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A STUDY F Tifl ATT TUBES OF
CEhTAIH STU BEES OF MICHIGAH STATE COLLEGE

TOWARD ARITHKBTIC

BY
FRANK CHARLES MADILL

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements

for the degree of

AASEEE OF ARTS

Department of Education

1951

 

M182.

ACKNOWLEDGERENTS

The writer wishes to express his sincere thanks and
to acknowledge his appreciation to his adviser, Dr. Harry
w. Sundwall, for the suggestions and the helpful criticisms
he has given in making this study. Other individuals who
gave invaluable aid were Dr. M. Isabelle Blyth and Mr. Lyle
Green of the Mathematic's Department. He is also indebted
to his wife, Caroline Jane Madill, for typing and technical
advice. Without such appreciated aid, the successful comple-

tion of this study would have been impossible.

TABLE OF CONTSNTS

CHAPTER
I. THE FROBLZH ADD DEFIEITIOES OF TIRES USED .
The problem . . . . . . . . . . . . . . .
Statement of the problem . . . . . . . .
Importance of the study . . . . . . . .
Definitions of terms used . . . . . . . .
Attitude . . . . . . . . . . . . . . . .
Arithmetic . . . . . . . . . . . . . . .
II. REVIEW OF TEL“: LITIGRlTURE . . . . . . . . . .
Literature on attitudes and attitude
measurement . . . . . . . . . . . . . .
Literature on attitudes toward arithmetic
An outlook in attitude measurement . . . .
III. THE KATEE ALS USHD AND GROUIS STUDIED . . .
Attitude Scale . . . . . . . . . . . . . .
The objective attitude scale . . . . . .
The subjective supplement . . . . . . .
Scoring of the scale . . . . . . . . . .
The groups studied . . . . . . . . . . . .
.IV. FIEDILGS OF THE STUDY . . . . . . . . . . .
Group variances . . . . . . . . . . . . .
Relationship of the two objective forms .
A quantitative analysis of the objective

scale 0 O O O O O O O O O O O O O O O 0

PAGE

O‘IIFD'P‘FP‘NNN

20
22

24
24

26

27
SO

30
37

59

CHM? TEL-R

A quantitative analysis of the subjective
supplement . . . . . . . . . . . . . .
A qualitative analysis of the subjective
supplement . . . . . . . . . . . . . .
An evaluation of the scale . . . . . . .
V. SUMIARY AND COICLUSIOIS . . . . . . . . . . .
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . .

HYPE-11131300000000000000000000.

PAGE

40

42
45
49
51
56

LIST OF TABLES

TABLE PAGE

I. Means and Standard Deviations of Arithmetic

AttitUd-e scores 0 O O O O O O O O O O I O O . 35

II. Critical Ratios of the Difference Eetween Group
heans and the Standard Error of the

Difference O O O O O I O O O O I O O O O O C O 36

FIGURE PAGE

1. Comparison of the Means of the Group Scores . . . . 51
2. Individual Averages of Form A,.Arrangcd in

Intervals of Width Three . . . . . . . . . . . . 52
3. Individual Averages of Form B, Arranged in

Intervals of Width Three . . . . . . . . . . . . 35

CRAP TI’LR I
THE IROBLRH AED DEFIKITIONS OF TERMS USED

Arithmetic is one of the basic three R's that has been
taught to American children for many years. Educators have
usually considered this basic subject as necessary as "read-
ing and writing" in anyone's education. Yet, the value of
arithmetic has been a controversial issue. Students, teachers,
and adults have been quite divided in opinion as to how
valuable, necessary, enjoyable, or useful arithmetic is.
Attitudes toward arithmetic have varied considerably. hany
peOple of both sexes and all ages have stated a personal
dislike for arithmetic and some of its related fields. Some
children have struggled through grade school arithmetic only
looking anxiously ahead to a high school course containing
no mathematics. Students have purposely elected to study
courses which they thought tended to shy away from the funda-
mentals or the applications of arithmetic. Adults who actually
use arithmetic in everyday life have condemned it as being
foolish, worthless, or unpractical. Many persons have attitudes
toward arithmetic that are unfavorable. Still many students,
teachers, and adults have had natural yearnings for arithmetic
and have had highly favorable attitudes toward tne subject.
They have learned to like arithmetic and many of its applications.

They have expended extra effort to study the more advanced

mathematical fields. They have put their knowledge of
arithmetic to work and in many cases it has helped to reap or
to increase profits. They have sought personal enjoyment from
the applications of arithmetic to a personal interest.

As a high school teacher of mathematics, the writer has
observed varied personalities which have had many diversified
attitudes toward arithmetic. It is possible that many of these
attitudes are carried on to college, sometimes changed and
often intensified. It is a personal interest of the writer
to investigate scientifically such attitudes and determine

to what extent they may be modified in a favorable direction.
I. THE PROBLEM

Statement gf'the problem. It was the purpose of this

 

 

study (1) to determine the attitudes toward arithmetic of
certain selected groups of students at Michigan State College
and to co pare the relationship of the attitudes of these
selected groups: (2) to test the reliability and the validity
of a specific attitude scale; and (5) to contribute some
information that may lead to the deveIOpment or an improved
arithmetic attitude scale.

Importance 2f the study. The develOpment of desirable

 

 

. . . . , ., - , .. l
attitudes is a maJor aim of the teacners of secondary mathematics.

 

l "The Place of Mathematics in Secondary Education"
Fifteenth Yearbook of the National Council of Mathematics,
(New York: Bureau of _Iublications, Teachers College, Columbia
University, 1940) p. 26.

 

 

 

3
Deepite this general objective, very little has been done in
the way of measuring such attitudes.

It was desirable to know just what Specific attitudes
toward arithmetic were existent; to what degree these atti-
tudes varied; how these attitudes develOped or changed;
and what factors had influenced these develOpments. It was
important to know the attitudes toward arithnetic of secondary
teachers who were aiming to develOp desirable attitudes topard
arithmetic and the attitudes toward arithmetic of the elementary
teachers who were building the foundations of secondary
education.

Very little information was found on the Specific
measurement of attitudes toward arithmetic. This study made
use of a general attitude scale which was converted into a
Specific attitude scale to neasure attitudes toward arithmetic.
It was heped that the study would Justify the use of such a
scale and possibly contribute information which would lead
to the development of an attitude scale for arithmetic and

indirectly to the deve10pment of desirable attitudes as

outgrowths from the teaching of arithmetic.

II. DEFIEITIOES OF TLRHS USED

Attitude. An attitude was interpreted as meaning the
state of mind of a person regarding some hatter which was
conceptual in nature. here an attitude differed from a
trait in that an attitude was more Specific and less general-

2 has quoted Thurstone by saying: ”an attitude

ized. Stagner
is a generalized reaction for or against a Specific psychological
object." The writer felt that the definitions of the terms
attitude, Opinion and interest were greatly different, eSpecially
when interpreted tecnnically and psychologically. This study

has assumed for practical pmrposes that a person's Opinion

of some conceptual object very nearly eXpressed either his

true attitude toward it or his interest in it. However, the

fact was recognized that one's opinion may not always eXpress

his true attitude or interest, but in such cases it expressed

the attitude or interest which the person desired other people

to think he held. So for this study, attitude toward arith—

metic would then reflect the opinion of or the interest in

arithmetic of the individual.

Arithwstic. The term arithmetic was interpreted to

 

mean the basic use of numbers and numerical computation

 

2 Ross Stagner, "Attitudes,“ The EncyCIOpedia 2;
Educational Research, 1950 edition, I, 77.

below the level of algebra. In a great majority of the

cases this meant all matheuatics below the high school

level where algebra was usually first taught. This dividing
line was purposely selected because it was the point where
most people either dropped all formal study of mathematics

or branched out into one of its many fields. It was believed
that arithmetic was a more general term than.mathematics and

more desirable for the purposes of’this study.

CHAPTFE II

EEVIEW OF THE LI BEATURE

iuch has been written concerning attitudes in general,
eSpecially when treated as a segment of the study of person-
ality. Literature on attitudes is rather abundant in the
fields of psychology and education. However, in the very
Specific field of attitude measurenent relatively less has
been written. Following is a brief sumzary of the findings
of some investigators who have written in various fields

related to attitude measurement.

Literature 9p attitudes and attitude weasurenent.

 

 

Stagner,l in his discussion of personality gives a detailed
summarization of attitudes in general. Be defines attitudes

as being an integration of Opinions. They are generalized
tendencies to approve or to disapprove or to approach or to
withdraw from a conceptual object. Attitudes are always
relative to the situation and are characterized by objectivity,
direction, and intensity. Attitudes are more Specific than
traits. He mentions a few methods of attitude measurement

both objective and subjective in nature, giving references

 

1 Ross Stagner, Psychology of Personality, (New York:

McCraw-Hill Book Company, 1948) p. 202.

 

to the writings of such authorities in the field as

Thurstone, Likert, and hemmers.

‘..
I)
N

Thurstone and Chave in measuring attitudes toward
the Church gave a good but highly technical discussion of
attitudes and their measurement. They discussed theories

of attitudes, gave an objective description of them, and
indicated the possibilities of their measurement. They told
how Thurstone selected a unit of measurement and with it
devised the now famous attitude scale of equal-appearing
intervals. In developing this scale, Thurstone collected

a series of statements about the Church and with the help

of other competent judges he arranged them in groups which
appeared to be equal distances apart. A scale value was

then determined for each statement. It was the work of

the reSpondent to choose from a lot of mixed statements the
statements with which he best agreed. The respondent's score
was the calculated median of the total scale value of his
chosen statements. Thurstone also defended this test of
attitude measurement by showing various tests for its relia-
bility and validity. he showed a method of the adjustment

of this test in order to eXperiment on different subjects

Concerning attitudes toward a Specific subject. Finally

 

2 L. L. Thurstone and B. J. Chave, The Measurement 2:
Attitudes. (Chicago: University of Chicago Press, 1329),
p. 1.96.

 

8
they explained the Simpler method of testing attitudes through
paired comparisons. In general, Thurstone and Chave have
written in a highly technical nature and they are often
difficult to understand.

Thurstone:5

in an address to the Midwestern Psychological
Association discussed the deve10pment of the neasurement of
social attitudes and some of the criticisms and questions
raised about this subject. He gave an example of a test of
paired comparisons involving attitudes toward twenty different
nationalities. The main criticism of such a test was the
variance of the wording of the Specific questions used. After
discussing the method of paired comyarisons, he branched into
the method of using a statement scale. here a series of
Opinions were presented to a subject for endorsement or
rejection. The statements were in no classified order and
each had been given a separate scale value. The subject was
allowed to select those statements with which he agreed. his
score was then the calculated median scale value of all of the
statements he had endorsed. The meanings of the scores were
then determined by the relative positions of any two or more
scores. The score indicated whether one's attitude was above,

eqLal to, or below the attitude of another individual. In

 

3 L. L. Thurstone, "The Leasurement of Social Attitudes,"
The Journal 9: Abnormal and Social I3sycholog:,:, 26: 249-69,
October-December, 1951.

 

9

his example on attitudes toward prohibition; if x, y, and 2
had medians equally Spaced apart, the increment in attitude
difference between x and y would have been just as great as
between y and z. Thurstone purposely avoided the discussion
of the methods by which such scales were constructed because
of the limitation of his time. He then gave several examples
of the measurement of Specific attitudes by the method of a
statement scale. he emphasized the fact that an attitude
cannot be entirely measured by any single device and, that most
measurehents only measure comparison of characteristics in
terms of "more" or "less". he defined an attitude as being
the effect for or against a psychological object. he discussed
the uncertainty of using an Opinion as an index of attitude.
In closing he advised that attitude scales not be used to
legally detect subversive attitudes toward patriotism, economic
beliefs, or religion or as a means of eliminating undesirable
students from institutions of higher learning.

Thurstone4 has shown that attitudes can be measured
as legitimately as many concrete objects. He made use of
an Opinion as a verbalized eXpression of an attitude and
consequently as a means of measuring attitudes. He discussed

the uncertainty of using an Opinion as an index of attitude

 

4 L. L. Thurstone, "Attitudes Can Be Measured" The
American Journal 2: Sociology, 55: 529-54, January, 1922.

 

 

10
and the probabilities of the slanting of attitudes by hiding
the truth. He then assumed that attitude scales would be
used only on those Specific situations that offered a mini-
mum of pressure on the attitude to be measured. he realized
the probable changes in attitude, and he also insisted on
making use of a calculated standard error of measurement to
offset errors in the measuring instrument itself. He
emphasized the restriction and the limitation of the attitude
to be measured and he Spoke of an attitude as a point or a
vicinity along the attitude continuum. Four types of descrip-
tion were possible by means of an attitude scale. These were
(1) the average attitude of a particular individual, (2) the
range of Opinion he is willing to tolerate or accept, (5) the
relative pOpularity of each.attitude of the scale for a
designated group, and (4) the degree of homogeneity or
hetrogeneity of a designated group on a Specific issue. He
gave a lengthy discussion of the construction of an attitude
scale by his method of equal appearing intervals nd several
tests for the validity of such a scale.

Attitudes are measured by presenting a list of state-
ments about a specific issue to a subject. The subject
endorses as many of these statements as he desires. His
score is the average Scale value of all of the statements

he has endorsed. His range of tolerance is the standard

ll
deviation of the value of all of his selected statement
values. The reliability of the scale can be determined by
presenting two parallel forms to the same individuals at the
sane time. The reliability then is indicated by the correlation
coefficient between the scores of the two forms. Group
attitudes can be compared by direct comparison of the group
means because of the fact that a rational base line had
been established. Such comparisons however are not possible
when the statements' scale differences have not been measured.
Changes in attitude can be measured by determining the
significance of the difference in attitude scores on the
same test, before and after. The most essential characteristic
of the present system of attitude measurement is the fact
that the steps or intervals on the scale appear or seem to
most peeple to be equally noticeable shifts in attitude.

H. H. Remmers and Ella Belle Silance5 discussed the
merits of Thurstone's method of equal appearing intervals
for the construction of an attitude measuring device. It was
concluded that such a system if applied to measure very
many Specific psychological characteristics would encompass
considerable time, labor, and eXpense. The possible number

of variations would be nearly infinite. It was the purpose

 

5 H. H. hammers and Ella Belle Silence, “Generalized
Attitude Scales," The Journal gf_Social Psychology, 5: 298-512,
August, 1954.

12

of the authors to construct a generalized scale which would
make possible the measurement of more attitudes without
sacrificing the theoretical rigor and precision underlying
the method develOped by Thurstone.

The construction of such a scale was done by Miss
Silance as a part of her master's thesis at iurdue University
under the direction of H. E. Remhers. The major part of the
problem was the finding of a number of verbally functional
statements and the allocation of each statement to a particular
point on the scale. She gathered 150 statements from
(1) college freshman themes written on school subjects liked
or disliked; (2) textbooks on methods of teaching and educational
literature; and (3) statements composed by herself. The
statements were then sorted according to Thurstone's methods
by 150 college and high school students. On the basis of
these sortings, two equivalent fbrms were constructed by
selecting 4b pairs of Opinions of which the medians and the
interquartile ranges were as nearly as possible identical.
These.opinions were arranged in descending order of magnitude
of the interquartile range value.

Although the scale was constructed to measure attitudes
toward four different high school subjects at one time, it
was heped that it would also lend itself to measure attitudes
toward college subjects and elementary-school subjects as well.

The author expressed hOpe of measuring the reliability

15
of the scale by measurin the correlation between the results
of the two forms. No measure of validity had been determined,
but hemmers suggested that validity be established by
measuring the attitude toward a Specific subject with both a
Specific scale and as a part of the general scale. The
correlation between the two series would constitute a measure
of validity of the generalized scale, using scores on the
Specific scale as tne criterion measure.

It was this particular article that provided the writer
with the basis for this entire study. The scale constructed
by Miss Silance was adapted and used in obtaining most of
the data for the study.

Goodenough6 in a chapter devoted to attitude measure-
ment eXplained how attitudes can differ along several
coordinates such as direction, strength or intensity, breadth,
and fixity or fluidity. She emplained how most measuring
devices measure only a specific attitude for a Specific time.
She discussed Thurstone's method of equal-appearing intervals
for the construction of attitude scales. She gave the charac-
teristics of good statements that make up such a scale and
mentioned the process of measuring attitudes by such a scale.
She advocated using the median of the scale values assigned

to statements chosen as the individual score. She then

 

5 Florence Goodenough, Mental Testipg, (New York: Rins-
hart and Company, 1949) p. 374-85.

 

14
explained Likert's more simplified method where the subject
scores ev ry statement according to a degree of agreehent
and his score is the sum of his ratings. Likert's method
requires more statements to check than does Thurstone's.
Thurstone has included neutral statements where Likert has
not. Likert's statements are equally weighted where Thurstone's
are unequally weighted according to scale values obtained
through sorting in the process of constructing the scale.
Although Thurstone's scales are considered more valuable, the
more simplified Likert scale has been tested by self-correla-
tions, and it is considered satisfactory enough for the purposes
for which scales are likely to be used. Miss Goodenough
touched upon Thurstone's method of paired comparisons
mentioning that it is laborious, but that it does have a
relatively small error of estimate. She noted some of the
practical applications of attitude measurement in psychological
research. She emphasized that the potential usefulness of
attitude measurement has not been completely realized. It
is not enough just to measure attitudes but more important
to ascertain the general trend of those attitudes. She
discussed the dependability of paper and pencil attitude tests
and the sources of error in such measurements. vBecause of
lack of evidence as to the usefulness of such scales for the

prediction of actual behavior, she concluded, "It might not

15
be a bad idea to put a ban on the construction of further
scales of this type until more has been learned about the
significance that may justifiably be attached to those now
available".

A. W. Graham7 in n attempt to measure a change in
democratic attitudes 0; students made use of an attitude
scale constructed on the principles of the Thurstone method.
In this study two groups of students were selected from a
Portland, Oregon grade school. One group was taught by
teachers who used democratic methods and whose primary concern
was personality develoPment. The other group was taught by
the more traditional autocratic methods where subject matter
was the primary concern. All of these eighth-grade pupils
were carefully matched as to sex, intelligence, and socio-
economic factors. An attitude scale was administered to the
children of both groups at the beginning of the experiment
and at its termination one semester later. The attitudes
measured were toward six Specific characteristics of democracy.
At the first administration there was a very close correspon—
dence between the two groups of pupils. There was no signifi-
cant difference between the means of the groups for any of

the factors measured. The second administration of the

 

#—

, 7 A. W. Graham "Do Teachers Hho Use Democratic Methods
Develop Democratic Attitudes?" The Elementary School Journal
,47: 24-27, September, 1946.

 

16
attitude scale resulted in mean differences that were small,
but all in favor of the group taught by democratic methods.
The differences were significant in four of the six areas
measured. However the ciiierences in attitudes toward
social studies and school were not significant. It was also
noted in this eXperiment that the attitudes of the group
taught by democratic methods Shifted in a positive direction
while in most cases the atLitudes of the other group shifted
negatively. The author concluded that (1) children are usually
lacking in many of the attitudes needed in a democratic
society; (2) typical teaching procedures do virtually nothing
to develop democratic attitudes and (5) superior teachers,
using democratic methods, do have a measure of success in
establishing democratic attitudes.

Engle8 in a discussion of social problems and attitudes
stresses the point that many attitudes are based on very
limited experience. he defines a prejudice as any attitude
that prevents one from considering or evaluating certain
new evidence. He mentions that an individual's attitudes
in general cannot be measured, but that indidividual attitudes
on Specific social questions can. To measure specific social

attitudes, an attitude scale is first constructed by competent

 

8 T. L. Engle, Isychologypirinciples and Applications.
(Yonkers-on-Hudson, New'York: world BookVUOmpany, 1949)
p. 432-35.

l7
personnel. The scale usually consists of a series of state-
ments that have been given numerical values according to their
relative position with one another. A subject is asked to
select all of the statements with which he agrees. The sum
total of the relative values of all of his selected statements
is the individual's score. Then this score indicates his
attitude toward the Specific social gueSLion and it can be
compared with the scores of other individuals or with any
group.

Corey and Beeryy attempted to measure the effect of
teacher pOpularity upon attitude toward school subjects.
College freshmen were asked to write the names of their high
school teachers who were liked best and least and to include
the high school subjects which each teacher had taught. A
month later they were asked to list only the high school
subjects liked beat or least. These school subjects listed
were followed by a check indicating Whether they were being
followed at the University and whether it was required or
elected. The authors compiled their data in a chart contain-
ing eight different combinations of answers. They gave no
statistical treatment of it, not even percentage calculations;

but, they concluded that there were twice as many pOpular

 

9 Stephen M. Corey and G. s. Beery, "The Effect of
Teacher Popularity Upon Attitude Toward School Subjects"
The Journal 9: Educational Research, 29: 665-70, December,

1558.

 

18
.teachers teaching pOpular academic subjects as there were
pepular academic subjects taught by unpopular teachers. If
the teacher was not liked, the attitude was very apt to be
carried over to the school subject. They concluded with the

10 "with the young, the

following quotation by John Dewey:
influence of the teadher's personality is intimately fused

with that of the subject; the child does not separate or

even distinguish the two."

Rose11 discussed the major requirements and uses of the

questionnaire for the process of attitude measurement and the
Opinion poll. He emphasized the importance of the minimum
of ambiguity in the individual sentences or words. All
questionnaires should be pretested before being allowed to
be used. Individual subjects to be measured must be approached
in the "right way" depending on the various situations.
Care must be taken to get the best possible representative
sample of a pOpulation. He suggests a few ways of getting
the best possible sample according to the pepulation being
measured. He told how attitude measurers must boil down

their data into a comprehensible reading form by the processes

of "coding" and "analysis". He showed how attitude measurement

 

10 John Dewey, How wg_mh1nk, (New York: D. 0. Heath
and Company, 1955) p. b9.

11 Arnold.M. Rose "Attitude Measurement and the Question-
naire Survey" The Scientific Monthly, 68: 97-101, February,
1949.

19
has now broadened out to detect attitudes and to reveal
eXpectations, wishes, activities, facts, and estimations.

He showed how such questionnaires were used to measure
changes in attitudes in social studies. he also discussed
questionnaires that were used to predict marital adjustment,
future consumers‘ marnets, and election results. he finally
explained that periodical literature in the field is some-
what behind and that the field promised to have a most
interesting future.

Katz and Allportlg in a report of the Syracuse University
Reaction Study made use of several methods of attitude
meaturement. They wrote in detail about different measurements
of attitudes toward religious beliefs, cribbing, moral
standards, etc.. In the latter section of their book they
gave a good summary of attitudes and their measurement.

They discussed the techniques of yes-no tests, multiple
choice tests, rating scales, ranking scales, and attitude
scales. They showed how each develOped from a preceding one
and briefly discussed the advantages and disadvantages of
each and their uses at Syracuse University. They then
discussed and evaluated the methods of Thurstone especially
showing the advantages over the a priori, logical scales.

They question the possible discrimination of the judges

 

12 Daniel Katz and r. B. Allport, Students‘ Attitudes
(Syracuse, New York: The Craftsman Press, Inc., 19517
p. 355—75.

 

20
even though Thurstone and his associates have found this
element to be of little practical consequence. Allport's
scales at Syracuse differed from Thurstones on one main issue.
Thurstone's statements were expressions which reflect the
underlying attitudes while Allport's statements were thought
to represent the attitudes themselves. Katz and Allport gave
a few techniques of the administration of attitude surveys.
They mentioned the importance of the arousing of interest,
the role of the personality of the administrator, and the
value of an added subjective device. In general this book
represented a sound practical application of the principles

of attitude measurement.

Literature on attitudes toward arithmetic. flank}3

 

 

observed the attitudes of young children toward mathematics.
Twenty children who varied in ages from 5.2 to 11.4 years

were studied. Their backgrounds were first investigated as

to their level of educational attainment, their arithmetic
acceleration or retardation, their socio-economic levels, and
their past school attendance. This was done through school
records and interviews with the children, teachers, and parents.
The children were divided into three groups according to their
mathematical abilities which were determined by several

standardized tests and their chronological ages.

 

15 Emma H.-Plank, "ObserVations of the Attitudes of
Young Children Toward Mathematics,"The Mathematics Teacher,
43-263, October, 1950.

21
Each group then was observed in a classroom. The
observer first established rapport with the children and
led them to select and to make use of the materials that
were present. These materials were sets of the hontessori
mathematical material for elementary schools which basically
were objects that offered young children a Chance for individual
eXperimentation on quantitative and Spatial relationships.
Such a test eXplores the reaction of a child when faced with
a baffling problem. The test denotes such qualities as
fluidity, flexibility, rigidity and persistence. The author
went into detail about each of the twenty subjects. She
told about the childYS background and the results of the
various tests, interviews, and her direct observations. She
gave interpretations of the results of the work with the
Montessori materials and many direct quotations fr m the
children which expressed Specific attitudes toward arithmetic.
She attempted to explain the Specific cases of retardation
in arithmetic by the comparison of certain personality
factors that were noticed in her observations. Some of her
conclusions were interesting. They were:
General ability is not always attended by mathematical
ability.

There is less relation between intelligence and achieve-
ment in arithmetic than in either reading or spelling.

Failure in arithmetic is traced to school absences and
defects in the educational process rather than in
the child's innate mental makeup.

22
Continual failure makes many children dislike arithmetic.

Some children fail in arithmetic because they dislike
Speed. '

In her review of the literature she mentioned that very little
was avilable.on the relations between personality and success
or failure in arithmetic.

Young14 in discussing the aims of mathematical instruc-

tion, listed the follOting paragraph as one of the aims.

The Acquisition 2: mental habits and attitudes which
will make mathematical training effective l2.£§2 life
gffgn indivi ual. Among such habitual reactions are
the following: a seeking for relations and their precise
eXpression; an attitude of inquiry; a desire to under-
stand, to get to the bottom of a situation; concentration
and persistance; a love for precision, accuracy, thorough-
ness, and clearness, and a distaste for vagueness and
incompleteness; a desire for orderly and logical organization
as an aid to understanding and memory. '

 

 

 

 

These represent Specific attitudes that an educator
thought ought to be develoPed in secondary mathematical
instruction. He did not offer a way of measuring these

attitudes.

 

£3 outlook in attitude measurement. This review has
presented only a few of the many publications in the field of

attitude measurement. It was unfortunate that so few articles

 

14 John W. Young The Reorganization 9; Mathematics in
Secondary Education. (new York: Boughton Liifi'lin Co. , 1927‘)’
171 pp.

 

 

25
were found in the specific field of attitudes toward arith-
metic. The whole field of attitude measurement is still
controversial and unlimited in its Opportunities for contri-
butions. The trend in the construction of attitude scales
is toward simplification and the uses of these scales are
gaining popularity. It would be of extreme importance if
a valid scale could be constructed through which attitudes
toward arithmetic and the changes in such attitudes could be
measured. It is possible that achievement in arithmetic
could be promoted indirectly through the deveIOpment of
attitudes more favorable to arithmetic. But first, some
better means of measuring such attitudes and their changes

is in demand.

CHATTER III

THE HATER 3L8 USED AID GROUES STUDIED

After considerable investigation of the publiShed liter-
ature of attitude measurement, the attitude scale was selected
as the most apprOpriate measuring device for this particular
study. A general subject matter attitude scale was adapted
to measure a Specific attitude and a subjective supplement

was added to it.

I. THE ATTITU E 50333

The objective attitude scale. An adapted form of a

 

scale designed to measure attitudes tovard general academic
school subjects was used in this study. It had been
constructed on the basis of Thurstone's method of equal
appearing intervals by Miss Elle Belle Silencel as a part
of her master's Thesis at Iurdue University in 1934. The
construction of this scale has been described in the survey
of the literature. The scale consisted of two forms which
were comparable in content. Silance constructed the scale

for the purgose of measuring several Specific attitudes at

 

I B. H. hammers and Ella Belle Silance, "Generalized
Attitude Scales" The Journal 2: Social Psychology,
5: 306-10, August, 19o4.

Asp.

 

 

25

a single sitting with one measuring instrument. She gave
no evidence of the degree of reliability or validity of the
scale. It was a purpose of this study to test an adaptation
of this scale for those qualities.

The general pattern of Miss Silanee's attitude scale
was retained for use in this study. There were two individual
Scales, each a list of forty-five statements labeled Form A
and Form B. The statements were arranged in the same
logical order from expressions that were unfavorable to
those that were highly favorable. The statements had been
revised chiefly by substituting the word "arithmetic" for
any word or group of words that originally had referred to
the general subject in hiss Silance's attitude scale.
Basically a general scale was converted into a Specific one
which was designed to measure only attitudes tetard arithmetic.
The scale was then entitled fig Arithmetic Interest Scale.
The two forms were printed on separate sheets of paper and
each contained the identical Specific instructions for self

administration.

The Subjective supplerent. Along with the two objective
scales was an answer sheet which permitted easier adminis-
tration and scoring of both of‘the forms. On this anewer
sheet was a Space for the subjective supplement. This space

vms preceded by instructions which asked for any reasons

26
why the individual subject may have liked or disliked
arithmetic. This answer sheet also contained general
instructions, a limitation of the term "arithmetic", and
lines for such information as name, class, date, etc., Whien
the writer may have desired in Specific instances for other
studies which grew out of this one. A c0py of the complete

attitude scale ap cars in the appendix of this study.

Scoring 9: the scale. The subjects were asked to

 

select five stateuents rather than one because a single
reSponse is not necessarily a reliable index of attitude,
while several reSponses tend more to indicate the presence
of an attitude.2 In order to avoid unnecessary anu cumber-
some totals of numbers, the average of the five numbers of
the selected statements was used as the indiviuual score

for each form of the objective attitude scale. No range of
tolerance was computed because the subjects had been limited
to five choices rather than being unlimited as in Thurstone's
scoring techniquefir5 The average of the averages for each
group was computed also. The standard deviations of the
frequency distributions for each group was computed as a

measure of variability in group attitude. The standard

 

2 I. D. HacCrone, hace Attitudes in South Africa,

(London: Oxford University Press, 1937) , p. 114.

 

3 L. L. Thurstone "Attitudes Can Be Measured" The
American Journal of Sociology, 53: 534, January, 1928.

 

27
error of the mean of each group was also calculated in order
to determine the significance of the differences in mean
attitudes for the various groups.

No scoring was applied to the subjective statements
given by the subjects but these statements were used to
provide further Specific attitudes toward arithmetic and to

establish the validity of the objective scales.
II. T18 GROUES STUDIED

It was desired to have several groups of subjects
whose ranges of formal mathematical training were quite varied.
It was suSpected that arithmetic attitudes may have corres-
ponded to arithmetic ability, achieverent, or attainment.
Five such distinct groups were selected from the students
who were in attendance at Michigan State College during the
winter quarter of 1951.

The first group consisted of thirty students who were
in the Arithmetic Clinic. These students were unable to pas
a simple standardized arithmetic test taken at the beginning
of their freshman year in college. The clinic had been set
up to help those students learn the fundamentals of arithmetic
below the level of algebra. These students were given periodic
examinations which if passed satisfactorily would release
them from the clinic. No students were ever compelled to

leave the clinic at any time. The students received no

28

credit for this course. This first group then was rather
obviously weak in arithmetic.

Next was a group of thirty-one students who were study-
‘ing high school algebra at the College without receiving
college credit for it. These students were exempt from the
Arithmetic Clinic because of their scores on the entrance
tests. The course was voluntary and the group consisted of
individuals Who were eager either to improve their knowledge
of mathematics or to study algebra for the first time. It
was thought that this group as a whole would be a step higher
in arithmetic attainment than the group comprising the Arith-
metic Clinic.

A third group consisted of 111 students who were majoring
in elementary education. They were students who had studied 4*
various levels of arithmetic training. They were future
teachers who were going to have to teach elementary arithmetic
whether they liked it or not. They did not take the arithmetic
entrance test since they entered college before the Arithmetic
Clinic was establiShed. This large group was studying a common
course in the methodology of teaching elementary arithmetic.
It was thought that this group might be about average for
college students in arithmetic attainment.

A fourth group was composed of 112 secondary education

majors. In this group were students who had taken little

29
or no mathematics on the college level. In addition there
were students Who had teaching majors that had required many
courses in mathematics on the college level. It was thought
'that this group would be about average in mathematical
attainment.

Finally, a small group of ten mathematic majors comprised
the fifth group. These students were studying college calculus
which.indicated mathematical attainment. They were taking
majors in some science, engineering, or business which involved
complicated mathematics. This group was considered to be

above average in mathematical attainment.

CEAI’ TIER IV

FILDILGS OF THE STUDY

It was the purpose of this study to determine the
different levels of attitude of five Specific groups of
students, to test the objective attitude scale used, and to
obtain statements of attitudes from college students toward

arithmetic. Data was collected from 294 students.

Group variances. The individual secres for both forms

 

of the objective attitude scale were compiled into their
apprOpriate groups. The arithmetic mean was computed for
both forms of each group as the major measure of central
tendency. This measure was the basis by which the various
groups were compared for corresponding levels of attitudes
toward arithmetic. This com_arison is best shown by the
bar graphs in Figure 1.

Because these groups varied considerably in size, the
standard deviation of the scores of each form far each group
was computed as the measure of diSpersion. With these
data, it was possible to compute the standard error of each
arithmetic mean. These in turn were used to test the signifi-
cance of the differences between the means of the various
groups and to determine the critical ratios for the differences

between the means.

;

{ Arithmetic Clinic

FORM A

 

 

 

Secondary Group

 

 

 

Elementary Group

1* x

 

 

i Algebra Group

 

31

 

: Math Majors

 

 

10 15

E

i Arithme tic Clinic

FORM B

25

 

 

4‘

i Secondary GrouP

L__._x

50

 

! Elementary Group

 

 

 

I Algebra Group ,/

 

 

__M._._..A_4A“_-_g

*-iii#‘—h_nm r._g_._._. ..i_ ii

 

Math Majors

FIGURE

1

A comparison of the means of

the group scores.

 

----—---”.om-.1
I

..----.——..___—. ....,_ .

 

 

 

 

 

 

 

 

 

 

 

LIT—l: lfi—WTTT‘VWFfi‘fiI—J 0

Averages

FIGURE 2

INDIVIDUAL AVERAGES OF FORM A ARRANGED
IN INTERVALS OF ‘s’I'ID‘I'H THREE

75
70
65
so
55
so
.45
'40
35
.50
25
'20
15
10

5

 

 

 

 

 

 

 

 

 

 

 

 

75 j

65

'60

-50

L20 1

 

:IINTBRVALS OF WIDTH THREE

-_‘. “—- -b

I
.0——--—- n.» n----—- p..--

.—--~ -

L—. .-—J

INDIVIDUAL AVERAGES OF FORM B ARHANGED IN

gen-~— .‘ --

' 14 17 20 23‘“§B' 29 32’ ”35 33' 41 ' 0
Averages '
FIGURE 8

7O

55

45

40

35

30

25

15

10

 

54

Figure 1 shows how the groups compared according to
their arithmetic means without any consideration for measures
of dispersion or standard error. The graphs Show a distinct
difference between.the means of the math majors and of the

other groups. They also show that the means of the other
four groups were comgaratively close together, with the mean
of the Arithmetic Clinic being the lowest. With these means,
the score of any individual can be compared with.the group
average and the average of the group can be Compared with that
of another group or any combination of groups. A score of
twenty—one would have represented a neutral attitude toward
arithmetic. All of the group scores were above this neutral
point on the scale. They differed in degree even though they
were all favorable toward arithmetic.

In Tables I and II the group means and variabilities in
attitude are presented for comparison and with indication of
the significance of differences in means for the various
groups. The critical ratios were computed by dividing the
actual difference between the means by the standard error of
that difference. The significance of the difference between
the averages of any pair of groups then depended upon the value
of this critical ratio. when this ratio was greater than 1.96,
the difference was significant at the five per cent level,
meaning that such a difference would have been due to chance

about five times in one hundred. If the critical ratio were

35

MEANS AND STANDARD DEVIATIONS OF ARITHMETIC ATTITUDE SCORES

 

 

 

 

 

 

 

 

 

 

GROUP Form A Form B
N 7 0" O; X 0- 0}

Hath

Majors 10 55.2 1.5 .45 55.0 2.09 .96
ZIgeEra fi—

Group 51 51.8 5.8 .69 52.2 5.5 1.00
El em silt ary

Group 111 51.0 4.8 .46 51.5 5.1 .49
SECOndary

Group 112 51.2 7.1 .67 50.1 6.7 .64
Arithmetic
_Group 50 50.5 5.2 .98 28.2 5.4 1.00
Total 294 50.7 5.8 .54 50.6 6.2 .56

qulz

Number of subjects in group

Arithmetic mean of group

Standard deviation of the group

= Standard error of the mean

36
TABLE II

CRITICAL RATIOS OF THE DIFFERENCE BETWEEN GROUP KEANS AND

THE STANDARD ERROR OF THE DIFFERENCE

 

 

Algebra Elementary Secondary Arithmetic

 

 

 

 

Group Group Group Clinic

Math

Majors 2.03%:- 3.h.3-::--;:- h.26-::—-::- n.89-zz-ez-
Algebra

Group .81 1.76 2.8h%*
Elementary

Group 1.h9 2.79**
Secondary

Group 1.60

 

* Significant at the five per cent level.

%* Significant at the one per cent level.

57
greater than 2.58, such a difference was significant at the
one per cent level and would have been due to chance once in
one hundred occuranees. All levels greater than the five
per cent level are considered to be negligible. Table II
shows that the group of math majors have attitudes that are
significantly different from the attitudes of the other feur
groups. The Arithmetic Clinic group has attitudes that are
significantly different from all groups excepting the second-
ary education group. Any other combination of pairs of groups

has differences that are insignificant.

Relationship_g§_tne'two objective forms. The two

 

 

objective attitude scales used on each subject were assumed
to be very similar in structure, appearance, and order. It
was a purpose of the study to compare these two different
forms and to call attention to any relationship that may
exist between them.

The scores of all 294 subjects were totaled and treated
exactly as were the scores for each group as previously
mentioned. Form A has a slightly higher arithmetic mean and
a slightly lower! standard deviation and standard error of
the mean than Form B. Table I shows that these differences
were exceedingly small. When tested statistically, the
actual difference between these means was .10, while the

standard error of this difference was .49 . The critical

58
ratio for this difference is .20. Such a difference is
insignificant.

The difference between the standard deviations when
divided by the standard error of that difference gives a
critical ratio of 1.15. This difference in the variability
between the two forms is insignificant. 0n the basis of
these two measures the two forms appear to be comparable.

The frequency distributions of the two forms were tested
for goodness of fit by a chi square comparison. This determines
the significance of the d fference between the actual distri-
bution of the data and the theoretical normal distribution
based on the actual mean and standard devhation. The chi
square value for Form A was 115.6 while that for Form B was
51.8. For 12 degrees of freedom such a chi square value
will exceed 26.2 only one percent of the time. Therefore,
it can be said that each of the distributions differed signifi-
cantly from normality.

0n the basis of the chi square values Form B may be
considered more normal than Form A. The chi square value
for Form A is twice as great as for Form B. This is
illustrated in the histograms in Figures 2 and 5.

Another means of showing a.relationship is the computa-
tion of a product moment coefficient of correlation. Such
a coefficient was computed on the basis of individual scores

which consisted of the mean of the five numbered statements

59
for each subject on each fbrnxof the scale. This particular
product moment coefficient was .78. For the number of scores
involved, the standard error of this coefficient was .04.

It waS'thought that the median of the three scores appearing
closest together would yield a more reliable attitude score
for any given individual. These median scores for individuals

+ .05 between Form A and Form B.

yielded a correlation of .65
The difference between this correlation and the one obtained
for the mean scores was found to be significant at the five
per cent level. From this it would appear that the median
score is not superior to the mean score. There is a probable

advantage in favor of the mean score.

é_guantitative analysis g: the opjective ggale. An

 

 

attitude scale gives material and data that can be treated
quantitatively.l Many statements, eSpecially at the extreme
ends of the objective scale, were not selected by any of the
subjects. Again certain more general statements were very
frequently chosen. For these and other reasons the average
of the five numbers of the statements was used as the
individual score for each subject. This does not mean that

this score corresponds to the attitude expressed by that

 

1 L. L. Thurstone and E. J. Chave, The Measurement
9; Attitudes. (Chicago: University of Chicago Press, 1929),
p. 10.

 

 

40
statement which is equal to the individual's average or score.
This score is a point on the scale which is relative to scores
associated with the other statements chosen by an individual.
A person whose score is thirty-seven has a more favorable
attitude toward arithmetic than a person whose score is thiry-
six or anything lower than thirty-seven. The statements in
each form were arranged in a logical order from low to high.
The average score on each form for the entire 294 subjects
is 51. An individual score could have been compared with
this score to determine whether the subject's attitude toward
arithmetic was above or below the group average. The averages
for each group were computed as shown in Table I. These
group means were used in making comparisons in attitude for

the different groups.

A Quantitative analysis gf_the subjective section.

 

 

The subjective section consisted of a request for a written
response to the question of why one may have liked or dis-
liked arithmetic. Instructions were printed on the answer
sheet for such a reSponse and the request was to have been
stated orally by the person who administered the attitude
scale. However, the reSponse was not mandatory and conse-
quently a small percentage of these responses were obtained.
Of the 294 subjects who completed the objective scale, only

104 reaponded with a written reason for liking or not liking

41
arithmetic. This represented thirty-five per cent participa-
tion. Of these 104 written responses, five were completely
neutral. Fifty-two subjects stated that they liked arithmetic
for some reason or another while forty-seven subjects eXpressed
reasons for disliking it. This gives a total of ninety-nine
who responded positively or negatively.

For each of these ninety-nine subjects the median of the
three closest scores was determined for both Form A and Form B.
Using these as the measures of central tendency, it was possible
to compare the Specific cases with.the averages of the different
groups and with that of the entire group. 0f the fifty-two
who stated a liking for arithmetic, eleven had median scores
below the group average while forty—one had median scores above
the group average on both forms of the scale. For those who
stated a dislike for arithmetic, thirty-three had median
scores below the group average and fourteen were above on
both forms of the scale. These figures were used in making
a comparison of individuals above or below average on the
attitude scale for both the "likes" and "dislikes". A test
of significance was applied to these results which yielded a
chi square value of 22.16. With a single degree of freedom
as was this case, sucn a chi square value is significant
at the one per cent level when it exceeds 6.65. This test
then shows that the group stating a liking for arithmetic

had objective attitude scores that were significantly

42
different from the objective attitude scores of the group
which stated a disliking for arithmetic. Such a difference
would occur by chance alone far less than once in one hundred
times. A tetrachoric correlation of .70 with a standard
error of .14 was computed from the percentages of the total
which were above and below the group average for both the
"likes" and the'dislikes". This showed that those who gave
reasons for disliking arithmetic tended to choose unfavorable
statements on the attitude scale, and those who expressed
reasons for liking arithmetic tended to choose statements on

the attitude scale which were favorable to arithmetic.

é qualitative analysis gf_the subjective section.

 

These subjective reSponses were not only valuable fer evaluating
the attitude scale, but they have given many additional verbal
eXpressions of attitudes toward arithmetic. many of these
staterents could be useful in the improvement or the develOpment
of a Specific arithmetic attitude scale. In fact, they have
already been considered by another investigator for such a
purpose. Several of these specific individual eXpressions
indirectly offer a severe challenge to the teachers of arith-
metic. The statemeits are listed in the order of the frequency
of their occurence. The number in parentheses at the end of
each statement represents the number of times the statement

(or one nearly the same in content) was mentioned.

43
The following statements were eXpressed by those students
who said they disliked arithmetic:

The teachers of arithmetic were never interesting. Their
methods were wrong. They progressed too fast. (16)

I do not understand arithmetic. (9)

I had a weak arithmetic foundation. (8)

Arithmetic was always too difficult. (8)

I am weak in arithmetic. (7)

Arithmetic has always been forced upon me. (7)

It requires too much work and time. (6)

I met constant failure in grade school arithmetic. (4)

It is a dry subject. (5)

It does not fascinate me. (2)

Arithmetic is boring. (2)

Arithmetic teachers have humiliated me before the class. (2)

Arithmetic requires unnecessary accuracy. (1)

I had too much long division forced upon me. (1)

I went to several different schools; the levels of arith-
metic were not standardized. (l)
The following were expressed by those who said they

liked arithmetic:

I have a feeling of satisfaction and accomplishment when I
work a problem. (9)

Arithmetic is useful in everyday life. (8)

Arithmetic is concrete. It is objective, being either
right or wrong with no subjective scoring of problems. (8)

44
It helps me to develop more accurate thinking. (7)
It is based on facts. (6)
I just enjoy arithmetic as pure fun. (5)
I enjoy solving arithmetic problems and puzzles. (5)
It is necessary for ordinary business transactions. (5)
Arithmetic is essential for my future work. (5)
It is a pleasant subject. (4)
Arithmetic is practical. (4)
It has many useful applications. (4)
Arithmetic was always taught in an interesting fashion. (4)
I like the logical order of arithmetic reasoning. (4)
I was given a good arithmetic foundation in grade school. (4)
Arithmetic is essential in everything we do. (5)
Arithretic is interesting. (3)
It is of use to the normal person. (2)
It fascinates me. (2)
I have excelled in it. (2)

It is the basis for abstract thinking and scientific
interpretation. (1)

I enjoy computing bowling averages. (l)
I enjoy computing baseball batting and fielding averages. (l)

I like it only because my present teacher is synpathetic
and very patient with me. (1)

45

Ag evaluation of the scale. Since the general scale

 

had been constructed with no reported evaluation, the writer
felt that an investigation of the reliability and validity of
the scale as applied specifically to arithmetic should be
made as a part of this study.

The test for reliability was the significance of the
reLationship between the scoring on forms A and B. As has
been previously'nermioned, two different coefficients of
correlation were found between these two forms. That based
upon the correSponding averages was .78. That based upon
the correSponding medians of the three closest choices was
only .65. Correlation coefficients for good attitude scales,
such as those of Thurstone, hammers and Allport, are .80 and
above.2 The findings of this study therefore indicate fair
reliability for the arithmetic attitude scale that was used.
The critical ratio of the actual difference between these
correlations of .78 and .65 and the standard error of this
difference was 2.22. This means that the difference between
these two correlation coefficients was significant at the
five per cent level but insignificant at the one per cent
level. Or, roughly Speaking, such a difference would occur
by chance only about three times in one hundred.

0n the basis of this significance, it appears that the

 

2 Ross Stagner, "Attitudes," The EncyclOpedia 3:
Educational Research, 1950 Edition, Walter S. honroe, Editor,
Thaw York: The Hacmillan Company, 1950), p. 82.

 

 

46
averages are superior to the medians for the measures of
central tendency. The writer feels that reliability coef-
ficients of .78 and .65 with standard errors of .04 and .05
and based on a population of 294 are not reliable enough for
predictive usage. But, these results tend to point in a
direction favoring the mean scores for individuals. A
reliability coefficient of .80 would have been considered good.
Consequently, the reliability of'the scale has been considered
to be fair. The scale may be considered useful for the comparison
of group attitudes tonard arithmetic.

Such a scale would be valid only if it measured what
it purported to Leasure. The scale attempts to identify the
attitudes toxard arithmetic and to determine the differences
in such attitudes for five Specific groups.

No effort to disguise the scale was made. Although it
was called an Interest Scale, the subjects were told that it
was an attitude scale.

It was realized that peOple can easily hide or falsify
a true attitude in such a written reSponse. It was heped
that such a practice did not occur in this study. A Special
emphasis was made to assure the subjects that all data would
be handled in a confidential manner. The subjects were given
both verbal and printed guarantees that their results would
not affect their grade in the course in which the scale was

administered. The attitude scale did not require a name or

47

any mark of personal identification. The subjects were also
told the purposes of the study and the importance of their
participation.

Only one-third of the subjects responded to the requested
subjective question. Of these responses, those who stated a
definite liking or a disliking for arithmetic tended to choose
statements corresponding generally in attitude on the attitude
scale. The Significance of the relationship of the objective
and subjective sections was beyond the one per cent level. The
validity coefficient as Shown by the tetrachoric correlation
coefficient might have been somewhat higher if the mean score
rather than the median score for individuals were used in this
comparison. Application of the scale indicated significant
differences between the average attitudes of groups differing
in level of training in mathematics. Two extreme groups were
obviously different in levels of mathematical attainment from
the other three groups. Correspondingly, these groups showed
differences in arithmetic attitude that were Significantly
different from those of all other groups except in a Single
case as shown in Table 11. Therefore it can be stated that
for the comparison of group attitudes the scale is valid.

This study Shows which statements on the attitude
Scale seem to represent attitudes toward arithmetic of students

with various sonool background.

48
The writer suggests that some statenents on the scale
be eliminated Since they were not chosen by any of the

subjects used in this study.

CHATTER V

SUHEARY AfiD CCECLUSIOKS

This study was a survey of the existing attitudes toward
arithmetic of five selected groups of Michigan State College
students during the winter quarter of 1951 and an evaluation
of a Specific attitude scale.

The subjects who provided.the data for the study were
the following five Specific groups: an Arithmetic Clinic
group, a group studying high school algebra, a group studying
elecsntary education, a group studying secondary eduCation,
and a group studying calculus.

A general subject-matter attitude scale was adapted
for particular use with arithmetic. This attitude scale
with a subjective supplement was used in obtaining the data
for the study. The subjects were given standardized instruc-
tions and allowed fifteen minutes to select statements on the
scale and to write reasons for liking or disliking arithmetic.

The distribution of'the selections of the objective
scale indicated which of the statements best expressed the
subjects' attitudes toward arithmetic. The responses to the
request for attitude reasons provided additional interesting
opinions of arithmetic.

The study shows that the average attitude score for all

five groups studied was favorable toward arithmetic. These

50
scores differed in theii'degree of favorableness. It was
found that those who had taken the more advanced courses in
college mathematics had attitudes which.were more favorable
toward arithmetic than those who had not studied advanced
college netheratics.

The general attitude scale adapted for this study
was shown to be satisfactorily valid and reliable for use
in measuring group attitudes toward arithmetic. Suggestions
for the improvement of the scale were given.

The study has led to the following problems: (1) by
what means and to what extent can changes in attitudes toward
arithmetic be measured, (2) how can teachers influence changes
in attitudes toward arithmetic at various age levels, (5) can
achievement in arithmetic be improved by improving attitudes

toward arithmetic.

BIBLIOGRAPHY

BIBLIOGRAPHY
A. BOOKS

Barr, A. 8., William H. Burton, and Leo J. Brueckner,
Supervision. New York: Appleton-Century-Crafts, Inc.
1947. 871 pp.

Bizler, H. H., Check Lists for Educational Research. New
York: Bureau of’Publications, Teachers College,
Columbia University, 1928. 160 pp.

Engle, T. L., Ps cholo Principles and épplications, New
York: World Book Company, 1945. 549 pp.

Garrett, H. E., and M; R. Schneck, Psychological Tests,

Methods, and Results. New'York: Harper and Brothers
PuBIIshers, 1953. 224 pp.

Goodenough, Florence, Mental Testing. New York: Rinehart
and Company, Inc., 1949. 541 pp.

Hildreth, G. H., A Biblio ra h of’Mentgl Test and Ratin
Scales, New York: The Psychological Corporation, 1953

 

Hunt, Thelma, measurement lg PS cholo , New‘York:
Prentice-Hall Inc., 1936, 4 pp.

 

Katz, Daniel and F. H. Allport, Students' Attitudes, Syracuse:
The Craftsman Press, Inc. 1931. 397 pp.

Lindquist, E. F., Statistical Analysis ig Educational
Research. Boston: Houghton Mifflin Company, 1940
257 pp.

 

 

MacCrone, I. D., Race Attitudes in South Africa. London:
Oxford University Press, 195?." “—2731 "pp""". "'

MhCall, William A., How §g_Measure in.Education, New York:
The Macmillan Company, 1923. 4T0 pp.

Murphey, G. and R. Likert, Pgblic gpinion and the
Individual. New‘York: Harper and Brothers Publishers,
$1938. 316 pp.

53

Mursell, James L., Psychological Testing. New York:
Longmans, Green and Company, 1947, 398 pp.

Otis, Arthur T., Statistical liethod in Educational Measure-
ment. New York: Worldeook Company, I926. 329 pp.

 

Stagner, Ross, Psychologygf Personalit . New‘York:
McGraw-Hill Book Company, 1948. 452 pp.

 

Symonds, Percival H., Diagnosi_g Personality and Conduct.
New'York: D. Appleton—Century. Company, Inc., I931.
569 pp.

Thurstone, L. T., and E. J. Chave, The Measurement of
Attitude. Chicago: University of CEIcago Press,
I929. 96 pp.

Young, J. T., The Reorganization of mathematics in Seconds
Education. New Ybrk: Houghtbn —MIffIin Company, 1927.

I71 pp.
B. PEIIODICAL ARTICLES

Bain, Read, "Theory and Measurement of Attitudes and.
Opinions," The Psychological Bulletin, 27:357-68,
May, 1930.

 

Corey, Stephen M. and G. S. Beery, "The Effect of Teacher
Pepularity upon Attitude toward School Subjects",
The Journal gf_Bducational Research, 29:665-70, December,
I938.

Droba, D. D. "Methods of’Measuring Attitudes, " The
Psyphological Bulletin, 29: 309-23, May, 1932.

Graham, A. T., "Do Teachers The Use Democratic Methods
Deve10p Democratic Attitudes?" The Elementary Journal,
47: 24-27, September, 1946.

 

Hinckley, E. C. "The Influence of Individual Opinion on
Construction of an Attitude Scale, " The Journal of
Social Psychology, 3: 283- 96, August, I932.

Laycock, Sam R., " A Time Saving Device for the Construction
of Attitude Scales, The Journal of Social Psyghology,
4: 366-72, August, 1933.

 

 

Sh

Likert, Rensis, Sidney Roslow, and Gardner Mnrphey,
" A Simple and Reliable Method of Scoring the Thurstone
Attitude Scales", The Journal g£_Socia1 ngcholgy,
2:228-38, May, 1934.

 

Pace, C. Robert, " A Study in Validity of Attitude Measure-
ment," Educational and ngchological Measurement,
10: 411-19, Afitumn, 1950.

Plank, Emma N., "Observations on Attitudes of Young
Children Toward Mathematics," The Mathematics Teacher,
43-252-63, October, 1950.

 

Remmers, H. B. and Ella Belle Silance, " Generalized
Attitude Scales,“ The Journal Lf Social Psychology,
5: 298-312, August, 1934.

Rose, Arnold M. "Attitude Measurement and the Questionnaire
Survey" The Scientific Monthgy, 68:97-101, February, 1949.

Symonds, Percival M., "A.Socia1 Attitudes Questionnaire,"
Journal gngducational Research, 16-316-22, May, 1925.

Thurstone, L. L. "Attitudes Can Be Measured," The American
Journal 9f Sociology, 33:529-54, January, 1928.

, "The Measurement of Social Attitudes,"
The Journal of Abnormal and Social Psycholgy, 26:247-69,
December, 1931.

 

Wang, Charles K. A., "Suggested Criteria for Writing
Attitude Statements," The Journal 2; Social Psychology,
3-367-73, August, 1932.

 

C. PUBLICATIONS OF LEAEHBD ORGANIZATIONS

Grice, H. H., "Construction and Validation of a Generalized
Scale Designed to Measure Attitudes toward Defined
Groups," In Remmers, H. H., (Editor) Studies in
Attitudes. Purdue University Bulletin, Vol. 35, No. 4.
Studies in HigherIEducation, No. 26, 1934, pp. 37-63.

"The Place ofHMathematics in Secondary Education," Fifteenth
Yearbook Lf the National Council Lf Teachers Lf Mathe-
matics. New York: Bureau of Publications, Teachers
College, Columbia University, 1940. 253 pp.

 

 

55

D. EECYCLOIEDIA ARTICLES

Commins, W. D., and B. B. Friedman, "Attitudes," The
Encyclopedia 9; Lodern Education, Harry N. Rivlin, editor,
new York: F. Hubner and Company, 1943. 899 pp.

 

 

Stagner, Ross, "Attitudes," The EncyclOpedia gglfiduational
Research, Walter 3. Monroe, editor, New York: The Lac-
millan Company, 1950. 1520 pp.

 

 

APPEII DIX

J

'- I. - c ‘ '.- . . .- ‘1. l" ‘ -7 E:
.u Arithmetic interdu scale 27

Following is a list of statements about arithmetic. They

refer to simple arithmetic or numerical computation below the

level of algebra. Choose the five statements With which
you agree. Your score will in no way affect your grade in any
course.

1. I hate arithmetic.

2.
3.
4.
5.
5.
7.
8.
9.
10.
ll.

12.

13.
14.
15.
15.
17.

18.
19.
20.
21.
22.

23.
24.
25.
25.
2?.
28.
29.
50.

31.
32.
33.
54.
33.
3d.
3?.
5H.
59,
40.
$1 .9
43.
45,
e e. i

.4 ,1

Arithmetic is the most undesirable subject taught.

I detest arithmetic.

I look forward to arithmetic with horror.

Arithmetic is disliked by all students.

It is a punishment for anybody to take arithmetic.
Arithmetic is a waste of time.

Arithmetic is based on "fogy" ideas.

I would not advise any one to take arithmetic.

I have seen no value in arithmetic.

I have no desire for arithmetic.

Arithmetic reminds me of Shakespeare‘s play—~"Much Ado
About Nothing."

Arithmetic is very drT.

Arithmetic does not teach you to think.

I am not interested in arithmetic.

The minds of students are not kept active in-erithmetic.
Mediocre students never take arithmetic so it should be
eliminated from schools.

I could do very well without arithmetic.

my parents never had arithmetic, so I see no merit in it.
Arithmetic will benefit only the brighter students.

I havenfit any definite like or dislike for arithmetic.

I am careless in my attitude toward arithmetic, but I
would not like to see this attitude become genernl.

I don"t believe arithmetic will do anybody any harm.
Arithmetic is a good pastime.

Arithmetic is not a bore.

Arithmetic saves time. -

Arithmetic is not receiving its due in public high schools.
I am willing to spend my time studying arithmetic.
Arithmetic is O.K..

All lessons and all methods used in arithmetic are clear

.and definite.

Arithmetic is a cultural subject
All of our great men studied ari
Arithmetic-is a good subject.
Arithmetic is a univereil subject.
Arithmetic teaches me to be aweurate.
Any student who takes arithmetic is bound to be benefited.
Arithmetic is very practical. -
Arithmetic develops seed reasoning ability.
Arithmetic is profitable to everybody who takes it.
I really enjoy arithmetic.‘
Arithmetic has an irreeietable attraction for me.
Arithmetic is of great value.
I love to study arithme31e.
I voild rather stud; arithmetic ttgn eat.

1 -._ i‘ . ‘

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. y 1 w . _. Q . \ \, ,_. m "f (3 a , ,. 1
‘1‘1' Ll ,‘ U '2)? 'tl:5~~..-I \J J-L-r) I." 1.-” ~) _ .31 9'. '51.:1‘-) » -.

thmetio.

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I235231-UVH-L“? i3 3 11.319 02!. 81:6. DUDUIIDS 8.1301113 Lfit..i'-b13 ”1.3.8:;

refer to simple arithmetic or numerical computation below the level
of algebra. Choose the five statements with which you agree
Your score will in no way affect your grade in any course.

31.

32.
33.
34.

36.
35.
37.

38.
39.
40.
41.
42.

43.
44.

45.

Arithmetic is the worst subject taught in school.

Words can't express my antagonism toward arithmetic.

No sane person would take arithmetic.

Arithmetic is all bunk.

Arithmetic is more like a plaque than a study.

Nobody likes arithmetic.

Arithmetic has no place in the modern world.

Arithmetic can t benefit me.

All of the material in arithmetic is very uninteresting.
The average student gets nothing worth having out of arithmetic.
Arithmetic does not hold my interest at all.

Arithmetic seezzs to be a necessary evil.

Arithmetic is dull.

Arithmetic interferes vith developing.

Arithmetic has numerous limitations and defects.
Arithmetic dose not motivate the pupil to do better work.’
He definite results are evident in arithmetic.

. To me arithmetic is more or 1683 boring.

No student should be concerned with the way arithmetic is
taught.

Arithmetic is all right but I would not take any more of it.
iiy'likes and dislikes for arithmetic balance one another.
Arithmetic does not worry me in the least.

Arithmetic might be worth while if it were taught right.
Arithmetic has its drawbacks, but I like it.

I think arithmetic is amusing.

Arithmetic is not based on untried theories.

. Arithmetic aims mainly at power of execution or application.

Every year more students are takingarithmetio.
Arithmetic has its merits and fills its purpose quite well.

. All methods used in arithmetic have been thoroughly tested

in the classroom by experienced teachers.

Arithmetic serves the needs of a large number of boys and
girls.

Arithmetic teaches methodical reasoning.

Arithmetic is interestirg.

There are more chances for development of high ideals in

arithmetic.

Arithmetic ms hes me efficient in school work

Arithmetic v.111 help pupils socially as well as intellectually-
Arithmetic gives pupils the ability to interpret situations
they will meet later in life.

The merits of arithmetic far outweigh the defects.

Aiithmetio fascinates me.

Arithmetic is one subject that all young Americans should know
I believe arithmetic is the basic one for all high school oourzes.
Arithmetic is one of the mos t useful subjects I know. '

If I had my way, I would eompel everybody to study arithmetic.
The very existence of humanity depends upon this subject.

I am "crazy" about arithu Ht 0. -

An Arithmetic Interest Scale ’

 

Instructions:

Circle five of the numbers which correspond to the five
numbers of the statements you have selected which best compare
with your Opinion(s) of arithmetic.

 

Form A Form B In the Space below, write any reason(s)
why you like or dislike arithmetic.
1 1
2 2
5 5
4 4
5 5
6 6
7 7
8 8
9 9
10 10
11 11
12 12
15 15
14 14
15 15
16 ~ 16
17 17
18 18
19 19
20 2O
21 21
22 22
25 25
24 24
25 2
26 26
27 27
28 28
29 29
5O 5O
51 51
52 52
55 55
54 54
55 55
56 56
57 57
58 58
59 59
4O 40
41 41
42 42
45 45
44 44

HICHIGRN STRTE UNIV. LIBRRRIES

 

HI MM
1

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