Group Title: relationship of self-paced individualized instruction to pupil achievement
Title: The Relationship of self-paced individualized instruction to pupil achievement when measured by pooling the probabilities of several independent samples
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Title: The Relationship of self-paced individualized instruction to pupil achievement when measured by pooling the probabilities of several independent samples
Physical Description: vii, 78 leaves ; 28 cm.
Language: English
Creator: Johnson, Paul Ivan, 1937-
Publication Date: 1979
Copyright Date: 1979
Subject: Individualized instruction -- Evaluation   ( lcsh )
Educational Administration and Supervision thesis Ph. D   ( lcsh )
Dissertations, Academic -- Educational Administration and Supervision -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 73-76.
Statement of Responsibility: by Paul Ivan Johnson.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098637
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000091159
oclc - 05859412
notis - AAK6554


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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.



ACKNOWLEDGMENTS ............................. ii

ABSTRACT ... ................................... v


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


Introduction ........................ 28

Miller's Review ...................... 28

Hirsch's Review ...................... 29

Schoen's Review ...................... 29

Summary ............................ 33


Introduction ...................... 35

Bull (1971) ...................... 37

Englert (1972) .................. 38



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


Summary .......................... 63

Conclusions ...................... 65

Implications ...................... 65

Suggestions for Further Research .. 67


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



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. Self-paced

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 self-paced individualized instruction have presented an

inconsistent and inconclusive data base from which to for-

mulate an objective decision.

Some researchers have reported that self-paced 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 chi-square 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 self-paced 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 self-paced

individualized instructional practices are not superior

to traditional instructional practices. In fact, the

reverse could be implied; that traditional instructional

practices are superior to self-paced 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.



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. Self-paced individual-

ized instruction was one such methodology. Some researchers

have reported that self-paced 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 self-paced 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 self-paced 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,


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 self-paced 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 self-paced

individualized instruction. A secondary analysis technique

called the aggregate chi-square 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).


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--
eventually--to 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


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
teacher-education faculty can employ
in planning or improving teacher-
education programs. (p. 657)

Twenty-one 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


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'


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


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).


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 chi-square described by Gage (Note 1). The

use of the aggregate chi-square 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. 8-9).


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,

self-paced 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

chi-square model to five clusters of studies that had been

done on different teacher behavior variables (Gage, Note 1,

p. 13). By applying the aggregate chi-square 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 process-product 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 self-paced individualized instruction presented the

same ambiguous picture as studies done on many other in-

structional variables, it was appropriate to apply the


aggregate chi-square procedure in a secondary analysis of

self-paced individualized instruction.

Administrator Need for Objective Data
Regarding Self-Paced Individualized Instruction

Individualized instruction (specifically, self-paced)

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 self-paced 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 self-paced 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 self-paced individualized instruction

were not insignificant in terms of finances, administrator

time, teacher time, in-service, curriculum changes, hardware,


software, e t c. Stronger empirical evidence regarding

self-paced individualized instruction was needed.

Statement of the Problem

The problem of this study was to determine the effec-

tiveness of self-paced individualized instruction by apply-

ing the aggregate chi-square statistical procedure to the

results of several, previously done, independent studies

which have analyzed self-paced 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.


1 The purpose of this study was to investigate the

aggregate main effect of self-paced 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).


1. The results of this study are not decisive or

final in the case of self-paced 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 self-paced individualized instruction at

the high school level in mathematics oriented


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 long-term results while others measure short-term

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, audio-visual aids, and grouping

practices are examples of process variables.

Process-product research. Process-product research is

the attempt of researchers to establish casual relationships

between specific process variables and specific product


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


One and two-tailed tests. The definition offered by

Huck et al., (1974, pp. 45-46) 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 two-tailed test. A
two-tailed 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 two-tailed
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 one-tailed 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 one-tailed test, the
same difference with a two-tailed test
would be significant at a level of sig-
nificance twice as large as that used
with a one-tailed 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
two-tailed 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 lecture-discussion mode, and teacher-

paced learning environments.


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


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


Biddle gave inconsistent results and presented a confusing

picture as to the effect of teacher praise on learning out-


Gage took the same set of studies collected by Dunkin

and Biddle and applied the aggregate chi-square 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. 99-101) which are necessary

when applying the aggregate chi-square model to a set of data.

As written previously, the aggregate chi-square model was


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 chi-square, the most signifi-

cant chi-square, and a mean (mean of the two or more results

chi-square (Gage, Note 1, pp. 14 & 33).


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 one-tailed probability value

associated with each result to be combined. Researchers, in

many cases, have reported statistical measures such as F

ratios, t-ratios, and chi-squares. In addition, those statis-

tics have been reported as being either significant (at a


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 two-tailed test for significance and some

hypotheses required a one-tailed test for significance.

If the hypothesis was non-directional, the exact prob-

ability value associated with the reported statistic was

divided by two (p/2). For example, if the hypothesis required

a two-tailed 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 one-tailed 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


one-tailed probability value was taken directly to the chi-

square table for two degrees of freedom developed by Gordon,

Loveland, and Cureton (1952).

The exact one-tailed probability was subtracted from

one (l-p) before going to the chi-square 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 one-tailed probability to a

two-tailed probability. The probability derived from the

aggregate chi-square statistic had to be doubled. The use

of one-tailed 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 chi-square 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 chi-square procedure.

1. The exact one-tailed 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?


If it was in the same direction as the majority,

the exact one-tailed probability was taken di-

rectly to the chi-square 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

(1-p). When the result was generated from a

two-tailed test, the probability was divided by

two (p/2) before going to the extended table of

chi-square developed by Gordon et al., 1952.

3. All of the independent chi-square statistics with

two degrees of freedom were added together to ob-

tain the aggregate chi-square statistic.

4. The aggregate chi-square statistic was taken to

a regular chi-square table of values and the

appropriate probability value was read from that


5. The derived probability was doubled to correspond

to the two-tailed 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-



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.




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 self-pacing had to be

an identifiable feature of each study. Self-pacing 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 forty-five

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).


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. Thirty-three 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 "Self-paced 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 (self-paced) mathematics programs at the secondary level

(pp. 353-354).

This was an appropriate set of studies to use to in-

vestigate (by applying the aggregate chi-square methodology)


the true strength of individualization as an educational

strategy. First, it was appropriate because Schoen (1976)

specifically identified self-pacing (p. 352) as a common

factor in all of the studies which he had included in the

review. Self-pacing was the key element because self-pacing

has been identified as "the only common universal attached

to the term individualization (Good et al., 1975,

p. 169).

The fact that self-pacing 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 chi-square 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 self-pacing) 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


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

self-pacing 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 pre-treatment group equivalency

(Schoen, 1976, p. 353).

The criterion measure used was either a standardized

test or a teacher-made 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 self-contained 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


this investigation would be defensible. That is, there

were enough separate measures included in the pooled data

to insure reliability of the aggregate chi-square statis-

tic. Fisher (1948, p. 100) used 3 independent samples in

arriving at an aggregate chi-square 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 chi-square 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.


The main criterion in choosing a set of studies for

use with the aggregate chi-square 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 self-pacing which was present in each study.

Self-pacing 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 (self-paced) instruction. Therefore, Schoen's

collection of studies provided an adequate set on which to

apply the aggregate chi-square 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 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 chi-square 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



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 non-directional. A non-direc-

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 chi-square.

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 chi-square 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 semester-long 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 self-choice 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 two-by-

two factorial design was used to analyze the effects of


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 Mid-Year Geometry Test. (p. 11)

Statistical Results

Bull applied the t-test to his non-directional hypoth-

esis and obtained the result t= 3.229.

Conversion to Chi-square

By applying the statistical techniques and extended

table of chi-square for two degrees of freedom outlined in

Gordon et al.. (1952), a chi-square 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 first-year algebra students. The first-

year algebra students were classified as low-achievers and

the group was comprised of pupils from grades ten

through twelve. The duration of the study was one


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. 9-10).


Englert hypothesised that "there is no difference in

algebra achievement levels between senior high school

students taught by the group-oriented approach" (p. 51).

The standardized Seattle Algebra Test was used as the

criterion measure. The hypothesis was non-directional 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 t-scores 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 Chi-square

Each of the teachers represented an independent sample

as defined in the Procedures section of Chapter I. There-

fore, each chi-square generated from each independent sample

(teacher A, teacher B, & teacher C) was entered in the

aggregate chi-square table as separate entries.

The original probability of each independent sample

was divided by two (p/2) because the hypothesis was non-


The independent probability associated with teacher B

was subtracted from one (1-p) because the result was not in

the direction of the majority of the cases in the total set

used for the aggregate chi-square 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

chi-square of the reported results.

Table 3

Conversion to Chi-square of Each Independent


Teacher t Exact One-tailed 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 lecture-demonstration 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


group consisted of all the students studying geometry

by a traditional geometry curriculum. The criterion

measure was the standardized Cooperative Mathematics Test

for Geometry.


Fisher tested the following directional hypothesis.

The computer-managed 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. 96-97)

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 Chi-square

The calculated value t(81) = -1.28, p >.05 was taken

directly to the extended table of chi-square for two degrees

of freedom (Gordon, et al., 1952). An exact one-tailed

probability of .20 was obtained. The probability of .20

yielded a chi-square of 3.2189.

The extended table was used directly in this case

because the test was one-tailed and the results favored the

control which was true for the majority of the independent

samples used to derive the aggregate chi-square 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 tenth-grade

geometry students.

Characteristics of Individualization

Individualization was characterized by self-pacing,

self-testing (pre and post) and learning activity packages

(LAP's). End-of-unit teacher designed tests were used as

the criterion measure (dependent variable).

Experimental and Control Groups

The experimental groups were made up of 45 tenth-grade

geometry students. The control groups were made up of 47

tenth-grade 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 lecture-discussion

and whole group pacing (all students received, performed,

and were tested on assignments at the same basic intervals).


Hanneman's hypothesis was stated in the following

manner. "The performance on end-of-unit tests of students

receiving instruction through independent study will not

differ from the performance of those receiving group in-

struction" (p. 25).


A two-tailed t-test was used as the measure of the

non-directional 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 t-scores 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 Chi-square

Table 5

Conversion to Chi-square 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 top-track

and 13 middle-track 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.


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)


Top track students and middle track students were

tested on the hypothesis (p. 49 & p. 54). Therefore,

the top-track and middle-track 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 t-ratios For Each Independent


Group Individualized Control df t

Middle-Track 54.00 61.96 94 5.92*

Top-Track 62.29 62.15 35 0.06'"

Note. The hypothesis was tested at p <.05.
*A <.001
** Not significant

Conversion to Chi-square

Table 7

Conversion to Chi-square of Each Independent Sample

Group t Exact Probability 2

Middle-Track 5.92 .001 13.8155

Top-Track 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 tenth-grade 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). Self-pacing was a key char-

acteristic of both instructional package treatments. Self-

pacing was not a characteristic of the guided discovery



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 sub-measures 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. 71-72). Many educators would agree that "initial

learning" and "retention" are measures of academic achieve-


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

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 F-test 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

chi-square 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

F-ratios for Each Independent Sample

Outcome Measure df F p

Initial Learning 2/208 9.19 <.01

Retention 2/208 2.76 <.05

Note. Both F-ratios favored the control (refer to Table 8).

Conversion to Chi-square

The outcome measures initial learning and retention

were considered to be independent samples. The conversion

to chi-square of the F-ratio for each outcome measure is

as follows.

Table 10

Conversion to Chi-square 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 non-directional.

Hirsch tested for post-hoc 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 F-ratio reflects the differ-

ence between the control and experimental groups.

Ludeman (1973)

Background Data

Ludeman investigated the effects of an individualized

program on ninth-grade 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 video-tape program

and continuous progress format. Individualized student-

teacher-parent contracts were developed and were the basis

of a self-paced 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 video-taped presentations.

The control group was the previous year's ninth-grade

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 ninth-graders. The previous year's ninth-

grade class had not received the individualized instruc-

tional program.


There were no written hypotheses for Ludeman's pro-

ject. An objective was established which was to increase

achievement levels by 5-10 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

t-ratio was not significant the mean scores did favor the


For Algebra 1 (after 20 weeks of instruction), Ludeman

reported a control group mean of 26.90 and an experimental


group mean of 30.50. The statistic generated from those

means was t(29) = 1.858, p >.05. Again, the t-ratio was

not significant, but the mean scores favored the exper-

imental group.

Conversion to Chi-square

Table 11

Conversion to Chi-square 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 one-tailed probabilities were subtracted
from one (1-p) as per Procedures section of Chapter I.

An assumption was made that Ludeman, in calculating

the t-ratio, used the one-tailed 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 one-tailed


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.


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 t-scores 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 Chi-square

Table 13

Conversion to Chi-square of Each Independent Sample

School t Exact One-tailed 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

(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, self-testing 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.


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 effects-TC vs. ILP

and preparer vs. non-preparer could not be rejected" (p. 12).

The hypothesis was clearly null and non-directional in


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 F-ratio. The F-ratio 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
Unit 1 7.7 9.5
Unit 2 8.4 8.7
Unit 1 7.4 6.9
Unit 2 7.5 8.1
Unit 1 14.3 12.4
Unit 2 13.2 11.2
Unit 1 13.3 15.7
Unit 2 15.2 15.2
Unit I 19.0 21.0
Unit 2 34.5 34.5
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 Chi-square

F(1/20) = .08
= .99/2 = .495
S2= 1.4064


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.

Twenty-five students were in the individualized class and

23 were in the lecture-discussion class. The experimenter

taught both classes. Assignments to classes were made by

a flip of a coin.


Taylor generated a non-directional null hypothesis

which was stated as follows. "There is no difference

between lecture-discussion 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 Chi-square

The reported results showed that t(22) = 1.1648.

In converting the t-ratio to an exact one-tailed proba-

bility, the following result was obtained.

p = .25/2 = .125

A probability of .125, when taken to the extended

table of chi-square, yielded a chi-square 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 self-pacing

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.


Thomas stated the hypothesis in the null form and

as non-directional 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 Chi-square

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 Chi-square

Table 15 shows the individual chi-square statistics

and the aggregate chi-square 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 chi-square was

very important in that 8 of the 11 original studies were

reported as not significant. By pooling the previously

reported non-significant studies into the aggregate chi-

square, the non-significant studies added data that pro-

duced a highly significant result.

Table 15

The Aggregate Chi-square 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 Chi-square 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


be in either direction (which requires a two-tailed 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 two-tailed test, the result had to be converted to

a one-tailed probability (as per procedures).

The problem (and therefore the reason why the pro-

bability of the aggregate chi-square 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 two-tailed 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 one-tailed result (using one-

tailed probabilities in the aggregate chi-square) with the

two-tailed assumption (willingness to consider a result in

either direction). The probability, then, is .002 and not



The aggregate chi-square statistic for the 24 inde-

pendent research samples of self-paced individualized

instruction showed that self-paced individualized instruc-

tional practices were not superior to traditional instruc-

tional practices. In fact, the reverse could be implied;


that is, traditional instructional practices were superior

to self-paced 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.




The problem of this study was to determine the effec-

tiveness of self-paced individualized instruction by

applying the aggregate chi-square statistical procedure

to the results of several, previously done, independent

studies which have analyzed self-paced 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 chi-square procedure was

applied to the results of a set of 11 previously reported

studies in an attempt to discern a consistent pattern.


The strength of the aggregate chi-square procedure

lies in the use that was made of non-significant 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 non-significant

studies were able to be included in a statistical procedure

such as the aggregate chi-square, 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 chi-square 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 self-paced

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 self-paced 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.


The results of this study indicated that traditional

classroom instructional practices were superior to indi-

vidualized (self-paced) 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 self-paced individualized instruction as

measured by that study.


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 chi-square 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 self-paced 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 self-paced 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. Self-paced 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). Self-paced 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 self-paced

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


Suggestions for Further Research

In this study, self-paced individualized instruction

was investigated. It might be very helpful if some


empirical research studies were done in which self-paced

and teacher-paced individualized instruction were com-

pared. The following comments will serve to strenghten

this point and also serve to clarify what is meant by


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


The significant point made by Taveggia was that the

feature termed "go-at-your-own-pace" (p. 1030) was one of

five features that seemed to make the PSI appraoch better

than conventional approaches. Also, that the term go-at-

your-own-pace 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
unit-perfection, forced-pacing, 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 unit-perfection, forced-pacing, and

monitored progression mentioned above, do indeed point to

a definite forcing of the pace which is in direct contrast

to the concept of self-pacing.

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 self-paced individualized

instruction did not fare well when submitted to the aggre-

gate chi-square 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




The aggregate chi-square procedure was developed by

Fisher (1948). The aggregate chi-square procedure was

based upon the fact that the distribution of the sum of

several values of chi-square was itself distributed as a

chi-square for two degrees of freedom was -2 times the

natural logarithm of the probability (pp. 99-101).

While other researchers have discussed the technique

and its possible applications (Lancaster, 1949 & Wallis,

1942), there have been few attempts to use the aggregate

chi-square 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 chi-square 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 chi-square

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 chi-square model was based on the proof that

any p value could be transformed to a chi-square value

with two degrees of freedom and that the sum of indepen-

dent chi-squares was distributed as a chi-square. Follow-

ing is an outline of the transformation procedure as

developed by Fisher (1948, pp. 99-101).

i. The transformation equation:

X2 = -2 log p (1)

2. Composite X2 is given by the formula:
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.


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to the reference section of this study.

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packages: Aid to student? or teacher? Unpub-
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Harold L. Schoen, W104A East Hall, University of
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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



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


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