. .r‘flm . - .-. . _-... __-,._.~..,....““

A STUDY OF THE EFFICACY for A‘GRADUATE ~   '
MATHEMATICS METHODS COURSE IN CHANGING m‘é ,
SERWCE ELEMENE'ARY TEACHERS’ ATTITUDESz. ‘ ' '

TOWARDS SCHOOL MATHEMATICS ‘

Thesis for the Degree of Ph.. D.
MICHIGAN STATE UNIVERSITY
CURTISS EDWEN WALL ,
1972

  
      

  

LIBRAR 3/

Michigan State
University

This is to certify that the

thesis entitled

A STUDY OF THE EFFICACY OF A GRADUATE
MATHEMATICS METHODS COURSE IN CHANGING
IN-SERVICE ELEMENTARY TEACHERS'

ATTITUDES TOWARDS SCHOOL MATHEMATICS

presented by

CURTI SS EDWIN WALL

has been accepted towards fulfillment
of the requirements for

PH.D. fiegree in ELEMENTARY EDUCATION

 

 

 

Date 10-11-72

 

0-7639

 

 

it”

r—JE

ABSTRACT

A STUDY OF THE EFFICACY OF A GRADUATE MATHEMATICS
METHODS COURSE IN CHANGING IN—SERVICE ELEMENTARY
TEACHERS' ATTITUDES TOWARDS SCHOOL MATHEMATICS

BY

Curtiss Edwin Wall

In recent years there have been three related
trends in elementary school mathematics: increasing
criticism of present practices, the use of activity
materials, and researching the role of attitudes. The
purpose of this study was to investigate the effects of
activity materials on attitudes of in-service elementary
teachers while enrolled in Methods and Materials of
Mathematics in the Elementary School, Education 830A, at
Michigan State University.

The course is offered for a three hour block of
time weekly for ten weeks. The contents of the course
may vary slightly from instructor to instructor but
includes content topics appropriate to the elementary
school and methods for teaching these topics. Three
instruments were used for measurement in this study:

(1) Dutton's Attitude Inventory, (2) an accompanying
informational questionnaire, and (3) M. J. Dossett's Test

of Mathematical Understanding.

l"

 

 

::e p:
"H
tor:

the a1

Studer
PIEpa:

Use ma

Curtiss Edwin Wall

Fall term, 1971, there were two intact groups
available for this experiment. During the first class
the pre-tests of attitude and mathematical understanding
(Form A) were administered. At the last class meeting
the attitude survey, the questionnaire, and Form B of the
test of mathematical understanding were administered.

The questionnaire contained information on the
student's teaching experience and level of mathematics
preparation. The attitude test was administered before
the mathematical understanding test to avoid the students'
feelings about the test to influence their answers on the
attitude scale. They were told that the results were for
the administrator's purposes only.

The concerns of the study were expressed in the
following hypotheses:

1. There is no difference in attitude between
pre—and post-tests of Dutton's attitude
inventory for the in—service teachers.

2. There is no difference in achievement
between pre—and post—tests of Dossett's.
test of mathematical understanding for in-
service teachers.

3. The attitudes of invservice elementary
teachers are not changed by an increase
in mathematical understanding on the
achievement and attitude pre— and post«

tests.

4. All groups are from the same population
on the attitude pre—tests.

5. All groups are from the same population
on the achievement pre—tests.

Curtiss Edwin Wall

6. There is no correlation between any of the
fourteen variables (sex, degree program,
age, years of teaching experience, years
of high school mathematics, credits of
college mathematics, grade level aSsignment,
mathematics vs. non-mathematics teaching,
years at present grade level assignment,
perceived attitude, grade level attitude
developed, average mathematics grades, atti-
tude, and mathematical understanding) taken
two at a time. ’

To test the first two hypotheses a repeated measures
design was employed. This analysis will allow the testing
of interaction effects as well as main treatment effects.
The third hypothesis was tested with a cross—lagged panel
correlation. Also four intact groups during early January
(Winter term) were tested with Dutton's Attitude Inventory,
the questionnaire, and Form A of Dossett's Test of Mathe-
matical Understanding. To test whether the sampled groups
were from the same population analysis of variance was used.
Since attitude and mathematical understanding are of
interest, equality of means for both attitude and mathe-
matical understanding were tested. If there is no differ—
ence then external validity will have been increased. In
order to assess any possible relationship between the
fourteen variables a Pearson product—moment correlation
coefficient was found for each pair of variables.

Data obtained from the various tests indicated
the following:

1. There was a significant positive change

in the attitude towards mathematics of
in—service elementary teachers.

Curtiss Edwin Wall

There was no significant change in the
mathematical understanding of in-service
elementary teachers.

There was no reason to infer that posi-
tive attitude promoted learning nor did
achievement promote more positive attitude.

There is no reason to suspect that the
experimental groups were in any way dif-
ferent from the population at the begin-
ning of the experiment.

A STUDY OF THE EFFICACY OF A GRADUATE MATHEMATICS
METHODS COURSE IN CHANGING INnSERVICE ELEMENTARY

TEACHERS' ATTITUDES TOWARDS SCHOOL MATHEMATICS

BY

Curtiss Edwin Wall

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Elementary and Special Education

1972

 

(£79574

Dedicated to my Wife,

Diane

ii

 

 

cornitt
VidEd If“;
The Wri
5731-536 by
were me:

their ti

Cole! RO
inuah
diSSerta.

ACKNOWLEDGEMENTS

The researcher owes a great debt to Dr. Perry E.
Lanier, whose knowledge, assistance, and encouragement
made this study possible. He also wishes to acknowledge
the special assistance of another member of the doctoral
committee, Dr. Calhoun C. Collier, who graciously pro—
vided materials, time, and his classes for this study.
The writer also wishes to acknowledge the contributions
made by Dr. Dale Alam and Dr. Edward A. Nordhaus, who
were members of his committee and gave generously of
their time and talents in guiding the study.

The writer is deeply appreciative of Drs. William
Cole, Robert W. Scrivens, and Waldeck Mainville who made
invaluable suggestions in the initial stages of the
dissertation. Dr. Mainville's editorial comments were
extremely helpful. Dr. Scrivens and Dr. Howard Hickey
were very helpful in allowing their classes to partici-
pate in the study. The writer also thanks the one hun—
dred twenty-two students who participated in the study.

Diane E. Wall, the researcher's wife, participated
in the editing and preparation of the manuscript and for

this the writer is deeply grateful.

iii

TABLE OF CONTENTS

DEDICATION O O O O O O O O O O O O O O

ACMOWIJEDGEMENTS O O C O O I O O O O O 0

LIST OF TABLES O O O O O O O O O O O O 0

LIST OF APPENDICES . . . . . . . . . . .

Chapter

I.

II.

III.

THE PROBLEM. . . . . . . . . . .

Definitions . . . . . . . . .
Need for the Study. . . . . . . .
Theory. . . . . . . . . .

Purpose of the Study . . . . .
Statement of the Problem. . . .
Hypotheses . . . . . . . .
Overview . . . . . . . . .

REVIEW OF LITERATURE. . . . . . . .

Historical Perspective on Attitude . .
Relationship of Mathematical Attitude,
Understanding, and Achievement . . .
Changing Attitudes Toward Mathematics .
Elementary School Classroom . . . .
Middle School or Junior High Classroom
High School Classroom . . .
College Students--Non Teachers
Pre-service Teachers .
In-service Teachers. . . .
Summary . . . . . . . .

DESIGN OF THE STUDY . . . . . . . .

The Sample . . . . . . . . . .
The Course . . . . . . . . . .
Measures . . . . . . . . . . .
Experimental Design . . . . .

Mathematical Understanding and Attitude
Equality of Groups . . . . . .
Relationship of Selected Variables. .
Basic Assumptions . . . . . .

' Limitations of the Study . . . . .
Testable Hypotheses. . . . . . .
Summary . . . . . . . . . . .

iv

Page
ii
iii
vi

vii

15
30
3O
31
33

35
35

46
57
57
61
62
63
63
77
80

81

81
82
84
86
87
91
92
93
94
94
96

Chapter

IV. ANALYSIS OF THE DATA. . . .

V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

REFERENCES

APPENDICES

The Sample . . . . . .
Assumptions . . . . . .
Change in Attitude. . . .

Change in Mathematical Understanding.

The Relationship of Attitude
Mathematical Understanding
Reliability . . . . . .
Equivalence of Groups. .
The Relationship of Selected
Summary . . . . . . .

Summary . . .
The Problem .
The Literature
The Sample. .
The Course. .
The Measures . .
Experimental Design.
Testable Hypotheses.
Results. . . . .

Conclusions . . . .

Recommendations. . .

and

Variables

Page
97

98
108
109
111

114
115
115
118
121

122

122
122
123
126
126
126
127
128
129
130
139

141

157

Table

15.

16.

17.

LIST OF TABLES

Repeated Measures Design . . . .
Cross-lagged Panel Correlation . .

Sex and Age . . . . . . . .

Term Hour Credits of College Mathematics

Years of High School Math and Grades

Teaching Experience . . . . .

Means and Values of Selected Variables

Grade in Which Attitude was Developed
Skewness and Kurotosis . . . .
Mathematical Understanding Pre-Tests
Analysis of Variance . . . . .
Analysis of Variance . . . . .

t-Test for Mathematical Understanding

Pearson Product-Moment Correlations for
and Post-Tests of Attitude and Mathematical

Understanding for Groups 1 and 2
Analysis of Variance for Attitude

Analysis of Variance for Mathematical
Understanding . . . . . . .

Pearson Product—Moment Correlations

vi

Pre-

Page
88
89

101
102
103
104
105
107
109
109
110
111
113

116

117

117

119

LIST OF APPENDICES

Appendix Page
A. Course Outline and List of Activity

Materials. . . . . . . . . . . . 157

B. Sample Activity Sheets . . . . . . . . 159

C. Instruments. . . . . . . . . . . . 166

vii

CHAPTER I

THE PROBLEM

In the years since Sputnik many new curriculum
programs in mathematics have been created. These programs
have been produced by many different people with different
philosophies for children, adults studying mathematics,
pre-service teachers, and in—service teachers. In all of
this mathematics educators seem to have forgotten to make
any direct provisions concerning attitudes. Yet attitudes
seem to be involved in every learning situation. Kinney
and Freeman (1945) wrote:

The importance of this aspect of the learning
situation may be seen when we consider the large
number of adults and pupils in the upper grades
whose emotional blocks render them incompetent
in the field of mathematics regardless of intel-
lectual level. The little that has been done
in tracing back these blocks to their causation
reveals that they stem in large part from a failure
of the school to recognize the importance of the
feeling tone which is attached to any classroom
activity (p. 385).

Additionally,it is self—evident that there is no choice
about teaching attitudes. In fact regardless of subject

matter attitudinal changes are taking place in the class«

room no matter what might be our personal conviction.

 

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ofour tee
Uohnson,
Te
professors
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Start

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The develOpment of attitudes is such an important aspect
of our teaching that it should be cultivated by design
(Johnson, 1957).

Teachers have long been admonished by education
professors to teach for creativity, appreciation, and
positive attitudes while at the same time (according to
their students) the professors have been singularly tedious
and uninteresting. This has inevitably led to the same
type of teaching in the schools. As Dienes (1970) noted:

On the whole, a teacher will teach as he was

taught himself. He remembers how he was taught
and even without thinking he will carry on in
the same way. If he was taught in school
through lectures he will tend to lecture to
children . . ., If we wish teachers to be able
to set up concrete problem situations that the
children can manipulate, then they must also
learn to set such concrete situations for them—
selves and to manipulate them themselves. They
must feel in their own skins what it is like to
start from scratch and learn something (p. 265).

Therefore, if efforts are made to change the type
of teaching in the schools, higher education personnel
must change their methods. Consequently, teachers must
experience in a methods class what educators would want
their pupils to experience. In this dissertation the
effects of one such methods course over parameterSv—
attitude toward mathematics and mathematical understanding—-
will be considered. Included will be a study of the rela—
tionship of selected variables to achievement and attitude.

.Learning can be thought of as a reorganization

(change) in behavior, and takes place by wholes (Kinney

and Freeman, 1945). Learning is colored by feelings and

 

ezotion
classro
id to

learnin

emotions. A student not only learns mathematics in the
classroom; he also learns to like or dislike mathematics
and to like or dislike school. This student is also
learning to like or dislike himself. He learns to be a
cooperative member of a group or to work by himself (p.
385).

Educators know very little about the role atti-
tudes play in the learning of mathematics. We are not
even sure whether we can change attitudes or what means
will bring about a change. Researchers in mathematics
education must begin to deal more effectively with the
affective domain. It would seem that this has not been
done so far since the Pennsylvania project (Suydam and
Weaver, 1970) reported inconclusive evidence on a number
of questions concerning attitudes.

This researcher values a positive attitude toward
mathematics. Educators should endeavor to develop "good"
attitudes toward mathematics early in school. It is
important that a student feel good about himself and what
he is doing as well as being able to do it. Even if a
positive attitude does not help students learn mathematics
it will help their self image.

In summary, the following is most appropriate:

The teacher's most important and most diffi—

cult problem is the removal of the child‘s
inhibitions and the restoring, building up, and
maintaining of the child's faith in his own
ability . . . the teacher himself will need a

new faith in the ability of the youth (Allen,
1937, pp. 322-323).

 

 

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E'EIIEriC a

Definitions

 

In mathematics we are used to dealing with a
quantity or a concept that is easily defined. Attitudes
do not lend themselves to definition. An attitude has to
be attached to something (mathematics) with a positive or
negative emotional meaning. In fact Aiken (1970) views
attitudes as being made up of both cognitive and affec—
tive components. An attitude might be described as a
conviction or a belief about something, e.g., to what
extent do I value mathematics? Shaw and Wright (1967)
believe that the proliferation of definitions was caused
in part by the "issue of specificity versus generality
. . . . In this context the issue concerns the degree to
which attitudes may be considered to have a specific
referent . . . . (Specificity) appears to have the
advantage of preventing the construct from becoming so
generic as to be valueless (p. 2)."

The problem of definition becomes apparent when
some theoretical definitions of attitudes are considered.
First, consider how Thurstone (1937) defined attitude.

He used the concept attitude to "denote the sum total of
a man's inclinations and feelings, prejudice or bias,
preconceived notions, ideas, fears, threats and convic—
tions about any specific attitude (pp. 6—7)." Donovan
Johnson (1957) defined attitudes in the following manner:
“An attitude is usually defined as an enduring emotional

set or predisposition to react in a characteristic way

toward a given person, object, idea, or situation
(p. 114)." More recently Aiken (1970) defined an attitude
as "a learned predisposition or tendency on the part of
an individual to respond positively or negatively to some
object, situation, concept, or other person (p. 551)."
Although there are many definitions , Shaw and Wright
(1967) believed there was at least one common element
whose statement they attributed to Cardno. "Attitude
entails an existing predisposition to respond to social
objects which in interaction with situational and other
dispositional variables guides and directs the overt
behavior of the individual (p. 2)." This view is reflected
by Stagner (1937) "the attitude is characterized always by
(1) an objective, (2) direction, and (3) intensity (pp.
167, 186)."
However, for the purpose of psychological testing
"attitudes" must be defined operationally. Consequently
the following definitions will be used:
1. Attitude toward mathematics will be
the score* obtained on the Dutton
Attitude Inventory.

2. A negative attitude in this study
will refer to a score less than 5
on Dutton's Scale.

3. A neutral attitude is a score greater

than or equal to 5 but less than or
equal to 5.9.

*For the method of computation of this score see
the section on Theory of Attitude Measurement.

4. A positive attitude is a score greater
than 5.9.

Similarly "mathematical understanding" must be defined
operationally.

Mathematical understanding is the total score

on Dossett's Test of Basic Mathematical Under-

standing.*

For the sake of clarity and completeness the
following comments on selected concepts are offered. The
Dutton Attitude Inventory is in some sense an opinion
questionnaire. In order to see the relationship between
attitude and opinion Thurstone's (1937) definition is
referred to-—an opinion is "a verbal expression of atti-
tude" i.e., "an opinion symbolizes an attitude (pp. 6—7).".
"Group instruction" will mean that a class has a common
goal which was determined for or by them. In contrast to
this, individual instruction will be taking place if
students have different goals. "Individualized instruc—
tion" will only mean that students are working individually
or in small groups with no reference to goals but at dif«
ferent speeds. The definition of activity learning is
"taken to mean school learning settings in which the
learner develops mathematical concepts through active
participation. This process may involve the manipulation
0f physical materials, the use of games, or partaking in

experiments with physical objects (Kieren, 1969, p. 509)."

¥L

*See Appendix C for Dossett's Tests.

- ‘21

 

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to models ‘
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active lea-
diverse in
I005, QEODC
devices, ar
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materials C
lend themse

criptions Q

This leads to a discussion of the last definition--
that of activity materials. "Activity materials" refers
to models and objects manipulated by students as well as
tools, teaching aids, and literature that is found in an
active learning situation. These materials are very
diverse in nature and include items such as: Cuisenaire
rods, geoboards, numerical games and puzzles, measuring
devices, and Dienes blocks. These materials are either
student produced or purchased commercially and vary in
cost, appropriateness, and objective.« Some of these
materials can be used in group instruction while others
lend themselves more to individual instruction. For des~
criptions of these materials and their use the following
may be consulted: Davidson (1968), Davidson and Fair
(1970), Fitzgerald (1968, 1970), Johnson (1967), Phillips

(1967), and Rosskopf and Kaplan (1968).

Need for the Study

Although mathematics educators may argue about the
exact percentages they do agree that there exist substan-
tial numbers of elementary teachers who have poor attitudes
toward mathematics. [This has been documented by Dutton
(1962), McDermott (1956), and Rice (1965). For instance,
Reys and Delon (1968) in a survey of University of
Missouri education majors found 40 percent had unfavorable
at‘ttitudes. Dutton (1951) found that of the 127 elementary
eci'ucation majors he surveyed 38 percent had poor attitudes

toward mathematics .

 

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is high st;
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Perhaps the dislikeiof mathematics by teachers is
only the tip of the iceberg. A distaste for doing mathema-
tics yet holding mathematicians in awe has been documented by
the Husen Study (1967). In countries where achievement
is high students have a greater tendency to perceive
mathematics as a fixed closed system, difficult to learn,
only for an intellectual elite, and important to the
future (Husen, 1967, p. 45).

Although mathematics educators should be concerned
about this dislike of mathematics by large segments of our
society, the need is even more pressing when considering
the effects upon children. A. S. Peskin (1965) found
correlations between teachers' and students‘ understand—
ings of algekra and geometry were significantly positive.
The correlations between teachers' understanding scores
and students' attitudes were also significantly positive.
Recently, a significant correlation (.05) between teacher
attitude and student attitude has been found by
Phillips (1970). Poffenberger and Norton (1956) have
concluded from their research that student attitudes and
achievement in arithmetic are affected positively by
teachers having the following characteristics: "a good
knowledge of the subject matter, strong interest in the
subject, the desire to have students understand the
material, and good control of the class without being
overly strict (p. 116)." Certainly a teacher with these

characteristics has a positive attitude toward mathematics.

In a study of Anglo—american and Latin American
pupils in first year algebra classes in Texas, Garner
(1963) reported that they achieved more under teachers
highly trained in mathematics. However, the students'
achievement was inversely proportional to the amount of
professional education of their teachers. More importantly
he found that pupils under teachers with favorable atti~
tudes achieved significantly more and had more positive
attitudes than their counterparts who had teachers with
poor attitudes.

In contrast to these positive results several
authors (Caezza, 1970: Wess, 1970; and Deighan, 1971) were
unable to find significant correlations. In a study of
the relationship of teacher attitude to student attitude
and achievement Keane (1969) was unable to find any relav
tionship between teacher attitude and student attitude,
and achievement. Also there was no correlation between
students' attitude and achievement. In fact the only
significant factor affecting attitude seemed to be the
economic area. Because of these negative results a few
words of caution are in order. It is a well known truism
of statistical studies that correlation does not neces-
sarily imply causation. It should also be noted with
respect to the negative results above that the absence of
a known correlation does not mean that one does not exist.

To summarize, Banks (1964) very appropriately

wrote:

10

An unhealthy attitude toward arithmetic may
result from a number of causes. Repeated failure
is almost certain to produce a bad emotional
reaction to the study of arithmetic. But by far
the most significant contributing factor is the
attitude of the teacher. The teacher who feels
insecure, who dreads and dislikes the subject,
forvfluxnarithmetic is largely rote manipulation,
devoid of understanding cannot avoid transmitting
her feelings to the children . . . . On the other
hand, the teacher who has confidence, understand—
ing, interest, and enthusiasm for arithmetic has
gone a long way toward insuring success (pp. l6e17).

It would seem suitable at this time to ask the
question: When are attitudes toward mathematics developed?
Most studies report that attitudes are formed throughout
the school experience. Morrisett and Vinsonhaler (1965,
p. 132) felt that attitudes were traceable to the early
childhood of the adult. However, most studies (Dutton,
1962; Smith, 1964; and White, 1964) report that the grades
four through six were most often cited.

Narrowing the field of concern even further,
consider the reasons why students dislike mathematics.
Interestingly many of the reasons given by students for \
disliking mathematics have remained constant over time.
Florence Young (1932) gave the following reasons for losing
interest in a subject: (a) failure to see need, (b)
uninteresting, and (c) lack of foundation. W. H. Dutton
(1951) found that in measuring attitudes of prospective
elementary teachers there was a high degree of emotionality
accompanying unfavorable attitudes, they stressed their

lack of understanding of the subject, boredom, poor

teaching, and fear of mistakes. Mildred Gebhard (1948 and

 

1949) f
aposit

followe

:udes a
he res
is it 9
traits?
to 346

liii-em: I
were:

rules,

St'ddent
the new
Side Of

11

1949) found that changes in attractiveness were greater in
a positive direction when the experience of success
followed the expectation of failure.
However, the problem of the relation between atti-
tudes and achievement is relevant again. Is motivation
the result of previous success and failure experiences or
is it governed by parental attitudes and personality
traits? Dutton and Blum (1968) submitted a questionnaire
to 346 sixth, seventh, and eighth graders. The most fre—
quent reasons for a poor attitude toward mathematics
were: frustrating word problems, homework, memorizing
rules, and the possibility of making mistakes. These
students felt that mathematics should be avoided and that
the new mathematics was not very useful. On the other
side of the coin some students felt that working with
numbers could be fun, presented a challenge, was practical,
and logical.
Schilhab (1956) found the following reasons for

poor attitudes in mathematics:

(1) Their arithmetic achievement.

(2) Poor foundation.

(3) Belief of no practical application (p. 83).
Also he felt that "students with low grades and low achieve-
ment scores have more negative attitudes, while the students
with high grades and high achievement scores tend to like
mathematics (p. 83)." Aiken and Dreger (1961) reported
that college men who disliked mathematics as contrasted

with those who liked mathematics stated that their previous

“a

 

tathernati

hostile .

the follc
arithmeti

mr' & h
floltlverl r

(1938) fe
by rote 1
F

Jects, Mc
facts-.fr
mi1themati
of falilur
aids Whlc
use of in
He also C
have com
This Cone

:Qllowing

12

mathematics teachers has been more impatient and
hostile.

The students in Smith's (1964) study gave some of
the following reasons for disliking arithmetic: fear of
arithmetic, inadequate teaching, failure, bored teachers,
written problems, and a lack of understanding. Zavitz
(1938) felt that failure to learn mathematics was caused
by rote learning rather than reasoning.

From case studies of his elementary school sub—
jects, McDermott (1958) distilled some enlightening
facts-—frustration, general dislike, and/or fear of
mathematics. He felt that the frustration was the result
of failure to learn fundamental skills, a dependence on
aids which should have been discarded, or the continuous
use of inefficient problem solving methods (pp. 185—186).
He also concluded that students who dread mathematics
have convinced themselves they can not learn mathematics.
This conclusion is arrived at by one or more of the
following roads:

(1) The belief that they are "behind" the

rest of the group therefore the
material is beyond their powers of
comprehension.

(2) Believing they can not fill in a gap

in a learning sequence.

(3) Belief they can not succeed.

Additionally students who fear mathematics doubt the use-
fulness or purpose of mathematics in other than commercial

Settings. It is certainly true, as noted by Billig (1944)

that if the ". . . content to be mastered exceeds the

 

potential a
definitely

In
certain an;
Other auth;
(1961). mi
that drill
learn the j
attitude,

The
effects dOE

 

t0 aVOid ma
much as pos‘
achange ir.
eXperienCeE

Hon
SEIiQUs ran
me" and t
n‘cber an)“.
Bergtein (J

Maintained
I

13

potential ability of the pupil the learning situation is
definitely unfavorable (p. 170)."

In order to master any fundamental process, a
certain amount of drill and rote learning is necessary.
Other authors--Bernstein (1955), Collier (1959), Clark
(1961), Wilson (1961), and Lyda and Morse (1963)--noted
that drill and rote learning beyond what is needed to
learn the fundamental process can produce a negative
attitude.

The most logical question to ask next is, "What
effects does a fear of mathematics have?" One obvious
answer that all can agree on is that students will tend
to avoid mathematics and taking mathematics courses as
much as possible. Donovan Johnson (1957) also noted that
a change in attitude will usually require long and intense
experiences.

However there exists other and potentially more
serious ramifications. In a study on number anxiety
Aiken and Dreger (1957) found that persons with a high
number anxiety tended to make lower mathematical grades.
Berstein (1955) concluded that if a certain feeling is
maintained, i.e., failure, it could lead to a particular
Self-image, in this case a poor self-concept. This self-
COncept will have impact on his actual performance as well
as influence his expectation of future performance.

Other authors agree with this analysis. Alpert

SE_£Q3 (1963) noted a kind of self-perpetuating cycle

l4

linking self-concept, attitude, anxiety, expectation, and
performance. Also, Shapiro (1962) found that elementary
school children who had positive attitudes were more
persevering in solutions to mathematical tasks than those
students who did not like mathematics. Furthermore,
McDermott (1958) stated that students will resort to dis—
honest means if they are sufficiently afraid and frus—
trated. Billig (1944) observed that the amount of effort
students are willing to use in learning tasks will vary
significantly in relation to their attitude. It should
be remembered that the differences in performance between
students will increase as they continue in school. Addi-
tionally, Lyda (1947) noted failure on the part of
students to solve problems results from manipulating
numbers without regard to meaning, not analyzing a prob-
1em, beginning problems before they have a method for
solution, and failure to consider whether or not they
have acquired a reasonable answer (pp. 387—388). In
relating personality variables to attitudes, individuals
with more positive attitudes and higher achievement tend
to have better personal and social adjustment than those
with negative attitudes and low achievement (Aiken, 1970,
p. 556). However, correlation does not imply causation.
These lengthy observations highlight the conclu—
sion that we must begin to deal more effectively with
the attitudes of teachers in order to influence the atti—

tudes of their students, i.e., we must learn how to change

15

attitudes. As Halloran (1967) suggested it is possible to
change attitudes.

According to Bassham, Murphy, and Murphy (1964)
research in the area of attitudes is "of far greater
importance than might be suggested by the rather low
correlations that have been found. Attitude and interest
appear to act as catalyst, making the efficient pupil
utilization of abilities and experience possible (p. 66)."
Also they recommended that research was needed in the
modification of negative attitudes as well as in other
affective variables. There has been little change in the
situation since then. In a recent review of the litera—
ture "Attitudes Toward Mathematics," Aiken (1970) pointed
out that "there has been only a small amount of research
on techniques for developing positive attitudes and

modifying negative attitudes towards mathematics (p. 591)."

Theory

Any educational theory must have a sound psycholo—
gical-philoSOphical basis. The following is based on the
philosophy of John Dewey and the third force psychology
of A. H. Maslow, Carl Rogers, Earl Kelley, and Arthur W.
Combs (Association for Supervision and Curriculum Develop-
ment, 1962). How we act and feel is in large part deter-
mined by our self-concept. Therefore, an individual's
attitude toward mathematics is determined by how he per-

ceives himself in relation to the object mathematics.

! . '1

 

Cur self-c
environmen
Pe
from t
by the
growi:
from t
(ASCD,
7.19. above J
Art.
interactic‘i
POSitive st
fin abilit:
the inlei<
the Characy
“d are dew
apersonal
"R'
Th
1e .

SS lmPOr
etc _
Essectlve
KSlgl‘iifiCa

.
5611.7

83 and

4&3

.aViOr.

(l

 

16

Our self-concept is determined by the manner in which our
environment impinges upon us.

People learn who they are and what they are

from the ways in which they have been treated
by those who surround them in the process of
growing up. People discover their self-concept
from the kinds of experiences they have had
(ASCD, 1962, p. 53).

The above has implications for how people learn.

Among other things this means that we need to have
interaction experiences with other people that are success?
ful. These successful interaction experiences with a
positive self-concept help produce individuals who have
the ability to identify with others. This in turn allows
the individual to have a certain freedom from threat which
enables him to be open and accepting of experience. Thus
the characteristics of Openness and acceptance are learned
and are developed through a positive self and identificav
tion with others. Then behavior can be affected through
a personally significant perception. If learning is seen
as change in behavior then learning should be the
"discovery of personal meaning (Combs, 1965, p. 115)."

Thus the content in a professional program is of
less importance than the student's using himself as an
effective instrument for change. The only important
(significant) experiences are those we discover for our-
selves and only this significant learning will change
behavior. It may be concluded from the above discussion:

(1) That the way humans learn and behave is

a product of the way he perceives the
world.

 

educatio
under hi
teaching

and disc

provided
11ng She
Planning
dSsignme

Work but

l7

(2) Behavior exists in the present and is

modifiable.

(3) A common characteristic of humans is

a basic drive to be healthy and adequate.

(4) Behavior is the product of self—concept

(ASCD, 1962, p. 67).

These conclusions have important implications for
education. Learning takes place inside an individual
under his control rather than that of a teacher. Thus
teaching is not telling but facilitating the deve10pment
and discovery of personal meanings. People will only i
change their behavior when they discover a new and better
way to accomplish their tasks. This comes about through
exploratory behavior (ASCD, 1962, p. 72).

For the classroom it means students need to be
provided with freedom to explore and work. Teacher plan—
ning should be replaced by teachervpupil and individual
planning with choices in tasks rather than typical
assignments. However it does not mean the end of group
work but there should be areas for individual work. It
also means that a wide variety of materials should be
available for student use. The learning situation should
be structured so that with a reasonable amount of energy
the student can succeed.

If John Dewey were alive he would find much in
this theory of learning to support his conclusions about
the way we learn. The following statement best illuSv
trates this:

. . . there is incumbent upon the educator the
duty of instituting a much more intelligent,

and consequently more difficult kind of plan“
ning. He must survey the capacities and needs

18

of the particular set of individuals with whom
he is dealing and must at the same time arrange
the conditions which provide the subject-matter
or content for experiences that satisfy these
needs and develop these capacities. The plan—
ning must be flexible enough to permit free play
for individuality of experience and yet firm
enough to give direction towards continuous
development of power (John Dewey, 1938, p. 58).

The place of the affective domain in this scheme
becomes clear when we consider the results of the clinical
studies by the Esalen Institute (1968) which are in agree—
ment with the statement made by Kinney and Freeman in the
introduction. Feeling is attached to cognitive learning—H
we can not learn without a feeling response being involved.
The intensity of this response is associated with the
amount of personal involvement in the learning, i.e.,

"the intensity of the feeling response determines the
relevancy of the cognitive content to the individual

. . . (Esalen Institute, 1968, p. 2—3)." Thus the role
of affective behavior in learning should not be denied.
In fact desirable affective behavior should be developed.
To do otherwise might limit or destroy desirable or
anticipated cognitive behaviors. Consequently, the most
desirable learning environment is one where learning takes
place through self-motivation and the learner continues
to be self-motivated (Esalen Institute, 1968, p. 2—3).
"The emotional dimension of this learning is one of

positive excitement and aliveness (Esalen Institute,

1968, p. 2-3)."

'-JM -

 

 

At
the reason;
(1951, 196;
prospective
because of
teachers' 6
cate the p:
drill and E
little hel;
$13, a;
9d fright»;
the direct;
toward the
1945' p. 3e
exprESSed E
ir‘portanCe
felt Reeds
the Student
in Order tc
non‘threate
for them nel

Nowl
esearchersl
ning (194:

”as affecte

I

at leaSt pr)

 

19

At this point it is appropriate to recall some of
the reasons for negative attitudes. From Dutton's studies
(1951, 1962, and 1965) on attitudes one can surmise that
prospective elementary teachers dislike mathematics
because of a lack of understanding blamed on their
teachers' explanations. Their teachers did not communi—
cate the practical aspects of mathematics, assigned boring
drill and pages of homework, were impatient, and gave
little help. The prospective teachers themselves feared
failure, and felt insecure. They remembered punishment
and frightening experiences. One factor in determining
the direction of an attitude is "evidence of his progress
toward the goal or failure to achieve (Kinney and Freeman,
1945, p. 385)." In support for the third force psychology
expressed earlier in this section they also note the
importance of having activities be related to the child's
felt needs and the degree of significance recognized by
the student. It seems obvious from this discussion that
in order to change teachers' attitudes toward mathematics
non-threatening experiences which will provide successes
for them need to be provided.

Now that a basis has been provided, consider what
researchers have written about success experiences.

Billig (1944) found that mastery of commercial arithmetic
was affected by the student's attitude toward mathematics.
It was necessary for a student to experience success or

at least prevent repeated failure in order to change

20

attitudes. Bassham, Murphy, and Murphy (1964) noted that
to change a pupil's attitude toward mathematics, his
self—concept in relation to mathematics materials must be
changed. To cope with such negative attitudes, the teacher
must provide success experiences for the learner; the
child should be taught to set reasonable goals that
culminate in the reward of success. The need to provide
for success experiences was also referred to by Proctor
(1965) in a discussion of techniques for giving selfv
confidence to slow learners and thus changing their
attitudes toward mathematics.

Howard F. Fehr (1956) described an experiment
where he felt success was a tremendous incentive. In this
experiment students were given problems and then after a
period of time had elapsed they were urged to proceed to
the next problem. It was noted that success increased
interest and provided a stimulus to proceed. Also if
several unsolvable problems were encountered in sequence
any incentive to continue disappeared. He concluded
intermittent rewards (success) seemed to be a very favor—
able factor in motivating further learning.

M. L. Hartung (1953) was also concerned that
repeated failure would produce unfavorable attitudes toward
mathematics. He felt that it was a well known fact that to
Produce a favorable attitude toward mathematics the student
muSt haVe repeated successful experiences. In separate

Studies Lerch (1961) and Tulock (1957) were also concerned

 

‘-

 

about cons
would res;
producing
that the c
ences at w.
point was i

able goals

 

21

about consistent failure. They felt that consistent failure
would result in the subject losing self—confidence and
producing a hostility toward mathematics. Both believed
that the child (or adult) should be provided with experi-
ences at which he can be successful. Additionally the

point was made that the teacher and child must set reason—
able goals resulting in success.

Among the seven needs listed by Hach (1957) to
improve the emotional climate of the classroom are: the
need for success, recognition, and sympathetic understand-
ing. From the School Mathematics Study Group research on
their mathematics programs Alpert et_al. (1963) felt that
textbook writers should pay more attention to those aspects
of school which will affect success in mathematics.

Donovan Johnson (1957) observed:

If our students are to learn to like mathematics
they must find pleasure in performing the learn“
ing activities in and out of the mathematics
classroom. And our students will find pleasure
in doing that which they can do successfully,
that which seems significant in meeting their

needs, that which gives them status they esteem
(p. 116).

He also noted that by increasing mathematical understand-
ings attitudes can be made more positive.

Dean Hendrickson (1969) investigated the teaching
Performance of teachers after a year of intensive content
Study.7 The content of those courses is unknown but he
reported that students were given the opportunity to engage
in open ended activities. The teachers put less reliance

‘31 the textbook (indicating more confidence in subject

 

that t<
natics
tainly
change.
felt ti

school

cornen:
ECOtiox
Probabj
SUCCESS
attituC
that a

diSCUSs
bY the
ProctOr
improve
FOngma
materia
am Wat
Piaget.

\
“d d-

‘eria;

22

matter). They also did more experimenting and demonstrat-
ing. What is important is the observation by Hendrickson
that to a significant degree their students liked mathe—
matics more and thought mathematics class was fun. Cer—
tainly teacher effectiveness and attitude is the key to
change. This is confirmed by Torrance gt_al. (1966) who
felt that teacher effectiveness had positively influenced
school climate, methods, and attitudes toward teachers.

This section is closed with some appropriate
comments by Allen (1937). He attributed favorable
emotional attitude toward both teacher and topic was
probably the greatest contributing factor to student
success and mental annoyance of unfavorable emotional
attitude contributes most to failure. Also he believed
that a growing interest in mathematics is accompanied by
success in the subject.

The next statement by Allen (1937) leads to a
discussion of the next topic. Student success is advanced
by the use of real and meaningful material. In agreement,
Proctor (1965) declared attitudes of her students were
improved by the use of real meaningful activity materials.
Fogelman (1970) discovered that boys need to manipulate
materials to be more successful than girls who can sit
and watch a demonstration. This experiment involved a
Piagetian test of conservation of quantity.

One of the benefits of working with activity

Inaterials is the reality brought to the situation (Davis,

23

1967a). In addition it provides an alternative to the
authoritarian type of teaching usually associated with
mathematics teachers. Davis (1967b) referred to a study
by Barrett in which Barrett concluded that in grades six
and seven pupils prefer to work with concrete objects and
dislike sophisticated mental tasks. Dienes (1967) has for
many years suggested that mathematics classes should use
games and concrete objects which embody a concept. Chil-
dren have perceptual needs which will be filled by concrete
objects. They also need a variety of encounters with a
concept in order to develop it. Biggs (1965) concluded
that traditional teaching develops high number anxiety
but good computational skills. She found that the use of
Cuisenaire rods helped develop a positive attitude and a
better understanding in boys with a high I.Q. Bruner
(1966b) by suggesting his optimal learning sequence
(enactive, iconic or image manipulative, and symbolic)
endorses working with concrete objects. He also noted
that although secondary level children could learn at the
third stage alone there was a danger in doing so.

Most teaching with activity materials is done in
a discovery mode. Openshaw (1962) concluded that in
changing attitudes method was more important than the
subject matter being taught. Students involved in an
inductive—discovery approach will feel successful in
mathematics (Donovan,Johnson, 1957). Also the student

will have a confident inquiring attitude even when his

 

tasks are
learning 1
necessary

Pr
classes we
an increas
control g:
contrast :

grade gent

anl’ Signif
W:
ing? Endc

getting te

them act-11d

with the 1’?

0.:
that attit
need to be
as the Cor.
tics' One
attitudes
Provide di
experience

Si
attitlIdes
dCCeptance
9: pl‘oblem

 

24

tasks are non—mathematical.” Davis (1966) affirmed that
learning by discovery in a group setting provides the
necessary rewards (success) for children.

Price's results (1967) in comparing algebra
classes were that the discovery and transfer groups had
an increase in positive attitudes and achievement. The
control groups exhibited a negative attitude change. In
contrast to this Berger and Horowitz (1967) using ninth
grade general mathematics students were unable to find
any significant attitude differences between groups.

What then are the implications for teacher train—
ing? Endorsing activity materials, Lola May (1971) urged
getting teachers to use the materials developed by giving
them actual experiences through the process of working
with the materials as the students would.

Obviously from the preceding remarks it is evident
that attitudes of elementary teachers towards mathematics
need to be changed. Attitudes are at least as important
as the content to be mastered in the learning of mathema-
tics. One vehicle suggested by research for changing
attitudes is the use of activity materials since they
provide direct, active, and personally meaningful success
experiences.

Since opinions are expressions of attitudes,
attitudes toward mathematics are measured through the
acceptance or rejection of these opinions! This presents

a problem "what if the person being tested is a liar?"

25

There is no solution to this problem. It serves as a
warning that measures of this type are only indices of
the truth. Only if the person is consistent in different
measures of his attitude are the results relatively
positive. Indeed it may be as Decker (1963, p. 14) sug—
gested that the act of testing (measurement) may itself
change a person's attitude. In responding to an item a
subject must put conscious thought and effort into the
response thereby altering what may have been true.

Next consider some of the assumptions in attitude
measurement which were recognized by Thurstone and Chave
(1937).

(1) An attitude scale is used only in those
situations in which one may reasonably

expect people to tell the truth about

their convictions or opinions.

(2) Man's attitude is subject to change.
(3) Attitude scales should be used only in

those situations that offer a minimum of

pressure on the attitude to be measured (p. 9).
Since the attitudes and opinions are such a complex
phenomena Thurstone and Chave suggest restriction to some
specified or implied continuum along which measurement is
to take place. Then along this continuum persons or
groups may be represented in the form of a frequency diSs
tribution. The base line is scaled to represent different
opinions. The height at any point is the frequency with
which the attitude occurs. This type of scaling presents
a problem since the same unit of measurement on the scale

separates two opinions at any place on the scale. This

may not be true in reality but the assumption must be made

 

 

3---.

 

I..-

 

”sat it :
remember
allocati<
scale is
indices

an estim.

point on
here is
freqUeno
attitude
a narrow
human an.

0!) the SI

26

that it is when using a Thurstone type scale. Also
remember the following point made by Thurstone "the true
allocation of an individual to a position on an attitude
scale is an abstraction. In allocating we may use various
indices . . . the true attitude of an individual is only
an estimate (1937, p. xii)."

An attitude is not necessarily represented by a
point on the scale but by a narrow range. When measuring
there is no continuous curve; there are intervals. The
frequency is counted in this interval. Then, when an
attitude is referred to; the reference is to a point or
a narrow range. This is entirely realistic since the
human animal is variable and will subscribe to opinions
within a range.

The following characteristics can be represented
on the scale for an individual:

(1) The mean position he occupies on the scale.

(2) The range of opinion he is willing to accept.

(3) That one opinion which he selects as the one
which most nearly represents his own attitude

on the issue at stake (Thurstone, 1937, p. 15).
In addition, for a group the following can be described:

(4) The relative popularity of each attitude of

the scale for a designated group as shown

by the frequency distribution for that

group.

(5) The degree of homogeneity or hetrogeneity in

the attitudes of a designated group on the

issue as shown by the spread or dispersion

of its frequency distribution (Thurstone,

1937, p. 16).

The first step in measuring attitude is to sample

our population and ask them for a list of statements

27

expressing their opinion about a particular subject. We
should also consult other sources such as relevant liter-
ature, groups, and individuals. These opinions will be
culled for a set of statements. Each statement will
represent a position on the base line. The following
requirements for good statements should have special

attention:

(1) It should be so stated that one can speak
of it in terms of "more" and "less."

(2) Neutral statements must be included in
the inventory, otherwise the scale will
not fpnction properly (Thurstone, 1937,
p. 12 .

The initial list might be refined by the following
criteria:

(1) Statements should be brief to lessen
fatigue.

(2) The statements should be such that they
can be endorsed or rejected in accord—
ance with their agreement or disagreement
with attitude of the reader.

(3) Every statement should be such that
acceptance or rejection of the statement
does indicate something regarding the
readers' attitude about the issue in
question.

(4) Because of their ambiguity, double
barreled statements should be avoided
except possibly as examples of neutrality
when better neutral statements do not seem
available.

(5) One must insure that at least a fair
majority of the statements belong on the
attitude variable that is to be measured
(Thurstone, 1937, p. 22).

Thurstone also suggested the following informal criteria:

(6) The opinions should reflect the present
attitude of the subjects rather than past
attitudes.

(7) Each opinion expresses as far as possible
only one thought or idea.

(8) Avoid statements applicable to a very
restricted range of endorsers.

 

U) P” ff) [“1" m

— '-—-

(10)

0)

(11)

$ij

'7.

distance '.
to submit
rents in K;
neqative.
Piles. T;
each pile
the deter:
A Cumulat;

Statement
the 3C a1 e

the first

 

1937, pp,

pr OPOrt 10'

28

(9) Each opinion should preferably be such

that it is not posSible for subjects

from both ends of the scale to endorse

it.

(10) Statements should be free from related

and confusing statements.

(11) Avoid slang (Thurstone, 1937, pp. 56-58).

The next step is to determine the position and
distance between the various statements. One method is
to submit them to a jury and have them arrange the state-
ments in an order from the most positive to the most
negative. The jury should sort the statements into eleven
piles. They should also choose the two statements from
each pile which is most representative. The last step is
the determination of the scale values for the final list.

A cumulative proportions graph is developed for each

statement from the jury placement of the statement. Then
Q1"Q3
the scale value is equal to ——————— . Q1 and Q3 represent
2

the first and third quartiles, respectively (Thurstone,

1937, pp. 36-39).

proportion

 

 

29

Questions may be evaluated on the basis of the shape
of the curve. The larger the slope the better the question.
After the scale value has been determined the test may be
administered to a new group of subjects. The individual
scores will be determined in the following manner. For
each question the subject indicated agreement we record
the scale value of the question. Then we sum these scale
values to obtain his score. The score is then divided by
the number of questions to obtain an average score. This
average score is between zero and eleven.

Dutton (1954, 1962) described how he developed his
scale. This is essentially the method employed by Thur-
stone and Chave described in this section. Dutton con—
cluded that the main result of his study was "attitudes
toward arithmetic may be measured objectively and that
significant data may be obtained helpful in the education
of prospective elementary teachers (1954, p. 30)."

However there are some problems associated with
the Dutton type questionnaire. First, the result of most
attitude instruments is a composite of attitudes toward
mathematics instead of an attitude toward a specific part
of mathematics. Therefore important aspects may be over-
looked (Aiken, 1970). The next reservation is suggested
by Bassham, Murphy.and Murphy (1964) that all responses
are considered as correct although no direct evidence is
available to indicate that self-rating ability does not
vary among individuals and between situations for the same

individuals.

 

Tr
effects of
elementary
of Mathene
at Michiga

investiga:

_—__——___

 

30

Purpose of the Study

 

The purpose of this study is to investigate the
effects of activity materials on attitudes of in-service
elementary teachers while enrolled in Methods and Materials
of Mathematics in the Elementary School, Education 830A,
at Michigan State University. The following areas will be
investigated:

(1) Attitude change toward mathematics.

(2) Change in mathematical understanding.

(3) The relationship of certain selected

variables.

A study of these relationships will provide mathe-

matics educators with some insights into the affective

domain, and the effect of activity materials on attitudes.

Statement of the Problem

 

The following question will be under consideration
in this study: Are the attitudes of in—service elementary
teachers toward mathematics modified through the success
experiences provided by activity materials in Education
830A?

In order to answer this question the experimenter
will participate in the design of and teach a curriculum
for a graduate mathematics methods class (Education 830A).
A test of mathematical understanding will be administered
as pre- and post—test in addition to a pre— and post—test

attitude survey.

31

Hypotheses*

 

The main concern is the efficacy of a mathematics
methods course for in-service teachers. Consequently of
interest is what changes take place during the course.
This suggests the following two hypotheses:

(1) There is a positive attitude change for
in—service teachers enrolled and parti—
cipating in Education 830A as measured
by the pre- and post-test of Dutton's
Attitude Inventory.

(2) There is a gain in achievement for in-
service teachers enrolled and partici-
pating in Education 830A as measured by
the pre— and post-test of achievement.

Another question directly related to the above is
whether attitudes are changed by achievement or vice versa.
Therefore the following hypothesis:

(3) The attitudes of in-service elementary

teachers enrolled and participating in
Education 830A are changed by an
increase in mathematical understanding
as tested by a crossed-lagged correla—
tion on the achievement and attitude
pre- and post-tests.

In order to increase external validity and control
for confounding variables analysis of variance will be

used to test the following hypotheses:

(4) All groups are from the same population
on the attitude pre—tests.

(5) All groups are from the same population
on the achievement preatests.

In addition to these major hypotheses there are

SeVeral minor hypotheses concerning the attitudes of

‘

*Hypotheses will be stated in testable form in
Chapter III .

32

in-service elementary teachers. There are fourteen
variables in this study: (1) sex, (2) degree program,
(3) age, (4) years of teaching experience, (5) years of
high school mathematics, (6) credits of college mathe—
matics, (7) grade level assignment, (8) mathematics vs.
non—mathematics teaching, (9) years at present grade level
assignment, (10) perceived attitude, (11) grade level
attitude developed, (12) average mathematics grades,
(13) attitude, and (14) mathematical understanding.
Consequently there is a total of 91 hypotheses concerning
the degree of relationship between any two of these vari—
ables. These can be condensed into one hypothesis:
(6) There is a correlation between any of

the variables (sex, degree program,

age, years of teaching experience,

years of high school mathematics,

credits of college mathematics,

grade level assignment, mathematics

vs. non—mathematics teaching, years

at present grade level assignment,

perceived attitude, grade level attie

tude developed, average mathematics

grades, attitude, and mathematical

understanding) taken two at a time.
Of course, this procedure is rather inefficient since some
hypotheses are more significant than others. Below is a
sampling of the more significant hypotheses:

(A) There is a correlation between attitude
and achievement as measured by the pre—

tests.

(B) There is a correlation between grade
level and achievement.

(C) There is a correlation between grade
level and attitude.

 

33

(D) There is a correlation between in-service
elementary teachers' attitudes as measured
by Dutton's questionnaire and their per-
ceived attitude.

(E) There is a correlation between in-service
elementary teachers' achievement on the
pre-test and their perceived attitude.

(F) There is a correlation between in—service
elementary teachers' attitude and credits
of college mathematics.

(G) There is a correlation between in-service
elementary teachers' attitude and years
of high school mathematics.

(H) There is a correlation between in—service
elementary teachers' attitude and average
mathematics grades.

All relationships will be measured by Pearson product—

moment correlation coefficient.

Overview

The thesis is composed of five chapters: introduc—
tion, a review of the related literature, design of the
study, an analysis of the data, and a summary and conclu—
sions.

The purpose of the first chapter is to acquaint
the reader with the need to change attitudes and to build
a theory upon which to base a study. An orientation to
the study is provided. Consequently definitions and a
statement of the problem are presented.

In Chapter II a review of the literature pertaining
to the study is presented. The chapter is divided into
four sections. The first is a historical perspective of

the related literature. Section two is concerned with

 

34

pre-service and in-service teachers' attitudes toward
mathematics. The relationship of mathematical understand-
ing, attitude, and achievement is covered in section three.
The fourth section contains change studies.

Chapter III has five sections. It contains a
description of the sample, the design, the testable hypo-
theses, the analysis, and procedures.

In Chapter IV the data is analyzed accompanied by

 

appropriate charts, tables, and explanatory material.
In Chapter V a summary is presented and accom—

panied by conclusions and recommendations. The results

are also discussed and implications for future research

are considered.

CHAPTER II

REVIEW OF LITERATURE

There has been extensive research of attitudes
towards mathematics. It has been reported in published
and unpublished research papers, journals, monographs,
periodicals, and professional books. Several leading
mathematics educators have at various times published
extensive reviews of this literature. They are: Lewis R.
Aiken's "Attitudes Toward Mathematics" (1970); Rosalind L.
Feierabend's "Review of Research on Psychological Problems
in Mathematics Education" (1960); and Marilyn Suydam and
J. Fred Weaver's "Attitudes and Interests" (1970).

Chapter II has been divided into the following
sections: (1) a historical perspective on attitude, (2)
pre-service and invservice elementary school teachers'
attitudes toward mathematics, (3) the relationship of
mathematical attitude, understanding, and achievement, and

(4) changing attitudes toward mathematics.

Historical Perspective on Attitude
The role of attitudes in the learning of mathema-

tics has probably been a matter of conjecture since the

35

 

first man
explain 1
articles
Education
contain a
there doe
0f attitu
comments
must have
A
words "mo
i1C1 used
of the me
authors 1
and Claso
mathemati
Ellery per
i.e., halls
wrote that
What he d1"
Part Of hi

 

36

first man formed a system for counting and attempted to
explain it to another man. This concern is evident in
articles from Readings in the History of Mathematics
Education (Bidwell and Clason, 1970). These articles
contain a number of unfounded opinions. Before 1936
there does not seem to be any research about the effect
of attitudes on learning mathematics. Consequently any
comments about attitudes in this text before that time
must have been based on observation and opinion.

Also it should be noted that in many articles the
words "motivation" and "attitude" are sometimes confused
and used interchangeably. The omission in certain articles
of the mention of "attitude" is important, e.g., those
authors interested primarily in drill. John Perry (Bidwell
and Clason, 1970) in his discussion on the teaching of
mathematics in 1904 believed that every elementary teacher,
every person, should feel positively towards mathematics,
i.e., have the feeling that he can learn mathematics. He
wrote that it was not really what the student was told but
what he discovered which would become a permanent valuable
part of his mind. Engineering friends thought Perry
exaggerated the importance of everyone loving mathematics.
His reply was that these people had not seen an affection
produced in the average boy by subjecting him to a delight-
ful training when he was quite young- Perry would have

teachers "refrain from prosing and the setting of tasks;

if they I
question:
(p. 238)
l
Readings
for a po:
burn (Bi.
drithmet.
it is me.
self, an,
make it '
seeley (:
Grube's 1
not be e:
PleaSUre
Only whe,
the deve;
begin. 5
Ms Self.
subjeCt.
V
freedOm i
teachErS
ExpresSin
ties);re to

T;
18) does 1

37

if they merely make timely suggestions and answer his
questions, and leave him to find out things for himself
(p. 238).

However, this is not the first reference in
Readings in the History of Mathematics Education to a need
for a positive attitude towards mathematics. Warren Col-
burn (Bidwell and Clason, 1970) in "The Teaching of
Arithmetic" argued ". . . that to teach a subject well,
it is necessary for the teacher to understand it well him—
self, and to take an interest in it; otherwise he will not
make it interesting to his scholars (p. 36)." Also, Levi
Seeley (Bidwell and Clason, 1970, p. 108) in discussing
Grube's method indicated that incentives and threats would
not be effective if part of the work itself was not the
pleasure of work. The child must desire to know a thing.
Only when the child is conscious of the continual unity in
the development of his powers can the impulse to know
begin. Then he will be driven to further development by
his self-activity. This method awakens a love for the
subject.

William Speer (Bidwell and Clason, 1970) connected
freedom in the classroom with attitudes. He admonished
teachers of the error that unnatural restraints in
expressing results in a lack of feeling which lessons
desire to see and to do (p. 188).

The next discussion of attitudes (remote though it

is) does not take place until 1920 in a National Education

38

Association report (Bidwell and Clason, 1970). It indicated
that when other factors are equal, learning a thing is
easier and quicker if its bearing and need are recognized.
"The felt need predisposes attention, calls into play
accessory mental resources, and in proportion to its
strength secures the necessary repetition (p. 369)."
More explicit references to attitudes are made in The
Reorganization of Mathematics in Secondary Schools (Bidwell
and Clason, 1970). The following was mentioned as comprising
the disciplinary and cultural aims of the study of mathe-
matics:
l. The acquisition of mental habits and attitudes

which will make the above training effective in

the life of the individual. Among such habitual

reactions are the following: (a) a seeking for

relations and their precise expression; (b) an

attitude of inquiry; (c) a desire to understand,

to get to the bottom of a situation; (d) concen—

tration and persistence; (e) a love for precision,

accuracy, thoroughness, and clearness, and a

distaste for vagueness and incompleteness; (f)

a desire for orderly and logical organization as

an aid to understanding and memory.

2. Appreciation of beauty in geometrical forms in
nature, art, and industry.

3. Ideals of perfection as to the logical structure and
precision of statement and of thought.

4. Appreciation of the power of mathematics
(pp. 393-94).

Before this period mathematics was being taught
badly by unskilled, untrained, personnel. According to this
report a successful teacher of mathematics must not only be
highly trained in his subject and have a genuine enthusiasm

for it. Also all the problems must be "real" to the pupil,

39

must connect with his ordinary thought, and must be within
the world of his experience and interest (pp. 399 and 409).

The only other statement about attitudes came from
William A. Brownell (Bidwell and Clason, 1970) who indi-
cated that it was a "wholesome situation" for children to
want to learn arithmetic because they like it. Also he
believed that "under the stimulation of such a motive it
is highly probable that learning will be economical and
thorough (p. 516)."

Five years later in 1940 a commission of the
National Council of Teachers of Mathematics and Mathemav

tical Association of America in a report on The Place of

 

Mathematics in Secondary Education (Bidwell and Clason,
1970) perpetuated some popular beliefs, perhaps myths,
about attitudes. One, that many important people seemingly
have little appreciation of mathematics even though they
live rich cultural lives. For some of these people,
unpleasant memories are evoked by the thought of their
study of mathematics, tempered only by the joy of finishing
it. The commission analysis was that in many cases the
"dislike for mathematics may have been created either by
the ineffectiveness or the personality of a teacher or by
the unsuitability of the material that a competent teacher
had been required to present . . . (pp. 590—91)." While
every effort should be made to reduce the number of those
who have a feeling of unhappiness and belief that no sub-

.stantial benefit is being obtained, the commission was

40

prepared to accept the possibility of conditioned anti-
pathy in some individuals. They would not force mathema-
tics beyond the elements of arithmetic either upon a

pupil having an entrenched distaste, a genius, or without
legitimate leisure for mathematical development. Desirable
attitudes, habits, and appreciations are developed from the
outcome of teacher procedures and of the resulting class—
room atmosphere. For lasting results continued attention
to their growth, and cooperation throughout the school is
necessary (p. 615).

The Commission on Postwar Plans (Bidwell and Clason,
1970), exhumed some well buried corpses when it began to
discuss drill. Drill was of course to be wisely used.

The commission felt that drill may alleviate or delay the
feelings of "insecurity, frustration, and fear which too
many now experience in the arithmetic of the upper grades
because of their lack of confident mastery (p. 629)."

Well, the pendulum swings back again. The commission con!
tinued by citing a survey in Fortune which said that mathe~
matics was the best—liked but also the.leastvliked subject
in high school (p. 632).

Conveniently, there are three surveys of the
research literature in this area. The most recent having
just appeared in Review of Educational Research (Aiken,
1970). It covered the literature appearing after the
first survey by Rosalind Feierabend. The'Feierabend

artzicle, "Review of Research on Psychological Problems in

41

Mathematics Education" (1960), reviewed the literature to
1958. The third review is "Attitudes and Interests" by
Marilyn Suydam and J. Fred Weaver (1970). One must be
very careful of works such as these since they often
reflect the bias of their authors. However, these reviews
have been most helpful in pinpointing articles for further
study.

These reviews stated the findings were mixed with
low correlations and differences that are not significant.
However, there was some very important information in these
articles: (1) attitudes are determined by success, and
(2) there are significant sex differences.

Pre- and In—Service Teachers‘ Attitude
Toward Mathematics

 

In this section the reader will be acquainted with
the status of pre-service and in-service elementary school
teachers' and pupils' attitudes toward mathematics. Also
information on selected factors affecting their attitudes
will be discussed. In Dutton's (1951) first attempt to
measure attitudes in a study of 211 prospective elementary
teachers, he found 74 per cent of the responses to be
unfavorable. He also felt the statements were highly
emotional. His method was to collect written statements
from the students in an elementary—curriculum methods
class. In 1954 he conducted a second study using 289
prospective elementary teachers. He used a Thurstone type

scale developed from the statements about mathematics by a

42

group of elementary education majors. He concluded that
this group had a favorable attitude towards mathematics.
In a study which contradicts Dutton's finding O'Donnel
(1958) using Remmers scale reported a favorable attitude
toward arithmetic by 109 seniors. In another study using
Remmers scale Virginia Stright (1960) found that 20 per
cent of her population of third, fourth, and sixth graders
and their teachers felt arithmetic was uninteresting.
Dutton (1961) conducted a study of 55 elementary
education majors. He found that one third had unfavorable
attitudes towards mathematics. He continued his investi«
gations of elementary teachers. In 1962 in a study of 127
women in an elementary curriculum class he found that:

l. The attitudes of students toward arithmetic
in 1954 were almost identical with attitudes
held by students in the 1962 sampling. Two
conclusions on this finding seem warranted:
(a) these students are the product of a type
of teaching which was based upon mechanical,
drill procedures; (b) instruction in the
teaching of arithmetic at the university
level (even when students identified their
attitudes toward arithmetic) did not change
the attitudes held by these students. Will
teaching experience and in-service educational
programs change the attitudes of teachers who
have unfavorable feelings toward arithmetic?

2. New applications of the attitude scale and
limiting the "free-response" statements have
caused the writer (Dutton) to qualify and
extend statements made in the 1943 study.

Many students (elementary education majors)
have ambivalent feelings toward arithmetic.
The extremes, students with either very posi-
tive or very negative attitudes toward arith-
metic, are exceptions to the rule.

43

While it is important to know that 38
percent of the students in this study dislike
arithmetic very much, it is also important to
know that 24 percent like arithemetic extremely
well and 38 percent like arithmetic fairly
well-~but not enthusiastically. Fifty-one
percent of the students like arithmetic, but
like other subjects just as well.

3. There was not enough evidence found in this
study to indicate any pronounced improvement
in the instructional programs of public and
private elementary schools directed toward the
development of positive attitudes of pupils
toward arithmetic. Prospective elementary
school teachers reflect attitudes developed
in a traditionally oriented arithmetic program.

4. Attitudes toward arithmetic, once developed,
are tenaciously held by prospective elementary
school teachers. Continued efforts to redirect
the negative attitudes of these students into
constructive channels have not been very effec~
tive. While the best antidote is probably
improved teaching in each elementary school
grade, continued study should be made of change
ing negative attitudes toward arithmetic at the
university level and through in-service instrqu
tion while doing regular classroom teaching.

5. The aspects of arithmetic liked and disliked
by prospective elementary school teachers
remained approximately the same between 1954
and 1962. Factors liked most were the chal—
lenge arithmetic presents, the usefulness of
the subject, fun working problems, and the
logical aspects of number. Students continued
to dislike impractical word problems, boring
work, long problems, and drill over work
already known (p. 424).

Brown (1961) reported that the attitudes of
in-service teachers are more positive than pre-service
teachers. However, the number of years experience did
not seem to affect attitudes. He also reported that
teachers who understand arithmetic have a more positive
attitude than those who have not taught. McDermott (1956)

in interviews found 34 of 41 college students feared

44

mathematics. Rice (1964) used a Likert type scale in a
study of teachers' attitudes toward modern mathematics
programs. He found that those teachers who have received
instruction in modern mathematics programs have a signifi—
cantly better attitude than those who have not had the
training. Teaching in a modern mathematics program also
made a significant difference. However it does not seem
to influence mathematics attitude in general (p. 1433A).

Wilson (1961) noticed that although modern curri-
cula had decreased the number of failures in mathematics
he doubted whether these new programs had improve attitudes.
In a study supporting Dutton's later findings, White (1963)
found that prospective elementary teachers had positive
attitudes toward mathematics. In contrast to this Smith
(1964) concluded that too many prospective teachers had
negative attitudes. The same conclusion was reached by
Shryock (1963). Sister Josphena (1959) also concluded that
too many teachers dislike mathematics.

The status of elementary teachers' attitude has
been the subject of investigation in recent years. Nugent
(1969) surveyed 670 elementary and junior high school
teachers to determine the status of their attitudes towards
the new mathematics. He used a Thurstone type of instru-
ment and found a slightly favorable attitude towards the
new mathematics programs. Also, the following factors
were found to be significantly related to attitude: "age,

total teaching experience, sex, training in mathematics,

  
  
  
  
  
  
  
  
  
  
  
  

training i
area from
recognized
sarily the
However wh
tude towar
training i
’w'as unable

Ra
Elementary
The author

and the Ca

45

training in the new mathematics, and the type of population
area from which the student came (p. 2665A)." It should be
recognized that attitude towards mathematics is not neces-
sarily the same as attitude towards the new programs.
However when Pitkin (1969) attempted to compare the atti—
tude towards mathematics of elementary school teachers with
training in modern mathematics as the dependent variable, he
was unable to find significant differences.

Raines (1970) studied the attitudes of 329 Virginia
elementary teachers towards the teaching of arithmetic.
The author used the Minnesota Teacher Attitude Inventory
and the California Arithmetic Attitude Scale 3 to assess
attitudes. He concluded that Virginia teachers tended to
have a favorable attitude toward the teaching of mathema—
tics. A favorable attitude was more likely if the teacher
had a generally favorable attitude towards teaching and
children or had additional training in mathematics. Sig-
nificantly Raines found that attitudes were developed in
the elementary and junior high years. Their attitudes
were changed little by the teaching experience.

The importance of attitudes towards mathematics
is emphasized by the results of the following studies.
Early (1970) stated that those teachers who preferred to
teach in the later grades had significantly better atti-
tudes than those who chose the primary grades. Crittenden
(1971) ascertained that the grade level of a teacher was

related to his attitude towards mathematics. Also he

46

found that secondary teachers believed that elementary
teachers had a greater dislike for mathematics than was
expressed by the elementary teachers.

Burbank (1970) used Dutton's questionnaire to study
the relationship of parental attitudes, student attitudes,
and student achievement. His sample consisted of 411
seventh graders. His results showed significant correla—
tion between parents' attitudes and student attitude and
achievement. He also found a correlation between student
attitude and achievement. Poffenberger and Norton (1959)
have also studied the effect of parental attitude on their
offspring's attitudes and have identified three factors
affecting attitude and achievement: parental expectation
of children's achievement, parental encouragement regard—
ing these subjects, and parents' own attitudes toward this
area of the curricula.

Relationship of Mathematical Attitude,
Understanding, and Achievement

The purpose of this section is to review the
literature dealing with the interrelation of mathematical
understanding, achievement, and attitudes. The status of
mathematical understanding will also be considered.

Most reviews (Suydam and Weaver, 1970, and Aiken, 1970)
report that there is no conclusive evidence in this area.
Suydam and Weaver reflect this View:
First of all, there is no consistent body of
research evidence to support the popular belief

that there is a significant positive relationship
between pupil attitudes toward mathematics and

 

pupil
resea
are c

However,

much int
he maths
A.C. She
that att;
Studies (
of SUbjec
fifth gr,
mathemat.
and HUgh.
USe in a.
StUdEnt

Variabil
reCOmmen

a.

elementa
sehOOl h

’1
(he atti

47

pupil achievement in mathematics. We have little
research basis for believing that these two things
are causally related (p. 2).

 

 

However, there are several articles of interest some of
which indicate that elementary teachers do not understand
the mathematics they are supposed to teach.

In an introductory psychology course Margaret and
A. C. Sherriffs (1947) conducted a study which indicated
that attitude is related to achievement.. In separate
studies Chase (1949) and Dean (1950) searched for patterns
of subject preferences. Chase found none in a sample of
fifth graders. Dean found that those who did well in
mathematics stated a preference for mathematics. Bendig
and Hughes (1954) developed their own attitude scale for
use in an introductory statistics courses They found that
student attitude accounted for 4 to 5 per cent of the
variability in achievement. Thus the following two
recommendations were made:

a. instructors of such courses should attempt

~to modify and allay the fears of their

student early in the course if maximum

student achievement is the academic goal,

and

b. any battery of measuring instruments devised

to predict achievement.in.aimilar courses

should include some personality or temperav

ment measuring device to tap this attitude

dimension (p. 274).

O'Donnell (1958) in a study of 109 prospective
elementary teachers found that liking mathematics in high

school has a positive effect on achievement in arithmetic.

The attitude scale used was developed by H. H. Remmers.

48

He also reported an improvement in the mathematical under-
standings of prospective elementary teachers which is
strongly related to the number of years of high school
mathematics. However, he found little relationship between
attitude and achievement. He concluded from this that
there is a difference between "attitude" and "liking"
mathematics. Other interesting conclusions were: (1)
attitude towards arithmetic scores are not affected signi-
ficantly by whether or not a student feels well prepared
to teach elementary school arithmetic, (2) there was no
evidence to suggest that the attitude scores were signifi-
cantly affected by the experience of the student, and (3)
at the .01 level of confidence the like or dislike of high
school mathematics is a factor in the ability of students
to understand the problem solving processes. O'Donnel is
unable to shed any light on the question of which came
first, achievement or attitude. A criticism of his study
is that he administered his attitude scale after his
achievement test possibly biasing his attitude scores.
Another criticism is his use of multiple t-tests which
increases the probability of a type II error. He should
have used analysis of variance with post hoc comparisons.
Aiken and Dreger (1961) reported that attitudes
made a significant contribution to the prediction of
achievement for girls but not boys. However attitudes
were significant (.02) predictors of gains in scores on

the achievement test. In support of this Garner (1963)

49

found that achievement was significantly correlated with
end of the course attitude. However, Garner's study can
be criticized for his use of the Chi squared technique,
sampling, and randomization.

Bassham, Murphy, and Murphy (1964) reported that
in sixth grade pupils there is a difference in level of
achievement associated with attitude. However, they do
not specify a level of confidence which subjects their
findings to a degree of uncertainty. However Campbell
(1971) found that achievement on a test of mathematical
concepts and attitude was highly correlated. Correlation
for the pre-test was .583 and .624 for the post-test.

The paper by Danial C. Neale (1969) is important
because it criticized the place of attitudes in learning
mathematics. This article involved several studies which
revolve around two issues: (1) the importance of attitudes
and beliefs about mathematics as instructional objectives,
and (2) the role of attitudes in causing students to learn
mathematics. Neale concentrated his efforts on the second
one.

The first study which Neale considered in order to
study the place of attitudinal objectives in mathematics
instruction was the "International Study of Achievement in
Mathematics." The following were his interpretations.
First, those attitudes listed as desirable by the Husen
study seem to be independent of mathematics achievement.

In fact, we produce students whose attitudes are very

 

50

undesirable but who have attained a high level of achieve-
ment. Secondly, in at least two areas we do rather well
in promoting a "good" attitude towards mathematics, This
section was concluded with the following statement:
The implication is clear. If certain attitudes
are important objectives of mathematics instruc-
tion, then such attitudes must be given deliberate
and separate attention, both in the development of
mathematics curricula and in curriculum evalua-
tion (p. 632).
Neale then considered the effect of attitudes towards
learning mathematics, i.e., does an interest or liking for
mathematics promote learning. First, Neale reviewed
studies by Dutton, Aiken, and Anttonen. According to
Neale the belief that a favorable attitude will maximize
mathematical learning has not been scientifically proven.
Although,this idea is a widespread belief and held by
many to be crucial. He suggested that the following
hypotheses are indicated from the above studies.
1. At present, students develop an increasingly
unfavorable attitude toward mathematics as
they go through school.
2. At present, the part played by such attitudes
in causing students to learn mathematics is
slight (p. 634).
Neale cited a study by Anttonen and others to support the
first hypothesis. He then examined several studies which
seem to support the hypothesis that attitudes play a
significant role in causing students to learn mathematics.
A criticism of these articles was that they test attitude

and achievement at the same time.. Consequently an alter—

native hypothesis is that learning will promote favorable

51

attitudes towards mathematics. The author did observe that
both hypotheses may be valid to some extent.

In order to resolve some of the above questions
Neale applied a multiple regression analysis to tests
given to 105 sixth grade boys. This sample was taken from
a suburban elementary school. The following tests were
administered at the beginning of the school year:

(1) the Lorge Thorndike Intelligence Test.

(2) S R A.Achievement Battery.

(3) Semantic Differential (a test for measuring
attitudes).

They were tested at the end of the year with an advanced
form of the achievement test. From the initial tests
achievement was to be predicted at the end of the year.
This achievement was then measured by an advanced form
of the S R A Achievement Battery.. The explained variation
is then broken into the variation explicable by either (1):
(3), above or in the possible combinations.

Neale reported that prior achievement (26%) is
the largest single contributor to postvtest variation.
However, most of the contribution to explained variation
is made by variables acting together. When the attitude
variable (12.5%) is partitioned 10 per cent is attributed
to prior achievement acting jointly with attitude.

Thus Neale concluded this section by stating that
the two rival hypotheses have still not been reconciled.
However he felt that the evidence points to a small role

for attitudes in mathematics achievement. He wrote, "In

(2),

52

short, positive or negative attitudes toward mathematics
appear to have only slight causal influence on how much
mathematics is learned, remembered, and used (p. 636)."
Neale then discussed the work of Cattell and others

(The Prediction of Achievement and Creativity) with the
following observation:

Of particular interest in the present discussion

are the correlations between the attitude factors

and achievement. The two factors with the strong-

est relationship are submissiveness (.50) and

superego (+.44). Curiosity has a negative rela—

tionship with achievement (-.20)! How different

this picture is from what mathematics educators

are recommending. What makes Sammy learn is not

so much that he enjoys discovering the orderli—

ness of mathematical relationships, but rather

that he wants to be an obedient person and do his

duty (pp. 637, 638).
From the Cattell study Neale concluded that school influ-
ences overpower attitudes as a characteristic of the
learning process.

Neale summarized with the following convictions.

We must make changes in the educational structure in order
that intrinsic interest will play a larger role in school
learning. Modern mathematics has not really changed the
essential feature of the school which determines learning.
During the education of mathematics teachers we should not
stress the intrinsic motivation in learning. Instead we
should tell them that children learn because they want to
be good, do their duty, and gain adult approval. However,

he approved of educators who wish to make mathematics fun

and use the natural curiosity of children to instill learning.

53

Since this was an expository article it is diffi-
cult to criticize the research done by Neale. He did not
report sampling procedures, control group data, and only
slightly referred to his analysis. Although this is quite
appropriate for the journal in which the article appeared
it does make the experiment slightly suspect. In order to
make his point the author referred to the Husen Study.

The validity and reliability of this study has been
seriously questioned by mathematics educators. Conse-
quently, any conclusions drawn from it are suspect. The
author missed an essential point in his discussion of a
longitudinal study by Anttonen. If students do not like
mathematics it is likely that they are not going to take
elective courses. Consequently their achievement will be
much lower than those who do and the achievement attitude
correlation might be much higher than thought by Neale.
This high drOp out rate in mathematics is attributable to
the attitude on the part of students that they can not be
successful in mathematics.

Turning to Neale's second hypothesis, he is cer-
tainly right when he said that both the hypothesis and
the alternative have some validity. Psychologists such
as Maslow, Rogers, and Coombs have long felt that positive
attitudes are formed through success experiences. Yet,
he continually states that educators feel attitudes cause

learning. A criticism of Neale's study on the validity

of the :

unexpla;

have wrj
of eleme
this unc'
of locat
rural ar
Standing
in-servi
The samp
teach in
S”will ci1
(p. 368).
the resul
or assign
““derstan
Which the
for teach
sch001$ w
teaChers

0f partic:

 

I
lfldiCates

C

StudieSI

54

of the second hypothesis is that he did not report how much
unexplained variation there was in his study.

Thomas Gibney, John Ginther, and Fred Pigge (1970b)
have written extensively on the mathematical understanding
of elementary school teachers and the factors influencing
this understanding. One factor under study was the effect
of location; i.e., large city, small city, small town, or
rural area. They constructed a test of mathematical under-
standing which they administered to 1,082 pre—service and
in—service teachers, five of which were termed unusable.
The sample was further broken down to "127 preferred to
teach in a large city, 438 in a medium city, 308 in a
small city, 161 in a small town, and 43 in rural areas
(p. 368)." Analysis of covariance was used to evaluate
the results. They concluded that preference for and area
or assignment in an area was not related to mathematical
understanding. However, the population of the area from
which they graduated was significant. "The mean score
for teachers who had graduated from mediumwcity high
schools was significantly greater than the mean score for
teachers from large—city high schools (pp. 369, 370)."

Of particular importance to this study was their classifi«
cation by preference and non-preference as the following
indicates: (1) teachers who most preferred to teach
mathematics did significantly better than teachers who
preferred to teach language arts, science, or social

studies, and (2) teachers who least preferred to teach

55

mathematics did significantly poorer than teachers who
least preferred to teach language arts, science, or social
studies.

In the second study (Gibney, et al., 1970a) the
authors compared mathematical understanding and the number
of mathematics courses taken. Their pOpulation was 1,050
elementary teachers. They concluded:

1. Elementary teachers, in general, with more
years of high school mathematics did pro—
gressively better (significant beyond .05
level) on a test of mathematical understanding.

2. Elementary teachers with more college mathe-
matics courses did progressively better
(significant beyond .05 level) on a test of
mathematical understandings.

3. Elementary teachers with more exposure to
"modern mathematics" did progressively better
(significant beyond .05 level) on a test of
mathematical understandings (p. 381).

These studies seem to confirm earlier findings.
Glennon (1968) was one of the first to investigate the
mathematical understanding of elementary teachers and the
test he developed has been widely used. In 1949 he found
44 per cent of 104 prospective freshman elementary teachers
understood the mathematics concepts in use through the
first seven grades. The study also dealt with 172 seniors
and only 43 per cent had the appropriate understandings.
When he tested in-service teachers he found that they
understood only 55 per cent of the material.

J. Fred Weaver (1956) administered Glennons list

to 348 prospective elementary teachers and found 44 per

cent as their average level of understanding. In (Bean,

56

1959; Carroll, 1961; Cresell, 1964; Melson, 1965; Orleans,
1953; Philips, 1953; Rodney, 1951; and Schaaf, 1953) the
authors have also reached the conclusion that the mathe—
matical understandings of teachers is rather low. Fulker-
son (1960) found that methods courses were being used to
improve mathematics competency rather than teaching
competency.

Kenney (1965) reported that even experience did
not contribute to understanding. These findings have been
supported recently by Carol Kipps (1969) who found in a
test of concepts that there was little difference between
elementary teachers with little training in modern mathe-
matics and those with substantial training.

Recently Whipkey (1970) studied the relationship
of attitude and achievement using regression analysis
techniques. His sample was a group of 175 prospective
teachers in a mathematics course at Youngstown State
University. He concluded that there was "a small but
important relationship between mathematical attitude and
mathematical achievement . . .(p. 3808)." His results
were not significant. However, significant correlations
have been found by Beattie (1970), B. L. Erickson (1970),
and Hilton (1970).

Two studies seem to indicate that only attitudes
at extreme ends of the scale may have an effect on achieve-
ment. Cristantiello (1962) reported that for students

having attitudes on the extremes their ability was less a

57

factor of achievement. Jackson (1968) found that attitude
had little effect on achievement for subjects in the
middle range of attitude scores. In conclusion, ability
rather than attitude is the best predictor of achievement

for students in the middle attitude range.

Changing Attitudes Toward Mathematics
This section will review selected articles that
deal with attitude change. Unfortunately most of these
articles have as populations pre-service teachers or
school children rather than in-service teachers. As
pointed out previously the literature of attitude change

is indeed scanty.

Elementary School Classroom

The following is a discussion of attitude changes
toward mathematics in the elementary school classroom.
The measurement of young children's attitudes toward
mathematics has been found to be difficult. However, a
method that has promise in this area has been developed
by Jonathan Knaupp (1971). Knaupp used pictures to
represent situations and asked the students how they would
feel if they were in the picture. The purpose of his study
was to compare teacher demonstrations with student activity.
The concept to be studied was the decimal numeration system.
Blocks and ice cream sticks were used to model the system

resulting in four groups.

58

There was a significant increase in understanding
for all groups but there were no significant changes in
attitude. In this study there was some other significant
results:

1. The student activity classes tended to be
more independent. This tendency was
statistically nonsignificant.

2. There was a greater preference for mani-
pulative materials by the teacher
demonstration group.

3. Preference for manipulative materials was
directly and significantly related to a
child's preference for teacher assistance.
1.0. was a significant factor inversely.

In another study with early elementary school
children Fedon (1958) used a color scheme and Dutton's
scale to determine the intensity of feeling towards
mathematics. His sample consisted of 32 children (ages
seven to nine) in Wilmington School. Unfortunately his
results were not very definitive. Ronshausen (1972) found
individualized instruction via programmed instruction and
para-professionals to be inconclusive as a means of
changing first graders' attitude toward mathematics. Also
there were no significant differences between groups on
achievement.

Dana F. Swick (1960) reported second and third
grade pupils had an improvement in their attitudes toward
mathematics. He used a multisensory approach to the

teaching of mathematics. Deighan (1971) found a low

correlation between attitude and achievement. He also

59

found a significant decrease in attitude in the beginning
of grade three.

There are several studies involving fourth grade
students. In the Gosciewski study (1970) parents and
teachers combined to give positive reinforcement to fourth
grade children with respect to performance in mathematics.
Unfortunately no significant differences were observed.
Lyda and Morse (1963) used Dutton's Attitude Scale, the
Stanford Arithmetic Achievement Test, and the Otis Quick
Scoring Mental Ability in a class of fourth graders and
found a gain in attitudes and achievement when "meaningful
methods of teaching arithmetic are used (p. 136).“ Their
method stressed understanding of the following concepts:
the use of fundamental operations, number, place value,
and numeration system.

Lerch (1961) found that there was no significant
difference in attitude change when students were grouped
for mathematics. In a study of the effects of attitude
and self-esteem on achievement with fifth graders, Moore
(1972) concluded that attitude and selfvconcept have a
significant effect on achievement although they are not
the main determiners of success. Care should be exercised
in interpreting this result since the author used intact
groups and teachers and correlation does not necessarily
imply causation.

The effects of homework on children has always

been of interest to mathematics educators. Maertens and

60

Johnston (1972) studied the effect of homeworkv—no homework
on the attitudes towards mathematics of 146 fourth graders,
137 fifth graders, and 134 sixth graders. They used a
Semantic Differential instrument designed by them to
measure attitude. Significant differences were not
reported.
Contrary to most mathematics educators' beliefs
Studer (1972) found that students in innervcity expository
mathematics classes had a better attitude toward mathemav
tics than non inner—city expository or discovery groups
and inner-city discovery classes. Her sample involved
both fourth and sixth graders. However she failed to
comment on the statistical significance of her results.
In an interesting study James Young (1970) used a computer
to help teachers plan their lessons with the specific
objective of changing student attitude. His abstract
failed to define his sample. Unfortunately, he was unable
to find significant differences.
Jack Wilkinson (1971) studied the effectiveness of

a laboratory approach in the teaching of mathematics. His
sample consisted of sixth grade classes to which subjects
were non—randomly assigned. He found only one significant
difference. This occurred among males whose experience in
the laboratory setting resulted in a more positive attitude.
He made the following conclusions:

. . . pupils in the experimental classes did as

well on the geography achievement tests as pupils

in classes which used the teacher—textbook method;
the laboratory materials used in this study did

61

not significantly affect pupils attitudes toward
mathematics; laboratory methods of teaching
sixth grade mathematics can be used by teachers
without prior in—service or pre-service training;
and classrooms can be modified to accommodate a
laboratory method of teaching elementary mathe-
matics (p. 4637A).
Middle School or Junior High Classroom
The next series of studies have as their populations
middle school or junior high students. In a study which
involved 94 students in a Mesa, Arizona intermediate school,
Frase (1971) was unable to find significant differences on
either attitude or achievement measures when he compared a
guided discovery unit with an individualized progress unit.
Using an activity approach Randall Johnson (1971)
also found no significant difference in class means of
seventh graders and cited confounding variables as a
possible source of what attitude improvement was recorded.
He also reported no difference in achievement except for
some low ability students on certain concepts. In a study
of ability grouping of seventh graders Robert E. Willcutt
(1969) found no significant difference in achievement.
However the experimental group did show more positive
attitudes toward mathematics. This contradicts earlier
findings.
Jon L. Higgins (1970) studied attitude changes in
29 eighth grade classes in Santa Clara County, California.
The treatment was a mathematics through science unit with

a laboratory experiment on "Graphing, Equations, and

Linear Functions," developed by S.M.S.G. In the total

62

group he found a significant decrease in mathematics self—
concept as correlated with attitude. The children also

found mathematics to be duller and more difficult.

High School Classroom

In a study of Allen Bernstein (1955) it was found
that ninth grade students in remedial arithmetic had an
increase in attitude toward mathematics after participating
in an individualized instruction program. In classrooms
using games and a modified programmed lecture Jones (1968)
found a significant improvement in achievement with an
accompanying change in attitude.

Burgess (1970) conducted an experiment in secondary
schools to determine the effect of the regular use of
mathematics games on low achievers. Significant differ—
ences in favor of the experimental group were found on
the attitude variable. However, achievement in certain
skill areas was significantly greater for the control
group. Olson (1971) studied the effect of pairing students
to help each other during study periods on attitude and
achievement in plane geometry. He found no significant
differences between control and experimental groups in
either achievement or attitude. The paired students
indicated they had learned more by being paired with
another student. The only significant variable affecting
attitude was that of the teacher.

Usiskin (1970) compared two methods of teaching

Euclidean geometry (transformation and the usual course)

63

and found for the year a decrease in attitude for both
groups, although a larger number of the experimental group
chose to continue to study mathematics. Their level of
comprehension was about the same. Urwiller (1971)
compared two types of homework assignments, spiral and
traditional, on groups enrolled in second year algebra.
There were 732 students from Iowa, Kansas, South Dakota,
and Nebraska. Students were tested in September, 1967;
May, 1968; and September, 1968. Results were a more
favorable attitude for the spiral groups but the differ—
ences was not significant. An unusual aspect of the study
was a non-significant decrease in attitude for both groups

when measured in May.

College Students--Non Teachers

There is a recent study involving college students
who were not becoming elementary teachers. Arcidiacono
(1971) found no significant difference in attitude between
experimental and control groups. The purpose of his
experiment was to compare team teaching with the lecture

method in a college intermediate algebra class.

Pre-service Teachers

There are many studies of attitudinal change toward
mathematics using pre-service teachers as a population.
The first two deal with attitude change of pre—service
elementary teachers in mathematics courses. Douglas B.

Aichele's (1971) study is especially interesting because

64

of its recency. The purpose of this study was to ascer-
tain whether attitudes are changed in a basic mathematics
course at the University of Missouri--Columbia. It is a
basic objective of the course to develop positive attitudes.
Attitudes were measured in the four areas below:

A. The learning of mathematics.

B. Mathematics as a process.

C. The place of mathematics in society.

D. School and learning generally.

The first step was the deve10pment of an attitude
scale to measure the four areas listed above. The Husen
study was used as a model for the statements. A numerical
scale from unfavorable to most favorable was calculated
for each statement using the method of paired comparison.
One hundred and seventeen students from classes at the
university determined the scale values. The directions to
the students were to select the most favorable statement
from each pair. After scaling.reliability was ascertained
through a testvretest procedure with an interval of one
week. A random sample of 71 students was selected for
this purpose. However, they were not from the same
population as that used for the experiment. The reliability
co-efficients were part A .3132, part B .5585, part C .5766,
and part D .5002.

The attitude scale was then administered to all 65
students taking the course in a pre-test post-test mode.
For each of the four parts students were assigned the

median score of the scale values of those statments with

which the student agreed. The pre- and post—test attitude

65

scores were compared by means of a sign test. Also items
were analyzed independently for any significant changes.
There were no Significant differences between medians in
any of the four categories. Although there were no signi—
ficant differences for categories there were some items
that did provide significant differences.

Aichele concluded that if attitudinal changes are
legitimate objectives of instruction then more emphasis
should be place on them in this course. This may mean
changing some aspects of the course. Also he recommended
that attitudinal objectives should be regularly measured.

Possible confounding variables such as instructor
and time were not mentioned. The results of his study
can not be inferred to any other pOpulation. The study
has a very narrow purpose, sc0pe, and audience. Any
results in category A are very suspect because of the low
reliability co-efficient. One wonders why the author did
not try to increase the reliability in this category.

William Campbell (1971) used pages of elementary
mathematics textbooks as a supplement to which he "added
questions, comments, and directions (p. 6448A)." His
sample was 134 students at Wisconsin State University——
Platteville. He used the Aiken Dreger Attitude Inventory
and a test of mathematics concepts. He found both groups
decreased in their attitudes towards mathematics. McNerney
(1970) used a slide presentation of current textbook pages

in a required general education mathematics course for the

66

purpose of changing attitudes. He found non-significant
differences in attitude and achievement.

In contrast to the above results are the following
studies. B. L. Erickson (1970) found that a two quarter
mathematics content course for prospective elementary
teachers resulted in significant change in attitude and
achievement. He also reported a significant correlation
between attitude and achievement. B. C. Gee (1966) inves-
tigated the effect of a Brigham Young University mathe-
matics course (Mathematics 305) on 186 prospective elemen-
tary teachers. This mathematics course was based upon the
C.U.P.M..recommendations. It included topics on logic,
sets and relations, structure of the whole numbers,
integers and rationals, standard topics in number theory,
systems of numeration, algorithim, and nondecimal numera—
tion systems.

Gee pre— and postvtested with Glennon's Test of
Arithmetical Understanding and Dutton's Attitude Scale.

He also obtained information on their A.C.T. scores. He
found a significant positive change in attitudes towards
mathematics. Also there was a significant gain in
mathematical understanding. If success is measured by
final grade then attitudes and mathematical understanding
are significantly related to success.

This conclusion has also been reached by Shatkin
(1969) who studied attitude change in pre—service elemen-

tary teachers at Ohio State University. The author

67

developed his own Likert Type Inventory for assessing
attitudes in a course. He found that success was signi-
ficantly correlated with the degree of attitude change.

Several scholars have studied attitude change in
methods courses as opposed to mathematics courses. Hilton
(1970) revealed a significant correlation between knowledge
and attitude as well as an increase in positive attitudes
of 72 juniors during their professional preparation.

The next article is of special importance since it
is one of the few and most recent attempts at changing
attitudes which produced positive results. Consequently
"Enrichment: A Method of Changing the Attitudes of Pros—
pective Elementary Teachers Toward Mathematics" by Bonnie
H. Litwiller (1970) will be reviewed in depth.

Litwiller began with a discussion 0f the role
of attitudes in learning mathematics as a rationale for
her study. She hypothesized that teachers transmit their
attitudes to their students. This is critical since in
her view many elementary teachers have a negative attitude
toward mathematics. The author cited references to
support a conclusion that attitudes can be changed even
though they have been formed over a long period of time.
It is unfortunate that Litwiller did not give more
references to support her rationale. In fact a perusal of
the literature in this area indicated that most of the

results are inconclusive.

68

The aim of the experiment was to see whether the
attitudes of prospective elementary teachers could be
changed toward mathematics through the use of enrichment
problems. Enrichment problems were defined as "extra
problems that were not necessarily related to the mathe-
matical concepts being studied at that time (Litwiller,
1970, p. 346)." These enrichment problems consisted of
reviews of previous concepts, introductions to concepts
that would be studied at a later date, and recreational
mathematics. These problems were designed to be challeng—
ing to the students; they quite often had more than one
possible solution. In the experimental sections enrichment
problems were considered each class session in addition to
the regular assignments.

The sample was a group of 195 prospective elementary
teachers at Indiana University. It consisted of six sec-
tions; four were designated experimental (95 students) and
two were control (50 students). Students were assigned
alphabetically to the sections by the university. These
sections were allotted to four instructors.

The pre- and post-test was Dutton Attitude Scale
which was administered the first and last weeks of school.
A multiple choice achievement test designed by the authors
of the textbook (Peterson and Hashisaki) was administered
to "insure uniformity in testing the achievement of
experimental and control groups (Litwiller, 1970, p. 346)."

The hypothesis to be tested was the following:

69

There is no significant difference between the

attitudes toward mathematics of the students who

were exposed to the enrichment problems and the

students who were not exposed to the enrichment

problems as measured by Dutton's Attitude Scale

(p. 348).
Analysis of covariance was used to test this hypothesis.
The reason given for this analysis was non—random assign-
ment of students to groups so as to compensate for initial
group differences. The difference was found to be signi—
ficant at the .001 level F (l, 142) = 17.90. It was noted
that teacher was a confounding variable and the results
must be interpreted in light of this fact.

The Peterson-Hashisaki test of arithmetic achieve-
ment was found to have a Kuder-Richardson Formula 21
reliability of .78 in the experimental and .85 in the
control. A t-test of uncorrelated means resulted in a
rejection of the hypothesis that "there is no significant
difference between the achievement of the experimental
group and the control group as measured by the achievement
test scores (Litwiller, 1970, p. 349)." The study con“
cluded that planned enrichment could be a means of improv~
ing prospective elementary teachers attitudes toward
mathematics.

The following criticisms can be leveled at this
article. First, as the author pointed out teacher is
confounded with treatment. The author did not indicate
the arrangement of teachers with sections. However, it

would seem that a small modification of the experiment

would result in three instructors each teaching an

70

experimental and control section. This modification would
control for this confounding variable as instructor would
be crossed with test and groups. Another possible con-
founding variable is time.

Secondly, Litwiller did not detail the method of
presenting the enrichment material. If students see
mathematics as being interesting and enjoyable because it
is presented in a certain manner, it should be expected
that attitudes will change. If the enrichment material
is not presented to the students in the same manner as
the required material, then method might be the real
factor in the change in attitude.

Another factor when considering confounding varian
bles is difference in type of material, e.g., in an algebra
class using geometry as enrichment. It is different, then
type of material is confounded with enrichment. Also if
this enrichment was class activity as the paper indicated
then classroom is the appropriate unit of observation
rather than individual. It becomes apparent that Litwiller
failed to fulfill the requirements of a good hypothesis
since she did not adequately define the variable enrichment.

Thirdly, Litwiller also violated the requirements
for a good hypothesis by failing to adequately define the
population to which the results may be inferred. She
indicated that it is elementary teachers. However, since
her example was a group of prospective elementary teachers

'at Indiana University during the first semester a

71

Cornfield-Tukey argument would allow inference to a popu-
lation with their characteristics but not to the population
of all elementary teachers.

Fourthly, Litwiller did not state the conditions
under which the tests were given. 'The Dutton Scale is a
type of self-administered test and is sensitive to outside
influences. If the students felt that their answers on
the Dutton Scale would influence their grade, the test
would be biased. Furthermore, grades (success) are an
important factor in the development of attitudes. Conse-
quently, the grades received or grades perceived by the
student are another confounding variable in the study.
This is especially true since they are determined by the
student's work in the ordinary portion of the classroom
activities. The order of administering the achievement
test and post-test should also be considered. If the
achievement test was administered prior to the post-test,
a student's results or anticipated results would affect
his answers on Dutton's Scale.

Design of the experiment is considered next. The
design violated by its non—random assignment the first
criterion for a good control group. The author endeavored
to correct this defect through the use of analysis of
covariance. She failed to supply data to support the
assumptions of ANCOVA. In fact the distribution of
students into the three categories of the Dutton Scale by

centile rank did not seem to be normal. Since the Dutton

72

Scale was administered to the students twice there is the
possibility of a carry over effect.

In another study of enrichment (geometry) for
prospective elementary teachers and Dutton's Attitude
Scale, Wardrop (1970) found no significant differences
between control and experimental groups on the variables
of attitude and achievement. Recently, Darrell Kilman
(1971) compared micro-teaching and lecturing on two
variables: attitude and achievement. He used Dutton's
scale and a test called Structure of the Number System as
pre- and post-tests on a group of prospective elementary
teachers at the State College of Arkansas. He found no
significant differences between groups on either attitude
or achievement. However, there were significant gains in
achievement and attitude for each group during the experiw
ment.

In 1965 Dutton conducted a study on attitude change.
Unfortunately he used a one group design in which he prev
tested and postvtested with his attitude scale. He also
pre- and post-tested with an arithmetic comprehension test.
His sample was 160 prospective elementary teachers. He
found a significant gain in mathematical comprehension
when teachers enrolled in a control course which put an
emphasis on understanding. He was unable to report a
significant gain in attitude. However, he still felt that
attitudes played a significant role in learning and con—

cluded with the following:

73

Those aspects liked most by students are the
general interest in and practical value of
arithmetic, enjoyment in working arithmetic
problems, the challenge presented and recog-
nition that arithmetic is an important sub-
ject. Dislike for arithmetic centered around
not being sure of oneself, fear of work
problems, and not being good with figures. To
this list of negative feelings must be added
boring drill and difficult impractical prob-
lems (Dutton, 1965, p. 364).

The next study with prospective elementary teachers
similar to the above seems to support the thesis about
success experiences. William Pureell (1965) measured
attitude change in relation to two things: (1) change in
the understanding of arithmetical concepts, and (2) grades
received in course work. At two levels the attitudes Were
measured with Dutton's Attitude Scale: (1) the freshman
content course and (2) a junior methods course. Purcell
reported that the understanding of arithmetical concepts
promoted positive attitudes and higher grades. On the
other hand a positive attitude did not result in higher
grades. He also found a tendency on the part of well
prepared students to underestimate their attitudes and
poorly prepared students to overestimate their attitudes.
Furthermore Purcell concluded that teaching methods is
important in trying to change attitudes.

Buckeye (1970) investigated the effect of a
mathematics laboratory on the creativity and attitude of
145 (four groups; 77 experimental and 72 control) pros-

pective elementary teachers at Eastern Michigan University.

These students were enrolled in a mathematics methods

74

course. The experimental group participated in discovery
experiences with concrete materials in addition to the
regular course. All groups were pre- and post-tested

with Dutton's Attitude Scale and a test of creativity.

They were also post-tested with an experimenter designed
achievement test using ANCOVA. The experimenter dis-
covered significant differences in attitude and creativity.
In interpreting this study one must be careful since the
author did not detail the enrichment experience. The
laboratory approach means different things to different
people. However, it certainly indicated that this approach
provides a method for increasing attitudes.

In another study of attitude change in prospective
elementary teachers Raymond Knodel (1971) compared a cone
trol group of 33 students who were taught "a two«quarter
program in which one quarter of mathematics content,
taught by a mathematics specialist, was followed by one
quarter of arithmetic methods, taught by a specialist in
arithmetic methods (p. 4010)," with an experimental group
of 32 students "a two-quarter program in which the material
taught in the separate mathematics content and arithmetic
methods courses was integrated and team taught, with the
mathematics specialist teaching the mathematics portion
and the methods specialist teaching the methods portion
(p. 4010)." To be included in the study, students must
have completed the course. This could result in a severe

bias in the experiment if any number of students dropped

75

a section. Pre- and post-test scores of Dutton's Scale
were evaluated using a t-test; a significant effect was
reported for both groups. However, there was no signi-
ficant difference between groups on attitude gains. The
use of ANCOVA to test gains in mathematical understanding
by the author resulted in a significant difference.

Rolf E. Muuss (1969) measured the attitude change
of prospective elementary teachers enrolled in elementary
education courses at Goucher College and then as interns
in the Baltimore and Baltimore county schools. The
instrument used was the Minnesota Teacher Attitude Inven—
tory (MTAI). He found that during the academic phase
their attitudes became more tolerant and child centered.
However, as interns there was a significant decrease in
attitudes. They became more teacher centered and tradi—
tional. It was found that frustration and difficulty in
the internship were correlated (significant at .05) with
this decrease in attitude. Instead of defining attitude
on a like—dislike dimension, Patrick Collier (1970)
considered the formal-informal dimension. He found that
high achievers have a more informal View of mathematics
and that the informality was developed in the methods
courses.

In a study of the effect of programmed instruction
on attitudes Beattie (1970) sampled 128 prospective elemen-
tary teachers. The programmed instruction was used as a

supplement to the usual methods course. He found a

76

positive correlation between mathematical understanding
and attitude toward teaching elementary school mathematics.
Also there was a significant improvement in attitude for
both experimental and control but there was no significant
difference between groups.

Several studies have attempted to study attitude F-
change in both mathematics and education courses and the

effect of combined courses. Taylor (1971) found a signi—

 

ficant positive change in attitudes toward mathematics in
both education and mathematics courses designed for pros—
pective elementary teachers. Wickes (1968) compared two
different course arrangements for the preparation of pre—
service elementary teachers at Brigham Young University.
He found significant gains in attitude and understanding
for both--the 104 students in the unified methods and
mathematics course (experimental) and the 65 students
enrolled in separated mathematics course and methods
(control). However, there was no significant difference
between groups. The control group exhibited significantly
greater understanding than the experimental group.

In another study involving a comparison of dif—
ferent types of methods courses for pre-service elementary
teachers, Norman Young (1969) was unable to ascertain
which approach was the most efficacious for all types of

students.

77

In-service Teachers

 

In the next series of studies research with
in—service teachers will be considered. Todd (1966)
evaluated a mathematics course for in-service teachers
on the basis of attitude change and understanding of
arithmetic. The course had as its basis the Committee
on the Undergraduate Program in Mathematics (CUPM);recom-
mendations. He pre- and post-tested with Glennon's Test
of Mathematical Understanding and Dutton's Attitude
Inventory. He reported significant changes in mathematical
understanding and attitude for those remaining in the
course. However 30 dropouts (out of 287) were significantly
lower in understanding.

In agreement Haynes (1970) concluded that in—service
courses can change the attitudes (as measured by MTAI) of
mathematics teachers. Bean (1959) concluded "a casual
relationship seems to exist between a teacher's arithmeti-
cal competence and her enjoyment in teaching arithmetic
(p. 708)." He also felt that an understanding of arithme—
tical concepts preceded the development of competence and
enjoyment of mathematics.

M. J. Dossett (1964) used Dutton's Arithmetic
Attitude Inventory, a test of Mathematical Understanding,
and the Texas Elementary School Mathematics Practices
Interview Scale in a study of the effects of workshops on
89 Missouri teachers. The workshops were of the lecture-

demonstration type. Two sessions were used for construction

78

and demonstration of exploratory materials. Although her
sample was randomly selected it was from a population of
volunteers introducing a serious defect in her experiment.

Dossett found no significant gains (t-test) in
attitude but when the control group (a group of in-service
teachers not in the workshop) was compared with the treat—
ment through the use of analysis of covariance a signifi-
cant difference was recorded. Her design allowed her to
test the effect that a pre-test might have on a post-test.
Contrary to what might be expected there was no signifi—
cant effect. A plausible explanation is the setting—~a
workshop in contrast to the usual classroom. Also there
was a significant difference, using analysis of covariance,
:himathematicalunderstanding between control and treatment
but again the gain was not significant through the use of
a t-test. There was no testing effect for the test of
mathematical understanding. She reported a significant
correlation between mathematical understanding and semester
hours of collegiate mathematics.

Through the use of the Texas instrument Dossett
found that there was a significant increase in the use of
instructional materials, discussion time, and non-textbook
activities. She was able to conclude that workshops helped
change the practices in the mathematics classrooms of the
participants.

These results were supported in the following

studies. Clarkson (1969) used a summer institute to study

79

the relationship of attitude, mathematics skills, and
mathematical understanding of 71 elementary school
teachers. He also used Dutton's scale as a measure of
attitude. The institute consisted of: course work,
seminars, and observations of "master" teachers. He
found significant differences in attitude, understanding,
and computational skill. He also cited a significant
relationship between increased understanding and attitude.
Schmelter (1970) used media (radio» television)
and Saturday class sessions to change the attitudes of
213 in—service teachers toward mathematics. Teachers
participated in a course called Teaching Contemporary
Mathematics in the Elementary School. The project was a
joint venture of the Wisconsin Department of Public
Instruction and the System of Wisconsin State Universities.
Strangely, attitude toward radio instruction deteriorated
significantly yet attitude toward mathematics increased
significantly. There are other plausible explanations
for this attitude change.7 The novelty presented by the
varied types of instruction might have induced a Hawthorne
type effect or there might have been a significant increase
in understanding among the teachers resulting in improved
attitudes.
In contrast to the positive results reported above,
Wess (1970) in a study of 22 elementary teachers who parti—
cipated in the "Woodbury County Elementary School Develop—
mental Mathematics" program found no significant difference

in teachers' attitudes.

80

Summary

The following conclusions are appropriate from the
literature reviewed in these first two chapters. There is
a crying need to find effective means of modifying
teachers' attitudes toward mathematics. Substantial num-
bers of teachers have poor attitudes. There is a sound
philosophical-psychological framework on which to base
decision-making for attitude change- This framework is
supported by the literature. A potential way to modify
attitudes is through the success experiences provided by
activity materials. However, various other types of
course work also holds hope for the modification of the
attitudes of students, prospective teachers, and teachers.
It would seem that the attitudes'of young children are not
very stable (Aiken, 1970, pp. 557—558). The so-called

modern curriculum does not seem to have promoted positive

 

attitudes among the population of pre—service and innservice

teachers.

CHAPTER III

DESIGN OF THE STUDY

The Sample

 

The concern of this study is to assess the rela—
tionship of activity materials used in a graduate mathe—
matics methods course to elementary teachers'attitudes~
towards mathematics. The sample pOpulation consisted of
all in-service elementary teachers registered and parti-
cipating in Education 830A, Methods and Materials of
Teaching Mathematics in the Elementary School, Fall and
Winter terms during the 1971-72 school year at Michigan
State University. Through the use of an argument proposed
by Cornfield and Tukey (1956) the results of this research
may be extended to a larger target population. This
population is all in—service elementary teachers in Michi—
gan who enroll in Methods and Materials of Mathematics in
the Elementary School (Education 830A) at Michigan State
University. The Cornfield and Tukey argument applies only
to those elementary teachers who in the future or past
experience the type of curriculum presently used in that

course .

81

 

82

The students in Methods and Materials usually are
teachers from all over the lower peninsula of Michigan.
Sections have been offered in Pontiac, East Lansing, Grand
Rapids, Berrien Springs, Alpena, Traverse City, Saginaw,
Flint, and Port Huron. Methods and Materials is elected
by many masters candidates in elementary education. The
course is also taken by other interested in—service elemen-
tary teachers, possibly to complete state certification
requirements.

In general students enrolled in Methods and Materials
are somewhat older and have more teaching experience than
the population of university students. Hopefully they have
a better understanding of the problems involved in the
teaching of mathematics than pre—service elementary
teachers. A more detailed examination of population para—

meters will be undertaken in Chapter IV.

The Course

 

Education 830A is entitled Methods and Materials
of Mathematics in Elementary Schools. The Michigan State
University catalogue (1972) described it in the following
manner:

Develops an understanding of the basic principles
and techniques of effective instruction in the
various subject-matter areas in the school curri—
culum. Students will be expected to investigate
research as it relates to the improvement of
instruction in a special field of study.

The course is normally taught by faculty members

of the Department of Elementary Education at Michigan

83

State University. It is usually offered for a three-hour
block of time weekly for ten weeks. The contents of the
course may vary slightly from instructor to instructor but
usually includes content topics appropriate to the elemen-
tary school and methods for teaching these topics. They
include sets, numbers and numeration, numbers and opera~
tions, geometry, and measurement. Accompanying these
topics are activity materials such as: attribute games
and blocks, Dienes' blocks, Cuisenaire rods, geoboards,
clay, weights, beans, and playing cards.* The text chosen

by the staff is Freedom to Learn by Biggs and MacLean

 

(1969).
The staff agreed on the following objectives for
students participating in Education 830A:
1. Students should become acquainted with
materials for teaching skills, concepts,
and applications in mathematics.
2. Students should acquire a greater under-
standing of the structure and uses of

mathematics.

3. Students should acquire a more positive
attitude toward mathematics.

4. Students should take back and use new
methods of teaching mathematics in their
classrooms.
As a result of objective four, students are asked

to design aids and teach lessons as part of the require-

ments of the course.

 

*For a complete outline see Appendix A.

84

Measures

Three instruments were used for measurement in this
study: (1) an attitude inventory developed by Dutton
(1962); (2) an accompanying informational questionnaire;
and (3) a test of mathematical understanding developed by
M. J. Dossett (1964).* Consider first Dutton's survey
and questionnaire.

There is an increase in validity by the choice of
Dutton's instrument over an experimenter constructed
instrument. It has been widely used and respected in
research of this type. Shaw and Wright (1967)-have a
telling argument on this point:

Research that has been done by different
investigators often is not directly comparable.
Second the quality of measuring instruments
often is poorer than it would have been if
existing scales had been used and improved

(p. ix).

Also a pilot study during the summer of 1971 at
Michigan State University was conducted to ascertain the
validity of the survey. From conversations with the
students it was concluded that the scale approximated
their attitudes toward mathematics. Thurstone (1937)
suggested the following criteria for validity:

l. A measuring instrument must not be
seriously affected in its measuring
function by the objects of measurement

. . . the scale must transcend the
group measured.

 

*Dutton and Dossett granted permission for the use

of their tests. See Appendix B for examples of these
measures.

 

85

2. The scale values of the statements should

not be affected by the opinions of the

people who constructed it (p. 91).

If properly administered Dutton's Scale satisfies these
requirements. In order to ensure the validity of our
scale,conditions must be such that the student has no
reason to lie. The procedures set up for the experiment
minimize this problem.

Also Dutton's survey has been found to be extremely
reliable by several authors. At the .01 level Litwiller
(1970, pp. 52-53) found that correlation coefficients of
.75 and .74 were significant thus indicating a high
degree of reliability. Dutton (1954) used the testvretest
method to estimate the reliability of his scale and found
a correlation of .94. In a later study (1962) he reported
a reliability of .84. In this same study he concluded
that the scale values were stable after a change in rank
order and placement of the questions.

Content validity of the test of mathematical under-
standing was assured by Dossett (1964, pp. 88—103) through
her careful construction of the test. She began by setting
objectives and listing the content areas that should be
included in such a test by reviewing the literature. The
test was then given to in—service elementary teachers for
their evaluation of the appropriateness of the items.

Items were then revised or eliminated. The test was then
given a trial administration. She also performed an item

analysis to guarantee internal validity.

86

From the trial administration Dossett was able to
determine a Pearson product-moment correlation coefficient
of .87 for the two forms of the test. This coefficient
was significant at the .05 level (1964, p. 96). In order
to ascertain the equivalency of the two forms she
examined item correlations within and between the two tests.
She was unable to find any significant differences in
correlations which implies that the two forms are considered
to be equivalent.

A test of mathematical understanding was chosen
because the literature seemed to indicate a difference
between understanding and achievement.* The literature
also indicated that a test of understanding might provide
greater correlations with Dutton's Attitude Inventory.

This test had been used with in—service teachers. Also

it was ready and available for examination and testing.

Experimental Design

Three principle questions were asked: (1) Does
the use of the mathematics activity materials used in
Methods and Materials increase in-service teachers' atti-
tudes toward mathematics? (2) Does the use of the
mathematics activity materials used in Methods and
Materials increase in—service teachers' understanding of
mathematics? and (3) Does an increase in mathematical under-

standing result in an increase in attitude or vice versa?

 

*In this study they will be taken to mean the same.

87

Mathematical Understanding and Attitude

In order to answer these principle questions data
was sought through administration of an attitude inventory and
tests of mathematical understanding. During Fall term,
1971, there were two groups available for this experiment.
The Fall term groups were designated "1" (East Lansing)
and "2" (Grand Rapids). At the beginning of the first
class in each of the locations the pre—tests of attitude
and mathematical understanding (Form A) were administered
respectively by a person having nothing to do with the
class. During the final exam period or last class as the
instructor preferred the attitude survey, the question—
naire, and Form B of the test of mathematical understanding
were administered respectively.

The completed questionnaire contained information
on the student's teaching experience and level of mathe~
matics preparation. The attitude test was administered
before the mathematical understanding test to avoid their
feelings about the test influencing their answers on the
attitude scale. In order to avoid the subjects biasing
the results because of a desire to do well in the course
or influence their grades an outside person administered
the tests. Students were told that the results were for
the administrator's purposes only.

Since the groups contained unequal numbers of
subjects the testing situation could be represented as

shown in Table 1.

88

TABLE l.-—Repeated Measures Design.

 

Groups Subjects Pre-test Post-test

 

 

15

 

The same design would also represent the situation for
mathematical understanding. This is a repeated measures
design with two unequal groups. A repeated measures
design involves the making of several observations on the
same subjects (Glass and Stanley, 1970, p. 469). This
analysis will allow the testing of interaction effects

as well as main treatment effects (Glass and Stanley,

1970, p. 470). This analysis is more powerful than the

89

t-test generally used for the comparison of two means
because it allows for the variation introduced by the
use of two groups.

In order to answer the third question an analysis
recommended by Aiken (1970) and Campbell and Stanley (1963)
was used. Aiken wrote that:

Their (Campbell and Stanley, 1963) proposal

for obtaining information concerning cause—

effect relations through correlations across

time (cross-lagged panel correlation) would

appear to be a potentially fruitful approach

to an analysis of the direction of cause and

effect in studies of teacher and pupil atti—

tudes and achievement (p. 588).
Since attitude and achievement data from the groups was
obtained during a ten—week experiment the cross-lagged

panel correlation seems appropriate. The situation is

represented in Table 2.

TABLE 2.--Cross-lagged Panel Correlation.

 

Pre Post
Attitude xll X12
Achievement X21 X22

 

90

From Stanley (1967) the following analysis is

obtained:

For large samples and making the assumption that
(xll,x12,x21,x22) are distributed multivariate

normal

/H[(r - r )- ( )J

D
x11X22 x12X21 X11X22
a
rX X - rX X
11 22 12 21

o
X12X21

%n(0,1) if HO: true

Note: if H : true (p - p ) = 0
° X11X22 X12X21

2 2
6 = (1 - 4 ) + (l - r ) + r r
X11x22 X12X21 X11X22 X12X21
11 12 21 22 11 21 12 22
+ 2Er x r x + rX x rX x
X12 11 X22 21 11 21 22 12

- 2[r + r

r r r r
X11X22 X11X12 X11X21 x11x22 X11X12 X22X21

+r +r

r r r r
x11x12 X22X12 X12X21 X11X21 X22X21 X12X21

If rX x > rX x , then achievement causes an
11 22 21 12

increase in attitude. On the other hand if r >
X21X12

r , then attitude causes a gain in achievement.
x11x22

 

91

Equalityfiof Groups

 

In a study of this sort an inevitable question is
asked: How do you know that your group is not in some way
special, i.e., different from all others? In order to
answer this question four intact groups were tested with
Dutton's Attitude Inventory, the questionnaire, and Form
A of Dossett's Test of Mathematics Understanding. They
were tested during the first class period using the same
procedure as the Fall groups. Were these groups sampled
from the same population? The most appropriate analysis
(Glass and Stanley, 1970, p. 353) for this situation is
analysis of variance. Since attitude and mathematical
understanding are of interest it is appropriate to test
equality of means for both attitude and mathematics under-
standing. If there is no difference then external validity
will have been increased.

This approach will have two beneficial side
effects if the null hypothesis is accepted. First, con“
founding variables such as history, maturation, and time
suggeSted by Campbell and Stanley (1963, p. 5) are less
likely to account for any significant results found in
the Fall experimental groups. The Winter groups will
serve as control groups for the experiment. In fact there
is only a short time lag (two and one half weeks) between
the post—testing of the Fall groups and the testing of the
Winter groups. Consequently this would allow the use of

the Winter data as a post-test.

 

92

Secondly, if equality of groups existed this would

allow the combining of data for the Fall and Winter groups

obtaining more accurate correlations.

The Relationship of Selected Variables

 

In this study data on twelve variables was
obtained from the administration of the questionnaire.
These variables are: (1) sex; (2) degree program; (3)
age; (4) years of teaching experience; (5) years of high
school mathematics; (6) credits of college mathematics;
(7) grade level assignment; (8) mathematics vs. non—
mathematics teaching; (9) years at present grade level
assignment; (10) perceived attitude; (11) grade level
attitude developed; and (12) average mathematics grades.

Information about attitude and mathematical understanding

 

were obtained from Dutton's Attitude Inventory and Dossett's

Test of Mathematics Understanding. An obvious question to

ask: What relationship if any exists between these four—

teen variables? In order to assess any possible rela—

tionships a Pearson product-moment correlation coefficient

was found for each pair of variables.
The overall design has three major advantages as
compared with other studies in this area:

1. A curriculum was designed for the express
purpose of changing the attitudes of
in-service elementary teachers.

2. Subjects in this experiment are in-service
teachers instead of the more accessible
pre-service teachers whose attitudes about
school mathematics are subject to greater
errors of measurement.

93

3. Extensive precautions are to be taken in
accordance with Thurstone's warnings to
insure that the subjects feel free to
record their true attitude on the survey.

Basic Assumptions

 

The above design necessitates that several under—
lying assumptions be recognized. There are five general
assumptions:

1. Mathematical understanding can be objec—
tively measured.

2. Attitude towards mathematics can be
objectively measured.

3. Both the test of mathematical understanding
and Dutton's Attitude Scale are valid and
reliable.

4. Any change in mathematical understanding or
attitude is the result of learning experi-
ences in Methods and Materials of Mathema—
tics in the Elementary School, i.e., the
result of the treatment.

5. The activity learning experiences were
appropriate for the sample and were
appropriate for the tOpics to be discussed
in the course.

In order for the analysis in the previous section to be
performed on the hypotheses certain assumptions must be
satisfied. Analysis of variance and repeated measures

have the following assumptions in common: normality of
sample populations, equality of variances, and independence
within and between groups (Glass and Stanley, 1970, pp.
340, 470). In addition a repeated measures design has

the assumption: "the correlations of all pairs of levels

of the fixed factor across the population of random factor

94

levels must be the same (Glass and Stanley, 1970, p. 470)."

These assumptions will be considered in Chapter IV.

Limitations of the Study

Although for the purposes of an experimental study
the above assumptions were made there may be no valid
measures of attitudes towards mathematics (Morrisett and
Vinsonhaler, 1965, p. 133). Also as pointed out by Aiken
(1970, p. 558) a score on an attitude inventory is a
composite of attitudes. It is not a measure of attitude
toward some specific part of mathematics. The following
limitations are also inherent in this study:

1. The limitations imposed by the use of
statistical methodology.

2. Only six sections of Methods and Materials
at Michigan State University participated
in this study during the Fall of 1971 and
Winter of 1972. Therefore any generaliza—
tions are limited to a population similar
to these samples.

Testable Hypotheses

The main concern is the efficacy of a mathematics
methods course for in-service teachers. Consequently of
interest is what changes took place during the course.
This suggests the following two hypotheses:

1. There is no difference in attitude as
measured by Dutton's Attitude Inventory
between pre— and post-test of groups
1 and 2.

2. There is no difference in mathematical
understanding as measured by Dossett's
Test of Mathematical Understanding

between pre- and post—tests of groups
1 and 2.

95

Another question directly related to the above is whether

attitudes are changed by achievement or vice Versa.

Therefore the following hypothesis:

3.

Let r represent the Pearson product-
momen correlation between the pre-
test of mathematical understanding as
measured by Dossett's Test of Mathema-
tical Understanding and the post-test
of attitude as measured by Dutton's
Attitude Inventory. Let r2 represent
the Pearson product-moment correlation
between the post-test of mathematical
understanding as measured by Dossett's
Test of Mathematical Understanding and
the pre-test of attitude as measured by
Dutton's Attitude Inventory. Then there
is no difference between r1 and r2 on
groups 1 and 2.

The next group of hypotheses pertain to the equivalence

the six groups involved in the study.

4.

There is no difference in mean attitude
scores as measured by Dutton's Attitude
Inventory of the six groups.

There is no difference in mean mathema—
tical understanding scores as measured
by Dossett's Test of Mathematical Under—
standing of the six groups.

In addition to these major hypotheses there are

minor hypotheses concerning the attitude of in-service

elementary teachers and other variables.

6.

Given the fourteen variables (sex, degree
program, age, years of teaching experience,
years of high school mathematics, credits
of college mathematics, grade level assign-
ment, mathematics vs. non—mathematics
teaching, years at present grade level
assignment, perceived attitude, grade level
atttiude developed, average mathematics
grades, attitude and mathematical under-
standing) there exists no correlation
between any of the variables taken two at

a time on all six groups.

of

 

96

All hypotheses will be tested at the a = .05
level of significance. The calculations will be performed
by the Michigan State Computer Center utilizing a C.D.C.

3600 and 6400.

Summary

This chapter contains the elements of the experi-
mental design for this research project. It began with a
discussion of the sample and population. The next section
contained information about the course which was used to
provide the sample population for this study. Section
three was devoted to an analysis of the instruments used
to collect the data for the study. In the fourth section
the experimental design including statistical and design
procedures were the focus of attention. Basic assumptions
and limitations were the focus of sections five and six,
respectively. The testable hypotheses of the study were

presented in the last section.

CHAPTER IV

ANALYSIS OF THE DATA

Data for the study were gathered from four sources:
a questionnaire, Dutton's Attitude Inventory, and forms
A and B of Dossett's Test of Mathematical Understanding.
All information was compiled during the two testing
sessions either on the first or last day of class. In the
first section of this chapter relevant bibliographical
data from the questionnaire will be presented. Also
under discussion will be some of the reasons given by the
students for liking or disliking mathematics. The second
section will focus on the assumptions of the study.

The next five sections will present relevant
information bearing on the hypotheses of the study.
Section three will discuss attitude change. In section
four change in achievement will be considered. The rela-
tionship of attitude and achievement will be discussed in
section five. Correlations between selected factors,
attitude, and achievement will be under study in section
six. In section seven the equivalency of groups will be

under consideration. The last section is a summary.

97

 

98

 

The Sample

In this section data gathered about population
parameters is presented. During Fall term, September 23
through December 10, 1971, Methods and Materials was
offered in Grand Rapids and East Lansing. The on campus
or East Lansing group (designated "1") met on Wednesday
evenings from seven to ten and consisted of 26 peOple-—
3 males and 23 females. Their ages ranged from 22 to 51.
All elementary grades (K-7) were represented. Five
students were non-degree. There were 19 Master of Arts,
one Master of Science, and one Master of Arts for Teachers
candidates in the group. This group originally contained
29 students. However two students were graduate assistants
and a third was not teaching in any capacity. Therefore
they did not fit the criterion of an in—service elementary
teacher and were eliminated from the study.

The Grand Rapids group (2) consisted of 15 people—-
2 males and 13 females. Their ages ranged from 24 to 46.
In contrast to group one there were no kindergarden or
middle sChool teachers in the sample. Nine people were
non-degree and six people were seeking a Master of Arts
degree. This group originally consisted of 20 people.
However, five were eliminated for the following reasons:
one student was an undergraduate and not teaching, two
students were not elementary teachers, and two peOple did
not take the post-test. When asked why they refused to
participate they declared that they could not devote any

more time to the class.

 

“'5‘ :1! ‘z
.

99

The next collection of groups met Winter term,
January 3 through March 10, 1972. These groups were
tested at the first class meeting. The Pontiac group
(3) met at Oakland University, where extension classes
of Michigan State University meet. This group had 18
members-~3 males and 15 females. Their ages ranged from
22 to 52. In this group all elementary grades were
represented. There were five non—degree students, twelve
Master of Arts candidates, and one Education Doctorate
candidate in the sample. Of the original 20 students,
one was eliminated because he was a principal and not
teaching and another was not presently employed.

The fourth group met in Saginaw and contained 21
people of which 2 were male and 19 were female. Their
ages ranged from 22 to 52. All elementary grades were
represented except the seventh grade. There were two
special education teachers who taught a multiplicity of
grades. The sample contained eleven non-degree students
and ten Master of Arts candidates. One person was
removed from the original group because he was an elemen-
tary principal.

Group five met in Grand Rapids and contained 2
males and 15 females. Their ages ranged from 22 to 34.
There were no kindergarten or seventh grade teachers in
this sample; all other elementary grades were represented.
There were eight non—degree students, eight Master of Arts

candidates, and one person on an Educational Specialist

100

program. From this group four persons were removed from
the study-—three because they were not presently employed
and another was employed as a secondary teacher.

The last group (6) met on the campus of Michigan
State University in East Lansing. This group contained
25 people—-5 males and 20 females. Their ages ranged
from 22 to 43. All elementary grades were represented
except kindergarten. In this group there were two non-
degree students, eighteen Master of Arts candidates, two
Master of Science candidates, one Master of Arts for
Teachers, and two Educational Specialist candidates. Of
the original group ten were not presently employed, one
person had never taught, and another was an elementary
principal. Consequently they were taken out of the study.

The six groups were compared from the questionnaire
data on the fourteen variables (sex, degree program, age,
years of teaching experience, years of high school mathe—
matics, credits of college mathematics, grade level
assignment, mathematics vs. non-mathematics teaching,
years at present grade level assignment, perceived atti-
tude, grade level attitude developed, average mathematics
grades, attitude, and mathematical understanding). It
is summarized in Tables 3-7.

The subjects' reported grade level where attitude
toward mathematics was developed spanned all levels
including college. However a majority of them reported

their attitude was developed in later elementary (fifth

101

 

 

 

 

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TABLE 7.-—Means and Values of Selected Variables.
Minimum Maximum
Value Value Mean s.d.
Fall
1a 23 51 30.45 6.66
2b 0 12 3.73 2.19
3C 1 4 2.68 1.04
4d 0 38 9.07 9.04
5e Kindergarten 9th 3.08 2.04
6f 0 6 2.32 1.39
7g c = 3 A = 5 4.23 .678
Winter
a
1 21 52 27.67 7.07
2b 0 17 3.80 3.47
3C 1 5 2.86 1.08
4d 0 45 10.3 10.29
5e Kindergarten 8th 3.99 2.16
6f 0 7 2.19 1.84
7g E = 1 A = 5 3.98 .74
Total

la 21 52 28.59 7.03
2b 0 17 3.78 3.09
3C 1 5 2.80 1.07
4d 0 45 9.89 9.87
5e 0 3.93 2.12
6f 0 2.23 1.69
79 E A 4.005 .721

aAge eAssigned Grade Level

erars at Present School
gAverage Math Grade

Years of Teaching Experience
CYears of High School Math
dCredits of College Math

106

and sixth grades) and junior high school (seventh, eighth,
and ninth grades). See Table 8. They were asked to list
two reasons why they liked mathematics and two reasons for
disliking mathematics. These reasons can be divided into
several categories and are considered in order from most
mentioned to least mentioned.

The reason most often mentioned for liking mathe-
matics was the usefulness or practicality of the subject.

The preciseness and logical structure of the subject was 4

 

the next most frequently mentioned category. This category
also included the following responses: the definiteness

of answers, competition, and flexibility in problem solv-
ing. Thirdly, the students seemed to feel that mathe-
matics was challenging, but they failed to expand on this
impression. Many of the sample liked teaching mathematics,
but failed to explain their reasons for this answer. Many
liked mathematics because they were successful at doing
mathematics problems; others felt that the game or puzzle
aspects of mathematics was their prime reason for liking
mathematics.

On the other side of the coin were the reasons for
disliking mathematics. Many mentioned specific subjects
(the subjects collectively were the most frequent response)
as reasons for disliking mathematics but they failed to
analyze these subjects for definitive answers. The single
response most often mentioned was story or word problems.

Many of the sample mentioned teacher related causes for

107

 

 

 

 

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108

disliking mathematics. They felt that their teachers had
stressed drill, memorization, and repetition. They also
felt that their mathematics teachers had been threatening,
unfriendly, and exhibited a superior attitude. Many of
the in-service teachers felt they could not comprehend
mathematics and therefore felt they were failures in the '3
subject. Another group disliked mathematics because its I

exactness made minor errors very important. Still others

 

felt that it was too abstract, even irrelevant.

‘9‘".

Assumptions

 

Before any statistical analysis can be performed
certain assumptions must be satisfied. They are the
assumptions of normality, equality of variance, and
independence within and between groups (Glass and Stanley,
1970, pp. 340, 470). Independence between groups is
easily satisfied by the design of the experiment. Inde—
pendence within groups is assumed. In Table 9 data is
presented to support the normality of groups. None of
these observed values are so different from the expected
values to present a serious invalidation of the assumption
of normality.

Also means and standard deviations were calculated
for both the attitudes and mathematical understanding of
the six groups. This information is contained in Table
10. As can be easily seen from the table both means and
standard deviations are quite close in value. Consequently,
there is no reason to reject the assumption of equality of

variances.

109

TABLE 9.--Skewness and Kurotosis.

 

 

 

 

 

 

 

 

Ideal Groups 1 & 2 Groups 3-6 Total

Attitude

Skewness O - .396 - .883 - .729

Kurotosis 3 2.83 2.902 2.850
Achievement

Skewness 0 - .098 - .526 - .48 H

Kurotosis 3 2.47 2.425 2.626
TABLE 10.--Mathematical Understanding Pre-Tests.

Attitude Achievement

Group Mean s.d. Range Mean ' s.d. Range

1 6.416 1.645 5.65 35.769 6.451 26

2 6.605 1.870 7.86 37.533 5.974 18

3 6.522 1.995 7.00 36.111 10.220 36

4 6.808 1.610 6.33 34.857 9.051 31

5 6.104 2.083 7.04 36.118 7.322 30

6 6.866 1.550 5.51 37.640 8.361 28

 

Change in Attitude

 

One of the major concerns of this study was the
efficacy of the activity learning and materials of the
course, Methods and Materials of Mathematics in Elementary
School, in changing the attitude of in-service elementary
This concern is stated in the following test-

teachers.

able null hypothesis HO:

110

H There is no difference in attitude as

measured by Dutton's Attitude Inventory
between pre— and post-tests on groups

1 and 2.

The statistic used to test this hypothesis was repeated
measures, a form of analysis of variance. This statistic
accounts for the two group design and tests for any inter-
actions that might be present. The data given in Table 11
presents the results of that analysis. This data is for
groups 1 and 2 only since postvtest data for groups 3—6
were not obtained.

A chi square test for homogeneity of variance for

pooled variances X = 1.96 d.f. = 1 and across groups

X = 4.867 d.f. 3 were both non—significant. Hence the

requirements of the chi square test were met.

TABLE ll.--Analysis of Variance.

 

 

Sources d.f. Mean Squares F
Groups 1 .270 .058
Subject /G 39 4.657
Repeated Measures 1 2.560 . 4.400*
R M - G l .093 .160
R M ° S/G 39 .582
Total __81

 

*Significant at the .05 level.

 

111

An F value of 4.09 was required for the hypothesis

to be significant at the .05 level with l and 39 degrees
of freedom. Therefore the observed F was significant and
the null hypothesis was rejected. An examination of the
other F values in Table 11 indicates that there were no

interactions on this measure.

Change in Mathematical Understanding

 

Another major concern of this study was change in
'mathematical understanding. The analysis was the same as
in the previous section. The null hypothesis is as
follows:
HO There is no difference in mathematical

understanding as measured by Dossett's

Test of Mathematical Understanding

between pre- and post-tests on groups

1 and 2.

Table 12 presents the data.

TABLE 12.--Ana1ysis of Variance.

 

 

Sources d.f. Mean Squares F
Groups 1 265.158 3.962
Subject /G 39 66.920
Repeated Measures 1 24.695 1.486
R M ° G . 1 73.774 4.440*
R M ° S/G 39 16.616
Total —8I_

 

*Significant at the .05 level.

 

112

The tests for homogeneity of variance across
groups X2 = 4.746 d.f. = 3 and pooled variances X2 = 17
d.f. = l, were both non-significant.

In this case the tabled F for l and 39 is 4.09.
Therefore the null hypothesis is not rejected. The other
F tests indicates an interaction across groups on the
tests of mathematical understanding. The following figure

aids in the illustration of this interaction.

39—

38— Group 2
37—
36—
35—
34—
33—
32—
31—
30-

Group 1

Pre Post

The result suggested the comparison of the groups
individually. A t-test for achievement was employed to
test Ho for groups 1 and 2 individually. The results of
this test are presented in Table 13.

For group 1 the probability that the observed
F = 5.27 is significant at 25 degrees of freedom is .03.
At the .05 level this value is significant. Therefore
group 1 had a statistically significant decrease in

mathematical understanding as measured by Dossett's Test.

 

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113

 

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114

In the case of group 2 an F of .82 with 14 degrees of
freedom had a probability of being significant of .38.
This value was not significant at the .05 level. Group 2
made slight but not significant gains in mathematical
understanding.

These results indicate the magnitude and direction

 

of the interaction. In other words the two groups acted

differently on the variable of mathematical understanding

 

over the treatment period. “a

The Relationship of Attitude and
Mathematical Understanding

 

 

A major question suggested by the literature was
one of causality. Does a positive attitude promote the
learning of mathematics or does learning promote a positive
attitude? Aiken (1970) suggested a possible analysis-—
crossed-lagged panel correlation. The null hypothesis
related to this question is:

Let r represent the Pearson product-
X21X12
moment correlation between the pre-test of
mathematical understanding as measured by
Dossett's Test of Mathematical Understand—
ing and the post-test of attitude as
measured by Dutton's Attitude Inventory.
Let r represent the Pearson product-
X11X22
moment correlation between the post—test
of mathematical understanding as measured
by Dossett's Test of Mathematical Under—
standing and the pre-test of attitude as
measured by Dutton's Attitude Inventory.
Then there is no difference between
r and r on groups 1 and 2.

X21x12 x11X22

115

Pearson product-moment correlations were calculated
and are reported in Table 14. A value of .308 is needed to
be significant with 39 degrees of freedom. In terms of the

hypothesis r x = .144 and rX x : .328. Substitution
X21 12 11 22

into the formulas on page 90 yielded the following:

6r - r = 2.663 then 2 = .442. This value is

x11X22 X12X21
not significant at the .05 level since 2 must be greater

than 1.96.

Reliability

 

From Table 14 estimates of the reliability of the
two measures can be obtained. The Pearson product-moment
correlation computed between pre- and post-tests of
Dutton's Attitude Inventory is .785. This value is
significant at the .05 level where the tabled r is .301.
The correlation for Dossett's Test of Mathematical Under-
standing is .603 which is also significant at the .05
level. These results are in substantial agreement with
those reported in the literature (Dossett, 1964, p. 96).

This indicates that both tests were extremely reliable.

Equivalence of Groups

 

A minor question of this study dealt with external
validity, confounding variables, and equality of groups.
These concerns are presented in the following two null

hypotheses:

 

116

 

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117

H There is no difference in mean attitude
scores as measured by Dutton's Attitude
Inventory of the six groups.
H There is no difference in mean mathemati-
cal understanding scores as measured by
Dossett's Test of Mathematical Understand-
ing of the six groups.

The statistic used to test these hypotheses was
analysis of variance. The data in Tables 15 and 16 present
the results of that analysis. As illustrated by the

probabilities neither of the F statistics were significant.

In fact both probabilities are quite large.

TABLE 15.--Ana1ysis of Variance for Attitude.

 

 

Source of Variance d.f. M.S. F P less than
Between Groups 5 1.546 .494 .780*
Within Groups 116 3.131

Total 121

 

*Not significant at .05 level.

TABLE 16.--Analysis of Variance for Mathematical Understanding.

 

 

Source of Variance d.f. M.S. . F P less than
‘Between Groups 5 23.988 .371 .867*
Within Groups 116 64.571

Total 121

 

*Not significant at .05 level.

 

118

The Relationship_of Selected Variables
In this section the relationship of fourteen vari-
ables (sex, degree program, age, years of teaching experi-
ence, years of high school mathematics, credits of college
mathematics, grade level assignment, mathematics vs. non-
mathematics teaching, years at present grade level assign—
ment, perceived attitude, grade level attitude developed,
average mathematics grades, attitude, and mathematical
understanding) will be considered. Data on these variables
was obtained through the administration of a questionnaire,
Dutton's Attitude Inventory, and Dossett's Test of Mathe—
matical Understanding. These correlations can be summed
in the following null hypothesis.
HO Given the fourteen variables (sex,
degree program, age, years of teaching
experience, years of high school mathe-
matics, credits of college mathematics,
grade level assignment, mathematics vs.
non-mathematics teaching, years at
present grade level assignment, per-
ceived attitude, grade level attitude
developed, average mathematics grades,
attitude, and mathematical understand—
ing) there exists no correlation between
any of the variables taken two at a time
on all six groups.
The statistic used to test this hypothesis was the
Pearson product-moment correlation coefficient. The data
in Table 17 presents the results of that analysis. For
any one of the 92 correlations to be significant at the
.05 level a value greater than .178 must be obtained based

on 122 degrees of freedom. There are 36 signi-

ficant correlations. However, there is a certain amount of

119

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120

redundancy built into the table. For instance, attitude and
perceived attitude are significantly and highly correlated
(.780). Therefore it is to be expected that variables
highly correlated with one will also be correlated with the
other. This is borne out since average mathematics grades,
years of high school mathematics, credits of college mathe—
matics, grade level assignment, and average mathematics
grades are each significantly correlated with attitude and
perceived attitude.

The following variables were found to be signifi-

cantly correlated.

The correlations are listed from

largest absolute value to smallest:

(1) Attitude and perceived attitude .780
(2) Years at present grade level assignment

and years of teaching experience .598
(3) Age and years of teaching experience .544
(4) Achievement and perceived attitude .447
(5) Achievement and average math grades .441
(6) Achievement and credits of college math .432
(7) Perceived attitude and years of high school math .428
(8) Attitude and average math grades .403
(9) Perceived attitude and average math grades .400
(10) Attitude and present grade level assignment .397
(11) Perceived attitude and present grade

level assignment .394
(12) Achievement and average math grades .390
(13) Attitude and years of high school math .383
(14) Achievement and years of high school math .381
(15) Perceived attitude and credits of college math .379
(16) Achievement and grade level assignment .377
(17) Grade level assignment and sex -.366
(18) Grade level assignment and attitude .365
(19) Credits of college math and years of

high school mathematics .365
(20) Average math grades and years of high

school math .361
(21) Grade level assignment and years of high

school mathematics .320
(22) Years at present grade level assignment and age .305

 

(23)
(24)
(25)
(26)
(27)

(28)
(29)
(30)
(31)
(32)
(33)

(34)
(35)

(36)
(37)

121

Achievement and sex

Credits of college math and sex
Achievement and years of teaching experience
Years of high school math and age

Years of high school math and years of
teaching experience

Perceived attitude and sex

Grade level attitude developed and sex
Achievement and age

Grade level assignment and years of high
school mathematics

Average math grades and years of teaching
experience

Average math grades and credits of college
mathematics

Attitude and sex

Math vs. non-math teachers and grade
level assignment

Years of high school math and sex

Credits of college math and years of
teaching experience

-.292
-.267
-.249
-.240

-.229
-.225
-.218
-.216

.216
—. 212

.208
-.206

..197
—.186

~.184

The implications of these correlations will be considered in

the next chapter.

been considered.
sample.

The remainder of the chapter was devoted to the testable

Summary

In this chapter the results of the hypotheses have

hypotheses.

The chapter began with a discussion of the

Then the validity of the assumptions was discussed.

Only one major hypothesis was found to be significant.

There was a change in attitude for groups 1 and 2.

In addi—

tion to this a few correlations were found to be significant.

In the next chapter implications and further research will

be the focus.

 

CHAPTER V

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

Summary

The effects of activity materials in a graduate
methods course over parameters-~attitude toward mathe—
matics and mathematical understanding—~were considered.
Included was a study of the relationship of selected
variables to achievement and attitude. In this section

a summary of the dissertation will be presented.

The Problem

 

In the years since Sputnik many new curriculum
programs in mathematics have been created. These programs
have been produced by many different people with different
philoSOphies for children, adults studying mathematics,
pre-service teachers,and in-service teachers. In recent
years portions of the mathematics education community have
become enamored with the active learning approach of some
British schools. In all of this mathematics educators
seem to have forgotten to make any direct provisions

concerning attitudes.

122

 

123

It is evident that there is no choice about teach-
ing attitudes. In fact regardless of subject matter
attitudinal changes take place in the classrooms no matter
what might be our personal conviction. The development
of attitudes is such an important aspect of teaching that
it should be cultivated by design (Johnson, 1957).

Educators know very little about the role atti-
tudes play in the learning of mathematics. We are not
sure that attitudes can be changed or what means will
bring about a change. Hence, it behooves researchers in
mathematics education to deal more effectively with the
affective domain.

In the last few years attitudes have increasingly
become the focus of educational research. The poor atti—
tudes of students and teachers have been well documented.
However, attitudes seem to be of only incidental interest
in many classrooms. It was decided to ascertain whether
these activity materials had any effect on the attitude
of in-service elementary teachers toward mathematics. The
purpose of the study was to investigate the effects of
activity materials used in a graduate methods course on

in-service elementary teachers.

The Literature

 

There is a great amount of theory and little fact
surrounding the effects of attitudes on the learning of
mathematics. Much of this theory was given the force of

educational fact by the pre-World War II publications of

124

educators and educational commissions. Beginning in the
thirties mathematics educators began to investigate atti-
tudes in a systematic manner. Many of the early instru-
ments for measuring attitude were rather crude. In the
fifties Wilbur Dutton began to investigate attitudes.
With the passage of time he developed an instrument along
the lines suggested by Thurstone and Chave (1937). Since
then he has continued to refine that instrument. Early
results began to show a pattern-—many students and teachers
disliked mathematics. They found it to be boring and
difficult. These results have been confirmed by other
researchers.

Many other people developed their own instruments
for measuring attitudes and the research consisted of
trying to relate attitudes to other variables such as
achievement, grade level, and grades (marks). The
results of this type of research were mixed. Many authors
were able to report significant but not large correlations
between attitude and achievement. On the other hand, a
substantial number of researchers were unable to report
significant correlations. In effect researchers were
asking: How does attitude affect achievement? Neale
(1969) questioned this approach and asked instead: How
does achievement affect attitudes? Aiken (1970) recom—
mended the crossed—lagged panel correlation across time

to study this dilemma.

125

It was not until the last decade that researchers
in mathematics education began to study how attitudes are
changed. Most of these studies had as their samples
kindergarten through twelfth grade students. In the lower
elementary studies most of the results were not signifi-
cant with an important exception—-manipulative materials.
Non-significant results might be attributed to the
unstableness of attitudes at this age level. At the
middle school level researchers were unable to find
significant differences including the use of activity
materials. At the secondary level the key to changing
attitude seemed to be some form of individualized instruc-
tion whether it was programmed instruction or games.

There were quite a number of studies which had
pre-service teachers as their sample. In these studies
success in the course seemed to be necessary for a change
in attitude. Another successful method was the use of
enrichment activities. However, it was impossible to
determine if change was promoted by enrichment or the
special attention given students. Activity materials also
promoted changes in attitude at this level.

There were only a few studies which used in—service
teachers for their sample. Most authors reported signifi-
cant relationships between a teacher's increased mathemati—
cal understanding and liking of mathematics. One

conclusion is that in-service courses can be used as a

126

vehicle to change attitudes. Materials were instrumental

in at least one study.

The Sample

 

The population of this study was all in-service
elementary teachers who registered and participated in
Methods and Materials of Mathematics in Elementary School
(Education 830A) Fall and Winter terms during the 1971-72
school year at Michigan State University. The larger
target population, through the use of the Cornfield and
Tukey argument (1956), was all in—service elementary
teachers in Michigan who enroll in Methods and Materials
of Mathematics in Elementary School at Michigan State

University.

The Course

 

The course was taught by faculty members of the
Department of Elementary Education at Michigan State
University. It was offered for a three hour block of
time weekly for ten weeks. The content of the course
varied slightly from instructor to instructor but
included content tOpics appropriate to the elementary

school and methods for teaching these topics.

The Measures

 

Three instruments were used for measurement in this
study: (1) an attitude inventory developed by Dutton
(1962), (2) an accompanying informational questionnaire,

and (3) a test of mathematical understanding developed by

127

M. J. Dossett (1964). The choice of Dutton's instrument

increased validity.

Experimental Design

 

Three principle questions were asked: (1) Does
the use of the mathematics activity materials used in
Methods and Materials increase in-service teachers'
attitudes toward mathematics? (2) Does the use of the
mathematics activity materials used in Methods and
Materials increase in-service teachers' understanding of
mathematics? and (3) Does an increase in mathematical
understanding result in an increase in attitude or vice
versa?

To test the first two questions a repeated
measures design was employed. This analysis allowed the
testing of interaction effects as well as main treatment
effects (Glass and Stanley, 1970, p. 470). It was more
powerful than the t-test, which is generally used for the
comparison of two means, because it allowed for the
variation introduced by the use of two groups.

In order to answer the third question, an analysis—v
cross-lagged panel correlation—erecommended by Aiken (1970)
and Campbell and Stanley (1963) was employed. Since
attitude and achievement data from the groups was obtained
during a ten week experiment the cross-lagged panel corre-
lation seemed appropriate.

In a study of this nature an inevitable question is

asked: How do you know that your group is not in some way

128

special, i.e., different from all others? In order to
answer this question four intact groups during early
January (Winter term) were tested with Dutton's Attitude
Inventory, the questionnaire, and Form A of Dossett's Test
of Mathematical Understanding. They were tested during
the first class period using the same procedure as for

the Fall groups. Were these groups sampled from the

same population? The most apprOpriate analysis (Glass

and Stanley, 1970, p. 353) for this question was analysis.
of variance. Since attitude and mathematical understand—
ing were of interest it was appropriate to test equality
of means for both attitude and mathematical understanding.
If there was no difference then external validity would
have been increased. Several correlations were also

calculated.

Testable Hypotheses
There were six testable hypotheses:

1. There is no difference in attitude as
measured by Dutton's Attitude Inventory
between pre— and post-tests of groups
1 and 2.

2. There is no difference in mathematical
understanding as measured by Dossett's
Test of Mathematical Understanding
between pre- and post-tests of groups
1 and 2.

3. Let r1 represent the Pearson product—
moment correlation between the pre—test
of mathematical understanding as
measured by Dossett's Test of Mathe~
matical Understanding and the post—test
of attitude as measured by Dutton's
Attitude Inventory. Let r2 represent

Results

129

the Pearson product-moment correlation
between the post-test of mathematical
understanding as measured by Dossett's
Test of Mathematical Understanding and
the pre-test of attitude as measured
by Dutton's Attitude Inventory. Then
there is no difference between r1 and
r2 on groups 1 and 2.

There is no difference in mean attitude
scores as measured by Dutton's Attitude
Inventory of the six groups.

There is no difference in mean mathemati-
cal understanding scores of the six
groups as measured by Dossett's Test of
Mathematical Understanding.

Given the fourteen variables (sex, degree
program, age, years of teaching experi-
ence, years of high school mathematics,
credits of college mathematics, grade
level assignment, mathematics vs. non-
mathematics teaching, years at present
grade level assignment, perceived atti-
tude, grade level attitude developed,
average mathematics grades, attitude,
and mathematical understanding) there
exists no correlation between any of the
variables taken two at a time on all six
groups.

Data obtained from the various tests indicated

the following:*

1.

There was a significant positive change in
the attitude toward mathematics of groups
1 and 2 in—service elementary teachers
enrolled in Methods and Materials.

There was no significant change in the
mathematical understanding of groups

1 and 2 in-service elementary teachers
enrolled in Methods and Materials.

 

*Each number corresponds to the appropriate
numbered hypothesis.

 

130

3. In this case there was no reason to
infer that positive attitude promoted
learning nor did achievement promote
more positive attitudes.

4. There is no reason to suspect that the
experimental groups were in any way
different from the population at the
beginning of the experiment.

Conclusions

 

The first major hypothesis concerned a gain in
attitude. In the null form it is:

Ho There is no difference in attitude as
measured by Dutton's Attitude Inventory
between pre- and post-tests of groups
1 and 2.

This hypothesis was rejected at the .05 level. Since the
null hypothesis has been rejected the alternative must be
true. The alternative hypothesis states that there is a
gain in attitude as measured by Dutton's Attitude Inven-
tory between pre- and post-tests. As in any statistical
research there may be a number of competing hypotheses
which could be used to explain the above result. How-
ever, in this study the use of groups three through six
in a comparison with groups one and two eliminated these
competing hypotheses.

The result was that the gain in attitude must be
attributed to the treatment experienced by groups 1 and 2.
The treatment was the activity materials that in-service
elementary teachers worked with in Methods and Materials.
These materials were organized around the following five

topic areas: sets, numbers and numeration, numbers and

operations, geometry, and measurement. Accompanying these

131

topics were activity materials such as: attribute games
and blocks, Dienes' blocks, Cuisenaire rods, geoboards,
clay, weights, beans, and playing cards. The use of
activity materials in a graduate methods course is an
effective tool for changing the attitudes of in—service
elementary teachers. When the present study is taken in
conjunction with previous research, it is reasonable to
conclude that activity materials such as those used in
the treatment of this study can be used to change the
attitudes of a wide range of children and adults toward
mathematics. Since positive attitudes are an important
objective of instruction, many types of mathematics
classes should begin to use these or similar materials.
It should be recognized that the use of activity
materials is only a portion of the treatment. These
materials can not be separated from other aspects of the
class which compose the treatment. One of these aspects
is the type of instructor who teaches Materials and
Methods. Both instructors in the experiment were
enthusiastic about and well versed in the use of activity
materials in the classroom. Certainly this enthusiasm
and knowledge was instrumental in helping in«service
teachers feel successful in the class. Another aspect of
the class that could account for the change in attitude
was a lack of tension and apprehension. There were no
tests given to determine grades and inability to perform
tasks or recall facts was not a reason for a reprimand.

A trusting, helping atmosphere was maintained in the

132

classroom. These possible confounding variables (teacher,
lack of negative reinforcement) were not separated from
the use of activity materials since the choice of activity
materials is an indication of the learning theory sub-
scribed to by the teacher. An unenthusiastic teacher who
uses negative reinforcement will alienate students whether
or not he uses activity materials.

In any interpretation of this study, there are
several factors and limitations that should be taken into
account. The nature of the various tests are of paramount
importance. Any self-reporting inventory is of a very
immediate nature, i.e., events immediately preceding it
may have an effect that is out of proportion to reality.
Also the very fact of taking the test may change one's
attitudes. Because the attitude inventory was used as a
pre-test the subjects may have been sensitized to the
treatment. Dutton's inventory measures a general attitude
toward mathematics and does not focus on specific aspects.
In this respect it is a rather crude instrument. However,
it is questionable whether better self—administering
inventories can be developed because of the very nature
of the subjects being measured. An important limitation
of this study was the use of intact groups. Although
(extensive precautions were taken to ameliorate this prob-
Jfinn, randomization is a much more appropriate method for

controlling confounding variables.

13.3

The second major hypothesis concerned gain in
mathematical understanding. The null hypothesis is:

HO There is no difference in mathematical

understanding as measured by Dossett's

Test of Mathematical Understanding

between pre— and post—tests of groups

1 and 2.
This hypothesis was not rejected at the .05 level. There-
fore, there is no reason to conclude that the activity
materials in Methods and Materials contributed to a gain
in mathematical understanding for in-service elementary
teachers. When the interaction effect is considered it
is quite clear why the hypothesis was not rejected. Group
1 (East Lansing) had a significant decrease (at .05 level)
in mathematical understanding as measured by Dossett's
Test of Mathematical Understanding. The probability was
less than .03. Group 2 (Grand Rapids) had a large gain
in mathematical understanding. This gain was not signi-
ficant with a probability less than -38. In a repeated
measures design these effects tended to cancel one
another.

There are two possible explanations for this
phenomena. (1) Data about the sample indicates non—
significant initial differences with regard to mathemati—
cal understanding and background. The experiment could
have enlarged these differences. (2) Although Dossett's
test was developed specifically for use with invservice
teachers there are some problems in using it with the

population of this study. First the test may be too

long and the subjects View it as boring and tedious. This

'134

could lead to inaccuracies in measurement. Secondly it is
a test of general mathematics understanding and is not
very specific. Actual gains may be masked by the type of
tests.

The third major hypothesis had as its focus the
effect of achievement on attitude and vice versa:

HO Let r1 represent the Pearson product-
moment correlation between the pre-test
of mathematical understanding as
measured by Dossett's Test of Mathema-'
tical Understanding and the postntest of
attitude as measured by Dutton's Attitude
Inventory. Let r2 represent the Pearson
product-moment correlation between the
post-test of mathematical understanding
as measured by Dossett's Test of Mathe-
matical Understanding and the pre-test of
attitude as measured by Dutton's Attitude
Inventory. Then there is no difference
between rl and r2 on groups 1 and 2.

This hypothesis was not significant at the .05 level. A
value of 1.96 is needed for significance. Non-significance
of this hypothesis may be attributed to the interaction
effect mentioned in the previous hypothesis. If there

was no interaction effect then the correlation r

 

xllX22

would have been larger. If r was larger, then the
x x
11 22
numerator of

VnI (r - r ) - (o '0 )1

z _ X11X22 X12X21 x11x22 X12x21

8r - r

x11X22 X12x21

‘would be larger and the denominator smaller resulting in

a larger observed value for 2. Consequently if there had

135

been no interaction the result may have been a significant
2.

In order to increase external validity and control
for confounding variables four Winter term groups were
pre-tested with Dutton's Attitude Inventory, Dossett's
Test of Mathematical Understanding, and the questionnaire.
This led to the following two hypotheses:

HO There is no difference in mean attitude

scores as measured by Dutton's Attitude
Inventory of the six groups.

H There is no difference in mean mathe-
matical understanding scores as measured
by Dossett's Test of Mathematical Underv
standing of the six groups.

Neither of these hypotheses were rejected at the .05 level.
Hence there is no reason to believe that any of the groups
were out of the ordinary. Non-rejection of these hypo—
theses has several important implications.

First, the successful results of the first major
hypothesis can be extended by a Cornfield Tukey argument
to include in-service elementary teachers taking Materials
and Methods in the future or past Winter term. In fact
there is reason to expect that the results could be
extended to any in-service elementary teacher taking
bkathods and Materials at any time as long as the course
cxnntent remained the same. Secondly, confounding variables
Stufli as time, maturation, and history are less likely to

account for experimental differences. Otherwise these

confounding variables would have also affected the Winter

136

term subjects during the Fall. The result would be
significant differences between Fall and Winter pre-tests
which did not occur.

Since the groups are the same they can be combined
for the purpose of finding correlations between the follow—
ing variables: sex, degree program, age, years of teaching
experience, years of high school mathematics, credits of
college mathematics, grade level assignment, mathematics
vs. non—mathematics teaching, years at present school
assignment, perceived attitude, grade level attitude
developed, average mathematics grades, attitude, and
mathematical understanding. The hypotheses for the corre-
lations of these variables can be combined into the follow-
ing single hypothesis:

HO Giyen the fourteen variables there

eXists no correlation between any
of the variables taken two at a time
on all six groups.

The following correlations were found to be sig—

nificant at the .05 level:

( l) Attitude and perceived attitude .780
( 2) Years at present grade level assignment

and years of teaching experience .598
( 3) Age and years of teaching experience .544
( 4) Achievement and perceived attitude .447
( 5) Achievement and average math grades .44l
( 6) Achievement and credits of college math .432
( 7) Perceived attitude and years of high

school math .428
( 8) Attitude and average math grades .403
( 9) Perceived attitude and average math grades .400
(10) Attitude and present grade level assignment .397
(ll) Perceived attitude and present grade level

assignment .394
(12) Achievement and average math grades .390

(13) Attitude and years of high school math .383

 

137

(14) Achievement and years of high school math .381
(15) Perceived attitude and credits of college math .379
(16) Achievement and grade level assignment .377
(17) Grade level assignment and sex —.366
(18) Grade level assignment and attitude .365
(l9) Credits of college math and years of

high school mathematics .365
(20) Average math grades and years of high

school math .361
(21) Grade level assignment and years of

high school mathematics .320
(22) Years at present grade level assignment and age .305
(23) Achievement and sex —.292
(24) Credits of college math and sex —.267
(25) Achievement and years of teaching experience —.249
(26) Years of high school math and age —.240
(27) Years of high school math and years of

teaching experience -.229
(28) Perceived attitude and sex —.225
(29) Grade level attitude developed and sex -.218
(30) Achievement and age —.216
(31) Grade level assignment and years of high

school mathematics .216
(32) Average math grades and years of

teaching experience —.212
(33) Average math grades and credits of college

mathematics .208
(34) Attitude and sex —.206
(35) Math vs. non—math teachers and grade

level assignment .197
(36) Years of high school math and sex «.186
(37) Credits of college math and years of

teaching experience «.184

At this point it is appropriate to note that there is
a difference between statistically significant correlations
and meaningfully significant correlations. It is also
important to remember that correlation does not imply
causation. Correlation only provides a direction in which

to search. With 122 subjects at d = .05 a value of .178

must be attained for a correlation to be significant. This

Value is not very large. Therefore some of the smaller

COrrelations may not be very meaningful.

-138

Certainly a value of .78 is a meaningfully signifi-
cant correlation between attitude and perceived attitude.
This value can be taken as an indication of the validity of
Dutton's Attitude Inventory. At least the test measured
what the people believed to be their attitude toward
mathematics.

The 34th, 23rd, and 17th correlations indicate
that sex differences exist. The data reveals that in—service
men have better attitudes and mathematical understanding
than in—service women. However, there is a smaller corre—
lation for attitude than mathematical understanding. Also
as expected women seem to develop their attitude toward
mathematics later than men.

Certainly correlations 8, 10, and l3 lend support
to the hypothesis that success experiences promote positive
attitudes toward mathematics. Of course this set of
correlations is related to the 14th, 6th, and 5th correla-
tions which are rather obvious kinds of correlations. The
more mathematics a subject enrolls in the more he should
gain in understanding and the more positive his attitude.

Correlations l9 and 16 indicate that teachers with
negative attitudes and poor mathematical understanding
seem to prefer teaching the early elementary grades.

This is a rather disturbing result. The early grades are
considered to be the most important in the development of
a child yet they have teachers who dislike and do not

understand mathematics. In addition to this, correlations

139
30 and 25 indicate that these teachers are older and more
experienced. These teachers (fifteen) also had poorer
grades in mathematics when they were in school.

From the correlations it appears that teacher
training institutions are developing teachers with better
attitudes and greater mathematical understanding than they
have in the past. This is a very hopeful sign and a con—
tinuation of this trend is very desirable. Activity
materials are an appropriate method of instruction for

improvement of mathematics attitude.

Recommendations

 

The results clearly indicate that if the education
faculty at Michigan State University wish to promote
positive attitudes of in—service teachers toward mathematics
an appropriate method in Methods and Materials is the use
of activity materials. A possible confounding variable in
this research has been the use of the attitude inventory
as a pre—test possibly sensitizing the subjects to the
treatment. An appropriate design to control for this
variable is the Solomon four-group design (Campbell and
Stanley, 1966). This design should be employed in further
research on attitudes.

Another advance would be the development of atti-
tudinal measuring instruments which focus on specific
aspects of mathematics rather than on mathematics in
general. These instruments could focus on specific aspects

of activity materials in order to discover which activity

'140

materials are responsible for changes in attitude. Another
instrument for measuring mathematical understanding should
be developed. This instrument should focus upon the con-
cepts learned with activity materials. Because of the
nature of the subjects involved, the test should be as
short as possible. Then it might be possible to obtain
significant results using the cross-lagged panel correla—
tion technique. This technique could be very useful in
attitudinal research.

Certainly the results of this study should be
duplicated at other institutions to ascertain whether the
(phenomena was particular to Michigan State University.
Also other populations should be considered for the
experiment especially pre—service elementary teachers.
Greater research is needed on attitudes~—how they are
developed and changed. The new instruments and statisti«
cal techniques recommended will provide additional answers

to the questions posed by Aiken (1970).

REFERENCES

141

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APPENDIX A

COURSE OUTLINE AND LIST OF

ACTIVITY MATERIALS

157

Day

Hour

WNH UNH wNH LONE-J MNH WNH WNH (AMI-J

UNI-4

Outline for 830A

Experimental Group

Mathematical Concepts

Test of Mathematical Understanding

Sets

Numbers and Operations
ll

Numbers and Numeration
ll

Place Value

Number
Patterns
Number Patterns in Mathematics

Geometry
Area

Symmetry
Similarity
II
Coordinate Geometry

Graphs
Relations
Functions

Measurement
II

Wrap up
Post-test

158

Materials

Attribute Games
and Problems

Cuisenaire Rods
H

" and Task Cards
Movie - Gattegno

Dienes Multibase
Blocks
Task Cards

Arrow Games
Pythagorean Square
Pascals Triangle

Geoboards

Mirror Cards
Tangrams and Cards
Movie - Geoboard

Picks Theorem - Geoboard
Madison
Shoeboxes

Counting blocks
Paper, clay,
scales, and weights

 

 

APPENDIX B

SAMPLE ACTIVITY SHEETS

159

MEASUREMENT OF AREA---ARBITRARY UNITS
(work in pairs if you wish)
Order 6 objects from least to most, according to area.

Choose any object you wish for your unit and use it to measure the area
of at least A other objects. Record your data in the table.

 

AREA MEASURE

 

object X unit object greater than less than closest to

 

YELLOW \
SQUARE 4 5 6 7 7

 

 

 

 

 

 

 

 

 

 

 

 

 

Find two different shapes with about the same area.

What would be an easy way to find the area of the outer surface of a
cylinder?

Find the pressure on each square inch of your footprint.

What is the surface area of your skin?

160

161

Some Activitig§_with Tangrams

Use the sets of Tangrams provided to answer the following: 7

s IMIIAR I}:

 

Are the statements true or false?

1. Figures A, B, and D form a figure similar to figure F.
2. Figures B, D, and E form a figure similar to figure A.
3. Figures A and B form a figure similar to figure E.

4. Figures B, D, and F form a figure similar to figure A.

 

5. Figures D, E, and F form a figure similar to figure A.

6. Figures G and C form a figure similar to figure E.

7. Figures A, B, D, E and F form a figure similar to figure C.
8. Figures B, C, D and E form a figure similar to figure F.

. All the figures form a figure similar to figure A.

10. All the figures form a figure similar to figure C.

AREA

1. Assume that the area of figure A is one. Determine the area of each
of the following figures. '

(a) Figure B (c), Figure E
(b) Figure C (d) Figure F

2. Assume that the area of figure D is one. Determine the area of each
of the following figures.

(a) Figure A (c) Figure E
(b) Figure C (d) Figure F
3. Put figures B, D and A together to form a parallelogram. Let the

area of this parallelogram be one and determine the area of
(a) Figure F, (b) Figure E, and (c) Figure G.

 

 

aaaaaaaa

 

 

 

163

Volume

PART I

1. Make successively larger models of a cube. What things
change?

2. What things stay the same?
3. Find some patterns.

4. Draw a graph of your results.

PART II
1. Investigate the soma cube.
2. A good way to start is the following:

a. Obtain 27 cubes and divide them into groups of 4. There
will be one group with only 3 in it.

b. Take a group of 4 cubes and arrange them in some form so
that the cubes are sharing sides (never just a corner).
The arrangements should not be symmetrical with respect
to a side of a cube.

c. Take another group of 4 and arrange them in a form
different from the first.

d. Do this for each group of 4, and finally the group of 3.

3. The groups of 4 can be glued together to preserve the
arrangement.

4. These 7 different arrangements can be used to make many

larger solids. For example, the 7 pieces could be arranged
to form a cube.

PART III

1. In how many ways can you measure a sphere?

 

 

164

THE GAME OF FRACTIONS
or
"Add to One"

Materials:
A deck of cards consists of 54 cards, marked with fractions from 1/2 to
9/10.

Object:

The object of the game is to draw cards from the deck, or discard pile
to form a set of cards, which add to one. (The game is similar to
Rummy games.)

Rules:

1. The dealer will deal 6 cards to each player, and place the remainder
of the deck in the center of the table, turning up the top card to
begin a'discard pile.

2. The first player to the left of the dealer draws one card, either
from the deck or the discard pile. During each playing turn, the
player must discard one card. The play goes counterclockwise,
around the table.

3. When a player gets a set of cards which add to "one," he may place
it in front of him, during his turn, and then he must discard one
card.

4. To "go out" and thus terminate a hand, a player must have a card
to discard. He will receive one extra point for going out.

5. A player may draw up to 3 cards from the top of the discard pile,
if he plays the bottom card he draws, as part of a "set of one"
during that turn.

6. If the deck is exhausted, the dealer will reshuffle the discard
pile and continue with the game.

7. If no one can go out, and the discard pile becomes so small it ap-
pears no combination would permit any of the players to go out,
the hand may be terminated by mutual consent.

Scoring
1. A player receives 1 point for each "set of one" he plays before
any player goes out, during a hand.

2. The player, who goes out, receives 1 additional point.

3. A game is won by the first player to score 5 or more points. (A
number larger than 5 may be used by mutual consent.)

(8) As students gain skill and confidence, it helps to add the rule,
"you may not use two cards, each equal to 1/2, to make a set of
"one." That is, if you have 3/6 and 4/8 in your hand, only one of
them may be played as a part of a single "set of one."

(9) The games goes best with three or four players, however, any
number from two to six may play. With six, you may wish to deal
only five cards.

 

 

APPENDIX C

INSTRUMENTS

165

(1)

(2)

(3)

(4)

(5)

Directions for Test Administration

The pre—tests should be administered as soon as possible

to avoid contamination of the results.
The attitude survey must be given first. Otherwise the
students' feelings about their success on the achievement

test will bias the attitude survey.

It should be indicated that the tests have no place in
evaluating their grades. Their purpose is to help evaluate

the course.

Furthermore, it should be explained that their honesty is

needed and appreciated, especially on the attitude survey.

Time: about one and a half hours.

166

 

167

 

 

QUESTIONNAIRE
Name Student Number
Ma1e_____ Fema1e______Degree Program: Non-degree MA MS MAT Special PhD EdD
Age Years of teaching rm»

Years of high school math

Term credits of college math

 

g“:
.

 

 

 

Present teaching assignment (grade and subject) ’~%

Years at present teaching assignment

1.

Place a circle around one number to show how you feel about
mathematics in general.

1 2 3 4 5 6 7 8 9 10 ll
Dislike Like

My feelings toward mathematics were developed in grades:

1 2 3 4 5 6 7 8 9 10 ll 12 Other (circle one)

My average grades made in mathematics were: A B C D (circle one)
List two things you like about mathematics.

A.

B.

List two things you dislike about mathematics.

A.

B.

168

Please read all statements before you begin. Place check (/)
before those statements which tell how you feel about mathematics.
Select only the items which express your true feeling.

1. I avoid mathematics because I am not very good with numbers.
2. Mathematics is very interesting.

3. I am afraid of doing word problems.

4. I have always been afraid of mathematics.

5. Working with numbers is fun.

6. I would rather do anything else than do mathematics.

 

7. I like mathematics because it is practical.
8. I have never liked mathematics.
9. I don't feel sure of myself in mathemtics.

10. Sometimes I enjoy the challenge presented by a mathematics
problem.

 

11. I am completely indifferent to mathematics.

 

12. I think about mathematics problems outside of school and
like to work them out.

 

13. Mathematics thrills me and I like it better than any other
subject.

 

14. I like mathematics but I like other subjects just as well.

 

15. I never get tired of working with numbers.

 

169
Dossett's Test of Math Understanding: Form A

When you write the. numeral "'3" you are writing
a. the number 5.

b. a pictorial expression.

(1. a symbol that stands for an idea.

(i. a Hindu-Babylonian symbol.

’1'

Bill discovered that > means is greater than" and < means "is

less than.H In which of the following are these symbols 119—t used

correctly?

21. The number of states in the United States < the number of
United States Senators.

b. The number of states in the United States > the number of
stripes in the flag.

c. 23>32 .

d. 3 + a < 5 + a

When two Roman numerals stand side by side in a symbol, their
values are added

a. always.

b. sometimes.

c. never.

d. if the base is X.

Which of the following describe/describes our own system of
numeration? \
a. additive
b. positional
c. subtractive
d. introduces new digits for numbers larger than 10
l) a and b are correct.
2) a and c are correct.
3) a and d are correct.

4) a, b, and d are correct.

Zero may be used

a. as a place holder.

b. as a point of origin.

C. to represent the absence of quantity.
(1. in all of the above different ways.

2,200. 02 is shown by
a. 2000 + 200 l- 20.

l). 2000 + 20 + Z/lO.
c. 2000 -i- 2.00 + 2/100.
(,1. 2000 + 2.00 + 200.

 

170

'38-'10 rearranged so that the 8 is 200 times the size of the 4 would
:1. 58‘10.

b. 8540.
c. 5048.
d. 5408.
Which of the following does not show the meaning of 423mm?

a. (4x100)+(2x10) + 3(l 2423

b. 42 tens + 3 ones 2 423

c. 423 ones 2 423

d. 4 hundreds + 42 tens + 23 ones = 423

A numeral for the X's in this example can be written in many
different bases. Which numerals’are correct?
,____,_-__.--_-..._.

I--- ....——-

8. 100mm 3 xx X xx xx
'3' l4twelve X X X X
C. 16tc11 X XX X X

—-—-.-

 

(l. 31f1V€
l) a and c are correct.
2) b and c are correct.

3) a, b, and care correct.
4) all four are correct.

A "2" in the third place of a base twelve number would represent
a. 2 x 12

b.12x23
c. 12x212
d. 2x12Z

In this addition example, in what base are. the numerals written?
a. base two

 

l). base three 128:
c. base four + '
(1. none of the above 200?

About how many tens are there in 6542?

it. 6540
l). 6541
C. 6‘3 , '
(l. 6. 5-2

Place or order in a series is shown by
5!. book no. 7.

l). three boxes of matches.

C. a dozen cupcakes.

(1. two months.

be

 

u, ,‘\ “.354-

ll.

‘ ,.
.0.

l7.

~§
ll).

1‘)

171

Which of the following indicates a group'.>
a. 45 tickets

b. track 45

c. page 54

d. apartment NO. 7

The sum of any two natural numbers

a. is not a natural number.

b. is sometimes a natural number.

c. is always a natural number.

(1. is a natural number equal to one of the numbers being added.

The counting numbers are closed under the operations of

 

a. addition and subtraction.

b. addition and multiplication.

c. addition, subtraction, multiplication, and division.
d. addition, subtraction, and multiplication.

If a and b are natural numbers, then a + b = b + a is an example of
a. commutative property.

b. associative property.

c. distributive property.

d. closure.

If a x b = 0 then

a. a must be zero.

b. b must be zero.

c. either a or b must be zero.
(1. neither a nor b must be zero.

When a natural number is multiplied by a natural number other
than 1, how does the answer compare with the natural number
multiplied?

a. larger

b. smaller

C. the same
(1. can't tell from information given

Which of the following is the quickest way to find the sum of
several numbers of the same size? ' d

«1. counting

1). adding

C. subtracting

0. multiplication

.3 -'l .

F»:

‘-

4

(Q
o

172

How would the product in this example be affected if you put the
29 above the 4306 and multiplied the two numbers?

a. The answer would be larger.

b. The answer would be smaller. 4306

e. You cannot tell until you multiply both ways. x29

d. The answer would be the same.

 

 

An important mathematical principle can be helpful in solving __
the following example. [3

28 + 659 + 72 = C] “It“
What principle will be of most help? i

a. the associative principle
b. the commutative principle 'fi

 

c. the distributive principle
(1. both the associative and distributive principles

The product of 356 x 7 is equal to
a. (300 x 50) x (6 + 7).

b. (3X7)+(5x7)+(6x7).

c. 300x50x6x7.

d. (3003’?) +(50x7)+(6x7).

Which of the following is not a prime number?

a. 271
b. 277
c. 281
d. 282

Which of the following numbers is odd?

a. 18 x ll

b. 11 x 20

c. 99 x 77

d. none of the above -

The inverse operation generally used to check multiplication is
a. addition.

l). subtraction.

C. multiplication.

d. division.

The greatest common factor of 48 and 60 is
E1. 2 x 3.

l). 2 x Z x 3. .
C.Z.\;Z.\'fo2x3x5.

(1. none of the above.

H0.

173

Look at the example at the right. Why is the "4"

in the third partial product moved over two 157
places and written under the 2 of the multiplier? X 246
.1, If you put it directly under the other partial 942
products, the answer would be wrong.- 628
I). You must move the third partial product two 314
places to the left because there are three 38622

C o

1
(1.

numbers in the multiplier.

The number 2 is in the hundreds column, so the third partial
product must come under the hundreds column.

You are really multiplying by 200.

Which of the fundamental properties of arithmetic would you
employ in proving that (a + b) + (a + c) = 2a + b + c?

a.
b.
c.
d.

Associative property

Commutative property

Associative and distributive properties
Associative and commutative properties

If N represents an even number, the next larger even number

can be represented by

a.
b.
c.
d.

N+1.
N+2.
N+N.
2xN+l.

Every natural number has at least the following factors:

a.
b.
c.
(1.

zero and one.
zero and itself.
one and itself.
itself and two.

It is said that the set of whole numbers has a natural order. To
find the successor of a natural number, one must

Cl.
b.
C.
(‘1.

add 1.

find a number that is greater.
square the natural number.
subtract 1 from the natural number.

The paper below has been divided into 6 pieces. ,It shows

_

 

 

 

 

 

Li ...... l _--_L______l.._._.__l

a,
b,
C.

d.

sixths.
thirds.
halves.

}ILrtS.

 

174

A fraction may be interpreted as:

,1. a quotient of two natural numbers.
1,, equal part/parts of a whole.

e. a comparison between two numbers.

(i. all of the above.

When a common (proper) fraction is divided by a common
fraction, how does the answer compare with the fraction divided?
a. It will be larger.

1.. It will be smaller.

c. It will be twice as large.

('1. There will be no difference.

Which algorithm is illustrated by the following sketch?

 

 

 

 

 

 

 

 

 

Mull..." j
1 3 " '///

ENE: ? ’ / / 1/2
1 3 ' ,»’»///

l). ,—+—-: ?
a 4 1/2
1 1 1 _- - J

t. __ + __ ._ ?
2 2+2“ 1/4 1/4 1/4 1/4
4 3

(i. -....- :_ ?
4 2

Another name for the inverse for multiplication of a rational
number is the

a. reciprocal.

l). opposite.

(\ reverse.

(h zero.

Examine the division example on the right. 3

Which sentense best tells why the answer 5 -°— -— = 6

is larger than the 5? ° 4

.2. Inverting the divisor turned the i3- upside down.

.. Multiplying always makes the answer larger.

V. The divisor isless than 1.

.. Dividing by proper and improper fractions makes the answer
larger than the number divided.

wIN

' value of a common fraction will not be changed if

we add the same number to both terms.

;,. We multiply one term and divide the other term by that same
number .

we subtract the same amount from both terms.

. we multiply both terms by the same number.

 

.31!

175

The nearest to 45% is
a. 44 out of 100.

b. .435.
c. 4. 5.
(l. .405.

The principal of a school said that 27 per cent of the pupils went

to the museum. Which statement best describes the expression

"27 per cent of the pupils went to the museum? "

a. It means that. 27 children out of every 100 children went to
the museum. .

b. It means that you must multiply the. number of children in the
school by 27/100 to find the number who went: to the museum.

c. If the children were divided into groups of 100 and those who
went to the museum were distributed evenly among them,
there would be in each group 27 who went to the inusemn.

d. 27 per cent is the same as . 27 -- a decimal fraction written
in per cent form.

Sally completed 2/3 of the story in 12 minutes. At that rate how
long will it: take her to read the entire story?

a. 18 minutes

b. 12 ininutes

c. 6 minutes

(1. 24 minutes

There were 400 students in the school. One hundred per cent of
the children had lunch in the cafeteria on the first day of school.
On the second day 2 boys were absent and 88 children went home
for lunch. Which of the following equations can be used to find
the per cent of the school enrollment who went home for lunch?
a. 400 -- 88 = X

b. .5. = 88
100 400

°° l = 400
88

What can be said about y in the following Open sentence if x is
:1 natural number? ‘

Fl.x<y. x+x+l=y
l). X>y
C. xzy

d- X r y

 

’i f).

-.
‘.
. l

176

Which one of the following fractions will give a repeating decimal?
a. 1/2

b. 3/4
c. 5/8
(1. 6/11

Which of the following is not an open sentence ?

:1. 7+2:
b.h-5=9
c. c/1-30=6
d. n-3

For a mathematical system consisting of the set of odd numbers
and the operation of multiplication,

a. the system is closed.

b. the system is commutative.

c. the system has an identity element.

d. all of the above are correct.

Measurement is a process which .
a. compares an object with some known standard or accepted unit.
b. tries to find the exact amount.
c. is never an exact measure. .
d. chooses a unit and then gives a number which tells how many
of that unit it would take.
1) a and b are correct.
2) a and c are correct.
3) a, b, and d are correct.
4) a, c, and dare correct.

The set of points sketched below represents a

 

I \

. \ /
a. line.
he ray.

c. line segment.
d. none of the above.

 

How many triangles does the figure contain?

do 4
b. 6
L. 8

 

 

 

"O-u | .,

rear .0 IV!!!" -_

4;”... «r 9‘ , ..

in}.

«J‘

53.

‘34,

J!

(\3

‘J1

177

The set of points on two rays with a common end-point is called
a. a triangle.

b. an angle.

c. a vertex.

d. a side of a triangle.

If a circle is drawn with the points of the compass 3 inches
apart, what would be 3 inches in length?

a. circumference

b. diameter

0. area

(1. radius

The solution set of an open sentence may consist of
a. two or more numbers.

b. no numbers.

c. only one number.

(1. any or all of these.

Consider a set of three objects. How many sub—sets or groups
can be arranged? '

a. nine

b. eight

c. seven

d. six

If two sets are said to be equivalent, then

a. every element in the first set can be paired with one and only
one element in the second set.

b. every element in one set must also be an element in the
second set.

c. they are intersecting sets.

d. one must be the null set.

 

178

Form B:

I

1, Which of the underlined words or signs in the following sentences
refer to symbols rather than the things they represent?
a. 4 can be written on the blackboard.
b. Regardless of what symbol we use, we are thinking about the
nwnber g.
c. A pencil is used for writing.
(1. The ninnber 16 is the same as the. number 7 + 9.

 

 

", When we use the = symbol between two terms (as Z + 2 = 4) we
mean that both terms represent the same concept or idea. Which
of the following is not correctly stated?

a. 3 + 4 = 5 + 2 “7'

b. 54-227 and 7=5+2

c. (8+2)x3=7x3

 

d. 7 s 7
1) a and b are correct.
2) a and c are correct.
3) a, b, and c are correct.
4) a, b, c, and d are correct.

3. If the Roman system of numeration were a ”place value system”
with the same value for the base as the Hindu-Arabic system, the
first four base symbols would be
a. I, X, C, and M.

b. l V, X, and L.

c. , L, C, and Ni.

d. X, C, L, and D.

( U

'3. Which of the following does not describe a characteristic of our

 

decimal system of numeration?

a. It uses zero to keep position when there is an absence of value.

b. It makes ten a standard group for the organization of all
numbers larger than nine.

C. It makes 12 the basis for organizing numbers larger than
eleven.

1

c. It uses the additive concept in representing a number of
several digits. -

JV

-. In the numeral 7, 843, how does the value of the 4 compare with
the. value of the 3?

a. 2 times as great

1). 1/2 as great

C. l/lO as great

(1. 1/20 as great:

N. a. .4,.~ o

"‘~a‘- '0

10

179

2;. the numeral 6, 666 the value of the 6 on the extreme. left as
:7. :npa red with the 6 on the extreme right is

2. . '6, 000 times as great.

':. l, 000 times as great.

c. the same since both are sixes.

ti. six times as much.

"Thich of the following statements best tells why we write a zero in
the numeral 4, 039 when we want it to represent "four thousand
thirty-nine? "
a. Writing the zero helps to fill a place which would otherwise be
empty and lead to misunderstanding.
b. The numeral would represent ”four hundred thirty-nineH if we
did not write the zero.
c. Writing the zero tells us not to read the hundreds' figure.
cl. Zero is used as a place-holder to show that there is no number
to record in that place.
1) a and b are correct.
2) a and c are correct.
3) a and d are correct.
4) a, b, and d are correct.
Below are four ntunerals written in expanded notation. ‘x'v'hich one
is not written correctly? '

 

a. ~l(ten)2 + 9(ten)1 + 3(ones) = 493ten

'3. 3(seven)3 + 6(seven)1 + 1(one) z: 363seven

e. 4(twelve)3 + 5(twelve)1 + e(one) = 450t\vel\re
( .

1:" you are permitted to use any or all of the symbols 0, l, 2, 3, 4
and 5 for developing a system of nume ration with a place value
system of numeration similar to ours, a list of all possible bases
would include:

a. base one, two, three, four, five, and six.

2). base two, three, four, five, and six.

c. base two, three, four, and five.

(1. base. one, two, three, four, and five.

About how many hundreds are there in 34,870?

:1 o 3 ’2
b. 35
c. 350

d. 3, 500

 

l:

 

180

11, In what base are the numerals in this multiplication example.

written? 349
a. base five 23%
b. base eight __
0. base eleven .1z4?
70?

(1. you can't tell

 

1024?

1.1. Which of the following are correct?
a. In the symbol 53, 5 is the base and 3 is the exponent.
b. In the symbol 53, 3 is the base and 5 is the exponent.

c.5325x5x5
d. 5J=3x3x3x3x3

l) a and d are correct.
2) b and c are correct.
3) a and c are correct.
4) b and d are correct.

 

H. In the series of numerals l, . . .17, 18, I9, 20, 21, . . . , what term
best applies to 19'? ‘
a. neininal
b. ordinal
c. composite
d. cardinal

14. Examine the following illustration:

1 2 3‘74 5 ti::>

Which of the following does the above best illustrate?
a. The idea of a cardinal number
b. The use of an ordinal number ..

 

c. A means for determining the cardinal number of the set by
counting with ordinal numbers
(1. None of the above

15. The. quotient of any two whole numbers
a. is not a natural number.
1). is sometimes a natural number.
C. is always a natural number.
(1. is a natural number less than one of the numbers being
divided.

\.. J“

“My M‘WMW‘ i

I
......~ u. v qa\W‘-m‘7\“~w Ifi~W€WsMuHWW“

'- tD‘H‘Q‘bWM‘ ‘Mnhomav‘ man-v".- u

_ ......N.~_ ”*v... ~t~ r-4 “nun ---cw-'~ soon—'9'.

.. .‘g—

16.

17.

18.

19.

.z o.

181

The integers are closed under the. operations of
a. addition.
b. subtraction.
c. multiplication.
d. division.
1) a and b are correct.

2) a and c are correct.
3) a, b, and c are correct.
4) a, b, c, and dare correct.

A student solved this example by adding down; then he checked his

work by adding up.
Add 34 34

T 2% ‘ 22
86 Check 86

It could be classified as an example of

a. the distributive principle.

b. the associative principle.

c. the commutative principle.
d. the law of compensation.

The statement “the quotient obtained when zero is divided by a
number is zero" is expressed as

a. 1=0
0

b. 9—20
a

c 9—.

° 0

d. 22:0
a

When a natural number is divided by a natural number other than 1,

how does the answer compare with the natural number divided?
a. larger

b. smaller
c. one -half as large
d. can't tell from information given

If you had a bag of 350 marbles to be shared equally by 5 boys,

which would be the quickest way to determine each boy‘s share?
a. counting

1). adding
C. subtracting
(l. dividing

 

.’.l.

2 4

f.)
9,,

.2 c',

182

If the multiplier is x, the largest possible number to carry is

a.
b.
c.
(1.

Which one of the following methods could be used to find the answer

x.
x + 1.
0.

x-l.

to this example?

a.
b.
c.
d.

17 l 612

Multiply 17 by the quotient.
Add 17 six hundred times.
The answer would be the sum.

Subtract 17 from 612 as many times as possible.

The answer

would be the number of times you were able to subtract.

Which one of the following would'give the correct answer to this
exalnple?

Which would give the

a.
b.
c.
d.

2.1
x 21
The sum of l x 2.1 and 21 x 2.1.
The sum of 10 x 2.1 and Z x 2.1.

The sum of l x 2.1 and 20 x 2.1.
The sum of l x 2.1 and 2 x 2. l.

 

correct answer to 439 x 563'?

Nlultiply 439 x 3, 439 x 60, 439 x 5 and then add the answer.
Multiply 563 x 9, 563 x 3, 563 x 4 and then add the answer.

Multiply 563 x 9, 563 x 39, 563 x 439 and then add the answer.
Multiply 439 x 3, 439 x 60, 439 x 500 and then add the answer.

Which of these numerals are names for prime numbers?

a.
b.
c.
(3.

LO 1'

3

4/2
12five
9 -Z
l) a is correct.
2) a and c are correct.

3) a, b, and d are correct.
)

4 a, b, C, and d are correct.

x represent: an odd number; let y represent an even number.

,
I‘henx + y must represent

a.
l).
C.
(J,

an even nun’iber.

a prime numbe 1'.

an odd number.

a composite'nuniber.

 

28.

29.

31.

183

The inverse Operation for addition is

a.
b.
C.
d.

addition.
subtraction.
multiplication.
division.

The least common multiple of 8, 12, and 20 is

a.
b.
c.
(1.

2x2.

2x3x5.

2x2x2x3x5.
ZxeZx2x2x2x2x3x5.

Which statement best tells why we carry 2 from the second
column? '

3..

b.

C.

If we do not carry the Z, the answer would

be 20 less than the correct answer.

Since the sum of the second column is more
than 20, we put the 2 in the next column.
Since the sum of the second column is 23
(which has two figures in it), we have room
for the 3 only, so we put 2 in the next column.
Since the value represented by the figures in
the second column is more than 9 tens, we
must put the. hundreds in the next column.

The operations which are associative are

addition.

subtraction.
multiplication.

division.

1) a and b are correct.

2) a and c are correct.

3) a, b, and c are correct.
4) a and d are correct.

Which of the following is an even number?

(100)
(100)1'ive
(100) seven
(zoom.

three

251
161
252
2.7.1.

935

 

 

 

184
)3

5.5. The fact that a + (b + c) is exactly equal to (c + b) + a is an
example of
a. distributivity.
b. commutativity.
c. closure.
(1. associativity.

53. Observe the drawing on the right. When the circle
is cut into equal pieces, the size of each piece
a. decreases as the number of pieces increases.
b. increases as the number of pieces decreases.
c. increases as the number of pieces increases.
(3. decreases as the number of pieces decreases.
l) a and b are correct. \.

 

 

2) a and c are correct.
3) b and c are correct.
4) b and d are correct.

34. The symbol 3/4 may be used to represent the idea that
a. 3 is to be divided by 4.
- b. 3 of the 4 equal parts are being considered.
0. 3 objects, are to be compared with 4 objects.

d. all of the above.

L4
U1
0

When a whole number is multiplied by a common (proper)
fraction other than one, how does the answer compare with the
whole number?

a. It will be larger.

b. It will be smaller.

0. There will be no difference.

d. You are not able to tell.

36. Which of the addition examples is be st represented by the shaded
parts of the diagram below?

+

 

 

 

+

l
///-f.4%/T'T/.) X - . M _
'II/I‘szgfll/p' A1, . ‘1':

 

 

 

 

 

 

 

 

 

 

 

 

+
wlw Jill—4 ub-[w WI“

0
+

 

l)

37.

39.

40.

41,

185

We can change the denominator of the fraction to the number

"1" without Changing the values of the

mIslulw

fraction by _
3. adding 5/4 to the numerator and denominator.

b. subtracting 5/4 from the nume rater and the denominator.
c. multiplying both the numerator and the denominator by 5/4.

d. dividing the numerator and the denominator by 5/4.

What statement best tells why we "invert the divisor and multiply"

when dividing a fraction by a fraction? '

a. It is an easy method of finding a common denominator and
arranging the numerators in multiplication form.

b. It is an easy method for dividing the denominators and
multiplying the numerators of the two fractions.

c. It is a quick, easy, and accurate method of arranging two
fractions in multiplication form.

d. Dividing by a. fraction is the same as multiplying by the
reciprocal of the fraction.

If the denominator of the fraction 2/3 is multiplied by Z, the
value of the resulting fraction will be

a. half as large.

b. double in value.

c. unchanged in value.

d. a new symbol for the same number.

45% may also be written as

a. .45

b. 45/100

c. 45 x 100%
d. .450

l) a and b are correct.
2) a and c are correct.
3) a and d are correct.
4) a, b, and d are correct.

. 5 and . 27 are illustrations of "decimal fractions. " They could

be written as "common fractions" in the form of 1/2 and 27/100

respectively. What is a decimal fraction?

a. It is another way of writing percentage.

l). It is an extension of the decimal number system to the right
of one‘s place.

C. A number like . 37... which has both a decimal and a fraction
as parts of it.

(1. A number like . 2 which is a fraction and has a decimal as

' 353'

either the numerator or denominator or both.

 

 

44.

45.

.17.

D . .~:‘_‘. _.

186

The ratio of x's in Circle A to " in Circle B can be shown by
a. 13/4.

b. 1/4. ,/

c. 1/2. ’ XX ;

d. 4/16. XX

 

Sue paid 20¢ for 4 apples. Which of the equations below could be

used to find the cost of 1 apple?

a .3. :3.
’ 20 x
b. x+4=20
Y
.4220
C 4

The decimal for the numeral 6/17 will
a. be a repeating decimal.
b. not repeat or end since 17 is prime.
c. repeat in cycles of less than 23 digits.
1) a is correct.
2) a and b are correct.
3) a and c are correct.
4) a, b, and c are correct.

Which of the following statements is not correct?
a. (—9) + 6 = -3

b. (-5) + (-5) = ~10

c. -8 + O = -8

d. (~8) + (9) = -l

 

Which of the following is a list of all of the factors of 12?
a. l, 2, 3, 4, 8 8:12

b. 1, 2, 3, 4, 6&12
c.1,2,3,4&
d.2_,3,4,6&12

Modular arithmetic is

a. commutative.

1). associative.

c. distributive with respect to multiplication over
addition.

(1. all of the above.

 

'19.

:32.

‘ ~~%.

187

Which of the following is an approximate measure?
a. 35 farms

1). 12 buttons

c. 7 l/Z inches

d. 15 heads

Which of the following does the sketch. below represent?

 

\
7

a. line
b. ray
c. line segment
d. set of points

1) a is correct.

2) a, b, and d are correct.
3) a, c, and d are correct.
4) b and cl are correct.
Which of these triangles are right triangles according' to the
length of the sides given?

\\

 

 

 

 

 

o \O \\
a. \9 b. ‘X
c. \\ ’l" d. o u
\o 5
4’" q"

A distinct point is

a. a point you can see.

b. a sharp object.

c. the intersection of two lines.
(1. a dot.

A clerk sold a lady a round tablecloth that had a radius of 14
inches. Which of the formulas can she use to determine the
length around the cloth?

a. Az’lrrz
b. C=¢rd
c. CIZfl'r
d. ArC/d

‘7' r——_ A ”I a wwmflhl . ‘FfiWmfi-“flwmnm-m‘- _-_-..._-____._,_ . _.

 

.‘n
U4
o

54.

55.

188

Which of the following best defines a solution set?

a. A solution set: is a set which includes each and every member
that gives a. true statement.

b. A solution set is a single sentence which identifies a variable
that will give a true statement.

c. A solution set is a set containing all the positive integers, zero,
and the negative integers.

d. A solution set is a set containing rational numbers.

Examine the following illustration.

S : {031: (‘1): 2) (‘2): 3:00-10}

Which one of the following is not a subset of S?
a. (+9, +10)

b. (0, (—2), 5

c. (3, (—3), 10)

d. (1, (-1), 6, 103

If we. use the set concept to define the operations for the counting
numbers, addition would be defined in terms of

a. the intersection of disjoint sets.

1). the union of intersecting sets.

c. the intersection of sets with common elements.

(1. the union of disjoint sets.

 

 

 

 

 
 
    

  
   

  
     

        
  

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till

till )7

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tillllllllll) t

t
31293 O 744

  

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