought need to be designed.

Procedures

 

Piaget's theory of cognitive development was used
as the basis for designing the course "Introduction to
Reasoning and Problem Solving," hypothesized as an effec-
tive treatment for facilitating formal thought development.

The sample included 11 control group and 16 exper-
imental group first-term minority engineering students

at Michigan State University. The control-experimental

group pr

 

the expe
posttest
respond

Pendulum
measures

1

 

effective
there are
abStract
and Vari
are trea
ses “586
(b) Pea]
and IQ)

of Vari

Jerry K. Stonewater

group pretest-posttest research design was used with only
the experimental group receiving the treatment. Pre- and
posttest measures were obtained by asking students to
respond to the Equilibrium in the Balance problem and the
Pendulum Problem, which are theoretically both valid
measures of abstract reasoning.

The study addresses whether (a) the treatment was
effective in increasing abstract reasoning ability, (b) if
there are significant correlations between both pretest
abstract reasoning ability or change in abstract reasoning
and various academic ability variables, and (CT if there
are treatment by ability interactions. Statistical analy-
ses used were (a) a two—sample, one-tailed t-test,

(b) Pearson product-moment coefficients of correlation,
and (c) Finn's multivariance technique for an analysis

of variance with unequal cell size.

Conclusions

 

The major conclusions of the study are:

1. Upon entry into the University, a high per-
centage of the minority engineering students in the sample
were functioning below the formal-operational level.
Eighty-five percent were not formal-operational on the
Balance test and 89% were not formal-operational on the

Pendulum Problem.

 

follow in
found:
(a
I

grOup 2

2.

Jerry K. Stonewater

For the Equilibrium in the Balance test the

following were found:

found:

to posttest

group:

(a)

(b)

(a)

(b)

4.

(a)

(b)

(c)

the change in abstract reasoning from pretest
to posttest was not significantly greater for
the experimental group as compared to the
control group;

the posttest level of abstract reasoning was
not significantly greater than the pretest
level of abstract reasoning for the control
group but was significantly greater for the
experimental group.

For the Pendulum Problem the following were

the change in abstract reasoning from pretest
to posttest was not significantly greater

for the experimental group as compared to

the control group;

the posttest level of abstract reasoning was
not significantly greater than the pretest
level of abstract reasoning for either the
control or the experimental group.
Descriptive data indicate that from pretest

the following are greater for the experimental

the change in average level of abstract
reasoning;

the decrease in percentage of students who
are not formal-operational;

the percentage of subjects who increased
in level of abstract reasoning.

I
I

ability ‘.

 

test are :-

 

(<

dCademic ‘

actiOn e?
lag Was n

within Or

percentii

s‘tudehts

5.

Jerry K. Stonewater

The significant correlations for academic

ability variables with the Equilibrium in the Balance

test are:

(a)

(b)

(C)

(d)

6.

high school grade point average, arithmetic
and mathematics placement scores, and ACT/SAT
mathematics percentile are all positively
correlated with entry-level abstract
reasoning ability;

ACT/SAT verbal percentile is positively cor-
related with amount of change in abstract
reasoning for the experimental group;

entry-level abstract reasoning ability is
positively correlated with first-term MSU
grade point average;

MSU grade point average and grade in first-

term mathematics course are both negatively

correlated with amount of change in abstract
reasoning for the control group.

There were no significant correlations for

academic ability variables with the Pendulum Problem.

7.

There were no treatment-by-ability inter-

action effects; the amount of change in abstract reason-

ing was not different for high or low ability students

within or between groups.

8.

High ability students on the ACT/SAT verbal

percentile changed significantly more than low ability

students.

H

in

Depart:

 

INSTRUCTION IN PROBLEM SOLVING AND PIAGET'S

THEORY OF COGNITIVE DEVELOPMENT

BY

Jerry K. Stonewater

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Administration and Higher Education

1977

To Barby and Benzy

ii

doctora.
in writi
Chairpez
and Norm
Upon my 1
like to a
Sweetland
T
of SuPpOr
of the enc
Staff, the
int/owed 1
CIation, Wh

Study grew

ACKNOWLEDGMENTS

I would like to express my appreciation to my
doctoral committee for their assistance and encouragement
in writing this dissertation. Drs. George Van Dusen,
Chairperson, Stephen Yelon, Director, Lou Stamatakos,
and Norm Kagan helped me build upon my strengths, improve
upon my weaknesses, and grow as a scholar. I would also
like to acknowledge my original chairperson, Dr. William
Sweetland; his untimely death was a great sadness to me.

The College of Engineering provided a great deal
of support for this work. I am especially appreciative
of the encouragement from the Engineering Student Affairs
staff, the time spent by the engineering students
involved in this study, and the Alfred P. Sloan Foun-
dation, which funded the project out of which this
study grew.

I am also grateful for my family's contribution
to my education. My parents, Jean and Stoney, taught me
to value learning; they sacrificed much so that I could
have the best education possible. For this I thank them.

I also thank my in-laws, Viv and Ray, for believing in me.

iii

daughter

we have ,

 

 

 

Lastly, I'm grateful to my wife Barb and our
daughter Jennifer for the love we share, for the fun

we have, and for life together.

iv

Chapter

II.

mm‘rve-Tm

REV

RE

 

Chapter

I.

II.

TABLE OF CONTENTS

THE PROBLEM I O O O O O O O O O O
The Problem-Intellectual Development . .
The Intellectually Developed Student

Defined I I 0 O O O I O O O
Piaget's Theory . . . . . . .
Intellectual Development: A Pre-

requisite to the Study of

Engineering . . . . . . . .
College Students and Formal Reasoning

Ability . . . . . . . . . .
Increased Minority Involvement in

Engineering . . . . . .
Instructional Strategies that Facilitate

Intellectual Development. . . . .

The Treatment. . . . . . . . . .
The Instructional Strategy: Content .

The Instructional Method . . . . .
Statement of the Problem . . . . . .
Terms and Their Definitions . . . . .
Purpose. . . . . . . . . .
Research Hypotheses. . . . . . . .
Summary. . . . . . . . . . . .
REVIEW OF THE LITERATURE. . . . . . .
Piaget' 5 Age Specification for Entry into

Formal Operations. . .

Research Concerning the Development of

Formal Reasoning . . . .

Pre-college Students-~General.
Pre-college Students—-Enrolled in a

Science Course . . . .
College Students . . . .

Page

12
13
14

15
19

22
23
24
25
28

30

31
32
32

35
37

—‘

Chapter

III.

IV,

 

 

Tl

Chapter

College-age and Older Subjects--Not

Necessarily Enrolled in a Uni—
versity . . . . . . . .
Minority Students . . . . .

Research on Effective Instructional
Strategies. . . . . . . .
Summary . . . . . . . . .
III. THE RESEARCH DESIGN . . . . . .

The Population and Sample . . .

Assignment to Control and Experimental

Groups . . . . . . . . .
The Treatment . . . . . . .
The Research Design . . . . .
Methodological Procedures . .

Pre- and Posttesting . . .
Instrumentation . . . . . .

Reliability and Validity. . . .

Reliability . . . . . . .
validity O O O O C O O 0
Internal and External Validity .

Hypotheses and Statistical Analysis
Statistical Analysis: Hypotheses
Statistical Analysis: Hypotheses

3-66 0 O O O O O O O 0
Statistical Analysis: Hypotheses

67-82. 0 o o o o o o 0

Summary . . . . . . . . .

IV. ANALYSIS OF THE RESULTS. . . . .

Analysis of the Effectiveness of the
Treatment . . . . . . . .

Statistical Results. . . . .
Descriptive Results. . . . .

1. Subjects' average Piaget-level .
2. Subjects functioning below the

formal-operational level on
pretests . . . . . .

vi

the

Page

40
41

42
46
51
51
51
53
58
59

59
60

63
63
68
75
76
83
84
87
89

89

9O
93

93

96

Chapter

 

Chapter

3. Number of Piaget-levels changed-—
Pre— t0 pOStteSt o o o o o 0

Summary . . . . . . . . . . .
Correlations . . . . . . . . . .

Correlations between Pretest Scores for
the Experimental Group and Perfor-
mance in the Treatment . . . . .

Correlations between Pretest Scores
(Both Groups Combined) and Variables
Measuring Academic Ability . . . .

Correlations between Change Scores for
Experimental Group and Performance
in the Treatment Course . . . .

Correlations between Change in Abstract
Reasoning for the Experimental Group
or Control Group Separately, and
Variables Measuring Academic Ability.

Summary of Correlations. . . . . .

Analysis of Variance . . . . . . .
Summary. . . . . . . . . . . .

V. SUMMARY AND CONCLUSIONS . . . . . . .

Purpose and Need for the Study . . . .
Limitations of the Study . . . . . .

Mortality of Subjects . . . . . .
Instrumentation Biases . . . . . .
Reliability . . . . . . . . .
Use of One-Item Tests . . . . . .

Conclusions . . . . . . . . . .

Levels of Abstract Reasoning Upon Entry
into the University . . . .
Effect of the Treatment--Statistica1
Results . . . . . . . . . .
Effect of the Treatment--Descriptive
Results . . . . . . . . . .
Correlations . . . . . . . . .
Interaction Effects-~Treatment vs.
Ability . . . . . . . . . .

vii

Page

100
102

103

104

106

108

108
111

112
116

119

119
120

120
121
122
122

125

125
126

127
127

128

Chapter

EFERENCI

Chapter Page

Implications for Further Research and

Instruction . . . . . . . . . . 130

Changes in the Instructional Format . . 130

Changes in the Research Design. . . . 131

Questions Left Unanswered . . . . . 132
Speculations . . . . . . . . . . 134

Summary . . . . . . . . . . . . 136
REFERENCES 0 O O O O O O O O O O O O O 139

viii

Table

10.

ll.

12,

 

LIST OF TABLES

Table Page
1. Summary of Piaget Studies . . . . . . . 49
2. Number of Subjects Assigned to Group and

Subsequent Mortality Prior to Testing . . 54
3. Grading Scale Used in the Treatment Course . 57

4. Mortality of Subjects Due to Incomplete
Testing 0 O O O O O I I O O O O 60

5. Criteria for Scoring Responses to the Pen-
dulum Problem . . . . . . . . . . 64

6. Criteria for Scoring Responses to the Equil-
ibrium in the Balance Test . . . . . . 64

7. Mean, Group Size, and Variance for Six
Measures of the Original Sample and
Final Sample. . . . . . . . . . . 71

8. T-test Values for Analysis of Differences
Between Original Sample and Final Sample . 72

9. Cut-Off Values by Academic Ability Variables
for Dividing Subjects into High and Low
Groups. 0 O O O O O O O I O O O 86

10. Values of t-tests for Analysis of Differ-
ences Between Groups' Mean Piaget-Level
by Test and by Measure . . . . . . . 91

11. Values of t-tests for Analysis of Differ-
ences Between Pretest and Posttest by
Group and by Measure . . . . . . . . 94

12. Number and Percentage of Subjects at Each
Piaget-Level by Group, Test, and Measure . 97

13. Percentage of Pre-Formal-Operational Sub-
jects by Test, Group, and Measure. . . . 99

ix

Table

14.

15.

l6.

17.

18.

 

 

Table Page

14. Number and Percentage of Piaget-Levels Changed
from Pretest to Posttest by Group and
Measure 0 C O O O I O O O O O O O 101

15. Pearson Product-Moment Correlations: Ability
Measure by Pretest . . . . . . . . . 105

16. Pearson Product-Moment Correlations: Ability
Measure by Change Score. . . . . . . . 109

17. F-Values for Analysis Using Change Scores:
Treatment by Ability Interaction. . . . . 114

18. Number of Ability Measures out of Eight for
Which Indicated Group's Change Score was
Greater . . . . . . . . . . . . . 115

Figure

 

In:

LIST OF FIGURES
Figure Page

1. Introduction to Reasoning and Problem
Solving: Course Objectives . . . . . . 56

xi

 

One
his
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as t

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deVElOpt
Surpris
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Such as
problem
riculum

studEnt

CHAPTER I

THE PROBLEM

One should conclude . . . we can aid a student in

his intellectual growth and that his intellectual

growth may be one of our ultimate accomplishments

as teachers. . . . (Beistel, 1975, p. 151)

American higher education is falling far short
of realizing its priority goal of educating intellectually
developed students. This failure is evidenced by the
surprisingly large percentage of college students who
cannot utilize strategies based upon logic and abstract
thought processes to solve problems basic to much of
their coursework. The inability to utilize such strate-
gies is especially troublesome for students in disciplines
such as engineering, where the ability to solve complex
problems is a prerequisite to a large part of the cur-
riculum. Success in engineering is most improbable for
students lacking these abstract reasoning skills.
Problem solving strategies based upon the work

of psychologists who have studied intellectual develop-
ment can be designed to promote the growth of logical,

abstract thought. Taught early in the curriculum, these

strategies can provide the necessary remediation for

 

 

 

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students deficient in reasoning skills so that they can
develop the intellectual capabilities required for
success in engineering.

In this chapter, it will be established that
although intellectual development is a major goal of
higher education in this country and although logical
and abstract reasoning capabilities are a prerequisite
to the study of engineering, a large percentage of
engineering students cannot reason on the basis of
logical, abstract thought. To remedy this situation,
it will be shown that, based upon the work of Swiss
psychologist Jean Piaget, it is possible to design
instructional strategies that directly teach students
to develop abstract reasoning and problem solving capa-
bilities.

The Problem-Intellectual
Development

 

 

Promoting the intellectual development of stu-
dents has long been a primary goal of higher education
in this country. A recent Carnegie Commission report
pointed out that one of the primary missions of universi-
ties should be to provide "opportunities for the intel-
lectual . . . development of individual students, and the
provision of campus environments which can constructively
assist students in their more general deve10pmental growth"

(”The Purpose and Performance," 1973, p. 7). Properly

select
inteli

The L
tude

—-—~

devot
ment.

as on

(n 5754-:r04-o

This
Opera
who c
entir
will
Opera

Opera

menta
The 1:

leel

selected, these "opportunities" should lead to the

intellectually developed student.

The Intellectually Developed
Student Defined

 

 

Jean Piaget, an eminent Swiss psychologist, has
devoted a lifetime to the study of intellectual develop—
ment. He defined the intellectually developed student
as one who can draw

. . . conclusions, not from a fact given in

immediate observation, nor from a judgment one

holds to be true without any qualifications, but

in a judgment which one simply assumes, i.e.

which one admits without believing it, just to

see what it will lead to. (Piaget, 1952, p. 69)
This type of reasoning is defined to be formal-
operational or abstract thinking. Thus, the student
who can reason from "A" to "B" even though "A" may be
entirely hypothetical, is intellectually developed and
will be referred to hereafter as an abstract or formal—

operational thinker. Students who are not formal-

operational will be referred to as pre-formal-operational.

Piaget's Theory

 

Piaget's theory is a description of the develop-
mental stages one moves through in cognitive growth.
The invariant sequence and the age associated with each

level are (Wadsworth, 1971):

 

 

It is
ment
of p]
an it
hiera

levei

the .
birt]
CORCI
beSII
find
two f
new 1
into

thOu:

thOuc
as We

to cc

USEd
e 0:
0f th

O. Sensori-Motor (0-2 years)
I. Preoperational thought (2-7 years)
II a-b. Concrete Operations (7-11 years)
III a-b. Formal operat1ons (ll-15 years)
It is these four stages that describe cognitive develop-
ment and level of abstract reasoning. Although the rate
of progress through these stages may differ according to
an individual's experience and heredity, the sequence is
hierarchical, i.e. level 0 is prerequisite to level I,
level I to IIa, etc.

During the sensori-motor period of development,
the infant moves from performing only reflex actions at
birth to coordinating vision and touch, developing the
concept of object permanence, becoming aware that objects
besides the self cause actions, and experimenting to
find new ways of solving familiar problems. At about
two years of age, the child begins to mentally invent
new means of solving problems, which permits movement
into the second stage of development: preoperational
thought.

The major development during the preoperational
thought stage is the acquisition of language. Language,

as well as behavior, moves from egocentric and nonsocial

to communicative and social. Thought is not restricted

 

1The numbering system 0, I, IIa-b, and IIIa—b is
used by Piaget and Inhelder in their descriptions. To
be consistent, they are used here and in the description
of the measurement instruments and their scoring.

 

by P
stag
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forn

chil

atta
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vert

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one
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is c
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by perceptions and motor activities as in the previous
stage, but is symbolic and representational, i.e.,
behaviors can be thought-out rather than actually per-
formed. Thought, however, is restricted, in that the
child cannot reverse operations or follow transformations.

During the concrete-operational stage, the child
attains the ability to reverse operations and attend to
transformations. Thought, for the first time, involves
logical operations, but they are limited to concrete
objects and problems rather than hypothetical or entirely
verbal ones. Further, the child develops the operations
of seriation and classification.

It is not until the formal-operational stage that
one can solve problems that are hypothetical and involve
the complete utilization of logical reasoning. Thought
is characterized by scientific reasoning and hypothesis
building and testing; problems are solved by logic
without relying upon the specific content. Logically
derived conclusions are seen as valid, independent of
actual truth. The child is able to derive general
theories and utilize them abstractly in solving problems.

These four stages of development are distinctly
different in terms of how children think and what their
abilities are. The process that brings about the changes
between stages is described by Piaget in terms of four

constructs: schema, assimilation, accommodation, and

eguilit
led to

are bri

structu
a perso:
new infc

schema d

 

 

 

if it i
ing one

tion. T
RECESSar
deveIOpm
Satisfie
balance ‘
A“ imbal
Provides
to OCCur
balanCe .
next

I as:

mUSt all

equilibrium (Wadsworth, 1971). These four constructs
led to hypothesizing the treatment under study. They
are briefly defined here and elaborated upon later.

Schema are defined by Piaget to be cognitive
structures that house all of the intellectual information
a person possesses. Assimilation is the process by which
new information is taken into existing schema. If a
schema does not exist for a new bit of information or
if it is incomplete, a new schema is added or an exist-
ing one is modified. This process is called accommoda-
tion. Together, assimilation and accommodation are
necessary but not sufficient conditions for cognitive
development to occur. The sufficient criterion is
satisfied when a state of equilibrium is reached and a
balance exists between assimilation and accommodation.
An imbalance between assimilation and accommodation
provides the internal motivation for cognitive growth
to occur because the child strives to reach a state of
balance. Thus, in order to move from one level to the
next, assimilation, accommodation, and disequilibrium
must all occur.

Intellectual Development: A Prerequisite
to the Study of Engineering

 

 

The ability to reason at the formal-operational
level is especially critical for students enrolled in

engineering curricula. As pointed out by Renner and

Lawson,

riculur

I

Thel
stu:
and

thi

We

can

fore
In c
forn
Opel
to j
leve
thej
lea;
18\n

IEdSOD

 

 

 

Thus ' m
of the

Courses

thinkin
the beg
summarj
finding
Subject
0f tho,
P. 262‘

I

SUQQeec

Lawson, successful mastery of a science-oriented cur-

riculum such as engineering requires the ability to

 

reason abstractly:

The maximum educational gain that comes from the
study of science is derived from the isolation
and investigation of a problem. Quite obviously
this involved the formulation and stating of
hypotheses and using a form of thinking which
can be described as, if . . ., then . . ., there-
fore. That is, of course, propositional logic.
In other words, science teaching should promote
formal thought. But it cannot do so if concrete
operational thinkers (i.e. pre-formal) are asked
to interact with science on a formal operational
level and their teacher teaches them as though
they think formally. Concrete operational
learners must interact with science at that
level; they cannot do otherwise. (1973a, p. 168)

Thus, mastery of engineering courses as well as mastery
of the prerequisite mathematics, physics, and chemistry
courses requires the ability to reason abstractly.
Students who have not yet developed formal
thinking capabilities are severely handicapped from
the beginning of their study of engineering. Kolodiy
summarized Griffith's study and concluded that "his
findings seem to suggest that if we are teaching formal
subjects to students who are only at the concrete level
of thought, basically nothing is getting through" (1974,
p. 262). Thus, concrete-Operational subjects cannot

succeed in engineering course work.

 

College
Reasoni

I
student
(Elkins

1974; ht

 

P9rcente
abstraci
(Elkind
SUbject.
AS Kohl,

Not
(65
the
dev
Thi
Hey
The
Of’
abs

Educat:

College Students and Formal
Reasoning AbiIity

 

 

Although no existing study addresses engineering
students in particular, it has been well documented
(Elkins, 1962; Karplus & Karplus, 1970; Lawson & Renner,
1974; McKinnon, 1970, 1971) that a surprisingly large
percentage of college students have not attained the
abstract level of reasoning. Studies report from 42%
(Elkind, 1962) to 78% (McKinnon, 1970) of the college
subjects studied were pre-formal-operational thinkers.
As Kohlberg and Gilligan point out in the summary of
their research:

Not until age 21-30 (does) . . . a clear majority
(65%) attain formal reasoning. . . . They (i.e.
the data) suggest that there is no further
development of formal reasoning after age 30.
This means that almost 50% of American adults
never reach adolescence in the cognitive sense.
The point, however, is that a large proportion

of Americans never develop the capacity for
abstract thought. (1971, p. 1065)

The literature clearly points out that higher
education's priority goal of intellectual development is
not being met. Because of the large percentage of
students who are pre-formal-operational, it is reasonable
to conclude that the "opportunities for intellectual
development" recommended by the Carnegie Commission and
presently utilized in higher education are extremely
inadequate.

Although a large percentage of college students

do not possess the abstract reasoning skills necessary

for sur
in its
reasoni
that th
between
Althougl

abstraci

 

study a,
A furthe
Of a grc
Course 5
Ahall/Sis
not $193
a large

formal

the for:
pOpulat

based u

for survival in engineering, the literature is limited
in its documentation of minority students' abstract
reasoning abilities. Karplus and Peterson (1970) found
that there was a large difference in reasoning abilities
between urban and suburban eleventh- and twelfth-graders.
Although 80% of the suburban students could reason
abstractly, only 9% of the urban students could. (The
study assumed the urban population was mainly minority.)
A further study by Griffiths (1973) found that only 39%
of a group of college students enrolled in a physics
course scored at the highest abstract reasoning level.
Analysis of racial subsamples indicated that they were
not significantly different from the total group, i.e.,
a large percentage of minority students were also pre-
formal thinkers.

Although no studies have been found which assess
the formal reasoning abilities of minority engineering
populations in particular, this research study is
based upon the assumption that minority engineering
students are deficient in abstract reasoning capabil-
ities. Although the study will actually determine
whether or not a group of minority engineering students
are formal thinkers, there is reason to believe they
are not. First, in 1972, 20 of 42 first-term minority
engineering students at Michigan State University were

enrolled in a remedial mathematics course, yet 40 of the

 

42 had

 

matherw
It is ;
tional

high so

Secondl

 

in Engi
student
Science
On Mine
if a la
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pefcent
in gene
centage

in fOn

centag,
OPQrat.
diSCip.
or the:

10

42 had completed at least three years of high school
mathematics (College of Engineering, MSU, 1974, p. 3).
It is doubtful that a student capable of formal-opera-
tional reasoning and who had completed three years of
high school mathematics would enroll in a remedial course.
Secondly, the nationally based Committee on Minorities
in Engineering has recommended that minority engineering
students need extensive instruction in "mathematics and
science learning techniques and processes" (Committee

on Minorities in Engineering, 1975, p. 23). Once again,
if a large percentage of these students were formal-
operational, they would probably not require such inten-
sive remedial instruction.

Because of these reasons and because of the large
percentage of pre-formal-operational college students
in general, it will be assumed that a significant per-
centage of minority engineering students are deficient
in formal reasoning skills.

The result of a situation in which a large per-
centage of minority engineering students are pre-formal-
operational is that either they leave engineering for
disciplines which require less rigorous reasoning ability,
or they manage somehow to develop the formal reasoning
skills necessary for survival. It is most probable that
students who lack formal reasoning abilities leave

engineering rather than develop the required skills.

 

Althou
of fac

. I
fami111
instrug
ment oi
that "a

lectual

 

instruc

levels"

ing
tha
abi
of
of

Hence,

facilit
Studem
P- 18).
than or
neering
reasonS
Sizes 1
to a la
likely
Skills

11

Although this cannot be officially documented, a number
of factors lead to this conclusion. First, educators
familiar with Piaget's theory claim that current
instructional procedures do not facilitate the develop-
ment of formal reasoning. Renner and Lawson stressed
that "at least some of this lack of development of intel-
lectual capabilities can be traced to inapprOpriate
instructional strategies at the secondary and college
levels" (1975, p. 1). Kolodiy concurred:

. . . present teaching techniques might be reach-

ing fewer than one-half of our students. All

that students are learning, apparently, is the

ability to parrot back material for the purpose

of attaining passing grades without learning any

of the concepts involved. (1974, p. 262)
Hence, current instructional practice does little to

facilitate the cognitive development of today's college

student. Second, Minorities in Engineering: A Blue-

 

print for Action (Alfred P. Sloan Foundation, 1974,

 

p. 18), pointed out that between one-third and more

than one-half of the minority students who enter engi-
neering do not earn a degree. Although there are many
reasons for this high attrition rate, this study hypothe-
sizes that at least part of the problem can be attributed
to a lack of formal reasoning ability. Thus, it is
likely that minority students lacking formal reasoning

skills leave engineering for other majors.

Increa:

1n Engg

neerin;
efforts
to incr

dation,

 

 

 

minorit-
1982.

by 1982
enginee
ing. N
rESOUrc
embarre
Stood E

of reme

minOrit

on Min:

12

Increased Minority Involvement
in Engineerifig,

 

 

It is estimated that 5.1% of freshmen in engi-
neering colleges in 1973 were minorities, while national
efforts in minority engineering education are striving
to increase this to 18% by 1982 (Alfred P. Sloan Foun-
dation, 1974). This represents an increase from 2,640
minority students in 1973 to approximately 13,500 in
1982. With an attrition rate of one-third to one-half,
by 1982, from 4,500 to 6,750 of the entering minority
engineering students will not earn a degree in engineer-
ing. Not only is this a tremendous waste of human
resources, time, and money, it is also an educational
embarrassment that the phenomenon is not better under—
stood and that at present there appears to be no means
of remedying the situation.

The problem of the academic preparation of
minority students was also pointed out by the Committee
on Minorities in Engineering:

What's more, as the most economically and edu-
cationally disadvantaged, these four minorities
(Black, American Indian, Mexican American and
Puerto Rican) often lack the rigorous academic
preparation for engineering school. . . . (1975,
PP- 1-2)
To address the problem, the committee recommended effec-
tive support programs, including methods to improve

academic course success, teaching learning skills that

relate to study habits, and teaching academic skills

 

|
relate
should

and pr

ation,

 

college
Prepar
have 6

P- Slc

dilenm
manr;
p001 ¢
POSse

engin

13

related to analytical processes. Included in this
should be "mathematics and science learning techniques
and processes . . ." (1975, p. 23).

Also addressing mathematics and science prepar-

ation, Minorities in Engineering: A Blueprint for

 

Action stated that "even the seemingly well-prepared
college-bound minority youth feel insecure about their
preparation in mathematics and science and expect to
have difficulty with these subjects in college" (Alfred
P. Sloan Foundation, 1974, p. 18).

Thus, engineering educators are faced with a
dilemma: national efforts are calling for expanded
minority involvement in engineering, yet of the potential
pool of minority students, a large percentage does not
possess the cognitive skills necessary for success in
engineering.

Instructional Strategies that

Facilitate Intellectual
Development

 

 

 

In order to address the problem of insufficient
minority involvement in engineering, the College of
Engineering at Michigan State University received a
three-year grant from the Alfred P. Sloan Foundation
to "address the problems of ethnic minority students
who have not mastered the technical skills necessary
to enter traditional engineering programs" (College of

Engineering, 1974, p. 3). The grant proposal stated:

 

'Tv"
J(I)
”
one:
-:3m
”I”

OH

to de
deve
rese
ins:
Of 1
ing.
Sci

thi

Saa

an

in

re,

as

an,

14

We have also experienced the problem of students
having inadequate technical backgrounds to enter
engineering directly from high school. In the
1972 fall term, twenty of the forty-two ethnic
minority engineering freshmen were enrolled in
remedial mathematics; forty of the forty-two had
completed a minimum of three years of high
school mathematics. Typically, a significant
number of such students . . . take much longer
in the technical foundation courses due to
repeats, and tend to become discouraged with

the academic rigor rather early in their college
careers. (1974, p. 3)

One of the major goals of the Sloan Project was
to develop instructional strategies to assist students in
developing abstract reasoning abilities. However, little
research has been conducted that isolates effective
instructional strategies that facilitate the development
of formal reasoning. McKinnon (1970) pointed to an
inquiry-oriented strategy for instruction in a general
science course and Renner and Lawson (1975) also found
this technique useful in a science education course.
Saarni's (1974) work, although not directly related to
an instructional strategy, indicated a high positive
correlation between problem-solving ability and abstract
reasoning, leading to the speculation that instruction

in problem solving may result in increased abstract

reasoning skill.

The Treatment

 

This section will describe a course developed
as part of the Sloan Project, "Introduction to Reasoning

and Problem Solving," the instructional strategy designed

 

 

COnten

Cation
varior

the OE

Ia) p
ing p
info:
drawi
apply
Prob]
Prob:
Stra
were
appi
conI
9&1

Pro,

15

to facilitate cognitive development. Both the content
and the instructional method will be described and a
theoretical basis for each will be discussed.

The Instructional Strategy:
Content

 

Saarni's work (1974) provided the primary moti-
vation for selecting problem solving as the particular
content area for the instructional strategy. The impli-
cation of her study was that direct instruction in
various problem-solving strategies should facilitate
the development of abstract reasoning.

The content of the course fell into two areas:
(a) preparing for problem solving, which included restat-
ing problems, defining unknown terms, specifying given
information, deciding what is to be solved for, and
drawing and labeling diagrams; and (b) learning and
applying specific strategies to.be used as methods in
problem solving, including the simplification, sub-
problem, contradiction, inference, and working backward
strategies. The preparation for problem-solving skills
were included because they were prerequisite to the
application of the specific strategies. Because the
course was designed for students with limited mathemati—
cal background, the content and problems did not involve

proofs or problems from theoretical mathematics, but

ratbe;
did n:

simple

define

it out

 

 

cation
inag
Proble
origin
includ
and se
Sizes
t0 the
a 2X2
Final.‘

genere

16

rather included the puzzle variety of problems, which
did not require mathematical manipulations beyond
simple arithmetic.

Each of the five problem—solving strategies was
defined and explained by listing the steps used to carry
it out as an algorithm. The algorithm for the simplifi-
cation strategy instructed students to reduce the numbers
in a given NxM array, attempt to solve the reduced
problem, and generalize this method of solution to the
original problem. For example, suppose the problem~
included a 5x7 array (i.e., a rectangle with five rows
and seven columns) and asked how many rectangles of all
sizes were included in the 5x7 array. First, according
to the algorithm, the numbers must be reduced to, say,

a 2x2 array, then this reduced problem is solved.
Finally, the method used to solve the 2x2 problem is
generalized to solve the 5x7 problem.

Thus, the algorithmic approach provided an
organized method of adapting to problem solving.
According to Wadsworth (1971, p. 9), Piaget views cog-
nitive acts as acts of organization and adaptation to
a perceived environment. By providing students with
an algorithmic strategy for solving a particular class
of problems, the course provided this organization.

In addition, the strategy is a schema in the Piagetian

sense and functions in one of two ways: either it fits

 

and

0f

for
Seq
prc

prq

atl
ad

17

into an existing schema or it creates a sense of dis-
equilibrium because it does not fit. In the latter case,
students are motivated to change and accommodate, thus
enhancing cognitive growth.

The subproblem strategy utilized an algorithm
that helped students identify parts of a given problem
that must be solved before the major variable could be
solved. The following is a simple example of this type
of problem:

What is the total income for the day if 3 hats

were sold in the morning grossing $18 and twice

as many hats were sold in the afternoon for $1

less than the morning price?
In order to solve this problem, the student must add
morning income to afternoon income to obtain the answer.
However, the value for "afternoon income" is not directly
available from the problem statement. One must first
figure out the cost per hat in the morning (X) and from
this determine the cost per hat in the afternoon (Y). X
and Y are each examples of subproblems. The major aim
of this strategy was to provide students with a procedure
for analyzing problems in order to identify unknowns and
sequence the order in which they must be solved. The
procedure results in an organized plan for solving the
problem.

The algorithm for the contradiction strategy

also provided students with an organized plan. In

addition, the strategy was based upon reasoning with

propc

 

regar
Accor;
reaso:

criti

 

algori
Solvir
what i
that
assun
leis
Opera
baSi:
real
PrOv
COnf
Crea

to O

0ft

algo
tc>

to

back

18

propositional logic on the basis of an assumption,
regardless of the truth or falsehood of that assumption.
According to Piaget (1952, p. 69), the ability to

reason on the basis of hypothetical assumptions is a
critical attribute of formal-operational reasoning. The
algorithm specified that the assumption to be used in
solving the problem should be the logical negation of
what was to be proved. Thus, if the problem was "Show
that all even integers are divisible by two," then the
assumption was "There is an even integer that is not
divisible by two." According to Piaget, the pre-formal-
operational thinker cannot accept this assumption as a
basis for reasoning to reach a contradiction because, in
reality, it is untrue. Hence, the contradiction strategy
provided the opportunity for the pre-formal thinker to be
confronted with hypothetical assumptions that would
create the disequilibrium necessary for cognitive change
to occur.

The inference strategy was primarily an extension
of the contradiction strategy. Although no specific
algorithm was presented, instruction was based upon how
to obtain (infer) additional information from the givens.

The working backward strategy taught students how
to solve problems starting at the solution and working

backwards to the givens as opposed to the traditional

 

 

COurs

catic
wOrkj
of Se
abi1:
In a:
acts
Prov
COHt
tioh

tiQr

T
‘33;

8.31

are:

19

approach of starting with the givens. For example, the
following problem is much easier to solve working back-
wards:

Describe a sequence of emptyings and fillings of

a 3 qt. jar and a 7 qt. jar to obtain 5 qts. of

water.
As opposed to starting with three and seven quarts of
water and describing how five quarts are obtained, this
algorithm directed students to begin with five quarts
and work backwards until three and seven quarts are left.

Thus, the major emphasis of the content of the

course was five problem-solving strategies: simplifi-
cation, subproblem, contradiction, inference, and
working backward. This content area was selected because
of Saarni's implications that increased problem-solving
ability leads to improved abstract reasoning ability.
In addition, Piaget maintains that cognitive acts are
acts of organization and adaptation; the algorithms
provided this necessary organization. Finally, the
contradiction strategy was included to provide instruc-
tion on reasoning on the basis of hypothetical assump-

tions.

The Instructional Method

 

The instructional method used was the mastery
self-paced model. The critical features of this model
are: (a) material to be learned was divided into small

learning units called modules, which can be completed

in a ;

 

incluo
mater:
allowe
Speed,
learni
Object
level

requiI
Prerec
modul,
achie.
Small

numbE
(Kulj
Self
lest
tjje
eQu]
is
Of
mat
in;

20

in a relatively short period of time. Each module
included objectives to be mastered, printed instructional
material and practice exercises. (b) Students were
allowed to progress through the modules at their own
speed, allowing for individual differences in rate of
learning. [(c) Students were expected to master the
objectives in each module by performing at a specified
level on each modular exam. Theoretically, this mastery
requirement insures that students understand all material
prerequisite to later modules. (d) Retesting on each
module was allowed without penalty until mastery was
achieved. (e) Extensive use was made of one—to—one or
small group tutoring which permitted diagnosis and
remediation of individual problems.

The mastery self-paced model was chosen for a
number of reasons. First, it has been well documented
(Kulik, 1975) that in 25 out of 31 studies, the mastery
self-paced form was statistically superior to traditional
lecture methods in terms of both student performance on
the final examination and long-term retention. If the
course "Introduction to Reasoning and Problem Solving"
is to have an impact on the formal reasoning abilities
of students, they must learn and retain the course
material. It is well established that self-paced learn-
ing increases significantly the probability that students

will learn and retain the material. K. Patricia Cross

 

put 1

 

criti

(1976-

was be

 

create
and tr
as Cr;
OCCUI
and.t;
Chile:
Ch3n<
017 at

21

put it bluntly: "I believe mastery learning is the
critical missing link in the education of low achievers"
(1976, p. 78).

The second reason the self-paced model was chosen
was because of its reliance on individual tutoring, which
creates an environment of increased social interaction
and transmission, one of four factors Piaget pointed out
as critical for cognitive development. Since tutoring
occurs between students, between students and tutors,
and between students and instructor, interaction with
others is increased. As a result, there is a greater
chance that more can be assimilated into existing schema
or accommodated into new schema. This occurs merely
because of the increased amount of information flow,
i.e., the greater the transmission of new information,
the greater the occurrence of assimilation and accommo-
dation. Accordingly, there is a greater chance for dis-
equilibrium to occur, and subsequently, cognitive change.
Hence, the self-paced approach, which permits increased
social interaction, should facilitate cognitive develop-
ment.

The third factor of the self-paced method that
led to its selection was that it provides the opportunity
for students to act upon objects in order to abstract
their properties. This factor was also pointed out by

Piaget as critical for the development of formal

 

reasor
defini

and nc

In add
chance
accept
0f the
cannot
someth
Strate
thESe

feeaba
Thus,

by kXD1

faci1:

ChOSe:
termS
it (Zr
and t

lems

22

reasoning. The self-paced model of instruction demands by
definition that the student act; there are no lectures
and no way to learn unless the student does something.

In addition, the repeated testing feature provided a
chance for continued action until the standard of
acceptable performance was met. Also, the very nature

of the subject matter induced students to act: problems
cannot be solved unless students act. They must do
something to the problem to solve it. The specific
strategies attempted to provide guidelines for what

these actions should be and the homework exercises and
feedback provided the Opportunity for students to act.
Thus, the opportunities to act upon objects brought about
by both the instructional format and the content should
facilitate cognitive growth.

In summary, the mastery self-paced model was
chosen because it is superior to the lecture method in
terms of student performance on final exams and retention;
it creates an environment conducive to social interaction
and transmission; and it forces students to act upon prob-

lems rather than passively receive information about them.

Statement of the Problem

 

It is a priority goal for higher education to
provide opportunities for students to develop formal
reasoning capabilities. For engineering students in

particular, formal reasoning skills are critical for

 

maste

succe
of mii
these
resea
that

Hence

ment .

Mich.
Nati

Unit

f
&
19V:

91.01

Ff
:r
p.

0
ET

23

mastery of an engineering curriculum; without them

success in engineering is improbable. A large percentage
of minority engineering students, however, do not possess
these necessary formal reasoning abilities. In addition,

research in the area of effective instructional strategies
that promote the development of formal thought is limited.
Hence, instructional strategies that facilitate develop-

ment of formal thought need to be designed and tested.

Terms and Their Definitions

 

Minority student--a student who is classified in

 

Michigan State University records as Black, Chicano,
Native-born or Asian-American and is a citizen of the

United States.

Level of abstract reasoning! level of cognitive

 

functioning or Piaget—level--refers to one of the four

 

levels in the following invariant hierarchy of cognitive
growth as defined by Piaget:
O. Sensori-motor
I. Preoperational thought
II a-b. Concrete operations

III a-b. Formal operations

A formal-operational student or a formal

 

thinker--refers to a student who functions at level IIIa

or b, formal operations.

studa

t0 t1”;

in th

tiOn,

 

 

 

 

inStrw

forma

an

24

Pre-formal-operational student—~refers to a

 

student who functions at levels 0, I, IIa, or IIb.

Abstract reasoning or abstract thinking—~refers

 

to the abilities of a formal-operational student.

Problem solving--refers to the behaviors specified

 

in the simplification, subproblem, inference, contradic-
tion, or working backward strategies utilized in the

instructional treatment.

Intellectually developed student--refers to a

 

formal-operational thinker.

Mastery self-paced instructional model (also

 

referred to as the self-paced or mastery model)—-a

 

method of instruction that divides the material to be
learned into small learning units such that students are
allowed to progress through each unit at their own speed.
A mastery level is set for performance on each unit exam,
and retesting is allowed without penalty until the
mastery level is attained. Extensive use is made of

tutoring.

Purpose
The purposes of this study are:
1. to design an instructional strategy based

on Piaget's theory of cognitive development such that

the i

forma

tiona

devel<

aSSESS

 

Cogni‘
data I
scoPe
fOr d
Suppo
wheth

thong

to RE
ment
Groug
(Expc
were
leve:
in a?

Ses

25

the instruction will facilitate the development of
formal reasoning;

2. to assess the effectiveness of the instruc-
tional strategy;

3. to provide recommendations for further
development of instructional strategies based on the
assessment in (2); and

4. to measure minority students' entry-level
cognitive abilities in order to (a) provide descriptive
data on this population that could be useful beyond the
scope of this study in providing the college information
for developing further curricular, advising and general
support programs for minority students, and (b) assess
whether or not the age criteria for entry into formal

thought as specified by Piaget is met by this population.

Research Hypotheses

 

In order to determine if the course "Introduction
to Reasoning and Problem Solving" is an effective treat-
ment for increasing abstract reasoning ability, a control
group (no exposure to the course) vs. experimental group
(exposure to the course) design was used. Both groups
were pre- and posttested to measure individual student's
level of reasoning according to Piaget's theory. "Change
in abstract reasoning ability" referred to in the hypothe-

ses is defined to be the difference between pretest and

postt

 

IGSEd

is 91

in th

 

stude1
measu
reasc
Chang

Of a:

foll

U)
/rr
C
n.

4
w
H

H
m
w

0
Sr
/.

26

posttest scores. A more complete description of the
research design, the pre- and posttests and the scoring
is given in Chapter III.

Three general research questions are addressed
in this study:

1. Was the course effective in increasing
students' abstract reasoning abilities?

2. Are there any relationships between various
measures of academic ability and level of abstract
reasoning upon entry into the university or amount of
change in abstract reasoning after one term of instruction?

3. Is there an interaction effect between level
of academic ability and treatment?

The null hypotheses for these research questions
follow. A complete listing of the hypotheses appears in

Chapter III.

Question l--Was the course effective in increasing

students' abstract reasoning abilities?

 

Hypotheses 1-2:

 

There will be no significant difference in change in
abstract reasoning between the control and experi—
mental groups.

Question 2--Are there any relationships between

 

various measures of academic ability and level of abstract

 

reasoning upon entry into the university or amount of

 

change in abstract reasoning after one term of instruction?

 

 

 

In:

0911mm

III?

dgnrfl

the

reas

 

27

Hypotheses 3-10:

 

For the experimental group only there will be no
significant correlation between pretest scores
measuring level of abstract reasoning and each of
four measures of performance in the treatment.

Hypotheses 27-34:

 

For the experimental group only, there will be no
significant correlation between amount of change in
abstract reasoning and each of the four measures

of performance in the treatment.

Hypotheses 11-26:

 

For both groups combined, there will be no signifi-
cant correlation between pretest scores measuring
level of abstract reasoning and each of eight
measures of academic ability.

Hypotheses 43- 50 and 59- 66 (Control) and 35- 42
and 31- -§9flTExper1menta1):

 

 

There will be no significant correlation between
amount of change in abstract reasoning and each
of eight measures of academic ability.

Hypotheses 3-10 and 11-26 provide an analysis of
the correlations between pretest level of abstract
reasoning ability and various ability measures. For
significant correlations, the ability measures could
be used as predictors of abstract reasoning ability.

The remaining hypotheses provide correlations that, if
significant, could be used to predict amount of change
in abstract reasoning after one term in college for both
the group receiving the treatment and the control group.

Such descriptive data are useful as a first step toward

 

idem:

abil:

 

 

&

28

identifying variables that predict formal reasoning
ability and can assist faculty in planning student
schedules that reflect the needs of each individual.
Below is a list of the performance variables
for the treatment course and a list of the measures of

academic ability:

 

 

Course Performance Academic Ability

1. grade received 1. high school grade point

2. total points earned 2. MSU first term grade

3. final exam score point

4. score on questions 3. MSU arithmetic placement
on final requiring score
application of a 4. MSU mathematics placement
problem-solving score
strategy. 5. ACT/SAT mathematics

percentile

6. ACT/SAT verbal percentile

7. ACT/SAT composite per-
centile

8. grade received in mathe-
matics course taken.

Question 3--Is there an interaction effect

 

between level of academic ability and treatment?

 

Hypotheses 67-82:

 

There will be no significant difference in change
in abstract reasoning between high and low ability
students in the control or experimental group.
Summary
The thesis of this research is that (a) a
priority goal of higher education in this country is

to provide students with opportunities for the develop-

ment of formal reasoning as defined by Piaget; (b) that

 

I
stude
ing;

engin

 

neces
instr'

ment

a Sou:
gies.
area

Hence
teat}
IEas(

to d.

a gr,

form
drug
tive
desc
the
datj
Iv'

29

students must be formal thinkers to succeed in engineer-
ing; (c) that a surprisingly high percentage of minority
engineering college students cannot reason at levels
necessary for success in engineering; and (d) that
instructional strategies that facilitate the develop—
ment of formal reasoning need to be designed.

Piaget's theory of cognitive development provides
a sound basis for the design of such instructional strate-
gies. In addition, there is evidence that a content
area for such instruction is problem-solving skills.
Hence, it is hypothesized that a course designed to
teach problem solving will increase students' abstract
reasoning abilities. The major purpose of this study is
to determine if such a course has the desired effect on
a group of first-term minority engineering students.

In Chapter II the literature related to the
formal reasoning ability of college students is reviewed
and instructional strategies that have been found effec-
tive in developing formal reasoning capabilities are
described. The design of the study, the analysis of
the data, and a summary of this research and recommen-
dations for further study are presented in Chapters III,

IV, and V, respectively.

 

 

deter
Reasc
ment

tarm

Univ:
deve.
engi
engi
reas
Engi
proh
rict
with

piac

CHAPTER II

REVIEW OF THE LITERATURE

One of the central purposes of this study is to
determine if the self-paced course "Introduction to
Reasoning and Problem Solving" facilitates the develop-
ment of formal-operational thought in a group of first-
term minority engineering students at Michigan State
University. Instructional strategies promoting the
development of formal reasoning capabilities of minority
engineering students are needed because (a) success in
engineering is improbable for the student without formal
reasoning abilities; (b) a large percentage of minority
engineering students are not formal thinkers and most
probably cannot survive the rigors of engineering cur-
ricula; and (c) limited research has been conducted
with the aim of isolating instructional strategies that
facilitate formal reasoning development.

In this chapter it will be shown that although
Piaget maintains that formal thought capabilities should

be developed sometime between the ages of ll and 15,

30

 

 

reVl!

t0 p

abi]
ACCC
Shox
mail
in

31

a large percentage of students are not formal thinkers
by age 15. In particular, it will be shown that a sur-
prisingly high percentage of college students are not
formal-operational. This section of the literature
review will be divided into five categories on the basis
of age: (a) pre-college students--general; (b) pre-
college students enrolled in science courses; (c) col-
lege students; (d) college-age and older subjects not
necessarily enrolled at a university; and (e) minority
students.

In addition, the last section of this chapter will
review the research on instructional strategies designed
to promote the development of formal reasoning.

Piaget's Age Specification for Entry
into Formal Operations

 

 

Success in an engineering program requires the
ability to reason at the highest level of abstraction.
According to Piaget, freshman engineering students
should have no difficulty with abstract reasoning; he
maintained that formal operations should be evidenced
in adolescents sometime between the ages of 11 and 15
years:

The same unity of behavior encountered earlier in
the various stages is found again between eleven
or twelve and fourteen or fifteen, when the sub—
ject succeeds in freeing himself from the concrete

and in locating reality within a group of possible
transformations. This final fundamental

 

 

forma

Of cc

Cifi
in a
grou

57%

32

decentering, which occurs at the end of childhood,
prepares for adolescence, whose principal charac-
teristic is a similar liberation from the concrete.
(1969, p. 130)
After age 15 at the latest, students should have the
mental prerequisites for engineering study. The follow-
ing review of the literature will indicate that at
least with respect to American students, very few are
formal thinkers by age 15 and that a large percentage
of college students cannot reason abstractly.

Research Concerning the Development
offiFormal Reasoning

 

 

Pre-college Students--
General

 

Evidence to refute Piaget's 11-15 year age spe-
cification for entry into formal thought was indicated
in a study by Higgins-Trenk and Gaite (1971). For a
group of 13-18 year olds, the researchers found that
57% were pre-formal thinkers. They concluded:

Most writers . . . have tended to tacitly
accept Piaget's descriptions of intellectual
development and have been content to describe the
adolescent as typically attaining the formal-
operational stage of thought during early
adolescence, and then to leave the reader with
the distinct impression that such adolescents
thereafter habitually function at that level.
However, . . . this is altogether too simplistic
a view of the typical adolescent's thought pro-
cesses and intellectual functioning.

Thus, . . . formal operational thought may be
a relatively foreign experience to the normal
adolescent.

What is clear is that . . . relatively few
adolescents seem capable, or inclined, to use
this formal mode of thought. This suggests that

 

 

M.V'——

1+ (W

 

to th

 

 

Stude
and 6
think
Studs

inte:

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33

there is need for considerable revision of the
commonly held view that the normal adolescent
attains the level of formal operations soon
after pubescence.

Indeed, it would seem that the normal

adolescent is unlikely to reach the level of
formal-operational thought until his late teens
or early twenties if he reaches it at all, and
these results suggest that he may well not.
(1971, pp. 201-202)

Lawson and Renner (1974) reported similar results

to those of Higgins-Trenk and Gaite in a study of Oklahoma

students. Seventy-one percent of the 99 eleventh graders

and 66% of the 97 twelfth graders studied were not formal

thinkers. To determine level of cognitive development,

students were administered the following tests during

interviews of approximately 20 minutes:

1.

tests to assess concrete operational students--
a) Conservation of Solid Amount--determined
understanding of the principle that a change

in shape does not alter the amount of

material; b) Conservation of Weight--determined
understanding of the principle that a change in
shape of one of two objects of equal weight
does not change weight of object; c) Conser-
vation of Volume using c1ay--determined under-
standing of the principle that a change in
shape of an object does not change its volume.
tests to assess formal-operational students--
a) Conservation of Volume using Metal Cylinders--
test for understanding of the principle that
water displacement is a function of volume, not
mass; b) Elimination of Contradiction--test to
determine if an apparent contradiction could be
explained; c) Exclusion of Irrelevant Variables--
test to determine if irrelevant factors could
be identified, controlled and excluded in
explaining results of experiment.

In an extensive study of six different groups of

subjects ranging from fifth and sixth grade students to

adult physics teachers, Karplus and Karplus (1970)

-_—

%

foun
subu
cour

was ‘

but I

Abili

 

 

 

 

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and I

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tes;
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‘31

34

found that 86% of the tenth through twelfth grade
suburban students enrolled in college preparatory
courses were pre-formal-operational. The test used

was the Island Puzzle: given four islands A, B, C, and
D, if planes could go between C and D and between B and D,
but not between A and B, can they go between A and C?
Ability to solve this problem required application of
the contradiction strategy (American Association of
Physics Teachers, 1975). The results were consistent
with the 57% pre-formal thinkers found by Higgins-Trenk
and Gaite and the 66% to 71% reported by Lawson and
Renner.

In contrast to these results, however, were
those reported by Karplus and Peterson (1970). They
used the ratio test to measure the formal thinking
ability to urban and suburban sixth graders and of urban
and suburban eleventh and twelfth graders. In the ratio
test subjects were asked to measure the height of an
object in a specified unit of measure if they knew both
the height of the object in a different unit of measure
and a conversion factor between the two units. Although
the difference in reasoning ability of the two sixth
grade groups was statistically nonsignificant, the
researchers found a very large difference between the
two eleventh and twelfth grade groups. Twenty percent
of the suburban students were classified pre-formal on

the basis of the ratio task, while 91% of the urban

35

students were pre-formal. The results for the suburban
students indicated a surprisingly lower percentage of pre-
formal—operational thinkers than the results reported by
the other studies mentioned. However, as will be explained
in Chapter II, these results are questionable due to the
use of the Ratio Test as the measurement instrument.
Utilizing principal components analysis, Lawson and Renner
(1974, p. 556) found that the Ratio Test as well as the
Island Puzzle possibly did not measure formal reasoning.
Therefore, it is likely that Karplus and Peterson's study
did not accurately report the formal reasoning ability of
their subjects. In addition, Karplus and Karplus' results
were questionable due to the use of the Island Puzzle.

Thus, three studies indicated that a large per-
centage of high school students had not mastered formal-
operational tasks by age 15, while a fourth study con-
curred with this conclusion for urban students, but
indicated a much lower percentage of pre-formal thinkers
for suburban students. This later study, however, used
the Ratio Test, a questionable instrument for measuring
formal reasoning.

Pre-college Students-~Enrolled
in a Science Course

 

 

An interesting subset of high school students
was the group enrolled in science courses. As was

mentioned previously, Lawson and Renner (1974) found

 

 

that
in c
cate
only
cour

were

 

cent.
large
Whom
stud
stud

form

test
Were
Cour
uan
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SCOI
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rela

rela

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36

that from 66% to 71% of eleventh and twelfth graders

in general were classified below the formal-operational
category. Additional research, however, indicated that
only 4% of the students who enrolled in a chemistry
course and 3% of those who enrolled in a physics course
were pre-formal-operational. A somewhat higher per-
centage (29.5%) was reported for biology students. This
large difference between students in general (some of
whom could have been enrolled in a science course) and
students enrolled in a science course indicated that
students enrolled in a science course perform better on
formal reasoning tasks.

To defend against the criticism that Piaget task
tests gave science students an advantage because the tests
were based upon the content of physics and chemistry
courses, Lawson and Renner (1974) referred to a 1974
unpublished study by Lawson, Norland, and Devito. The
study showed low correlations between Piaget task test
scores and CEEB Achievement exams in science, math, and
English. For the Conservation of Volume test the cor-
relations were 0.32, 0.30, and 0.38 respectively, and
for the Exclusion of Irrelevant Variables the cor—
relations were 0.32, 0.37, and 0.33. These results
supported the assertion that at least these two Piaget

tests were content free.

 

 

print

Laws
Rati
tior
Sin.

it

On

En:
Ra.
in

0f

ix

ft

37

In contrast to Lawson and Renner's results,
Karplus and Karplus (1970) found that 72% of suburban
twelfth graders enrolled in a physics course were pre—
formal thinkers. The discrepancy between the results
of these two studies can also be explained by the
principal components analysis results of Renner and
Lawson (1974). As was pointed out, in addition to the
Ratio Test, the Island Puzzle was found to be a ques-
tionable instrument for measuring formal reasoning.
Since Karplus and Karplus used only the Island Puzzle,
it is doubtful that their results were accurate.

In summary, although the two studies reporting
on formal reasoning ability of pre-college students
enrolled in a science course were contradictory, the
Karplus and Karplus study was questioned because of
inadequate testing procedures. Therefore, on the basis
of Lawson and Renner's results, it can be concluded that
a very large percentage of pre-college students enrolled

in a science course were formal-operational.

College Students

 

In a study involving college students, Elkind
(1962) found that 42% were pre-formal thinkers as
measured by the Conservation of Volume task. When the
initial group was subdivided by sex, a significant dif-
ference was found between the performance of men, 26%

of whom were pre-formal thinkers, and the performance

 

 

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and

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38

of women, of whom almost twice as many (48%) were pre-
formal. In addition, the percentage of subjects who
could successfully conserve volume increased regularly
with age for women, but not for men. This large dif—
ference between male and female subjects, which is found
consistently throughout the literature, could be
explained by our sex-stereotyped culture, in which males
are rewarded for successful performance on mathematical
and scientific problems while females are consistently
not rewarded.

Tower and Wheatley (1971) attempted to replicate
Elkind's 1962 study on college students, but their sample
included only females. They found that 39% of the 71
females aged 17-27 could not conserve volume and were
therefore considered pre-formal. These results are very
close to the 42% pre-formal thinkers reported by Elkind
for both men and women, but somewhat less than the 48%
he found for the women-only subgroup.

As was the case with high school students, an
interesting subgroup of college students was those
enrolled in a science course. Griffiths (1973) studied
chemistry and physics students at a state university and
an inner-city community college and found that 61%
scored below level IIIb, the highest of the two formal
reasoning classification levels. He concluded that

a majority of students enrolled in introductory science

COU.

 

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def '
the
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and
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MC]
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39

courses have a major handicap in the development of
their cognitive structures.

McKinnon (1970) reported interesting results on
college freshmen enrolled at the University of Oklahoma.
Of 143 students tested, he found that only 22% were
definitely formal thinkers, 27% were "probably" beyond
the concrete level, and 51% were concrete. When the
groups were divided by sex, he found that 44% of the
males were classified formal, while only 8% of the
females were formal. Approximately the same percentage
were "probably" beyond concrete--27% of the females
and 28% of the males, while 28% of the males were clas-
sified concrete and 65% of the females were concrete.
McKinnon's finding that 28% of the males were not formal
thinkers was very similar to Elkind's result of 26%.
However, McKinnon's results for the percentage of pre-
formal-operational females (65%) was much higher than
either Elkind's (48%) or Tower and Wheatley's (39%).

In summary, the research on the formal reasoning
abilities of college students clearly indicated that a
large percentage were not formal-operational. Studies
involving both males and females reported from 42% to
51% of the subjects were not formal-operational and
Griffiths showed that 61% could not reason at the
highest formal reasoning level. Studies by sex reported
the range of pre-formal thinkers for females was 39% to

65% and 26% to 28% for males.

 

 

 

 

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repc
majc
Thu:
rea
of
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fr
02:
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40

College-age and Older Subjects--
Not Necessarin Enrolled”1n a
University

 

 

Two additional studies researched the formal
reasoning capabilities of adults and young adults.
Subjects were not identified on the basis of college
attendance. Kohlberg and Gilligan (1971, p. 1065)
reported that not until the ages of 21 to 30 was a clear
majority (65%) of subjects classified as formal thinkers.
Thus, 35% of the subjects were deficient in abstract
reasoning. They also reported no further development
of formal reasoning after age 30 and concluded that
"the point, however, is that a large proportion of
Americans never develop the capacity for abstract
thought" (1971, p. 1065).

In addition, two groups analyzed in the Karplus
and Karplus (1970) study cited earlier revealed that
from 60% to 84% of the subjects were not formal-
operational. In a group of participants at a conference
on Piaget's work, 84% of the subjects were found to lack
formal reasoning capabilities. For the other group--
participants at an American Association of Physics
Teachers conference--60% were not formal. A surprisingly
high percentage of people who must teach physics, a
subject which is based primarily upon formal thought,
cannot reason at levels expected of the students enrolled

in their courses.

 

 

you

ing"

 

and

him

 

out
has
par
es"
StL‘
anc‘
ti:
91!
St1
ea:
in.
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to
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Wa:
re‘
a 1

Was

41

Hence, the two studies reported on adult and
young adult populations revealed a lack of formal reason-
ing abilities. For the three groups analyzed, 35%, 60%,

and 84% of the subjects were not formal-operational.

Minority Students

 

McKinnon and Renner (1971, p. 1048) have pointed
out that there is "no indication that his [Piaget] work
has been extended to include entering college students,
particularly American students." This statement is
especially true for minority college students; only two
studies reported findings concerning "urban" students
and in neither case was the sample defined to be dis-
tinctly minority. Karplus and Peterson (1970) reported
91% of the urban eleventh and twelfth graders in their
study were not formal-operational, but as was discussed
earlier, these results were questionable due to an
inadequate measurement instrument. Griffiths (1973)
reported that 61% of the college subjects in introduc-
tory chemistry and physics were not functioning at
level IIIb, the highest formal-operational level. The
only relationship to minority students in this study
was that analysis on the basis of "racial" subsamples
revealed no difference from the original sample, i.e.,

a large percentage of students in the racial subsamples
was also performing below the highest formal-operational

level.

 

 

reas

tan;

of t

 

 

 

tior

for:

42

The research to date only addressed the formal
reasoning capabilities of minority college students
tangentially. There is a clear need for an analysis
of the abstract reasoning abilities of this population.

Research on Effective Instructional
Strategies

 

 

Unfortunately, the research exploring instruc-
tional strategies designed to facilitate development of
formal reasoning is extremely limited.

First, it is important to note that Piaget's
major reason for researching cognitive development was

to describe its sequence rather than to specify practical

 

applications to improve learning. In their introduction

 

to The Growth of Logical Thinking_from Childhood to

 

Adolescence, Inhelder and Piaget stated:

 

This book has two aims: to set forth a description
of changes in logical operations between childhood
and adolescence and to describe the formal struc-
tures that mark the completion of the operational
development of intelligence. (1958, pp. xxiii-xxiv)
They were describing the thinking of adolescents, not
specifying ways to improve it. Further, Wadsworth, when
discussing what he saw to be the implications of Piaget's
work for education, stated, "It must be kept in mind that
Piaget has not directed his efforts toward the solution
of educational problems" (1971, p. 120). Thus, it is
plausible that one of the reasons for the lack of research

into the applications of Piaget's work is that Piaget

himself did not interpret his work in this way.

 

der
MCK.
ins‘

|
reas

forn

 

 

 

Pro:
ing
ing
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or .
emp
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43

A few educators, however, have attempted to
derive practical applications from Piaget's theory.
McKinnon attempted to determine if inquiry-based
instruction has any impact on the development of formal
reasoning. He found that an inquiry-oriented course
in science had positive effects on the development of
formal reasoning:

The Forum for Scientific Inquiry course was
designed to bring about a change in students'
capacity to think logically; thus, in the short
time of one semester, students did begin to
accept the responsibility for their learning
experiences; they did begin to contribute more
mature views and judgments to the questions
being evaluated. This effect was felt when

the two groups were evaluated at the conclusion
of the course as shown by the net increase in
scores in the five Piagetian tests. (1970, p. 37)

By "inquiry," McKinnon meant that the course
provided the opportunities for "questioning, hypothesiz-
ing, verifying, restructuring, interpreting, synthesiz-
ing, and predicting . . ." (1970, p. 37). Students met
in small groups to discuss and research a particular
topic and were asked to examine a particular aspect
of a problem, to find out what was known, and to suggest
ways to interpret data to either arrive at a solution
or better understand the problem. Thus, the inquiry
emphasis of McKinnon's course was not oriented to facts

about science, but rather oriented to the process of

problem solving in science. Although not designed to

 

tea
HCK

pro

an

wh:

de
in
an

kn

IE

91
t]

t<

16
tb

as

44

teach problem solving pggfgg, the objectives for
McKinnon's course were certainly elements involved in
problem solving.

Renner and Lawson (1975) studied the effects of
an experimental program for elementary school teachers
which attempted to provide an integrated science exper-
ience via an inquiry approach. The curriculum was
designed to provide students with laboratory experiences
in science that were concrete (in the Piagetian sense)
and the emphasis on these experiences was "How do I
know?" "Why do I believe?" and "What is the evidence?"

Using an experimental-control group design, the
researchers found that the inquiry strategy of instruction
produced significant movement into the formal stage for
the experimental subjects. They concluded that these
results support "the hypothesis that curricular materials
which confront students with concrete materials and
problems can promote the development cf formal thinking
abilities" (Renner & Lawson, 1975, p. 9).

Although not directly describing an instructional
strategy, an interesting study by Saarni (1974) attempted
to correlate problem-solving ability with Piaget-level
of reasoning. She found that the higher the Piaget-
level, the better the student was on productive
thinking problems: " . . . individuals classified

as formal-operational (or transitional) were generally

 

mo

pr
ati
int
be‘
ab

in

a}

H:

Al

rn

m

45

more competent problem solvers on the productive thinking
problems than those who were classified as concrete-oper-
ational" (1974, p. 342). Her research does not allow an
interpretation that deduces a cause-effect relationship
between levels of abstract reasoning and problem-solving
ability: i.e., it cannot be determined if an increase
in abstract reasoning causes better problem-solving
performance or if an increase in problem solving
abilities causes an increase in abstract reasoning.
However, her research does raise the question of whether
or not such a cause-effect relationship exists. She
maintained, however, that an individual's problem-solving
strategies can be understood in terms of developmental
capacities as defined by Piaget and that Piaget's theory
"appears to satisfy some of the more crucial demands of
a theory of human problem solving as suggested by Newell,
Simon, and Shaw, and Simon and Newell" (1974, p. 338).
Thus, she definitely left open the possibility of a
cause-effect relationship or at least an interaction
effect between problem solving and development of
abstract reasoning. The study being submitted here
addresses this issue.

A number of other researchers have suggested
various instructional strategies that might increase
students' abstract thinking, although none have been

specifically studied. Beistel (1975, p. 151) suggested

46

that a concrete student can develop formal thinking
abilities through careful experimentation on the stu-
dent's part and extensive critique of this experimen-
tation by the instructor. As a result of this procedure,
the student deduces the relevant principles involved.

This approach seemed to involve the inquiry approach

found helpful by McKinnon (1970) and by Renner and Lawson
(1975) and also an extensive feedback mechanism. Further,
Beistel claimed that in a lecture, intellectual develop-
ment can be facilitated by using techniques such as
introducing new topics in concrete terms, occasionally
using an unsatisfactory hypothesis or incorrect con-
clusion for student evaluation, creating discrepant
events, or describing how a process or concept is reversed
(i.e., Piaget's reversibility concept). Kolodiy (1974,

p. 262) hypothesized that programmed instruction spe-
cifically designed for developing formal thinkers may also
be an effective mechanism for improving abstract reason-

ing.

Summary

A number of conclusions can be drawn about the
level of abstract reasoning ability of various groups
of students. First, a large percentage of pre-college-
age students are pre-formal-operational. Two studies

(Higgins-Trenk & Gaite, 1971; Lawson & Renner, 1974)

47

indicated from 57% to 86% of the subjects studied were
pre-formal thinkers. An additional study by Karplus

and Peterson (1970) was not valid because of questionable
measurement procedures.

In contrast to these studies were those analyzing
pre-college students enrolled in a science course. Law-
son and Renner (1974) reported only 3% to 29.5% of these
subjects were not functioning at formal-operational
levels. (A second study by Karplus & Karplus [1970]
is discounted because of measurement problems.) These
results indicated that either students enrolled in
science-related courses are already formal-operational
or learn formal-operational thinking in the course.

The third conclusion from the literature is
that a large percentage of college students function
at pre-formal-operational levels (from 42-60%). In
addition, the literature consistently pointed out that
a larger percentage of college females than college
males were pre-formal-operational. When adult popu—
lations were studied, it was found that a large per-
centage (from 35-84%) were also pre-formal-operational.

The last population for which a review of the
research was conducted was minority students. No studies
reported on this particular population, indicating a

need for analysis of minority groups.

48

Thus, it can be concluded that a large percentage
of subjects from high school age to adult do not function
at formal-operational levels. The only exception is
students enrolled in a high school science course; they
were primarily formal—operational. Hence, the research
clearly pointed out that Piaget's age specification of
15 years for entry into the formal-operational stage is
not accurate for American subjects. Effective means to
remedy this situation are needed.

Unfortunately, the literature is limited in
specifying effective instructional strategies that
facilitate the development of formal reasoning.

McKinnon (1970) and Renner and Lawson (1975) pointed

to an inquiry approach as effective. Based upon syn-
thesizing and evaluating science-type problems to pre-
dict Solutions, the approach included skills necessary
for problem solving. Saarni's (1974) results, indicating
a high positive correlation between problem-solving
ability and level of thinking, led to the speculation
that improvement in problem-solving ability will result
in an improvement in abstract reasoning. These studies
pointed to direct instruction in problem solving as a

strategy that should be tested.

49

Table 1

Summary of Piaget Studies

 

Group

Author/Date

Percentage
Pre-Formal

 

Pre-college

 

 

 

 

 

 

 

 

 

(General)

13-18 years Higgins-Trenk & Gaite (1971) 57

11th grade Lawson & Renner (1974) 66

12th grade Lawson & Renner (1974) 71

10-12th grade Karplus & Karplus (1970) 86

ll-12th grade Karplus & Peterson (1970) 20a

(suburban)

ll-12th grade Karplus & Peterson (1970) 91a

(urban)

Pre-colle e

(Sc1ence Course)

11-12th grade Lawson & Renner (1974) 4

(chemistry)

11-12th grade Lawson & Renner (1974) 3

(physics)

ll—thh grade Lawson & Renner (1974) 29.5

(biology)

12th grade Karplus & Karplus (1970) 72a

(physics)

College Students

Male and female Elkind (1962) 42

Male and female McKinnon (1970) 51

Male and female Griffiths (1973) 61
(below
level

IIIb)

 

 

50

Table 1 (Continued)

 

 

 

 

 

 

 

 

 

 

 

 

Percentage
Group Author/Date Pre-Formal

College Students
(continuedT
ale Elkind (1962) 26
Male McKinnon (1970) 28
Female Elkind (1962) 48
Female Tower & Wheatley (1971) 39
female McKinnon (1970) 65
Adult and
oungyAdult
21-30 years Kohlberg & Gilligan (1971) 35
dult Karplus & Karplus (1970) 84
(Piaget conf)

Adult Karplus & Karplus (1970) 60
(physics
teachers)
inority
Students
11-12th grade Karplus & Peterson (1970) 91a
(urban)

Joollege Griffiths (1973) 61
(racial sub- (below
samples) level

IIIb)

 

 

 

 

aResults questionable due to invalid

testing.

 

CHAPTER III

THE RESEARCH DESIGN

In this chapter the research and statistical pro-
cedures for analyzing the data will be outlined. The
procedures include a description of the population, the
sample selection, details of the treatment, methodologi-
cal procedures, reliability and validity concerns, and‘
the research design and statistical procedures derived

from the hypotheses.

The Population and Sample

 

The population in this research is defined to be
all first-term minority engineering students in colleges
of engineering. The sample includes all 60 first-term
minority students enrolled in the College of Engineering
at Michigan State University.

Assignment to Control and
ExperImental Groups

 

 

Of the 60 students in the sample, 38 were
identified in June, 1975 prior to Michigan State Uni-

versity's Summer Orientation program. The remaining

51

52

22 students were admitted to the university during the
summer. Because new freshmen must attend the orientation
program to enroll for Fall courses, it was necessary to
assign the original 38 students to the control or experi-
mental group prior to orientation. This was accomplished
by numbering the students alphabetically, selecting
numbers from a table of random numbers, and assigning
students who corresponded to the random number to the
experimental group.

The remaining 22 students were alternately
assigned to each group as they appeared at orientation
sessions.

From the June, 1975 list of 38 students, 19 were
assigned to the control group and 19 to the experimental
group. Of the 19 assigned to the experimental group,
three did not enroll in the university, two requested
that they not be associated with the Sloan Project or
its courses, and two had scheduling conflicts and could
not enroll in the experimental section. These seven
students were deleted from the study.

Of the 19 students assigned to the control
group, six did not enroll for Fall term, two requested
they not be associated with the Sloan Project, and two
had scheduling problems forcing them to enroll in the

experimental course but were not included in the data

53

for the experimental group. Thus, 10 of the original
control subjects were deleted from the study.

Therefore, on the basis of the original random
assignment in June, 1975, 12 students were counted in
the experimental group, 9 in the control group, and 17
were deleted from the study.

For those students who were admitted during the
summer, 10 were placed in the experimental group and 12
in the control group. The final sample included 22
students in the experimental group and 21 in the control

group. Table 2 summarizes this assignment procedure.

The Treatment

 

As was discussed in Chapter I, it is hypothesized
that the treatment "Introduction to Reasoning and Problem
Solving" will facilitate development of abstract reason-
ing. The content of the course included two areas:

(a) Preparation for Problem Solving, which included
restating problems, defining unknown terms, specifying
given information, deciding what is to be solved for,
and drawing and labeling diagrams, and (b) learning and
applying the following problem-solving strategies:
simplification, subproblem, contradiction, inference,
and working backward.

Once these content areas were identified, the
design of the course followed the principles of

instructional design specified in Davis, Alexander,

54

 

 

 

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55

and Yelon's Learning System Design (1974). This procedure

 

included specifying and sequencing behavioral objectives
(see Figure 1 for a complete listing), developing written
instructional material for each objective, and writing
evaluative questions to test accomplishment of each
objective. In addition, the written materials were
reviewed by an engineering professor for a critique of
the content and its relationship to engineering, a
mathematics professor for a critique of the content as

it relates to mathematics, and an instructional develop-
ment specialist for analysis from the viewpoint of
instructional design. The reviews included suggestions
for making the text clearer, ideas for alternative ways
to explain a difficult point, and suggestions for reorga-
nizing the order of the material. The reviews also indi-
cated weaknesses in the text; changes were generally made,
unless they mandated extensive reorganization of the
course, which time did not permit. One such weakness was
the lack of sufficient practice exercises throughout the
modules.

Upon completing one of the modules in the course,
students took exams based upon the specific objectives for
the module. Students were required to repeat exams until
70% mastery was achieved. There was no penalty for re-
peated exams except after mastery was achieved; in this
situation, students were required to keep the repeated

grade.

56

Module 1--Preparing for Problem Solvipg: Understanding the Problem

1. State in proper order the steps used in the Preparation phase of
problem solving.

2. Given a problem statement, properly define all terms used in the
problem, recognize appropriate and inappropriate examples of the
terms, and develop your own example of each term. (Criteria for
accomplishing this specified in the module)

3. Given the statement of a problem, recognize proper and improper
restatements of the problem. For improper restatements, tell why
they are improper.

4. Given the statement of a problem, restate the problem in your own
words according to the criteria specified in this module.

Module 2-—Preparing for Problem Solving: Translation

5. Given a problem statement, determine the "Given” information.

6. Given a problem statement, write a Solution Statement according to
the criteria listed in this module.

Module 3--Drawing Diagrams

7. Given a problem statement, determine which of the following diagrams
is appropriate: (a) Given Situation Only, (b) Given and Solution
Situation Same, (c) Given and Solution Situation Different, or
(d) Two (or more) Givens and Solution.

8. After deciding which diagram is appropriate for a given problem
statement, draw and label the diagram.

Module 4, 5,6,7, 8--Simplification, Subproblem, Drawing Inferences,

Contradiction, Working Backward

9. List the steps to use in applying the strategy.

10. Given a problem that can be solved by the strategy, use the strategy
to solve the problem.

11. List the criteria used to determine if the strategy should be tried
in solving the problem.

12. Given a list of problems, select those for which a given strategy
~ would be appropriate to try.

Figure 1. Introduction to Reasoning and Problem Solving: Course Objectives

57

In addition to the eight modular exams (each
worth 50 points for a total of 400 points), a compre-
hensive final examination was administered during the
university scheduled final exam session. No retests
were allowed and the final counted 100 points or 20%
of the total course grade.

Grades were assigned according to the point dis-
tribution in Table 3. The only exception made to this
scale was when the final examination grade was higher
than the grade corresponding to total points. In this
case, the student received the final examination grade
for a course grade. Final examination grades were based

upon the same percentage range as total points.

Table 3

Grading Scale Used in the Treatment Course

 

 

Total Course Course Percentage
Points Grade Range
470-500 4.0 94-100
440-469 3.5 88-93
410-439 3.0 82-87
380-409 2.5 76-81
350-379 2.0 70-75
320-349 1.5 64-69
290-319 1.0 58-63

0-289 0.0 0-57

 

 

 

In addition to the instructor, two senior under-

graduate engineering students were involved as tutors.

58

Their function was to assist in one-to-one or small
group instruction and in the administration of exami-
nations.

The course met two days per week for two hours
each day; there were 19 class sessions during the term
for a total of 38 contact hours. Each student received

two credits upon satisfactory completion of the course.

The Research Design

 

The experimental design for this research is
the control-experimental group pretest-posttest form

suggested by Campbell and Stanley (1963):

R C1 C2

R El T E2
R stands for random assignment, CI for the control group,
and E1 the experimental group. T is the treatment, in‘

this case exposure to the course, and the subscripted
notation refers to pretest (l) and posttest (2). Sub-
jects were randomly assigned to the control or experi-
mental group, both groups were pretested to measure
their abstract reasoning ability upon entry, the
experimental group received the treatment, and both
groups were posttested to measure abstract reasoning

ability upon exit.

m

 

 

 

Methodological Procedures
This section will explain the procedures used
to analyze the abstract reasoning abilities of the
subjects in the study, including descriptions of the
pre- and posttesting, the instruments used, and how

they were scored.

Pre- and Posttesting

During the first seven days of the term, students
in the experimental and control groups were asked to make
one-half-hour appointments with the researcher for an
"interview." When students appeared for the appointment,
they were told they would do two experiments but that the
results of their performance would not be discussed with
them until the end of the term. Students who missed an
appointment were rescheduled two additional times; those
failing to appear at any of the appointments were deleted
from the study.

Posttesting was conducted eight weeks later during
the last week of Fall term classes. Once again, students
scheduled one-half-hour interviews and were asked to
repeat the same two experiments used on the pretest.
After completing the posttest, the correct answer was
explained to the student as well as the purpose of the
research. They were asked not to discuss it with their
friends. A follow-up of those who did not appear for

testing was conducted the same way as during the pretesting.

60

Of the 22 experimental group and 21 control group
subjects, 16 experimental subjects and 11 control sub-
jects completed all testing. Table 4 indicates how

many subjects were lost during the testing.

Table 4

Mortality of Subjects Due to Incomplete Testing

 

 

 

 

 

Grou Original No No Completed
p Number Pretest Posttest All Testing
Control 21 6 4 ll
rxperimental 22 3 3 16
Instrumentation

 

In order for the researcher to measure level of
cognitive development, each student was asked to perform
two experiments described by Inhelder and Piaget. These
were the Pendulum Problem (1958, pp. 67-79) and the
Equilibrium in the Balance problem (1958, pp. 164-181).

In the Pendulum Problem, the student was con-
fronted with three different length strings hanging from
the ceiling. Attached to the end of each was a small
;pan on which weights could be attached. Each of the
three pans were identical in size, shape, and weight.
{The student was also shown a number of 50 gram and
.100 gram weights. The concept of frequency was then
ciefined for the student as "the number of times the

lpendulum swings back and forth in a given time period."

61

With this information the student was told to experiment
with the equipment in order to determine "what thing

or things (if anything) affects the frequency of the
pendulum."

In the Equilibrium in the Balance problem, the
student was shown an apparatus that consisted of a yard
stick with a balance point or fulcrum at its center.

At each inch marking there was a small hole from which
weights could be hung. The weights consisted of paper
clips and it was explained that each clip constituted
one unit of weight; thus two clips weighed two units,
three clips weighed three units, and four clips weighed
four units. The experiment proceeded as follows:

1. The experimenter balanced the yardstick and
asked the subject if it was in balance.

2. The experimenter handed the subject two
equal weights and asked if the subject agreed they were
equal.

3. Next, the experimenter placed a weight at
some point on the balance and handed an equal weight to
the subject. The subject was then told, "Place this
‘weight on the stick so it balances." Once the subjects
did this, they were asked to explain why the weight was
so placed.

4. In the second phase, the experimenter placed

-two equal weights at equal distances from the fulcrum

 

 

 

62

and asked the subject to agree the stick was in balance.
Then the experimenter moved one weight toward the fulcrum
and without letting go of the stick asked the subject
"What happens when I let go of the stick and why?"

5. In the third part of the test the experimenter
placed two unequal weights at unequal distances from the
fulcrum in such a way that the stick balanced. The
subject was then shown that the stick balanced and told
"Take the weight on the right side of the fulcrum and
place it on the left. Do the same with the weight on
the left so that the stick balances." Once the subject
did that, the experimenter held the stick and asked
the subject to explain why the weights were so placed.
After the subject gave a reason, the experimenter
released the stick so the subject could see if the
stick balanced.

6. In the last part of the experiment, the
subject was given two unequal weights and told to place
them on the stick so it balanced. Once the subjects did
this, they were asked to explain why the weights were so
placed. Then the experimenter released the stick.

Once the experiments were completed by each
student, responses were translated into a score that
indicated level of abstract reasoning ability. The
criteria for each experiment, the associated level in

Piaget's hierarchy, and the corresponding numerical

 

 

 

63

score are listed in Tables 5 and 6. A score of one
indicates that the subject should be able to perform
the criteria at the indicated level; a zero indicates
performance is not expected.

Thus, the score received indicated the level of
abstract reasoning, i.e., a score of 2 indicates the
subject is at level IIB, while a score of 2.5 on the
Pendulum Problem indicates the subject is in transition
between IIB and IIIA. Hence, each subject was scored
according to whether or not each criteria in each

experiment was accomplished.

Reliability and Validity

 

The concepts of test reliability and validity
are of importance to researchers using any type of test-
ing instruments. Both of these concepts will be defined
and discussed in terms of this study. In addition,
Campbell and Stanley's (1963) criteria for internally

and externally valid designs will be discussed.

Reliability

 

According to Anastasi, "reliability refers to
the consistency of scores obtained by the same indi-
viduals when re-examined with the same test on different
occasions . . ." (1968, p. 71). In addition, she points
out that "test reliability indicates the extent to which

individual differences in test scores are attributable to

64

Table 5

Criteria for Scoring Responses to the Pendulum Problem

 

Piaget-Level

 

Criteria
I IIA IIB IIIA IIIB

 

l. Orders weights serially 0 0 1 1 l

2. Discovers relationship between _
frequency and length (inverse) 0 1 l 1 l

3. Excludes variables when given com-
bination in which only one

 

 

 

 

variable varies 0 0 0 0.5 0.5
4. Produces combinations of variables
in a systematic way 0 0 0 0.5 0.5
5. Varies only one variable while
holding others constant 0 0 0 0 1
Total Points (score) 0 l 2 3 4
Table 6

Criteria for Scoring Responses to the Equilibrium in the Balance Test

 

Piaget-Level

 

Criteria
I IIA IIB IIIA IIIB

 

1. Discovers by trial and error relation-

ship between length and width 0 l l 1 l
2. Gives a qualitative explanation of
relationship 0 0 l l 1
3. Gives a quantitative explanation of
relationship 0 0 0 1 l
4. Explains relationship in terms of
height compensation 0 0 0 0 1
Total Points (score) 0 l 2 3 4

 

 

 

 

Note. The use of 0.5 for 3 and 4 is necessary in order to create a
discrimination between the two criteria for level IIIA, B (i.e., criteria
3 a 4) and still maintain equal intervals between levels I through IIIB. In
‘this way it is possible numerically to discriminate between a subject who
«can do criteria 1, 2, and 3 but not 4 (or 1, 2, and 4 but not 3) and is
scored 2.5, indicating the subject is in transition between levels IIB and
IIIA.

65

'true' differences in the characteristics under consid-
eration and the extent to which they are attributable
to chance errors" (1968, p. 71). Thus, a highly reliable
test will produce scores which measure actual differences
rather than scores that measure differences due to chance.
Traditionally, to determine test reliability,
two equivalent forms of the same test are administered
and a test-retest correlation coefficient is computed to
provide an index of the reliability of the test. This
procedure was not followed because equivalent forms of the
tests used had not been identified in previous research.
For tests where the scorer must interpret the
behavior of the subject in order to arrive at a score,
Anastasi (1968, p. 86) recommends assessing the inter-
rater reliability by having subject responses indepen-
dently scored by two examiners. To obtain a measure of
reliability, the resulting scores are correlated by use
of one of the standard reliability coefficient techniques.
In this study a second scorer was not available;
hence, an inter-rater reliability measure was not com-
puted. This omission was due to the lack of availability
of a trained scorer who could devote approximately 40
hours to analyzing subject responses to the tests. The
lack of reliability information will be discussed in

Chapter V as a limitation of this study.

66

Validity

 

In general, the concept of validity refers to
"what the test measures and how well it does so" (1968,
p. 99). The type of validity most important to this
study is construct validity, which is defined as the
extent to which a particular test measures a theoretical
construct or trait. The Equilibrium in the Balance test
and the Pendulum Problem have construct validity if they
in fact measure formal-operational reasoning according
to the theory.

There is growing evidence that these two tests
do measure formal-operational reasoning and thus do have
construct validity. In their 1974 study, Lawson and
Renner utilized the multivariate statistical technique
of principal components analysis to examine the extent
to which these and other tests actually measure formal
reasoning. In principal components analysis, first the
major factors which account for a significant amount of
the variance in scores are identified. In the Lawson and
Renner study there were two factors: formal- and concrete-
operational reasoning. Hopefully, if the construct being
analyzed hypothesizes N factors, then the analysis will
identify exactly N factors. Next, correlations are
computed between each of the tests being analyzed and
each of the factors found. If all tests that are

hypothesized to measure one of the factors have

 

67

significantly large correlations with that factor, then
the test is said to have construct validity.

As a result of a principal components analysis,
Lawson and Renner (1974) concluded that the Pendulum
Problem does measure formal reasoning. They analyzed
six Piaget-tests that were hypothesized to measure only
two factors--formal- and concrete-operational reasoning.
Results indicated that only these two factors were found,
they accounted for a significant amount of the variance
in scores (55.2%), and all correlations were as predicted.
The correlation for the Pendulum Problem was 0.72 with

the hypothesized factor, formal-operational reasoning.

In the same study, subsequent analyses of various
combinations of Piaget-tests yielded results similar to
the above: the hypothesized number of components was
found, they accounted for a significant amount of the
variance of the scores, and correlations between test
and principal component were as predicted by the theory.
(The Pendulum Problem had a correlation with the hypo-
thesized component of 0.68 on one analysis; the Equil-
ibrium in the Balance test had a correlation of 0.80
with the hypothesized component.) Thus, the authors
concluded that "these results indicate that all the
tasks . . . are measuring the same thing, i.e.: formal-

operational thinking" (1974, p. 556).

iiilllll
I'll

 

 

 

 

 

68

Similar results using principal components
analysis are also reported by Lovell and Butterworth

and Hughes, in Measurement and Piaget (Green, et al.,

 

1971, pp. 85-87).

Thus, there is evidence that both the Pendulum
Problem and the Equilibrium in the Balance test do
indeed measure what Piaget intended them to measure:
formal-operational thought.

Internal and External
Validity

 

 

Internal validity is defined to be the "basic
minimum without which an experiment is uninterpretable"
(Campbell & Stanley, 1963, p. 5). To be internally
valid, an experiment must control for (a) history--what
happens between first and second measures in addition
to the experimental variable, (b) maturation of subjects,
(c) testing--the effects of taking a first test upon a
second test, (d) instrumentation--changes in observers
who interpret tests, (e) statistical regression--oper-
ating when groups are chosen on the basis of extreme
scores, (f) selection biases--differential selection
for groups, (g) mortality of subjects--differential
loss of subjects, and (h) selection-maturation inter-
action--differentia1 maturation in multiple group designs.

External validity asks the question "To what

populations . . . can this effect be generalized?"

69

(Campbell & Stanley, 1963, p. 5). The factors that
must be controlled in order that an experiment be
externally valid are: (a) reactive effect of testing--
pretest sensitizes subjects to experimental variable,
(b) interaction of selection biases and experimental
variable, (c) reactive effects of experimental arrange-
ments--the experiment itself creates the effect, and
(d) multiple-treatment interference-~multiple treatments
affect subjects.

According to Campbell and Stanley, there are
three sources of concern for the internal validity of
this study: mortality of subjects, selection biases,
and instrumentation biases.

There are a number of factors which contributed
to the mortality of subjects from the original random
assignment to control or experimental group. Nine sub-
jects did not appear for classes during Fall term, four
had scheduling difficulties, and four refused to par-
ticipate resulting in 43 subjects remaining in the two
groups out of the original 60. In addition, nine sub-
jects from the control group and six from the experi-
mental group did not complete all testing. Thus, there
is a high mortality rate of subjects from the original
random assignment to the final group which completed all
testing. Since Campbell and Stanley pointed out that

it can be assumed the mortality rate is randomly

70

distributed over the two groups, a post-hoe analysis
was conducted to determine if in fact this was the case.
Two-sample, two-tailed t-tests (p 5 .05) were conducted
to determine if (a) the original control group differed
significantly from the original experimental group or
from the final control group; (b) if the original
experimental group differed from the final experimental
group; or (c) if the final experimental group differed
from the final control group. Both "original" groups
were defined to be all 60 students who were randomly
assigned to groups minus the nine students who did not
enroll for Fall term, as data were not available for
them. The "final" groups were comprised of only those
students who completed all testing. The t-tests were
conducted on the following variables: (a) ACT/SAT math
percentile, (b) ACT/SAT verbal percentile, (c) ACT/SAT
composite score percentile, (d) high school grade point
average, (e) MSU arithmetic test score, and (f) MSU
mathematics test score. The ACT or SAT percentiles were
localized Michigan State University norms. Table 7
reports means, N's, and variances for the various
groups on the six measures.

As Table 8 indicates, there was no significant
difference between any two groups for any of the six
variables. Thus, it can be concluded that although

mortality of subjects was a troublesome factor in this

71

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73

study, the mortality was randomly distributed between
control and experimental groups with respect to the
six variables analyzed. Hence, the final control and
experimental groups on which data were collected and
analyzed were not significantly different from the
original randomly selected control and experimental
groups. Further, the final control group was not sig-
nificantly different from the final experimental group
on the basis of the six variables. Thus, the contami-
nating effects of mortality are not a limitation of this
study and do not affect its internal validity.

The second area of concern for the internal
validity of this study is selection biases due to the
four students who refused to participate in the study.
It is possible that their lack of involvement skewed
the results in one direction or the other. Although
this is a contaminating factor, the evidence indicating
no significant differences between original and final
groups minimizes the effects of the selection biases.

The internal validity of this study could have
been affected by instrumentation biases created by the
researcher conducting all testing. Knowledge of who was
in the control group and who was in the experimental
group could have an effect upon how student performance
was analyzed and scored. It is possible that the

researcher compensated for student performance by

 

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(
I
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74

either scoring the experimental subjects higher than the
control subjects--thus biasing the results in favor of
the experimental group--or by scoring the experimental
subjects lower than the control subjects--thus biasing
results in favor of the control group. Because of the
researcher's involvement in pre- and posttesting, instru-
mentation biases may have been introduced.

Of concern for the external validity of this
study are interaction of testing and treatment, inter-
action of selection and treatment, and reactive arrange-
ments.

In any design using pretests, the risk is present
that subjects will be sensitized to the treatment, i.e.,
there will be an interaction of testing and treatment.
According to Campbell and Stanley the effects of pretests
sensitizing the subjects should be equal on both the
experimental and control groups, thus the effects
cancel out one another. In addition, the pretest, which
involved performing specific Piaget experiments, had no
content relationship to the treatment, which involved
learning specific problem-solving strategies. Therefore,
interaction of testing and treatment is not a significant
problem in this study.

Campbell and Stanley also suggest that selection
and treatment interaction could affect the external

validity if a significant number of subjects refused to

75

cooperate in the study. In this study, four students
refused to be involved in the study, introducing inter-
action of selection and treatment biases. However,
results of the post-hoc analysis indicate that this
effect is minimal.

The term "reactive arrangement" refers to a
situation in which subjects know they are involved in an
experiment. As Campbell and Stanley pointed out, this
effect is almost always present: attempts should be
made to reduce its effects. In this research, subjects
were aware that they were involved in an experiment,
but did not know what the experiment was about nor whether
they were in the control or experimental group. Thus,
any reactive arrangement effects should be randomly dis-
tributed over all subjects cancelling out any particular

effect in favor of either group.

Hypotheses and Statistical Analysis

 

There are three major research questions this
study addresses. Each is listed below, followed by the
specific hypotheses and statistical procedures relevant

to the research question.

Question l--Was the course effective in increas-

 

ing_students' abstract reasoning abilities? These

 

hypotheses were stated to determine whether the amount

of change from pretest to posttest was significantly

 

 

 

 

76

greater for the experimental group as compared to the

amount of change for the control group.

Hypothesis 1:

 

Ho: There will be no significant difference in
change in abstract reasoning ability as
measured by the Equilibrium in the Balance
test between the control and experimental
groups.

There will be a significant difference in
change in abstract reasoning ability as
measured by the Equilibrium in the Balance
test in favor of the experimental group.

Hypothesis 2:

 

H - There will be no significant difference in
change in abstract reasoning ability as
measured by the Pendulum Problem between
the control and experimental group.

H : There will be a significant difference in
change in abstract reasoning ability as
measured by the Pendulum Problem in favor
of the experimental group.

Statistical Analysis:
Hypotheses l-2

 

 

Change in abstract reasoning (A) will be measured
by the difference between the posttest score (P0) and

the pretest score (P ):

for each subject.

77

The average change score (ii) for each group
will be computed and a two-sample, one-tailed t-test

. < . . . . .
with p - .05 Will be conducted to determine Significance,

 

 

where:
r-az
t = l 2
l 1
_._+__
8p n1 n2
where:
2 512 (nl-l) + 322 (nz-l)
S =
and the degrees of freedom are n + n -2.

1 2
The null hypothesis is rejected when the computed

t-value is greater than or equal to t + n _2.

.05, n1 2

Otherwise, the null hypothesis is not rejected.

Question 2--Are there any relationships between

 

various measures of academic ability and either level

 

of abstract reasoning upon entry into the university or

 

amount of change in abstract reasoning after one term

 

in collgge? The hypotheses are stated to determine what,

 

if any, relationships exist between each of the measures
of academic ability and (a) level of abstract reasoning
upon entry into the university, (b) amount of change in
abstract reasoning after one term of instruction without

exposure to the treatment, and (c) amount of change

 

 

78

in abstract reasoning after one term of instruction
with exposure to the treatment.

The academic ability measures are listed below.
They were chosen because they are standard measures of
academic ability and commonly used:

1. GRADE--the grade received in the treatment
course; computed on the basis of total points received
in the course (see Table 3). Thus, the grade was com-
posed of 50 points each for the eight module exams
(400 points) plus 100 points for the final. Grades
are on a four-point system and include 4.0, 3.5, 3.0,
2.5, 2.0, 1.5, 1.0, and 0.0.

I 2. TOTPTS--total points earned in the treatment
course.

3. FINAL--points earned on the final examination
for the treatment course; maximum possible is 100 points.

4. OBJPTS--points earned on the final exami-
nation questions requiring application of a problem
solving strategy; students selected three of four
problems on the final; each was worth 16 points for
a total of 48 points on the final.

5. HSGPA--high school grade point average; used
on a 0 to 4.00 scale as reported in the student's per-

manent university records.

79

6. MSUGPA--first-term MSU grade point average;
a 0 to 4.00 scale is used; includes all courses student
took during Fall, 1975.

7. ARITHP--MSU Arithmetic Placement score; an
MSU exam administered to all incoming students; used
in conjunction with the Mathematics Placement score to
place students in appropriate math courses; scored from
0 to 40 points.

3, MTHPLC--MSU Mathematics Placement score; an
MSU exam administered to all incoming students; used in
conjunction with the Arithmetic Placement score to place
students in appropriate math courses; scored from 0 to
30 points.

9-11. MTHPCT, VRBPCT, CMPPCT--ACT or SAT
mathematics, verbal and composite percentiles; ACT or
SAT raw mathematics, verbal or composite scores were
converted to a percentile score. Percentiles are
localized MSU Fall, 1975 norms.

12. MTHGRD--grade received in mathematics
course taken during Fall, 1975; four-point scale used

(same as for GRADE).

80

Hypotheses 3-6:

 

Ho: For the experimental group only, there will be
no significant correlation between the Equil-
ibrium in the Balance pretest and each of the
following variables:

3. GRADE
4. TOTPTS
5. FINAL
6. OBJPTS

H1: For the experimental group only, there will be
a significant correlation between the Equil-
ibrium in the Balance pretest and each of the
variables 3-6.

Hypotheses 7-10:

H : For the experimental group only, there will be
no significant correlation between the Pendulum
Problem pretest and each variable in Hypotheses
3-6.

H1: For the experimental group only, there will be
a significant correlation between the Pendulum
Problem pretest and each variable in Hypotheses
3-6 0

Hypotheses ll-18:

 

Ho: For the experimental and control groups combined,
there will be no significant correlation between
the Equilibrium in the Balance pretest and each
of the following variables:

11. HSGPA
12. MSUGPA
l3. ARITHP
14. MTHPLC
15. MTHPCT
16. VRBPCT
l7. CMPPCT
18. MTHGRD

H : For the experimental and control groups com-
bined, there will be a significant correlation
between the Equilibrium in the Balance pretest
and each of the variables 11-18.

81

Hypotheses 19-26:

H : For the experimental and control groups combined,
there will be no significant correlation between
the Pendulum Problem pretest and each variable
listed in Hypotheses 11-18.

H1: For the experimental and control groups combined,
there will be a significant correlation between
the Pendulum Problem pretest and each variable
listed in Hypotheses 11-18.

Hypotheses 27-30:

 

H0: For the experimental group only, there will be
no significant correlation between change in
abstract reasoning ability as measured by the
Equilibrium in the Balance test and each of
the following variables:

27. GRADE
28. TOTPTS
29. FINAL

30. OBJPTS

H1: For the experimental group only, there will be a
significant correlation between change in
abstract reasoning ability as measured by the
Equilibrium in the Balance test and each
variable 27-30.

Hypotheses 31-34:

 

H : For the experimental group only, there will be
0 no significant correlation between change in
abstract reasoning ability as measured by the
Pendulum Problem and each variable listed in
Hypotheses 27-30.

H : For the experimental group only, there will be
a significant correlation between change in
abstract reasoning ability as measured by the
Pendulum Problem and each variable listed in
Hypotheses 27-30.

82

Hypotheses 35-42:

 

H :
o

For the experimental group, there will be no
significant correlation between change in
abstract reasoning as measured by the Equil-
ibrium in the Balance problem and each of the
following variables:

35. HSGPA

36. MSUGPA
37. ARITHP
38. MTHPLC
39. MTHPCT
40. VRBPCT
41. CMPPCT
42. MTHGRD

For the experimental group, there will be a
significant correlation between change in
abstract reasoning as measured by the Equil-
ibrium in the Balance problem and each
variable 35-42.

Hypotheses 43-50:

 

H :
o

For the control group, there will be no sig-
nificant correlation between change in abstract
reasoning as measured by the Equilibrium in

the Balance Problem and each variable listed

in Hypotheses 35-42.

For the control group, there will be a sig-
nificant correlation between change in abstract
reasoning as measured by the Equilibrium in

the Balance Problem and each variable listed

in Hypotheses 35-42.

Hypotheses 51-58:

 

H :
o

For the experimental group, there will be no
significant correlation between change in
abstract reasoning as measured by the Pendulum
Problem and each of the variables listed in
Hypotheses 35-42.

For the experimental group, there will be a
significant correlation between change in
abstract reasoning as measured by the Pendulum
Problem and each of the variables listed in
Hypotheses 35-42.

83

Hypotheses 59-66:

Ho: For the control group there will be no signifi-
cant correlation between change in abstract
reasoning as measured by the Pendulum Problem
and each of the variables listed in Hypotheses
35-42.

H1: For the control group there will be a significant
correlation between change in abstract reasoning
as measured by the Pendulum Problem and each of
the variables listed in Hypotheses 35-42.

Statistical Analysis:
HypothesesI3-66

 

A Pearson product-moment coefficient of cor-
relation (r) will be computed. To test the significance
of the correlation, the following one-tailed t-test

statistic is used (p 5 .05)

where the degrees of freedom are n-2. The null hypothe-
sis is rejected when the computed t-value is greater
than or equal to t 05. n-2' Otherwise the null hypothesis

is not rejected.

Question 3--Is there an interaction effect between

 

level of academic ability and treatment? These hypotheses

 

were stated to determine if there was a difference in
the amounts of change in abstract reasoning for high
ability vs. low ability students within or between

groups.

 

 

84

Hypotheses 67-74:

 

Ho: There will be no significant difference in
change in abstract reasoning as measured by
the Equilibrium in the Balance test between
high or low ability students in the experi-
mental or control group. Ability level will
be measured by:

67. HSGPA

68. MSUGPA
69. ARITHP
70. MTHPLC
71. MTHPCT
72. VRBPCT
73. CMPPCT
74. MTHGRD

H : There will be a significant difference in
change in abstract reasoning as measured by
the Equilibrium in the Balance test between
high or low ability students in the experi-
mental or control group for each ability
measure listed in variables 67-74.

Hypotheses 75-82:

 

Ho: There will be no significant difference in
change in abstract reasoning as measured by
the Pendulum Problem between high or low
ability students in the experimental or
control group. Ability level will be
measured by the variables listed in
Hypotheses 67-74.

H : There will be a significant difference in
change in abstract reasoning as measured by
the Pendulum Problem between high or low
ability students in the experimental or
control group for each ability measure
listed in variables 67-74.

Statistical Analysis:
Hypotheses 67-82

 

 

A two-way analysis of variance was used for

each hypothesis. Because of unequal cell size, Finn's

85

(Scheifley & Schmidt, 1973) multivariance technique was
used; it utilizes an exact least squares analysis. The
following matrix represents the manner in which the

data were classified for each academic ability variable:

 

 

 

 

’ Academic Variable
Group HI LO
Control x11 x12
Experimental x21 x22

 

 

 

 

where iij represents the mean change score for the i-th
row and j-th column. An F-test was computed and the
null hypothesis was rejected for values greater than or
equal to F 05; n1, n2, where n1 and n2 are the degrees
of freedom; otherwise the null form was not rejected.
The degrees of freedom were 1 and 21 for "grade in math
course taken" and were 1 and 23 for the remaining
variables.

Each of the eight academic ability variables
was split into a high and a low group. Table 9 indi-
cates the cut-off value and how many subjects in each
group were classified high and low. Subjects with
scores greater than or equal to the cut-off value were
included in the high group and subjects with scores less

than the cut-off value were included in the low group.

For cases where there was more than one subject with the

86

 

 

 

 

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87

cut-off value for a score, the subjects were randomly

placed in the high or low group.

Summary

The original sample for this study was comprised
of 60 first-term minority engineering students, while
the final sample included 11 students in the control
group and 16 in the experimental group. The control-
experimental group pretest-posttest research design was
used with only the experimental group receiving the
treatment. Pre- and posttest measures were obtained
on the basis of student responses to both the Equilibrium
in the Balance test and the Pendulum Problem, which
are both valid measures of level of abstract reasoning.

A limitation of this study is that no attempt
was made to assess either test-retest reliability or
inter-rater reliability for the tests used. There is,
however, reasonable evidence from other research to
conclude that the study has construct validity.

The concerns for the internal validity of this
study are mortality of subjects, selection biases, and
instrumentation biases. A post-hoe analysis of six
entry-level variables indicated that the mortality of
subjects was randomly distributed between groups.

The external validity concerns are interaction
of testing and treatment, interaction of selection and

treatment, and reactive arrangements.

88

This study addresses whether (a) the treatment
was effective in increasing students' abstract reasoning
ability, (b) if there are any significant correlations
between both pretest abstract reasoning ability or
change in abstract reasoning and various academic ability
variables, and (c) if there are any treatment by ability
interactions. Statistical analyses used were (a) a two-
sample, one-tailed t-test, (b) Pearson product-moment
coefficients of correlation, and (c) Finn's multi-
variance technique for an analysis of variance with
unequal cell size.

Chapter IV will report on the results of the

analysis of the data.

CHAPTER IV

ANALYSIS OF THE RESULTS

The analysis of the data with respect to the
hypotheses listed in Chapter III is reported in this
chapter. The discussion is organized into three sections:
(a) the results of the analysis for determining if the
treatment, "Introduction to Reasoning and Problem Solving,
facilitates the development of abstract reasoning ability;
(b) the results of correlations between either pretest
scores or change scores and numerous ability measures;
and (c) the results of the analysis of variance for
determining if there is an interaction effect between
treatment (experimental or control) and ability level
(high or low).

Analysis of the Effectiveness
oflthe Treatment

 

 

The extent to which the treatment is an effective
vehicle for improving abstract reasoning is addressed in
two parts. The first part reports statistically sig-

nificant or nonsignificant changes in abstract reasoning

89

90

caused by the treatment, while the second part describes
the magnitude of any changes observed but is not intended

to be interpreted statistically.

Statistical Results

 

The main criterion used to assess the effective-
ness of the treatment is the difference between the change
in abstract reasoning for the experimental group and the
change in abstract reasoning for the control group. Two

null hypotheses are Specified:

Hypothesis 1:

 

There will be no significant difference in change
in abstract reasoning ability as measured by the
Equilibrium in the Balance test between the control
and experimental groups.

Hypothesis 2:

 

There will be no significant difference in change
in abstract reasoning ability as measured by the
Pendulum Problem between the control and experi-
mental groups.

The two-sample, one-tailed t-tests need to compare
the mean change in abstract reasoning from pretest to
posttest for the experimental group to that of the control
group are not statistically significant for both the
Equilibrium in the Balance test and the Pendulum Problem
(Table 10). The calculated t-values are t = 0.41
(p < .35) for the Balance test and t = 0.46 (p < .33)

for the Pendulum Problem. Thus, the null forms for

both Hypotheses 1 and 2 are not rejected for p i .05,

9].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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92

indicating that for both measures, the amount of change
for the experimental group was not statistically dif-
ferent from the amount of change for the control group.

This analysis compares the amount of change for
the experimental group relative to the amount of change
for the control group. Since this relative difference
between amounts of change for the two groups is not sig-
nificant, further analysis was conducted to determine if
either the control or experimental group increased in
abstract reasoning relative to itself. That is, (a) are
the experimental group's posttest scores statistically
different from its pretest scores, and (b) are the control
group's posttest scores statistically different from
its pretest scores?

To answer these questions, a one-sample, repeated
measures t-test is used. For each group separately,
posttest scores minus pretest scores were computed for
each subject. The average of these differences (5) is
the unit of analysis for the dependent measure t-test.1

This analysis reveals that for the experimental
group, the posttest scores are significantly greater
than pretest scores for the Equilibrium in the Balance
test (t = 2.42; p < .02), but not for the Pendulum

Problem (t 1.40; p < .09). For the control group,

 

H
0|

is the standard deviation of
the difference scores.

t = , where s
sd//E'

 

d

93

posttest scores are not significantly different from
pretest scores for either measure (Balance: t = 1.39,
p < .10; Pendulum: t = 0.71, p < .25). These results
are summarized in Table 11.

Hence, the statistical results indicate that for
both measures, the amount of change from pretest to post-
test for the experimental group is not statistically dif-
ferent from the amount of change for the control group.
However, for the Equilibrium in the Balance test only,
the posttest scores are significantly greater than the
pretest scores for the experimental group, but not for

the control group.

Descriptive Results
1 The descriptive results to be discussed are:

l. The average Piaget-level at which subjects
are classified on the pretest and on the posttest;

2. The percentage of subjects functioning below
the formal-operational level on the pretest (groups com-
bined) and on the posttest;

3. The number of Piaget-levels changed from

pre- to posttest.

1. Subjects' average Piaget-level. The sta-

 

tistical analysis is based upon an average level of
abstract reasoning for each group and changes in

these averages from pre- to posttest. These same

94

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95

averages are useful in describing the average level of
abstract reasoning of the subjects in each group. The
individual scores from which the averages are computed

and their corresponding Piaget-levels are:

Score Level

0 I Preoperational

1 IIA} Concrete-O erational
2, 2.5 IIB p

3, 3.5 IIIA _ .

4 1113} Formal Operational

As Table 10 shows, the pretest levels for both
groups and for both tests are between 1.91 and 2.22,
indicating the average Piaget-level for all subjects
on the pretest is close to level IIB--the higher of the
two concrete-operational levels. The average level for
the control group is 1.91 on the Equilibrium in the
Balance test and 2.05 on the Pendulum Problem; the
experimental group scores are 1.94 and 2.22 on the two
tests. As subjects were randomly assigned to group, it
is expected that the pretest mean for the experimental
group is not statistically different from the pretest
mean for the control group on either measure used. This
is in fact the case; Table 10 summarizes the two-sample
independent t-test results, which were used to determine
if pretest differences existed between groups (Balance:
t = 0.12, p < .40; Pendulum: t = 0.74, p < .24).

Table 10 also indicates subjects' average Piaget-

level on the posttest. The control group means are 2.18

96

on the Equilibrium in the Balance test and 2.27 on the
Pendulum Problem; the experimental group means are 2.31
and 2.66 on the two tests respectively. These data indi-
cate that according to each measure, both groups are well
into the concrete-operational stage on the posttest, but
still functioning below the formal-operational level.

The amount of change from pretest to posttest
for the control group is 0.27 on the Equilibrium in the
Balance test and 0.23 on the Pendulum Problem and the
amount for the experimental group is 0.38 and 0.44 on
the corresponding tests (see Table 10). Thus, the amount
of change for the experimental group is 1.41 times
greater than the amount of change for the control group
on the Equilibrium in the Balance test and 1.91 times
greater on the Pendulum problem. However, as has been
discussed, the amount of change for the experimental
group is not statistically greater than that of the con-
trol group on either teet.

Thus, the descriptive results indicate that both
groups increased in abstract reasoning from pretest to
posttest and that the amount of increase is greater for
the experimental group. Statistically, however, only the
change from the Balance pretest to posttest for the

experimental group is a significant increase.

2. Subjects functioning below the formal-

 

operational level on the pretests. According to Piaget's

 

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98

theory, it is expected that all subjects in this study
should be formal-operational. However, a surprisingly
large percentage of subjects are not formal-operational
upon entry into the university (pretest data). According
to the Equilibrium in the Balance pretest, pre-formal-
operational subjects comprise 91% of the control group,
81% of the experimental group, and 85% of both groups
combined. For the Pendulum Problem, 82% of the control
group, 94% of the experimental group, and 89% of both
groups combined are not formal-operational thinkers (see
Table 13). Posttest results indicate that on the basis
of the Equilibrium in the Balance test, the percentage
of students who are not formal-operational is 82% for
the control group and 69% for the experimental group.
For the Pendulum Problem, posttest results indicate 91%
of the control group and 69% of the experimental group
are not formal-operational. Scores reporting the per-
centages for the control and experimental groups combined
are meaningless because of the possible treatment effect
on the experimental group; hence, they are not reported.
For the Balance test, the number of subjects not
functioning at formal-operational levels decreased nine
percentage points from pretest to posttest for the con-
trol group and 12 percentage points for the experimental
group. For the Pendulum Problem, the number of pre-

formal operational thinkers actually increased nine

99

 

 

 

 

 

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100

percentage points for the control group and decreased 25
percentage points for the experimental group (see Table 13).
Thus, on both measures, the percentage-point decrease in
pre-formal-operational subjects is greater for the experi-
mental group. Once again, it should be kept in mind that
these percentage-point differences are only descriptive;

the mean changes between the two groups are not statis-

tically significant.

3. Number of Piaget-levels changed--Pre- to

posttest. In order to provide a more complete analysis

 

of how subjects changed from pre- to posttest, Table 14
summarizes the number of Piaget-levels subjects changed.
The table indicates that a large percentage of subjects
in each group and on each test do not change at all,
although fewer subjects in the experimental group show
no change than in the control group. That is, on the
Equilibrium in the Balance test 64% of the control group
did not change, while only 50% of the experimental group
did not change. For the Pendulum Problem, 64% of the
control group and 44% of the experimental group remains
unchanged.

In addition, change in the positive direction is
observed in the control group for 27% of the subjects on
both tests, while a somewhat higher percentage is
observed for the experimental group: 44% (Equilibrium

in the Balance) and 30% (Pendulum). Conversely, higher

101

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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102

percentages of subjects who either do not change or
change in the negative direction are observed for the
control group (73% and 73% on the two tests) over the

experimental group (56% and 69%).

Summary

The descriptive results of the analysis of the
effectiveness of the treatment indicate that on both
measures used (a) both the control and experimental
groups increased in abstract reasoning from pre- to
posttest and (b) the magnitude of the increase is greater
for the experimental group than the control group. None
of these changes, however, reach statistical significance
for p E .05, except for the amount of change from pretest
to posttest for the experimental group when the measure
is the Balance test. The remaining changes observed are
attributed to chance.

The results do indicate that a very large per-
centage of the sample entered the university at pre-
formal-operational thinking levels. Depending upon the
test used, from 85% to 89% of the entry group are pre-
formal-operational. Their average Piaget-level is
concrete-operational.

After one term at the university, from 82% to
91% of the control group and 69% of the experimental
group are still not formal-operational thinkers. Their

average Piaget-level is concrete-Operational.

103

Correlations

 

To determine if there are any variables correlated
with either pretest level of cognitive functioning or
amount of change, a number of correlation coefficients
were computed. First (a) the correlation between the
pretest score for the Equilibrium in the Balance test

and each of the following variables was computed:

a) Grade received in the treatment (GRADE)
b) Total points received in the treatment (TOTPTS)
c) Final exam points (FINAL)

d) Points received on final exam questions
requiring application of a problem-

solving strategy (OBJPTS)
e) High school grade point average (HSGPA)
f) First term MSU grade point average (MSUGPA)
g) MSU arithmetic placement score (ARITHP)
h) MSU mathematics placement score (MTHPLC)
i) ACT or SAT mathematics percentile (MTHPCT)
j) ACT or SAT verbal percentile (VRBPCT)
k) ACT or SAT composite percentile (CMPPCT)
1) Grade received in MSU Math Course(s) (MTHGRD)

Subsequently, each variable was correlated with (b) the
pretest score for the Pendulum Problem, (c) the Equil-
ibrium in the Balance change score (posttest minus pre-
test), and (d) the Pendulum Problem change score.

All correlations involving pretest data from the
Balance and Pendulum Problems were computed on the com-
bined experimental and control groups in order to increase
the size of N and provide for a more stable and reliable
correlation. This procedure is valid due to the random
assignment of subjects to groups, which insures similarity

in performance between groups. The change score

104

correlations were computed separately for each group
because of the possible effects of the intervening
treatment on the experimental group.

The only exception to using both groups for pre-
test correlations occurred for variables (a) through (d),
for which only the experimental group was used. These
variables measure performance in the treatment course
which was taken by the experimental group only.

Correlations were not computed for posttest
scores, since they provide no information about either
level of COgnitive functioning upon entry into the uni-
versity or amount of cognitive change after one term,
which are the major areas of interest of this study.
Therefore, posttest correlations were omitted.
Correlations between Pretest Scores

for theERperimentgl:Group and7
Performance in the Treatment

 

 

Hypotheses 3-10 specify the possible correlations
between pretest scores and four variables measuring per-
formance in the treatment course: grade received (a),
total points earned (b), score on the final exam (c),
and ability to apply problem-solving strategies on the
final exam (d). Table 15 reports these correlations
and their corresponding probability (p) values for the
two tests used.

None of the eight correlations are significant

for p 5 .05, indicating that no relationships exist

105

 

 

 

 

 

 

 

 

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106

between abstract reasoning ability and performance in
the course as measured by these four variables (see .
Table 15). It should be noted, however, that for p 5 .10,
the Equilibrium in the Balance test is significantly cor-
related with each of the four variables measuring course
performance. Although these correlations are not large,
the general result for .05 < p 5 .10 is that the higher
the level of abstract reasoning upon entry into the
course (as measured by the Equilibrium in the Balance
test only), then the higher the course grade, the higher
the total points earned, the higher the final exam score,
and the better the ability to apply problem-solving
strategies on the final. For p 5 .10 no correlations

were significant for the Pendulum Problem.

Correlations between Pretest Scores
Tfioth Groups Combined) and
Variables Measuring Academic

Ability
Null Hypotheses 11-26 state there is no sig-

 

nificant correlation between pretest scores for both
groups combined and eight different academic ability
variables: high school grade point average (e), first-
term MSU grade point average (f), arithmetic placement
score (9), mathematics placement score (h), ACT/SAT
mathematics percentile (i), ACT/SAT verbal percentile
(j), ACT/SAT composite score (k), and grade in math

course taken (1).

107

The null form is rejected at p 5 .05 for
Hypotheses 11-15 and not rejected for Hypotheses 16-26
using the Equilibrium in the Balance measure (see
Table 15). Thus, there are significant positive cor-
relations between pretest level of abstract reasoning
using the Equilibrium in the Balance test and high
school grade point average, first-term MSU grade'point
average, the arithmetic placement score, the mathematics
placement score, and the ACT or SAT Mathematics percentile.
There are no significant correlations between Equilibrium
in the Balance pretest scores and the ACT or SAT verbal
percentile, composite percentile, or grade received in
math course taken. In addition no correlation is sig-
nificant when the Pendulum Problem was the measure used.

These results indicate that the higher the high
school grade point, the higher the MSU first-term grade
point, the higher the arithmetic or mathematics placement
score, or the higher the ACT/SAT mathematics percentile,
the higher the subject's level of abstract reasoning as
measured by the Equilibrium in the Balance pretest.

In addition, the composite ACT/SAT percentile
and the grade received in math course taken are sig-
nificantly correlated with the Balance test for p < .10
and p < .06 respectively, but not for the criterion level

of p 5 .05.

108

Cortelations between Change Scores
for Experimental Group and Per-
formancelin the Treatment Course

 

 

 

No correlation between change scores and the four
variables measuring course performance is significant
for either measure (see Table 16). Since this indicates
there is no relationship between success in the course
and how much abstract reasoning changed, null Hypotheses
27-34 are not rejected for p E .05. However, for p 5 .10
there is a significant positive correlation (r = .34)
between the grade received in the course and the amount
of change in abstract reasoning as measured by the Pen-
dulum Problem.
Correlations between Change in Abstract
Reasoning f6} the Experimental—Group_

or Control Group Separately, and?
Vafiabies Measuring Academic Ability

 

 

For the experimental group, null Hypotheses 35-42
state that the correlations between change in abstract
reasoning as measured by the Equilibrium in the Balance
test and each of the eight academic ability variables
(e through f) are not significant. Null Hypotheses 51-58
specify that the correlations between these variables and
amount of change as measured by the Pendulum Problem are
also not significant. None of these hypotheses are
rejected for p S .05, except for Hypothesis 40, indi-
cating that the amount of change in abstract reasoning

as measured by the Equilibrium in the Balance test is

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110

positively correlated with ACT/SAT verbal percentile
(r = .50, p < .02). See Table 16 for a summary.

For the control group, null Hypotheses 43-50
specify nonsignificant correlations between change in
abstract reasoning as measured by the Balance test and
the eight academic ability variables; null Hypotheses 59-
66 specify nonsignificant correlations between change in
abstract reasoning as measured by the Pendulum Problem
and these same variables. Of these, only null Hypotheses
44 and 50 are rejected for p 5 .05: the amount of change
as measured by the Balance test is negatively correlated
with first-term MSU grade point average (r = -.69),
significant at p < .01, and also negatively correlated
with grade in math course taken (r = -.70), significant
at p < .01. No other correlations for either measure
are significant. Thus, for the control group, the
greater the amount of change according to the Balance
test, the lower the MSU grade point average or the lower
the grade in math course taken.

Although not significant for p 5 .05, the amount
of change in abstract reasoning measured by the Equilibrium
in the Balance test is positively correlated with ACT/SAT
mathematics percentile (r = .50), significant at p < .06,
and ACT/SAT composite percentile (r = .43), significant

at p < .10.

111

Summary of Correlations

Of the 64 correlations computed, 8 are significant
for p 5 .05 (Hypotheses 11, 12, 13, 14, 15, 4o, 44, and
50); an additional 9 are significant for p 5 .10
(Hypotheses 3, 4, 5, 6, 17, 18, 31, 47, and 49).

The correlations significant for p i .05 are
as follows: for both groups combined, the greater the
high school grade point average, the greater the first-
term MSU grade point average, the greater the arithmetic
and mathematics placement scores, and the higher the
ACT/SAT mathematics percentile, then the higher the level
of abstract reasoning on the Balance pretest. Also, for
the experimental group, the higher the ACT/SAT verbal
percentile, the more the change in abstract reasoning
as measured by the Balance test. However, for the con-
trol group, the higher the MSU grade point average or
the higher the math grade, the less the amount of change
in abstract reasoning on the Balance test.

For those correlations significant for p f .10
but not for p 5 .05, it was found that the higher the
level of abstract reasoning on the Balance pretest, the
better the performance in the course according to all
four measures. In addition, for both groups combined,
the higher the ACT/SAT composite score or the higher the
math grade, the higher the level of abstract reasoning

on the Balance pretest. For the experimental group only,

112

the greater the amount of change in abstract reasoning
as measured by the Pendulum Problem, the higher the
course grade. For the control group only, the cor-
relations between ACT/SAT mathematics percentile or
ACT/SAT composite percentile and amount of change on

the Balance test are positive.

Analysis of Variance

The last part of the analysis of the data
(Hypotheses 67-82) addresses the question of whether
there is an interaction between treatment (control or
experimental) and level of academic ability (High or
Low). Separate analyses were conducted for each of the
following academic ability variables: high school grade
point average (e), MSU first-term grade point average
(f), arithmetic placement score (g), mathematics place-
ment score (h), ACT/SAT mathematics percentile (i),
ACT/SAT verbal percentile (j), ACT/SAT composite per-
centile (k), and grade in math course taken (1). A
significant interaction effect indicates that the mean
change in abstract reasoning scores for at least two
of the following groups are significantly different:
(a) experimental-high, (b) experimental-low, (c) control-
high, and (d) control-low.

Analysis of the data reveals that none of the

hypotheses are rejected for p S .05, indicating that mean

113

change scores are not significantly different for the
four groups. Table 17 summarizes these results per
hypothesis.

Secondary analyses were conducted to determine
if any of the high ability groups had significantly dif-
ferent change scores from any of the corresponding low
ability groups. The data were divided into either the
high or the low group without regard to membership in
the experimental or control group. The only significant
difference in change in abstract reasoning between high
and low groups occurs for ACT/SAT verbal percentile for
the Equilibrium in the Balance test. For this variable,
F = 4.81 for p < .04. Thus, the mean change in abstract
reasoning for those who scored high on the ACT/SAT verbal
percentile (0.62) is significantly greater than the mean
change for those who scored low (0.07).

For p 5 .10 the mean change in abstract reasoning
is greater for the low group for the variable MSU grade
point average (low = 0.57, high = 0.08) on the Equil-
ibrium in the Balance Problem (F = 3.94, p < .06).

Two addition descriptions are helpful in synthe-
sizing these data. First, it is interesting to observe
whether the high or low ability group has the greater
change in abstract reasoning score, irrespective of
membership in the control or experimental group.

Column 3 in Table 17 reports which group has the

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115

higher change score. It indicates that the low group
has a higher mean change score six out of eight times
on the Equilibrium in the Balance test and only three
out of eight times on the Pendulum Problem.

The second descriptive summary that is useful is
to determine which group, high or low, has the greater
change when subjects are divided by experimental and
control group (columns 4-9, Table 17). On the Equil-
ibrium in the Balance test, the experimental group-low
scores are greater than the experimental group-high
scores six out of eight times. The control group-low
scores are greater than the control group-high scores
only four out of eight times. For the Pendulum Problem,
the experimental group-low subjects score higher four
out of eight times, while the control group-low subjects
score higher only three out of eight times. These data

are summarized in Table 18.

Table 18

Number of Ability Measures out of Eight for Which
Indicated Group's Change Score was Greater

 

 

 

 

Measure
Group
Equilibrium

in the Balance Pendulum

Low High Low Hig
Control 4 4 3
Experimental 6 2 4
Both 6 2 3

 

 

 

116

Summary

The analysis of the data addressed three major
issues:

1. Is the treatment effective in increasing the
abstract reasoning of experimental group subjects?

2. What are the correlations between either
pretest scores or change scores and various ability
variables?

3. Is there an interaction between treatment and
ability level?

The results of the analysis of the effect of the
treatment were divided into two parts: statistical results
and descriptive results. The statistical results indi-
cate that (a) the change in abstract reasoning for the
experimental group as measured by both the Equilibrium
in the Balance test and the Pendulum Problem is not sig-
nificantly different from the change in abstract reason-
ing for the control group; (b) for both measures, the
posttest level of abstract reasoning for the control
group is not significantly greater than its pretest
level, but that (c) the posttest level of abstract
reasoning for the experimental group is significantly
greater than its pretest level for the Balance test,
but not significantly different for the Pendulum Problem.

The descriptive results, which report the magni-

tudes of differences between the two groups rather than

 

 

 

117

statistically significant differences, indicate that

(a) the average pretest level of cognitive functioning
for both groups is concrete-operational, (b) the average
posttest measure is higher than the pretest measure for
both groups and for both tests, but it remains below the
formal-operational level, (c) the change from pre- to
posttest is greater for the experimental group on both
tests used, (d) the percentage of students who are not
formal-operational decreases more from pretest to post-
test for the experimental group according to both tests,
and finally (e) a larger percentage of students in the
experimental group increase in level of abstract reason-
ing from pre- to posttest on both measures used.

The results of the pretest correlations show that
high school grade point average, MSU first-term grade
point average, arithmetic placement score, mathematics
placement score, and ACT/SAT mathematics percentile are
all positively correlated with the Equilibrium in the
Balance pretest and are significant for p f .05.

The significant correlations for the change
score are ACT/SAT verbal percentile for the experimental
group (r = .50), MSU grade point for the control group
(r = -.69), and grade received in mathematics course
taken for the control group (r = -.70). These are all
significant for p 5 .05 for the change score as

measured by the Equilibrium in the Balance test.

118

The analysis of variance results indicate that
there is no significant treatment-by-ability-level
interaction, indicating that the mean change in abstract
reasoning is not different for ability level by group.
Irrespective of group, however, students with high
ACT/SAT verbal percentiles change significantly more
than those with low percentiles for the Equilibrium in
the Balance test.

Chapter V will summarize this study, discuss its
limitations, and offer recommendations for further

research.

CHAPTER V

SUMMARY AND CONCLUSIONS

In this chapter the purpose and need for this
research are summarized, the limitations of this study
are discussed, the conclusions drawn from the data
analysis are presented, and implications for further
research are outlined.

Purpose and Need for
the Study

 

 

The purpose of this study is to determine if a
course, "Introduction to Reasoning and Problem Solving,"
increases the Piagetian-defined level of abstract reason-
ing of a group of first-term minority engineering stu-
dents. The design of the course is based upon impli-
cations for instruction derived from Piaget's work. The
major assumption underlying this research is that it is
necessary for students to function at the highest levels
of abstract reasoning in order to succeed in engineering.
However, the literature clearly documents that a sur-
prisingly high percentage of college students do not

meet this criterion. In addition, limited work has

119

120

been conducted to describe effective instructional strate-

gies for increasing abstract reasoning.

Limitations of the Study

 

The limitations of this study are: (a) the mor-
tality of subjects resulting in the small sample size,
(b) instrumentation biases, (c) the lack of data concern-
ing the reliability of the tests used to assess formal-

operational thinking, and (d) the use of one-item tests.

Mortality_of Subjects

 

Of the 60 students in the original sample, 27
completed all pre- and posttesting. The mortality of
subjects was due to (a) students who did not attend MSU
during Fall term (N=9), (b) students who had scheduling
problems (N=4), (c) those who declined participation
(N=4), and (d) those who did not complete all pre- or
posttesting (N=l6). The resulting sample size of 27
students is small and limits the scope of this study.

In retrospect, it would have been difficult to
decrease the magnitude of the mortality of subjects.
However, if the random assignment to the experimental
course had been delayed until the day before classes
began, the nine students who did not enroll for Fall
could have been excluded entirely. In addition, this
procedure would have enabled the researcher to conduct

only one random assignment of subjects to group, instead

121

of the procedure used of assigning students randomly
before summer orientation, then alternating new students
between groups as they appeared at orientation during the
summer. Conducting only one random assignment would have
introduced administrative problems for registration, but
it would have provided a "cleaner" procedure.

Decreasing the mortality due to incomplete test-
ing would also have been difficult, but a more persistent
follow-up of students would have helped.

Although it was shown in Chapter III that there
were no statistical differences between groups before
mortality or after mortality on any of six variables,
the mortality of subjects must be considered a limitation
of this study because of the resultant small sample size

of 27 students.

Instrumentation Biases

 

Biases due to instrumentation procedures are the
second limitation of this study. Because the researcher
conducted and interpreted all pre- and posttesting with
knowledge of which subjects belonged to which group,
there is a possibility that the procedure caused either
over- or under-compensation for the experimental group,
thus biasing the data.

The obvious means of avoiding this difficulty is
to employ a person to administer and interpret all tests.

This person should not be otherwise involved in the study

122

and should not know to which group each subject belonged.
This procedure was not used because of the lack of
trained personnel to devote 40-60 hours testing stu-
dents. Thus, the experimenter's involvement in the
testing procedures could have biased the results and

is a limitation of this study.

Reliability

 

The testing of students to assess level of
abstract reasoning did not include a second scorer,
whereby estimates of the inter-rater reliability of the
tests used could be determined. The second scorer was
not used because of the lack of trained personnel and
money. Additionally, no test reliability data are
available due to the use of one-item tests. None the
less, since reliability information is unavailable, a
limitation of this study is that it is not known if
the tests are a reliable measure of abstract reasoning,

nor if the rater was reliable in scoring.

Use of One-Item Tests

 

Both the Equilibrium in the Balance test and the
Pendulum Problem can be considered one-item tests, that
is, each presents only one problem for the student to
solve. Students are classified as formal or pre-formal
thinkers on the basis of their solution to the one

problem. Generally, one-item tests are not sufficient

123

to assess the presence or absence of such a complex con-
struct as formal reasoning; one-item tests also tend to
reduce reliability. Therefore, the one-item tests used
in this study limit the sc0pe of the interpretations of
the results. Conclusions can only be specified with
reSpect to the test used and should not be generalized
to refer to the overall construct of formal-operatiOnal
reasoning.

Unfortunately, acceptable alternatives to the
one-item tests are not available. It is possible to use
more than a single one-item test and sum the scores on
each test. Many researchers employ this technique to
avoid one-item test limitations. This is, however,
questionable procedure and was not used in this study
because summed scores are difficult to interpret. For
example, a summed score of six from two one-item tests
could be obtained by a score of three on each test or
by a score of two on one test and four on the other. In
the first case, a score of three on each test corresponds
to the first level of formal-operational thought and is
not difficult to interpret; the scores of two and four,
on the other hand, refer to concrete-operational thought
and formal thought, respectively. It is unclear how the
summed score of six should be interpreted in the later
case. Thus, summed scores are ambiguous and are not used

in this study.

124

Although not a limitation of the study, it should
be noted that the experimental design did not control for
the possible contaminating effects of self-paced instruc-
tion. It was pointed out in Chapter I that there are a
number of features of self-paced instruction that are
theorized to facilitate the development of abstract
reasoning (it is superior to the lecture method in terms
of final examination performance, it increases the proba-
bility of occurrence of social interaction and transmis-
sion, and it forces students to be active rather than pas-
sive learners). These features were evident in the treat-
ment course. However, students from both the experimental
and control groups were taking as many as two additional
self-paced courses. The possible contaminating effects
of the other self-paced courses are of course unknown.
Theoretically, an alternative experimental design could
have been used by introducing a second control group that
was neither taking the problem-solving course nor any
other self-paced course. This would have provided a
comparison that eliminated the self-paced effect. How-
ever, in reality this was not possible as the mathematics
courses required of everyone were self-paced. Thus, the
possible effects of self-paced instruction upon abstract
reasoning must be left for further research.

In conclusion, the limitations of this study are

the mortality of subjects, instrumentation biases, lack

125

of reliability information, and the use of one-item
tests. Further, because of the experimental design used,
a possible contaminating effect due to the self-paced

mode of instruction was introduced.

Conclusions

 

The conclusions of this study will be presented
in five parts:

1. For the students in this study, what is the
average level of abstract reasoning upon entry into the
university?

2. What are the results of the statistical
analysis of the data to determine the effect of the
treatment upon abstract reasoning?

3. What are the results of the descriptive
analysis of the effects of the treatment?

4. Are there any correlations between various
academic ability measures and either level of cognitive
ability upon entry into the university or amount of
change after receiving the treatment?

5. Are there any interaction effects between
treatment and ability?

Levels of Abstract Reasoning

Upon Entryginto the
UniVersity

 

 

 

The literature pointed out that from 42% to 51%

of college students have not developed formal-operational

126

thinking abilities. At least with respect to the stu-
dents in this study, a much larger percentage was clas-
sified at levels below formal-operational upon entry
into the university: for the Equilibrium in the Balance
test, 85% of all subjects were not formal—operational
thinkers, while on the Pendulum Problem, 89% were not
formal-operational. In addition, when the average pre-
test Piaget-level for all 27 subjects is considered,
the Equilibrium in the Balance test score is 1.93 and
the Pendulum Problem score is 2.15. Both of these
scores put the average level of cognitive ability at
II-B, or the higher of the two concrete-operational
levels. Thus, these results indicate that this sample
is extremely ill-equipped to handle an engineering

curriculum.

Effect of the Treatment--
StatisticaI'Results

 

The results of the analysis of the data indicate
that when the amount of change in abstract reasoning is
compared between groups, no statistically significant
treatment effect is found. However, when the amount of
change within each group is analyzed, the experimental
group did change significantly from pretest to posttest
on the Equilibrium in the Balance test but not on the
Pendulum Problem. The control group did not change sig-

nificantly on either test.

127

Effect of the Treatment--
Descriptive Resfilts

 

 

The descriptive analysis of the treatment reports
magnitudes of change rather than statistically significant
or nonsignificant changes. The magnitudes of all descrip-
tive results are greater for the experimental group:

(a) the amount of change in average level of abstract
reasoning from pretest to posttest is greater for the
experimental group on both tests used; (b) the decrease
from pretest to posttest in percentage of students who

are not formal-operational is greater for the experimental
group, or conversely, the percentage increase from pre-
test to posttest of formal-operational students is

greater for the experimental group; and (c) a larger
percentage of the experimental subjects increased in

level of abstract reasoning from pretest to posttest.

Correlations

 

Sixty-four correlations were computed to deter-
mine if there are any relationships between various aca-
demic ability variables and either level of abstract
reasoning upon entry into the university or amount of
change observed. It was found that high school grade
point average, arithmetic placement score, mathematics
placement score, and ACT/SAT mathematics percentile were
all positively correlated with the Balance pretest

(significant for p 5 .05). Thus, these scores could

128

be used to predict level of cognitive ability upon entry
as measured by the Balance test. First-term MSU grade
point average was also positively correlated with the
Balance pretest (significant for p E .05), but obviously
cannot be used as a predictor of pre-first-term cognitive
ability.

For the experimental group, ACT/SAT verbal per-
centile was significantly correlated in the positive
direction with amount of change according to the Balance
test and could be used as a predictor. Only MSU grade
point average and grade in first-term math course were
negatively correlated for the control group on the
Balance test (significant at p 5 .05). Once again,
these grades cannot be used for pre-term predictors.
This negative correlation could be an artifact due to
a ceiling effect: the better math students were also
those who scored higher on the pretest; hence, the
amount of change possible was limited by high pretest
scores.

Interaction Effects--Treatment
vs. Ability

 

There were no significant differences between
treatment-by-ability means, indicating that there was
no difference in change in abstract reasoning regardless
of ability level or treatment. However, it was found

that for those subjects who scored high on the ACT/SAT

129

verbal percentile (irrespective of group), their mean
change in abstract reasoning on the Balance test was
significantly greater than the mean change for those
who scored low.

One possible explanation for the lack of dif-
ferences in the four means (experimental-high,
experimental-low, control-high, and control-low) is
that, in general, the ability measures on these students
were low to begin with. Thus, splitting the already low
ability groups into high and low subgroups produced an
artifact: the high students were not that much aca-
demically superior to the low students. Thus, it remains
possible that for a group of students with a wider range
in academic ability, a treatment-by-ability interaction
might occur. I

The significant difference between amount of
change on the Balance test between high and low ACT/SAT
verbal percentile subgroups is surprising in light of
the fact that for all of the other ability measures, change
scores between the high group and the low group were not
significantly different. It is possible that since most
classroom learning is dependent upon the ability to read
and comprehend the written word, that those students with
the high verbal percentiles were better able to assimilate
new information into existing schema and thus had a

greater probability for cognitive change. Therefore,

130

their change scores were significantly higher than the
change scores of those who were less able to read and
comprehend. It is suggested that an area for further
study address possible changes in abstract reasoning

between high and low verbal ability students.

Implications for Further Research and Instruction

 

Recommendations for further research and instruc-
tion focus upon three issues: (a) changes in the instruc-
tional format, (b) changes in the research design, and
(c) questions left unanswered by this study.

Changes in the Instructional
Format

 

Changes in the instructional format of the
problem-solving course should be explored. Based on
McKinnon's (1970) findings that inquiry-oriented
instruction facilitates the development of abstract
reasoning, an attempt should be made to introduce inquiry-
oriented sessions into the course. Although the course
is self-paced, one class hour per week could be used as
an instructor-lead inquiry session, in which the instruc-
tor could utilize inquiry techniques to assist students
in discovering processes of solving different classes
of problems. Not only would students discover solutions
to problems, but they would also see how others attempt
to solve them. In the event of a correct solution, stu-

dents would have a model to follow; for incorrect

 

millinfl, I

 

 

131

procedures students could see where the errors were
and possibly receive feedback on their own methods of
attack.

A second change based upon Renner and Lawson's
(1975) findings would be to introduce laboratory-type
"hands-on” experiences. Many of the problems in the
course are the kind that can actually be worked as an
experiment, providing the student with the opportunity
to handle and manipulate concrete rather than hypothe-
tical or verbal problems.

A third change would be to add to the text more
intermittent practice frames with immediate feedback,
increasing the amount of time students spend practicing
the skills that are descriptive of the formal stage of
reasoning (e.g., hypothesis building and testing, ”If
. . ., then . . ., therefore" reasoning, etc.). The
immediate feedback would function as a catalyst to
create the disequilibrium necessary for cognitive growth
to occur.

It is suggested that inquiry sessions, "hands-on"
experiences, and additional practice frames with feedback

should be included in the course.

Changes in the Research Design

 

The major question that was left unanswered by
this study concerns the possible effects self-paced

instruction has upon the development of abstract

t... .mlIIIIIIHIM

 

132

reasoning. To research this issue, it is suggested that
an analysis of variance design be used with four groups:
(a) students taking the problem-solving course self-
paced, (b) students taking the problem-solving course

by the lecture method, (c) students not taking the
problem-solving course but one other self-paced course,
and (d) students taking neither the problem-solving
course nor a self-paced course. This type of design
would provide information about the change in abstract
reasoning between all combinations of these four groups.
Conclusions could be drawn about the effects of problem-

solving instruction and the self-paced format.

Qgestions Left Unanswered

 

An additional question that was not addressed in
this study concerns the general rate of cognitive
development and whether or not university instruction
in general has an impact. For example, it was noted in
this study that the magnitude of the mean Piaget-level
increased for both the control and experimental group on
both tests from the pre- to post-measures. The question
to be answered is whether or not university students
change any more or any less than a comparable group of
subjects not enrolled in a university. Answers to this
question would provide information concerning the impact
of university instruction in general on the development

of abstract reasoning.

133

A second issue that should be addressed concerns
the low correlations between scores on the Equilibrium
in the Balance test and the Pendulum Problem. The cor-
relation between the two tests is 0.27 for the pretest
scores and 0.11 for the posttest scores. Since both tests
do in fact measure formal-operational reasoning, it is
expected that each correlation would approach one. Since
the correlations do not approach one, it is suggested
that students may not have completely understood all
terminology used in the tests. Without understanding
the problem to be solved, it is doubtful that it would
have been solved, resulting in classifications of less
than the formal category. It is suggested that research
be conducted to determine if this in fact was the case.

Another area that should be addressed in more
detail is the interaction effects between treatment and
ability. As was pointed out previously, a possible reason
for the lack of differences between the mean abstract
reasoning changes between treatment by ability groups
is that the average score on all ability measures was
relatively low. Thus, dividing an already low group
into high and low subgroups did not produce any real
contrast between high and low ability students, as they
were all "low." Further research should address the

issue of treatment-by-ability interactions with groups

134

that range over the entire continuum of academic ability,
rather than only considering the lowest percentiles.

The last question that needs to be answered con-
cerns the measurement of abstract reasoning. Because
Piaget specified one-item tests to use in assessing
abstract reasoning, traditional means of assessing the
reliability of the tests are not adequate as they depend
upon utilizing multiple tests that are equivalent. It
is not clear at the present time that all Piaget tests
are equivalent for reliability purposes. In fact, the
correlations between the two tests used in this study
were low. Although this low correlation could have been
due to students not completely understanding the problem,
it could also have been due to the tests measuring dif-
ferent aspects of the construct formal reasoning.

Thus, additional research needs to be conducted
on the reliability of the Piaget tests and the equivalency

of various tests.

Speculations

 

To some extent the results of this study are dis-
appointing--there is only slight evidence that the
problem-solving course provided a catalyst for intel-
lectual development. Aside from the careful statistical
analysis presented here, the author's personal reaction
to the effects of the course are mixed. On one hand,

some students were never able to transcend their

 

 

 

135

concrete-operational mode of thinking. To them the
techniques of problem solving were something to be
memorized and strategies remained foreign to their mode
of thinking. On the other hand, some students did
appear to be able to internalize the various techniques
and in the process were able to accept and experiment
with hypothetical and abstract thinking.

One can speculate as to the reasons why some stu-
dents changed and others did not. On a purely intuitive
level, the author feels that the major differentiating
factor between those students who changed and those who
did not was the extent to which students had mastered
the abilities that define concrete-Operational thinking
(decanters perceptions, attends to transformations,
reverses operations, etc.). The more of these abilities
that were mastered, the greater the probability that
change occurred; conversely, students who had not
mastered sufficient prerequisite abilities did not
change. In short, the course did not remediate for
insufficient knowledge of the prerequisite abilities
from the concrete-operational stage.

The obvious course of action required to remedy
this situation is that a diagnostic system needs to be
developed that not only identifies at which level a
student is, but also specifies what skills within the

level the student has mastered. Once students are

136

specifically diagnosed, instruction can be prescribed

specific to the diagnosis.

Summary

The limitations of this study are the mortality
of subjects resulting in a limited sample size, instru-
mentation biases, the lack of information about the
reliability of the tests and scores, and the use of
one-item tests. Keeping these in mind, the major con-

clusions of this study are:

1. Upon entry into the University, an extremely
high percentage of the minority engineering students in
this sample are functioning below the formal-operational
level. Thus, these students are not equipped with the
abstract reasoning capabilities needed for success in
engineering. Eighty-five percent are pro-formal-
operational on the Balance test and 89% are pre-formal-
operational on the Pendulum Problem.

2. For the Equilibrium in the Balance test, the
following findings apply:

(a) the change in abstract reasoning from pre-
test to posttest is not significantly greater
for the experimental group as compared to
the control group;

(b) the posttest level of abstract reasoning is
not significantly greater than the pretest
level of abstract reasoning for the control
group; but

(0) the posttest level of abstract reasoning is
significantly greater than the pretest level

of abstract reasoning for the experimental
group.

 

apply:
(a)

(b)

(C)

(a)

(b)

(e)

5.

137

For the Pendulum Problem, the following

the change in abstract reasoning from pretest
to posttest is not significantly greater for
the experimental group as compared to the
control group;

the posttest level of abstract reasoning is
not significantly greater than the pretest
level of abstract reasoning for the control
group; and

the posttest level of abstract reasoning is
not significantly greater than the pretest
level of abstract reasoning for the experi-
mental group.

Descriptive data indicate that:

the change in average level of abstract
reasoning from pretest to posttest is
greater for the experimental group on both
tests used;

the decrease from pretest to posttest in per-
centage of students who are not formal-
operational is greater for the experimental
group; and

a larger percentage of the experimental sub-
jects increase in level of abstract reasoning
from pretest to posttest.

The significant correlations for academic

ability variables with the Equilibrium in the Balance

test are:

(a)

(b)

high school grade point average, arithmetic
and mathematics placement scores, and ACT/SAT
mathematics percentile are all positively
correlated with entry-level abstract reason-
ing ability;

ACT/SAT verbal percentile is positively cor-
related with amount of change in abstract
reasoning for the experimental group;

138

(c) entry-level abstract reasoning ability is
positively correlated with first-term MSU
grade point average;

(d) MSU grade point average and grade in first-
term mathematics course are both negatively
correlated with amount of change in abstract
reasoning for the control group. (As was
pointed out, this may be due to a ceiling
effect on the better students.) ‘

6. There are no significant correlations for

academic ability variables with the Pendulum Problem.

7. There are no treatment-by-ability interaction
effects; the amount of change in abstract reasoning is not
different for high or low ability students within or
between groups.

8. High ability students on the ACT/SAT verbal
percentile change significantly more than low ability
students.

A number of suggestions have been presented to
improve the course, to improve the research design, and
to answer questions left unanswered by this research.
When implemented, these recommendations may provide a
more definitive understanding of changes in abstract

reasoning.

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