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MURPHY, John Thor-s, 1943BELIEFS AND PRACTICES:
A STUDY OF MODERNNESS
IN MATHEMATICS TEACHING IN SELECTED MICHIGAN
PUBLIC ELEMENTARY SCHOOL CLASSROOMS.
Michigan State University, Ph.D., 1972
Education, theory and practice

U n ive rsity M icro film s, A XEROX C o m p a n y , A n n A rb o r, M ich ig an

©

1972

JOHN THOMAS MURPHY

ALL RIGHTS RESERVED

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED

BELIEFS AND PRACTICES:

A STUDY OF MODERNNESS

IN MATHEMATICS TEACHING IN SELECTED MICHIGAN
PUBLIC ELEMENTARY SCHOOL CLASSROOMS
By
J. Thomas Murphy

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
College of Education

1972

P LE ASE NOTE:

Some pages may h a v e
i nd i s t I n e t p r i n t .
F i l m e d as r e c e i v e d .

University M icrofilm s,

A X e ro x E d u c a t i o n Company

ABSTRACT
BELIEFS AND PRACTICES: A STUDY OF MODERNNESS
IN MATHEMATICS TEACHING IN SELECTED MICHIGAN
PUBLIC ELEMENTARY SCHOOL CLASSROOMS
By
J . Thomas Murphy

Introduction
At this date, after many years of the "modern"
approach in the teaching of elementary school mathematics,
the extent to which modernness has been implemented by the
classroom teacher appears to be unc l e a r .
Purpose
The purposes of this study were:

(1) to develop a

definition of the "modern mathematics approach" reflecting
a pooling of recent writing and research and validated by
expert opinion;

(2 )

to determine the extent to which there

is a difference between w hat the theorists say and what the
teachers believe, as relates to elementary school m a t h e ­
matics;

(3) to ascertain the practices of respondents and

the occurrence rate of those practices in elementary class­
rooms; and

(4) to analyze the data collected by a survey

questionnaire concerning elementary classroom te a c h e rs'

J . Thomas Murphy
utilization of validated modern mathematics teaching strat­
egies, in light of specific variables.
Design
Eight mathematics education experts were polled and
asked to validate

(as modern or non-modern)

items on a list

of mathematics teaching principles that were drawn from
current research and writing.

Based on this list of vali­

dated principles, a questionnaire instrument was developed
that consisted of two parts.

Part I asked the respondent

what he or she believed about mathematics ideology and
Part II asked what the respondent actually did on a day-today basis in the classroom.

The responses on the two parts

of the instrument were analyzed statistically to measure
the significance of the four research hypotheses that the
study sought to investigate.

A Jennrich two-way analysis

of variance with repeated measures technique, as well as a
one-way analysis of variance with unequal subclasses, was
applied to the returned data.

In both statistical pro­

cedures the level of significance was

.025.

Sample
Each of the mathematics experts utilized in this
study was a faculty member of a different teacher certifi­
cate granting university in the State of Michigan.

The

random sample of classroom teachers used in the study were

J. Thomas Murphy
all elementary classroom teachers who taught in one of
sixteen southwestern Michigan school districts.

These

districts were chosen for inclusion in this study because
of their size and per pupil expenditure.

The questionnaire

instrument was mailed to the home address of each respond­
ent selected.

The response rate was 9 3.3%.
Conclusions

On the basis of the data collected and analyzed in
this study, the following conclusions seem warranted:
1.

The responses of a group of mathematics ex ­

perts and a randomly selected sample of one hundred eighty
elementary school classroom teachers in Michigan to a list
of contemporary mathematical principles of instruction,
revealed a significant difference at the .025 level between
the reported beliefs of the two groups as to the consist­
ency of a modern approach to teaching mathematics to ele­
mentary school learners.
2.

The sample of one hundred eighty elementary

classroom teachers were also asked to respond to a list of
practices correlated to the set of contemporary mathemati­
cal principles of instruction.

A significant difference

was indicated between what the teachers indicated they
believed to be a modern approach to teaching mathematics,
and what they reported as actually practicing in their
classrooms.

J . Thomas Murphy

3.

When a score of "modernness" was established

for each of the respondents in the study, there was no
significant difference indicated when the scores were
grouped, a mean taken, and compared across the variable of
sizes of school district populations in the districts in­
cluded in this study.
4.

When a score of "modernness" was established

for each respondent in the study, there was no significant
difference indicated when the scores were grouped, a mean
taken, and compared across the variable of general fund
expenditure per pupil in each of the districts included
in this study.

ACKNOWLEDGMENTS

This study would not have been executed or brought
to fruition, without the assistance and guidance of numer­
ous individuals, most of whom must necessarily go unnamed,
but not unremembered.
The writer wishes to express his deepest apprecia­
tion to Dr. Calhoun C. Collier, chairman of the guidance
committee.

His suggestion brought me into the field of

elementary education, and his tutorage has immensely facil
itated the furtherance of my professional education, and
the completion of this graduate p r o g r a m .
In addition,

I would like to thank the other m e m ­

bers of the guidance committee, Dr. Janet Alleman, Dr.
Richard Marquard, and Dr. Donald Olmsted for their encour­
agement, understanding, and assistance.

Their help was

extended in unstinted amounts, not only in the completion
of this w r i t i n g , but from the beginning of my doctoral
program at Michigan State University.
To all of the many fine members of the College of
Education and especially those directly related to mathe­
matics education and the Elementary Intern Program, credit
must be given for stimulating and enriching my under­
standing of Education.

A special thanks is extended to friends and
acquaintances in Pontiac.

Thanks go to Larry, Margaret,

Doug, Susan, Mike, Penny, and all of the kids for granting
the writer the opportunity to relax and maintain a less
than completely serious frame of mind throughout the com­
pletion of my graduate program.
A special thank you to Edward Ajoian, who will never
receive a formal Ph.D., but who none the less remains the
"smartest man I k n o w . ”
The Eisenhour Construction Company of East Lansing
must be thanked for allowing me the opportunity to finan­
cially maintain this graduate program.
Finally, recognition was earned by those who,
perhaps more than any others, realize the sacrifices and
great patience necessary to complete even such a modest
work as this.

Heartfelt gratitude is due my parents and

relatives, but a special thanks is due to my wife, Cassie,
and our boys, Damon and Jason.

i i i

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS

........................................

ii

LIST OF T A B L E S ............................................... vi
LIST OF. E X H I B I T S .......................................... vii
Chapter
I.

I N T R O D U C T I O N ..........................

1

Conceptual Framework of the Study
. . .
Significance of the Study .................
Statistical Hypotheses to be Tested.
. .
Assumptions Upon Which the Study is Based.
General Procedures of the Study
. . . .
Definition of Terms U s e d ........... 10
Scope and L i m i t a t i o n s ...............11
Organization of the S t u d y ........... 11
II.

REVIEW OF RESEARCH AND RELATED LITERATURE.

.

Influential Factors on Organization and
T e c h n i q u e s .......................... * .
Attitudes-Interests and Mathematical
L e a r n i n g ............................. .
Modern Versus Traditional Organization
and Learning......................... 21
Techniques of Modern and Traditional
Organization and Achievement
. . . .
Inquiry-Discovery Learning and
Mathematical Growth
....................
Media-Materials of Traditional and
Modern Mathematics.................. 45
A S u m m a r y ............................. .
III.

RESEARCH METHODOLOGY AND TECHNIQUES.

. . .

Introduction
..............................
Establishing a B a s e .................. 57
Identification of the S a m p l e ........ 59

3
4
6
7
9

13
14
17

29
38
52
56
56

Page

Chapter

Development of the Questionnaire
I n s t r u m e n t .................................. 63
Size and Expenditure........................... 65
Analysis of the Data.
.................... 65
S u m m a r y .........................................71
IV.

PRESENTATION AND ANALYSIS OF DATA

. . . .

72

The Data Gathering T e c h n i q u e ................. 72
Hypothesis Testing
.......................
72
S u m m a r y .........................................84
V.

SUMMARIES, CONCLUSIONS, AND RECOMMENDATIONS

.

86

Summary of the Purpose and Design
. . .
86
Summary of the Results of Hypothesis
T e s t i n g ..................................... 88
Summary of the Findings........................ 90
Conclusions..................................... 91
Implications from the Findings
. . . .
91
Recommendations for Future Investigations.
95
BIBLIOGRAPHY

...........................................

99

A P P E N D I C E S ................................................. 113

v

LIST OF TABLES

Table

Page

1 . Summary of Statistical Tests Results Hypotheses Two, T h r e e , Four
.............

74

Statistical Summary of Hypothesis O n e .

75

Statistical Summary of Hypothesis T w o .

78

Statistical Summary of Hypothesis Three
Statistical Summary of Hypothesis Four

.

80
82

LIST OF EXHIBITS

Exhibit
A.
B.

Page
School District Variables:
Size and
Expenditures ...................................

62

Statistical Design— Measures

67

vii

...................

CHAPTER I
INTRODUCTION

The enigma which so often confronts those who attempt
to improve educational practices is the wide divergence
between theoretical ideas of effective teaching strategies,
as included in the literature and current research, and
the beliefs of classroom teachers relating to appropriate
instruction with a particular group of learners.

Perhaps

in no other professional vocation, does there exist the
volume of research as exists in education.

Perhaps further,

because of the fact that so many theories have been developed,
they are sometimes often conflicting with one another, with
the result being only a limited acceptance of any particular
theory by classroom teachers in g e n e r a l .

The complaint is

often heard from teacher-educators and school administrators,
that classroom teachers seldom utilize the findings in the
literature available to them, which in reality could have
dramatic impact upon the effectiveness of their teaching.
Assuming that the problem is crucial in the improve­
ment of education, there are direct implications for profes­
sionals of teacher-education, professionals in public school
classrooms, and all other professionals in the field of
1

2

education.

If education is going to continue to expend

vast sums of money, time, and effort for the purpose of
research, opinion or editorial writing,

in-service education,

workshops, client-institution relationships, university
course work, etc., then it is imperative,
opinion,

in the researcher's

that the findings of such research and the results

of such workshops and classes become actualized in belief
and practice among classroom teachers, to a far greater
extent than currently exists.
The advent of "modern mathematics" upon the ele­
mentary education scene was initially, and still remains,
one of the most misunderstood and unjustifiably attacked
pedagogical developments of recent y e a r s .

Unlike other

new innovations in other disciplines of elementary education,
which are recognized and utilized by only a few educators,
(let alone those outside the profession-such as parents)
modern math has become almost a household word.

Much of the

theoretical base from which current mathematics a r o s e , was
research oriented and corroborated by expert opinion.

The

typical elementary teacher was first made aware of the
"change" through in-service education which in the researcher's
experience was an ineffectual method of creating real change
in classrooms.

This "single-shot" indoctrination with

little or no follow up procedures, coupled with an existing
skepticism of research resulted in poor initial implementation.

3
At this date/ after many years of the "modern" approach/

the

extent to which "modernness" has been accepted as it con­
cerns elementary mathematics instruction appears to be
u n c lear.
The researcher views the improvement of education
as an important undertaking and with this criteria in mind
this study assumes the following purposes:
1.

To develop a definition of the

"modern mathe­

matics approach" reflecting a pooling of recent
writing and research and validated by expert
opinion.
2.

To analyze the data collected by a survey
questionnaire concerning elementary classroom
teachers utilization of validated modern m a t h e ­
matics teaching strategies/ in light of specific
variables.

3.

To determine the extent to which there is a
difference between what the theorists say and
what the teachers believe/ as relates to elementary
school mathematics.

4.

To ascertain the practices of respondents and the
occurence rate of those practices in elementary
classrooms.
Conceptual Framework of the Study

The modern approach to the teaching of elementary
mathematics is broad, generalized, and not very well defined.

4

In fact, this is part of the problem.

It is necessary to

limit the conceptual framework with which this study will
deal to specific aspects of a modern approach to instruction.
This study therefore, limits its emphasis to modern mathe­
matics as it relates to (a) teacher-student
interest;

attitude and

(b) discovery-inquiry versus expository learning;

(c) influential factors of organization;
traditional organization;
traditional approaches;

(d) modern versus

(e) techniques of modern versus

(f) media-materials of modern versus
*

traditional instruction.
Significance of the Study
There is often a gap between understandings of
elementary mathematics principles, as developed and exposed
by literature and research, and the daily practice taking
place in the elementary classroom.

It is a fair assumption

that the practices of a given classroom are formulated by
the teacher of that classroom, who has the greatest impact
upon the scope of what is to be learned by that particular
group of learners.

In the case of elementary school

mathematics, and in particular the recent development of
"modern" mathematics techniques, the attitude of the class­
room teacher was necessarily changed from one of dogmatic
computational skill development,

to a broader conceptual

framework of mathematical understandings and applications
on the part of the learner.

A significant aspect of this

5
study is to ascertain the degree of skepticism and acceptance
of modern approaches on the part of classroom teachers.

This

is not to say that all or even most of the literature and
research is applicable to the majority of classroom teachers.
It is important, however, that if in fact, modern mathe­
matics is a valuable technique in facilitating academic
growth in children to a fuller understanding of mathematics
in particular and education in general, then some change
should take place in the dissemination techniques of modern
thought, as well as a different organization of implementation
controls at the public school level.

To this point, the

significance of this study might be considered appropriate
only to existing classroom teachers, who through many months
and years of experience have developed techniques unique to
themselves.

The significance of this study also should

have relevance to the beginning or entering teacher
through new organizational methods in teacher training.
Combs1 asserts that "it is at the source of the supply— our
teacher preparation programs that review and innovation are
most critically called for if we are to bring about the
improvements we need in e d u c a t i o n ." (P. v)

He calls for a

reexamination of teacher education in the context of a
framework evolving from "our changing social needs and
purposes on the one hand, and our new understandings about

(Boston:

. W . C o m b s , The Professional Education of Teachers
Allyn and Bacon, I 9 6 M .

6
behavior and learning on the other."

(P. vi)

His belief

is particularly apropo to " m o d e m mathematics" and its
concept based understanding because he further believes
an individual's learning is affected only in the degree to
which he has discovered its personal meaning for him.
A number of other writers also see education as
needing conceptual frameworks.
(Goodlad, Howard, Smith,
2
3
Wattenberg).
Wiles expresses a similar opinion on
dissemination of educational ideas by calling for an emphasis
on theories of teaching, rather than on a particular model
or pattern of teaching.

If in fact, we create "thinking

mathematical" instructors, the result should be more
"mathematical thinking" on the part of the learner, as
opposed to responding in a programmed manner following
extensive formal analysis procedures of a particular skill.
Statistical Hypotheses to be Tested
Hypothesis I

H

There is no significant difference
° between expert opinion on the con­
sistency of a modern mathematics
program and the opinion of all the
teachers included in this study.

2

John I. Goodlad, "Next Steps in Cooperation in
Teacher Education:
By Who and for What?" Educational Research,
1962, 4 3, 223-27? E. z. Howard, "Needed:
A Conceptual Scheme
for Teacher Education."
School Review, 1963, 71, 12-26? E. R.
Smith, (ed)., Teacher Education:
A Reappraisal, (New York:
Harper and Row, 1962)? and W. wT Wattenberg, E v i d e n c e and the
Problems of Teacher Education,” Teachers College Record, 1963,
64, 374-80.
3

Kimball Wiles, "The Teacher Education we Need,”
Theory into Practice, 1967, 6, 260-65.

7
Hypothesis II

H

There is no significant difference
between what the respondents say
they believe constitutes an effec­
tive mathematics program and w h a t
they say they do in their day to
day activities in the classroom.

Hypothesis III

H

There is no significant difference
in "modernness" between teachers
of varying school district sizes.

Hypothesis IV

H

There is no significant difference
in "modernness" between teachers
from school districts that have
varying per pupil general fund
expenditures.

Assumptions Upon Which the Study is Based
1.

It is assumed that the development of current
pedagogical practices on the part of the class­
room teacher is essential to effect maximal
mathematical learning growth on the part of
the student.

2.

It is assumed that there exists a lack of
implementation on the part of classroom teachers,
of research, workshops,

literature, etc. as

they relate to elementary mathematics instruction
3.

It is assumed that the general fund per pupil
expenditure of a school district is divided
proportionally and equally across all academic
areas within the curriculum of that school
district.

8
4.

It is assumed that the attitudes of teachers
about mathematics affords a key to the indi­
cation of the type of pupil acceptance and
achievement in mathematics.

5.

It is assumed that the self-reports of teachers
and experts are valid representations of their
true feelings, belief systems, and practices
within the delimitations of any self-report
(Combs, 4 Combs and Syngg 5 ).

6.

It is assumed that the sample selected from
southwest Michigan is representative of class­
room teachers within a similar population
regardless of geographical location.

7.

It is assumed that content of elementary
mathematics can be considered dichotomous with
instructional strategies of elementary mathe­
matics .

8.

It is assumed that the analysis of variance
correlation coefficients are appropriate
statistical treatments for this exploratory
s t udy.

4

Op. c i t .

~*A. W. Combs and D. Snygg, Individual Behavior:
A Perceptual Approach to Behavior, (New York:
Harper~and
Row, 1^59).

9
General Procedures of the Study
In order to investigate the preceding hypotheses,
the researcher conducted a mailed questionnaire technique
among a large number of public elementary classroom teachers.
The items included on the questionnaire were drawn from
recent literature and research as being part of a modern
approach to mathematics instruction.

The items were validated

by a panel of mathematics educators from the various state
institutions of higher learning in Michigan.

After field

testing the original instrument, so as to allow the researcher
the opportunity to refine the inquiry technique, the final
copy of the questionnaire was sent to randomly drawn class­
room teachers, as well as back to the same panel of experts.
This allowed the researcher the opportunity to compare
classroom teacher perceptions and beliefs about modern
approaches to elementary mathematics with expert's beliefs
on the same pr i n c i p l e s .
The study also sought to ascertain if there was
any difference in "modernness" when teachers in districts
of varying wealth and size were compared.

Each district

selected for inclusion in the study was chosen on the
basis of geographic location, per pupil expenditure, and
enrollment size.

This allowed for a population of districts

to be chosen for statistical purposes.

10
Upon the complete collection of the data as returned
by the questionnaire, statistical analysis was performed
for the purpose of testing the hypotheses stated earlier.
Definition of Terms Used
Attitude— is defined as the degree of positive or
negative affect, or feeling, associated with some psychological
or real object.
Base— is defined as selected items of mathematical
instruction in elementary school mathematics,

that are drawn

from literature and research and validated by expert opinion.
Elementary Teacher— is defined as a classroom
instructor in grades Kindergarten through six in any public
school selected for inclusion in this study.
Expert— is defined as a member of a University staff
in the field of elementary mathematics education in the
State of Michigan, or other individual who has exhibited
high competance in the field.
Client-institution relationship— is defined as the
situation in which an individual teacher or a single school
district utilize personnel from either public or private
institutions, such as a University,

for the purposes of

consultation on a particular concern.
Modern Mathematics Approach— shall mean the approach
to mathematics organization and understanding reflected
in the base of this study.

11
Modernness— is defined as the numerical mean score
of the means of beliefs and practices, as established by
responses to the b a s e .
Large--is defined as a school district with more
than 10,000 K-12 students.
Mid-size— is defined as a school district with more
than 2,500, but less than 10,000 K-12 students.
Small— is defined as a school district with less
than 2,500 K-12 students.
High Expenditure— is defined as a school district
spending more than $895 per pupil in general fund expendi­
tures .
Middle Expenditure--is defined as a school district
spending more than $800 per pupil in general fund expendi­
tures, but less than $895.
Low Expenditure— is defined as a school district
spending less than $800 per pupil in general fund expendi­
tures .
Scope and Limitations
This study is designed to explore the beliefs and
practices of elementary classroom teachers as they relate
to mathematics instruction and organization,

and to compare

those beliefs and practices with a group of mathematics
education experts.

It further compares the beliefs and

practices of the teachers among two selected variables—

12
size and wealth of district.

This will allow the researcher

to test the widely voiced belief that rich-large or richmid-size school districts are more "modern" in their
approach to mathematics than poorer-small districts.
Organization of the Study
The study is composed of five chapters.

Chapter I

presents an introduction and overview of the problem and
the study, along with a rationale for undertaking the study.
The second chapter contains the related research on the
topic of mathematics education, and it is from this re­
search that the base was developed.

The third chapter

describes the research procedures of the study, with
Chapter IV being devoted to a statistical analysis of the
returned data.

A summary, conclusion, implications and

recommendations are presented in Chapter V.

CHAPTER II
REVIEW OF RESEARCH AND RELATED LITERATURE

The research and literature related to current
elementary mathematics education techniques, strategies,
and organizational patterns is extensive.

In order to

select the literature and research most pertinent to this
study, careful attention was given to research that involved
itself with the aspects of elementary mathematics contained
in the conceptual framework of this study.

The importance

of this related research cannot be overemphasized, due to
the unique nature of the way it was utilized in this study.
The base of this study, as defined in Chapter I, was taken
from a direct synthesis of this research, which was in turn
validated by experts in mathematics education, and then
submitted in a questionnaire format to the random sample of
elementary classroom teachers.
This chapter presents both negative and positive
research and literature on each of the aspects of elementary
mathematics that were part of the study.

Because the base

of this study may be debatable among educators, it was felt
that both pro and con arguments and points of view should
be included in the study, so as to allow fair representation

13

14

of all approaches to the topic.

The concepts for discussion

in this chapter were selected because they represent the
major concerns of current elementary mathematics thinking,
as distinguished from organizational patterns and inodes of
thinking that are of a more traditional point of view.
With this as the criteria in mind, the chapter is divided
into concerns of modern mathematics and the research and
literature of each discussed separately.
Influential Factors on Organization
and Techniques
Education remains a profession that demands and
receives a variety of inputs for its effectiveness.

The

dissemination of academic principles, as well as sociological,
moral, and philosophical ideas to a group of learners, each
of whom is characterized by his/her uniqueness and individ­
uality, over a period of years through formal school experi­
ences, is a fine and difficult art.

Inherent in that d i s ­

semination is the uniqueness and individuality of the educa­
tor, who is continually bombarded with mechanisms of change
in the form of literature, in-service education, personal
experience, etc.

These factors of influence have an effect

upon the educator as well as the learner, and for this
reason the relationship between influential factors and
achievement among learners is a most important consideration.

15
Reid^ In a study of sources of information used by teachers,
cited professional publications, professional preparation,
and colleagues in that order, as having the greatest input
for teachers.
Gibney 2 looked at demographic factors and the
influence they had on mathematics understandings of elementary
school teachers, and concluded that there were no significant
differences among teachers when they were grouped by size
of community in which they were teaching;

but that there

was a difference over the variable of experience.
As will be demonstrated later in this chapter,

the

attitude of the teacher can be considered an important
factor in pupil achievement; and the factors of influence
will mold the attitude.
As an example, Hollander 3 in a
study

of incentives to arithmetic achievement among fifth

and sixth grade learners,

found that pupils worked faster

when told they could earn a candy bar if they improved
their scores on a second test, and with greater accuracy
when told they had performed exceptionally well.

On the

other hand, those reproved by being told their scores were

^Isaih Reid, Diffusion of Innovations in Elementary
S c hools: A Study of the Primary Sources of Information Used
by Teachers, (Cornell University, 1969), DAO 30A:3697-3698,
March 1970.
2

Thomas C. Gibney, et a l ., What Influences the Mathe­
Elem.
Sch. Journal 70:367-372, April 1970.
3
Elaine Kind Hollander, The Effects of Various
Incentives on Fifth and Sixth Grade Inner-City Children's
Performance on an Arithmetic Test, (The American University,
1968) Diss Abst. 29A:1130, October 1968.
m a t i c s Understanding of Elementary School Teachers,

16
very low, attempted fewer items and made more errors than
were made under any other condition.
Other writers and researchers have lessened the
importance of the teacher upon mathematics achievement
in elementary schools and concluded that other factors
were of most significance.
Wrigley 4 for example, concluded
that high intelligence was the most important single factor
for success in mathematics.

Unkel

5

found that socio­

economic status has a significant effect on achievement
in elementary mathematics, at all intelligence levels.

. .

as economic level increased so did the achievement in
mathematics.

In a similar study done later, McKee6 found

that elementary mathematics achievement was not affected
by the social level of community so much, as it was purely
by the economic level.

It appears obvious that there do

exist factors that influence learning,

that direct their

attention to both teachers and learners, and that even
though no consensus can be reached as to which ones are
4
Jack Wrigley, The Factorial Nature of Ability in
Elementary Mathematics, British Journal of Educational
Psychology, 28:61-78, February 19 68.
5
Esther Unkel, A Study of the Interaction of
Socioeconomic Groups and Sex Factors with the Discrepancy
Between Anticipated Achievement and the Actual Achievement
in Elementary School Mathematics, Arithmetic Teacher, 13:
662-670, December 1966.
6Marjorie A. McKee, The
Success Studies in Seventy-Five
A Descriptive Study of Selected
University, 1969) DAI 30A:30 59,

Components of Academic
Educable Retarded Children:
Factors, (Wayne State
March 19 70.

17
most influential#

it becomes necessary to acquaint oneself

with the potential of these influences if mathematics
learning is to be increased.
Much additional research has shown that age and
intelligence are highly related to the ability to learn
7
various specific mathematical ideas.
Westbrook noted that
the intellectual factors of reasoning and verbal meaning
were related to achievement in mathematics in grades four#
five, and six.

Meconi

g

found that pupils with high ability

were able to learn under any method he investigated, while
Ebert

9

in an older study

(194 6) found large variations in

generalization ability, depending on the mathematical concept
involved, the intelligence level of the learner, and the
visual pattern within which the concept was presented.
Attitudes-Interests and Mathematical Learning
Perhaps due to the nature of attitude development
and because attitude is a difficult entity to evaluate or
7
Helen Rose Westbrook, Intellectual Processes Related
to Mathematics Achievement at Grades Four, Five and Six,
(University of Georgia, 1965) D iss. Abst. 26:6520, May 1966.
0
L. J. Meconi, Concept Learning and Retention in
Mathematics, Journal of Experimental Education 36:51-57,
Fall, 1967.
9
Rueben S. Ebert, Generalization Abilities in
Mathematics, Journal of Educational Research 39:671-681,
May 1966.

18
test, it is not completely accepted by writers and
researchers that is has the effect upon achievement that
it might be thought to have, especially in the area of
elementary mathematics.

Caezza and W e s s 10 reported that

low correlations existed between pupil achievement in
elementary mathematics and pupil or teacher attitude
toward mathematics.
same year, C o x ^

In another study done later in the

found that pupil achievement was not

found to be affected by the teachers knowledge of or
interest in mathematics.
Pinetz, Cheikin and Bassham

12

found no significant

difference between teacher attitude and achievement among
students.

Many other authors have found similar results

concerning teacher attitude and student performance.

Some

John F. Caezza, A Study of Teacher Experience,
Knowledge of and Attitude Toward Mathematics and the
Relationship of these Variables to Elementary School Pupils
Attitudes Toward and Achievement in Mathematics, Diss. Abst.
31A:921-922, September 1970; Roger G. W e s s , An Analysis of
the Relationship of Teachers Attitudes as Compared to Pupil
Attitudes and Achievement in Mathematics, Diss. Abst. 30A:
3844-3845, March 1970.
^ L i n d a A. Cox, Study of Pupil Achievement in
Mathematics and Teacher Competence in Mathematics, (University
of Kansas, 1970), DIA 31A:2767-2768, December 1970.
12

Mildred C. Pinetz, "The Relationship Between Teacher's
Attitudes and Effectiveness in the Classroom", Diss. Abst.
24:234 0, No. 6, 1963; Martin L. Cheikin, "An Investigation
of the Effect of Measured Teacher Attitude on Selected Eighth
Grade Students", Diss. Abst. 28:4042-A, No. 10, 1967; Harrell
Clark Bassham, "Relationship of Pupil Gain in Arithmetic
Achievement to Certain Teacher Characteristics", Diss Abst.
21:1839, No. 7, 1960.

19
researchers have accepted the findings of Pinetz and others
and addressed themselves to pupil attitude and its relation­
ship to achievement.
Research:

In a publication entitled Using

A Key to Elementary School Mathematics, written

in 1970 and edited by Marilyn Suydam at The Pennsylvania
State University, the following was concluded:
Many people believe that mathematics is disliked
by most pupils. . .It is true that in some surveys
a significant proportion of pupils rated mathe­
matics as the least liked subject. . .however,
there is no consistent body of research evidence
to support the popular belief that there is signi­
ficant positive relationship between pupil attitudes
toward mathematics and pupil achievement in mathe­
matics.
Relating this statement to the recent developments
of modern mathematics, Abrego 14 found no relationship between
achievement and pupil attitude in either traditional or
newer mathematics programs for elementary pupils.

Though

this seems to be a fairly large and representative sample
of the research and literature that negates the causal
relationship of attitude and achievement, there is equally
large and representative literature and research of the
opposite view.
13

Marilyn N, Suydam, Interpretive Study of Research
and Development in Elementary School Mathematics, A Closer
View. . .Attitudes and Interests, Center for Cooperative
Research with Schools, (The Pennsylvania State University,
1970), p. 4.
14
Mildred B. Abrego, Children's Attitudes Toward
Arithmetic, Arithmetic Teacher 13:206-208, March 1966.

20

Whipkey^

found admittedly a small, but in his opinion

important relationship between mathematical attitude and
achievement.

P h i l l i p s ^ found that the attitudes toward

mathematics on the part of pupils was significantly related
to the attitude of their previous grade teacher, and that
in fact, the type of teacher attitude encountered by the
pupil was significantly related to his present attitude
and achievement.
Sex differences may play an important role in estab­
lishing attitude and achievement relationships.

Boys seem

to prefer mathematics slightly more than do girls, especially
toward the upper elementary grades.
Chase and Wilson;
17
Stright and Neale
found that attitudes and arithemtic
achievement are significantly correlated for boys only.
Kenneth L. Whipkey, A Study of the Interrelation­
ship Between Mathematical Attitude and Mathematical Achieve­
ment, (Case Western Reserve University, 1969) DAI 30A:380 3,
March 19 70.
*6Robert B. Phillips, Teacher Attitude as Related to
Student Attitude and Achievement in Elementary School
Mathematics, Diss Abst. 30A:4316-4317, April 1970.
17
W. Linwood Chase and Gilbert Wilson, Preference
Studies in Elementary School, Journal of Education, 140:2-8,
April 1968; Virginia M. Stright^ A Study of the Attitudes
Toward Arithmetic of Students and Teachers in the Third,
Fourth and Sixth Grades, Arith. Teacher 7:280-286, October
1960; Daniel C. Neale, et a l ., Relationship Between Attitudes
Toward School Subjects and School Achievement, Journal of
Educational Research 63:232-237, January 1970.

21
Two authors address themselves to the question of
pupil interest and the modern mathematics approach.

Feldhake18

reported that high achieving pupils found new mathematics
programs more interesting than did low achievers, and
19
Dutton
observed that fewer pupils were afraid of mathe­
matics today, enjoyed it more, and achieved better on
mathematical problems than ten years ago.

Dutton went on

however, to discover that fewer pupils today see the prac­
tical use of mathematics than ever before.
In summation, teacher and pupil attitude toward
elementary mathematics and the resultant pupil achievement
may or may not be significantly correlated on the basis of
current research and literature.
Modern Versus Traditional
Organization anct learning
The research completed on the topic of comparisons
between newer or modern mathematics and older or traditional
methods of elementary mathematics instruction,
varied.

is large and

The variables one researcher seeks to control in

18 Herbert J. Feldhake, Student Acceptance of New
Mathematics Programs, Arithmetic Teacher 13;14-19, January
1966.
19
Hilbur H. Dutton, Another Look at Attitudes of
Pupils Toward Arithmetic, Elem. School Jour. 68:265-268,
February 1968.

22

his/her study, are of little importance to another researcher,
and the conditions in which a particular author or editor
finds himself vary greatly oftentimes from conditions found
by another.

The literature and research included in this

study reflects this dichotomy in research findings on the
topic of modern versus traditional organizations and their
effects.
Many authors and researchers have found that no
differences exist in achievement and attitude on the part
of the learner regardless of the type of instruction he
receives.
Hendrickson 20 found no differences in achieve­
ment between groups using a mathematics laboratory, an
enrichment problem technique, or a conventional approach,
though all groups gained.

The study further reports that

attitude improvement was found in the conventional group
more than any other.
graders, Bartel

21

In an experimental study among fourth

established a control group in which

instruction was traditional in nature, was not individual­
ized, and did not contain "new mathematics” content.

He

also established an experimental group in which instruction
was highly individualized and included "new mathematics"
20 Arthur D. Hendrickson, A Study of the Relative
Effectiveness of Three Methods of Teaching Mathematics
to Prospective Elementary School Teachers (University of
Minnesota, 1969) DAI 31A:1117, September 1970.
21

Elaine V. Bartel, A Study of the Feasibility
of an Individualized Instructional Program in Elementary
School Mathematics (University of Wisconsin, 1965) Diss
Abst. 26:5284, March 1966.

23
content.

The finding was that no difference in achieve­

ment was found when evaluation was measured by a standardized
test.
The question of what the student learns as a result
of modern programs as compared to traditional p r o g r a m s,
is the confounding variable in many studies.
literature conducted by Suydam

22

In a review of

it was found that generally,

modern programs are as effective as traditional programs in
developing traditional mathematical skills.
In an earlier
23
study, Payne
used an experimental approach and reported
24
similar findings.
Wright
went a step further and stated
that he found no significant differences in learning tradi­
tional concepts between those using a traditional program
and two separate modern programs, but that those individuals
in the modern programs achieved higher scores on a test of
25
modern concepts.
Gardner
made the assumption that
standardized arithmetic tests could measure program effective­
ness and on that basis concluded that there existed no
22

Marilyn N. Suydam, Interpretive Study of Research
and Development in Elementary School Mathematics, Ove rview. . .
Planning and for Instruction. Center for Cooperative Research
with Schools. (The Pennsylvania State University, 1970), p. 2.
23
Holland Payne, What About Modern Programs in Mathe­
matics? Mathematics Teacher 58:422-424, May 1965.
24
R. E. Wright, Something Old, Something New, School
Science and Mathe 70:707-712, November 1970.
25
Harvey F. Gardner, A Comparison of the Modern and
Traditional Curriculums and their Effects on Standardized
Test Scores (Southern Illinois University, 1969) DAI 30A:
2847-2848, January 1970.

24
significant growth differences between modern and traditional
mathematics curriculums in grades three and four.
Hungerford 26
27
and Graft
studied fourth, fifth, and sixth graders, some of
whom were taught utilizing Science Mathematics Study Group
(SMSG) materials and some using conventional text materials
and concluded that the SMSG group understood principles of
multiplication better than the conventional groups, yet no
significant difference was found in computation.

It was

shown that the data significantly favored the non-SMSG group
on a test of conventional arithmetic and the SMSG group on
a test of contemporary mathematics.
Crowder

28

In a more current study,

worked with fourth, fifth, and sixth graders and

concluded that pupils in the modern program achieved above
the normal rate of achievement in arithmetic computation
and concepts, but not in applications.
Questions often arise as to the necessary organizational
patterns and classroom management procedures that are suppos­
edly inherent in modern elementary mathematics instruction.
26

Ann D. Hungerman, Achievement and Attitude of SixthGrade Pupils in Conventional and Contemporary Mathematics Program!
Arith. Teacher 14:30-39, January 1967.
27
William D. Graft and Arden K. Ruddell, Cognitive
Outcomes of the SMSG Mathe Program in Grades Four, Five and
Six, Arith Teacher 15:161-165, February 1968.
28
Richard P. Crowder, The Effect of a Modern Mathematics
Program on Achievement in Arith and on Intelligence Test
Scores of Children in Grades Four thru Six, (University of
Arkansas) Diss. Abs. 30A:2901, January 1970.

25
As in -the case of discussing the effects of modern and
traditional mathematics instruction, organization of the
instruction and the classroom, has been similarly researched
with little or no concensus readily apparent to the reader.
29
As an example, Sherer
found that low achieving pupils a.n
grades three through seven, taught by author developed
materials, using instructional aids such as drawings,
counters, number lines, and charts, showed significantly
greater gains in achievement than those taught by a tradi­
tional approach.

Lindgren30 on the other hand, reported

that no significant differences in achievement existed for
the learner between conventional teaching and any other
organizational pattern.
Further research on modern versus traditional
organization leaves the reader free to use his/her own
belief system in accepting or rejecting modern or tradi31
tional ideologies. Ferney
looked at the Program for
29

Margaret T. Sherer, A n Investigation of Remedial
Procedures in Teaching Elementary School Mathematics to Low
Achievers, Diss. Abst. 28A:4031-4032, April 1968.
■^Richard Francis Lindgren, A Comparison of Team
Learning with Learning Through Conventional Teaching as
Methods in Teaching Arithmetic Reasoning in Grades Four
and Five, (The University of Connecticut, 1965) Diss. Abst.
28A :3369, March 1968.
31
Gary A. Ferney, An Evaluation of a Program for
Learning in Accordance with Needs, Diss. Abst. 30A:4327,
April 1970.

26
Learning in Accordance with Needs

(PLAN) organizational

structure, and found that fifth graders not using PLAN
achieved significantly higher on arithmetic reasoning tests
than the PLAN group, and that girls scored higher than boys
on PLAN organization*

From this he concluded that PLAN is

most appropriate to the learning styles of girls.

Also on

the topic of developmental reasoning abilities, Moore and
32
Cain
used a standardized test of logical reasoning and
another of creative thinking, and claim that there is
support for the idea that new approaches to teaching mathe­
matics help develop reasoning, and that some of the creative
abilities tested were fostered by new mathematics programs.
33 .
In 19 69, Carry and Weaver
reviewed the National
Longitudinal Study of Mathematical Abilities

(NLSMA) which

had completed extensive studies of mathematics textbooks
and their relationship to the modern versus traditional
dilemma.

They compared achievement patterns based on

thirty-eight measures over a three year period involving
six textbook series— three of which were modern and three
of which vrere conventional.

The conjecture of the study

32William J. Moore and Ralph W. Cain, "The New Mathe­
matics and Logical Reasoning and Creative Thinking Abilities",
School Science and Mathematics 68:731-33, November 1968.
33
Ray L. Carry and J. Fred Weaver, Patterns in Mathe­
matics Achievement in Grades Four, Five and Six:
X-Population
NLSMA Reports, No. 10.
(Stanford, California:
The Leland
Stanford Junior University, 1969).

27
was that achievement patterns would be more similar within
textbook classifications than across.

This hypothesis

was not supported at the completion of the study.

NLSMA

found large differences in achievement among m o d e m textbook
groups and also among conventional textbook groups.
One of the major tenets of the modern program is
that students should have options in their instruction and
because they have a voice in the selection p r o c e s s , achievement should increase.
Snyder 34 however, found no significant
difference in achievement between learners who were allowed
to select the mathematical topics they would study and those
who could choose from a limited assigned option list.

Both

groups gained more in reasoning tests, but less on skill
tests than a third group receiving "regular" instruction.
In a report on low achievers in mathematics,
35
DeVenney
found that special programs developed for low
achievers did not improve p u p i l s 1 computational skills to
as high a degree as a traditional program, but it did result
in an attitudinal change among learners from negative to
much more positive.
34 Henry Duane Snyder, J r . , A Comparative-Study of
Two Self-Selection-Pacing Approaches to Individualized
Instruction in Jr. High School Mathe, (University of
Michigan, 1966) Diss Abst. 28A:159-160, July 1967.
35 William S. DeVenney, Final Report on an Experiment
with Junior High School Low Achievers in Mathematics SMg?
Reports,
No. 7 (Stanford:
Stanford University Press, i&?9),
1
p. 53.
■■

*

■

■

11

28
Hoover

36

worked with learner independence among

fourth graders and found no meaningful differences for
pupils using programmed self-paced, write-in materials
and those having a conventional hardbound text instructional
mode.
Ward

37

addressed himself to departmentalized versus

self-contained organization.

In a study of fourth, fifth,

and sixth graders he found no difference in achievement
between pupils in self-contained classes with teachers of
limited in-service training, and in departmentalized classes
where intensive in-service activities were conducted with
the teacher.
The diversity of the literature and research as it
relates to the effectiveness of modern versus traditional
instruction, would seem to indicate a need for further
exploration in this area.

Although findings are conflicting,

the literature seems to point to no significant difference
in cognitive gain on the part of the learner regardless of
the teaching method, or pattern of organization.
^ G a i l L. Hoover, Learning to Learn Mathematics
Independently:
A Study at the Fourth Grade Level (Syracuse
University, 1969) DAI 31A:293, July 1970.
37

Paul E. Ward, A Study of Pupil Achievement on
Departmentalized Grades Four, Five, and Six, (University
of Georgia, 1969) DAI 30A:4749, May 19 70.

29
Techniques of Modern and Traditional
Organization and' Achievement
For a long period of time it was doubted that
children needed to understand what they learned in e l e ­
mentary mathematics.

It was enough simply,

developed high degrees of skill.

if they

It was believed that to

take time to give detailed exploration and develop under­
standing was wasteful, as well as confounding to the learner.
Then came the realization that if instruction were to be
meaningful, and if the teacher was concerned about increas­
ing the number of students who attained a high level of
skill, then certain things were to be gained if the content
of mathematics made sense to the learner.

The problem

remained one of defining meaningful instruction.
In 1955,
38
Dawson and Ruddell
summarized many studies by other authors
on the topic of meaningful mathematics instruction and
concluded that meaningful instruction leads to greater
retention, greater transfer, and an increased ability to
solve problems independently.
Since the Dawson and Ruddell study of over fifteen
years ago, many approaches or techniques have been attempted
to aid in developing the meaningfulness of elementary
mathematics instruction.

One such technique involves

38 Dan T. Dawson and Arden K. Ruddell, The Case of
the Meaning Theory in Teaching Arithmetic, Elem. Sch. Jour.
55:393-399, March 1955.

30
improving pupil attitude and therefore increasing pupil
achievement

(it is assumed) by establishing classroom

criteria for selection of the various mathematical concepts
that are to be studied.
Greathouse 39 experimentally selected
three groups of learners and taught one with a conventional
drill-computation method, a second with a group-selection
or group-oriented meaningful method, and a third with an
individually-selected or individually-oriented technique.
His findings were that the group-oriented method resulted
in significantly greater achievement than the individuallyselected, and that both the individual and group
selection groups scored better than the drill-computation
group.
Gleason 41 conducted a study which revealed no
significant difference in factual knowledge gain when
pupils were either self-directed or teacher-led.

Four

self-directed study techniques were tried on one hundred
twenty eight students enrolled in six fifth grade classes,
while the control group consisted of one hundred thirty two
fifth graders in seven classes of a more traditional nature.
39

Jimmie Joe Greathouse, An Experimental Investiga­
tion of Relative Effectiveness Among Three Different Arithmatic Teaching Methods (The University of New Mexico, 1965)
Diss. Abst. 26:5913, April 1966.
40Walter P. Gleason, An Examination of Some Effects
of Pupil Self-Instruction Methods Compared with the Effects
of Teacher Led Classes in Elementary Science on Fifth
Grade P u pil s ” , D i s s . Abst. 27:1656; No. 6, 1965.

31
The upper I.Q. subjects in the teacher-led group did better
in general knowledge than the upper I.Q. students in the
self-investigation group.

On the other hand, students in

the self-investigation group liked studies better than
students in the teacher-led group.
The question of individualization in elementary
mathematics has been widely researched and discussed.
41
Wolff
found that among third graders in Oregon, an
individualized approach to arithmetic resulted in no
significant differences in achievement when compared to a
non-individualized approach.

In any discussion of individual­

ization, careful attention must be paid to exactly how the
concept of individual differences is being defined and
exactly what it is that the researcher or author is
attempting to do.

What is considered as an individual

technique with one researcher does not fit the definition
of another.

The clear understanding that can be drawn from

research is however,

that some type of differentiation be

made in instruction between and among learners in a given
classroom.

The techniques can vary, but the concept is

well defined.
41Bernard Ryan Wolff, An analysis and Comparison of
Individualized Instructional Practices in Arithmatic in
Graded and Non-Graded Elementary Classrooms in selected
Oregon School Districts (University of Oregon, 1968) D iss.
Abst. 2 9 A :4397, June 1969.

32
In 1959 after extensive review of research,
42
Bernstein
concluded that differentiated (individualized)
instruction was more effective than total class instruction,
for the teaching of mathematics both of a general and
remedial nature.
Singh 43 suggested that the use of individualized
enrichment homework resulted in significant achievement
on most arithmetic subtests among fourth, fifth, and sixth
graders.
Looking at the idea of individualization in total,
44
Suydam
concluded that all in all, there is little sub­
stantial evidence to date that indicates that programs of
individualized instruction will lead to higher levels of
pupil achievement, when compared with non-individualized
programs.

She speculates that perhaps how each teacher

teaches is the most significant factor and obscures any
significant differences between the two types of programs.
It is this uniqueness of teacher instructional
style that Suchman 4 5 addressed himself to.
Suchman conducted
42

Allen Bernstein, Library Research— A Study of
Remedial Arithmetic, School Science and Mathematics, 59:
185-195, March 1959.
43
Jane M. Singh, An Investigation of the Effect of
Individualized Enrichment Homework upon the Academic Achivement of Children in Grades 4 thru 6 (Arizona State University,
1969) DAI 30A:
3203-3204, February 1970.
44
Suydam, o p . c i t .
45
Richard Suchman, "The Illinois Studies in Inquiry
Training," Journal of Research in Science Teaching, 2:320-32,
1964.

33
the Illinois Studies in Inquiry Training Project at the
University of Illinois.

He identified what he feels are

necessary conditions for self-directed, individualized
learning to occur in the elementary classroom.

These

conditions include:
1.

The need for some kind of focus for the
learners attention.
The need to present a
problem of stimulus that cannot be readily
assimilated at that moment by the learner.

2.

The need for a responsive environment.
The
child needs to have an environment where,
when seeking data or information, he receives
from the teacher only the data which he needs
for testing the theory with which he is engaged.

3.

The need for some condition of freedom.
An
external freedom which enables the child to
seek out desired data and information and
to acquire it at any rate, and in any sequence
that child wishes.
He can satisfy his own
needs for gathering the kinds of information
he wants and exercise freedom in trying out
new ideas.
In other words, the child becomes
involved in inquiry and in an individualized
approach to learning and is stimulated by the
p rocess.

Suchman envisions the teachers role as making the
learner aware of an individual-inquiry process.

Mathematics

educators agree that such a role involves the presentation
of material in such a way that the pupil gains skill in
working independently in similar situations.

The pupil needs

to know

what questions to ask, when to ask them, and where

to find

the

answer.

A teacher's question should not be

designed to discover whether the pupil knows the answer,
but to exemplify to the pupil the types of questions he must
ask of the materials he studies and how to find the answers.

34

The individualization of mathematics instruction requires
this type of sensitivity on the part of the teacher.
Individualized instruction can lead to inquiry and further
student motivation and as Schwab 46 hypothesized, "the art
of conducting discussions is not easy, but it is through
discussion that the skill of individualized inquiry can
be conveyed."
Another technique commonly thought of as part of
a modern mathematics program is the emphasis on grouping
47
of learners for instruction.
Koontz
found that fourth
graders who were heterogeneously grouped by achievement
achieve significantly higher scores in mathematics than
48
those homogeneously grouped by achievement.
Dewar
looked
at the heterogeneous classroom and with an extensive treat­
ment in sixth grade classes, concluded that providing
three intraclass groups in elementary mathematics benefited
high and low achieving learners more than did total-class
instructions.
46

Joseph J. Schwab, The Teachxng of Science as
Enquiry, (Cambridge:
Harvard University Press, 196?)
p. 62.
47 William F. Koontz, A Study of Achievement as a
Function of Homogenous Grouping, Journal of Experimental
Ed, 30:249-253, December 1961.
48
John A. Dewar, Grouping for Arithmetic Instruc­
tion in Sixth Grade, Elementary School J o u r n a l , 63:266-269,
February 1963.

35
When grouping is done on the basis of overall academic
ability, a different set of findings is discovered.
Provus 49
and Balow and Ruddell 50 found that ability grouping was
especially effective for those learners with high I.Q. *s.
Balow and Ruddell went further however, and concluded that
the "decreased-range" grouping was more effective than either
heterogeneous or homogeneous achievement grouping for all
51
students.
Savard
found just the opposite from Provus and
Balow-Ruddell.

He found that ability grouping tended to be

effective for lower ability pupils and of less advantage for
upper ability pupils.
To take the discussion full circle,
52
Balow and Curtin
reported that grouping by ability did
not significantly reduce the range of achievement any more
than any other type of grouping.
49
Malcolm M. Provus, Ability Grouping in Arithmetic,
Elem. Sch. Journal, 60:391-398, April 1960.
50Irving H. Balow and Arden K. Ruddell, The Effects
of Three Types of Grouping on Achievement, California
Journal of Educational Research, 14:108-117"^ May l£63.
51
. .
William G. Savard, An Evaluation of an Ability
Grouping Program, California Journal of Educational
Research, 11:56-60"^ March 1960.
52
Bruce Balow and James Curtin.
Ability Grouping
of Bright Pupils, Elementary School Journal, 66:321-326,
March 19 66.

36
Davis and Tracy

53

reported in their article that

pupils in grades four, five and six who were taught in
self-contained heterogeneous classrooms, scored significantly
higher on factors such as verbal and quantitative ability,
self-concept, anxiety, and attitude,

than did those grouped

by both ability and achievement across classrooms at each
grade level.
Terry

54

took the self-contained idea further and in

a study of sixth grade students in New Y o r k ,

reported no

significant differences in achievement was found between
groups taught by specialists in mathematics education and
those in a self-contained room with a regular teacher.

As

in the case of the effects of individualization, as well
as various organizational schemes already mentioned earlier
in this chapter,

the effects of grouping are similarly hard

to defend or reject due to the vast discrepancy of research
findings.
Some other techniques of modern organization demand
attention as well as grouping and individualization.
question of peer instruction,

The

that is, allowing students to

5 3O. L. Davis and Neal H. Tracy, Arithmetic A c h ieve­
ment and Instructional Grouping, Arithmetic Teacher, 10:12-17,
January 196 3.
54
Arthur F. Terry, An Investigation of the Relation­
ship Between the Mathematics Achievement of Sixth Grade
Students Under the Direction of the Teacher in the SelfContained Classroom and the Teacher Specialist, (The Univer­
sity of Rochester, 1969) DAI 30A:3355-3356, February, 1970.

37
instruct other students, was experimentally tested among
55
third graders by Ackerman
in 1970.
He found that pairing
low achieving third graders with either high or low achiev­
ing sixth graders as tutors, resulted in computation scores
which were significantly higher than the scores of those
third graders who simply talked to sixth graders on nonmathematical activities or who had no tutor contact at all.
56
Gray
looked at the effects of discipline integra­
tion and its relationship to acquisition and retention of
mathematical behaviors in grade five.

His conclusion was

that a higher achievement level was attained when unspecified
mathematical content was integrated with unspecified science
content, than when each was presented independently.
Morrell 57 conducted an experimental study with
fifth graders that compared four methods of teacher feed­
back from worksheets in arithmetic.

His finding

was that

pupils who corrected errors or who were retaught frequently
missed problems, either with or without written comments,
55

Arthur P. Ackerman, The Effect of Pupil Tutors on
Arithmetic Achievement of Third Grade Students, P i s s . A b s t .
31A:918, September 1970.
■^William L. Gray, The Effects of an Integrated
Learning Sequence on the Acquisition and Retention of
Mathematics and Science Behaviors in Grade Five, (University
of Maryland, 1970) DAI 31A:200 4, November 1970.
57 James E. Morrell, A Comparison of Four Methods of
Instructional Teacher Feedback from Practice Worksheets on
Fifth Grade Arithmetic (LeHigh University, 19 70) DAI 31A:
2 794, December 1970.

38
retained more than pupils who only had written comments on
their practice work.

In addition, Morrell found that high

achieving boys scored higher on the practice work than high
achieving girls, but that girls retained better at medium
and low achievement levels.
In summarizing the literature and research on
techniques of modern and traditional mathematics for
elementary school,
ness.

the important variable is teacher unique­

All of the literature validity and all of the effec­

tiveness of the research is contingent upon the ability of
the classroom teacher to disseminate knowledge among learners
in a manner that is compatable with individual learning
styles as found then and only then in that classroom.
Whether a technique is modern or n o n —modern is not the
crucial issue.

The manner of implementation by the class­

room teacher is critical in the opinion of researchers and
writers, either stated or implied.
Inguiry-Discovery Learning and
Mathematical Growth
The significance of the inquiry-discovery learning
style can be seen in the extent to which it is included
in elementary mathematics textbooks and supplemental
materials,

in use in elementary classrooms today.

The

theory behind the inquiry method as opposed to more formal
expository techniques is that, if the learner is interested

39
enough in a topic he will inquire into it further and hence
"learn" more about it, than if he simply is told about a
topic with little or no attention paid to whether or not
the learner is genuinely interested.

This cycle of interest

breeding inquiry, which breeds greater academic growth, which
then in turn produces further inquiry,

is basic to the

pedagogical belief in discovery learning.
Support for the inquiry-discovery technique comes
from many sources.

Fortson 5_8 found that when working with

youngsters of a very young age, those children experiencing
an instructional approach that involved multiple-stimuli
activities in a discovery mode, were significantly higher
on tests of readiness than those using a non-discovery
approach.

It was further noted that creativity among

discovery learners was much more prevalent than among
learners of other groups.
In a well documented article on expository versus
discovery modes and problem solving capabilities, Scandura 59
found that pupils taught by a discovery method were both
better able to handle problem tasks and appeared much more
self-reliant than groups taught by non-discovery techniques.
58
Laura R. Fortson, A Creative-Aestetic Approach to
Readiness and Beginning Mathematics in the Kindergarten,
(University of Georgia, 1969) DAI 30A:5339, June 19 70.
59
.
Joseph M. Scandura, An Analysis of Exposition
and Discovery Modes of Problem Solving Instruction, Journal
of Experimental Education 33:148-159, December 1964.

40
Other studies and articles dealing with the relative
effects of inquiry-discovery teaching, as opposed to other
inodes of instruction, cover a wide realm of concerns.
According to studies by Kellogg, Means and P a r k s a s
well as o t h e r s , teachers can enhance their own problem­
solving skills through practice and as a result they may
become more proficient in setting the stage for inquirydiscovery learning in the classrooms.
With the interest in modern mathematics upon concept
development as well as skill in computation, a study by
F l e c k m a n ^ is particularly pertinent.

It was reported that

fifth and sixth graders taught by a "guided discovery"
method learned more concepts than classes taught by con­
ventional textbook p r o c e d u r e s , at the same time that
computation ability was equivalent in either method.
Perhaps the greatest test of any method of instruc­
tion is in the degree of retention the child possesses of
60
Maurice G. Kellogg, "The Effect of Laboratory
Discovery Method and Demonstration Discovery Methods Upon
Elementary Science Methods Students' Ability to Analyze
and Interpret Graphs", P i s s . A b s t . 27:3345-A, No. 9, 1966;
Gladys H. Means, "The Influence on Problem Solving Ability
of Following an Efficient Model," P i s s . A b s t . 27:2892-A,
No. 9, 1966; Marshall E. Parks, "An Analysis of a Method
for Improving Science Problem Solving Ability Possessed
by Prospective Elementary Teachers", Diss. Abst. 29:3505-A,
No. 10, 1968.
^ B e s s i e Fleckman, Improvement of Learning Division
Through Use of the Discovery Method, (University of Georgia,
1966) Diss. Abst. 27A:3366-3367, April 1967.

41
what he learned through the method.

Worthen

62

conducted a

study of over five hundred fifth and sixth graders that
supported the claim of discovery method advocates.

He

concluded that the inquiry and discovery techniques result
in more learning and retention than expository methods, and
that a discovery instructional method facilitated retention
for a longer period of time.

Bassler

however, distinguished

between two types of discovery and found differing
abilities.

retention

He established an "intermediate guidance"

technique in which pupils were led to a desired behavior
through "guided discovery" with directed questions being
asked by the teacher.

The second type of instruction was

termed "maximal guidance" in which the teacher specifically
told students what they were to do and followed it by
practice on the part of the pupil.

His findings differ

from Worthen in that he reports the "intermediate guidance"
group had higher scores on a test immediately following
instruction, while the "maximal guidance" group had higher
scores on retention tests given much later.
64
Sowder
worked in
a study that was smaller than
Worthen*s or Bassler*s and
62

reported that "discovery techniques

Blaine R. Worthen, "A Study of Discovery and
Expository Presentation", Implications for Teaching, Journal
of Teacher Education 19:223-42, Summer 1968.
63
Otto
C. Bassler, Intermediate Vs. Maximal Guidance—
A Pilot
Study, Arithmetic Teacher 15:357-362, April 1968.
64
Larry Sowder, Discovery Learning:
A Status S t u d y ,
Grades 4-7 and an Examination of the Intleunce of Verbaliz­
ing Mode of Retention, Technical Report No. 99, Wisconsin
Research and Development Center for Cognitive Learning,
(Madison:
The University of Wisconsin, 1969), xv, p. 140.

42
are most efficient when used after grade five and then
only when an accuracy in computation has been developed
by the learner."
It is this prior ability concern that gives rise to
a negative attitude toward discovery and inquiry.
Although
65
Scandura
reported positively on discovery, he did observe
that it took longer for discovery taught learners to reach
the desired level of facility, especially learners of high
intellectual ability.

H a r m o n ^ went a step further and

studied just the topic of ability among learners and the
effects of discovery.

In her study of sixth graders she

concluded that "an expository approach was more effective
than an inquiry approach for problems involving concepts,
especially for children of average and high intellectual
ability."
65„
Scandura, oj>. c i t .
66
Adelaide T. Harmon, Problem Solving in Contemporary
Mathematics: The Relative Merits of Two Methods of Teaching
Problem Solving in the Elementary School, Diss. Abst. 30B:
3748, February 1970.

43
Heckman, Montague, Lisonbee and Fullerton, and
Tanner 6 7 among others, revealed that inquiry was no more
effective than other approaches when achievement was the
tested variable.
Tomlinson, May, and Adams

68

studied the variable

of retention as it relates to discovery techniques and
unlike studies previously mentioned found that inquirydiscovery methods were not significantly better than other
methodologies when retention was the criterion under
inves tigation.
6 7Maurice A. Heckman, "The Relative Merits of Two
Methodologies for Teaching Verbal Arithmetic Problems to
Undergraduate Elementary Education Majors," D iss. A b s t .
23:379 7, No. 10, 19 62; Earl J. Montague, "Using the College
Chemistry Laboratory to Develop an Understanding of Problem
Solving in Science," D i s s . A b s t . 24:2815, No. 7, 1963;
Lorenzo Lisonbee and William J. Fullerton, "The Comparative
Effect of BSCS and Traditional Biology A c h ," School Science
and Mathematics 64:594-98, October 1964; Richard T . T ann e r ,
"Expository-Deductive versus Discovery-Inductive Program­
ming of Physical Science Principles," Diss. Abst. 29:1480-A,
No. 5, 1968.
68

Robert M. Tomlinson, "A Comparison of Four
Methods of Presentation for Teaching Complex Technical
Material," D i s s . A b s t . 23:2813, No. 8, 19 62; Lola J. May,
"A Statistical”Comparison of the Effectiveness of Teaching
Per Cent by the Traditional, Ratio, and Discovery Method,"
D i s s . Abst. 26:3109, No. 6, 1965; Barbara J. Adams, "A
Study of the Retention of Information by SBCS Students and
Traditional Biology Students," Diss. Abst. 29:2599-A, No. 8,
1968.

44

There is conflicting evidence as to whether inquirydiscovery methods foster more favorable attitudes on the
part of the learner, than other techniques of instruction.
Charen and Coulter 6 9 were among those
investigators who
found that inquiry-discovery techniques fostered more
favorable attitudes, while Butts and Rawn and Inventasch

70

were among those w h o revealed that inquiry was no more
effective them any other approaches with regard to attitude
improvement.
In general, the conclusion may be made that although
inquiry seems to be no more effective than other inodes of
instruction for most educational criteria, this method is
at least equal to other methodologies for most criteria
investigated.

The key to the effect of discovery learning

is dependent upon many variables, some of which appear to
be controllable and some that are not.

The essential

criteria for its utilization is whether or not it appears
to be working for a particular learner or not; and if not,
something else should be used.
69
George Charen, "A Study of the Effect of Openended Experiments in Chemistry on the Achievement of Certain
Recognized Objectives of Science Teaching," D i s s . A b s t . 23:
2815, No. 8, 1962; John C. Coulter, "The Effectiveness of
Inductive Laboratory, Inductive Demonstration, and Deductive
Laboratory in Instruction," Journal of Research in Science
Teaching 4:185-86, 1966.
70
David P. Butts and Chester E. Rawn, "The Relation­
ship Between the Strategies of Inquiry in Science and Student
Cognitive and Affective Behavioral Change," Journal of Research
in Science Teaching 5:261-68, 1967-68; Harvey Inventesch,
irA Comparison of the Effects of Teacher Directed and Self­
directed Problem Solving on Attitudes and Understanding in
Science, " D i s s . Abst. 29:497-A, No. 2, 1968.

45

Media-Materials of Traditional
and Modern Mathematics

A visable characteristic of the modern approach to
elementary school mathematics, is the wide spread avail­
ability and usage of media and materials of instruction
on nearly all elementary concepts of mathematics.

This is

not to say that the availability and the usage is entirely
a positive thrust, but rather a statement of present condi­
tions.

In point of fact, a significant amount of research

has belittled manipulative aids and other media to the
point of their being ineffectual and a waste of valuable
resources.
In 1964, Jamieson 71 conducted an experiment
utilizing a multi-base abacus.

He formulated three groups

of learners each having a different mode of instruction on
the same concept.

One group watched the teacher demonstrate

on a large variable-base abacus; a second group also watched
the teacher, but in addition each learner had a small abacus
at his desk.

The third group received instruction without

any abacus in the room, having only the teacher and a
chalkboard for instructional purposes.

Jamieson's finding

was that there were no significant differences between mean
per pupil gains across the three methods of instruction.
71
King w. Jamieson, An Experiment with a Variable
Base Abacus, Arith Teacher 11:81-84, February 1964.

46
Whether or not teacher demonstration, with materials
facilitates learning to a greater degree than pupil manipula­
tion of materials is a question of debate.

It becomes a

very important discussion when a cost factor is considered.
72
Trueblood
concluded that pupils in elementary mathematics
classes who watched the teacher demonstrate and manipulate
materials scored higher on a posttest of skills than pupils
who manipulated materials themselves.
Toney 7 3 on the other
hand, studied the effectiveness of individual manipulation
of instructional materials when conqpared to teacher demon­
stration and concluded that there were no significant
differences among fourth graders using manipulative
materials and those seeing only a teacher demonstration
technique.
Pella and Sherman

74

used manipulative versus non-

manipulative procedures in a study conducted with ninth
graders.

Their approach was similar to Toney's in that

individual manipulation of instructional materials was
72

Cecil R. Trueblood, A Comparison of Two techniques
for using Visual-tactial Devices to Teach Exponents and
Non-decimal Bases in Elementary School Mathematics, Arithmatic Teacher 17:338-340, April 1970.
73
Jo Anne S. Toney, The Effectiveness of Individual
Manipulation of Instructional Materials as Compared to a
Teacher Demonstration in Developing Understanding in
Mathematics, (Indiana University, 1968) D i s s . A b s t . 29A:
1831-1832, December 1968.
74
Milton O. Pella and Jack Sherman, "A Comparison
of Two Methods of Utilizing Laboratory Activities in Teaching
the Course IPS," School Science and Mathematics 69:30 3-14,
April 1969.

47
compared to teacher demonstration without manipulation by
pupils.

In this investigation, the criteria were expanded

to include critical thinking, understanding of concepts,
interest, achievement, and laboratory skill.

Again,

there

were no significant differences resulting from the treat­
ments employed when understanding and achievement were the
criteria.

In addition, Pella and Sherman found similar

results when critical thinking and subject interest were
the criteria.

The manipulative method did however, prove

to be significantly better than the non-manipulative method
for the development of selected laboratory skills.

Sex and

I.Q. were used as variables, but did not prove to be
significant within the scope of the study.
This last fact of the relationship of sex to
mathematics understanding and interest has long been a
question of concern and debate among educators.

Fogelman 75

reported that boys who manipulated materials demonstrated
conservation more frequently than boys who watched the
experimenter manipulate the materials.

For girls, he found

just the opposite was true.
Pella and Sherman*s findings were corroborated
later in a study by Yager, Engen, and Snider

76

among high

school science classes.
75

E. R. Fogelman, Piagetian Tests and Sex DifferencesII, Educational Research 63:455-458, July/Aug. 1970.
Robert G. Yager and Harold B. Engen, and William C.
Snider, "Effects of the Laboratory and Demonstration Methods
Upon the Outcomes of Instruction in Secondary Biology,"
Journal of Research in Science Teaching 6:76-86, 1969.

48
The question of the worth of materials in mathematics
instruction is debatable due to the diverse opinions and
findings on the topic in current literature and research.
77
Harshnan, Wells, and Payne
in a study of manipulative
materials in grade o n e , reported among other findings, that
high expenditure

for manipulative materials did not seem

justified when inexpensive and/or teacher developed materials
worked as well, especially when both were evaluated over
the criteria of achievement.
In a one year study of one hundred students conducted
78
in 1967, Devine
found significant differences in achievement favoring a conventionally taught classroom when compared
to another utilizing programmed materials.

His findings

went further to illustrate that the teacher atttiude about
a particular approach had significant impact upon student
growth, and that teacher attitude about programmed materials
79
was quite negative.
In a follow up study by Sneider
77
Hardwick W. Harshman, David W. Wells and Joseph N.
Payne, Manipulative Materials and Arithmetic Achievement
in Grade One, Arithmatic Teacher 9:188-191, April 1962.
78
Donald F. Devine, "Student Attitudes and Achieve­
ment:
A Comparison Between the Effects of Programmed
Instruction and Conventional Classroom Approach in Teaching
Algebra, Mathematic Teacher 61:296-301, March 1968.
79
Sr. Mary Joetta Sneider, "Achievement and Program­
med Learning," Mathematics Teacher 61:162-64, February 1968.

49
Devine's findings were corroborated with a second group of
students.
Fennema

80

found no significant differences in overall

learning of a principle when learning was facilitated by
either a meaningful concrete or a meaningful symbolic
model and concluded that use of concrete materials in
elementary school mathematics may not always be as essential
to the development of meanings and skills as has been
hypothes i z e d .
In an older study. Sole

81

concluded that the use

of a variety of materials did not produce better results
than the use of only one material, technique, or approach.
He further concluded that the learning of mathematics
depends more on the teacher than on the materials used.
There have been many studies that defend m a n i ­
pulative materials and their utilization in elementary
82
mathematics.
Golsby
worked with first grade pupils
using experimental manipulative materials designed to
"promote readiness and enhance the curriculum" for low
80

Elizabeth H. Fennema, A Study of the Relative
Effectiveness of a Meaningful Concrete and a Meaningful
Symbolic Model in Learning a Selected Mathematical Principle,
Diss. A b s t . 30A:5338-5339, June 1970.
81
David Sole, The Use of Materials in the Teaching
of Arithmetic (Columbia University, 1957) Diss. Abst. 17:
1517-1518, July 1957.
82
Thomas M. Golsby, et al., Effect of Massive
Educational Intervention on Achievement of First Grade
Students, Journal of Experimental Education 39:46-52, Fall
1970.

50
achievers.

His treatment resulted in significantly greater

achievement than that attained by students using conventional
non-manipulative materials.
Weber,8 3 though finding no significant differences
between reinforcement of concepts through paper and pencil
activities or with manipulative materials, did discover a
trend which favored the use of manipulative ma t e r i a l s,
especially for students with a low socio-economic status.
Goebel84 was concerned with materials and teacher
time utilization.

He reported that teachers using programmed

instructional and manipulative materials, devoted sixtyeight per cent of their time to working with individuals,
while teachers of conventional classes devoted less than
five per cent of their time to individuals.
One of the most widely acceptable mathematical aids
being used in classrooms and spoken of in literature and
research is the Cuisenaire materials and method.

There is

very little negative research directly relatable to the
Cuisenaire method specifically.

All of the criticisms of

83
Audra W. Weber, Introducing Mathematics to First
Grade Children:
Manipulative vs Paper and Pencil, (Univer­
sity of California, Berkeley, 1969) DAI 30A:3372-3373,
February 19 70.
8 4Laurence G. Goebel, An Analysis of Teacher-Pupil
Interaction when Programmed Instructional Materials are
Used (University of Maryland, 1966) D iss. A b s t . 27A:982,
October 1966.

51
manipulative materials in general are applicable to Cuisenaire,
but any criticisms directly applied to this method are not
85
easily found.
Hollis
found that children learned tradi­
tional subject matter with the Cuisenaire program as well
as they did with the conventional method and that pupils
taught by the Cuisenaire program acquired additional concepts
and skills beyond the ones taught in the conventional program.
86
In a related study in 1966, Crowder
found that
Cuisenaire users learned more conventional subject matter
and more mathematical concepts and skills than non users,
at the first grade level.

He reported that pupils with

average and above average achievement patterns profited
most from Cuisenaire programs, and that while sex was not
a significant factor in relation to achievement, socio­
economic status w a s .
The studies and literature on materials in elementary
mathematics are varied.

The important fact that each

stresses is the proper utilization of any material and that
given improper use, no material usage would be most profit­
able.
85

Loye Y. Hollis, A Study to Compare the Effect of
Teaching First and Second Grade Mathe by the CuisenaireGattegno Method with a Traditional Method, School Science
and Mathe 65:683-685, November 1965.
86
Alex B. Crowder, A Comparative Study of Two
Methods of Teaching Arithmetic in First Grade (North Texas
State University, 1965} D i s s . A b s t . 26:3778, January 1966.

52
A Summary
The purpose of Chapter II was to present both
positive and negative research on the topic of elementary
school mathematics instruction.

Because of the breadth of

the topic of modern approaches to instruction,

the literature

and research in the chapter was limited to six aspects of
current literature on mathematics approaches.

The six

aspects are from the conceptual framework established in
Chapter I.

A summarization of the research findings is

as follows:
Influential Factors of Mathematics Organization and
Achievement.— The research and literature is not in agree­
ment as to the effect of any given influence upon mathe­
matics achievement.

Agreement can be found however, in the

fact that specific internal and external factors such as
age, attitude, literature, sex, socio-economic status,
material rewards, and previous experience are some of the
factors that can have dramatic affect upon the manner in
which mathematics instruction is organized and disseminated
to a group of learners.

These, as well as others, can

also have dramatic affect upon the degree of learning that
takes place within a learner.

The effective educator is

one that is aware of the influential factors that are
constantly at work in a classroom and works to make their
influence positive for the learners in the class.

53
Attitude-Interest and Mathematics Instruction and
Achievement.— As it relates to mathematics instruction and
pupil achievement, the impact of student or teacher attitude
upon academic growth, can be considered questionable.

There

is limited evidence to show that the modern approach to
mathematics facilitates greater interest for the learner
than did the traditional.

Evidence can be found to support

the commonly held belief that boys are more interested and
have a more positive attitude about mathematics than do
girls.

Teacher and pupil attitude about elementary mathe­

matics and the resultant pupil achievement may or may not
be significantly corrolated on the basis of current research.
Modern versus Traditional Organization of Mathematics
Instruction.— A significant body of knowledge was found that
reported academic growth in elementary mathematics is not
significantly different when either a modern or a traditional
approach is utilized.

It has been shown that modern programs

are as effective as traditional in developing traditional
skills, and perhaps more effective in developing modern
concepts.

It is a validated fact that no one specific

pattern of organization for instruction in elementary school
mathematics

(such as a team approach, grouping schemes,

text-no-text etc.)

is any better in increasing academic

growth than any other.

The key to effective instruction is

still the teacher, regardless of whether a modern or a
traditional approach is being used.

54
Techniques of Modern versus Traditional Organization,—
It is an accepted ideal of elementary mathematics instruction
that some type of differentiated instruction should be
established for each particular learner.

The problem is

that there is disagreement over the correct method of
actualizing this belief.

It has been illustrated that

individualized programs of elementary mathematics will not,
simply because they are individualized,

lead to greater

levels of pupil achievement when compared with non-individualizee
approaches.

The single best technique that should be utilized

is the one that works with a particular child on a particular
concept or skill.

The teacher is the critical variable and

the art of asking probing questions and leading meaningful
dialogue is essential for adequate elementary mathematics
instruction.
debatable.

The idea of grouping for instruction is highly
Ability and achievement grouping, or any other

homogenous technique, results in achievement levels not
significantly different in most cases, from heterogeneous
organizations.

This is particularly apropo in terms of

low or below average learner abilities.
Discovery-Inquiry versus Expository Learning.— There
is general consensus among the researchers and writers
that expository methods of instruction alone will not result
in pupil attitudes and interests of a positive nature, nor
will achivement be as great, as will take place in a setting
of some expository instruction supplemented by a discoveryinquiry mode.

There remains considerable question as to

55
exactly what effect certain psycho, socio, and emotive
variables have on a pupils capability in, or acceptability
of, discovery-inquiry inodes of instruction in elementary
mathematics.
Media-Materials of Mathematics Instruction.— Although
elementary mathematics instruction today is heavily accented
with media and manipulative materials, a large body of
literature and research negates the essentialness or w o rth­
iness of the approach.

There is basic question as to

whether teacher demonstration with manipulative devices
alone, or child manipulation by himself/herself results in
increased academic retention and knowledge.

It was noted

that high expenditure for media might not be as important
as once thought in curriculum planning, due to the fact
that teacher developed materials were as effective as
commercially prepared.

Research has shown that manipulative

materials may be of greatest benefit to the child of lower
socio-economic status.

Cuisenaire materials have proven

to be very effective in the presentation of both modern and
traditional concepts and skills.

C H A P T E R III

RESEARCH METHODOLOGY AND TECHNIQUES
Introduction

This chapter addresses itself to the purposes for
which this study was undertaken, and seeks to illustrate
how each of those individual purposes were investigated.
As stated in Chapter I, there are four purposes for this
study.

(1) To establish a working base that defines a

"modern" mathematics ideology.
differences,

(2) To ascertain the

if any, between what experts in mathematics

education at the University level believe is current
thinking in elementary mathematics, as compared to public
school teachers.

(3) To investigate whether a difference

exists between what elementary school teachers believe
about modern mathematics and what they do in their class­
rooms.

(4) To determine whether or not the variables of

school district size and wealth are correlated with a
more "modern" concept of mathematics instruction among
elementary classrooms in public school districts.
The techniques of investigating each of the above
purposes are discussed in this chapter.

56

Information is

57
presented as to how the sample was selected, how the
questionnaire was developed, how the base was arrived at,
and how the data was analyzed.
Establishing a Base
The first purpose of this study was to adequately
define what constituted "modern" thinking as it related
to elementary mathematics instruction.

To do this, careful

and extensive review of the current literature and research
was conducted.

The research in Chapter II of this study

encompasses the majority of the literature upon which the
base was ultimately drawn.

The researcher synthesized

the pros and cons of the literature and research findings
related to the topic, and arrived at a master list of
thirty principles of current thinking in the field of
elementary mathematics.

This compiled list was mailed to

a panel of eight mathematics experts in the State of
Michigan.

Each of these experts represented a different

major state institution of higher learning and was felt to
be significantly aware of mathematics principles in educa­
tion, to be considered an expert in the field.

The experts

(see Appendix A) were asked to agree or disagree with the
principle as stated, disallowing any inherent and obvious
questions they might have as to definitions, meanings, etc.
It was believed that because of the number of experts used,
and because each was asked to respond with an "off-the-top"

58
type of response,

that, the pooled responses would be a

valid assessment of the "modernness" of the set of principles.
Upon the return of all eight experts responses to
the list of thirty principles of mathematics in elementary
education, a tabulation was made on each item to determine
the extent to which any one particular item could be
considered acceptable and therefore modern in current think­
ing in the field.

For the purposes of this study the follow­

ing criteria were used to determine if an item was modern
or non-modern thinking in the field of elementary mathematics
instruction:

if a particular item received more than five

"agree" responses from the eight experts, it was recorded
as " m o d e m "

thinking; if five or more of the experts checked

the "disagree" response to an item,

it was considered n o n - _

modern ideology in today's mathematics.

All items receiving

either four or five checks in the "agree" column were
eliminated entirely because it was believed that such
responses indicated significant debate or indecision among
the experts, therefore not allowing a clear determination
of modernness to be made.

The list of thirty principles

submitted to the experts and the tabulation of responses
by which modernness was determined, can be found in Appendix
B.
The base of this study

(as defined in Chapter 1)

was developed by making a random selection of ten items
from all of the items that were designated either modern

59
or non-modern by the above criteria.

This list of ten

items became Part I on the questionnaire and included both
modern and non-modern principles.

A calibrated response

scale was developed for this list of ten items.

Part I of

the instrument was subsequently sent back to the same panel
of experts, and to a randomly selected sample of elementary
classroom teachers.

The responses of the experts on the

calibrated scale was necessary to facilitate investigation
of the first hypothesis concerning the difference between
expert and classroom teacher beliefs about principles of
elementary mathematics instruction.

Originally,

the experts

simply agreed or disagreed with the principle, and this
second inquiry from them sought to ascertain the degree of
acceptance or non-acceptance on each item held by each
expert.

Part I of the questionnaire became the measure

that was utilized to investigate the first hypothesis con­
cerning the difference between expert and classroom
teacher beliefs about mathematics instruction.
Identification of the Sample
Once the base was established and a list of ten
validated principles representing modern and non-modern
thinking concerning elementary mathematics was determined,
the researcher began to investigate the first hypothesis.
Is there a difference between what experts and teachers
believe constitutes a modern approach to mathematics in

60
elementary schools?

The researcher wanted to explore this

question on two levels.

First, among school districts

selected because of their varying size and wealth, and
second, among a random sample of heterogeneous elementary
classroom teachers from those same districts as compared
to the panel of experts discussed earlier in this chapter.
For this reason, some criteria of geography became essen­
tial.

The five county region in Southwest Michigan com­

prised of Berrien, Calhoun, Kalamazoo, Kent, and Van Buren
counties was chosen because this region, unlike other
regions in the s t a t e , has many representative districts in
categories of varying size and wealth.

The southeastern

region of Michigan is predominately urban and suburban,
thus seriously limiting the possibility of securing
responses from small size districts.

The northern portion

of the lower peninsula and the entire upper peninsula lack
districts that can be considered large by the definition
this study used of large in Chapter I.
Due to the large number of public school teachers
trained in institutions of higher learning in the State of
Michigan, and hence the large number of professional edu­
cators involved in teacher training in the state, it was
believed that Michigan teachers and experts could be con­
sidered as representatives of all teachers and experts in
elementary education.

61
Each individual respondent was an elementary class­
room teacher in southwestern Michigan.

Respondents were

randomly chosen from the selected school districts within
the five counties included in the study.

The districts

from which the respondents were drawn were not randomly
selected, but rather constituted a total population of
school districts in southwest Michigan given the two major
variables in the study.

The districts selected for inclu­

sion in the study as diagrammed across the variables of
size and per pupil expenditure, appear in Exhibit A.

This

information about size and per pupil expenditure was taken
from the Michigan Department of Education Bulletin 1012
of December, 1970.
The selected school districts constitute a total
population.

As such, it can be argued that the randomly

drawn teachers from these districts, are also representa­
tive of any other populations of similar characteristics,
because within a population the distribution of the ran­
domly drawn respondents scores can be considered as norm­
ally distributed.

It can be noted from Exhibit A that the

districts are grouped in cells and from those districts in
each cell a list was made of all of the elementary class­
room teachers.

From this master list of all of the ele­

mentary teachers in the selected districts, a sample was

EXHIBIT A.— School District Variables:
Large

Size and Expenditures.
Mid-size

Small
Berrien Springs Pub. Schs,

High
($895 + )

Battle Creek Cty.
Kalamazoo City Schs,

St. Joseph Cty Schs.
Comstock Public Schs
Godwin Heights Pub.

Middle
C$800 to
894)

Grand Rapids Cty.

East Grand Rapids Pub. Covert Public Sch.
Springfield Cty. Sch,

Portage Pub. schs.
Benton Harbor Cty.

Gull Lake Comm. Schs.
Niles Comm. Schs.

Low
($799 -)

Godfrey-Lee Pub. Sch.
Athens Area Schs.

63
taken utilizing a table of random n u m b e r s .x

For stability

and reliability of statistical tests, twenty respondents
in each cell were utilized.

This resulted in approximately

an eight per cent sampling of the teachers in the popula­
tion which has twenty-two hundred elementary classroom
teachers and a K-12 population of one hundred twenty-one
thousand students.
Development of the Questionnaire Instrument
The questionnaire instrument was developed in two
parts.

Part I was the base of modernness as validated by

expert opinion, and sought to elicit the beliefs of teachers
and experts as discussed earlier.

Part II of the instru­

ment concerned itself with that which a teacher does in
actual practice on a day to day basis in his/her classroom.
It was this portion of the instrument that, when compared
to Part I, allowed the researcher the opportunity to address
himself to another of the stated purposes of this study-whether or not a difference existed between what teachers
said they believed and what they actually did in their
classroom, or as stated in null form:

"There exists no

significant difference between what teachers believe consti­
tutes a modern mathematics program, and what they do in
their day to day activities in their classroom."
^Paul Blommers and E. F. Lindquist, Elementary
Statistical Methods (New York:
Houqhton Mifflin, 1960),
pp. 512-517.--------

64
Each of the items in Part II of the instrument is
correlated with the item in Part I having the same number.
As an example, item one in Part II is a question of actual­
ization
Part I.

(what do you do)

that relates to question one in

This pattern continues throughout the instrument.

As a test of its reliability, coherence, and validity, the
instrument was field tested on seventy-five randomly chosen
elementary teachers and necessary adjustments made.
The questionnaire was developed in a manner that
attempted to insure a response rate that would allow stable
statistical measures to be utilized, and that would give
validity to the findings.

Because questionnaires that are

sent through the mail have had a history of immense non­
response on the part of the sample, every effort was taken
to control the variables that could cause such an occur­
rence.

The questionnaire was anonymous.

In no place did

it call for the respondent to indicate any demographic or
personal information.

It was not important in the analysis

of the data, who the respondent was, but rather, only where
he or she taught.

For this reason, the return envelopes,

included in the original mailing, were coded so as to allow
the researcher the ability to record the responses in the
correct cell in Exhibit A.

Besides being anonymous, the

questionnaire was simple in response procedure, asking only
for the circling of a word.
rather than duplicated.

It was also short and printed

Finally, the questionnaire was

mailed to home addresses and in addition was mailed at a

65
time just prior to the Easter break of the respondents, so
that in most cases the respondent would have, and would take,
the time to respond.

A copy of the complete instrument and

the cover letter are included in Appendix C.
Size and Expenditure
The fourth purpose of this study was to ascertain
whether or not size and per pupil expenditure in the dis­
trict within which a person teaches, has any correlation
with his/her modernness in mathematics instruction.

As

will be discussed in the next section of this chapter, the
investigation of this question is done through an analysis
of variance procedure.

The analysis of variance with re­

peated measures techniques was chosen because of its assump­
tion that the variance within any one cell is equal to the
variance of any other cell and therefore will allow for
valid comparisons to be made among all of the cells in
Exhibit A.

Given the instrument and the population dis­

cussed earlier in this chapter, this statistical procedure
facilitated those statistical tests necessary to accept or
reject the hypotheses.
Analysis of the Data
To analyze the data collected during the investiga­
tion of this study, the researcher selected two statistical
treatments for the purposes of clarifying some aspects of
the study and to test the hypotheses as stated in Chapter I .

66

A two-way analysis of variance with repeated measures
2
technique, as suggested by Winer,
was utilized in deter­
mining whether or not significant differences existed be­
tween a teacher's beliefs and his/her actions, as well as
the differences between school districts according to size
and per pupil expenditure.

This statistical treatment was

chosen because of its ability to minimize the experimentwise error which has a cumulative effect within the alpha
levels upon the res u l t s .

Seven statistical tests were

conducted with the alpha level being set at .025.
In the first hypothesis, dealing with expert
versus teacher belief differences, a t test procedure was
utilized.

This hypothesis could not be tested by an

analysis of variance procedure because the panel of
experts were not selected from the same population that
the teachers were and were therefore separate from the
statistical design as illustrated in Exhibit B.

To test

the hypothesis that no significant difference existed
between expert and teacher beliefs as to the consistency
of a modern program of mathematics instruction, a numerical
value of “modernness" was established for each teacher.
This was accomplished by computing the mean score for
the teacher in Part I of the instrument.
2

This mean,

B. J. Winer, Statistical Principles in Experimental
D e s i g n , (New York:
McGraw Hill Book Co. , 1^62)
p. 105.

67
EXHIBIT B.— Statistical Design— Measures
Belief

Large

T-l
T-2
T-3
11
II
II

If
T-20
T-21
T-22
II
II
II

Mid-size

II
M
H
II

High
per pupil
expenditure

T-40
T-41
T— 42
If
II

Small

II
II
If
If
If

T— 60
T-61
T-62
It
II

Large

If
II
II
II

T— 80
T— 81
T— 82
tf

Mid-size

It
M

Middle
per pupil
expenditure

»
II

Practice

68

EXHIBIT B . — Continued.
Belief
II
fl

ir

Small

"
II
II

H
II

T —120
T —121
la

ll

Large

n
It
II
II
II

T-140
T-141
II
II
II

Mid-size
Low
per pupil
expenditure

"
„
ft

„
II
It
II

Small

„
II
II
tl

ll
T — 180

Practice

69
along with the other nineteen teacher means for Part I
in that same cell, were summed and a grand mean for the
cell was computed.

Comparisons with the experts mean score

were then made between and across all cells.
Hypothesis two was tested by determining a cell
mean for belief and a cell mean for practice within each
cell and then by utilizing the analysis of variance pro­
cedure discussed previously, determining the validity of
the hypothesis that no significant difference existed.
The format used for testing hypothesis three
was also utilized with hypothesis four.

For hypothesis

three, a mean value was determined for each level of the
variable of size--large, mid-size, and small.

This was

done by summing the teacher means for belief and practice
and arriving at a cell mean for all respondents in that
cell.

This summing was carried out across the expenditure

levels as well.

Comparisons were then made through the

analysis of variance procedure that determined the impact
of size of school districts upon modernness.
Hypothesis four utilized the same format as
hypothesis three except the summing was conducted for
each level of the variable of per pupil expenditure--high,
middle, and low.

This was carried out across the other

variables of size as well as belief and practices,

(measures).

70
The statistical format for the analysis of the
data in this study was as follows:

X
o to

Sources of
Variation
Measures*

H o Size
Ho

Levels

d.f.

(M)

2

1

if computed F>F

(.025)

1, 171**

(S)

3

2

if computed F>F

(.025)

2, 171

3

2

if computed F>F

(.025)

2, 171

Expenditure(E)

Criteria of Significance

denotes the grand mean numerical value for individuals with
the means for beliefs and practices added together.
**
171 degrees of freedom reflects twenty individuals per
cell minus the total degrees of freedom for all sources
of variation.

The following interaction effects are also tested:

Expenditure- ExS
Size

significant if

computed F>F (.025)

4, 171

Expenditure- ExM
Measure

significant if

computed F>F (.025)

2, 171

Size-Measure SxM

significant if

computed F>F (.025)

2,

ExpenditureMeasureSize
ExMxS

significant if

computed F>F (.025)

4, 171

171

This technique of analysis of variance tests all the
sources of possible variation to see if respondents vary

71
more due to these effects
vs. practice)

(size, expenditure, and belief

than they vary within any given cell.

It

is actually a comparison of between cells versus within
cells.
Summary
A panel of experts in the field of mathematics
education was selected to validate as modern or non-modern,
a list of specific items the researcher drew from current
literature and research.

From this validated list, a

questionnaire was developed and mailed to a random sample
of elementary classroom teachers in southwest Michigan.
The responses of the teachers, as well as the experts,
were converted into a numerical value of "modernness" as
regarded elementary mathematics instruction.

This "modern­

ness" value was then compared across the independent
variables of school district s i z e , as well as per pupil
expenditure.

Tests were also conducted to determine the

extent to which the teachers differed from the experts
as to a modern mathematics ideology, and also to what
extent there was a difference between what a particular
teacher said he/she believed and what he/she actually
did in class on a day to day basis.

CHAPTER IV
PRESENTATION AND ANALYSIS OF DATA

This chapter presents a statistical analysis of
the scores on the questionnaire instrument for both the
teacher and expert groups.

An analysis of variance-

repeated measures technique was utilized,

as well as a

standard t^ test for the purposes of establishing the
significance of differences among the relationships with
which this study was concerned.

A summary of the analysis

of data collected for the study concludes the chapter.
The Data Gathering Technique
Using a mailed questionnaire technique to a large
size sample of randomly selected school teachers, historically
has resulted in minimal numbers of returns.

In this study

however, it should be noted that the technique utilized
and discussed earlier concerning the process and develop­
ment of the instrument, was

most beneficial.

In the

investigation conducted in this study a total of two
hundred twenty-five questionnaires were mailed and a total
of two hundred eleven or 93.3% were returned.

A random

selection of one hundred eighty was made from the returned
72

73
instruments for validity, reliability, and equal cell
size as discussed in Chapter III.

This one hundred eighty

constituted the random sample that was utilized in the
investigation of all four of the research hypotheses
investigated in this study.
researcher,

In the opinion of the

the single most important factor contributing

to the large size of the return was the utilization of
the respondents home address for mailing purposes.

This

information was difficult to obtain, in many cases, due
to the reluctance of certain school districts to furnish
such information.

Of the sixteen school districts in the

study, as found in Exhibit A, thirteen responded to the
researcher's request for a district directory or list.
The Portage Public Schools, Springfield Public Schools,
and the Gull Lake Community Schools did not respond to
the request for information and for this reason, teachers
in those school districts were not included in the final
sample that was used in this study.
Hypothesis Testing
A total of eight statistical tests were taken to
ellicit the validity of the four major hypotheses
investigated in this study.

Hypothesis One was investi­

gated using a single test, while seven tests were con­
ducted in testing hypotheses two, three, and four.

Table

I presents a summary of the test results utilized for
validating the final three hypotheses.

TABLE 1.— Summary of Statistical Tests Results Hypotheses Two, Three, Four.
Sources of Variation

Degrees of Freedom

Mean Square

F Statistic

P
n. s .

Expenditure (E)

2

.054

.228

Size (S)

2

.382

1.599

1

1.332

5.260

171

.239

E x S

4

.098

.411

E x M

2

.254

1.004

S x M

2

.014

.054

n.s.

E X S X M

4

.234

.925

n.s.

171

.253

359

.246

Measures
Teachers:

(M)
E x S

M x Teachers:
Total

E x S

n.s.
>.025

(sig)

n. s .
n.s.

Key— n.s. denotes no significance at p <.025
p
denotes the probability of a false rejection of the null hypothesis
(M) denotes the hypothesis concerning beliefs versus practices

75
Ho^

There is no significant difference between expert
opinion on the consistency of a modern mathematics
program, and the opinion of all the teachers
included in this study.
The mean score of the eight elementary mathematics

experts on Part I of the questionnaire was 3.538, while
the mean of the

one hundred eighty randomly selected

respondents was

2.90 3.

Part I (beliefs)

of the questionnaire.

Hypothesis One concerned only
The test of

significance was conducted at the .025 level with one
degree of freedom.

The mean square difference was 3.156

and the computed F value was 13.02 3 which was greater than
the table value of F with one degree of freedom at the
.02 5 level of confidence.
Finding
The hypothesis as stated Ho^ was rejected.
Summary

Teachers

Experts
Mean Val u e s :
Test Data:
Source of Variance
Between Cells
Within Cells

3. 538

d.f

MS

1

3. 156

366

.242

2.903

F
13.023

p
>.025

76

Meaning
Given the finding of this investigation and based
upon the assumptions and procedures it has utilized-there
is a significant difference between what experts in mathe­
matics education believe is m o d e m ideology concerning
classroom instruction in mathematics, and what a randomly
selected sample of elementary classroom teachers believe.
Interpretation
Hypothesis One was rejected*

The analyzation

of the data clearly indicated that a significant difference
existed concerning the beliefs of experts as compared to
teachers.

The large F statistic was the result of two

major factors:

first, the items of Part I of the

questionnaire were originally sent to and validated by the
same panel of experts whose scores were eventually used in
establishing the expert's mean score.

Therefore,

the

experts were agreeing and disagreeing to items they them­
selves had validated as being modern and n o n - m o d e m .
resulted in a high mean value for experts.

This

Second, the

large difference between the size of the random sample
and the size of the panel of experts affected the degrees
of freedom and therefore created a tendency to inflate the
mean square values from which the F statistic was directly
computed.

It perhaps should also be noted that experts in

the field of mathematics education spend a significant

77
amount of their time involved with research and literature.
Teachers in public school classrooms on the other hand,
spend proportionally less time with research and literature,
therefore, it could be expected that the two groups would
react differently upon the base on this study

(as defined

in Chapter I) from each other.
This hypothesis was not developed specifically to
determine the modernness of the teachers or the experts.
To do so, would involve a subjective evaluation on the part
of the researcher as to what mean score level would
constitute modernness in a continuum from one to five.
Such a task was not undertaken.

The purpose of this

investigation was only to ascertain if a difference existed
between the two groups.

A significant difference was

discovered and the implications of that difference will be
discussed in more detail in Chapter V.
Ho 2

There is no significant difference between what
the respondents say they believe constitutes an
effective mathematics program, and what they say
they do in their day-to-day activities in the
classroom.
The mean score of the one hundred eighty respondents

for Part I of the questionnaire was 2.9 33, while the mean
value for the same sample on Part II of the questionnaire
was 2.842.

The test of significance was conducted at

the .025 level with one degree of freedom.

The mean square

of the difference was 1.332 which resulted in an F value

78
of 5.260 which was greater than the table value of F given
the aforementioned values.
Finding
The hypothesis as stated in Ho2 was rejected.
Summary

Teachers
Mean V a l u e s :

2.842

2.963

Test Data:
Source of Variance
Measures

(M)

d. f.
1

MS
1. 332

F
5.260

P
>.025

Meaning
Given the findings of this study and based upon the
assumptions and procedures it has utilized-there is a
significant difference between what elementary classroom
teachers believe is a modern mathematics program and that
which they do in their classroom on a day to day basis.
Interpretation
Hypothesis two was rejected.

The evidence as

presented in the data, clearly indicates that a s i g n i f i c a n t
difference existed between what a teacher says he or she
believes is a modern approach to the teaching of mathematics
in contemporary elementary education, and that which he
or she actually does in the classroom.

It was also

79
significant, that the score of the respondents on Part II
of the questionnaire was lower than on Part I, thereby
indicating that teachers have beliefs that are more modern
them the actual practices that they employ.

Teachers in

elementary grades do not use modern mathematical techniques,
nor exhibit modern mathematical ideologies in their
instruction,
in them.

to the same degree that they say they believe

In effect, teachers think more modernly,

than

they function in the classroom.
The researcher believes that the nature of the
construction of the questionnaire instrument allowed for
a one-to-one mean relationship between Parts I and II to
develop, if in fact that relationship actually existed
among the randomly selected respondents.

As stated earlier,

each item of Part II was correlated with an item in Part I
and the scoring scheme remained constant for both items.
For this reason, the researcher interprets the statistical
finding as valid and concludes that teachers function in
a manner that is different from what they themselves believe
to be modern, as related to elementary mathematics instruc­
tion.

The implications of such a finding will be discussed

in Chapter V.
Ho^

There is no significant difference in "modernness**
between teachers of varying school district sizes.

80
The mean score of the respondents who taught in the
school districts listed in the column named "large" in
Exhibit A was 2.837 when the scores from Part I and Part II
of the questionnaire were combined.

This compared with a

total mean value of 2.9 38 for teachers in the "mid-size"
column and 2.9 32 in the "small."

The test of significance

was conducted at the .025 level with two degrees of freedom.
The mean square difference value between the column values
was

.382 and the computed F statistic was 1.5 99 which is

less than the table value of F at the .025 with two degrees
of freedom level.
Finding
The hypothesis as stated in Ho^ was not rejected.
Summary

Mid-size

Large
Mean Values:

2 .837

Small
2.932

2.938

Test Data:
Source of Variation
Size(S)

d. f .
2

MS
. 328

F
1.599

P
n.s.

Meaning
Given the findings of this study and based upon the
assumptions and procedures it has utilized-there is no
difference in modernness between teachers working within

81
districts of varying population size that can be said to
be attributable to that varying size.
Interpretation
Hypothesis three was not rejected.

The analyzation

of the data indicated that statistically no significant
difference existed among teachers of elementary school
mathematics, when compared across the variable of the
size of the school district population.
however,

(non-statistically)

It is significant

given the belief possessed by

many that small school districts do not provide as modern
an approach to school in general and mathematics in
particular, as do mid-size or large school centers.
Had the analysis in fact illustrated that modernness of
instruction and belief concerning mathematics education
was significantly different in rural areas as opposed to
urban, then credence would be established for such an
argument.

Such was not the finding and the researcher

accepts the validity of the analysis.

The implications

of this finding are many and will be discussed in Chapter
V.
Ho^

There is no significant difference in "modernness"
between teachers from school districts that have
varying per pupil general fund expenditures.

82
The mean score of the respondents who taught in the
school districts listed in the row named "high per pupil
expenditure" in Exhibit A f had a total mean of 2.921 over
both Part I and Part II of the questionnaire combined.
This compared with a total mean value of 2.879 for teachers
in the "middle per pupil expenditure" row and 2.907 for
those in the "low" category.

The test of significance

was conducted at the .025 level with two degrees of freedom.
The mean square difference value between the row values
was .054 and the computed F statistic was

.22 8 which is

less than the table value of the F at the .025 with two
degrees of freedom level.
Finding
The hypothesis as stated in Ho^ was not rejected.
Summary

Middle Exp.

High Exp.
Mean V a l u e s :

Low Exp.

2. 897

2.921

2.907

Test Data:
Source of Variation
Expenditure

(E)

d.f.

MS

F

2

.054

.228

P
n.s.

Meaning
Given the findings of this study and based upon
the assumptions and procedures it has utilized-there is

83
no difference in modernness between teachers working within
districts of varying per pupil expenditure that can be said
to be attributable to that expenditure difference.
Interpretation
Hypothesis four was not rejected.

The analyzation

of the data indicated that statistically no significant
difference existed among teachers of elementary school
mathematics when compared across the variable of per pupil
expenditure within the various districts included in this
study.

As described earlier in this study, the researcher

believes that the districts involved in the investigation
of this hypothesis were representative of other districts
of similar expenditure levels within Michigan and other
locales.

For this reason, the researcher believes that

the implication of the findings is in the statistical
non-significance that was found.

Although statistically

significance was not found, the researcher feels it
very important that no difference in modernness according
to expenditure amount was discovered.

This finding negates

the belief held by many educators that if a district, or
an individual teacher within that district, simply had or
spent more money on the program, his or her instructional
approach could be "more modern."

The statistical test

of

this hypothesis indicates that such a belief is not wellfounded.

As with hypothesis three, the implications of

this finding will be discussed in fuller detail in Chapter V.

84
Summary
In this chapter the data collected rrom the mailed
questionnaire instrument was presented,
interpreted.

analyzed, and

Consideration was also given to the data

collection procedures that were utilized.

It was concluded

than an anonymous, short, printed, and home addressed
technique of development and dispersion of the data collect­
ing instrument, resulted in the high percent of return of
questionnaires sent.
The chapter addressed itself to discussing each of
the research hypotheses separately, by stating the findings,
the statistical summaries, and a semantic interpretation of
the results.
Hypothesis One concerning the experts beliefs as
compared to the teacher beliefs, was rejected.

A

significant difference did exist between the beliefs or
ideologies of these two groups concerning the consistency
of a modern elementary mathematics program.
Hypothesis Two was similarly rejected.

There was

a clear significant difference established between what
a teacher said he or she believed and what that same
teacher actually did in his or her classroom.

This difference

existed across all of the independent variables.

85
Hypothesis Three was not rejected.
case of per pupil expenditure,

As was the

the size of a school

districts student population was not significantly
correlated with the modernness of the elementary class­
room teacher's mathematics instruction.
Hypothesis Pour was not rejected.

The per pupil

expenditure within varying school districts could not
be said to be correlated with the degree of modernness
exhibited by the elementary classroom teachers in those
districts.
rejected,

Though statistically the hypothesis was not
it was significant to the researcher that

wealth and modernness are not necessarily correlated.

CHAPTER V
SUMMARIES, CONCLUSIONS, AND RECOMMENDATIONS

This chapter presents a summary of the study, a
summary of the research findings, conclusions,

implica­

tions, and recommendations for possible action and future
research*
Summary of the Purpose and Design
The purpose of this study was to determine if a
panel of mathematics experts, drawn from the major teacher
preparation institutions in the State of Michigan, and
a random sample of public elementary school classroom
teachers agreed as to the consistency of a list of con­
temporary mathematical ideals.

A second purpose of this

study was to determine the consistency among teachers in
public schools between what they believe about mathematics
instruction and that which they actually perform in the
classroom in the way of techniques.

Finally,

this paper

sought to discover a correlation between an elementary
school teacher's modernness in mathematics instruction and
the student population size of the teacher's district, or
the per pupil general fund expenditure of the district.
86

87

The literature and research contained m

Chapter II

of this investigation, reflects pro and con arguments on
six major areas of interest in contemporary mathematics
instruction.
1.

Discovery-inquiry learning and mathematical
growth.

2.

Influential factors of organization in mathe­
matics instruction.

3.

Attitudes and interests and elementary mathe­
matics learning.

4.

The effects of modern versus traditional
instruction and organization.

5.

The techniques of traditional and modern mathe­
matics approaches.

6.

Media and materials of traditional versus
modern mathematics.

Within these six areas, a synopsis of ideas was
taken by the researcher, submitted to a panel of experts
in mathematics education for validation, and recorded.
From this validated list of principles, a questionnaire
instrument was developed and mailed to a random sample of
elementary classroom teachers in Michigan.
The response rate was 9 3.3% and the analysis of
the returned data was carried out upon a random sample
of one hundred eighty respondents of those returning the
instrument.

A Jennrich program of analysis of variance-

repeated measures technique at the .025 level of confidence.

83
and the CDC 6500 computer were used in the analyzation
procedure-

A second statistical test called a one-way

analysis of variance with unequal subclasses was utilized
for Ho^ and was carried out at the .025 confidence level
on the CDC 3600 computer.
Summary of the Results
of Hypothesis Testing
Hypothesis One:

There is no significant difference
between expert opinion on the con­
sistency of a modern mathematics p r o ­
gram and the opinion of all of the
teachers in this study.

Result:
The null hypothesis was rejected at the .025 level
of confidence on the basis of the one way analysis of
variance

(t test) discussed earlier.

statistic was 13.023.

The computed F

The direction of the difference

indicated that experts could be considered more modern than
elementary classroom teachers concerning mathematics
principles.
Hypothesis Two:

There is no significant difference between
what the respondents say they believe
constitutes an effective mathematics
program and what they say they do in
their day-to-day activities in the
classroom.

Result:
The null hypothesis was rejected at the .025 level
of confidence using a two way analysis of variance-repeated
measures technique.

The computed F statistic was 5.260.

89
It could therefore be inferred,

that teachers function in

a manner quite separate from a list of modern principles
that they agree with.
Hypothesis Three:

There is no significant difference in
"modernness" between teachers of vary­
ing school district sizes.

Result:
The null hypothesis was not rejected because of
a computed F value of only 1.59 9 using a two way analysis
of variance-repeated measures technique at the
of confidence.

.025 level

Therefore, the student population size was

not a significant variable when correlated to teacher
modernness in elementary mathematics instruction.
Hypothesis Four:

There is no significant difference in
"modernness" between teachers from
school districts that have varying per
pupil general fund expenditures.

Result:
The null hypothesis was not rejected due to a
computed F value of only .228 utilizing a two w a y analysis
of variance-repeated measures technique at the .025 level of
confidence.

The inferrence can therefore be made that per

pupil expenditure within a school district was not a
significant variable when correlated to teacher modernness
in elementary mathematics instruction.

Summary of the Findings
A significant, difference was found

how experts

responded to a list of contemporary mathematics
principles as taken from current research and litera­
ture, and how a randomly selected sample of public
elementary classroom teachers responded.

The direction

of the difference indicated a higher mean score for
experts than teachers.
A significant difference was found among elementary
classroom teachers concerning their beliefs about the
consistency of a modern elementary mathematics program,
and the practices they utilized frequently in their
classroom.

The direction of the difference indicated

that teachers believe more modernly than they actually
perform in their class.
There was no significant difference in modernness
among teachers randomly selected from school districts
of varying student population size.
therefore,

The inference

is that size of district and the modernness

of instruction in elementary mathematics are not
correlated.
There was no significant difference in modernness
among teachers randomly selected from school districts
of varying per pupil general fund expenditures.

The

inference thus becomes that per pupil spending and
modernness in elementary mathematics instruction are not
correlated.

91
Conclusions
Based on the findings of this study, the following
conclusions would seem to be justified:
1.

Mathematics education experts and elementary
classroom teachers differ in their beliefs
as to the consistency of a modern elementary
mathematics program.

2.

Teachers of elementary mathematics have belief
systems concerning mathematics instruction,
that differ from the practices they actually
employ.

3.

Student population size and modernness of
elementary mathematics instruction are not
correlated.

4.

Per pupil expenditure and modernness of
elementary mathematics instruction are not
correlated.
Implications from the Findings

Based on the data collected and the findings of
this study, the following implications are presented for
consideration:
1.

Based upon the findings of the investigation
of the first hypothesis, contemporary mathe­
matics principles are not held by classroom

92
teachers to the extent that they are supported
by mathematical education experts.

Because

these principles take their form from research
and literature, and because mathematics experts
have a large part to play in the writing of
research and literature,

it becomes necessary

for newer and more effective methods of dis­
seminating principles of mathematics to the
teaching profession at large.

It appears of

questionable value to the teaching profession
for researchers to continue vast and expensive
research and writing endeavors, if and when the
findings of that effort are not utilized or
even discovered by the practitioners of the
teaching profession to a greater degree than
currently exists.
2.

The implications of the finding concerning the
beliefs of classroom teachers as opposed to
their reported daily classroom practices, fur­
ther points up the lag in research implementa­
tion.

Clearly it was shown that teachers func­

tion in a manner quite different from their
belief patterns concerning elementary m a the­
matics.

To the end that effective instruction

instruction is built upon solid and reliable
philosophical and psychological foundations,

93
careful attention in methods training should
be given to new teachers concerning the princ­
iples of mathematics instruction, as well as
techniques.

More importantly, to the teacher

already in the field, a stronger and more mean­
ingful in-service or retraining program should
be established, that is both on-going and evalu­
ative.

If modern mathematics is to continue to

succeed, greater efforts need to be taken to
insure that the pedagogical techniques with a
class of learners correlates with modern
principles of elementary mathematics to a
greater degree than presently exists.
3.

The finding of this study relative to school
population size and teacher modernness in ele­
mentary mathematics, indicates a fallacy in the
viewpoint, held by many, that mathematics educa­
tion in smaller districts is somehow more
"basic" or remedial in nature than in larger
districts.

Clearly, the teacher in the small

consolidated rural school setting needs the same
concerted effort to maintain contemporary ideals
as much as any other elementary educator.

This

study indicates that he or she is at the present
time functioning as "modernly" as any other
teacher in an urban s e t t i n g .

For the researcher

this points up the generally accepted belief that

94
the teacher is still the single most signifi­
cant aspect in learning and the kev to the
extent to which any subject, such as mathematics,
is learned.

By and large, rural districts

cannot afford extensive in-service and retrain­
ing programs on a wide scale, and therefore it
is necessary for mathematics experts and univer­
sities to develop methods of instruction and
re-instruction that can aid the smaller districts
as well as the larger*
4.

The fourth hypothesis allowed for a difference
to appear between the modernness of a teacher
in an affluent school district and another from
a poorer school district,

if in fact such a

difference actually existed among the respon­
dents.

Such a difference did not materialize

and therefore, the commonly held ideal that
instructional modernness and school district
affluence are correlated can be considered
questionnable.

The implication from this is

that similar mathematical proficiencies can be
expected across the broad spectrum of all school
d i s t r i c t s , regardless of the relative wealth of
any one district.

The teacher, and the beliefs

and practices of that teacher have greater
impact on the achievement of a learner in ele­
mentary mathematics than how much money is

95
actually spent on the mathematics program.

It

would appear from the findings of this study
that it is no longer an acceptable tenet to
maintain;

"give me more money for instructional

purposes, and I ’ll do a better job."
Recommendations for Future Investigations
Based on the findings, conclusions, and implications
from this study, the following recommendations would seem
justified:
1.

One of the findings in the related research
in this study concerned the need for educators
in elementary mathematics to possess positive
attitudes toward mathematics and the teaching
of mathematics.

Further surveys of current

practices in elementary mathematics instruction
are needed to determine the validity of this
finding.
2.

Programs of pre-service for elementary teachers
must concern themselves to a larger extent
with the psychological principles upon which
modern pedagogical practices in mathematics
instruction are based.

A need remains to

develop this set of principles as they relate
specifically to elementary mathematics, as
compared to broader psychological principles
that cut across all elementary disciplines.

96
3.

The questionnaire instrument developed for
and utilized in this investigation, could be
expanded to include other demographic variables
such as age, years of experience, major and
minor etc.

In this manner further isolation

could be attempted to uncover the variables
that may be significant in indicating a new
approach to the in-service and pre-service of
teachers.

This study indicated that they were

not size and/or expenditure, but what variables
might serve as indicators are still unknown.
4.

The researcher would suggest that a study
similar in scope to this one be attempted in
another geographical area.

It is further

suggested that any new geographical area of
investigation include a "metropolitan" area
or configuration.

This would allow for differ­

ences to develop among neighboring contiguous
school districts that would in all likelihood
reflect nearly similar sizes and expenditures
on the one hand, but allow for the ascertaining
of other suspected variables of significance.

97
5.

An in-depth study should be made of the effective­
ness of elementary mathematics methods courses.
Such a study should follow the development of
an evaluative model that would enable the
researcher to maintain contact on a long range
scale of time, with both the teacher and that
teacher's methods instructor.

Implementation

of methods learned in the university methods
courses in elementary mathematics , lag behind
what could and should be expected.
6.

Due to the large size of the volume of research
conducted in mathematics education, and because
concensus in that research is difficult to
ascertain, a further research project should be
conducted that isolated a particular aspect of
mathematics instruction and developed a synopsis
of all current research on the topic.

Although

this has been done many times over, the uniqueness
of this study would be a new and innovative
method of disseminating the findings of the
project to the practioners of the profession.
Developed concisely and with clarity of
meaning, such a project would not necessarily
need to take a written form, as in a piece of
literature that only a relative few would see.

98
The innovativeness of this aissemination would
lend the project its needed credibility.
7.

Because of the apparent difference between the
practices of elementary classroom teachers in
mathematics instruction and the beliefs those
same teachers maintain, concerning an effective
mathematics program,

it is recommended that

careful attention be given to development of
a more adequate implementation model for pre­
service and in-service education.

If this

difference exists, as indicated in this study,
then something must be done to eliminate the
discrepancy between belief and practice if
meaningful instruction in elementary mathematics
is to take place.

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----------------Schwab, Joseph J.
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APPENDICES

APPENDIX A
MATHEMATICS EDUCATION EXPERTS

113

Panel of Mathematics Experts

Dr. Joseph Payne----- University of Michigan
Dr. William Fitzgerald
Dr.

Herbert Hannan

Michigan State University
Western Michigan University

Dr. William Swart----- Central Michigan University
Dr.

Donald Buckeye

Dr.

Ted Eisenberg

Dr. Eugene Smith

Eastern Michigan University
Northern Michigan University
Wayne State University

Dr. David Wells----- Oakland University and Oakland
Intermediate Schools

114

APPENDIX B
PRINCIPLES OF ELEMENTARY MATHEMATICS AND
TABULATION OF THE EXPERT'S RESPONSES

115

INSTRUCTIONS:

Please check the appropriate blank to show whether you agree
or disagree with each specific Item as It relates to contemporary
elementary school mathematics instruction.
AGREE
DISAGREE

1.

There is a significant positive relationship between pupil
__
attitude toward mathematics and pupil achievement in mathematics.

2.

No one single organizational pattern for mathematics in­
struction can be said to be better than any other in developing
_
mathematical growth among elementary school learners.

3.

"Modern" programs are as effective as "traditional" programs
in developing "traditional" mathematical skills.

4.

Chronological age of the learner is positively correlated
with the ability to learn various mathematical concepts

3.

Whether the instruction given is inductive or deductive In
nature, has no significant affect on mathematical learning.

6.

In an attempt to facilitate retention, intensive and specific
review and practice should be provided in elementary math.

7.

Socioeconomic status influences mathematical achievement.

8.

A testing-reteaching-retesting strategy will help decrease
the errors most pupils make.

9.

A correct diagnosis
mentary mathematics
the specific errors
specifically how he
remedial help.

procedure for individualizing in ele­
is one in which the teacher ascertains
that a pupil is making, determines
works, and gives specific personalized

10.

Whether a learner is taught through a "guided discovery"
technique or an "expository" technique, computational
skill ability remains constant.

11.

Grouping on the basis of ability in mathematics is of
greatest benefit for pupils of higher ability.

12.

Programs of Individualized instruction in elementary
mathematics do not lead to higher levels of pupil achievement,
when compared with non-individualized programs.

13.

Programmed materials are as effective in promoting mathematical
growth as are textbooks, when either is used in a meaningful
way.

14.

Programmed instruction is most effective for learners of
low or below average mathematical abilities.

13.

The learning of elementary mathematics is more dependent upon
possession of adequate materials, than any other single factor.

16.

"New mathematics" has more to offer the high achiever than
the low achiever.

116

3

8

8

8

117
17.

18.

AGREE

Having pupils manipulate materials is a much more effective
method of facilitating mathematical growth, than merely
watching teacher demonstrations with materials.

DISAGIU

&

Computer-assisted instruction is limited in its scope to
drill and practice activities, and hence to computational
concerns as opposed to concept development.

4

1

19.

Providing children many opportunities to solve problems In a
_
variety of ways is more helpful than formal analysis procedures.

20.

The primary concern of elementary mathematics should be the
development of the ability of computational proficiency on
the part of the learner.

6

21.

Pupils achieve more in mathematics when working independently
than when working in groups of either small or large size.

6

22.

The intelligence level of the learner is positively correlated
with his/her ability to learn various mathematical concepts.

1

23.

Verbal problem solving is as adequate a method in assessing
mathematical growth as is pencil and paper techniques.

2

Heterogeneous classroom composition in mathematics is the
most effective organizational scheme for attaining maximal
growth for each learner in elementary school mathematics.

6

Instruction in "new mathematics", much more than in "tradi­
tional math", concerns itself with patterns and relationships
of numbers and operations in a continuum of mathematical
experiences.

0

Mathematical learning is not significantly affected by the way
in which the curriculum is organized (by topic or area).

3

27.

The attitude of the teacher about mathematics contributes
significantly to the achievement of the learner.

3

28.

Expenditure for manipulative materials to supplement the
teacher-textbook technique, should be a high priority item
when considering elementary mathematics organization.

3

24.

25.

26.

29.

The "new mathematics" results in greater concept development
by the student, while the "traditional" technique results in
greater skill or computational understandings.

30.

Specialist teachers for elementary mathematics is a more
desirable technique for instruction than self-contained
organizations with a single teacher.

5

3

APPENDIX C
THE QUESTIONNAIRE AND COVER LETTER

118

M IC H IG A N

fo iiii.i:

STATE

o r M M jr a t io n

-

UNIVERSITY

d f p /i n t m i n i

of

elem entaky

ea st

and

l a n s in g

s p e c ia l

* Mic h ig a n

e d u c a t io n

•

e a ic x s o n

hall

March 23, 1972
Dear Educator:
I am a graduate assistant In the Department of Elementary and Special
Education at Michigan State University and am In the process of completing
my dissertation for a Ph.D. degree.
My major field of Interest Is
elementary mathematics education.
1 am In great need of your help and
would appreciate it if you would take five to ten minutes to respond
to the attached questionnaire.
This study is centered in southwest Michigan and your name was selected
at random from a compiled list of all elementary classroom teachers in
the vast geographical area included in the study.
In a few cases, some
of your fellow workers may have also been chosen by the random process,
and in other cases, your entire building because of its relatively small
size or location.
The study simply seeks to ascertain what teachers in
the field believe and are experiencing in the realm of "modern" mathematics.
I am concerned with whether or not mathematics instruction, as conceived
at the University level, differs from what you, an experienced in the
field educator, feel to be the situation.
If so, the implications for
the University are immense.
Please note that this questionnaire is completely annonymous.
Therefore,
I am not seeking any personal information about you.
Hence, your responses
to these questions will not be used in any personal evaluative sense,
but will enable me to perceive how a large sample of randomly selected
elementary public school classroom teachers feel about mathematics.
For economical reasons, please return the questionnaire even if you
decide not to respond to it.
I have enclosed an envelope for you to
return your answered or unanswered form and certainly appreciate your
help in this matter.
Without your help it will not be possible.
Would you please try to complete this within the next week, as I am
working under a deadline for completion of my degree requirements.
Thank
you again and feel free to contact me if you have any questions or concerns.

Sincerely,

^ M

S'/* y ?

Thomas Murphy
{/
Graduate Assistant
301 E Erickson Hall
Michigan State University
(517) 353-4398

119

PART

I

rN S T R U G T T O N S : P l e a s e cf r c l r ilie n p p r o p r i a l t> w o r d (s3 that r e f l e c t s
a bo ut e a c h o f the F o l l o w i n g items, ns t h e y r e l a t e to
e l e m e n t a r y sc ho ol m a t h e m a t i c s i n s t r u c t i o n .

1.

I n an a t t e m p t to f a c i l i t a t e
p r a c t i c e s h o u l d he p r o v i d e d
Strongly

2.

agree

Agree

agree

Agree

in m a t h e m a t i c s

Undecided

agree

Agree

agree

Agree

Strongly

Strongly
upon

Disagree

agree

Agree

agree

Agree

disagree
as

disagree

the possession

Strongly

disagree

is a m o r e e f f e c t i v e m e a n s o f c r e a t i n g
t he t e a c h e r d e m o n s t r a t e w i t h m a t h e r l a l s .

Undecided

Disagree

Strongly

Undecided

Undecided

Undecided

disagree

s o l v e p r o b l e m s in a v a r i e t y
analysis procedures.

V e r b a l p r o b l e m s o l v i n g is as a d e q u a t e a m e t h o d
g r o w t h as art* p a p e r a nd p e n c i l t e c h n i q u e s .
Strongly

Disagree

Strongly

Disagree
of

disagree

than when

Strongly

disagree

assessing mathematical

Disagree

Strongly

disagree

I n s tr u ct ion in " n o w m n t h e m a t i c s " m u c h m o r e t h a n in " t r a d i t i o n a l m a t h " is c o n ­
c e r n e d w i t h p a t t e r n s and r e l a t i o n s h i p s o f n u m b e r s a nd o p e r a t i o n s i n a c o n t i n u u m
of m a t h e m a t i c a 1 e x p e r i e n c e s .
Strongly

0.

Agree

for

in m a t h e m a t i c s

P u p i l s a c h i e v e m o r e in m a t h e m a t i c s w h e n w o r k i n g i n d e p e n d e n t l y
w o r k i n g in g r o u p s of e i t h e r s m a l l o r l a r g e size.
Strongly

9.

agree

disagree

benefit

Strongly

Disagree

Undecided

6.P r o v i d i n g c h i l d r e n w i t h m a n y o p p o r t u n i t i e s to
of w a y s is m o r e h e l p f u l t h a n t h r o u g h f o r m a l

8.

is o f g r e a t e s t

in p r o m o t i n g g r o w t h
a m e a n i n g f u l way.

r e v i e w a nd

Strongly

Disagree

Undecided

Having pupils manipulate materials
m a t h e m a t i c a l growth, than w a t c h i n g
Strongly

7.

Disagree

T h e l e a r n i n g of e l e m e n t a r y m a t h e m a t i c s is m o r e d e p e n d e n t
of a d e q u a t e m a t e r i a l s , t h a n a ny o t h e r s i n g l e f a c t o r .
Strongly

3.

Undecided

P r o g r a m m e d m a t e r i a l s a r e as e f f e c t i v e
arc t e x t b o o k s , w h e n e i t h e r is u s e d in
Strongly

U.

Agree

G r o u p i n g o n the b a s i s o f a b i l i t y
p upils of hi g h e r ability.
Strongly

3.

agree

r e t e n t i o n , i n t e n s i v e a nd s p e c i f i c
in e l e m e n t a r y m a t h e m a t i c s .

y o u r he lief
contemporary

agree

Agree

Undecided

Disagree

Strongly

disagree

T h e " n e w m a t h e m a t i c s " r e s u l t s i n g r e a t e r c o n c e p t d e v e l o p m e n t b y t h e st ud en t,
w h i l e t he " t r a d i t i o n a l " t e c h n i q u e r e s u l t s in g r e a t e r skill o r c o m p u t a t i o n a l
development.
Strongly

agree

Agree

U n d e c i ded

Disagree

Strongly

disagree

PART I I
INSTRUCTIONS;

).

(s)

that m os t n e a r l y a n s we r s

(Vice a w e e k

W ee k l y

B i - w ee k ly

In the m a t h e m a t i c s i n s t r u c t i o n in your c l a s s ro o m,
students g r o u p e d by a bi l it y for instruction?
Daily

i.

the w o r d

the q u e s t i o n for you.

How f re qu en t ly is a typical l e ar ne r in y ou r c l a s s r o o m i nv ol v ed
practi ce a c t i v i t i e s in m a t h e m a t i c s ?
Daily

7.

Circle

T wice a w e e k

Weekly

How ofte n do you use p r o g r a m m e d m at er ial s
Daily

Twice a w e e k

in re vi ew or

M o n t h l y or r ar e ly

h o w f r e q u e n t l y are the

Bi-weekly

M o n t h l y or

r arely

in y o u r c l a s s r o o m in m a t h em a ti cs ?

We ek ly

B i- we e kl y

M o n t h l y o r r ar e ly

How freque ntl y do learners in y o u r c l a s s ro om use v a r i e d s u p p l e m e n t a l m a t e r i a l s
for m a t h e m a t i c a l instru ct io n?
Daily

T.

W ee kl y

M o n t h l y or r ar e ly

B i- w e e k l y

in

M o n t h l y or rarely

in y o u r c l a s s r o o m so lve p ro b l e m s u t i l i z i n g mo re

TWice a w e e k

T w ic e a w e e k

We e kl y

Bi-weekly

than

M o n t h l y or r ar el y

Weekly

TVice a w e e k

TVi ce a we ek

Weekly

TVice a w e e k

Bi-weekly

im m a t h e m a t i c s

Weekly

How frequently does v o u r m a t h e m a t i c a l
as opposed to skill, d e v e l o p m e n t ?
Dai ly

B i - w ee k ly

M o n t h l y or

rarely

f o r m a l l y assess m a t h e m a t i c a l g r o w t h w i t h p a p e r and pe nc il

How often d oe s y ou r i n s t r u c t i o n
re la ti on sh ip of o pe ra t io n s?
Daily

0.

learners

How freque ntl y do von
techniques ?
Daily

9.

Bi-weekly

How f re que nt ly is a typical l e ar ne r in y ou r c l a s s r o o m w o r k i n g i n d e p e n d e n t l y
in m at h em a t i c s , away from a g r o up or the c lass?
Daily

R.

IVicc a w e e k

How frequently do
one method?
Daily

7.

Weekly

How frequently do learners in y o u r c l a s s r o o m use m a n i p u l a t i v e m a t e r i a l s
the a c q u i s i t i o n of m a t h e m a t i c a 1 concepts an d skills?
Daily

b.

Twice a w e e k

W ee k l y

M o n t h l y or

rarely

de al w i t h n u m b e r pa tt e rn s and

B i - w e ek l y

instruction concern

Bi-weekly

the

M o n t h l y or ra rely

itself w i t h concept,

M o n t h l y or r arely