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12-1978
A Meta-Analysis of the Effects of Modern Mathematics in Comparison with Traditional Mathematics in the American Educational System
Kuriakose K. Athappilly Western Michigan University
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Recommended Citation Athappilly, Kuriakose K., "A Meta-Analysis of the Effects of Modern Mathematics in Comparison with Traditional Mathematics in the American Educational System" (1978). Dissertations. 2693. https://scholarworks.wmich.edu/dissertations/2693
This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. A META-ANALYSIS OF THE EFFECTS OF MODERN MATHEMATICS IN COMPARISON WITH TRADITIONAL MATHEMATICS IN THE AMERICAN EDUCATIONAL SYSTEM
by
Kuriakose K. Athappilly
A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the Degree of Doctor of Education
Western Michigan University Kalamazoo, Michigan December 1978
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS
The present study as well as my entire doctoral program could not
have been accomplished without the assistance and encouragement of
many people. I wish to gratefully acknowledge my appreciation to the
following people.
Dr. Uldis Smidchens, chair of my doctoral committee, Professor,
Department of Educational Leadership, contributed most generously of
his time, energy, and expertise as an educationist, researcher, and
above all as a person of utmost understanding and singular dedication
to the cause of the students. Dr. John Kofel, my former advisor and
committee member, Director of International Education Center, had truly
been to me "a friend, philosopher, and guide" from the beginning to
the end of my doctoral program. Dr. James Riley, Professor, Department
of Mathematics, provided his professional expertise for the research.
To these members of my doctoral committee a debt of gratitude is most
willingly, but inadequately, acknowledged.
I am extremely grateful to the authorities of Western Michigan
University General Library. Although they must remain anonymous, it
must be acknowledged that this research could not have materialized
without their generous financial cooperation. I thank also my friends
and acquaintances— students, faculty members, and administrators— who
gave me their encouragement, concern, and understanding.
My mother, Mrs. Monica Athappilly, has immeasurably provided her
moral and physical support to all my educational ventures. I am
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. grateful for her continued interest and prayers; for, in a sense, this
work represents a fulfillment of one of her dreams.
My deepest appreciation goes to my wife, Sossa, and children, Geo,
Geena, and Geetha. Sossa has for the past few years abandoned her pro
fession to free me for this work. In addition to being wife and mother,
she has most willingly assumed responsibilities that both parents should
share. The cooperation, encouragement, and sacrifice of my family made
this study a work of love.
This dissertation is dedicated to the memory of my father,
Kunjuvarkey Athappilly, 1907-1967.
Kuriakose K. Athappilly
i i i
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ATHAPPILLY, KURIAKOSE KUNJUVARKEY A META-ANALYSIS OF THE EFFECTS OF MODERN MATHEMATICS IN COMPARISON WITH TRADITIONAL MATHEMATICS IN THE AMERICAN EDUCATIONAL SYSTEM.
WESTERN MICHIGAN UNIVERSITY, ED.D., 1978
University Microhms International 300 n. zeeb road, ann arbor, mi 4bio6
@ 1978
KURIAKOSE KUNJUVARKEY ATHAPPILLY
ALL RIGHTS RESERVED
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS
LIST OF T A B L E S ...... , . . . vl
LIST OF F I G U R E S...... viii
CHAPTER
I INTRODUCTION AND BACKGROUND INFORMATION .... 1
Purpose of the S t u d y ...... 1
The Evolution of Modern Mathematics .... 2
The Role of Developmental Projects ...... 4
Toward Definitions ...... 10
The Statement of the P r o b l e m ...... 11
Significance of the S t u d y ...... 12
Research Questions ...... 13
Topics Excluded ...... 14
Organization of Dissertation ...... 15
II RATIONALE FOR THE S T U D Y ...... 17
Research Studies on Modern Mathematics .. . 17
Techniques of Aggregation ...... 28
S u m m a r y ...... 35
III DESIGN AND DATA A N A L Y S I S ...... 37
Concept of Meta-analysis ...... 37
Effect S i z e ...... 39
Techniques of Measuring Effect Sizes .... 45
Multiple Effect S i z e ...... 47
Sources of the Stud i e s ...... 48
Locating and Selecting Studies ...... 49
iv
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CHAPTER
Identifying and Coding Variables ...... 53
Analysis of Data . , ...... 55
Procedures for Judging the Magnitude and Significance of Effect S i z e ...... 56
S u m m a r y ...... 57
IV R E S U L T S ...... 59
Description of the Studies Included .... 59
Overall Effects of Modern Mathematics . . . 62
Specific Effects by Subject Characteristics ...... 64
Specific Effects by Treatment F a c t o r s ...... 69
Specific Effect Size by Study Variables...... 77
Specific Effects by Design Variables .... 85
S u m m a r y ...... 88
V INTERPRETATIONS, IMPLICATIONS, GENERAL RECOMMENDATIONS, AND SUMMARY ...... 92
Interpretations ...... 92
Implications ...... 100
General Recommendations ...... 104
S u m m a r y ...... 107
BIBLIOGRAPHY ...... 112
APPENDICES ...... 132
Appendix A: The List of the Variables Identified...... 133
Appendix B: Code Sheet of the Variables Identified ...... 137
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES
1. Overview of the Development of Mathematics Programs Since 1951 ...... 5
2. Study I ...... 31
3. Study I I ...... 31
4. Studies I and II C o m b i n e d ...... 32
5. Studies Classified by Source ...... 60
6. Studies Classified by Y e a r ...... 61
7. Length of Treatment and Number of Subjects Per S t u d y ...... 62
8. Grand Mean Effect Sizes in Achievement and A t t i t u d e ...... 63
9. Effect Size by Levels of Grade and Ability in Achievement ...... 67
10. Effect Size by Socio-economic Status and Sex in A c h i e v e m e n t ...... 70
11. Effect Size by Content Area in Achievement .... 72
12. Effect Size by Type of Program in Achievement...... 75
13. Effect Size by Length of Treatment in Achievement...... 76
14. Effect Size by Number of Subjects Per Study in A c h i e v e m e n t ...... 78
15. Effect Size by Study Approach in Achieve ment and Att i t u d e ...... 80
16. Effect size by Source in Achievement...... 82
17. Effect Size by Subject Area of Investigation in Achievement ...... 83
18. Effect Size by Affiliation of the Researcher in A c h i e v e m e n t ...... 84
vi
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19. Effect Size by Design Variables in Achievement...... 87
20. Effect Sizes by All Categories of Variables in Achievement and/or Attitude ...... 90
vii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES
1* Overview of the cluster approach ...... 34
2. Normal curves illustrating the effect of modern treatment in relation to tradi tional treatment— a hypothetical illustration ...... 41
3. Normal curves illustrating the effects of three different treatments of modern mathematics (modern algebra, modern geometry, and modern arithmetic) in relation to untreated traditional group— a hypothetical illustration ...... 44
4. Normal curves illustrating the aggregate effect of modern mathematics in achieve ment and attitude in relation to tradi tional mathematics group ...... 65
viii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I
INTRODUCTION AND BACKGROUND INFORMATION
Many innovations in the content, methodology, and materials
related to mathematics education at the elementary, secondary, and
post-secondary levels were introduced during the past three decades
in an effort to make the American mathematics curriculum relevant to
current needs. Studies on the effects of these innovative efforts
were found primarily in dissertations, journals, and technical
reports. Many curriculum changes in mathematics education were intro
duced under a common label called "new mathematics" or "modern mathe
matics." The terms "new mathematics" and "modern mathematics" were
used interchangeably, since most of the literature related to the
changes which occurred in mathematics curricula did not differentiate
between the two terms.
Purpose of the Study
The studies under the titles "new mathematics" or "modern mathe
matics" were numerous and their findings were often dissimilar and
conflicting (Dieudonne, 1973; Thom, 1971). Some attempts had been
made to summarize and extract meaning from portions of the vast body
of research data in the area of new mathematics. Recently, Hartley
(1977) investigated the summative results of research pertaining to
the effects of individually paced ins* -.-uctions in mathematics, which
is a later offshoot of modern mathematics. No attempt had been made,
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 however, to summarize the body of data related to the various con
cepts encompassed within new mathematics. The purpose of this study,
therefore, was (a) to summarize the effects which were unique to the
new mathematics curricula and (b) to examine whether or not the new
mathematics curricula led to improved academic achievement and atti
tude over "traditional mathematics."
The Evolution of Modern Mathematics
Toward the middle of the twentieth century, educational leaders
in the colleges and universities, high school curriculum supervisors,
mathematical organizations, and other interested individuals began
challenging the traditional sequential pattern of mathematics teach
ing. In 1940, Betz cited several factors, such as general unawareness
of the significance of mathematics, one-sided emphasis on the doctrine
of "social utility," and mechanistic methods of teaching, as contri
buting to the poor status of mathematics in the schools.
In the same year, the Progressive Education Association (PEA)
published a report, Mathematics in General Education, which aimed to
develop a mathematics curriculum to meet the needs of the students
(Bidwell & Clason, 1970, p. 531). Another report of the same year
was the Place of Mathematics in Secondary Education, published by the
joint commission of the Mathematical Association of America (MAA) and
the National Council of Teachers of Mathematics (NCTM). The report
emphasized "the goals of mathematics and the role of individual dif
ferences among students" (Bidwell & Clason, 1970, p. 532). Reports
and publications of this nature asserted that the curriculum changes
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in mathematics, which emphasized concepts, understanding, and insight
without neglecting basic skills, were imperative to cope with the
rapid advances in the field of science and technology.
World War II and Sputnik
Two critical events gave great stimuli for reform in mathematics
education: World War II and the Russian launching of the first satel
lite (Sputnik) in 195 7. World War II created a perceived need for
trained manpower in scientific technology which, in turn, placed a
new emphasis on improved mathematics education. The Education for
All American Youth Report of 1945, the three reports of the Commis
sion on Post-War Plans of the National Council of Teachers of Mathe
matics (NCTM) (1944, 1945, 1947), and The Steelman Report of 1947 were
typical products of the times. All five reports reflected an increas
ing dissatisfaction with the content and approach of school mathemat
ics, while calling for innovation in mathematics education.
The launching of Sputnik by the Russians and the consequent com
petition of the United States with Russia over space superiority
created even greater pressure than World War II for the production of
large numbers of qualified scientists, engineers, and mathematicians.
As a result, the direction of mathematical research was toward the
objective of developing mathematics curriculum adequate for the tech
nological age rather than predictive studies toward success in the
former sequential mathematics courses (Anglin, 1966).
National foundations, the federal government, and private organ
izations provided financial aid for curriculum improvement in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4
mathematics. As a result, mathematics scholars— in cooperation with
educational specialists, supervisors, and teachers— began developing
new mathematics programs for elementary and secondary schools.
Sherman (1972) reviewed the purposes and accomplishments of all major
mathematics programs established since 1951. An overview of those
programs is shown in Table 1, including the year, range, and essential
characteristics of 10 prominent programs.
The Role of Developmental Projects
Most of the developmental projects as seen in Table 1 were ini
tiated primarily to correct perceived weaknesses in school programs.
Consequently, the initial concerns were to produce adequate educa
tional materials and to establish appropriate teacher training pro
grams. The University of Illinois Committee on School Mathematics,
"the progenitor of all current curriculum projects in mathematics"
(NCTM, 1970, 32nd Yearbook, p. 257), was founded in 1951 to investi
gate problems concerning the content and teaching of high school math
ematics and to correct weaknesses in that area. As a first step
toward that goal, it produced materials which were pilot tested in
classrooms, and it was ascertained that the success of the materials
depended on the skills of the teacher.
In addition, the Madison Project placed a major emphasis on
teacher training through demonstration centers and in-service pro
grams. The three main objectives of the Greater Cleveland Mathematics
Program were production of ample materials, teacher training, and
evaluation. The Minnesota Mathematics and Science Teaching Project
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. teachers point of view student background and experience bers Correspondence course with graduate credit for Stress Stress mathematical on ideas Intuitive and axiomatic approaches Cultural aspects man'sof experience with num Precise terminology Nongraded units be to used in accordancewith Deemphasis on social arithmetic Logical development Table 1 Table Support of of Illinois patterns USOE Carnegie Consistency; Early verbalization precise terminology discouraged University Logical structure mathematics; of study of Ball State Structure of mathematics NSF Learning through discovery University Interrelatedness of principles NSF Structure mathematics of from the historical Secondary Secondary Elementary Grades 7,8 Elementary 1957 1957 Secondary Year Overview of Developmentthe of Mathematics Programs Since 1951 (1953) (1953) College Started Program (BCMI) (BSU) Institute of of Illinois Committee on School Mathe University Mathematics Ball State University matics (UICSM) Boston College
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CTv Essentials own own solutions problems to of of arithmetic erately omitted program ously encouraged of of project the ber systems Concept of set and operations on set Integration of arithmetic, algebra and geometry Inductive and deductive reasoning Discovery of structure through finding one's Discussion through careful questioning Unstructured tasks; social applications delib Ungraded material intended as a supplementary Fundamental learning processes major a concern Relationship between set theory and foundations Mathematics as a language; precise terminology Unifying concepts; particular emphasis num on Learning through discovery Verbal and operational components simultane Logic Properties mathematical of a structure Precise language; mathematical laws Algebraic and geometric principles texts sale of Carnegie Carnegie USOE Funds from Table 1 1 (continued) Table NSF NSF USOE NSF Range Support Present Secondary Elementary Elementary Elementary 1959 Grades 7,8 1957 1959 Year Started Program (SUSNP) of Marylandof Stanford Sets and Project Numbers Project Madison University University Mathematics
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Essentials tion things out tant mathematicaltant ideas opportunity for conventional practice and review course of study; no grade levels mathematics not just some to subdivision mathematics concept formation K-12 Concepts mathematics of as part of whole the of Intuitive thinking; guessing; inventing; trying Search for patterns and relationships Structure mathematics of Spiral approach New content wellas as conventional topics; No strong position on verbal-operational ques Developing a feeling mathematicalfor ideas Topics with "travel"— an adventure Logical structure mathematicsof Emphasis a on continuous and systematic flow of No stress verbalization; on not a complete Project seeks teachable alternatives for impor Discovery approach Mathematical exploration Mathematical laws Support of of Illinois Carnegie Table 1 1 (continued) Table The Council NSF University NSF Year Present Program Started Range (UIAP) (SMSG) (GCMP) School Study Group 1958 Secondary Greater Cleveland 1959 Secondary Elementary Project Mathematics Elementary Arithematic 1958 Elementary Program Mathematics
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Their Origins and Evolutions, by Helene tive tive science curriculum K-9 and undergraduate courses for cultural and historical aspects of mathemat number system, Euclidean space, a withspace ics pre-service education of teachers measure Concepts of an algebraic structure and a deduc Connections between mathematics and science; Three specific mathematics structures; the real Ungraded units Working toward a coordinated mathematics-science Table 1 1 (continued) Table Range Support Essentials Present 1961 Year (1958) (1958) Elementary Started Note. Note. From Common Elements New in Mathematics Programs: Program (MINNEMAST) and and Science Teaching Project Sherman. Sherman. NewYork: Teacher's College Press, 1972. Minnesota Mathematics
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 started with the goal of finding out what children could learn and
then preparing teachers to teach it.
The Boston College Mathematics Institute was organized with the
purpose of reeducating high school teachers in the elements of con
temporary mathematics. The Ball State University Project was created
to improve materials in secondary school geometry, and later in grades
7-12 and then in grades K-6. Summarily, improving materials and
training teachers were the first steps toward introducing new mathe
matics in almost all the developmental projects.
Behavior modification in the minds of the pupils was effected by
placing emphasis on logical thinking and structure instead of drill
and rote. These concepts as described in Table 1 were introduced
through different terminologies by different projects. The UICSM
Project introduced the term "discovery approach." The University of
Illinois Arithmetic Project stressed the ideas of thinking and con
cept development. The SMSG Project advocated a deeper understanding
of basic concepts, structure of mathematics, and precise and sophis
ticated mathematical language.
The Madison Project recommended the unstructured task and the
discovery approach. The purpose of the Greater Cleveland Mathematics
Program was to develop a curriculum which could be presented in a
logical, articulated, and sequential manner. It focused on patterns
of relationships and the logical structure of mathematics. The BSU
Project also gave prominence to the structure of mathematics. Crea
tive thinking and structure of mathematics went hand in hand in almost
all the developmental projects in the process of introducing new
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 mathematics.
There was a basic similarity in the outcomes of the various pro
jects, although the different projects were not provided with any
common set of objectives prior to the formulating of the modern math
ematics programs.
Toward Definitions
Schaaf (1964) made a list of factors common to all the innovative
projects and other activities: (a) increased emphasis on abstract
ideas, (b) increased attention to logical region, (c) the use of con
temporary terminology, (d) the insistence upon precise language, and
(e) new "mathematical ideas" or "innovations of content."
The new ideas and content listed by Schaaf included the concepts
and language of sets, Venn diagrams, systems of enumeration, base
system, the real number system of algebra, modular arithmetic, ine
qualities, relations and properties, functions, axioms and postulates
in geometry, the elements of logic, and the nature of measurement.
Ferguson (1964) pointed out that the modern mathematics programs
had built-in motivation factors in that students experience the thrill
of discovering mathematical principles, for example, in algebra seeing
structure, order, and beauty.
According to a comprehensive analysis of six major programs for
the elementary grade levels (Sherman, 1972), the common elements of
new mathematics were a numeration system, measurement, geometric
ideas, algebraic ideas, structure (laws and systems) sets, statistics,
probability, number theory, and logic. Kline (1973) identified a set
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 of 12 major elements pertaining to the concepts of new mathematics.
They were set theory, bases of number systems, the congruences, ine
qualities, matrices, symbolic logic, Boolean algebra, relations and
functions, abstractions, structure, group and field, and statistics
and probability.
It was, therefore, conceivable that another analysis of modern
mathematics programs could produce another set of elements which were
not identical with the previous sets. Hence, for the purposes of this
study, the term "new mathematics" was used in referring to new devel
opments in mathematics at the elementary, secondary, and post-secondary
levels. The term applied to both new subject matter and new approaches
to topics, such as those discussed above. "Traditional mathematics"
referred to the content and teaching method prior to the introduction
of the developmental programs.
The Statement of the Problem
"There can be little question that following a fifteen year
period of unprecedented favor and prominence in the curricula of
American schools, mathematics teaching today faces the future with
less certainty of its goals and less optimism about its potential
effectiveness," observed Hill (1969, p. 440), who chaired the
National Advisory Committee on Mathematics Education.
The uncertainty about mathematics curricula stems from strong
public criticisms and from the abundance of research findings of
which some are dissimilar while others are conflicting in nature
(Norland, 1971; Yasui, 1967). Because of the dissimilar and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 conflicting nature of study findings, decision making at the levels
of resource allocation, curriculum development, and classroom teaching
becomes difficult and often misguided. Moreover, the money, time, and
thought expanded on new mathematics have been considerable— perhaps
even enormous (Kline, 1973).
These facts about the conflicting nature of research findings
and public opinions about the outcomes of new mathematics, coupled
with the scarcity of summative research resolving dissimilar and con
flicting findings, formulate the problem for which this study was
intended.
Significance of the Study
The conflicting nature of past findings made the study highly
relevant and important. The study, which was a meta-analysis— an
analysis of analyses— was precisely designed to resolve outcomes which
were not in agreement with each other. Conclusive findings about the
outcomes of modern mathematics were focused on three major areas.
These areas were (a) the subject— the importance of an appropriate
mathematics curriculum, (b) the methodology— the utilization of a
meta-analysis technique to discover the message of the new mathematics
in the midst of conflicting findings, and (c) the two levels of deci
sion making— balanced resource allocation decisions and well-founded
curriculum decisions yielding increased learning.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Research Questions
In order to summarize the effects of new mathematics and to exa
mine whether or not the new mathematics curriculum led to improved
academic achievement, the research sought answers to the following
questions:
1. Did the aggregation of studies show any evidence of change
in academic achievement attributable to new mathematics programs?
This question was concerned with the overall impact of the modern
mathematics as evidenced by the mean effect size of achievement when
all achievement studies were combined.
2. Did the aggregation of studies show any evidence of atti-
tudinal change attributable to new mathematics when compared to tradi
tional mathematics programs? This was concerned with the grand mean
effect size of attitude toward mathematics when all attitudinal stud
ies were combined.
3. What was the differential effect of the new mathematics pro
grams with respect to the following variables?
Subject characteristics:
Grade Socio-economic status Ability Sex
Treatment factors:
General content area Specific subject area Type of program Length of treatment Number of subjects per study
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Study variables:
Study type Study approach Author Source of study Affiliation of researcher Involvement of researcher General subject area
Design variables:
Type of test Quality of design Novelty effect
Topics Excluded
In addition to the previously discussed projects, there were
other developmental efforts that introduced innovative concepts in
mathematics education. Some of these projects were excluded from this
study. Primarily, they fell into one of the three following catego
ries .
1. In the first category were projects which pertained to pro
grammed instruction, individual learning packets, tutoring, and com
puter assisted instruction. The emphasis of these projects was
purely on the methodology of teaching with little or no relationship
to the new mathematics concepts as discussed in the definitions.
2. In the second category of excluded studies were those which
posed a problem of definition. They pervaded the area of cognitive
learning styles, pedagogical techniques, and teacher training. Exis
tential approach, self supervision, hierarchical components, modeling
theory, and resource personnel were some of them.
3. In the third category were the studies which were strictly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 within the area of new mathematics, but were excluded for technical
reasons, such as inadequate methodology, inappropriate design, and
lack of pertinent quantitative data which were necessary for the pur
poses of comparison.
Organization of Dissertation
Chapter I discussed the purpose and significance of the study.
It traced the evolution of modern mathematics and the role of the
developmental projects. The chapter also addressed itself to the
definition of terms, statement of the problem, research questions,
and the topics excluded.
Chapter II reviews research studies on modern mathematics and
techniques of aggregation. The section on research studies on modern
mathematics has been subdivided into three categories: (a) appraisals
of modern mathematics, (b) research studies on modern mathematics by
primary analysis, and (c) research studies on modern mathematics by
aggregate analysis.
Chapter III presents detailed description of the concept of
meta-analysis, sources of the study, methods of locating and collect
ing studies, the process of identifying and coding the variables, and
the techniques of measuring effect sizes. It also deals with the
question of multiple effect sizes and the method of analysis of data.
Chapter IV reports the results of the data analysis. The results
are in two major categories: (a) the overall effects and (b) the
specific effects by different groups of variables, such as subject
variables, treatment variables, study variables, and design variables.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16
Chapter V interprets the results of the study reported in
Chapter IV, discusses the implications and the major recommendations
based upon the research data, and gives a summary of the study.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER II
RATIONALE FOR THE STUDY
The purpose of the chapter is to present the logical basis for
the study through the review of the literature which focuses upon the
area of the investigation. In this light, it may be expedient to
devote a separate section of the chapter to the justification of the
methodology, since the study utilized a new technique of aggregation—
a meta-analysis procedure— as its research tool. Subsequently, the
rationale for the study focuses upon two categories of topics: (a)
research studies on modern mathematics and (b) the techniques of
aggregation.
Research Studies on Modern Mathematics
To begin then, the effects of the new mathematics curriculum
were reported in literally hundreds of research studies. A comprehen
sive treatment of the total studies conducted was not attempted in
this chapter due to the number of studies being prohibitively large;
however, a summary description of the studies on modern mathematics
is given in the ensuing sections. The description consists of three
parts: (a) appraisals of modern mathematics, (b) studies of modern
mathematics by primary analysis, and (c) studies of modern mathematics
by aggregate analysis.
1-7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 Appraisals of Modern Mathematics
Mathematics educators were divided in their opinions, points of
view, and observations regarding the efficacy of the new mathematics
programs. Supporters of modern mathematics advocated the relevance,
appropriateness, and the distinct advantages of the new programs.
Fehr (1953) urged the adoption of the new mathematics because of
its relevance to understanding the universe. In the same vein,
MacLane (1954), Hartung (1955), Price (1957), Kennedy (1961), Mueller
(1962), and Davis (1963) advocated the introduction of new mathematics
programs at the elementary and secondary levels in order to meet the
changing needs created by the advancement of science and technology.
Writers have also observed more specific advantages of the new
programs. For example, Clark (1961) pointed out that according to the
findings of Begle, Glennon, Gundlach, and others, the new mathematics
programs permitted and even encouraged children to learn more mathe
matics at an earlier age. Similarly, Grossnickle (1964) suggested
that the advantage of modern mathematics was in the introduction of
new and more profound topics at all grade levels. Furthermore, some
writers were of the opinion that slow learners benefitted more through
increased learning when utilizing new mathematics programs rather than
the traditional programs (Cunningham, 1965).
Conversely, the opponents of the new mathematics programs
expressed different and opposing points of view. For example,
Rosskopf (1954) objected to the new movement for a complete reorgani
zation from traditional to contemporary mathematics instruction
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19
primarily because he believed very few teachers were competent to
teach it. Similarly, Pingry (1956) and Kline (1959) criticized the
outright revision of the mathematics content as the proper approach
for improving student learning. Additionally, Kline (1958) echoed
these sentiments when he said, "The modern mathematics stressed during
the past few years is trivial in importance compared to topics cur
rently taught and is hopelessly beyond young people, because of its
abstractness and rigor" (p. 422).
About the same time other writers like Hannon (1959) believed
that the traditional approach to teaching and the related materials
had to form the basic core of the elementary curriculum. Several
years later Rappaport (1962) expressed alarm over the haste with which
untrained educators desired to teach modern mathematics. Also,
Rappaport held a concern for the impracticality of the new math pro
grams for the majority of the children.
It was apparent from the preceding sampling of viewpoints that
there were conflicting appraisals regarding the worth of the modern
mathematics programs. Again, the primary purpose for the study was
to seek a resolution to the maze of conflicting and dissimilar find
ings about modern mathematics as illustrated by the examples which
were cited.
Studies on Modern Mathematics by Primary Analysis
A review of the research on modern mathematics revealed two
important aspects pertaining to the nature of the studies. One, in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20
attempting to study the effectiveness of new mathematics programs,
investigators have adopted diverse methodologies for trying to gain a
clearer perspective. Two, in spite of several attempts to gain a
clear perspective, the study findings showed sharp contrasts and dis
similarities. The ensuing section seeks to point out the conflicts
and disagreements among the study findings of the several different
designs.
The most common methodologies utilized to study new mathematics
ranged from simple descriptions to true experimental research designs.
For example, Davis (1965) and Johntz (1967) used the descriptive proc
ess to investigate the effectiveness of the Madison Project. They
concluded that the creative classroom experience of the new programs
was superior to the traditional mathematics experience. Most of the
studies compared the new programs with traditional ones by reviewing
data from tests given to both experimental and control groups.
Biddle (1967) used a course in programmed instruction and Berger and
Howitz (1967), a discovery program for general mathematics students.
After one year of study, both investigations found that there was "no
significant difference" between the experimental and control groups.
Some researchers compared the effects of the experimental and
the traditional programs over a longer period of time. Hungerman
(1967) found that the School Mathematics Study Group (SMSG) students
were superior to traditional students on a conventional test. Graft
and Ruddell (.1968) found a significant difference favoring the SMSG
students in understanding, thought process, and transfer of learning.
Several studies compared the effectiveness of the treatment at
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various levels in relation to factors such as ability, intelligence,
and sex. Williams (1963) found no significant difference in achieve
ment, in grade nine, between the traditional algebra text and the
SMSG materials when used by average, high-average, and high ability
students. Shuff (1962) noted that at the upper and middle ability
levels, the students who received the traditional mathematics treat
ment in the seventh grade made better achievement gains than the stu
dents who received the SMSG instruction. Conversely, at the lower
level, the SMSG students performed better than the traditional group.
Peck (1963) divided the sixth grade students studying modern
mathematics and traditional mathematics into two groups. The high
intelligent group consisted of students with IQ’s of 115 or more and
the low intelligent group consisted of students with IQ's less than
115. The findings indicated that there was no significant difference
in achievement between the students who received modern mathematics
and traditional mathematics at the high IQ levels or at the low IQ
levels. About the same time, Harshmon (1962) made a comparison of
programs, using modern manipulative materials and traditional mate
rials. Groups receiving each program were subdivided into three
intelligence levels: (a) those with IQ's of 99 and below, (b) those
with IQ's between 100 and 125, and (c) those with IQ's of 125 and
above. The results significantly favored the traditional program in
the IQ subgroup of 100-125, but the opposite was true for the IQ
ranges of 125 and above and 99 and below.
Wozencraft (1963) carried out a comprehensive study to determine
the difference between boys and girls in arithmetic ability. She
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 found that there was a significant difference in arithmetic ability
which favored the girls in total and average groups of children; how
ever, the groups with high or low intelligence quotients did not show
significant sex differences. Northcutt (1964) examined the compara
tive achievement of fifth grade boys and girls in two teaching
approaches, a traditional approach and a programmed (modern) approach.
He found no significant difference in gain between the sexes using
either approach.
Longitudinal studies were yet another category of research in
this area. Yasui (1967) reported that the mean achievement score of
modern mathematics was significantly higher than the mean score of
students of traditional mathematics. Conversely, the findings of
Osborne (1965) revealed that the introduction of SMSG materials and
their study for increasing period of time had "no significant effect"
on arithmetic skills, algebra skills, and mathematics reasoning skills.
The findings of Austin and Prevost (1972) confirmed Osborne's findings,
but contradicted those of Yasui: After a four year experiment, the
traditional mathematics group scored higher than the transitional and
modern mathematics groups on the Metropolitan Achievement Test of
Mathematics Computation.
In another longitudinal study, Norland (1971) observed that after
four years of instruction in modern mathematics, achievement in arith
metic computation declined significantly in four of the five schools
which participated in the study. Conversely, in 1973 the National
Assessment of Educational Progress reported quite different findings.
One question about the new mathematics programs of the
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1960's has been their impact on pupil's ability to compute .... But the evidence, limited though it may be, sug gests that Johnny can add: computation with whole numbers is far from being a lost art. The performance of the thir teen year olds on each of the four operations with the whole numbers was almost as good as the performance of the seventeen year olds and the adults .... The thirteen year old's responses indicate that computational algorithms for all four operations are well developed by the latter part of elementary school. (p. 457)
Thus, the findings of the research studies fell into three cate
gories: (a) findings showing no significant difference between the
traditional and modern mathematics groups, (b) findings favoring the
traditional mathematics, and (c) findings favoring the modern mathe
matics. The findings, therefore, were so conflicting and dissimilar
in nature that one could not make a sound generalization about the
effects of new mathematics programs without a further investigation
into the results of those studies conducted with a more appropriate
technique of analysis.
Studies on Modern Mathematics by Aggregate Analysis
Studies on modern mathematics by aggregate analysis were exceed
ingly small in number. Some attempts were made in the generic area
of mathematics to aggregate the studies and classify them under one
or two major categories according to source, topic, or period of pub
lication; however, no analysis was made in any of the attempts to
summarize the findings or draw any conclusions. Several of these
attempts are summarized in the following paragraphs.
In the April 1957 issue of the Arithmetic Teacher, a bibliography
appeared which included six years of research on arithmetic teaching
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 (1951-1956). The annual listing of research pertaining to elementary
school mathematics continued in the Arithmetic Teacher for the next
14 years. In 1971 the Journal of Research in Mathematic Education
began publishing a comprehensive listing of research which had been
conducted in elementary and secondary mathematics education.
School Science and Mathematics began publishing listings of per
tinent research from published sources and dissertations from 1965 to
1972 (Burns & Dessart, 1965, 1966; Mangram & Knight, 1970, 1971, 1972;
Mangram & Morris, 1968, 1969; Summers & Hubrig, 1965, 1968).
In 1972, Suydam published a Review of Research on Secondary
School Mathematics. The review consisted of 780 research reports and
770 dissertations published between 1930 and 1970. The combined 1,550
documents were each categorized and annotated, but no effort was made
in the reviews to summarize the findings nor draw any conclusions.
The annual updates of the review's original bibliography appeared in
the subsequent editions of the Journal for Research in Mathematics
Education (Suydam & Weaver, 1973, 1974, 1975, 1976, 1977).
Collectively, the published annual bibliographies were— for the
purposes of this study— the most valuable source for locating the
majority of previous research studies. At the same time, it should be
noted that no attempt was made in any of the publications to analyze
the aggregate results of the research.
The first known attempt to analyze and synthesize the results of
studies pertaining to selected areas in the teaching of arithmetic was
made by Bartram in 1956. Some of the relevant areas of his investiga
tion were problem-solving procedures, general teaching methods,
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instructional materials, and provision for individual needs. Bartram
selected 200 studies out of the 600 which were available. Bartram's
criteria for selection of the 200 studies were (a) the restriction of
experimental factors; (b) the control of variations in pupil factors;
(c) the control of nonexperimental factors, such as instructional
methods and materials; (d) the validity and reliability of the meas
uring instrument; (e) the adequacy of the sampling technique; and (f)
the statistical significance of the findings. The selection criteria,
employed by Bartram, especially statistical significance, resulted in
the exclusion of several studies which contained valuable information.
Bartram followed three procedures in treating each area: (a)
identification and analysis of points of view concerning the area,
(b) detailed presentation of research, and (c) forming conclusions and
suggesting teaching practices. His more significant conclusions in
these areas are summarized as follows:
1. Arithmetic problem solving was learned better through the
medium of an activity program than through one more rigidly organized;
also, computational skills were learned equally well. In addition,
activity programs contributed more to the child's social and emotional
adjustment.
2. Inductive methods were superior to deductive for teaching
process and generalizations.
3. Children profitted from meaningful practice. Such practice
should be motivated by a social need and should follow the acquisition
of understanding.
4. A combination of the activity method and grouping within the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 classroom appeared to be the most satisfactory means of meeting the
needs of the individual children.
Bartram's (1956) research focused on a critical examination of
teaching practices, rather than on an evaluation of a new program.
Consequently, Bartram's findings were not directly related to the
present study.
A second effort was undertaken by Hartley in 197 7, as mentioned
in Chapter I. Hartley (1977) collected and synthesized 153 experi
mental studies which pertained to the efficacy of four techniques of
mathematics instruction. Hartley utilized a meta-analysis methodology
for conducting an aggregate analysis of the results.
The four techniques Hartley examined were computer-assisted
instruction, cross-age and peer tutoring, individual learning packets,
and programmed instruction. The effectiveness of each technique was
determined by comparing the resulting mathematics achievement of stu
dents taught by the particular technique with the achievement of stu
dents taught by a traditional method of instruction. In addition to
teaching methodology, other variables such as design characteristics,
subject variables, and attributes of instructional technique were
quantified. The resulting relationships to effectiveness of each
technique were also examined.
Briefly, Hartley found, on the basis of the studies collected,
that tutoring was the superior technique for increasing mathematics
achievement. Peer tutoring, in which classmates tutor one another,
was as effective as paid adult aides; however, cross-age tutoring, in
which older students tutor younger students, was slightly better than
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 the peer and adult tutoring. Also, Hartley discovered that intensive
supervision and instruction of the student tutors did not result in
an increase in effectiveness, although some instruction was beneficial.
In addition, Hartley noted that computer-assisted instruction was
less effective than tutoring in increasing the effectiveness of mathe
matics instruction, although it was considerably more effective than
either individual learning packets or programmed instruction. Indi
vidual learning packets and programmed instruction were essentially
comparable to traditional instruction; that is, these two techniques
resulted in little or no achievement gain over traditional instruction.
In many cases, traditional instruction was definitely superior to
individual learning packets and programmed instruction.
The Hartley (1977) study was a later off-shoot of the innovative
programs in mathematics and it did not strictly come under the defini
tion of new mathematics given in Chapter I. Consequently, the results
of the study were not directly applicable to the purposes of the
present study.
It was clear, therefore, that no comprehensive attempt had been
made to review the large volume of research studies on the effective
ness of modern mathematics by any aggregate technique of analysis.
The present study attempted to utilize an appropriate technique of
analysis to resolve the conflicting and dissimilar findings on modern
mathematics programs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28
Techniques of Aggregation
One of the early techniques of aggregation of studies was the
narrative. The narrative was mainly a chronologically arranged verbal
description; however, this technique failed to portray the accumulated
knowledge of large numbers of studies due to the cumbersome length of
the resulting narrative. As a consequence, the review of literature
was created as an improved technique of extracting information from
groups of studies.
The review of literature usually involved three processes. They
were: (a) gathering the relevant empirical studies, (b) discarding
studies with inadequate empirical designs, and (c) drawing conclusions
from the remaining studies. In order to draw conclusions, the
reviewers often made use of a listing of studies with a tally kept of
the number of significant and nonsignificant results. Conclusions
favored the results found in the larger number of studies in case of
dissimilar or conflicting findings.
One of the drawbacks of the review of the literature was that it
did not take into account the influence of the variables that could
affect the results of a study in a systematic way. For example, some
of these variables were the quality of design, ability level, age and
sex of subjects, length of treatment, and date and source of study. A
second drawback was that charts and tables were often used by
reviewers to differentiate the results; however, when encountering
large numbers of studies, the charts and tables proved to be inadequate
means to fulfill the purpose.
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A third drawback of the review of research was that it often
eliminated the "poorly done" studies. Glass (1976) said that it was
an empirical question whether the so-called "poorly done" studies
gave results significantly at variance with those of the best designed
studies. He believed that because the difference was so small, the
integration of research results through eliminating the "poorly done"
studies would result in discarding a vast amount of important data.
In addition to the narrative and the review of the literature,
other techniques of aggregation have been utilized for summarizing
accumulated data. Averaging, voting, and the cluster approach were
the three major methods Light and Smith (1971) explained in their
paper, "Accumulating Evidence." These three methods are briefly dis
cussed in the ensuing paragraphs.
The Averaging Method
The averaging method was a more systematic approach than mere
listing and simple narratives of the studies. The approach consisted
of computing overall means for relevant statistics for a large number
of studies. The measures of central tendency were used to compute the
overall average which was used in the averaging method. The method
could be considered to be the first attempt of quantification, but the
weakness was that it threw away precisely the information needed most,
namely, a true estimate of the treatment effect.
The Voting Method
A more systematic and widely used approach to aggregation was the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 voting method. In the approach, all studies showing a relationship
between a particular dependent variable and an independent variable
were examined for their significance. The studies were classified
under one of three categories of relationship— significantly positive,
significantly negative, or neither significantly positive nor signif
icantly negative. Tallies were made for each of the three categories.
The category that had the maximum number of tallies was assumed to
give the best estimate of the direction of the true relationship
between the two variables under consideration.
The weakness in the method was threefold. First, valuable
descriptive information was disregarded. Secondly, the method did not
integrate measures to determine the strength of experimental effects
or relationships among variables (according to whether the problem
was basically experimental or correlational). Third, aggregation by
voting did not take into account the possibility of Simpson’s Paradox
(Glass, 1977). Perhaps an example would illustrate the paradox.
Assume two studies on the effects of individualized instruction
compared with the traditional method were made in regard to achieve
ment. In Study I, 50 students received individualized instruction
and 40 students received traditional instruction. In Study II 50 sub
jects were in the individualized instruction group and 300 were in the
traditional instruction group. Tables 2 and 3 show the number of
passes, failures, and the pass rates in each study. The pass rate
for individualized instruction exceeded the traditional by 42% versus
40% in Study I. In Study II the pass rate for individualized instruc
tion exceeded the traditional by 56% versus 55%. Utilizing the voting
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method, the score would be 2-0 in favor of individualized instruction;
however, an opposite result would be attained if one aggregated the
raw data.
Table 2
Study I
Individualized Traditional Total
Pass 21 16 37
Failure 29 24 53
Total 50 40 90
Pass Rate 42% 40%
Table 3
Study II
Individualized Traditional Total
Pass 28 165 193
Failure 22 135 157
Total 50 300 350
Pass Rate 56% 55%
In Table 4 the pass rate was reversed. The traditional exceeded
the individualized instruction by 53% versus 49%. The cause of this
paradoxical situation was the problem of unbalanced experimental
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design. Since the voting method did not pay attention to the sample
size, there existed the possibility of misinterpretation of results.
Table 4
Studies I and II Combined
Individualized Traditional Total
Pass 49 181 230
Failure 51 159 210
Total 100 340 440
Pass Rate 49% 53%
The Cluster Approach
As a more refined technique, the cluster approach had been elab
orated upon by Light and Smith (1971) in an attempt to devise a tech
nique for resolving contradictions among different studies. The
approach stemmed from a pragmatic insight that many populations could
be broken down into small, identifiable subpopulations which were
called clusters. The clusters were not random samples from a popula
tion. Rather, they were natural aggregations within the population.
The clusters usually differed in broad and systematic ways. Thus, a
study could contain several clusters according to the various aggrega
tions within the population.
The main difference between the cluster approach and the voting
method was in the selection of the unit of analysis. In the voting
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method the unit of analysis was a complete study. Conversely, in the
cluster approach the unit of analysis was a cluster which could be a
part of a study or a complete study.
The primary steps involved in the method of aggregation consisted
of access to the original data, review of the quality of the studies,
tests for differences among clusters, and classification and combina
tion of the studies according to the availability of the explanation
of differences.
Figure 1 gives an abbreviated overview of the basic logic of the
cluster approach. The first block identified the five kinds of cluster
differences for which one tested. If all the five tests failed to find
differences, data from the several clusters could be combined. If few
of the tests indicated differences, an explanation of these differ
ences was sought. The data were combined after adjustments had been
made to eliminate the differences. If no explanation was found for
the differences, data could not be combined.
The cluster approach presented problems in its application.
Since the cluster approach required access to the raw data, the study
became overly restrictive. The approach also eliminated several
studies because of the statistical and methodological restrictions.
The subjects had to be selected from a known and precisely definable
population. The dependent variables and the independent variables
which were measured had to have a common measure or be subject to
suitable conversion methods. The overall quality of the studies
selected had to be comparable. Another serious problem was the
inability to draw any conclusion in the absence of explanations for
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34
Test for differences among clusters in: (a) Means (b) Variances (c) Covariate relations (d) Subject-treatment interactions (e) Contextual effects differences found
One or more differences found
Search for explanation: (a) Selection effects (b) Amplification effects (c) Sensitization (d) Different proportions of types of subjects in clusters (e) Contextual differences (f) Unmeasured variables (g) Other
No explanation found Explanation found Adjust away explained cluster differences
Cannot Combine combine data data from from cluster clusters
Figure 1. Overview of the cluster approach. (From "Accumulating Evidence: Procedures for Resolving Contradictions Among Different Research Studies" by R. J. Light and P. V. Smith, Harvard Educational Review, 1971, _41, 463.)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35
differences. These restrictive criteria, therefore led to a loss of
information.
Conversely, the present study attempted to include as many
empirical findings as possible in order to analyze and summarize the
voluminous body of research data pertaining to new mathematics cur
ricula. The intended result was to determine the meaning of these
findings without the loss of information. In this regard, a meta
analysis technique, as explained by Glass (1976), was adopted for the
study. The primary reason for utilizing meta-analysis was that it
diminished the weaknesses contained in the aforementioned techniques
in many respects. The concept of meta-analysis is thoroughly dis
cussed in Chapter III.
Summary
Chapter II dealt with rationale for the present study. The
focus of the rationale was on research in modern mathematics and
techniques of aggregation. The section pertaining to research on
modern mathematics was subdivided into three areas: (a) appraisals
of modern mathematics, (b) studies on modern mathematics by primary
analysis, and (c) studies on modern mathematics by aggregate analysis.
The discussion on appraisals of modern mathematics revealed that there
existed conflicting viewpoints regarding the efficacy of modern math
ematics. Since the studies on modern mathematics by primary analysis
were numerous and their findings were diverse, an aggregate analysis
was necessary to draw a conclusive meaning from the findings. The
number of studies on modern mathematics by aggregate analysis was
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small. Among the few existing ones, none focused upon the area of
investigation of the present study.
In the discussion of the techniques of aggregation five methods
were identified. They were narrative, reviews, the averaging method,
the voting method, and the cluster approach. Strengths and weaknesses
of each technique were discussed. The examination of these techniques
revealed the necessity for a better technique of analysis for the pur
poses of this study. The present attempt, therefore, was the first
aggregate analysis of the effects of modern mathematics by using a
meta-analysis technique.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER III
DESIGN AND DATA ANALYSIS
The purpose of Chapter III is to describe the theoretical and
practical aspects of meta-analysis methodology. The theoretical
aspect consists mainly of the explanation of the effect size and the
techniques of calculating effect sizes. The practical aspect deals
with identifying the sources, locating and selecting the studies,
identifying and coding the variables, and the procedures for analyzing
the data.
Concept of Meta-analysis
An aggregate analysis was deemed to be appropriate to resolve
the dissimilar and conflicting nature of the study findings about the
new mathematics. Many of the techniques of aggregation used in the
past were inadequate for the purposes of the present study. A meta
analysis procedure, introduced by Glass (1976), was found to be an
improvement on the techniques discussed in Chapter II.
In an attempt to explain the concept of meta-analysis, Glass
(1976) made a distinction among primary, secondary, and meta-analysis.
Primary analysis was defined as the original analysis of data in a
research study. Secondary analysis was thought of as the reanalysis
of data for the purposes of answering the original research question
with better statistical techniques. Also, secondary analysis could
be utilized for answering new questions with old data. Conversely,
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38
meta-analysis was the analysis of analyses. In the present study,
meta-analysis, as defined by Glass, was adopted as an alternative
technique to analyze the aggregate findings on the effects of modern
mathematics.
The literature on meta-analysis consisted mainly of the writings
of Glass. Two of his papers which specifically focused on meta
analysis were "Primary, Secondary, and Meta-analysis of Research"
(Glass, 1976) and "Integrating Findings: The Meta-analysis of
Research" (Glass, 1978). The former provided an overview of the meta
analysis methodology and the latter pertained to the different tech
niques involved in the meta-analysis procedure.
Glass explained meta-analysis as a statistical technique which
analyzed a large collection of analysis results from individual studies
for the purpose of integrating the findings. The method connoted a
rigorous alternative to the casual narrative discussions of research
studies which typified attempts to make sense out of the rapidly
expanding body of research literature. A special feature of Glass'
meta-analysis technique was that it statistically aggregated, rather
than merely listed, the results of a large number of studies. Also,
it used summary statistics instead of raw data for aggregation.
One of the advantages of meta-analysis was that conflicting or
dissimilar studies were not eliminated on the basis of a lack of
explanations for differences. Large numbers of conflicting studies
were combined to yield an overall conclusion by calculating what
Glass called the "effect size" (Glass, 1976). Theoretically, effect
size as used in this study was a standardized measure of the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. effectiveness of the treatment when compared to an untreated control
group.
Effect Size
To elaborate, the key concept in Glass' meta-analysis method
ology was the effect size. The effect sizes were calculated for each
study in a measure that was common for all studies. Then the meas
ures were combined to yield more information than just a significant-
not-significant count. The result indicated the difference between
the effect of a treatment and its comparison treatment. Thus, the
effect size yielded information pertaining to the specific treatments
in the study. Specifically, each effect size measured how much more
or less effective the modern mathematics was than the traditional
mathematics counterpart.
Effect size was defined as the difference between the means of
the experimental and control groups, divided by the standard devia
tion of the control group (Glass, 1978). The formula was as follows:
(mean of treatment) - (mean of control) Effect size (standard deviation of the control group) ' '
Symbolically, the formula can be written as:
X X E C ES S (2) X
where XE the mean of the experimental group
X C the mean of the control group
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40
= the standard deviation of the control group
and ES = the effect size
A positive measure of effect size indicated that the findings
favored the treatment group while a negative measure indicated that
results supported the control group. For example, if the effect size
was k standard deviations, then an average person in the control
group was k standard deviations below the average person in the
treatment group.
By dividing the difference of means by the standard deviation,
the effect sizes were standardized and were readily converted into
z-score units. Consequently, the effect sizes were appropriate for
comparison even though they were derived from different studies.
Also, corresponding to each effect size, a percentile score could be
obtained from the table which indicated the areas and ordinates of
the unit normal distribution.
For example, let the effect size comparing the effectiveness of
two treatments— a traditional and a modern— be 0.49. Then, the value
0.49 indicated that an average person receiving the modern treatment
had been raised from the 50th percentile to the 69th percentile of
the traditional group, as depicted in Figure 2.
In Figure 2, the normal curve of the dotted lines represented
the control group or the traditional group. An average student in
this group occupied the midpoint of the baseline. This point coin
cided with the zero point of the z-score. The average person, there
fore, was said to be at the 50th percentile of this normal curve,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41
CONTROL GROUP TREATMENT GROUP (TRADITIONAL) (MODERN)
50th Percentile 69th Percentile
Average Effect Size is 0.49
Figure 2. Normal curves illustrating the effect of modern treatment in relation to traditional treatment— a hypothetical illustration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 which represented the traditional mathematics population. The normal
curve of the solid line represented the treatment group.
The assumed value of the effect size in Figure 2, namely 0.49,
being a measure of the difference between the means, was represented
by the distance between the midpoints of the two normal curves. The
midpoint of the curve which represented the modern group coincided
with the 69th percentile of the traditional group. (The percentile
value corresponding to the z-score of 0.49 was 69.)
Thus, given an effect size of 0.49, a person who was at the 50th
percentile of the modern group was, at the same time, at the 69th
percentile of the traditional group. In other words, the treatment
of the modern program under the average conditions could be expected
to move an average student from the 50th to 69th percentile of the
traditional mathematics population, if the mean effect size was 0.49.
The same explanation was applicable to all specific mean effect sizes,
except for the fact that the effect sizes were different in each par
ticular case.
The mean effect size was a powerful tool in summarizing the
results. The findings could be summarized at the general as well as
specific levels into which the independent variables of this study
were categorized, by using various measures of mean effect size. At
the most general level, average of all the effect sizes were found.
This was called the "grand mean effect size" or the "overall mean
effect size." The terms "mean" and "average" were used interchange
ably.
In the present study, modern mathematics population represented
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 the treatment group and the traditional mathematics population, the
control group. The outcome variable was either achievement or atti
tude measured in terms of effect size.
The hypothetical effect sizes of three different subject areas
are compared in Figure 3. In all three cases, a modern mathematics
group was compared with the traditional mathematics group. Since the
traditional mathematics group was not given any treatment in modern
mathematics, it was considered to be an untreated control group whose
average effect size was always zero. The four normal curves, there
fore, represented the typical three treated populations in relation
to the untreated control subjects in the subject areas of arithmetic,
algebra, and geometry. For example, 102 effect size measures from
modern algebra studies averaged about 1.18 standard deviation.
The major impression derived from Figure 3 was that modern alge
bra and geometry were not substantially different from each other in
their average impact on learning. Also, one could deduce that both
modern algebra and geometry groups performed better than the tradi
tional mathematics group.
In addition, data from Figure 3 illustrated that the children
who received the modern arithmetic treatment performed worse than the
traditional mathematics group. Thus, the various measures of effect
size illustrated the comparative effectiveness of different modern
mathematics treatments among themselves as well as their comparative
effectiveness with the traditional mathematics treatment.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TRADITIONAL EFFECT = 0 MATHEMATICS 50th NUMBER OF EFFECT Percentile SIZES = 150
MODERN EFFECT =1.18 ALGEBRA NUMBER OF EFFECT centile SIZES = 102
MODERN EFFECT =1.28 GEOMETRY 90th NUMBER OF EFFECT Perce itile SIZES = 205
MODERN EFFECT =0.25 ARITHMETIC NUMBER OF EFFECT SIZES = 180 /40 th Peroentile
Figure 3. Normal curves illustrating the effects of three different treatments of modern mathematics (modern algebra, modern geometry, and modern arithmetic) in relation to untreated traditional group— a hypothetical illustration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 Techniques of Measuring Effect Sizes
Different study designs required different techniques to measure
the numerator and the denominator in the equation of the effect size.
Glass (1978) discussed many techniques in his paper, "Integrating
Findings: The Meta-analysis of Research." In the ensuing sections
the techniques which were used for the present study are enumerated.
Techniques for determining the numerator were as follows.
1. The numerator of a typical experiment (in which the subjects
were randomly assigned to treatment and control groups, or the pre
test scores indicated that the groups were equivalent) was the differ
ence between the means of the posttests. The pretest scores did not
enter into the computation.
2. If the design made use of analysis of covariance, and if the
adjusted means were reported, then the numerator was the difference
between the adjusted means.
3. If a design reported the gain scores— pretest scores sub
tracted from the posttest scores— then the numerator of the effect
size was the difference between the averages of the two gain scores.
4. If a design reported only grade equivalent scores from a
standardized test, then the numerator was calculated by the length of
the treatment. This procedure assumed that theoretically an untreated
average group should gain one point for each month in the school.
Thus, if a study indicated a pre-post difference of 2.0 grade equiv
alents after the treatment of eight months, the numerator of the
effect size was 2.0 - (1 x 0.8) = 1.2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 Correspondingly, the methods for determining the denominator of
the effect size were as follows.
1. Whenever it was possible to obtain the standard deviation of
the control group, then that was used as the denominator of the effect
size.
2. If only the means based on scores from a standardized test
were given, the standard deviation of the norm group was taken as the
denominator of the effect size.
3. If only the t-statistic was available, then effect size was
derived from the formula:
1 1 ES n + n (3) 1 2
where n = the number of subjects in the control group 1 the number of subjects in the treatment n2 = group
The equation was derived from the formula (2) given earlier and
from the equation for the t-statistic, which was given by the formula:
X E t (4)
Ct * Formula (2) was:
ES
By combining formulas (2) and (4), formula (3) was derived.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 4. If a report contained only the information that the mean of
the experimental subjects exceeded the mean of the control subjects
at certain Of-level significance, then an approximate value of the
effect size was calculated, assuming the t-value for the particular
©(-level. For example, if Gf = .05, then the most conservative value
of t was equal to 1.96. The value 1.96 was obtained from the table
showing the percentile points of t-distributions in standard textbooks
dealing with statistics. By substituting this value for t in formula
(3) the effect size was obtained from the following formula:
5. If the report indicated only an F-value and there were only
two groups, the positive square root of F had to be substituted for t.
Glass (1978) discussed other techniques, such as probit analysis
and logit transformation, in his paper, "Integrating Findings: The
Meta-analysis of Research." However, these techniques were not
required in the present study since all of the data reported were
readily convertible to effect size by the previously discussed tech
niques .
Multiple Effect Size
In the process of calculating effect sizes, care was taken to
prevent the problem of multiple effect size. The problem of multiple
effect size arose when a study had more than one effect size. When
a study had more than one effect size, certain effect sizes could be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48
repeated either in part or whole. For instance, if a study dealt
with two grade levels, then effect sizes could be found for each
grade level separately and for the entire group combining the two
grades. Thus, three effect sizes were possible from a study consist
ing of two grade levels. If all three effect sizes were considered
for the grand mean effect size, the study would be contaminated by
the multiple effect size, because the effect sizes of each grade level
and the effect size of the combined group represented the same value.
The problem could be avoided by choosing either the two effect sizes
obtained from each grade levels— one from each grade— or the effect
size of the combined group, but not both sets.
The problem of multiple effect sizes became more acute when the
design utilized was a factorial study involving sex or ability. For
example, if a study included four grades and sex as a factor, then it
would have four effect sizes on grade, one for each grade, and two on
sex. In the overall analysis of studies, either the first four effect
sizes or the last two would be included, and not both sets.
Sources of the Studies
The success of meta-analysis depended on collecting large num
bers of research and evaluation studies from as many sources as
available. The first task, therefore, was to identify the sources
where studies had been done in the area of interest. Accordingly, a
wide variety of sources were examined in the search for appropriate
studies to be utilized in the meta-analysis. Information pertaining
to potential studies to be included was obtained primarily from the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49
reference entitled Dissertation Abstracts. Index to Education, Edu
cation Resources Information Center (ERIC) documents, mathematics
education and psychological journals, research reviews, and govern
ment publications were also helpful in providing information about
research studies which pertained to the area under investigation.
Locating and Selecting Studies
The sources helped advance progress toward the general goal of
conducting a meta-analysis— a quantitative aggregation of a large
number of research studies. After the primary sources of studies
were identified, the next task in the process was to locate specific
studies which pertained to new mathematics. This was accomplished
through a systematic and thorough search of the key sources— Index to
Education, Dissertation Abstracts Index, and ERIC for the years from
1951 to 1977. To systematize the process of locating and collecting
studies, the researcher adopted certain criteria for study selection.
The studies had to satisfy all criteria.
Broadly, the criteria for study selection were divided into pri
mary and secondary. The primary set of criteria were the following:
(a) the studies had to be experimental as opposed to descriptive or
theoretical, (b) the dependent variables were to be limited to achieve
ment or attitude, (c) tests of achievement or attitude were either
standardized or experimenter developed or teacher developed, and (d)
the area of experimentation had to be within the limits of the range
of elements discussed in the definition of new mathematics in Chapter I.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50
The primary set of criteria was intended to select studies from
abstracts and summaries of research studies in the areas of investi
gation; however, the criteria were not adequate to detect other
characteristics which were necessary for the final selection of
studies. Hence, a set of secondary criteria were established for the
final selection process. The secondary criteria were the following:
(a) studies comparing two or more groups had a control group which
received traditional mathematics treatment, (b) studies adopted ade
quate method of analysis, and (c) studies related to the concepts
introduced into mathematics instruction during the 1950's and 1960's.
Once the criteria for selection were established, the search for
locating the studies started. The search was conducted in two stages.
Initially, a variety of approaches was utilized to identify studies
from different sources. For example, separate computer searches
were made of the ERIC documents and Dissertation Abstracts by using
appropriate descriptors. The searches yielded approximately 300
studies which were potentially usable.
In addition to dissertations and ERIC, a genuine attempt was
made to locate and collect studies or results of studies from text
book publishers. Parenthetically, textbook publishers who played an
active role in the production and promotion of modern mathematics
materials frequently conducted research on the methods and techniques
incorporated within their textbooks. The publishers were identified
from the list of modern mathematics in EL-HI Textbooks in Print 1978.
Unfortunately, the search elicited responses which were predominantly
unusable.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 The search for studies on modern mathematics programs from
developmental projects was revealing on several dimensions. One,
many of the developmental projects were no longer in existence.
Among those which are in operation, ironically, no project indicated
that it possessed any studies which measured the effectiveness of
that particular program. A few of the research centers, such as the
Ohio State University ERIC Center for Science-Mathematics and Envi
ronmental Education, proved to be useful in citing some of the rele
vant studies and locating their sources. Approximately 50 studies
were located through this source.
At the end of the search through research centers, the investi
gator made a thorough search of the Dissertation Abstracts. In the
Dissertation Abstracts more than 300 studies were identified which
pertained to the area of investigation. Ninety-two dissertations
were selected from the 300 studies for meta-analysis in accordance
with the primary criteria. Copies of the 92 dissertations were
obtained from the University Microfilms, Ann Arbor, Michigan.
As mentioned, a second search was completed for the additional
studies. The search was based on the information gathered from the
bibliographies and references of the previous set of studies. In the
second search, 68 additional dissertation studies were identified and
the copies of these dissertations were obtained from the University
Microfilms, Ann Arbor, Michigan.
Many additional studies were located through a variety of
sources. Some of them were identified in more than one source.
Eliminating the repetitious studies, more than 200 studies were
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 originally identified through the brief descriptions given in the
form of abstracts or summaries as potentially providing relevant data
for the present analysis. The identification of these studies was
accomplished by the primary set of criteria mentioned earlier.
While the investigator reviewed the original studies, it was
necessary to eliminate those which met all the primary criteria, but
did not satisfy some of the secondary criteria. The studies which
were excluded fell into one of the following categories.
1. Studies comparing two or more groups which received new
mathematics treatments, but did not have a control group which
received traditional treatment. For example, a study which compared
a guided discovery group with a hinted discovery group belonged to
this category.
2. Studies which adopted an inadequate method of analysis. A
study which showed the comparative effect only by the difference in
percentage of pass or failure of the two groups typified this cate
gory.
3. Studies which belonged to the later developments in mathe
matics instruction were also excluded. This category included studies
pertaining to the existential approach, self-supervision, individual
ized instruction, television-aided instruction, computer-assisted
instruction, etc.
In the final screening, 134 studies were selected for meta
analysis from three different sources. The sources were the ERIC
documents, journals, and doctoral dissertations. Of the three
sources, doctoral dissertations contributed more than three-fourths
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 of the studies.
Identifying and Coding Variables
After the studies were selected through the processes of locating,
collecting, and screening, information about the individual studies
was extracted. Compilation of information about the studies was
carried out through an identification and coding process for the rele
vant variables. Since meta-analysis research is concerned with the
description of the relationships among findings and the characteris
tics of the studies, care was taken to identify as many characteris
tics as possible and classify them as specific variables.
Identification of the variables was a continuous process. Prior
to reviewing each study in depth, the descriptive variables, such as
number of students, grade level, type of treatment, length of treat
ment, year of study, rating, and the quality of design were identified.
Later these variables were classified under four categories: the
subject characteristics, treatment factors, study variables, and
design variables. Identification of a variable consisted of three
steps: (a) naming the variable, (b) defining the purpose the variable
serves, and (c) assigning values to the variable.
Originally, 31 variables were identified for coding. While
reviewing the studies, it was necessary to make modifications in the
original list of variables. Major modifications were introduced in
the range of values for certain variables. Two new variables were
added to the original list: the generic content area and the sub
achievement area. Variables such as retention, time, transfer time,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 the dependent variable, the length of treatment per day, the length
of the treatment per minute, subachievement area, and new versus
field-tested, were not considered for analysis, since the number of
effect sizes in those cells was too small. The list of variables
and the coding sheet are presented in Appendices A and B, respectively.
The list of variables provided the name, purpose, and range of the
values, while the code sheet consisted of the code, the name of the
variable, the columns, and the values.
Coding the necessary information from the selected studies
involved a great deal of concentration. For convenience, the col
lected studies were first of all arranged in a chronological order.
Then, each study was reviewed in its entirity to identify and assign
the proper values for the quantitative and descriptive characteris
tics which were listed in the code sheet. The descriptive data were
spread throughout the study, while the quantitative data were mostly
located in chapters dealing with methodology and results. Occasion
ally, tables of basic summary statistics, such as mean, standard
deviation, and sample size, were provided in the appendix.
Each code sheet contained one effect size and all other variables
related to the effect size. Consequently, the data in one code sheet
formed the unit of analysis. From most of the studies more than one
effect size was calculated. When there were many effect sizes in one
study, the photostat copies of the pages in the original studies,
which contained the necessary data for the calculation of effect
sizes, were obtained. The copies were attached with the respective
sets of code sheets which comprised the study. The procedure
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 facilitated the calculation of effect sizes and further reference.
Analysis of Data
After the information about the studies was coded in a systematic
manner, the findings were analyzed. The analysis of the results of
the present study was expected to fall under four major categories.
The first category was the overall effect of new mathematics pro
grams. This was divided into two subcategories: (a) the effect in
terms of achievement and (b) the effect in terms of attitude. The
overall effect size was measured by calculating the average effect
size of all the studies taken together.
The second category of results consisted of effect sizes which
were computed for each relevant independent variable. These effect
sizes were calculated to show the relative significance of the dif
ferent characteristics under consideration.
The calculations involved in the computation of effect sizes
were completed with the use of a hand calculator. The analysis of
data, such as the mean effect size, correlations, and regressions
were done by computer. The computer analysis cards were punched from
the code sheets. The data were entered into the computer from the
cards. The computer analysis was performed by using Statistical
Package and Bank Data Management Package, prepared and programmed by
Houchard (1974) for the Western Michigan University Computer Center.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Procedures for Judging the Magnitude and Significance of Effect Size
As previously discussed, the main tool for the analysis of the
results was the mean effect size. Consequently, judgments on the
results of the findings were heavily dependent upon the mean effect
sizes. In order to make sound judgments about the results, it was
necessary to have a clear perception of the magnitude and signifi
cance of the mean effect sizes.
Procedures to judge the magnitude and significance of effect
size of a group of studies were not detailed in literature on meta
analysis. The most obvious and direct meaning of the effect size was
the impact of a new treatment in terms of a shift from the mean point
toward the positive or negative direction of the unit normal curve.
The shift was measured in standard deviations which were readily con
vertible to percentiles. Any effect size which corresponded to a
percentile greater than the 50th suggested a positive impact or an
improvement by the new treatment over the untreated group; con
versely, any effect size which corresponded to a percentile less than
the 50th indicated a negative impact or a deterioration. The greater
the average effect size was, the greater the impact was.
To determine whether the effect size of a group of studies was
a chance factor or not, other procedures were necessary. In the
present study, t-values and the exact probabilities were calculated
corresponding to each average effect size of a group of studies.
This was done in order to determine the significance of the effect
size. The t-values were calculated under the assumptions that studies
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 on modern mathematics formed a sampling distribution and the hypo
thetical mean of the traditional mathematics population was at the
zero point. Thus, the t-values were obtained from the formula:
(Effect size) - (Zero) t ~ /Standard error of mean of') (the effect sizes J
The degrees of freedom needed in addition to the t-value for the
calculation of the exact probability were obtained by subtracting one
from the number of effect sizes. Since the studies on modern mathe
matics indicated positive as well as negative impacts, the probabil
ities corresponding to the t-values were calculated for the two-
tailed test. The effect sizes were then considered to be significant
or not significant at a certain level of significance. The probabil
ity of committing type I error for the present study was taken to be
0.05.
To help visualize more of the nature of the central tendency
and of the dispersion of the distribution of the effect sizes, the
values of the median and of the standard deviation were provided .
along with the average effect size.
Summary
The discussion of the methodology for the present study focused
mainly in two areas: the theoretical aspects of meta-analysis and
the practical steps involved in its implementation. The theoretical
section consisted mainly of an explanation of the concept of meta
analysis, the effect size, and the description of different techniques
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 of measuring effect size. The practical steps involved in the imple
mentation of meta-analysis consisted of identifying the source of the
study, locating and collecting studies, identifying and coding the
variables, and the analysis of the data.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV
RESULTS
The purpose of Chapter IV is to present the results of the anal
ysis of the data. The presentation of the results consists of a
brief description of the studies included and a detailed discussion
of the effects of the variables for which the effect sizes were cal
culated. The effects were classified into five groups: (a) the over
all effects of modern mathematics in achievement and attitude, (b) the
specific effects by subject characteristics, (c) the specific effects
by treatment factors, (d) the specific effects by study variables,
and (e) the specific effects by design variables.
Description of the Studies Included
A description of the studies was intended to give a picture of
the general nature of the studies included in the present analysis.
The factors that describe the general nature are the sources from
which the studies were collected, the year in which the studies were
published, the length of treatment of the studies, and the number of
subjects per study.
Source of Study
One hundred and thirty-four studies were identified from three
different sources. From the 134 studies, 810 effect sizes were cal
culated. The exact number of studies and effect sizes corresponding
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 to each source is shown in Table 5.
Table 5
Studies Classified by Source
Number of Number of Source Studies Effect Sizes
ERIC Documents 4 39
Journal Articles 16 127
Doctoral Dissertations 114 644
Total 134 810
Year of Study
The studies utilized came from the period between 1951 and 1977.
The majority of the studies were conducted during the second half of
the 1960's. The median year for the studies and the effect sizes was
1968. In Table 6, the number of studies and the number of effect
sizes corresponding to each year are listed.
Length of Treatment and Number of Subjects Per Study
The length of treatment of studies ranged from 1 week to 144
weeks. The average length of treatment was approximately 33 weeks.
The number of subjects per study ranged from 4 to 2,862. The average
number of subjects per study was 108. In order to obtain a clearer
perspective of the central tendencies along with the mean of the two
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61
Table 6
Studies Classified by Year
Year of Number of Number of Publications Studies Effect Sizes
1955 1 12
1956 1 4
1958 1 1
1959 6 20
1960 2 19
1961 4 11
1962 4 20
1963 5 59
1964 5 19
1965 10 47
1966 9 96
1967 13 73
1968 10 83
1969 13 70
1970 14 107
1971 8 47
1972 5 30
1973 2 9
1974 3 25
1975 6 26
1976 7 20
1977 5 12
Median Year: 1968 1968
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. variables, the measures of the median and mode are provided in Table
7. The maximum and minimum values of the variables are also provided
in the same table. This was done in order to indicate the direction
of the range of observations under consideration.
Table 7
Length of Treatment and Number of Subjects Per Study
Number of Maxi Mini Observa Variable Mean Median Mode mum mum tions
Length of treatment 33.84 24 36 144 1 810 (in weeks)
Number of subjects 206.90 108 24 2,862 4 810 per study
Overall Effects of Modern Mathematics
Description of the studies provided £i picture of the general
nature of the studies included. The ensuing sections will deal with
the quantified measures of the effects of modern mathematics in terms
of achievement and/or attitude.
Chapter III discussed the concept that the "effect size" was a
common measure of treatment effectiveness. This measure was defined
as the difference between the means of the modern mathematics group
and the traditional mathematics group divided by the standard devia
tion of the traditional mathematics group. Based on the preceding
definition, 660 effect sizes in achievement and 150 effect sizes iri
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 attitude were calculated from 134 controlled outcome studies. It
must be remembered that in some studies more than one effect size was
calculated.
The grand mean effect size in achievement was 0.24 and the grand
mean effect size in attitude was 0.12. Table 8 shows the average
effect sizes in achievement and attitude.
Table 8
Grand Mean Effect Sizes in Achievement and Attitude
Statistic Achievement Attitude
Mean 0.24* 0.12*
Median 0.09 0.04
Standard deviation 0.77 0.56
Standard error of mean 0.03 0.05
Number of effect sizes 660 150
t-value 8.00 2.40
Probability 0.00 0.018
Note. Degrees of freedom = Number of effect sizes - 1.
*The null hypothesis, stating the mean effect size is zero, is rejected with the probability of committing type I error being less than or equal to 0.05.
This meant that the average person receiving some form of modern
mathematics treatment was 0.24 standard deviation more improved on
the achievement measure and 0.12 standard deviation more improved on
the attitude measure than the average traditional mathematics group
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64
member (see Figure 4).
The interpretation of Figure 4 followed the reasonings presented
while discussing Figure 2 in Chapter III. The dotted lines repre
sented the normal distribution of the traditional mathematics popula
tion and the solid line represented the normal distribution of the
modern mathematics population. Since the effect size in achievement
was 0.24, a student at the 50th percentile of the modern mathematics
group was equivalent to a student at the 59th percentile of the
traditional mathematics group. Similarly, a student at the 50th per
centile of the modern mathematics group was equivalent to a student
at the 55th percentile of the traditional mathematics group in atti
tude. The effect size of attitude was 0.12. In other words, the
modern mathematics program raised an average student from the 50th to
the 59th percentile in achievement and from the 50th to the 55th per
centile in attitude over the traditional methematics population. The
basic assumption was that an average student of the traditional math
ematics population occupied the 50th percentile in a normal curve.
It is interesting to note that the improvements, reported in
terms of effect sizes which resulted from the treatment of modern
mathematics, indicated a great difference in scores on the achieve
ment and attitude dimensions. In the following sections, treatment
of the specific effects are provided.
Specific Effects by Subject Characteristics
The subject variables described the important characteristics of
the students who were involved in the studies that were utilized.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65
0.24
TRADITIONAL MODERN
50th Percentile 59th Percentile of Control of Control
Average Effect Size in Achievement: 0.24
0.12
50th Percentile 55th Percentile of Control of Control
Average Effect Size in Attitude: 0.12
Figure 4. Normal curves illustrating the aggregate effect of modern mathematics in achievement and attitude in relation to traditional mathematics group.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Four main subject characteristics were analyzed. They were the
grades which were chosen for the experiments, ability levels, the
levels according to socio-economic status, and sex. Classification
of the subjects on the basis of the variables discussed was intended
to enable the reader to form an idea of the effectiveness of modern
mathematics under varying conditions and circumstances. Since the
classification of levels within ability and socio-economic status was
arbitrary in nature, the effect sizes in those subcells were less
specific and reliable than the effect sizes of grade levels and sex.
Effect Size by Grade
Grade levels considered for the present study ranged from kinder
garten to post-secondary. The post-secondary group consisted mostly
of students of the community colleges and their freshmen and sopho
more counterparts at four year colleges and universities. Among the
five grade levels, the elementary had the largest average effect size
in achievement. It was 0.32. An average effect size which was close
to the elementary grades was from the senior high grades. Conversely,
the average effect size at the junior high grades was remarkably low
in comparison to the elementary and senior high grade levels. Table
9 indicates that the average effect size of the kindergarten and the
post-secondary grade levels in achievement was insignificant. This
was possibly due to the limited number of effect sizes in each cate
gory.
An attempt was made, in observing the average effect sizes, to
find the relationship between the significant effect sizes and grade
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.095 0.001* 4.67 0.000* 3.57 t-value Probability 79 88 2.63 0.010* 404 7.33 0.000* Sizes Effect Number of 0.140.04 110.06 249 0.43 140 4.25 0.676 0.000* 0.13 89 2.69 0.009* 0.08 Error of Meanof Standard Table 9 Table 0.47 0.58 0.63 0.03 Standard Levels of Grade Deviation Levels Ability of 0.02 0.09 0.47 0.07 44 1.71 0.17 0.06 0.00 0.28 0.18 0.77 0.35 0.00 1.25 Effect Size by Levels Grade of Ability and Achievementin Mean Median *The null hypothesis, stating mean the effect size is zero, is rejected with the probability of Note. Note. Degrees of freedomNumber = of effect sizes - 1. Senior Senior high Junior high Kindergarten Post-secondary 0.12 committing type error I being less than or equal to 0.05. "J Elementary 0.32 0.14 0.96 0.07 216 4.57 0.000* Low Unspecified 0.22 0.14 MiddleHigh 0.25 0.21 0.07 0.00 0.64 0.79 0.07
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68
levels. This was done by fitting a parabola with a downward bend.
The multiple correlation coefficient of the curvilinear regression
was only 0.09. The value of the multiple correlation coefficient
indicates that there is no significant relationship between the
levels of grades and the effect sizes in achievement.
Effect Size by Ability Level
Most of the studies which compared the effectiveness of modern
mathematics based on ability divided the subjects into three groups,
such as low, middle, and high. In a few instances, the comparison
was made between the high and low ability levels only. In this
study, all three levels of ability were considered. Effect sizes
were classified into low, middle, or high levels of ability according
to the designation in the original studies. A fourth group, desig
nated as "unspecified" ability level, consisted of studies which did
not make a comparative analysis by ability levels.
The average effect sizes were considered to be statistically
significant at = 0.05. Table 9 indicates that the average effect
size in achievement was the largest for the low ability group, which
was 0.35. This value was substantially higher than that of the grand
mean effect size in achievement which was 0.24 (see Table 8). The
average effect sizes of the middle, high, and the unspecified ability
levels did not deviate from the grand mean effect size in any signifi
cant manner. In general, Table 9 shows a decrease in mean effect size
as one goes from the low to the high level of ability.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 Effect Size by Socio-economic Status and by Sex
Socio-economic status and sex were the other two variables which
belonged to the category of subject characteristics. There were only
72 effect sizes out of 660 in achievement where the socio-economic
status of the students was identified (see Table 10). The 72 effect
sizes were not representative of the total population of studies
insofar as the mean of the 72 effect sizes was not equal to the grand
mean effect size. The only effect size which was significant at the
= 0.05 level was that of the students belonging to middle class
parents. The effect size of the middle group was 1.05 which was
conspicuously higher than the grand mean effect size of the present
study.
As in the case of the analysis of socio-economic status, the
cell sizes in achievement were quite small when the data were analyzed
by sex. As Table 10 indicated, the effect sizes of each category in
this particular subset did not represent the total population, since
the average mean for either sex was substantially lower than those
studies not grouped for sex.
Because of the small number of studies in the socio-economic
status and sex categories, caution must be exercised in the interpre
tation of the results.
Specific Effects by Treatment Factors
The previous section dealt with the characteristics of the stu^
dents who were involved in the studies that were utilized; whereas,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.67 0.108 28 0.25 0.805 Sizes t-value Probability Effect Number of 0.12 0.090.31 25 33 3.39 0.002* 0.03 588 7.00 0,000* Error of Meanof Standard in in Sex Table 10 Table 1.77 Standard Deviation of of Socio-economic Status Difference Levels 0.42 Median Effect Size by Socio-economic Status and Sex Achievementin 0.15 0.01 0.47 Mean *The null hypothesis, stating mean the effect size is zero, is rejected with the probability of Note. Note. Degrees of freedom = Number of effect sizes - 1. committing type errorI being less than or equal 0.05. to Female 0.03 -0.05 0.66 Unspecified 0.21 0.78 0.66 Low High 0.22 0.35 0.50 0.13 14 1.69 0.115 Male 0.11 -0.05 0.59 0.12 26 0.92 0.366 Middle 1.05
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71
the treatment factors described the type and content of application
of modern mathematics in the various experiments selected for the
study. Two factors which related to the content of application were
the general content area and the specific subject area. There were
three variables which described the type of treatment. They were the
type of program that had been selected for the treatment of modern
mathematics, the length of treatment, and the number of subjects per
study.
Effect Size by General Treatment Area
Table 11 reveals that 304 effect sizes in achievement were iden
tified as belonging to one of the three selected categories. The
categories were in accordance with the emphasis given in the treat
ment of modern mathematics in the generic area. The three areas of
emphasis were concepts, computation, and application. Studies which
could not be identified as belonging to one of these three categories
were treated in the fourth category, designated as the "combination."
Among the four categories, the highest average effect size was
that of the concepts. Almost similar achievement gain was evidenced
in computation. While the mean effect sizes of the concepts and of
computation were 0.36 and 0.31, respectively, the mean effect size of
the application was only 0.06. In other words, the improvement attrib
utable to modern mathematics in the area of application of the mathe
matical concepts was next to zero, whereas the improvements in the
areas of concepts and of content was quite substantial.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.006* 0.000* 0.000* 1.20 0.234 6.25 0.000* 85 1.75 0.084 Sizes t-value Probability Effect Number of . 1 0.07 134 5.14 0.04 3560.05 5.50 305 0.000* 4.20 0.000* 0.04 196 Error of Mean Standard Table 11 Table 0.88 Standard Deviation General Content Area Specific Subject Area of of effect sizes - Number Effect Size by Content Area in Achievement 0.31 0.08 1.05 0.11 95 2.82 0.21 0.05 Mean Median *The null hypothesis, stating mean the effect size is zero, is rejected with the probability of Note. Note. Degrees of freedom = Combination 0.22 0.09 0.71 Computation Concepts 0.36 0.27 0.78 committing type I error being less than or equal 0.05. to Application 0.06 0.00 0.45 0.05 75 Geometry 0.14 -0.01 0.70 0.08 Arithmetic Other 0.25 0.11 0.62 Algebra 0.43 0.31 0.68 0.08 74 5.38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73
Effect Size by Specific Subject Area
An attempt was made to classify effect sizes into specific sub
ject areas, such as arithmetic, algebra, and geometry. The fourth
section consisted of treatments pertaining to sets, groups, fields,
logic, and some specific areas of mathematics, such as calculus,
statistics, and probability, which were introduced in the language of
modern mathematics.
Among the specific subject areas, algebra had the highest average
effect size with 0.43 (see Table 11) in achievement. The average
effect size in arithmetic was almost close to the grand mean effect
size of the total number of studies in achievement (0.21 versus 0.24).
Table 11 also points out the average effect size of achievement in
geometry (0.14) was lower than that of arithmetic (0.21) and of
algebra (0.43). The mean effect size in achievement of the combina
tion group was typical of the grand mean effect size in achievement
as reported in Table 8. Generally, the mean effect sizes of arith
metic and of the combination group represented the grand mean effect
size. The larger mean effect size of algebra was offset by the
smaller mean effect size of geometry.
Effect Size by Type of Program
Initially, an attempt was made to identify studies according to
the specific development programs discussed in Chapter I; however, the
number of studies belonging to each specific development program other
than the School Mathematics Study Group (SMSG) was either very small
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or nil. The type of program therefore, was broadly divided into SMSG
programs and non-SMSG programs. The average effect size in achieve
ment of studies belonging to SMSG was considerably lower than the
average effect size in achievement of all other studies put together
(see Table 12). Other studies denoted as non-SMSG, included a few
studies on modern mathematics as introduced by other developmental
programs such as the University of Illinois Committee on School Math
ematics Program, the Ball State University Program, the Greater
Cleveland Mathematics Program, and a number of studies on modern
mathematics as introduced by innovators other than the reformers in
the developmental programs.
Effect Size by Length of Treatment
Table 13 reports the average effect size in achievement according
to the length of treatment. Studies with a length of treatment which
was less than 9 weeks and studies with a length of treatment which
was greater than or equal to 36 weeks had the same average effect
size (0.25). The average effect size reached the maximum point,
which was 0.48 in this context, when the length of’"treatment was
between 9 and 18 weeks. Studies with the length of treatment ranging
from 18 to 36 weeks had average effect sizes as low as 0.09. Thus,
the effect sizes indicated a gradual increase until the length of
treatment became 18 weeks or more. After 18 weeks, the effect sizes
gradually decreased until the length of treatment was 36 weeks or
greater. Thereafter, the effect sizes in achievement rose and
remained steady at 0.25, which was close to the grand mean effect
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 2.80 0.006* Sizes t-value Probability Effect Number of . 0.041 533 6.75 0.000* 0.05 Error of Mean of Standard Standard Table 12 0.52 Standard Deviation = Number of effect sizes - Effect Size by Type of Program Achievementin Degrees of freedom *The *The null hypothesis, stating mean the effect size is zero, is rejected with probability the of Note. SMSG 0.14 0.00 Othercommitting type I error being less than or equal to 0.05. 0.27 0.10 0.81 Program Mean Median
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.000* Probability 1.50 0.136 124 Sizes t-value Effect Number of . 0.05 133 5.00 1 0.04 334 6.25 0.000* Error Error of Meanof Standard Standard - - Table 13 0.57 Standard Deviation 0.17 Effect by Size Length of Treatment Achievement in Mean Median *The *The null hypothesis, stating meanthe effect size is zero, is rejected with the probability of Note. Note. Degrees of freedom = Number of effect sizes (weeks) Treatment Length of 36 more36 or 0.25 0.09 0.66 committing type I error being less than or equal to 0.05. Less Less than 9 0.25 Between 9 and 18Between 18 and 36 0.48 0.09 0.00 0.04 1.43 0.64 0.17 0.06 69 2.82 0.006*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77
size in achievement.
Effect Size by Number of Subjects Per Study
As indicated in Table 14, the number of subjects per study was
categorized into five groups. The upper limits of each class interval
of the distribution of number of subjects per study were in geometric
progression with common ratio equal to 2 and the base equal to 50.
With this arrangement, the effect sizes were relatively evenly dis
tributed in each subgroup. The average effect size in achievement
was fairly high for the studies in which the number of subjects was
less than 50. Studies in which the number of subjects was greater
than 200 and less than 400 had a typical average effect size (0.26).
It was only 0.02 standard deviation greater than the grand mean effect
size. The average effect sizes in the other cells of this subgroup
ranged from 0.13 to 0.16.
Specific Effect Size by Study Variables
The study variables outlined characteristics such as the empha
sis in the approach of the study, the source of the studies, the
affiliation of the researcher, and the author who introduced an
innovation. The variables in this category were not manipulative in
nature, but they were related to the outcomes. Collectively, they
described those characteristics of the study which were not treated
under the subject characteristics, treatment factors, or design vari
ables .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.002* Probability 3.75 0.000* 3.25 t-value 118 4.33 0.000* Sizes Effect Number of . 1 0.06 Error of of Mean Standard Table 14 0.67 0.53 0.04 178 Standard Deviation 0.08 0.34 0.04 68 0.07 0.05 0.78 0.06 153 2.67 0.008* Effect by Size Number of Subjects Per Study Achievementin of freedom = Number of effect sizes - *The *The null hypothesis, stating mean the effect size is zero, is rejected with the probability of Note. Note. Degrees Subjects Per Study Mean Median 50 50 or lessBetween 50 and 100 0.16 0.49 0.25 1.10 0.09 143 5.44 0.000* Between 100 and 200 0.15 Between 200 and 400 0.26 0.10 committing type I error being less than or equal to 0.05. More than 400 0.13
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 Effect Size by Study Approach
The variable of study approach allowed the studies to be distin
guished according to one of four areas of emphasis: methodology
approach, content approach, materials approach, and combination
approach. The combination approach consisted of studies with no
special emphasis on any of the three areas previously discussed. The
majority of the studies which emphasized the methodology belonged to
the discovery approach, heuristic method, and deductive reasoning.
Studies which emphasized the content dealt mostly with topics such as
sets, logic, structure, and nondecimal bases. Most of the studies
which emphasized the materials belonged to experiments with manipula
tive materials, the Cuisanaire approach, and the mathematics labora
tory.
Table 15 shows the average effect sizes of each of the study
approaches in achievement and attitude. Obviously, there were large
achievement gains in all the approaches except that of methodology.
Conversely, there were no substantial attitudinal gains in any of the
four approaches except that of methodology. In the methodology
approach the achievement gain was only 0.04, which was the least of
all the mean effect sizes. Conversely, the attitudinal gain in the
methodology approach was 0.17, which was the largest mean effect size.
The best result in achievement was that of the materials approach
with a mean effect size of 0.51. The content approach was second
with an average effect size of 0.35.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.319 0.017* 0.001* 0.312 2.43 1.33 0.202 1.00 3.64 -1.04 19 17 229 5.83 0.000* Sizes t-value Probability Effect Effect Number of . 1 0.07 84 0.05 30 1.60 0.12Q 0.06 0.05 0.09 Error of Mean of Standard Standard - - Table 15 Table Attitude 0.26 Achievement Standard Deviation 0.16 0.36 0.00 = Number of effect sizes Effect Sizeby Study Approach in Achievement and Attitude 0.17 0.21 0.70 0.08 0.35 0.10 0.94 0.51 0.21 0.86 0.14 39 Mean Median of of freedom *The null hypothesis, stating mean the effect size is zero, is rejectedwith the probability of Note. Note. Degrees Approach Content Combination 0.25 0.05 0.61 0.04 211 6.25 0.000* Content -0.07 0.00 0.21 Combination committing type I error being less than or equal to 0.05. Material Methodology 0.04 0.08 0.59 0.04 181 Methodology Material 0.12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 Effect Size by Source of the Study
Three different sources from which the studies were gathered
were the ERIC documents, journal articles, and doctoral dissertations.
Table 16 lists the effect sizes by each source. The ERIC documents
consisted of school district reports, final reports from federal
grants, and other miscellany. The studies from the ERIC documents
had the Largest average effect size in achievement— 0.77— an effect
size which was three times bigger than that of the grand mean effect
size. The journal articles had an effect size (0.42) which was about
twice the grand mean effect size in achievement. The mean effect
size of the doctoral dissertations was the least in the category of
source (0.16).
Effect Size by Other Descriptive Variables
Other descriptive variables analyzed for the present study were
the subject area of investigation, the affiliation of the researcher,
and the author. The subject area variable had two categories: (a)
mathematics alone and (b) mathematics and other subjects combined. In
Table 17, the average effect sizes by subject area are reported. The
average effect sizes in both categories were almost identical.
"Affiliation of researcher" identified the organization the
researcher had an association with while conducting the study. The
variable was broadly divided into affiliation with a university and
affiliation with any organization other than a university. Table 18
indicates that the average effect sizes in achievement were almost
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8.40 0.000* 39 2.85 0.000* 121 500 5.33 0.000* Sizes t-value Probability Effect Number of . 0.05 1 Error of Meanof Standard - - Table 16 0.67 0.03 Standard Deviation effect sizes 0.34 0.52 Effect Size by Source in Achievement “The “The null hypothesis, stating mean the effect size is zero, is rejectedwith the probability of Note. Note. Degrees of freedom Number = of Source Mean Median dissertations 0.16 0.04 Journal articles 0.42 ERIC documentsDoctoral 0.77 0.02committing type error I being less than or equal 0.05. to 1.70 0.27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sizes t-value Probability Effect Number of . 1 0.03 609 8.00 0.000* Error of of Mean Standard Standard - - Table 17 Standard Median Deviation Effect Size by Subject Area of Investigation Achievementin 0.23 0.09 0.61 0.08 51 2.88 0.006* Mean *The null hypothesis, stating meanthe effect size is zero, is rejected with the probability of Note. Note. Degrees of freedom Number = of effect sizes Area of Investigation Other Mathematics 0.24 0.09 0.78 committing type I error being less than or equal 0.05. to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 583 8.00 0.000* 0.03 Standard Number of Table 18 0.79 Standard Error Effect 0.09 0.25 0.09 0.49 0.06 77 4.17 Effect Size by Affiliation of Researcher the inAchievement *The *The null hypothesis, stating mean the effect size is zero, is rejected with probability the of Note. Note. Degrees of freedom Number = of effect sizes - 1. committing type I error being less than or equal to 0.05. Other University 0.24 Affiliation Mean Median Deviation Mean of Sizes t-value Probability
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85
identical for both the studies affiliated with a university and those
affiliated with other organizations. The similarity was also true for
the subject area of investigation.
An attempt was made to identify the author who introduced either
a new content, a new methodology, a new material, or any combination
of the three. The purpose of this variable was to identify the direc
tion of bias in the results due to the influence of personal interest in
the topic of experimentation. For this purpose, the variable (author)
was subdivided into four categories (commercial, school districts,
experimenter, and university); however, it was impossible in most of
the cases to identify the real author of the innovations.
An attempt was made to determine the extent of each researcher's
personal involvement in the experiment, but it was uncertain in most
of the cases exactly how involved the researcher was in the results of
the study. In some cases the researcher was either the director of a
project or a member of the staff of a project; however, information
of this nature was available in only a few cases. Consequently, a
determination could not be made about the extent of the researcher’s
actual involvement in the study.
Specific Effects by Design Variables
The fourth category of the specific variables dealt with the
design variables. The function of the variables in this category was
to measure the different characteristics of the research design used
in each study. Three variables were selected for this purpose. They
were the type of test used to measure the achievement and/or attitude,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86
the quality of the design used for each experiment, and the novelty
effect which was assumed to be a factor that depends on the length of
treatment.
Effect Size by Type of Test
Basically, there were only two categories of tests that were
used in the studies. One category consisted of a variety of standard
ized norm-referenced mathematics tests. The other category consisted
of teacher or experimenter developed tests. Table 19 shows that about
two-thirds of the effect sizes in achievement were measured by com
mercial tests. The average effect size in achievement of the commer
cial test was only 0.15 which was lower than the grand mean effect
size in achievement; however, the average effect size of the experi
menter developed test in achievement was about three times greater
than that of the commercial tests.
Effect Size by Quality of Design
The quality of design variable had two categories: studies where
subjects were randomly assigned to treatment and studies where sub
jects were not randomly assigned to treatment. As indicated in Table
19, 552 effect sizes in achievement were obtained from studies with
random assignments. The average effect size in achievement of the
studies with random assignment was 0.26, whereas the average effect
size of studies without random assignment was only slightly more than
half (0.14) of the former.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.000* 0.000* 5.00 0.000* 438 108 2.80 0.006* Sizes t-value Probability Effect Number of 0.03 0.05 Error of of Mean Standard Table 19 Table 0.62 0.53 Types of Test Standard Novelty Effect Deviation Quality of Design 0.13 0.96 0.07 202 4.71 Effect by Size Design Variables Achievementin 0.15 0.05 0.14 0.05 0.33 Mean Median *The *The null hypothesis, stating mean the effect size is zero, is rejected with probability the of Note. Note. Degrees of freedom Number = of effect sizes - 1. developed 0.42 0.25 0.97 0.07 222 6.00 0.000* Commercial Experimenter assignment Random assignment 0.26 0.09 0.80 0.03 552 8.67 No random Present committing type I error being less than or equal to 0.05. Absent 0.20 0.08 0.66 0.03 458 6.67 0.000*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Effect Size by Novelty Effect
Since the novelty effect variable depended on the length of
treatment, a critical measure was arbitrarily established to measure
the novelty effect. For the purposes of the present study, the nov
elty effect was assumed to be present whenever the length of treatment
was less than 18 weeks. For those studies in which novelty effect
was judged to be present, the average was significantly higher than
that of the studies not having novelty effect. Table 19 lists the
effect sizes in achievement by novelty effect. About one-third of
the number of effect sizes had a novelty effect. The average effect
size in achievement of the studies without novelty effect was less
that two-thirds of the effect size in achievement of the studies with
novelty effect.
Summary
Table 20 gives an overview of the mean effect sizes in achieve
ment and/or attitude of all the variables analyzed. In this section
only the major results are highlighted.
When the subjects were divided into five grade levels, the ele
mentary and the senior high had the larger effect sizes in achieve
ment, which were respectively 0.32 and 0.28. Among the low, middle,
and high ability groups, the low ability had the largest effect size
(0.35) in achievement. In a similar grouping based on the socio
economic status, the effect size of the middle group in achievement
was 1.05 which was conspicuously higher than that of the low and high
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89
Effect Sizes by All Categories of Variables in Achievement and/or Attitude
Variable Effect Size Variable Effect Size
Achievement by grade Achievement by number of subjects Kindergarten 0.06 50 or less 0.49* Elementary 0.32* Between 50 and 100 0.16* Junior high 0.17* Between 100 and 200 0.15* Senior high 0.28* Between 200 and 400 0.26* Post-secondary 0.12 More than 400 0.13*
Achievement by ability Achievement by study approach Low 0.35* Methodology 0.04 Middle 0.25* Content 0.35* High 0.21* Material 0.51*
Achievement by SES Attitude by study approach Low 0.15 Methodology 0.17* Middle 1.05* Content -0.07 High 0.22 Material 0.12
Achievement by sex Achievement by source Male 0.11 ERIC document 0.77* Female 0.03 Journal article 0.42* Dissertation 0.16* Achievement by general content area Achievement by area of Concepts 0.36* investigation Computation 0.31* Mathematics 0.24* Application 0.06 Other 0.23* Achievement in specific Achievement by affiliation subject area of the researcher Arithmetic 0.21* University 0.24* Algebra 0.43* Other 0.25* Geometry 0.14 Achievement by type of test Achievement by program Commercial 0.15* SMSG 0.14* Experimenter 0.42* Other 0.27* Achievement by design Achievement by length Random assignment 0.26* of treatment in weeks Other 0.14* Less than 9 0.25* Between 9 and 18 0.48* Achievement by novelty effect Between 18 and 36 0.09 Present 0.33* 36 and more 0.25* Absent 0.20*
Grand mean achievement 0.24* Grand mean attitude 0.12*
*The null hypothesis, stating the mean effect size is zero, is rejected with the probability of committing type I error being less than or equal to 0.05.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 groups.
In the general content area, the effect sizes of concepts (0.36)
and of computation (0.31) were very large in achievement, while that
of the application was as low as 0.06. Among the specific subject
areas, algebra had a high effect size of 0.43 in achievement; however,
geometry had a poor effect size of 0.14.
Of the three approaches such as methodology, content, and mate
rials the effect sizes in the content was 0.35 and in the materials
0.51. The methodology had the lowest effect size (0.04) in achieve
ment. Conversely, methodology had the highest effect size in atti
tude .
The effect sizes when averaged according to the different sources,
studies from dissertations showed the least effect size (0.16). The
journal articj.es and the ERIC documents indicated superior effect
sizes (0.42 and 0.77, respectively).
Caution must be exercised in the interpretation of the nonsigni
ficant results reported in the tables presented in this chapter. Two
or more effect sizes having the same magnitude may differ in their
significance. For example, in Table 20, the grand mean effect size
in attitude (0.12) is significant, whereas the mean effect size of
the post-secondary grades in achievement (0.12) is insignificant. It
must be noted that in many instances, it is very likely that the
results are insignificant due to small number of effect sizes.
Achievement by sex and by socio-economic status are other striking
instances of the previously discussed phenomenon.
In total, the grand mean of 660 effect sizes in achievement was
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91
0.24 and the grand mean of 150 effect sizes in attitude was 0.12. In
other words, an average student who received some form of modern math
ematics instruction was raised from the 50th to the 59th percentile
in achievement and from the 50th to the 55th percentile in attitude,
over the traditional mathematics population.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER V
INTERPRETATIONS, IMPLICATIONS, GENERAL RECOMMENDATIONS, AND SUMMARY
Interpretations
The interpretations generally followed the sequence of the dis
cussion of the results in Chapter IV. Some of the mean effect sizes
reported in Chapter IV were not significant. As a rule, interpreta
tions were focused primarily upon results which were statistically
significant.
The Grand Mean Effect Sizes
The obtained results revealed that there were improvements in
achievement and attitude which were attributable to modern mathemat
ics. Improvement in achievement of an average student was almost
double the improvement in attitude. The discrepancy between improved
rates of achievement and attitude is typical of innovations where the
appeal is intellectual in nature and demanding in its application.
New mathematics fits this mode in that it is mostly abstract. Con
sequently, modern mathematics requires a strict discipline of the
mind in order to comprehend its operations. It is consoling to note
that the effects of modern mathematics are worth its demands. Also,
attitude toward an innovative program is often a result of the per
ceived success or failure of the innovation. Dissimilar findings in
the absence of summarized results helped maintain a low morale toward
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93
modern mathematics for the past 30 years. People were uncertain
about whether or not "new math worked." Hopefully, this study has
revealed those areas where modern mathematics is effective. Conse
quently, a more positive attitude toward new mathematics should occur,
an attitude which will be comparable to the achievement level.
Subject Variables
Of the four variables identified as subject characteristics,
three lend themselves to interpretations. The three variables are
grade level, ability level, and socio-economic status. In general,
the findings in this group were not in agreement with the expecta
tions and viewpoints of the critics of new mathematics. Consequently,
interpretations of these findings must be interesting to the oppo
nents as well as proponents.
Grade level. Among the different grade levels, the mean effect
size in achievement of the junior high pupils was the least signifi
cant. One explanation of the low mean effect size of the junior high
grades is obvious. Confusion may have entered the minds of pupils
due to the sudden transition in their thinking patterns as they
switched from traditional mathematics in the elementary grades to
modern mathematics in the junior high.
The greater mean effect size of the elementary grades where the
students were introduced to only one kind of treatment (new mathe
matics) explains the clear perception of new mathematics concepts in
the absence of treatments antithetical in nature. In the case of the
senior high students, the large effect size could be assumed to be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 the result of a development of minds into a stage in which they could
discriminate one treatment from another without confusion.
Ability level. Summarily, the greater the effect, the lower the
ability level was. This finding explains the comparative effective
ness of the two treatments of mathematics. High ability students,
because of their innate intellectual acumen, do not depend upon the
kind of treatment as much as the low ability students do in the
understanding of mathematics concepts. Consequently, the introduc
tion of new mathematics effected a comparatively smaller gain in
achievement among the high ability groups while the low ability
groups experienced a large growth in achievement.
Knowing this trend, one would not be tempted to conclude that
modern mathematics was an easy approach. More likely, it was a mean
ingful approach which produced better results than the traditional
mathematics among all ability levels, especially at the lower levels
of ability.
Socio-economic status. The striking result in the application
of modern mathematics among different levels of socio-economic status
was the outstanding effect size of the pupils coming from middle
class parents. One of the main reasons for the high achievement of
pupils from middle socio-economic levels could be parental help. It
must be pointed out that among the parents who are most attuned to
the trends in mathematics, many are also teachers who generally
belong to the middle socio-economic class. These parents are often
more aware of the need for providing help to their children in under
standing modern mathematics.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95
Treatment Variables
The interpretations in the ensuing sections might help to find
some answers to the heated criticisms levelled against certain
aspects of modern mathematics. These interpretations pertain to the
general content area, specific subject area, type of program, and the
length of treatment.
General content area. In the general content area, which con
sisted of concepts, computation, and application, the highest mean
effect size was that of the concepts. Obviously this is a likely
outcome of the treatment of modern mathematics because one of the
unique characteristics of modern mathematics is the emphasis on the
understanding of concepts. The weakest score in the general content
area was in application. The poor result in this area perhaps
explains a lack of adequate time for the pupils to exercise their
minds. Possibly, a longer treatment of modern mathematics with suf
ficient emphasis on the concrete presentation of abstract concepts
would strengthen the students in the area of application of mathe
matical concepts.
Specific subject area. The specific content area was divided
into arithmetic, algebra, and geometry. A fourth category was
included to accommodate subject areas other than those previously
specified. Among the four subject areas, algebra had the highest
mean effect size in achievement. The explanation for this result is
subject to speculation. It could be that among all the specific sub
ject areas investigated, algebra was the most expressive medium to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 bring out the message of modern mathematics. Similar results can be
expected in other subject areas, especially in geometry, if the mode
of presentation of those subject areas are more attuned to the axio
matic approach of modern mathematics.
Type of program. The types of programs were grouped in School
Mathematics Study Group (SMSG) and non-SMSG. This was due to the
small number of studies in all developmental programs except SMSG.
The average effect size in achievement of the non-SMSG group was
almost twice that of the SMSG group. A number of studies in the non-
SMSG group were conducted by individual experimenters. In those
experiments, factors such as enthusiasm of the experimenter, attrac
tion of a new theme, and pride of the students in belonging to a
selected group, might have been favorable for the great success in
modern mathematics treatment. Conversely, in the treatment of SMSG
programs, most of the factors mentioned did not exist because they
were generally adopted on a massive scale by school districts; there
fore, the effect size of non-SMSG programs explains more of an initial
enthusiasm. Conversely, the results of the SMSG program indicate a
conservative, but steady and longstanding, effect of modern mathema
tics treatment.
Length of treatment. The effect sizes, when averaged according
to the length of treatment, demonstrated an interesting pattern. The
average effect in achievement was maximum when the length of treat
ment was between 9 and 18 weeks and minimum for a period between 18
and 36 weeks. The maximum and minimum values were 0.48 and 0.09,
respectively. In the studies with the length of treatment greater
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 than 36 weeks, the effect sizes were averaged to be 0.25, the mid
value between the two extremes. Therefore, the pattern was a rise,
fall, and a gradual steadiness toward the mean of the two values.
The rise possibly demonstrates the initial enthusiasm and fascination
for the new mathematics programs. The fall manifests the setback the
pupils experienced when they were confronted with the rigor and
abstractness of the modern math treatment. Once the pupils got over
the period of a confrontation, they began making steady improvements
which were reflected in the effect sizes of treatments over 36 weeks.
Most of the criticisms that were levelled against modern mathe
matics during the past decades could have been based on the results
obtained during the time of the initial setback. The ebb and tide is
over. Now, the "new math" flows steady. This is the time for a
mature judgment on the effect of modern mathematics.
Study Variables
The interpretations of the study variables highlight some pos
sible explanations of the differences in effectiveness in three
major areas. The areas are the study approach, author, and source of
the study.
Study approach. A polarity existed between the scores in
achievement and attitude in the four different types of study
approach. Among the four types, namely, methodology, content, mate
rials, and combination, all types except methodology had significantly
larger effect sizes in achievement. On the contrary, effect sizes of
all types of approaches, except that of methodology, were relatively
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98
small in attitude.
The polarity reveals a striking point. What worked better with
modern mathematics wjs due to an appeal to reason, rather than an
appeal to the senses. The methodology approach, which mostly con
sisted of discovery approaches and its subcategories, might have
appealed to the susceptibilities of the young minds; however, little
was accomplished in terms of achievement through the new methodologies.
What made modern mathematics effective in terms of achievement was its
content area characterized by the structure and abstractness.
The mean effect size on the materials is even more revealing.
The high achievement effect size of the materials approach makes it
explicit that the abstract concepts of modern mathematics, when pre
sented to the students in concrete forms through adequate materials,
produced the best results in achievement.
Author. The effect sizes in this group were associated with
school districts, universities, and private experimenters according
to the affiliation of the researcher. A comparatively large mean
effect size in achievement for the experimenter group indicates, per
haps, the direction of bias caused by self-interest.
The experimenter group consisted mainly of textbook publishers
and other commercial agencies. This group possibly tried to present
the most positive results for monetary purposes. Consequently, they
would be the most vulnerable in the direction of bias. To a lesser
degree, the school districts might have been affected by their pre
occupation to build up an exemplar. The results of the studies whose
authors were affiliated with universities indicated the least effect
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99
size in achievement. On the whole, effect sizes in that group repre
sented the most conservative and most dependable figures.
Source of the study. Sources of the study revealed a similar
situation. ERIC documents which consisted of school district reports,
final reports from federal funds, etc., had the highest average effect
size in achievement. The reason for the high effect size could be
that studies from the above sources are often attempts to convince
the funding agencies to continue funding of a specific program by
presenting the results as attractively as possible. The journal
articles also had a large effect size in achievement. Perhaps the
journals reveal the increased tendency to publish studies favoring
the new approach. Conversely, dissertations with no such self-motives,
contained more balanced and neutral findings.
Since the present analysis utilized 85 percent of the total num
ber of studies from complete dissertations, the findings presented
here would be considered to be highly conservative and most dependable
in nature.
Design Variables
Interpretations of the design variables deal with the type of
test and the quality of design. The elucidations will help one eval
uate the comparative effectiveness of different types in tests and
designs.
Type of test. The types of tests utilized were commercial and
experimenter developed. Effect sizes in achievement of studies meas
ured by commercial tests were very small in comparison with effect
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sizes in achievement measured by experimenter developed tests. One
of the weaknesses of the standardized tests was that they were not
constructed specifically for a study to test the objectives of a par
ticular treatment. As a rule, most of the commercial tests were more
suited to measure the objectives of the traditional treatments than
those of the modern treatment of mathematics. Consequently, the
effect size of the commercial test group did not adequately represent
the achievement of students in modern mathematics. It is obvious,
therefore, that the overall mean effect size would have been signifi
cantly higher than the present one, if all the studies utilized tests
which adequately measured the performance according to the objectives
specified for the particular treatment.
Quality of design. In the present study, the experiments in
which the subjects were randomly assigned to treatments had a higher
effect size in achievement than the experiments without random assign
ment. According to the methodologists, experiments with random
assignment should yield results with less bias than the experiments
without random assignment. On that basis, the greater effect size in
achievement obtained from the random assignment group should be con
sidered to be more objective than the smaller effect size obtained
from the studies without random assignment.
Implications
The data presented in Chapter IV and their interpretations in
the early sections of Chapter V have several important implications
for sound decision making and needed improvements in mathematics
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101
programs.
The aggregate findings of the experiments with modern mathematics
for the last 30 years show the critics and opponents wrong with regard
to the detrimental effects of the "new math." On the contrary, the
evidence shows the beneficial effects in terms of the improvement in
overall achievement and attitude. In the light of the findings of
the present study, it would be unwise to advocate the superiority of
the traditional mathematics over the new mathematics. Subsequently,
the current retrenchment in mathematics under the banner of the back-
to-basics philosophy can be regarded only as an aberration brought
about by the short sightedness of the doomsayers and the misjudgments
of the decision makers in the absence of reliable research findings
about the effectiveness of new mathematics.
The beneficial effects of new mathematics call for the promotion
of the new programs throughout the nation. The findings should
encourage the higher authorities throughout the country, on the
federal, state, and local levels of mathematics education, to re
introduce new mathematics curriculum in order to meet the changing
needs of the society.
The overall findings also indicated that the improvement of
attitude did not compare with that of achievement. The discrepancy
between the gains in achievement and attitude has serious implications
for the implementation of the new mathematics programs. In the first
place, the decision makers should be aware of the existence of the
discrepancy. Secondly, in assessing the new mathematics programs,
the decision makers should take the achievement score as the index of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102
effectiveness. Thirdly, in the decisions regarding the implementa
tion of the new mathematics program, they should not be guided by the
score in attitude which partly echoes the deteriorating influence of
the back-to-basics movement.
The implications based on the results which pertained to the
specific variables considered in the study are numerous. An attempt
is made in the following sections to discuss the major considerations
of the important findings. Among the grade levels, the explicitly
superior gains in achievement in the elementary and senior high, in
contrast with the junior high, have a serious implication. The
superior scores suggest that in the process of introducing new mathe
matics, the most appropriate grade levels are either the elementary
or the senior high. Unfortunately, in the past the majority of the
experiments conducted with modern mathematics were at the junior high
level. Much of the "anti-feelings" about new mathematics might not
have come to the surface, if the new programs were introduced either
at the elementary or senior high levels.
One of the allegations against modern mathematics has been that
it is good only for the highly motivated students who will go on to
major in mathematics in college. The findings of this study proved
the allegation wrong. The average mean scores, when categorized
according to the ability levels, show that although new mathematics
was beneficial for bright students, it was more beneficial for the
slow learners. It will be a serious mistake, therefore, to teach the
new mathematics only to a chosen ability group— a practice that has
been prevalent in the past decades. The consequences would be to
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deprive the slow learners of the benefits of the new mathematics.
Moreover, when a choice is available regarding the levels of ability
and introducing new mathematics, the decision should be to support the
low ability groups utilizing new mathematics.
The findings in the general content area are capable of solving
much of the heated controversy over the concept approach and the com
putational skills. The results indicated remarkably superior achieve
ment attributable to new mathematics in concepts and computation.
The criticism that modern mathematics is hopelessly beyond young
people because of its abstractness and rigor carries no weight in the
light of these findings. Also, computation with whole numbers is far
from being a lost art. More surprisingly, now Johnny adds better.
Knowing that Johnny adds better makes one skeptical of the back-to-
basics movement.
As the back-to-basics movement gains momentum, one should be
deeply concerned about the dangers it can hold. The movement, insofar
as it emphasizes the computational skills in isolation, might elimi
nate teaching for mathematical understanding. What good can computa
tional skills in isolation do, if a student does not know what to
compute when? The modern math has been addressing this problem by
placing the computational skill within a total mathematics program.
The findings of this study supported the assertion that this approach
has been successful, and that it is far superior to the traditional
approach. In reality, going back to basics demarcates a deterioration
in the learning of mathematics.
The results pertaining to application revealed that the
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improvement attributable to new mathematics in the area of applica
tion was remarkably low in comparison with the scores in concepts
and computation. Unless and until mathematics can operate to clarify
and solve human problems, it has indeed a narrow value. Hence, major
emphasis must be placed to strengthen the area of application in the
new mathematics curriculum. As a result, the students should be able
to apply the concepts and skills they learned in the practical field.
The relatively low geometry score in the specific subjects
revealed another area which requires serious consideration. Since
geometry occupies a central position in mathematics, according to
many mathematicians, presenting geometry as meaningful and under
standable as the rest of the subject areas is a proven necessity to
fulfill the goal of total mathematics program.
On the whole, the new mathematics has been proven to be superior
in all areas investigated in this paper. In a few instances, such
as geometry, application in the generic content area, and methodology
in the general approaches, the improvement attributable to new mathe
matics is negligible; however, "new math" is still on its way. Fur
ther, it is hoped that the leaders in the new mathematics curriculum
will recognize these areas as their unfinished task and propose
alternatives for improvement.
General Recommendations
The discussion of recommendations pertains to three points. They
are recommendations for utilization of the study findings, recommend
ations for further meta-analyses in the field of mathematics, and
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recommendations for improved techniques for meta-analysis.
Utilization of the Study Findings
The present study attempted to resolve several problematic issues
pertaining to modern mathematics education. In general, the study
tried to find an answer to the central issue: whether the treatment
of new mathematics resulted in an improvement or decline of mathemat
ical achievement. The overall analysis of 810 conflicting and dis
similar findings showed that the "new math" raised a typical student
from the 50th to the 59th percentile of the traditional mathematics
group in achievement. Though to a lesser degree, the introduction of
new mathematics also improved the attitude toward mathematics. The
attitude of a typical student rose from the 50th to the 55th percen
tile.
The study also showed the relative strengths and weaknesses of
new mathematics in several specific areas. These were categorized
under four sets of variables: subject characteristics, treatment
factors, study attributes, and design variables. The overall find
ings and the findings in those specific areas will enable mathematics
educators, educational specialists, curriculum developers, classroom
teachers, commercial agencies, and other interested persons or agen
cies to make sound decisions pertaining to such things as grade,
ability level, socio-economic status, sex, general content area,
specific content area, different subject areas, and type of program
in the process of implementing modern mathematics programs.
In summary, the authorities in mathematics education should
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realize that a great deal of improvement has been made in the learning
of mathematics since World War II. Sputnik is due credit for having
the greatest influence on the new mathematics. It would be unwise to
ignore these benefits derived from new mathematics and rush "back-to-
basics" which holds a promise for only a retarded growth.
Further Meta-analyses on Mathematics
There are many areas in mathematics in which there exists an
abundance of studies with dissimilar findings. Meta-analysis would
prove to be extremely useful for extracting the message of those
diverse findings. A meta-analysis would be indicated in such ques
tions as the relative effectiveness of the application of Piaget's
theories and the traditional treatment. Likewise, the literature on
instructional television, use of calculators, and such other innova
tions hold promise for statistical meta-analysis.
Improvements in the Techniques of Meta-analysis
Improvement in the techniques of meta-analysis is another area
of exploration. For example, the mean effect size as defined and
applied in the literature does not enable one to evaluate its signif
icance. A smaller mean effect size could be of a greater significance
than a larger mean effect size if the number of effect sizes is sig
nificantly small or the standard deviation is significantly high. An
attempt is made in the present study to evaluate the relative merits
of the mean effect sizes. The t-values and the corresponding proba
bilities were calculated to find the significance of each mean effect
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107
size. Since the concept of meta-analysis is relatively recent,
improvements in the techniques such as those discussed above are
quite possible and certainly rewarding.
Summary
During the last three decades many innovations in mathematics
education were introduced under a common label called "new mathemat
ics." The studies on new mathematics were numerous and their find
ings were conflicting and dissimilar in nature. No attempt had been
made to summarize those studies in order to extract the message of
the diverse findings about "new mathematics." The present study
attempted to summarize 134 studies on new mathematics, which were
located mainly from dissertations, abstracts, journal articles, and
ERIC documents.
In order to explain the term "new mathematics," the author
traced the origin and evolution of the movement of modern mathematics.
There were two critical events, World War II and the Russian Sputnik,
which gave great stimuli for reform in mathematics education. The
reform movement was initiated mostly by developmental projects which
emerged during the 1950's and early part of the I960's. The role of
the developmental programs was to correct perceived weakness in
school programs, such as to improve materials, to train teachers, and
most importantly to introduce new concepts in the area of content,
methodology, and materials in the treatment of mathematics. The new
concepts placed great emphasis on logical thinking, structure, and
abstractedness, instead of drill and rote of the traditional
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mathematics. Innovations of these developmental programs were known
under a common label, "new math."
The facts about the conflicting nature of research findings and
public opinions about the outcomes of new mathematics, coupled with
the scarcity of summative research resolving the dissimilar and con
flicting findings, presented the problem for which the study was
intended. Research questions were formulated to answer the problem
atic issues in the new mathematics curriculum. The issues pertained
to the examination of overall outcome— decline or growth in the learn
ing of mathematics— attributable to modern mathematics.
Since the study adopted a new technique for summarizing the
aggregate findings about new mathematics, the rationale for the study
focused on two categories of topics: the justification of the research
topic and the justification of the research tool. To justify the
topic, three major areas in the literature of modern mathematics were
examined. They were: (a) appraisals of modern mathematics, (b)
studies on modern mathematics by primary analysis, and (c) studies on
modern mathematics by aggregate analysis. Treatment on appraisals of
modern mathematics disclosed conflicting viewpoints regarding the
efficacy of modern mathematics. To make a sound judgment on the effi
cacy of modern mathematics, research studies were utilized. An over
view of the research studies on modern mathematics by primary analysis
revealed that there were numerous studies in that area; however, the
findings were diverse. Hence, an aggregate analysis was deemed to be
necessary in order to resolve the conflicting findings. The existing
aggregate analysis did not strictly pertain to the area of investigation
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109
of the present study. Therefore, the researcher attempted to utilize
an aggregate analysis methodology specifically to resolve the con
flicting findings about new mathematics.
The literature search for techniques of aggregation exposed five
different methodologies used in the past. They were narratives,
reviews, the averaging method, the voting method, and the cluster
approach. The drawbacks of the review methodology were: (a) it did
not evaluate the influence of the independent variables, (b) it
adopted inadequate techniques as means of aggregation, and (c) it
eliminated the so called "poorly done" studies. The weakness of the
averaging method was that it threw away precisely the information
needed most, namely, a true estimate of the treatment effect. The
defects of the voting method were: (a) it disregarded valuable des
criptive information, (b) it did not integrate the measures of the
strength of experimental effects, and (c) it did not solve the prob
lem of unbalanced experimental design. The major drawbacks to the
cluster approach were that the study became overly restrictive
because: (a) it required access to raw data, (b) it eliminated
studies which were not measured in a common scale, and (c) it elimi
nated studies which lacked explanation for differences.
Because of the drawbacks mentioned previously, none of the tech
niques was found to be adequate for the present analysis. Glass
(1976) introduced another technique of aggregation called meta
analysis. Meta-analysis, in many respects, was an improvement on the
previously discussed techniques.
Meta-analysis methodology stood for a statistical aggregation of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110
a large body of similar, dissimilar, and conflicting findings without
loss of information. The key concept in the technique of meta
analysis was the effect size, which was defined as the difference
between the means of the experimental and control groups measured in
standard deviations of the control group. Since the effect sizes
were standardized measures, comparison of different effect sizes were
possible. Practical steps to conduct the present study consisted of
the identification of the sources of the study, locating and collect
ing the studies, identifying and coding variables, and methods of
analyzing the data.
The overall mean effect sizes of 810 substudies revealed that
modern mathematics showed evidence of change attributable to new math
ematics programs. The change was reflected in the improvement of
academic achievement and in the improvement of attitude toward math
ematics. The results also revealed differential effects of specific
characteristics of subject variables, treatment variables, study
variables, and design variables. Spectacular improvements attribut
able to modern mathematics have been made in the understanding of the
concepts and computational skills. Among the specific subject areas,
algebra had an outstandingly superior achievement score. Of the dif
ferent approaches of modern mathematics, the one that emphasized the
concrete presentation of abstract ideas was indicated to be the most
rewarding. The major areas which revealed a need for substantial
improvements were geometry among the specific subjects and application
in general content area.
The findings of the study have several important implications for
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill sound decision making and needed improvements in many specific areas.
On the whole, the findings of the study strongly encourage the pro
motion of the new mathematics throughout the nation. At the same
time, the back-to-basics movement should certainly be regarded as an
aberration in the advancement of the learning of mathematics.
In summary, the present study did show evidence that the experi
ments with modern mathematics of the past three decades in its con
tent, methodology, materials, or in any combination, have, in general,
led to improved learning of mathematics. This is revealed by the
progress in achievement and attitude toward mathematical concepts,
computation, and application on all grade and ability levels. There
fore, findings of the present study should motivate the decision
makers to dispel their unfounded uncertainties and misguided appre
hensions about the effectiveness of new mathematics and to bring
forth the new mathematics to a respectable place in the curriculum
for the benefit of the students and for the betterment of the
society.
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*Gopal Rao, G.S., Penfold, D. M., & Penfold, A. P. Modern and tradi tional methematics thinking. Educational Research, 1970, 13, 61-65.
*Gore, J. A. Development and preliminary evaluation of a logic unit for college freshmen mathematics courses (Doctoral dissertation, University of Georgia, 1969). Dissertation Abstracts Interna tional, 1969, 30, 5342A. (University Microfilms No. 70-10,190)
*Grafft, W. D., & Ruddell, A. K. Cognitive outcomes of the SMSG mathematics program in grades 4, 5, and 6. The Arithmetic Teacher, 1968, 15, 161-165.
*Gravel, H. Teaching mathematical relations at the grade six level (Doctoral dissertation, Columbia University, 1968). Dissertation Abstracts, 1968, 2 3 _ , 1473A. (University Microfilms No. 68-11,133)
*Greabell, L. The effect of three different methods of implementa tion of mathematics programs in children's achievement in mathe matics. The Arithmetic Teacher, 1969, 1 6 , 288-291.
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*Greathouse, J. J. An experimental investigation of relative effec tiveness among three different arithmetic teaching methods (Doctoral dissertation, The University of New Mexico, 1965). Dissertation Abstracts, 1966, 2 6 , 5913A. (University Microfilms No. 66-4465)
*Greitzer, S. L. A comparison of the postulational approach and the traditional approach in teaching selected topics in algebra to above average students (Doctoral dissertation, Yeshiva University, 1959). Dissertation Abstracts, 1960, _21, 1137A. (University Microfilms No. 60-02108)
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*Hall, W. D. A study of relationship between estimation and mathe matical problem solving among fifth grade students (Doctoral dissertation, University of Illinois, 1976) Dissertation Abstracts International, 1977, 3 7 _ , 6324A. (University Microfilms No. 77-9014)
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*Harper, E. H. The identification of socio-economic differences and their effect on the teaching of readiness for "new math concepts" in the kindergarten. Final Report, 1970, The University of Wisconsin. (ERIC Document Reproduction Service No. ED 040 852)
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*Heine, B. An investigation of the effect of teaching elementary mathematical logic in problem solving ability of the fifth grade students (Doctoral dissertation, Temple University, 1972) . Dissertation Abstracts International, 1972, 33, 1587A. (Univer sity Microfilms No. 72-27,196)
*Herbeck, C. A. Experimental study of the effect of two proof for mats in high school geometry on critical thinking and selected student attitude (Doctoral dissertation, Ohio State University, 1972). Dissertation Abstracts International, 1973, 33, 4243A. (University Microfilms No. 73-2013)
^Hershberger, L. D. A comparison of two methods of teaching selected topics in plane and solid analytic geometry (Doctoral dissertation, Florida State University, 1970). Dissertation Abstracts Inter national, 1971, _31, 4622A. (University Microfilms No. 71-7030)
*Higgins, J. E. The effects of non-decimal numeration instruction on mathematical understanding (Doctoral dissertation, Indiana University, 1970). Dissertation Abstracts International, 1970, 31, 2791A. (University Microfilms No. 70-23,366)
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*Hollis, L. Y. A study to compare the effects of teaching first grade mathematics by the Cuisenaire-Gattegno method with the traditional method (Doctoral dissertation, Texas Technological College, 1964). Dissertation Abstracts, 1965, 2 6 , 904A. (Univer sity Microfilms No. 65-1458)
*Huber, J. C. A comparative study of traditional and transformational approaches to selected topics in trigonometry (Doctoral disserta tion, University of Houston, 1977). Dissertation Abstracts International, 1977, _38, 2628A. (University Microfilms No. 77- 24,431)
*Hungerman, A. D. Achievement and attitude of sixth grade pupils in conventional and traditional mathematics programs. The Arithmetic Teacher, 1967, 14, 30-39.
*Jackson, R. L. Numeration systems: An experimental study of achievement on selected objectives of mathematics education resulting from the study of different numeration systems (Doctoral dissertation, University of Minnesota, 1965). Disserta tion Abstracts, 1966, 2 6 , 5292A. University Microfilms No. 65- 15,267)
*Johnson, R. E. The effect of activity oriented lessons on the achievement and attitudes of seventh grade students in mathemat ics (Doctoral dissertation, University of Minnesota, 1970). Dissertation Abstracts International, 1971, 3 2 , 305A. (University Microfilms No. 71-18,755)
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*Kavett, P. F. A study of the teaching of non-decimal system of numeration in the elementary school (Doctoral dissertation, Rutgers State University, 1969). Dissertation Abstracts Inter national, 1969, 30, 1468A. (University Microfilms No. 69-16,492)
*Keese, E. E. A study of the creative thinking ability and student achievement in mathematics using discovery and expository methods of teaching (Doctoral dissertation, George Peabody College for Teachers, 1972). Dissertation Abstracts International, 1972, 33, 1589A. (University Microfilms No. 72-25,392)
*Keller, J. R. The effect of studying a unit of finite fields on the performance in intermediate algebra at the college level (Doctorate dissertation, West Virginia University, 1976). Dissertation Abstracts International, 1977, 3 ] _ , 2038A. (Univer sity Microfilms No. 76-22,435)
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*King, R. The theory of denominator sets as a new mode in the presentation and explanation of fractions (Doctoral dissertation, Saint Louis University, 1976). Dissertation Abstracts Interna tional, 1977, T7, 7592A. (University Microfilms No. 77-12,111)
*Kirkpatrick, J. E. The use of finite mathematical systems to achieve selected mathematics objectives in grade six (Doctoral dissertation, University of Minnesota, 1970). Dissertation Abstracts International, 1971, 3 2 _, 306A. (University Microfilms No. 71-18,761)
*Kleckner, L. G. An experimental study of discovery type of teaching strategies with low achievers in basic mathematics (Doctoral dissertation, Pennsylvania State University, 1968). Dissertation Abstracts International, 1969, 30, 1075A. (University Microfilms No. 69-14,534)
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*Klingbeil, D. H. An examination of the effects of group testing in mathematics courses taught by the small group discovery method (Doctoral dissertation, University of Maryland, 1974). Disserta tion Abstracts International, 1975, jS6, 849A. (University Microfilms No. 75-17,886)
*Kort, A. P. Transformation vs. nontransformation of tenth grade geometry effects on retention of geometry and on transfer in eleventh grade mathematics (Doctoral dissertation, Northwestern University, 1971). Dissertation Abstracts International, 1971, 32, 3157A. (University Microfilms No. 71-30,860)
*Kratzer, R. 0. A comparison of initially teaching Greenwood algorithm with the aid of manipulative material (Doctoral disserta tion, New York University, 1971). Dissertation Abstracts Interna tional, 1972, 3 2 , 5672A. (University Microfilms No. 72-11,464)
*Krogness, J. A comparison of three sixth grade programs of mathemat ics instruction (Doctoral dissertation, University of Minnesota, 1967). Dissertation Abstracts International, 1967, 30, 1242B. (University Microfilms No. 69-15,162)
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*Lackner, L. M. The teaching of the limit and derivative concepts in beginning calculus by combinations of inductive and deductive approaches (Doctoral dissertation, University of Illinois, 1968). Dissertation Abstracts, 1969, 29, 2150A. (University Microfilms No. 69-1378)
*Larsen, C. M. The heuristic standpoint in teaching of elementary calculus (Doctoral dissertation, Stanford University, 1960). Dissertation Abstracts, 1961, 21, 2632A. (University Microfilms No. 60-6740)
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*Maetsky, E. M. The relative merits of several methods of teaching probability in elementary statistics (Doctoral dissertation, New York University, 1961). Dissertation Abstracts, 1961, 2 2 , 503A. (University Microfilms No. 61-2559)
Mangrum, C. T., II, & Knight, C. W., II. Doctoral dissertation research in science and mathematics reported for volume 30 of dissertation abstract. School Science and Mathematics, 1972, _72, 505-534.
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*Mastain, R. K. A comparison of the effect of two mathematics curricula on the mathematical understanding of fourth grade pupils (Doctoral dissertation, University of Southern California, 1964). Dissertation Abstracts, 1964, 25, 3439A. (University Microfilms No. 64-13,531)
*McAloon, M. D. An exploratory study on teaching units in logic to grades three and six (Doctoral dissertation, University of Michigan, 1969). Dissertation Abstracts International, 1969, 30, 1918A. (University Microfilms No. 69-18,055)
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*McLure, C. W. Effectiveness of laboratories for eighth graders (Doctoral dissertation, Ohio State University, 1971). Disserta tion Abstracts International, 1972, 32., 4078A. (University Microfilms No. 72-4565)
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*Miller, H. F. The comparison of guess-and-check and multi-equation methods for deriving the equations for verbal problems in ele mentary algebra (Doctoral dissertation, Ohio State University, 1959). Dissertation Abstracts, 1959, 20, 2180A. (University Microfilms No. 59-05866)
*Moliver, M. A program in probability for non-college bound students in the tenth grade general mathematics (Doctoral dissertation, Temple University, 1977). Dissertation Abstracts International, 1977, J38, 1955A. (University Microfilms No. 77-21,777)
*Moray, J. Effects of curriculum on mathematics achievement in grades six (Doctoral dissertation, University of California, Berkeley, 1967). Dissertation Abstracts, 1968, JL8, 4538A. (University Microfilms No. 68-5681)
*Morris, J. K. A comparison of an inductive and deductive procedures of teaching in a college mathematics course for prospective elementary teachers (Doctoral dissertation, North Texas State University, 1973). Dissertation Abstracts International, 1974, 35, 308A. (University Microfilms No. 74-14,824)
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*Muller, D. J. Logic and the ability to prove theorems in geometry (Doctoral dissertation, The Florida State University, 1975). Dissertation Abstracts International, 1975, 3 6 , 51A. (University Microfilms No. 75-17,944)
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*Nichols, D. E. Comparison of two approaches to the teaching of selected topics in plane geometry (Doctoral dissertation, Univer sity of Illinois, 1956). Dissertation Abstracts, 1956, 1(5, 21906A. (University Microfilms No. 18,178)
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Northcutt, M. P. The comparative effectiveness of classroom and programmed instruction in the teaching of decimals to fifth grade students. Unpublished doctoral dissertation, George peabody College for Teachers, 1963. Dissertation Abstracts, (June, 1964) 24, pp. 5091-5092.
*Nystrom, N. K. An experimental study to compare the relative effects of two methods of instruction on learning of intermediate algebra (Doctoral dissertation, Arizona State University, 1969). Disserta tion Abstracts, 1969, 2 9 _ , 3532A. (University Microfilms No. 69- 5715)
*0'Brien, T. C. An experimental investigation of a new approach to the teaching of decimals (Doctoral dissertation, New York Univer sity, 1967). Dissertation Abstracts, 1968, 2 8 , 4541A. (Univer sity Microfilms No. 69-6164)
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*Pate, R. T. A study of transactional pattern differences between school mathematics study group classes and traditional mathemat ics classes (Doctoral dissertation, The University of Oklahoma, 1964). Dissertation Abstracts, 1964, _25, 3442A. (University Microfilms No. 64-13,333)
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*Pavlick, F. M., Jr. The use of advanced sets in the tea-hing of limits: A comparative study (Doctoral dissertation, Florida State University, 1968). Dissertation Abstracts, 1968, 2 9 , 518A. (University Microfilms No. 68-11,683)
*Payne, H. I. A study of students using SMSG and conventional approaches in first year algebra (Doctoral dissertation, Oklahoma State University, 1963). Dissertation Abstracts, 1965, 25, 4481A. (University Microfilms No. 64-8941)
*Peck, H. I. An evaluation of topics in modern mathematics. The Arithmetic Teacher, 1963, 10, 277-279.
*Peterson, J. M. A comparison of achievement in traditional mathe matics skills of seventh grade students using three different types of materials— traditional, transitional, and modern (Doctoral dissertation, Utah State University, 1966). Dissertation Abstracts, 1966, T j _ , 2790B. (University Microfilms No.
*Pettofrezzo, A. J. A comparison of the relative effectiveness of two methods of teaching certain topics in solid analytic geometry to college freshmen (Doctoral dissertation, New York University, 1959). Dissertation Abstracts, 1960, 2 0 , 4604A. (University Microfilms No. 60-1096)
*Phelps, J. A study comparing attitude towards mathematics of SMSG and traditional elementary school students (Doctoral dissertation Oklahoma State University, 1963). Dissertation Abstracts, 1964, 25, 1052A. (University Microfilms No. 64-8942)
*Phillips, J. W. Small group laboratory experiences as an alternative to total group instruction for college low achievers in mathe matics (Doctoral dissertation, Indiana University, 1970). Dissertation Abstracts International, 1971, _31, 5945A. (Univer sity Microfilms No. 71-11,406)
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*Possien, W. M. A comparison of the effects of three teaching methodologies on the development of the problem solving skills of sixth grade children (Doctoral dissertation, University of Albama, 1964). Dissertation Abstracts, 1965, 2 ! 5 , 4003A. (Univer sity Microfilms No. 64-12,774)
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*Post, T. R. The effects of the presentation of a structure of the problem solving process upon problem solving ability in seventh grade math (Doctoral dissertation, Indiana University, 1967) . Dissertation Abstracts, 1968, 28, 4545A. (University Microfilms No. 68-4746)
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*Price, J. S. Discovery: The effect on the achievement and critical thinking abilities of tenth grade general mathematics students (Doctoral dissertation, Wayne State University, 1965). Disserta- tion Abstracts, 1966, 2 f i , 5304A. (University Microfilms No. 66-1246)
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*Reimer, D. D. The effectiveness of a guided discovery method of teaching in college mathematics course for non-mathematics and non-science majors (Doctoral dissertation, North Texas State University, 1969). Dissertation Abstracts International, 1969, 30, 626A. (University Microfilms No. 69-13,220)
*Retzer, K. A. The effect of teaching certain concepts of logic on verbalization on discovered mathematical generalization (Doctoral dissertation, University of Illinois, 1967). Dissertation Abstracts, 1967, 28, 1351A. (University Microfilms No. 67-11,909)
*Rich, L. W. The effects of manipulative instructional mode in teaching mathematics to selected 7th grade inner city students (Doctoral dissertation, Temple University, 1972). Dissertation Abstracts International, 1972, J33, 330A. (University Microfilms No. 72-20,209)
*Robertson, H. C. The effects of the discovery and expository approach of presenting and teaching selected mathematical principles and relationships to fourth grade pupils (Doctoral dissertation, University of Pittsburgh, 1970). Dissertation Abstracts International, 1971, _31, 5278A. (University Microfilms No. 71-8785)
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*Rowlett, J. D. An experimental comparison of direct detailed and direct discovery methods of teaching orthographic projection principles and skills (Doctoral dissertation, University of
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*Roy, G. R. The effect of the study of mathematical logic on student performance in proving theorems by mathematical induction (Doctoral dissertation, State University of New York at Buffalo, 1970). Dissertation Abstracts International, 1971, _31, 4045A. (University Microfilms No. 70-22,109)
*Ruddell, A. K. The results of a modern mathematics program. The Arith- metic Teacher, 1962, 9 _ , 330-335.
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*Schlinsog, G. W. The effects of supplementing sixth grade arith metic instruction with a study of other number bases (Doctoral dissertation, University of Oregon, 1965). Dissertation Abstracts, 1966, 2 6 , 5307A. (University Microfilms No. 66-629)
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*Scott, L. F. Summer loss in modern and traditional elementary school mathematics programs. California Journal of Educational Research, 1967, 18, 145-151.
*Scrivens, R. W. A comparitive study of different approaches to teaching the Hindu-Arabic numeration system to third grades (Doctoral dissertation, University of Michigan, 1968). Disserta tion Abstracts, 1968, _29, 839A. (University Microfilms No. 68- 13,400)
*Settle, M. G. The relative effects of introduction in the guess and test procedure on writing relevant equations to verbal problems in algebra I (Doctoral dissertation, The Florida State University, 1977). Dissertation Abstracts International, 1977, 38.> 2633A. (University Microfilms No. 77-24,804)
*Shelton, R. M. A comparison of achievement resulting from teaching the limit concept in calculus by two different methods (Doctoral dissertation, University of Illinois, 1965). Dissertation Abstracts, 1965, 2 6 , 2613A. (University Microfilms No. 65-11,868)
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*Simmons, S. V. A study of two methods of teaching mathematics in grades five, six, and seven (Doctoral dissertation, University of Georgia, 1966). Dissertation Abstracts, 1965, J26, 6566A. (University Microfilms No. 66-2,500)
*Smith, C. W. A comparison of the effect of the activity approach and the lecture approach used in teaching general college mathe matics to risk students and the relationship between the two methods and learner characteristics (Doctoral dissertation, Temple University, 1975). Dissertation Abstracts International, 1975, J)6, 3476A. (University Microfilms No. 75-28,299)
*Smith, M. A. Development and preliminary evaluation of a unit on probability and statistics at the junior high school level (Doctoral dissertation, University of Georgia, 1966). Disserta tion Abstracts, 1966, 2 J _ , 1723A. (University Microfilms No. 66- 13,620)
*Smith, M. L. A comparison of three methods of teaching freshman mathematics: Lecture, guided discovery and programmed (Doctoral dissertation, University of Pittsburgh, 1975). Dissertation Abstracts International, 1976, 3 6 ^ , 5879A. (University Microfilms No. 76-5479)
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*Solheim, J. H. The effect of the study of the transformations of the plane on the attitudes of secondary school geometry students (Doctoral dissertation, Indiana University, 1971). Dissertation Abstracts International, 1972, _32, 3165A. (University Microfilms No. 72-1572)
*Steadman, D. G. An evaluative study comparing achievement and atti tude between the SSMGS unified mathematics programs and tradi tional accelerated mathematics programs in the Utah county public
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*Strickland, J. F., Jr. A comparison of three methods of teaching selected content in general mathematics (Doctoral dissertation, University of Georgia, 1968) . Dissertation Abstracts, 1969, 29, 4392A. (University Microfilms No. 69-9525)
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*Trafton, P. R. The effects of two initial instructional sequences on learning of the subtraction algorithm in grade three (Doctoral dissertation, University of Michigan, 1970). Dissertation Abstracts International, 1971, _31, 4049A. (University Microfilms No. 71-4753)
*Tryon, L. A. A comparison of a modern mathematics program and traditional arithmetic in the intermediate grades (Doctoral dissertation, University of Arizona, 1967). Dissertation Abstracts, 1968, 2^8, 3792B. (University Microfilms No. 68-3581)
*Tucker, J. A junior high school level experiment on developing algebra readiness by the use of finite non-numerical algebraic systems (Doctoral dissertation, Auburn University, 1969)
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*Volchansky, P. R. The effects of two mathematical instruction approaches on analytical cognition (Doctoral dissertation, The University of New Mexico, 1968). Dissertation Abstracts, 1969, 29, 4396A. (University Microfilms No. 69-9273)
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*Wilkinson, G. G. The effect of supplementary materials upon academic achievement in and attitude toward mathematics among eighth grade students (Doctoral dissertation, North Texas State University, 1971). Dissertation Abstracts International, 1971, 32, 1994A. (University Microfilms No. 71-25,378)
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*Yasui, R. V. An analysis of algebraic achievement and mathematical attitude between the modern and traditional mathematics programs in senior high school: A longitudinal study (Doctoral disserta tion, University of Oregon, 1967). Dissertation Abstracts, 1968, 28, 4967A. (University Microfilms No. 68-4017)
*Ziebarth, R. A. The effect of experimental curricula on mathematics achievement in high school (Doctoral dissertation, University of Minnesota, 1963). Dissertation Abstracts, 1964, 2 A , 4593A. (University Microfilms No. 64-4114)
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Appendix A
The List of the Variables Identified
Variable Name Purpose
1. Study number To group the effect One assigned to each sizes by category study
2. Study approach To specify the mod 1 = methodology ern mathematics 2 = content treatment in accord 3 = materials ance with the emphasis 4 = combination placed on the approach
3. Study type To differentiate 1 = achievement between achievement 2 = attitude and attitude studies
To identify the commercial innovator school district experimenter university other
5. Year of study To determine the year of publication
6. Source of study To determine where : public school report study was located ‘ ERIC : journal articles : dissertations
7. Affiliation of To specify the organi- : staff of a project researcher zation the researcher : school district worked for evaluation staff ; university ; other
8. Involvement of To determine researcher’s : evaluating on researcher personal dependency on material or project the outcome of the study
9. Area of study To define the area of mathematics investigation other subject areas included
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Appendix A— continued
Variable Name Purpose Values
10. Specific sub To describe what spe 1 = arithmetic ject area cific subject was taught 2 = algebra 3 = geometry 4 = sets 5 = logic 6 = statistics 7 = calculus 8 = trigonometry 9 = other
11. Type of pro To identify the spe 1 = UICSM gram cific innovative pro 2 = SMSG gram 3 = GCMP 4 = MADISON 5 = BCMI 6 = BSU 7 = UMMaP 8 = MINNEMAST 9 = OTHER
12. Length of To determine the dura Number of weeks until treatment tion of the treatment posttest
13. Length of To determine the dura Days per week treatment tion of the treatment
14. Length of To determine the dura Minutes per day treatment tion of the treatment
15. Time To find the total num Weeks x Days x Minutes ber of instruction 60 hours
16. Number To find the sample The number of subjects size that means are based on
17. New versus To determine if the 1 = new— never used field-tested treatment was field- before tested or new 2 = field tested
18. Grade To determine the grade 1 = KG or subjects 2 = Elementary 3 = JH 4 = SH 5 - Post-secondary
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Appendix A— continued
Variable Name Purpose Values
19. SES To determine the 1 =unknown socio-economic status 2 = low of subjects 3 = middle 4 = high
20. Ability To determine the ability 1 = unknown levels of subjects 2 = low 3 = middle 4 = high
21. Sex To determine the sex 1 = unspecified of subjects 2 = female 3 male
22. Test To determine the type 1 = commercial of test 2 = experimenter developed 3 = teacher developed 4 = other
23. Retention To identify the length Weeks between end of of retention treatment and testing
24. Effect size To calculate measures effectiveness
25. Method for To determine the tech 1 = control group effect sizes nique to be used to 2 = ANOVA measure effect size 3 = gain scores 4 = grade equivalent scores 5 = standardized tests 6 = t-test 7 = F-test 8 = ANCOVA
26. Random assign To determine the 1 = random assignment ment quality of design 2 = no random assignment
27. Novelty effect To determine the 1 - present novelty effect 2 = not present
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Variable Name Purpose Values
28. Dependent vari To identify overlap 1 = unique data able ping studies 2 = subtests when total included 3 = subtests when total not included 4 = total when subtests included
29. General content To determine the gen 1 = concepts area eral content area 2 = computation emphasized 3 = application 4 = combination
30. Nature of test To determine the cate 1 = retention gory of test 2 = recall 3 = transfer
31. Pool To determine the num ber of effect sizes in each study
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Appendix B
Code Sheet of the Variables Identified
Reference
CODE COLUMN VARIABLE VALUE
1- 4 STUDY NO 5 STUDY TYPE 1 Met 2 Cont 3 Mat 4 Comb 6 SUBSTUDY NO 1 Ach 2 Att 7 AUTHOR 1 Com 2 Sc D 3 Exp 4 Uni 5 Ot 8- 9 YEAR OF STUDY 10 SOURCE 1 P Sc 2 ERIC 3 JA 4 Diss 5 Ot 11 STUDY RATING 1 Rand 2 ECG 3 NECG 4 N CG 12 AFFI OF RES 1 SP 2 Sc ES 3 Uni 4 Ot 13 INV OF RES 1 0 wn 2 None 14 SUB AREA 1 Math 2 Sc 3 SS 4 Lang 5 Comb 15 METHOD 1 Reas 2 Act 3 Gru 4 Ot 16 CONTENT 1 Arit 2 Alg 3 Geo 4 Set 5 Log 6 Stat 7 Cal 8 Trig 9 Ot 17 TYPE OF PGM .1 UICSM 2 SMSG 3 GCMP 4 MAD 5 BCMI 6 BSU 7 UMMaP 8 MIN 9 Ot 18-20 LENGTH OF TREATMENT WEEKS 21 LENGTH OF TREATMENT DAYS 22-24 LENGTH OF TREATMENT MINUTES 25-27 TIME WEEKS X DAYS X MINUTES 60 28-31 NUMBER AVERAGE 32 FIELD-TESTED 1 New 2 Field T 3 Unk 33 GRADE 1 KG 2 Ele 3 JH 4 SH 5 Post Sec 6 Ot 34 SES 1 Unk 2 L 3 M 4 H 35 ABILITY 1 Unk 2 L 3 M 4 H 36 SEX 1 Both 2 Male 3 Female 37 TEST 1 Comm 2 Ex D 3 Tea D 4 Ot 38-40 RETENTION TIME WEEKS 41-48 EFFECT SIZE 49 METOD ES 1 CT 2 ANOVA 3 GAIN SC 4 GES 5 ST 6 t-T 7 F-T 8 ANCOVA 9 Ot 50 ASSIGNMENT 1 Rand 2 No Rand 51 NATURE OF TEST 1 Ret 2 Rec 3 Tran 52 NOVELTY EFFECT 1 Yes 2 No 53 DEP VAR 1 Uni 2 STWTI 3 STWTNI 4 TWSI 54 CONT GEN 1 Con 2 Comp 3 Appl 4 Gen 55 TRANSFER TIME 56-60 POOL
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