THE RELATIONSHIP OF SELFPACED INDIVIDUALIZED INSTRUCTION
TO PUPIL ACHIEVEMENT WHEN MEASURED BY POOLING THE
PROBABILITIES OF SEVERAL INDEPENDENT SAMPLES
By
PAUL IVAN JOHNSON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
ACKNOWLEDGtIENTS
Grateful acknowledgment is made to the many indi
viduals who made this study possible. Sincere appreciation
is extended to Dr. Ralph B. Kimbrough (chairman), Dr. Robert
S. Soar, Dr. Michael Y. Nunnery, Dr. Herbert Franklin, and
Dr. Paul S. George who made many valuable suggestions as
members of the dissertation committee.
To Dr. Ralph B. Kimbrough, Dr. Robert S. Soar, and
Dr. Michael Y. Nunnery the writer owes a special debt of
gratitude for their advice which was always clear and
directly to the point.
Finally, the writer would like to acknowledge his
indebtedness to his wife, Dona, without whose understanding,
encouragement, and inspiration, this dissertation probably
would not have been completed.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ............................. ii
ABSTRACT ... ................................... v
CHAPTER
I. INTRODUCTION ........................ 1
Need for the Study ............... 1
Statement of the Problem ......... 15
Delimitations and Limitations .... 15
Definition of Terms .............. 17
Procedures ........................ 20
Organization of the Study by
Chapters ........................ 26
II. SELECTING A SET OF STUDIES ......... 28
Introduction ........................ 28
Miller's Review ...................... 28
Hirsch's Review ...................... 29
Schoen's Review ...................... 29
Summary ............................ 33
III. REPORTED RESULTS OF THE INDIVIDUAL
STUDIES AND CONVERSION TO CHISQUARE 35
Introduction ...................... 35
Bull (1971) ...................... 37
Englert (1972) .................. 38
TABLE OF CONTENTS (con't)
CHAPTER Page
III. Fisher (1973) ..................... 41
Hanneman (1971) .................... 43
Herceg (1972) ....................... 45
Hirsch (1972) ....................... 47
Ludeman (1973) ................ 50
Penner (1972) ......... ........ .. 52
Schoen and Todd (1974) .............. 54
Taylor (1971) ..................... 57
Thomas (1971) ..................... 58
Summary .......................... 61
IV. SUMMARY, CONCLUSIONS, IMPLICATIONS AND
SUGGESTIONS FOR FURTHER RESEARCH .. 63
Summary .......................... 63
Conclusions ...................... 65
Implications ...................... 65
Suggestions for Further Research .. 67
APPENDIX HISTORY OF AGGREGATE CHISQUARE AND
TRANSFORMATION PROCEDURES ...... 70
REFERENCE NOTES ............................. 72
REFERENCES ... ................................ 73
BIOGRAPHICAL SKETCH ......................... 77
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE RELATIONSHIP OF SELFPACED INDIVIDUALIZED INSTRUCTION
TO PUPIL ACHIEVEMENT WHEN MEASURED BY POOLING THE
PROBABILITIES OF SEVERAL INDEPENDENT SAMPLES
By
Paul Ivan Johnson
June 1979
Chariman: Dr. Ralph B. Kimbrough
Major Department: Educational Administration & Supervision
The problem of inconsistent results in educational
research has been pervasive in the education profession.
Administrators and other educational decision makers have,
under the pressure of economic, political, and social forces,
often been pressured to adopt educational innovations with
out prior validation through empirical research. Selfpaced
individualized instruction was an example of an instructional
innovation which has been widely adopted by school decision
makers without such validation. Research results reported
on selfpaced individualized instruction have presented an
inconsistent and inconclusive data base from which to for
mulate an objective decision.
Some researchers have reported that selfpaced individ
ualized instruction is superior to traditional classroom
instructional practices. Other researchers have reported
the opposite result and yet others have reported "no
difference" in the two methodologies when those method
ologies were comapred by their effect on student achievement.
The concept of secondary analysis promised a solution to
the problem of inconsistent results thereby making it
possible for decision makers to base their decisions on
reliable research. In secondary analysis, the researcher
gathers data from previously completed research studies.
The researcher reorganizes and reanalyzes the data through
a variety of possible techniques.
In this study, the aggregate chisquare statistical
procedure was applied to the reported results of a pre
selected set of independent studies. By pooling the sta
tistical results of 24 independent samples taken from 11
independent research studies, it was shown that traditional
classroom practices are superior to selfpaced individual
ized instruction when pupil achievement is used as the
criterion. The derived statistic was aggregate X2 (48)=
101.9880, p <.002.
The above derived statistic shows that selfpaced
individualized instructional practices are not superior
to traditional instructional practices. In fact, the
reverse could be implied; that traditional instructional
practices are superior to selfpaced individualized
instructional practices. The probability of .002 for
this combined significance test indicated that in 998
of every 1,000 cases this result would be replicated.
In other words, it was very unlikely that this result
came by chance.
CHAPTER I
INTRODUCTION
Need For The Study
The school administrator has, among others, the
functions of decision maker and instructional leader. The
accountability movement has brought considerable pressure on
the local school administrator to justify the programs and
practices which have been and are being used in the class
rooms. The pressure exerted by the accountability movement
has caused administrators to scrutinize more carefully the
instructional practices used in their schools. In the past,
many instructional innovations have been advanced and adopted
because of emotional or philosophical commitments made by
educational leaders and not because of evidence garnered
through the empirical research process.
The results of educational research have also provided
a tenuous base upon which to build classroom practices. One
of the reasons for this was that classroom practices have
often been predicated on the basis of one or two research
studies or on no research at all. Very often one researcher's
conclusions have contradicted another's with all of this pro
ducing a state of confusion among educational decision makers
and practitioners.
Pressure from the demands of the accountability move
ment, coupled with the historically capricious results of
educational research and with the tendency of educators to
emotionally commit themselves to new ideas, has resulted in
many educational methodologies having been adopted apart from
any consistent empirical validation. Selfpaced individual
ized instruction was one such methodology. Some researchers
have reported that selfpaced individualized instruction was
superior to traditional classroom practices. Others have
reported the opposite result and many more have reported that
there was no difference in the two methodologies in terms
of enhancing student achievement.
Obviously, new or better techniques were needed to help
resolve the issue. Several researchers (Gage, Note 1; Glass,
1976; Light & Smith, 1971) have suggested secondary analysis
techniques as a means of resolving the dilemmas that pre
vailed in educational research results. Secondary analysis
was needed in the case of selfpaced individulaized instruc
tion because of two inherent problems that plagued self
paced individualized instruction research studies.
The problems of low expected relationships and small
sample size have been pervasive problems in all classroom
research as well as in the case of selfpaced individualized
instruction. The school classroom has been noted for its
multiplicity of variables. All these variables could have an
effect on the results of instruction. Therefore, the ex
pected relationship of any one variable (whether textbook,
3
teacher style, room temperature, instructional method,
etc.) to a learning outcome (grade or test score) was
automatically a low relationship. Generally, no one variable
would have a high correlation with any other variable. A
second problem plaguing all classroom research has been that
of small sample size. It has been difficult to generalize
the results of classroom research to larger populations be
cause the original studies were carried out with small
samples. There were other problems generated by the use of
small samples and these were discussed later in this study.
Studies completed on selfpaced individualized instruc
tion seemed to fit the pattern which had been evident in
other educational research. Secondary analysis techniques
were needed in order to resolve the inconsistent pattern of
research results coming from studies done on selfpaced
individualized instruction. A secondary analysis technique
called the aggregate chisquare was suggested by Gage (Note 1)
as a possible solution to the pervasive problems affecting
classroom research. The following sections are a more de
tailed discussion of the accountability movement and its
effect on classroom practices, the weakness of past educa
tional research, the tendency of educators to commit them
selves to innovative ideas before or without empirical vali
dation, the need for secondary analysis in educational re
search, the two pervasive problems in educational research,
the potential solution, and why administrators need a more
sound base from which to make decisions.
Accountability Demands Justification
of Classroom Instructional Strategies
The "formidable force" (Sciara & Jantz, 1972, p. 3)
of accountability has been a part of the educational scene
since 1963. Darland (1970) called it a "national crisis"
and asserted that "the American teacher has become a most
likely candidate for scapegoat of the 1970's (p. 41).
But, according to Briner (1969), it is the school admin
istrator who has been and will continue to be the scapegoat.
Teachers, parents, and others apparently
don't want to be fully accountable for
improving education. They tend to be
consumed by their typical social and
professional roles. On the other hand,
administrators must be directly account
able. It is their essential reason for
being. They are the ones upon whom the
hands of approval or disapproval will be
laid. Accordingly, their roles require
orientation and commitment to educational
success and the elimination of failure.
In their performance, they can be expected
to explain both success and failure; they
must be capable of proposing educational
improvements to the satisfaction of stu
dents, teachers, parents, and others.
(p. 205)
The move to accountability in education has produced
an "imperative" (Hostrop, Mecklenbarger, & Wilson, 1973,
preface) force upon educators (whether teacher or adminis
trator) to show "proof of results" (Lessinger, 1971, p. 13).
In addition, Lessinger asserted that general results will
not be proof enough. "We must go beyond such general out
lines of general results and find out what specific factors
produce specific education results" (p. 13).
5.
Authors of articles and books on educational account
ability have stressed two important factors. First, that
educators must be able to show which instructional methods
and programs are successful and second, (and more specific
ally) success of educational programs will be measured in
terms of pupil achievement.
Thus, for our purposes, classroom
activities will be deemed successful
if they induce desired changes in
pupils. Such a position requires us
eventuallyto validate the practices
of teaching with product variables
[product variables are evidences of
change in students as a result of their
involvement in the classroom].
(Dunkin & Biddle, 1974, p. 48)
The forces of accountability have demanded and will
continue to demand specific results from specific programs
(Berry, 1977, p. 4). Educators have not been and are still
unable to document with empirical evidence the merit of class
room practices. The "bottom line" in education is proof of
results. Educators have been unable to offer the proof
demanded.
The Weakness of Educational Research
In 1953, the Committee on Teacher Effectiveness of the
American Educational Research Association reported the
following conclusions.
The simple fact of the matter is that,
after 40 years of research on teacher
effectiveness during which a vast num
ber of studies have been carried out,
one can point to few outcomes that a
superintendent of schools can safely
employ in hiring a teacher or granting
his tenure, that an agency can employ
in certifying teachers, or that a
teachereducation faculty can employ
in planning or improving teacher
education programs. (p. 657)
Twentyone years later, Dunkin and Biddle (1974) con
cluded that "those who are seeking simple answers to the
problem of teacher effectiveness are only slightly better
off today than they were twenty years ago" (p. 16). Two
years after Dunkin and Biddle, the following observation was
made.
The status of numerous innovative in
structional strategies is extremely
tenuous, with many being perpetuated
by a false sense of empirical security.
Teachers are implementing a variety of
techniques which they assume to be
superior but which have not been proven
in practice, at this point in time.
(George & Maxwell, 1976, p. 56)
Education has had a history of weak empirical valida
tion of its programs and practices. "Educational research
typically follows innovation and hence has little construc
tive effect on educational practice" (Good, Biddle, & Brophy,
1975, p. vii).
The preceding comments were made by professional
educators all of whom are deeply committed to achieving
quality in education through empirical research. If, in
their judgment, educational practices have been and still
are based on a tenuous empirical foundation, is it any wonder
that the educational establishment has been forced into a
defensive position by the accountability movement.
The "Commitment" of Educators
to Individualized Instruction
Individualized instruction is an educational practice
that needs empirical justification. Educators have made a
heavy commitment to individualized instruction. Dunkin and
Biddle (1974) defined commitment as "advocating innovative
ideas for improving education" when those ideas are "attrac
tively argued but unsupported by data" (p. 51).
Individualized instruction became one of the basic
commitments of the education profession under the pressure of
accountability. Morris (1971) in commenting on "ten things
accountability will require" stated that one of those require
ments will be a commitment to individualized instruction.
Education must become, not only in
theory but in fact, child centered.
We will be forced to write programs
for each child based on extensive re
sults of highly sophisticated diag
nostic instruments. (p. 326)
Another writer (Davies, 1970) supported Morris'
prediction.
Teachers and all school personnel are
involved in the search for answers to
critical questions such as these:
How do we move from a mass approach
to teaching and learning to a highly in
dividualized approach?
How do we go about the "simple"
task of treating each child as an individ
ual human being? (p. 129)
Davies' and Morris' comments were made in response to
the accountability movement which was beginning to have a
strong impact (Hostrop et al., preface) on education at the
time of their writings.
Professional educators have more recently affirmed
their commitment to individualized instruction. Saylor
(1977) wrote in the guest editorial of the January issue of
Educational Leadership the following statement:
Judgments about the quality of edu
cation must be rendered in terms of
how well the school is developing
the respective set of talents, capa
bilities, and potentialities of each
student for living a life of personal
satisfaction and compassionateness in
our society. (p. 245)
The subsequent February issue of Educational Leadership
was devoted to the topic of individualized instruction.
Anderson, in the guest editorial to that issue, wrote that
"the resolve of most educational leaders is to pursue the
path toward individualization" (p. 324). It was apparent
that educators have made the commitment to individualized in
struction. Have the results of classroom research supported
this commitment? It was unlikely that a commitment to in
dividualized instruction was justifiable.
Educational Leadership, as noted above, devoted its
February, 1977, issue to individualized instruction. Follow
ing are some of the comments by different authors in that
issue.
Individualized instruction is an ex
cellent example of the truth that
educational innovations should be
tried cautiously, with proof demanded
that they actually produce better re
sults. (Weber, p. 328)
I am pessimistic that the concept will
survive [that is, the concept of mas
tery of learning] unless the practice
of individualized instruction proves to
be more effective. (Block, p. 341)
The results have been conclusive. Imple
mentation of the ACIL [Arizona Consortium
for Individualized Learning] process works
for students, teachers, administrators,
and parents. (Webb & Howard, p. 356)
In a time which has been described as
education's Age of Individualized In
struction there is a pressing need for
a comprehensive evaluation of all in
structional strategies [instructional
strategies pertaining to carrying out
the individualized instruction concept].
(George & Maxwell, p. 57)
Secondary Analysis: A New
Trend in Educational Research
The pressure of the accountability movement and the
historical unreliability of educational research has caused
educators to seek new ways to conduct research on instruc
tional strategies.
In educational research, we need more
scholarly effort concentrated on the
problem of finding the knowledge that
lies untapped in completed research
studies. . The best minds are
needed to integrate the staggering
number of individual studies. This
endeavor deserves higher priority
now than adding a new experiment or
survey to the pile. (Glass, 1976, p. 4)
Glass (1976) was referring to a relatively new approach
(of which lie was a forerunner) in educational research. The
new approach has been called "secondary" or "meta" analysis
(Glass, 1976, p. 3). The basic thrust of secondary analysis
is that the researcher analyzes previously done analysis on
a given topic or variable (analysis of analysis).
10
Two conditions in educational research have led to the
need for secondary analysis. One condition is that there
is a proliferation of studies on a wide variety of educa
tional topics, but little has been done to integrate the re
sults of the many individual studies.
A second condition is that the "fragile and confusing
findings" which has been characteristic of all educational
research (Glass, 1976, p. 3). "Where ten studies might
suffice to resolve a matter in biology, ten studies on com
puter assisted instruction or reading may fail to show the
same pattern of results twice" (p. 3).
The key problem was that of integrating the findings.
How shall educational researchers integrate the varied re
sults? There have been at least three approaches to the
problem of integration of research findings.
One attempt at integration has been made by simply
discarding all of the studies of a given set which have not
been done with acceptable designs or systems of statistical
analysis. This method of integration leaves the researcher
with a set of studies which are technically near perfection
in terms of design and analysis. Glass decried this method
and commented "that this approach takes design and analysis
too seriously" (Glass, 1976, p. 4). Glass further commented
th a t "eliminating the 'poorly done' studies is to dis
card a vast amount of data" (p. 4).
1 1
A second method of integration has been to collect a
significant number of studies done on a given variable and
compare the reported statistics in terms of the direction
of significance. Dunkin and Biddle (1974) used this method
in their comprehensive review of the research on teaching.
The inconclusive results reflected in the Dunkin and Uiddle
book are what led Gage (Note 1) to pursue a secondary
analysis of the same data.
Secondary analysis, then, is the third method which has
been used to integrate research data. Glass (1976); Light
and Smith (1971); and Gage (Note 1) have suggested various
approaches and statistical techniques for secondary analysis.
The method of secondary analysis used in this study follows
the aggregate chisquare described by Gage (Note 1). The
use of the aggregate chisquare procedure was especially
appropriate because of two problems that are inherent in
educational research.
Two Pervasive Problems of Classroom Research
Why have educators been unable to substantiate class
room practices through empirical research? Gage (Note 1)
identified two factors that continually plague educational
research. One factor was that of the "expected relationship"
of any single teaching variable to the effect on students
who are exposed to that variable. A second factor was the
effect of small sample size (Gage, Note 1, pp. 89).
12
Gage has pointed out that, first, the relationship of
"any single variable of teacher behavior" to that of pupil
achievement was "probably low"
On the face of it, the teaching
learning process is so complex that
any single variable of teacher be
havior should have only a low corre
lation (ranging from about + .1 to
about + .4) with student achievement
or attitude. (Gage, Note 1, p. 8)
A second problem that plagued educational researchers
was the problem of small sample size. Most educational re
search was based on relatively small numbers of teachers.
For a sample of the median size,
namely, 15 teachers, it is necessary
that a correlation coefficient equal
.51 if it is to be significant at
the .05 level. The coefficient must
equal .64 if it is to be significant
at the .01 level. (Gage, Note I, p. 8)
When the two problems, low expected relationships and
small sample size, were combined, it was not unexpected
that most of the studies in educational research would not
attain statistical significance.
Is it possible to gain significant knowledge from
classroom research that has the inherent problems of low
expected relationships and small sample size? Can classroom
research, faced with the two inherent problems just des
cribed, justify the programs and practices being used in
American classrooms through empirical research (specifically,
selfpaced individualized instruction)?
A Potential Solution
Gage followed his description of the problems with a
potential solution. It is a method for testing the sig
nificance of combined results. Gage applied the aggregate
chisquare model to five clusters of studies that had been
done on different teacher behavior variables (Gage, Note 1,
p. 13). By applying the aggregate chisquare model, Gage
was able to show that teacher variables (such as "praise")
had significant relationships to student achievement. The
importance of those findings is that in most previous re
search on those same variables, those variables did not
prove to have a significant relationship to student achieve
ment. Gage summarized the importance of his findings in the
following paragraph:
These tests of combined results do in
deed reveal some significant relation
ships between types of teacher behavior
and student achievement or attitude.
Thus, the results suggest that seeking
such processproduct relationships is
not altogether fruitless. They bear
out the widely held and hard to surrender
intuition that how teachers behave makes
a difference in what students learn.
(p. 15)
The method used by Gage and the resultant findings
indicate that research on teaching may not be as ambiguous
as previously believed.
Because the results of independent research studies
done on selfpaced individualized instruction presented the
same ambiguous picture as studies done on many other in
structional variables, it was appropriate to apply the
14
aggregate chisquare procedure in a secondary analysis of
selfpaced individualized instruction.
Administrator Need for Objective Data
Regarding SelfPaced Individualized Instruction
Individualized instruction (specifically, selfpaced)
was a classroom instructional practice widely adopted and
touted as the answer to the demands of accountability. In
dividualized instruction was also advanced as the means
through which the individual could reach his maximum poten
tial. Furthermore, it was written in a leading educational
journal that "the resolve of most educational leaders is to
pursue the path toward individualization" (Anderson, 1977,
p. 324).
Did the accumulated empirical evidence from research
done on selfpaced individualized instruction justify such
a strong commitment? The evidence did not justify such a
commitment. However, the empirical evidence did not pre
clude such a commitment either. As was mentioned earlier in
this study, the research on selfpaced individualized in
struction was inconsistent and therefore inconclusive.
In the face of the pressures from educational consumers
(accountability) and from educational professionals, an ad
ministrator needed a more objective data base from which to
generate a viable decision. Decisions involved with an
innovation such as selfpaced individualized instruction
were not insignificant in terms of finances, administrator
time, teacher time, inservice, curriculum changes, hardware,
15
software, e t c. Stronger empirical evidence regarding
selfpaced individualized instruction was needed.
Statement of the Problem
The problem of this study was to determine the effec
tiveness of selfpaced individualized instruction by apply
ing the aggregate chisquare statistical procedure to the
results of several, previously done, independent studies
which have analyzed selfpaced individualized instruction in
its relationship to student achievement. What would be the
result when these several independent results were pooled to
derive a single statistic?
Delimitations and Limitations
In a study of this nature there are certain constraints
on any results that might be obtained. The following de
limitations and limitations are explanations of the con
straints applicable to this study.
Delimitations
1 The purpose of this study was to investigate the
aggregate main effect of selfpaced individualized
instruction on pupil achievement. Consequently,
interaction effects from within or across the in
dividual studies were not investigated.
2. The test of combined significance bore upon a
question being asked about the cluster of
studies as a whole and consequently did not bear
on any of the individual studies in
cluded in the cluster. That is, the com
bined result does not change and cannot be
construed to change the previous results of
the independent studies.
3. It was beyond the scope of this study to make
an assessment of the experimental studies
analyzed except that they met the criteria
outlined in the procedures section.
4. The set of 11 studies used to obtain the
pooled statistic was not exhaustive (it did not
include all possible studies done on that subject
or that grade level).
Limitations
1. The results of this study are not decisive or
final in the case of selfpaced individualized
instruction and its effect on pupil achievement.
Replications of this type of study, other secon
dary analysis techniques, or other original re
search would be necessary to support the con
clusions reached in this study.
2. The results of this study were not generalizable
beyond selfpaced individualized instruction at
the high school level in mathematics oriented
subjects.
Definition of Terms
For the purposes of this study, the following terms
were used as defined below.
Control group. In experimental studies, "the control
group does not receive the experimental treatment [inde
pendent variable]" (Huck, Cormier, & Bounds, 1974, p. 245).
The control group must be equivalent to the experimental
group in respect to all crucial variables. Equivalency of
control and experimental groups is usually achieved by random
assignment of subjects to groups. Other equalization tech
niques are often necessary to support random assignment and
to insure equalization of control and experimental groups.
Dependent variable. "The researcher in experimental re
search must first identify those dependent variables which
will, taken together, make a reasonable test of the independ
ent variable? (Fox, 1969, p. 460). The selection of dependent
variables is determined by the research problem, stated hy
pothesis, and the class or type of dependent variable or
variables being measured. Some dependent variables may be
measures of achievement (as one class or type) while others
may be associated with attitude. Some dependent variables
measure longterm results while others measure shortterm
results. The dependent variable must be consistent with the
purposes of the research problem, measure results that cor
respond to the hypothesis being tested, and be measured with
instruments that are valid for the particular subjects being
used in the study. The goal of the researcher is to show
a casual relationship between the independent and dependent
variable or variables.
Experimental group. In experimental studies, "the
group that receives the treatment [independent variable]
is called the experimental or treatment group" (Huck et al.,
1974, p. 245).
Independent variable. The independent variable is the
condition or conditions that an experimenter can manipulate.
The experimenter attempts to show a casual relationship be
tween the independent variable and some outcome measure
(dependent variable). The independent variable is sometimes
called the "experimental, treatment, or intervention variable"
(Huck et al., 1974, p. 224).
Individualized instruction. Most authors agree that
there is no one definition for individualized instruction.
"The only common universal attached to the term is that stu
dents generally will be able to proceed at their own pace in
some areas" (Good et al., 1975, p. 169).
Process variable. Process variables are the conditions
in a given classroom that would be expected to effect the
performance of the pupils in that classroom. A teacher's
personality, the textbooks, audiovisual aids, and grouping
practices are examples of process variables.
Processproduct research. Processproduct research is
the attempt of researchers to establish casual relationships
between specific process variables and specific product
variables.
Product variable. Product variables are the outcome
measures associated with given classroom groups or with
individual pupils. A pupil's test score or a group mean de
rived from a set of test scores are examples of product
variables. Test scores used as outcome measures are usually
a measurement of pupil achievement or pupil attitude toward
school.
One and twotailed tests. The definition offered by
Huck et al., (1974, pp. 4546) was accepted for this study
and it is as follows:
When using some tests of signifi
cance, the researcher must also de
cide whether the test will be a one
tailed test or a twotailed test. A
twotailed test is sensitive to sig
nificant differences in either direc
tion (i.e., greater and less); the one
tailed test is sensitive to differences
in only one direction (i.e., greater or
less). Suppose, for example, that a
researcher compares achievement test
scores of a group of students exposed
to a new method of instruction to the
scores of another group instructed by
the traditional method. If a twotailed
test is used to compare the scores of
both groups, the researcher can answer
two questions: (1) Do students under the
new method score significantly higher?
(2) Do students instructed with the
traditional method score higher? On the
other hand, if a onetailed test is used,
the researcher can answer only one ques
tion: (I) Do students under the new
method score significantly higher?
Also, if differences are found to be
significant at a certain level of sig
nificance with a onetailed test, the
same difference with a twotailed test
would be significant at a level of sig
nificance twice as large as that used
with a onetailed test. For example,
if the researcher found a significant
difference at the .025 level with a one
tailed test, the same data used with a
twotailed test would be significant
only at the .05 level.
Traditional instruction. The researchers whose data
were pooled in this investigation used the term "traditional"
to define a mode of instruction which contrasted to individ
ualized instruction. "Traditional" was synonymous with large
group instruction, the lecturediscussion mode, and teacher
paced learning environments.
Procedures
The general approach was to use the procedures developed
by Gage (Note 1) to reanalyze the relationship of individualized
instruction to pupil achievement. The studies identified by
Schoen (1976) were selected for analysis. The aggregate chi
square statistic was applied to the data as demonstrated by
Gage.
Source of Data
The procedure and rationale for selecting the source of
data for this study followed closely that of Gage (Note 1).
Gage chose a process variable which had been reported on in
the Dunkin and Biddle text (1974) and which had inconclusive
results when a number of independent studies, reporting on the
same process variable, were compared. For example, teacher
praise was one process variable on which Dunkin and Biddle
gathered evidence. The studies reported on by Dunkin and
21
Biddle gave inconsistent results and presented a confusing
picture as to the effect of teacher praise on learning out
comes.
Gage took the same set of studies collected by Dunkin
and Biddle and applied the aggregate chisquare statistic
to pool the independent probabilities into an aggregate state
ment of significance. As was reported earlier, the pooled
statistic revealed a consistent relationship between the use
of praise and pupil achievement.
The purpose of this study was to ascertain the relation
ship of individualized instruction to pupil gain in academic
achievement by using the pooled probabilities of several in
dependent samples. A survey of the literature revealed that
several reviews of research had been published with regard to
individualized instruction. Schoen's (1976) review was the
one chosen. In Schoen's review, one researcher's results
favored the experimental group (individualized instruction),
three other researchers reported results favoring the con
trol group (traditional instruction), and eight other re
searchers reported results as not significant (Schoen, 1976,
p. 354).
Requirements and Statistical Procedures
This section is an explanation of the requirements and
procedures (Fisher, 1948, pp. 99101) which are necessary
when applying the aggregate chisquare model to a set of data.
As written previously, the aggregate chisquare model was
22
employed to reanalyze data from several selected studies.
By following the steps outlined below, the reader should
be able to duplicate this type of study.
Independence of the separate results being combined.
This means that each result used was obtained from a dif
ferent sample and that only one set of results for any single
sample of individuals was usable.
Some experimental studies were done using several
schools and/or teachers. When one or more schools or teachers
were treated as separate experimental units in a given study,
then each school and/or teacher was a source of an independent
result from an independent sample. For example, if in a cer
tain study, teacher A, teacher B, and teacher C were each
assigned to a control group and to an experimental group, the
statistical results were independent for each teacher. The
sample of students for teacher A was separate from each other
set of control and experimental groups. Likewise, the sample
associated with teacher B and the sample associated with
teacher C was separate and independent of each other.
For another example (which is a somewhat different case
than the one noted above), if in a given study there is more
than one dependent variable (criterion measure), the inde
pendence of the statistical results can be maintained by re
porting the least significant chisquare, the most signifi
cant chisquare, and a mean (mean of the two or more results
chisquare (Gage, Note 1, pp. 14 & 33).
23
Rational basis for combining the studies. All of the
studies must investigate the same variable. An example
would be a set of studies all of which investigated the re
lationship of time in class to student achievement. In this
study, the rational basis was the use of the individualized
instruction approach and its effect on student achievement.
The rational basis for this particular study was further de
fended in Chapter II.
All results in the set being analyzed must be included.
An objective means of achieving the above criterios was to
set specific limits on what studies were included. The re
searcher could use criteria such as subject matter, grade
level, studies done only within a given span of time (1960
1970, for example), and/or a criterion such as experimental
studies as opposed to field studies.
Another procedure (the procedure chosen for this study)
was to choose all of the studies included in some review of
research as when Gage (Note 1) chose only the studies reported
by Dunkin and Biddle (1974) in their review of classroom re
search. The important consideration was that the researcher
avoid the bias that would occur if he chose only those studies
which offered near significance.
Knowledge of the exact onetailed probability value
associated with each result to be combined. Researchers, in
many cases, have reported statistical measures such as F
ratios, tratios, and chisquares. In addition, those statis
tics have been reported as being either significant (at a
24
given level of probability) in favor of the control group,
significant in favor of the experimental group, or not
significant (NS).
Some studies were set up on the basis of an hypothesis
which required a twotailed test for significance and some
hypotheses required a onetailed test for significance.
If the hypothesis was nondirectional, the exact prob
ability value associated with the reported statistic was
divided by two (p/2). For example, if the hypothesis required
a twotailed test and the reported F for that hypothesis was
associated with a .30 probability value, then the exact one
tailed probability value is .15 (.30/2). When the reported
probability value was associated with a onetailed test,
dividing by two was unwarranted.
Determination of the direction of the statistical results.
Did the reported statistic favor the control group or the ex
perimental group? The researcher indicated which group was
favored when the statistic was significant. In many cases,
the reported statistic was reported as not significant. When
this happened, the actual results (the group means) were ob
tained from the body of the original research report. The
group means indicated which group did better on the criterion
measure or dependent variable.
If the reported statistical result and the exact one
tailed probability associated with that result were in the
same direction as the majority of cases in the set of studies
whose statistical results were being pooled, then the exact
25
onetailed probability value was taken directly to the chi
square table for two degrees of freedom developed by Gordon,
Loveland, and Cureton (1952).
The exact onetailed probability was subtracted from
one (lp) before going to the chisquare table with two de
grees of freedom if it was not in the same direction as the
majority of cases in the set of studies whose statistical re
sults were being pooled.
Ad ustement of _ag_g gate onetailed probability to a
twotailed probability. The probability derived from the
aggregate chisquare statistic had to be doubled. The use
of onetailed probabilities to derive the aggregate statistic
and probability was inconsistent with the assumption that the
procedure allows for a result in either direction. This in
consistency is easily remedied by doubling the probability
derived from the aggregate chisquare statistic (Gordon et al.,
1952, p. 315).
Summary of Procedures for Treatment of the Data. The
following summary was presented to help clarify the individual
steps in the aggregate chisquare procedure.
1. The exact onetailed probability associated with
the reported statistic was found for each in
dependent sample.
2. The direction of the result was determined. Was
it in the same direction as the majority of the
cases included in the set of studies being pooled?
26
If it was in the same direction as the majority,
the exact onetailed probability was taken di
rectly to the chisquare table with two degrees
of freedom (Gordon et al., 1952). If the result
was not in the same direction as the majority,
the probability value was subtracted from one
(1p). When the result was generated from a
twotailed test, the probability was divided by
two (p/2) before going to the extended table of
chisquare developed by Gordon et al., 1952.
3. All of the independent chisquare statistics with
two degrees of freedom were added together to ob
tain the aggregate chisquare statistic.
4. The aggregate chisquare statistic was taken to
a regular chisquare table of values and the
appropriate probability value was read from that
table.
5. The derived probability was doubled to correspond
to the twotailed assumption which allowed for a
result in either direction.
Organization Of The Study By Chapters
Chapter I was an introduction to the study. Topics
discussed in Chapter I were need for the study, the problem,
delimitations and limitations, definition of terms, and pro
cedures.
27
Chapter II is a discussion of the rationale for choos
ing the set of 11 research reports included in this study
which were all from the Schoen (1976) review.
Chapter III is a presentation of each of the 11 in
dividual research reports (in abstract form) with their
statistical results and the aggregate statistic derived by
pooling the individual statistics.
Chapter IV is a summary of this research study as well
as a discussion of the conclusions, implications, and sug
gestions for further research.
CHAPTER II
SELECTING A SET OF STUDIES
Introduction
To accomplish the purpose of this study, a set of
independent samples had to be identified. Reading and
mathematics have traditionally been the most individualized
of all school subjects. Therefore, a set of studies was
sought where each investigator had done an experiment on
individualizing and where all of the experiments in the set
were on reading or math, but not both.
An additional requirement was that selfpacing had to be
an identifiable feature of each study. Selfpacing has been
identified as the only common denominator among various in
dividualized programs (Good et al., 1975, p. 169 and Miller,
1976, p. 345).
Miller's Review
Miller (1976) produced a review of research on individ
ualized mathematics programs. One hundred and fortyfive
studies were included in Miller's review. The studies in
cluded were done over a wide range of grade levels (kinder
garten through college level) and over a wide range of time
(17 of the studies were done before 1960).
29
Primarily, because of these wide ranges in grade levels
and time, Miller's review was not used. An additional rea
son was that the sources of data were so varied that a great
amount of time and expense would have been necessary to sort
out and track down a representative set of studies.
Hirsch's Review
Hirsch (1976) reviewed research on individualized in
struction in secondary mathematics. Thirtythree studies
were reviewed. The grade level range of the studies was from
grade seven through grade twelve. The time range was from
1967 through 1974.
Although Hirsch had narrowed the field considerably
from that of Miller (1976), the grade level span was not ideal.
Time and expense were also a factor in deciding against using
Hirsch as a source.
Schoen's Review
The set of studies chosen was taken from Schoen's (1976)
review which he titled "Selfpaced Mathematics Instruction:
How EFfective Has It Been in Secondary and Postsecondary
Schools?" Schoen's review contained within it a subset of
studies in which the researcher had investigated individual
ized (selfpaced) mathematics programs at the secondary level
(pp. 353354).
This was an appropriate set of studies to use to in
vestigate (by applying the aggregate chisquare methodology)
30
the true strength of individualization as an educational
strategy. First, it was appropriate because Schoen (1976)
specifically identified selfpacing (p. 352) as a common
factor in all of the studies which he had included in the
review. Selfpacing was the key element because selfpacing
has been identified as "the only common universal attached
to the term individualization (Good et al., 1975,
p. 169).
The fact that selfpacing was a unifying factor supported
the requirement that there be a rational basis for combining
the studies. The fact that all of the studies included in
Schoen's review of research were further narrowed to include
only secondary level studies and only mathematics programs
helped to support the rational basis for their inclusion in
an aggregate chisquare methodology.
Second, Schoen's review was appropriate because each
study included in the review had many characteristics which
have been commonly identified with the individualized model
and thus support (in addition to selfpacing) the rational
basis for establishing them as a set to be pooled by the chi
square model.
First, they were based on a specific
set of behavioral objectives. Second,
the mathematics content to be learned
was divided into small modules or units.
Third, learning packets were written for
each unit; the learning packets served
as guides for the students, enabling
them to proceed more or less independ
ently through the content at their own
pace. Fourth, for the most part the
students learned independently from
31
textbooks and worksheets, through some pro
grams included other media. Fifth, each
packet contained protests and posttests:
the student was required to pass one or both
before proceeding to the next unit. The
teacher's role was that of manager, record
keeper, individual tutor, and sometimes cur
riculum developer. (Schoen, 1976, p. 352)
The five general characteristics just quoted plus the
selfpacing characteristic established the fact that the
studies included in Schoen's review contained studies which
were in fact models of individualization and thus met the
rational basis requirement.
Third, the studies included in Schoen's review were
appropriate because of the definite ambiguity of the results.
Of the twelve studies included in the review, one (Bull,
1971) concluded that students studying mathematics with the
individualized approach scored better than students studying
with a traditional approach. Conversely, Fisher (1973),
Herceg (1972), and Hirsch (1972) all concluded that the
traditional approach was better.
In contrast to the four previously named researchers
who found a significant difference in mathematics achievement
depending on the method of instruction used, eight research
ers reported no significant difference in either method
(Englert, 1972; Hanneman, 1971; Ludeman, 1973,
Penner, 1972; Schoen & Todd (Note 2); Schoen & Todd, 1974;
Taylor, 1971; Thomas, 1972).
Fourth, the general quality of the research designs and
statistical analyses were of high quality. Random assignment
of students or classes to treatments was the basic way in
which group equivalency was obtained. Analysis of
covariance and analysis of change from a pretest to a
posttest score were used to support random assignment in
attempting to insure pretreatment group equivalency
(Schoen, 1976, p. 353).
The criterion measure used was either a standardized
test or a teachermade test. Many of the studies measured
criteria other than but including achievement. Attitude
toward the content being taught or learned was the crite
rion most often measured other than achievement.
Fifth, another potentially important factor was that
the collection of studies in Schoen's review was relatively
recent. Bull's 1971 study and Schoen's 1974 study represent
the extremes on a time continuum.
Sixth, the studies were relatively easy to acquire.
Nine of the studies were doctoral dissertations and were
readily available through University Microfilms of Ann Arbor,
Michigan. One study was reproduced by the United States
Educational Resources Information Center (ERIC).
Seventh, by using the selfcontained set produced by
Schoen (1976), this researcher was able to avoid the re
searcher bias (Gage, Note 1, p. 13) of choosing a self
selected set. Only one study from the Schoen review was
not used and that was due to its unavailability and not to
any arbitrary decision by this researcher.
Eighth, the set of studies reported on by Schoen pro
vided enough independent samples so that the results of
33
this investigation would be defensible. That is, there
were enough separate measures included in the pooled data
to insure reliability of the aggregate chisquare statis
tic. Fisher (1948, p. 100) used 3 independent samples in
arriving at an aggregate chisquare statistic. Gage (Note
1, p. 33) used 18 independent samples and Gordon et al.
(1952, p. 315) used 13 independent samples. In this inves
tigation, 24 independent samples were used to arrive at
the aggregate chisquare statistic.
As was noted earlier in this chapter, Schoen and Todd
had two separate investigations which were both done in 1974.
Only one of Schoen and Todd's (Note 2) studies was available
in a form which included group means. The Schoen and Todd
(Note 2) study with group means was available from the
authors. The second Schoen and Todd study which was included
in the "Research Reporting Sections, Annual Meeting of National
Council of Teachers of Mathematics (1974)" has not been used
in this study. The group means were not available in the
ERIC document or from the authors.
Summary
The main criterion in choosing a set of studies for
use with the aggregate chisquare model was that of estab
lishing the rationale basis for inclusion of each study.
In this case, the rational basis for inclusion was the char
acteristic of selfpacing which was present in each study.
Selfpacing was focused on because it is the only common
element found in individualized programs (Good et al.,
1975, p. 169).
In addition, there were five other characteristics
of individualization cited by Schoen and quoted above
which supported the establishment of a rational basis for
inclusion in the set.
Ambiguity of results was another feature associated
with the studies reported i n Schoe n's review. Ambiguity
of results was consistent with other research done on
individualized (selfpaced) instruction. Therefore, Schoen's
collection of studies provided an adequate set on which to
apply the aggregate chisquare model.
The studies in Schoen's review had been done over a
relatively short time span, were narrowly focused in terms
of grade level and subject matter (when compared with other
reviews), and were relatively easy to acquire in terms of
time and cost.
CHAPTER III
REPORTED RESULTS OF THE INDIVIDUAL STUDIES
AND CONVERSION TO CHISQUARE
Introduction
Chapter III is a presentation of each of the II
individual research reports with their statistical results
and the aggregate statistic derived by pooling the individ
ual statistics.
Six elements of each study were identified as being
appropriate to include. The 6 elements are the background
data, the characteristics of individualization, a defini
tion of experimental and control groups as it applied to
each of the studies, the hypothesis tested, the reported
statistical results, and the conversion to chisquare of
the reported results.
In some of the studies more than one hypothesis was
being tested. Where it was appropriate (and it was appro
priate in all but one case), only one hypothesis and the
statistical results for the test of that hypothesis were
reported. In the one case (Hirsch) where each of two
hypotheses was appropriate to be included, the results of
each test of the hypothesis was reported as an independent
sample.
36
Procedures for carrying out the conversion to chi
square varied. The variation in procedure depended on
two factors. One factor was whether the hypothesis was
stated as directional or nondirectional. A nondirec
tional hypothesis required that the probability derived
from the test of the hypothesis be divided by two.
A second factor that had to be considered was whether
the reported statistical result was in the same direction
as the majority of the cases in the set of results being
pooled to determine the aggregate chisquare.
The set of independent results used was divided in
terms of favoring experimental or favoring control as
shown in Table 1.
Table 1
Delineation of Reported Results According
to Favoring Experimental or Control Groups
Experimenter Experimental Control
Bull (1971) 1
Englert (1972) 1 2
Fisher (1973) 1
Hanneman (1971) 1 4
Herceg (1972) 1 1
Hirsch (1972) 2
Ludeman (1973) 2
Penner (1972) 3 2
Schoen & Todd (1974) 1
Taylor (1971) 1
Thomas (1971) 1
Totals 10 14
The fact that the majority of the results (as shown
in Table I) favored the control groups was an important
result and was a determining factor in deriving the aggre
gate chisquare statistic (refer to Procedures section of
Chapter I). The direction of the results reported as "non
significant" was determined by comparing group means
reported in each researcher's original data.
The following sections are in alphabetical order by
the author's last name.
Bull (1971)
Background Data
Bull conducted a semesterlong study in which he
compared the achievement of two individualized geometry
classes against the achievement of two comparable classes
taught in a traditional manner.
Bull did the study at Camelback High School, Phoenix
Union High School District, Phoenix, Arizona.
Characteristics of Individualization
The characteristics of individualization were self
pacing, use of behavioral objectives, and selfchoice of
learning experiences.
Experimental and Control Groups
The individualized groups were classed as experimental.
There were 34 students in each of the two experimental
groups and in each of the two control groups. Two teachers
taught both an experimental and control group. A twoby
two factorial design was used to analyze the effects of
38
individualization and time of day. Students were ran
domly assigned to the respective groups.
Hypothesis Tested
The hypothesis tested was stated by Bull in the
following manner.
There is no difference in the mean test
score of geometry students taught by the
traditional method and the mean test score
of geometry students taught by the indi
vidualized instruction method as measured
by the MidYear Geometry Test. (p. 11)
Statistical Results
Bull applied the ttest to his nondirectional hypoth
esis and obtained the result t= 3.229.
Conversion to Chisquare
By applying the statistical techniques and extended
table of chisquare for two degrees of freedom outlined in
Gordon et al.. (1952), a chisquare of .0201 was obtained.
The transformation was as follows.
t(67) = 3.229, p <.05
p = 1 (.001/27 = .9995
X2= .0201
Bull reported the t ratio to have been significant in
favor of the experimental group.
Englert (1972)
Background Data
Englert's study was conducted at the Cleveland Heights
High School in Cleveland Heights, Ohio. The subjects used
in the study were firstyear algebra students. The first
year algebra students were classified as lowachievers and
the group was comprised of pupils from grades ten
through twelve. The duration of the study was one
semester.
Characteristics of Individualization
Englert defined individualization in the following
manner. "The emphasis in this approach is upon the indi
vidual as he learns and proceeds at his own pace" (p. 9).
Experimental and Control Groups
The experimental group was that group of students
which proceeded through the text book at its own pace.
There were three experimental groups taught by three dif
ferent teachers. For comparison, three control groups
were established. Each control group was taught by one of
the experimental group teachers. Thus, three teachers each
taught an experimental and a control group for a total of
six groups (pp. 910).
Hypothesis
Englert hypothesised that "there is no difference in
algebra achievement levels between senior high school
students taught by the grouporiented approach" (p. 51).
The standardized Seattle Algebra Test was used as the
criterion measure. The hypothesis was nondirectional and
tested at the .05 level.
Statistical Results
A comparison of group means from the arithmetic pre
test to the algebra posttest yielded the following results.
Table 2
Group Means and tscores for Each Independent Sample
Teacher Experimental Control df t
A 18.05 22.10 39 2.05"
B 22.33 21.30 42 .59
C 20.67 21.13 39 .23
Note. *p <.05.
Conversion to Chisquare
Each of the teachers represented an independent sample
as defined in the Procedures section of Chapter I. There
fore, each chisquare generated from each independent sample
(teacher A, teacher B, & teacher C) was entered in the
aggregate chisquare table as separate entries.
The original probability of each independent sample
was divided by two (p/2) because the hypothesis was non
directional.
The independent probability associated with teacher B
was subtracted from one (1p) because the result was not in
the direction of the majority of the cases in the total set
used for the aggregate chisquare statistic. The direction
was determined by comparing the group means in the original
data as shown in Table 2. Table 3 shows the conversion to
chisquare of the reported results.
Table 3
Conversion to Chisquare of Each Independent
Sample
Teacher t Exact Onetailed Probability X2
A 2.05 p = .05/2 = .025 7.3778
B .59 p = 1(.55/2) = .725 .6432
C .23 r = .80/2 = .40 1.8326
Total 9.8536
Fisher (1973)
Background Data
Fisher studied the difference in effect on achieve
ment in geometry when using an individualized approach
versus a lecturedemonstration approach. Fisher's study
was conducted at Albert Einstein Senior High School,
Montgomery County, Maryland. The duration of the study
was one year. The subjects in the study were eleventh
grade geometry students.
Characteristics of Individualization
The computer was used (computer managed instruction)
to supply daily monitoring of each student's progress
through his program. The individualized design used in
Fisher's study permitted "a student to progress through
the material at his own rate" (p. 81).
Experimental and Control Groups
The sample was divided into two groups. One group
(experimental) consisted of all students studying geometry
through the computer managed instruction plan. The control
42
group consisted of all the students studying geometry
by a traditional geometry curriculum. The criterion
measure was the standardized Cooperative Mathematics Test
for Geometry.
Hypothesis
Fisher tested the following directional hypothesis.
The computermanaged behavioral objective
instructional curriculum is more effective
than the traditional curriculum in develop
ing the basic skills, concepts, and logical
reasoning skills of geometry as measured
by the Cooperative Hathematics Test for
Geometry. (pp. 9697)
Statistical Results
Fisher's reported statistic of t(81) = 1.28, p >.05
was not significant. However, when the group means were
compared, the control group mean was greater (control group
mean 12.23 & experimental group mean 10.77) than the exper
imental group mean. Therefore, the results were taken as
favoring the control group.
Conversion to Chisquare
The calculated value t(81) = 1.28, p >.05 was taken
directly to the extended table of chisquare for two degrees
of freedom (Gordon, et al., 1952). An exact onetailed
probability of .20 was obtained. The probability of .20
yielded a chisquare of 3.2189.
The extended table was used directly in this case
because the test was onetailed and the results favored the
control which was true for the majority of the independent
samples used to derive the aggregate chisquare statistic.
Hanneman (1971)
Background Data
Hanneman conducted his study at Mankato (Minnesota)
High School. The duration of the experimental study was
14 weeks. The subjects of the study were tenthgrade
geometry students.
Characteristics of Individualization
Individualization was characterized by selfpacing,
selftesting (pre and post) and learning activity packages
(LAP's). Endofunit teacher designed tests were used as
the criterion measure (dependent variable).
Experimental and Control Groups
The experimental groups were made up of 45 tenthgrade
geometry students. The control groups were made up of 47
tenthgrade geometry students. The five experimental groups
were exposed to the program of individualization as de
scribed in Characteristics of Individualization. The five
control groups remained in traditional classrooms. Tradi
tional classrooms were characterized by lecturediscussion
and whole group pacing (all students received, performed,
and were tested on assignments at the same basic intervals).
Hypothesis
Hanneman's hypothesis was stated in the following
manner. "The performance on endofunit tests of students
receiving instruction through independent study will not
differ from the performance of those receiving group in
struction" (p. 25).
44
A twotailed ttest was used as the measure of the
nondirectional hypothesis.
Statistical Results
Hanneman organized the experimental and control groups
around five learning activity packages (LAP's). The LAP's
were designed to be equivalent. The statistical results
were reported as independent samples in terms of each of
the five LAP's. Table 4 shows the reported results.
Table 4
Group Means and tscores for Each Independent Sample
LAP Control Experimental t
1 86.3 84.3 .870
2 80.0 76.5 1.357
3 79.9 71.8 2.769*
4 79.4 76.5 1.184
5 84.7 85.1 .194
Note. <.01, df was 90 for all LAP's
Conversion to Chisquare
Table 5
Conversion to Chisquare of Each Independent Sample
LAP t Exact Probabilities X
1 .870 .40/2 = .200 3.2189
2 1.357 .20/2 = .100 4.6052
3 2.769 .01/2 = .005 10.5966
4 1.184 .30/2 = .150 3.7942
5 .194 .87/2 = .435(1.435)=.565 1.1419
Total 23.3568
Note. The probability for LAP 5 was subtracted from I
because the experimental group scored higher than the
control group (refer to Table 4).
Herceg (1972)
Background Data
Herceg investigated individualization by comparing
the achievement of 16 Algebra 2 classes. Three toptrack
and 13 middletrack Algebra 2 classes were randomly assigned
to three treatment groups. The investigation was conducted
in the Gateway School District at Monroeville, Pennsylvania.
The duration of the study was not precisely stated.
Characteristics of Individualization
The students in Group A studied Algebra 2 by using
computers, behavioral objectives, and were in an "individ
ual rate of learning setting" (p. 49).
Experimental and Control Groups
Group A students used the computer in an individual
rate of learning setting with formally presented behavioral
objectives. Group A was designated as the [experimental]
group. The [control] group, Group B, also used computers
and formally presented behavioral objectives, but Group B
students remained in a traditional classroom setting.
Hypothesis
Herceg stated the hypothesis in the following manner.
Students in an individual rate of learning
setting who are aware of the behavioral
objectives for a unit in CAM [Computer
Assisted Mathematics] will score as high
as or higher than students in a traditional
classroom setting who are aware of the be
havioral objectives for the same Computer
Assisted Mathematics unit. (p. 49)
46
Top track students and middle track students were
tested on the hypothesis (p. 49 & p. 54). Therefore,
the toptrack and middletrack groups of students repre
sent independent samples. The hypothesis was stated as
a directional hypothesis; therefore the exact proba
bilities will not be divided by two.
Statistical Results
Table 6
Group Means and tratios For Each Independent
Sample
Group Individualized Control df t
MiddleTrack 54.00 61.96 94 5.92*
TopTrack 62.29 62.15 35 0.06'"
Note. The hypothesis was tested at p <.05.
*A <.001
** Not significant
Conversion to Chisquare
Table 7
Conversion to Chisquare of Each Independent Sample
Group t Exact Probability 2
MiddleTrack 5.92 .001 13.8155
TopTrack 0.06 .90(1.90)=.10
Total 18.4207
Note. p = .90 was subtracted from 1 because the exper
imental group mean was greater than the control group
mean (refer to Table 6).
Hirsch (1972)
Background Data
Hirsch compared the effects of guided discovery and
individualized instruction on several outcome measures.
The duration of the study was one semester. The subjects
were tenthgrade Algebra 2 students. The schools were
located in Cedar Rapids and Iowa City, Iowa.
Characteristics of Individualization
Individualization was characterized by the use of
learning activity packages and progression through the
material at each student's own rate (p. 60).
Experimental and Control Groups
Three intact groups of students were assigned three
treatments. The three treatments were specified as guided
discovery, instructional packages (expository format),
and instructional packages (programmed format).
The treatments specified as instructional packages
were each individualized as defined above. The instruc
tional package treatments differed "only with respect to
programming style" (p. 64). Selfpacing was a key char
acteristic of both instructional package treatments. Self
pacing was not a characteristic of the guided discovery
treatment.
Hypothesis
Hirsch developed four hypotheses each one of which
corresponded to a single outcome measure. An assumption
was made that the outcome measures of initial learning and
retention were submeasures of academic achievement.
According to Hirsch, initial learning "was specifically
designed for this study . to provide a measure of stu
dent achievement" (p. 67).
Hirsch designed the "Retention Test to provide
a measure of student retention of complex number concepts"
(pp. 7172). Many educators would agree that "initial
learning" and "retention" are measures of academic achieve
ment.
Conversely, for the purpose of this study, it was not
assumed that "lateral transfer" and "vertical transfer"
were measures of academic achievement.
Therefore, only hypotheses "l" and "IV" are referred
to in the section Statistical Results and which are quoted
in the following text.
Hypothesis 1: There are no significant
differences among the adjusted group
initial learning means for the three
treatments.
Hypothesis IV: There are no significant
differences among the adjusted group re
tention means for the three treatments.
(p. 83)
Statistical Results
Hirsch used the Ftest to compare the three treatment
group's by the group's respective means. Table 8 shows the
comparative mean scores and Table 9 shows the obtained values
of F for the two outcome measures used in the aggregate
chisquare statistic.
Table 8
Adjusted Mean Scores
for Three Treatment Groups and Two Outcome Measures
Treatment Initial Learning Retention
Guided Discovery 17.87 9.06
Instructional Packages (expository) 15.75 7.92
Instructional Packages (programmed) 15.78 8.09
Note. Guided Discovery was the control (traditional)
group and Instructional Package groups were the exper
imental (individualized).
Table 9
Fratios for Each Independent Sample
Outcome Measure df F p
Initial Learning 2/208 9.19 <.01
Retention 2/208 2.76 <.05
Note. Both Fratios favored the control (refer to Table 8).
Conversion to Chisquare
The outcome measures initial learning and retention
were considered to be independent samples. The conversion
to chisquare of the Fratio for each outcome measure is
as follows.
Table 10
Conversion to Chisquare of Each Independent Sample
Outcome Measure F Exact Probability X2
Initial Learning 9.19 .0005/2 = .00025 12.0238
Retention 2.76 .0250/2 = .0125 8.7657
Total 20.7895
Note. The exact probabilities were divided by 2 because
the hypothesis was nondirectional.
Hirsch tested for posthoc comparisons among means
and found a significant difference between the control and
each experimental adjusted group mean. No significant
difference was found between the two experimental group
means and thus the reported Fratio reflects the differ
ence between the control and experimental groups.
Ludeman (1973)
Background Data
Ludeman investigated the effects of an individualized
program on ninthgrade algebra and basic mathematics stu
dents. The investigation was carried out at the Arnold,
Nebraska,public schools. The investigation took place over
the course of one school year.
Characteristics of Individualization
Individualization was based on a videotape program
and continuous progress format. Individualized student
teacherparent contracts were developed and were the basis
of a selfpaced schedule (p. 16).
Experimental and Control Groups
In both the Basic Mathematics and Algebra I classes,
the experimental groups were those that received individ
ualized mathematics and algebra instruction supplemented
by videotaped presentations.
The control group was the previous year's ninthgrade
class. Ludeman explained that this was due to the fact
that Arnold Public Schools were a small rural district.
Consequently, there were not enough students to form an
experimental and control group from among the current
enrollment of ninthgraders. The previous year's ninth
grade class had not received the individualized instruc
tional program.
Hypothesis
There were no written hypotheses for Ludeman's pro
ject. An objective was established which was to increase
achievement levels by 510 percentile points (p. 1), but
an hypothesis was not generated or if it was it was not
in the reported material.
Statistical Results
For Basic Mathematics, Ludeman reported a control group
mean of 1.667 and an experimental group mean of 2.333.
The results yielded t(22) = 1.580, p >.05. While the
tratio was not significant the mean scores did favor the
experimental.
For Algebra 1 (after 20 weeks of instruction), Ludeman
reported a control group mean of 26.90 and an experimental
52
group mean of 30.50. The statistic generated from those
means was t(29) = 1.858, p >.05. Again, the tratio was
not significant, but the mean scores favored the exper
imental group.
Conversion to Chisquare
Table 11
Conversion to Chisquare of Each Independent Sample
Group t p 2
Basic Mathematics 1.580 1.20 = .80 .4463
Algebra 1 1.858 1.10 = .90 .2107
Total .6570
Note. The exact onetailed probabilities were subtracted
from one (1p) as per Procedures section of Chapter I.
An assumption was made that Ludeman, in calculating
the tratio, used the onetailed distribution of t. As
was stated earlier, Ludeman reported no written hypothe
sis. However, when the original data which he did report
(t(22) = 1.580,p >.05 & t(29) = 1.858,p >.05) were compared
with the critical values which he also reported (2.074 &
2.045, respectively), the conclusion was reached that
Ludeman had worked from the assumption of a onetailed
hypothesis.
Penner (1972)
Background Data
Penner studied the effect of individualization on
achievement in trigonometry. The subjects used in the study
were seniors in the public schools of Omaha, Nebraska.
The study was conducted during the first semester.
Characteristics of Individualization
Penner used the term "individual progress approach"
(p. 4) and stated that this meant an approach "which al
lowed the students, with the aid of a syllabus, to progress
at their own rates" (p. 4).
Experimental and Control Groups
There were five experimental classes (individualized)
and five control classes. Five schools participated in
the study with a total of 212 students. Each participating
teacher taught one experimental and one control class. The
students in the experimental groups progressed at their
own rates while the students in the control groups were
taught in a traditional manner.
Hypothesis
Penner stated the hypothesis in the following manner.
There is no significant difference in stu
dent achievement in trigonomentry between
those students who use the individual pro
gress approach and those who use the tra
ditional approach. (p. 5)
Statistical Results
Penner used each of five schools (Benson, Burke,
Central, North, and South) as independent samples. Table
12 shows the reported results.
Table 12
Group Means and tscores for Each Independent Sample
School Experimental Control t
Benson 18.826 17.903 .545
Burke 20.364 20.192 .098
Central 25.211 21.588 2.073
North 13.538 16.080 1.409
South 15.235 25.158 4.331
Conversion to Chisquare
Table 13
Conversion to Chisquare of Each Independent Sample
School t Exact Onetailed Probability X
Benson .545 1 (.60/2) = .7000 .7133
Burke .098 1 (.90/2) = .5500 1.1957
Central 2.073 1 (.05/2) = .9750 .0506
North 1.409 .175/2 = .0875 4.8723
South 4.331 .001/2 = .0005 6.9078
Total 13.7397
Note. The exact probabilities were subtracted from 1
because the group means for those schools (Benson, Burke,
& Central) favored the experimental.
Schoen and Todd (1974)
Background Data
Schoen and Todd investigated two questions.
(a) Does the detailed preparation of a con
cept centered individualized learning pack
age (ILP) by a teacher improve the teacher's
ability to teach that concept using either
a teacher centered (TC) approach or a learn
ing package approach (as measured by student
achievement)?
(b) Is there a difference in student achieve
ment scores on concept taught by ILP as com
pared to a TC approach? (p. 2)
The results of the investigation of the second
question "b" are the results reported on in this study.
Characteristics of Individualization
The characteristics of individualization included the
use of behavioral objectives, selftesting on protests and
posttests, learning activity packages, and individual pro
gress through the learning activity packages (p. 3).
Experimental and Control Groups
Six mathematics teachers were paired on the basis
of comparability of classes.
Each teacher in each pair had two classes
of mathematics at a certain level and his
'mate' had two classes at the same level.
In particualr, each teacher in pair one
taught two ninth grade General rfathematics
classes, pair two taught two Algebra 1
classes and the third pair taught eighth
grade General Math classes. Thus, six
teachers and twelve classes were involved
in the experiment. (p. 2)
The treatment for the experimental group was the use
of ILP's and the control group teachers used a "lecture
discussion" (p. 3) methodology.
Hypothesis
Schoen and Todd did not write out their hypothesis in
a standard way. However, from the following quote, the
hypothesis can be discerned. "The hypotheses of no differ
ences in achievement scores on the main effectsTC vs. ILP
and preparer vs. nonpreparer could not be rejected" (p. 12).
The hypothesis was clearly null and nondirectional in
nature.
Statistical Results
The statistic reported by Schoen and Todd was F
(1/20) = .08 >.05. Schoen and Todd did not treat each
teacher as an independent sample. Therefore, the statis
tical results were reported by Schoen and Todd as a
composite Fratio. The Fratio was taken as favoring the
control groups because, when the group means were com
pared for each teacher (Table 14), the control groups had
higher scores.
Table 14
Group Means for Schoen and Todd
Teacher Experimental Control
1
Unit 1 7.7 9.5
Unit 2 8.4 8.7
2
Unit 1 7.4 6.9
Unit 2 7.5 8.1
3
Unit 1 14.3 12.4
Unit 2 13.2 11.2
4
Unit 1 13.3 15.7
Unit 2 15.2 15.2
5
Unit I 19.0 21.0
Unit 2 34.5 34.5
6
Unit 1 15.0 11.1
Unit 2 28.7 31.1
Note. Six of the group means favored the control and four
favored the experimental while two were identical.
Conversion to Chisquare
F(1/20) = .08
= .99/2 = .495
S2= 1.4064
57
The probability for F(1/20) = .08 was not recorded
in a standard table. Therefore, the conservative proba
bility of .99 was used.
Taylor (1971)
Background Data
Taylor conducted the study at Crestmoor High School
in San Bruno, California. The duration of the study was
one semester. The subjects were ninth, tenth, and eleventh
graders who were studying Algebra 1.
Characteristics of Individualization
Independent study was defined to be a learning situa
tion in which the students studied alone or in small groups
with a minimum amount of help from the teacher. The stu
dents used a conventional textbook and progressed indi
vidually by completing assignments and tests associated
with a given chapter before continuing to the succeeding
chapter. "The students progress at their own rate through
a specified course of study" (p. 10).
Experimental and Control Groups
There were 48 students used as subjects in the study.
The students were enrolled in two classes of Algebra 1.
Twentyfive students were in the individualized class and
23 were in the lecturediscussion class. The experimenter
taught both classes. Assignments to classes were made by
a flip of a coin.
Hypothesis
Taylor generated a nondirectional null hypothesis
which was stated as follows. "There is no difference
between lecturediscussion and independent study with
respect to growth in achievement in Algebra 1" (p. 13).
Statistical Results
The reported results showed that t(22) = 1.1648, p
>.05. The result favored the control group but was not
significant. The control group mean was 9.600 while the
experimental group mean was 7.522.
Conversion to Chisquare
The reported results showed that t(22) = 1.1648.
In converting the tratio to an exact onetailed proba
bility, the following result was obtained.
p = .25/2 = .125
A probability of .125, when taken to the extended
table of chisquare, yielded a chisquare of 4.1589.
Thomas (1971)
Background Data
Thomas studied the effects of individualization on
five classes of Advanced Algebra students in two high
schools of the Lincoln Public Schools at Lincoln, Nebraska.
The research covered a period of one school year.
Characteristics of Individualization
The individualized algebra program was characterized
by the use of learning activity packages (LAP's) and self
pacing. Thomas specifically describes the selfpacing
feature as "an opportunity for each student to proceed at
his own rate of speed, commensurate with ability, interest,
and motivation" (p. 5).
Experimental and Control Groups
The experimental group was divided into five classes
with a total of 102 students. Two teachers were assigned
to teach the five classes of continuous progress mathematics
(experimental group). The five classes of traditional
advanced algebra were taught by three different teachers.
There were 122 students enrolled in the traditional classes.
Hypothesis
Thomas stated the hypothesis in the null form and
as nondirectional in the following manner. "There is no
significant difference between the achievement posttest
mean of the continuous progress advanced algebra classes
and the achievement posttest mean of the traditional ad
vanced algebra classes" (p. 6).
Statistical Results
The results were reported as not being significant.
However, the reported group means indicated that the exper
imental group achieved more than the control group. The
experimental group mean was 29.91 and the control group
mean was 28.35.
The derived statistic was F(1/149) = .203, p >.05.
Conversion to Chisquare
F(1/149) = .203
p = .99
Reversed p = 1(.99/2) = .505
X = I 3664
The probability for F(1/149) = .203 was not recorded
in a standard table. Therefore, the conservative probability
of .99 was used.
Computation of the Aggregate Chisquare
Table 15 shows the individual chisquare statistics
and the aggregate chisquare statistic. By pooling the
individual research results, this researcher was able to
use the data from the studies previously reported as not
significant. This feature of the aggregate chisquare was
very important in that 8 of the 11 original studies were
reported as not significant. By pooling the previously
reported nonsignificant studies into the aggregate chi
square, the nonsignificant studies added data that pro
duced a highly significant result.
Table 15
The Aggregate Chisquare Statistic
Experimenter Nr. of Independent Samples Total of X2
Bull (1971) 1 5.0201
Englert (1972) 3 9.8536
Fisher (1973) 1 3.2189
Hanneman (1971) 5 23.3568
Herceg (1972) 2 18.4207
Hirsch (1972) 2 20.7895
Ludeman (1973) 2 .6570
Penner (1972) 5 13.7397
Schoen & Todd (1974) 1 .4064
Taylor (1971) 1 4.1589
Thomas (1971) 1 1.3664
Aggregate Chisquare 101.9880
Note. Aggregate X2(48) = 101.9880, p <.001.
The df (48) results from multiplying 2 times n (the number
of independent samples), (Gordon, Loveland, & Cureton, 1952,
p. 314).
The probability .001 had to be adjusted to be consis
tent with the underlying assumption that the results could
61
be in either direction (which requires a twotailed test).
Each independent statistical result used to achieve a com
bined statement of significance was generated from one
tailed probabilities. Even when the original researcher
used a twotailed test, the result had to be converted to
a onetailed probability (as per procedures).
The problem (and therefore the reason why the pro
bability of the aggregate chisquare statistic has to be
adjusted) came because of the fact that the aggregate chi
square procedure allowed for a result in either direction.
When the procedure allows for a result in either direction,
a twotailed probability is required.
"The tabled significance value must therefore be
doubled" (Gordon et al., 1952, p. 315) in order to avoid
the contradiction of the onetailed result (using one
tailed probabilities in the aggregate chisquare) with the
twotailed assumption (willingness to consider a result in
either direction). The probability, then, is .002 and not
.001.
Summary
The aggregate chisquare statistic for the 24 inde
pendent research samples of selfpaced individualized
instruction showed that selfpaced individualized instruc
tional practices were not superior to traditional instruc
tional practices. In fact, the reverse could be implied;
62
that is, traditional instructional practices were superior
to selfpaced individualized instructional practices. The
probability of .002 for this combined significance test
indicated that in 998 of every 1,000 cases this result
would be replicated. In other words, it was very unlikely
that this result came by chance.
CHAPTER IV
SUMMARY, CONCLUSIONS, IMPLICATIONS,
AND SUGGESTIONS FOR FURTHER RESEARCH
Summary
The problem of this study was to determine the effec
tiveness of selfpaced individualized instruction by
applying the aggregate chisquare statistical procedure
to the results of several, previously done, independent
studies which have analyzed selfpaced individualized
instruction in its relationship to student achievement.
What would be the result when these several independent
results were pooled to derive a single statistic?
The accountability movement and a philosophical/
emotional commitment of professional educators have exerted
considerable force on local school administrators to adopt
individualization as the dominant (if not only) instruc
tional strategy. This pressure has been very forceful in
spite of the fact that empirical research has not supported
the claims made for individualized instruction. Empirical
research studies done on individualized instruction have
presented an inconsistent and therefore an indicisive
pattern of results. The aggregate chisquare procedure was
applied to the results of a set of 11 previously reported
studies in an attempt to discern a consistent pattern.
64
The strength of the aggregate chisquare procedure
lies in the use that was made of nonsignificant data.
Schoen (1976) reported on the results of 12 independent
researchers. Of the 11 studies included from Schoen's
review in this study, 8 of the researchers reported non
significant results. Unless some secondary analysis was
done, the review was not very enlightening or helpful in
educational decision making that was concerned with self
paced individualized instruction.
However, when the results of the 8 nonsignificant
studies were able to be included in a statistical procedure
such as the aggregate chisquare, a completely new and much
stronger picture was obtained. This result was consistent
with the kinds of results that Gage (Note 1) obtained when
applying the aggregate chisquare procedure to sets of in
dependent studies which had been previously done on several
teacher variables. Instead of a capricious and inconsistent
picture, a clear indication of the strength of the vari
ables on the outcomes measured was clearly evident. Speci
fically, in this study, the result was aggregate X (48) =
101.9880, e <.002. This result indicated that selfpaced
individualized instructional practices were not superior
to traditional instructional practices. In fact, the reverse
could be implied; that is,traditional instructional practices
were superior to selfpaced individualized instructional
practices. The probability of .002 for this combined sig
nificance test indicated that in 998 of every 1,000 cases
this result would be replicated. In other words, it
was very unlikely that this result came about by chance.
Conclusions
The results of this study indicated that traditional
classroom instructional practices were superior to indi
vidualized (selfpaced) instructional practices. In evalu
ating this conclusion, it should be recognized that the
procedure used in this study did not take into account the
relative merit of the individual research studies in terms
of design, interaction effects within studies and a host
of other technical considerations. The basic assumption
was that the independent statistical results generated by
each researcher were adequate statistical statements of
the strength of selfpaced individualized instruction as
measured by that study.
Implications
There were two important implications derived from
this study. One implication was in the area of research
techniques. The emphasis that Gage (Note I), Glass (1976)
and Light and Smith (1971) have placed on secondary anal
ysis of educational research seemed completely justified.
The weak and vascilating research results often reported
in educational research reviews have not been a true account
of the status of educational practices. This study support
ed the notion that what is done in classrooms is important
and does effect a student outcome such as achievement.
The viability of the aggregate chisquare as a tool for
secondary analysis was also supported.
A second implication was in the area of decision
making regarding educational programs and practices.
Educational leaders have been susceptible to the political,
philosophical, and social forces which constantly impinge
on the life of the schools. Accountability has been a
force that has vitally affected the programs in the schools
(Darland, 1970; Davies, 1970; Lessinger, 1971; Morris, 1971;
Sciara & Jantz, 1972).
Educational leaders have been pressed to defend their
programs and practices with measureable results (Dunkin &
Biddle, 1974; Lessinger, 1971). More specifically, they
have been pressed to defend their programs and practices
with measureable results focused on the achievement of
individual students (Davies, 1970; Morris, 1971). The
practice of selfpaced individualized instruction was a
logical response to the demand for measureable results. The
progress of individual students was thought to be a more
precise gauge of success in the classroom than group norms
produced by standardized norm referenced tests.
The program of selfpaced individualized instruction
was also a logical response to another force that was
beginning to make a strong impact on the educational commu
nity. As Davies (1970) and Morris (1971) indicated, there
was a need to find an educational vehicle through which
and by which the maximum potential of the individual
student could be realized. Selfpaced individualized
instruction seemed to offer the solution to the problem
of accountability and to the need for individualized
plans of instruction.
It appeared that there had been a happy marriage of
the "commitments" to accountability which would be measured
by individual student performance and the commitment to the
development of "individual human beings" (Davies, 1970,
p. 129). Selfpaced individualized instruction provided
a common ground for these two commitments and practicing
educators have felt the persistent weight of the force
generated by these two commitments.
The results of this study indicated that selfpaced
individualized instruction was not the teaching/learning
strategy that would satisfy the demands of the accounta
bility movement nor was it able to meet the idealistic
demands of developing every individual to his fullest
potential (at least not academic potential). Perhaps some
other form of individualized instruction could meet those
demands and herein lies a hint at one direction further
research on the topic of individualized instruction might
take.
Suggestions for Further Research
In this study, selfpaced individualized instruction
was investigated. It might be very helpful if some
68
empirical research studies were done in which selfpaced
and teacherpaced individualized instruction were com
pared. The following comments will serve to strenghten
this point and also serve to clarify what is meant by
"teacherpaced.'.'
Taveggia (1976) reviewed 14 separate studies each of
which compared the "learning outcomes of a new instruc
tional procedure, the 'Personalized System of Instruction'
(PSI), with the learning outcomes of conventional approaches
to college teaching" (p. 1028). Taveggia made a potentially
significant observation when he pointed out that "five
features probably account for the superiority of PSI over
conventional methods" (p. 1030). The Personalized System
of Instruction was superior to conventional methods of
college instruction in all 14 studies reported in Taveggia's
review.
The significant point made by Taveggia was that the
feature termed "goatyourownpace" (p. 1030) was one of
five features that seemed to make the PSI appraoch better
than conventional approaches. Also, that the term goat
yourownpace was a misnomer. "A more appropriate desig
nation would be monitored pacing or forced pacing" (p. 1030).
The following excerpt presents a more complete picture of
the forced pacing concept of the PSI approach.
A second, less obvious option suggested
by the explanation developed above for the
superiority of PSI is to reorganize one's
conventional courses, "building in" the
unitperfection, forcedpacing, and moni
tored progression features of PSI. The
available evidence suggests that these
probably are the features which account
for the superiority of PSI over conven
tional methods. Thus, to the degree that
these features are incorporated into con
ventional courses, student mastery of
course content material probably will be
enhanced. (Taveggia, 1976, p. 1031)
The concepts of unitperfection, forcedpacing, and
monitored progression mentioned above, do indeed point to
a definite forcing of the pace which is in direct contrast
to the concept of selfpacing.
There does seem to be a significant relationship be
tween what kind of "pacing" is used and the strength of an
individualized methodology in terms of that methodology's
ability to change student outcomes. While it was recog
nized that the studies reviewed by Taveggia were at the
college level, the results may have important implications
for individualized instructional techniques at all levels
and may be revealing as to why selfpaced individualized
instruction did not fare well when submitted to the aggre
gate chisquare procedure. In the future, when studies of
individualized instruction are conducted, there should be
an effort to more clearly define the "pacing" function and
to attempt to establish its direct relationship to student
outcomes.
APPENDIX
HISTORY OF AGGREGATE CHISQUARE
AND TRANSFORMATION PROCEDURES
The aggregate chisquare procedure was developed by
Fisher (1948). The aggregate chisquare procedure was
based upon the fact that the distribution of the sum of
several values of chisquare was itself distributed as a
chisquare for two degrees of freedom was 2 times the
natural logarithm of the probability (pp. 99101).
While other researchers have discussed the technique
and its possible applications (Lancaster, 1949 & Wallis,
1942), there have been few attempts to use the aggregate
chisquare method in published research studies. Perhaps
the reason for the few applications of the aggregate chi
square has been due to the fact that an extended table of
values for chisquare with two degrees of freedom was not
available until 1952. It was in 1952 that Gordon et al.
developed the extended table which could be used in com
bining probabilities from independent samples. However,
it has to be noted that there were no published studies
known to this researcher using the aggregate chisquare
even after the Gordon et al. (1952) extended table was
published. There was, of course, one exception to the
previous assertion and that exception was Gage (1977).
The chisquare model was based on the proof that
any p value could be transformed to a chisquare value
with two degrees of freedom and that the sum of indepen
dent chisquares was distributed as a chisquare. Follow
ing is an outline of the transformation procedure as
developed by Fisher (1948, pp. 99101).
i. The transformation equation:
X2 = 2 log p (1)
2. Composite X2 is given by the formula:
k
X2= 2 Z loge pi (2)
3. Degrees of freedom:
2k degrees of freedom where k is the
number of independent probability values
to be combined.
4. Joint probability of k independent results:
The product of the k separate p values.
REFERENCE NOTES
1. Gage, N. L. Four cheers for research on teaching.
Unpublished research report, 1976. (Avaliable
from N. L. Gage, College of Education, Stanford
University, Palo Alto, California, 94305). The
Gage report was later published and was added
to the reference section of this study.
2. Schoen, H. L., & Todd, R. 11. Teacher prepared learning
packages: Aid to student? or teacher? Unpub
lished research report, 1974. (Available from
Harold L. Schoen, W104A East Hall, University of
Iowa, Iowa City, Iowa, 52242).
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BIOGRAPHICAL SKETCH
Paul Ivan Johnson was born October 9, 1937, at Boise,
Idaho. In May, 1955, he was graduated from Gooding High
School at Gooding, Idaho. In January, 1956, he enlisted
in the United States Navy and served four years in the
Pacific Area as a Communications Technician. Following
his discharge, he entered a training program with a
Christian service organization.
In February, 1961, he enrolled at Bethel College in
St. Paul, Minnesota,and graduated from that institution
with an elementary teaching major in 1967. From 1967
through 1972 he taught at the elementary level in the public
schools of Camarillo, California. In 1972 he received the
degree of Master of Arts in Educational Administration and
Supervision for the elementary school level. The Master
of Arts degree was received from San Fernando Valley State
College at Northridge, California. From 1972 until the
present he has served with the Wycliffe Bible Translators
as a school administrator. He currently holds the position
of International Coordinator for Children's Education. In
1974 he was granted a study furlough and subsequently en
rolled in the Graduate School of the University of Florida,
where he began his work toward the degree of Doctor of
Philosophy.
78
Paul Ivan Johnson is married to the former Dona
Jean Wilson. He is a member of the American Educational
Research Association, Association for Supervision and
Curriculum Development, National Association of Elementary
School Principals, National Association of Secondary
School Principals, and National Educators Fellowship.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
7. > 
Ralph B. Kimbrough, Professor
of Educational Administration
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Robert S. Soar, Professor
of Foundations in Education
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Michael Y. Nunnery, Profes or
of Educatio6al Administration
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Paul S. George, Ass ciate ~
Professor of General Tea(her
Education 
This dissertation was submitted to the Graduate Faculty of
the Department of Educational Administration and Supervision
in the College of Education and to the Graduate Council, and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
June 1979
Dean, Graduate School
UNIVERSITY OF FLORIDA
3 1262 08553 0797
