AN INVESTIGATION OF THE RELATIVE EFFECTIVENESS
OF THREE METHODS OF UTILIZING LABORATORY ACTIVITIES
IN SELECTED TOPICS OF JUNIOR COLLEGE MATHEMATICS
By
JOAN MARIE GOLLIDAY
A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1974
To
Mother, Daddy, and Bruce
ACKNOWLEDGMENTS
At this time, it is my privilege to publicly thank
the individuals who helped to make the attainment of my
doctorate a reality. I would like to express my deepest
thanks to Dr. Kenneth P. Kidd, chairman of my supervisory
committee, for his guidance and assistance during my gradu
ate studies. The many suggestions offered during the plan
ning and organization of the research have been instrumental
in its successful completion. His assistance is sincerely
appreciated.
My thanks also go to Dr. Elroy J. Bolduc, Jr.,
cochairman of the committee,' for his suggestions on the
writing and rewriting of the manuscript.
To Dr. Charles W. Nelson, I express my appreciation
for serving on the supervisory committee, and for the many
hours spent in reading and criticizing the final manuscript.
To Dr. Vynce A. Hines, I extend my deepest appreci
ation for his advice regarding the statistical aspects of
the study in both its preliminary and final stages.
Finally, I wish to thank Bruce Walek for his constant
support and encouragement.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS .......... ... ........................ iii
LIST OF TABLES ........................................ vi
ABSTRACT ............................................ x
Chapter
I. INTRODUCTION .................................... 1
General Background of the Problem ......... 1
Statement of the Problem ................. 3
Definition of Terms ........................ 6
Need for the Study .... ................... 7
Organization of the Study ................ 9
II. REVIEW OF RELATED RESEARCH .................... 11
InductiveDeductive Studies ............... 11
Laboratory Studies ........................ 15
Summary .................................. 19
III. THE EXPERIMENTAL DESIGN ....................... 21
Statement of Hypotheses ................... 21
Description of Procedures and Design ...... 26
Instrumentation ......................... ... 31
MyersBriggs Type Indicator .......... 31
PretestPosttest ..................... 33
Experiment Materials .................. 34
Statistical Treatment ..................... 34
IV. ANALYSIS OF DATA ......................... ...... 35
V. SUMMARY, CONCLUSIONS, LIMITATIONS, AND
IMPLICATIONS ................................ 61
Summary ..................................... 61
Conclusions ...................... ......... 63
Limitations .............................. 66
Implications .............................. 67
TABLE OF CONTENTS  Continued
Page
APPENDIX A  Performance Objectives and
PretestPosttest .................... 69
APPENDIX B  Experiments .......................... 76
BIBLIOGRAPHY ........................................ 118
BIOGRAPHICAL SKETCH ..................... .. .......... 121
LIST OF TABLES
Table Page
I Subject Distribution by Treatment and
Achievement Level 30
II Subject Distribution by Treatment and
Personality Type 31
III Subject Distribution by Achievement Level
and Personality Type 31
IV Mean Error Scores for Subjects in the
ExploratoryDiscovery Group, the
VerificationApplication Group and
Combination Group on the Pretest and
Posttest 35
V Analysis of Covariance 36
VI 95 Percent Confidence Intervals for
Comparisons Among the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group 38
VII Mean Error Scores of Sensing Subjects and
Intuitive Subjects on the Pretest and
Posttest 38
VIII Mean Error Scores for HighAchievers in the
ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and
Posttest 39
IX Mean Error Scores for AverageAchievers in
the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and
Posttest 40
LIST OF TABLES  (Continued)
Ta ble Page
X Mean Error Scores for LowAchievers in the
ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and
Posttest 40
XI 95 Percent Confidence Intervals for
Comparisons Among the AverageAchievers in
the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group 43
XII 95 Percent Confidence Intervals for
Comparisons Among the LowAchievers in the
ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group 43
XIII Mean Error Scores for Sensing Subjects in
the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and
Posttest 44
XIV Mean Error Scores for Intuitive Subjects
in the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and
Posttest 45
XV 95 Percent Confidence Intervals for
Comparisons Among the Sensing Subjects in
the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group 47
XVI Mean Error Scores for Sensing High
Achievers in the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group on the Pretest
and Posttest 47
XVII Mean Error Scores for Sensing Average
Achievers in the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group on the Pretest
and Posttest 48
LIST OF TABLES  (Continued)
Table Page
XVIII Mean Error Scores for Sensing LowAchievers
in the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and
Posttest 48
XIX Mean Error Scores for Intuitive High
Achievers in the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group on the Pretest
and Posttest 49
XX Mean Error Scores for Intuitive Average
Achievers in the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group on the Pretest
and Posttest 49
XXI Mean Error Scores for Intuitive Low
Achievers in the ExpLoratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group on the Pretest
and Posttest 50
XXII 95 Percent Confidence Intervals for
Comparisons Among the Sensing High
Achievers in the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group 54
XXIII 95 Percent Confidence Intervals for
Comparisons Among the Sensing Average
Achievers in the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group 54
XXIV Questionnaire Summary for the Exploratory
Discovery Group 57
XXV Questionnaire Summary for the Verification
Application Group 58
XXVI Questionnaire Summary for the Combination
Group 59
XXVII Summary of Comments for the Exploratory
Discovery Group 60
vi i
LIST OF TABLES  (Continued)
Table Page
XXVIII Summary of Comments for the Verification
Application Group 60
XXIX Summary of Comments for the Combination
Group 60
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
AN INVESTIGATION OF THE RELATIVE EFFECTIVENESS
OF THREE METHODS OF UTILIZING LABORATORY ACTIVITIES
IN SELECTED TOPICS OF JUNIOR COLLEGE MATHEMATICS
By
Joan Marie Golliday
August, 1974
Chairman: Dr. Kenneth P. Kidd
Cochairman: Dr. Elroy J. Bolduc
Major Department: Curriculum and Instruction
Purpose of the Study
The purpose of the study was to investigate the
relative effectiveness of the mathematics laboratory when
used in three different ways in conjunction with a tradi
tional lecturediscussion approach to teach community college
freshmen enrolled in a required mathematics course.
Procedures
The sample population for the study consisted of
seven sections of a required mathematics course at Santa Fe
Junior College, Gainesville, Florida. On the basis of their
scores on the MyersBriggs Type Indicator the subjects were
classified as sensing or intuitive. They were also classi
fied as high, average, or low achievers on the basis of
their college grade point average. One group, known as the
exploratorydiscovery group, received sixty minutes of labo
ratory experiences followed by thirty minutes of discussion.
The second group, known as the verificationapplication
group, received thirty minutes of lecture followed by sixty
minutes of laboratory. The third group, known as the combi
nation group, received thirty minutes of laboratory exper
ience both before and after the thirty minutes of lecture
discussion. All groups studied the topic of ratio and simi
larity for two and one half weeks. They were given both a
pretest and a posttest. The resulting mean error scores
of the 94 subjects were compared in a 3x3x2 factorial design
using analysis of covariance. The pretest error scores were
used as a covariate. Scheffe's Method was used to determine
the significance of the reductions in mean error score for
the various subcategories.
Conclusions
In comparisons of sensing and intuitive subjects
without regard to achievement level or method, the sensing
subjects did significantly better than the intuitive
subjects. Comparisons for all other main effects and
interactions were nonsignificant.
The investigation of reductions in mean error scores
indicated that the exploratorydiscovery group attained a
significantly greater reduction than the other two groups.
Within the categories of averageachievers and sensing
students, the exploratorydiscovery group also achieved a
~ ~
significantly greater reduction than the other two groups.
Lowachievers in the combination group achieved a signifi
cantly greater reduction in mean error score than those in
the exploratorydiscovery group. Finally, sensing high
achievers in the combination group achieved a significantly
greater reduction in mean error score than those in the
verificationapplication group.
CHAPTER I
INTRODUCTION
General Background of the Problem
In recent years, the general concern over the educa
tive process has produced a number of innovative teaching
techniques. Some of these innovations are genuinely new
while others are old techniques which have merely been reno
vated, energized, and'generally modernized. Regardless of
which technique one chooses to investigate, it soon becomes
apparent that all have suffered from the same malady 
namely, little or no experimental documentation of their
worth and validity.
At the present time the concept of a mathematics
laboratory is again appearing upon the educational horizon.
This particular teaching technique has had periods of popu
larity at least twice within the last century. The initial
appearance of this teaching approach is generally associated
with the English mathematician John Perry. Perry first
promulgated his revolutionary ideas in 1901 in a report on
the "Teaching of Mathematics" which he presented to the
British Association for the Advancement of Science. His main
concern was that too much emphasis was being placed on the
theoretical aspects of mathematics. He proposed that a more
meaningful approach would be to teach a combined physics and
mathematics course putting the physical or applied aspects
first. "Perry favored a laboratory approach, including
greater emphasis on experimental geometry, practical mensu
ration, the use of squared paper to plot statistics, inter
polate, discuss slope, and find maximum and minimum values,
easy vector algebra, more solid geometry, and the utilitarian
parts of geometry" (Mock 1963, p. 131). Apparently, the vast
majority of the mathematics teachers in England agreed with
Professor Perry and the Perry Movement was soon spreading
across England and to America. During the next couple of
years, articles dealing with the pros and cons of the labo
ratory approach abounded. It seemed that the laboratory
concept was here to stay. But Perry and his followers had
not reckoned with the rigid, unchanging testing system of
England's school system. As student scores fell so did
support for the Perry Movement. Its popularity lasted less
than ten years.
The second emergence of a laboratorytype of instruc
tion came in the early 1940's. This was an era of multi
sensory aids. Topics were taught using movies, film strips,
slides, and overhead and opaque projectors. The major
shortcoming in this approach was the passive role of the
student. All these multisensory aids were used by the
teachers to demonstrate principles which they expounded
rather than as a means of handson discovery by the student.
After approximately five years, the novelty of this approach
began to subside and the teachers gradually returned to
their traditional methods of instruction.
The most recent reemergence of the mathematics labo
ratory began in the early 1960's. Its spread across the
United States has been more gradual than in the past and
this in itself may be a healthy sign. The bandwagon approach
which has meant disaster in the past has been avoided. Today
the mathematics laboratory is viewed as an adjunct to rather
than a replacement for the more traditional forms of
instruction. Past studies have dealt with the question of
whether or not the laboratory method is better than the tra
ditional lecturediscussion technique. The results of most
of these studies have been inconclusive. What is needed, in
view of today's educational philosophy, are studies to deter
mine how the laboratory may be most effectively used in con
junction with the traditional lecturediscussion method. It
was with this in mind that the present study was undertaken.
Statement of the Problem
The purpose of this study is to investigate the rel
ative effectiveness of the mathematics laboratory when used
in three different ways in conjunction with a traditional
lecturediscussion approach to teach community college
freshmen enrolled in a required mathematics course. The
laboratory experience will be used as an introduction to a
~
topic, as a reinforced, or as both. All subjects are clas
sified by achievement level and personality type, as deter
mined by the MyersBriggs Type Indicator, so that the in
teraction of these factors with the various laboratory ap
proaches may be assessed. In particular, as a prelude to
the formal null hypotheses found in Chapter III, the
following research questions are of interest:
1. Do college freshmen studying ratio and similarity under
one sequencing pattern of laboratory experiences and
discussion do significantly better than those studying
the same topics under different sequencing arrangements?
2. Do highachieving college freshmen studying ratio and
similarity perform significantly better under any one of
the three sequencing arrangements?
3. Do averageachieving college freshmen studying ratio and
similarity perform significantly better under any one of
the three sequencing arrangements?
4. Do lowachieving college freshmen studying ratio and
similarity perform significantly better under any one of
the three sequencing arrangements?
5. Do college freshmen who have been identified as sensing
personality types on the basis of the MyersBriggs Type
Indicator perform significantly better under any one of
the three sequencing arrangements when studying ratio
and similarity?
6. Do college freshmen who have been identified as intui
tive personality types on the basis of the PyersBriggs
5
Type Indicator perform significantly better under any
one of the three sequencing arrangements when studying
ratio and similarity?
7. Do highachieving college freshmen who have been iden
tified as sensing personality types perform significantly
better under any one of the three sequencing arrangements
when studying ratio and similarity?
8. Do highachieving college freshmen who have been iden
tified as intuitive personality types perform signifi
cantly better under any one of the three sequencing
arrangements when studying ratio and similarity?
9. Do averageachieving college freshmen who have been
identified as sensing personality types perform signif
icantly better under any one of the three sequencing
arrangements when studying ratio and similarity?
10. Do averageachieving college freshmen who have been
identified as intuitive personality types perform sig
nificantly better under any one of the three sequencing
arrangements when studying ratio and similarity?
11. Do lowachieving college freshmen who have been iden
tified as sensing personality types perform signifi
cantly better under any one of the three sequencing
arrangements when studying ratio and similarity?
12. Do lowachieving college freshmen who have been iden
tified as intuitive personality types perform signif
icantly better under any one of the three sequencing
arrangements when studying ratio and similarity?
In order to test the series of null hypotheses gen
erated by these research questions, 94 college freshmen en
rolled in an introductory mathematics course at a Florida
community college were identified as sensing highachievers,
sensing averageachievers, sensing lowachievers, intuitive
highachievers, intuitive averageachievers, or intuitive
lowachievers on the basis of their college grade point
average and the MyersBriggs Type Indicator. Each student
was administered a pretest and a posttest on ratio and
similarity. The resulting mean error scores of the eighteen
groups were compared in a 3x3x2 factorial design using the
methods of multiple linear regression with the pretest
scores as a covariate.
Definition of Terms
The following terms will be used throughout the
study:
Sensing Subjecti a subject who has been classified as a
sensing personality on the basis of his MyersBriggs Type
Indicator score.
Intuitive Subject: a subject who has been classified as an
intuitive personality on the basis of his MyersBriggs Type
Indicator score.
HighAchiever: a subject whose grade point average at his
current community college is greater than or equal to 3.35.
AverageAchiever: a subject whose grade point average at
his current community college is between 2.65 and 3.35.
LowAchiever: a subject whose grade point average at his
current community college is less than or equal to 2.65.
ExploratoryDiscovery Method: a method of using the
mathematics laboratory as an introduction to a new topic
followed by class discussion of what was observed in the
laboratory.
VerificationApplication Method: a method of using the
mathematics laboratory to illustrate and verify topics which
have been taught in the classroom.
Combination Method: a method of using the mathematics
laboratory both before and after class discussion so that it
both introduces and verifies the classroom material.
Mathematics Laboratory: a mode of instruction which uses
experiments to aid students in the discovery and/or
verification of mathematical concepts.
Need for the Study
An area of concern for teachers of mathematics has
been that of helping the student to obtain a better under
standing of the mathematics he is studying. It was this
concern which produced modern mathematics. With the advent
of modern mathematics there were many teaching innovations
such as teamteaching, modular scheduling, discovery learn
ing, and the mathematics laboratory. At first these were
advocated as replacements for the traditional modes of
instruction. But as researchers found, students did not do
significantly better, or worse for that matter, under the
new methods of instruction. The unfortunate part is that
many of these innovations were abandoned because they did
not produce better results than the traditional methods.
The fact that they were at least as good as the old
techniques was completely overlooked.
There have been several studies in which the
mathematics laboratory has been compared to traditional
methods of instruction. See Wilkinson (1970), Cohen (1970),
Phillips (1970) and Bluman (1971). In all four of these
studies there were no significant differences between the two
methods of instruction; that is, they were equally effective.
Since the laboratory approach appears to be as good
as the traditional lecturediscussion method, it seems rea
sonable to use both. It was with this premise in mind that
this study was conceived. The questions which immediately
came to mind were as follows:
1. Is there a best sequence for using both the mathematics
laboratory and the lecturediscussion?
2. If there is a best sequencing pattern will it be the
same for all achievement levels?
3. Would the best sequencing pattern be related to
personality type?
A search of the literature found only two studies
which had considered this question of sequencing. See
Reuss (1970) and Emslie (1971). Reuss did his work in
biology while Emslie did his in physics. There was no exper
imental research into these questions using mathematics as
the vehicle of study.
Since many school systems are committing themselves
to the operation of mathematics laboratories, it is essen
tial that the above questions be answered. This study is
designed to investigate the role of the mathematics labora
tory at the community college level. It is hoped that this
study will stimulate the further research at the elementary,
middle, and secondary school levels which is needed.
Organization of the Study
Chapter I has been an introduction to the study,
including some general background information, a statement
of the problem, definitions, and an explanation of the need
for the study. Chapter II is devoted to a review of related
research. The results from five studies comparing the
inductive method of instruction to the deductive method are
reported in the first section of the chapter. In the second
section, the results from five studies comparing the mathe
matics laboratory to traditional methods of instruction are
examined, along with two studies that dealt with the
sequencing of a laboratory experience with traditional
lecturediscussion instruction. The final section is a
summary of the first two sections. Chapter III contains the
formal null hypotheses, along with a description of the
design, the sample population, and the procedures involved
in gathering the data. Information about the pretest, the
posttest, the experiments used and the MyersBriggs Type
Indicator is presented along with an explanation of the
statistical treatment. Chapter IV is devoted to a presen
tation and analysis of the data. It also includes the
results of a questionnaire completed by the subjects in the
study. Chapter V contains a brief summary of the study
10
together with a list of the conclusions reached. Several
limitations are cited, and some implications for instruction
and future research are discussed.
CHAPTER II
REVIEW OF RELATED RESEARCH
For nearly twenty years, educators and mathemati
cians have been concerned with the question of whether the
traditional deductive method of instruction is better than
the inductive approach. There have been studies which found
the deductive method to be better, while others found the
inductive approach to be significantly better. The vast
majority of the research into this area, however, found no
significant differences. Most recently, this question has
reemerged with reference to the mathematics laboratory. For
this reason, this chapter has been divided into three parts.
The first section contains a few representative samples of
the research done on the inductivedeductive question. The
second section deals strictly with research relating to the
laboratory approach to instruction, while the final section
will be a summary of the results reported herein.
InductiveDeductive Studies
One of the earliest studies to investigate the rela
tive effectiveness of the inductive and deductive methods of
instruction was conducted by Dr. Max Sobel (1956). In this
1
study, Dr. Sobel investigated the effectiveness of the in
ductive method of teaching algebra as compared to the tradi
tional deductive method. In order to do this he used four
teen ninthgrade algebra classes in Newark and Patterson,
New Jersey. Seven classes were taught by each method and
every class except two had a different instructor. The
teachers using the inductive approach were given a manual of
instruction, an explanation of the study, and numerous il
lustrative examples to be used. The deductive group used
the normal textbook. At the end of four weeks the students
were given a test which had been developed by the researcher.
A review of I.Q. scores for all students indicated that sub
grouping by intelligence level was also possible. An anal
ysis of the data found that bright students learned and
retained skills better when taught by the inductive method.
For the average intelligence level, there were no significant
differences.
In 1965, Krumboltz and Yabroff conducted a study to
determine the teaching efficiency of inductive and deductive
sequences of instruction with varying frequencies of alter
nation between problemsolving and rulestating frames.
They also investigated the interaction of these factors with
intelligence levels. The experimental sample consisted of
272 students enrolled in an introductory education course at
the University of Minnesota. Each student was given the
Miller Analogies Test and was categorized as high or low in
intelligence on the basis of his score. Four forms of
programmed materials on elementary statistics and test in
terpretation were randomly distributed to all students. Two
forms were inductive with different frequencies of alterna
tion while the other two were deductive with differing
frequencies of alternation.
Each student was given a test two weeks after the
end of the instructional period. An analysis of the data
using analysis of variance found the following results
1. The highintelligence group completed their work in
significantly less time than the lowintelligence group.
2. The inductive group made significantly more errors than
the deductive group.
A similar study to that of Krumboltz and Yabroff was
conducted by Koran (1971). Her sample population consisted
of 167 students enrolled in an introductory education course
at the University of Texas. Each student was given selected
measures from the Kit of Reference Tests for Cognitive
Factors. Koran also used programmed materials dealing with
selected areas of elementary statistics and test
interpretation. There were four forms of the programmed
material  two inductive and two deductive with differing
frequencies of alternation. These were distributed randomly
to the students.
Each student was given a test two weeks after the
completion of the programmed material. An analysis of the
data showed no significant differences in the time required
to complete the material. However, subjects in the inductive
treatment made significantly more errors than those in the
deductive treatment.
Becker (1967) investigated the interaction of two
instructional treatments with two aptitude variables. His
subjects were students enrolled in an Algebra I class in
San Carlos, California. All students were given multiple
choice tests to determine their mathematical and verbal
aptitude. On the basis of these tests 35 matched pairs were
obtained. Subjects in each matched pair were randomly
assigned to treatments. The two treatments were programmed
instruction  one inductive and the other deductive in
arrangement. The data collected were subjected to a multi
ple regression analysis. There were no significant
differences.
Tanner (1968) studied the relative effectiveness of
an expository treatment as compared to a discovery approach
to teaching physical science. The experimental population
consisted of 389 ninthgrade students enrolled in a general
science course. These subjects were randomly assigned to
three groups. One group received materials programmed in
an expositorydeductive format. The second group received
materials programmed in a discoveryinductive format. The
third group received materials containing the same program
frames but in a random order. An analysis of posttest
scores found no significant differences among the three
groups.
Laboratory Studies
In the last five years, nearly all research of an
experimental nature dealing with the laboratory approach to
instruction has been undertaken by doctoral students. The
intent of the studies has varied widely as have the popula
tions sampled. Wilkinson (1970) was interested in deter
mining whether the laboratory approach to teaching geometry
to sixth graders would be more effective than the traditional
teachertextbook approach. He used two experimental groups
and one control group. One experimental group used manipu
lative materials and worksheets to guide them in collecting
and generalizing their data. The second experimental group
received verbal instructions, by means of tape cassettes, in
addition to the written worksheets. All three groups were
given protests and posttests dealing with their attitude
toward mathematics, achievement in geometry, and nonverbal
intelligence. An analysis of the data showed no significant
differences for the three groups in any of these areas.
In another study, Cohen (1970) investigated the rel
ative effectiveness of the mathematics laboratory on under
achieving seventh and eighth grade boys. Two groups of four
teen boys each with average or aboveaverage ability, but
with belowaverage achievement were used. One group was
taught fraction concepts and computation with fractions
using the traditional textbookdiscussion approach. The
second group was taught the same material in a laboratory
setting using manipulative devices and multisensory
materials. A comparison of achievement scores, computa
tional skills scores, and attitude scores for the two groups
showed no significant differences.
Three studies dealt with the use of the laboratory
approach at the college level. Phillips (1970) conducted a
study at Oakland City College, Oakland City, Indiana, to
determine the effect of the laboratory approach on the
achievement and attitude of lowachiever mathematics students
enrolled in a developmental mathematics course. All subjects
were given a pretest and two posttests to determine attitude
and achievement. One posttest was given at the end of the
course while the second was given at the end of a second re
quired mathematics course. The experimental group was taught
the developmental mathematics by means of a mathematics
laboratory. The control group received the traditional lec
ture type of instruction. Both groups received the tradi
tional approach in the sequential course. An analysis of the
data showed no significant difference in achievement for the
two groups. The laboratory group did show a significant im
provement in their attitude immediately following the labo
ratory experience but this gain was no longer apparent fol
lowing their reencounter with the traditional approach.
In the second study, Smith (1970) investigated the
effectiveness of the laboratory in teaching abstract algebra
to college students. He used two classes of college students
enrolled in a required course in abstract algebra. Each
class was halved so that there were four groups with twelve
students in each. The control group received four lecture
sessions with no laboratory. The other three groups re
ceived varying laboratory sessions. One group had one lec
ture session and three laboratory sessions; the second had
two lecture sessions and two laboratory sessions; the third
group had three lecture sessions and one laboratory session.
The laboratory experience dealt with the manipulation of
concrete models relating to the materials taught in the lec
ture sessions. The material dealt with systems of numeration
and bases other than ten. The analysis of the data indicated
that those receiving laboratory experience scored signifi
cantly higher than the control group in both comprehension
and retention.
The third study, conducted by Bluman (1971), was to
determine whether the laboratory method of instruction in
mathematics would be more effective than the traditional
approach. For the purposes of this study, four intact
classes of freshmen enrolled in a collegelevel introductory
mathematics course were selected. Two classes acted as con
trol groups and received the traditional instruction. The
other two classes received their instruction by means of
filmstrips, experiments, demonstrations, overhead projector,
and problem sessions. Two teachers were used to teach these
four classes. Each teacher had an experimental and a control
group. The analysis of the data indicated that there was no
significant difference between the two treatments in either
attitude or achievement. There was, however, a significant
interaction between teacher and method.
In all of the above studies, the general purpose was
to determine whether the laboratory approach should be used
in place of the traditional approach. As stated in Chapter I
this eitheror approach is inappropriate in view of today's
educational philosophy. Instead, we need to ascertain in
what way the laboratory can best be employed to complement
the traditional approach. It is this question which needs
to be answered.
In researching the literature, two studies were
found which closely resemble the present study. The first
study was conducted by Reuss (1970). Reuss used three
groups of biology students all receiving laboratory
experiences. The control group used experiments employing
the traditional deductive approach. One experimental group
used experiments which were of the guided inductive type,
while the other group used materials written in the open
inductive style. All students were pretested on attitude
and basic knowledge of the topic to be studied. Posttests
were given. The data were analyzed with the class as the
basic statistical unit and again taking ability into
consideration. In all cases, there were no significant
differences among the three approaches.
In the second study, Emslie (1971) sought to deter
mine the relative effectiveness of two sequencing procedures
in the teaching of a unit on molecules and the atom.
Method I was a laboratorytheory sequence while Method II
was a theorylaboratory sequence. Method I was used with a
sample of 99 fourth and sixth graders in one school while
Method II was used with a sample of 158 fourth and sixth
graders in a school in another district. The criterion
variable was the score on a standardized test designed for
use with a sixth grade science textbook. The data were
analyzed using analysis of covariance with I. Q. and general
science achievement scores as covariates. This analysis re
sulted in no significant differences for the two methods
although the fourth graders appeared to score higher under
the laboratorytheory approach.
Summary
In general the studies comparing the inductive with
the deductive approach have been inconclusive. Both
approaches seem to have merit and would lead one to believe
that the mathematics laboratory could logically precede or
follow a lecturediscussion presentation. Experimental
testing of this assumption is needed, however.
The studies dealing with the laboratory approach
generally found it to be as effective as more traditional
approaches. These results should guarantee the laboratory
method a slot in every educator's repertoire. The issue
which has not been answered, at least for the mathematics
20
laboratory, is how most effectively to combine the laboratory
approach with the traditional lecturediscussion approach.
The studies by Reuss (1970) and Emslie (1971) illustrate that
this concern is shared by other sciences. Although their
studies found no significant differences for different
sequencing patterns, they have served to make us aware of the
need for further research in other areas and at other grade
levels. In the present study, the role of the mathematics
laboratory at the college level has been investigated. It
is hoped that this study will contribute additional informa
tion about the mathematics laboratory and its relation to
more traditional modes of instruction.
CHAPTER III
THE EXPERIMENTAL DESIGN
Statement of Hypotheses
As the preceding chapter shows, there is a dearth of
research dealing with the laboratory as an adjunct to more
traditional modes of instruction. Although the two studies
that dealt with this question had no significant results,
there were trends within intelligence levels which indicate
that further research might be informative.
A factor which was not considered in either of these
studies was personality type. It is conceivable that the
inductive nature of a laboratory experience might cause cog
nitive dissidence with certain personality types and hence
have an effect on the results of the study.
In the present study, personality type and achieve
ment level will both be taken into consideration and their
effects, if any, determined. In order to do this, the
following null hypotheses will be investigated:
HI. There is no significant difference between the
mean posttest score of students in the
exploratorydiscovery group and the mean posttest
score of students in the verificationapplication
group.
H2. There is no significant difference between the
mean posttest score of students in the
exploratorydiscovery group and the mean posttest
score of students in the combination group.
H3. There is no significant difference between the
mean posttest score of students in the
verificationapplication group and the mean post
test score of students in the combination group.
H4. There is no significant difference between the
mean posttest score of sensing students and the
mean posttest score of intuitive students.
H5. There is no significant difference between the
mean posttest score of highachieving students in
the exploratorydiscovery group and the mean post
test score of highachieving students in the
verificationapplication group.
H6. There is no significant difference between the
mean posttest score of highachieving students in
the exploratorydiscovery group and the mean post
test score of highachieving students in the
combination group.
H7. There is no significant difference between the
mean posttest score of highachieving students in
the verificationapplication group and the mean
posttest score of highachieving students in the
combination group.
H8. There is no significant difference between the
mean posttest score of averageachieving students
in the exploratorydiscovery group and the mean
posttest score of averageachieving students in
the verificationapplication group.
H9. There is no significant difference between the
mean posttest score of averageachieving students
in the exploratorydiscovery group and the mean
posttest score of averageachieving students in
the combination group.
H10. There is no significant difference between the
mean posttest score of averageachieving students
in the verificationapplication group and the
mean posttest score of averageachieving students
in the combination group.
H11. There is no significant difference between the
mean posttest score of lowachieving students in
the exploratorydiscovery group and the mean
posttest score of lowachieving students in the
verificationapplication group.
H12. There is no significant difference between the
mean posttest score of lowachieving students in
the exploratorydiscovery group and the mean
posttest score of lowachieving students in the
combination group.
H13. There is no significant difference between the
mean posttest score of lowachieving students in
the verificationapplication group and the mean
posttest score of lowachieving students in the
combination group.
H14. There is no significant difference between the
mean posttest score of sensing students in the
exploratorydiscovery group and the mean posttest
score of sensing students in the verification
application group.
H15. There is no significant difference between the
mean posttest score of sensing students in the
exploratorydiscovery group and the mean posttest
score of sensing students in the combination
group.
H16. There is no significant difference between the
mean posttest score of sensing students in the
verificationapplication group and the mean post
test score of sensing students in the combination
group.
H17. There is no significant difference between the
mean posttest score of intuitive students in the
exploratorydiscovery group and the mean posttest
score of intuitive students in the verification
application group.
H18. There is no significant difference between the
mean posttest score of intuitive students in the
exploratorydiscovery group and the mean posttest
score of intuitive students in the combination
group.
H19. There is no significant difference between the
mean posttest score of intuitive students in the
verificationapplication group and the mean post
test score of intuitive students in the
combination group.
H20. There is no significant difference between the
mean posttest score of highachieving sensing
students in the exploratorydiscovery group and
the mean posttest score of highachieving sensing
students in the verificationapplication group.
H21. There is no significant difference between the
mean posttest score of highachieving sensing
students in the exploratorydiscovery group and
the mean posttest score of highachieving sensing
students in the combination group.
H22. There is no significant difference between the
mean posttest score of highachieving sensing
students in the verificationapplication group
and the mean posttest score of highachieving
sensing students in the combination group.
H23. There is no significant difference between the
mean posttest score of averageachieving sensing
students in the exploratorydiscovery group and
the mean posttest score of averageachieving
sensing students in the verificationapplication
group.
H24. There is no significant difference between the
mean posttest score of averageachieving sensing
students in the exploratorydiscovery group and
the mean posttest score of averageachieving
sensing students in the combination group.
H25. There is no significant difference between the
mean posttest score of averageachieving sensing
students in the verificationapplication group
and the mean posttest score of averageachieving
sensing students in the combination group.
H26. There is no significant difference between the
mean posttest score of lowachieving sensing
students in the exploratorydiscovery group and
the mean posttest score of lowachieving sensing
students in the verificationapplication group.
H27. There is no significant difference between the
mean posttest score of lowachieving sensing
students in the exploratorydiscovery group and
the mean posttest score of lowachieving sensing
students in the combination group.
H28. There is no significant difference between the
mean posttest score of lowachieving sensing
students in the verificationapplication group
and the mean posttest score of lowachieving
sensing students in the combination group.
H29. There is no significant difference between the
mean posttest score of highachieving intuitive
students in the exploratorydiscovery group and
the mean posttest score of highachieving intui
tive students in the verificationapplication
group.
H30. There is no significant difference between the
mean posttest score of highachieving intuitive
students in the exploratorydiscovery group and
the mean posttest score of highachieving
intuitive students in the combination group.
H31. There is no significant difference between the
mean posttest score of highachieving intuitive
students in the verificationapplication group
and the mean posttest score of highachieving
intuitive students in the combination group.
H32. There is no significant difference between the
mean posttest score of averageachieving intui
tive students in the exploratorydiscovery group
and the mean posttest score of averageachieving
intuitive students in the verificationapplication
group.
H33. There is no significant difference between the
mean posttest score of averageachieving intui
tive students in the exploratorydiscovery group
and the mean posttest score of averageachieving
intuitive students in the combination group.
H34. There is no significant difference between the
mean posttest score of averageachieving intui
tive students in the verificationapplication
group and the mean posttest score of average
achieving intuitive students in the combination
group.
H35. There is no significant difference between the
mean posttest score of lowachieving intuitive
students in the exploratorydiscovery group and
the mean posttest score of lowachieving intuitive
students in the verificationapplication group.
H36. There is no significant difference between the
mean posttest score of lowachieving intuitive
students in the exploratorydiscovery group and
the mean posttest score of lowachieving intuitive
students in the combination group.
H37. There is no significant difference between the
mean posttest score of lowachieving intuitive
students in the verificationapplication group
and the mean posttest score of lowachieving
intuitive students in the combination group.
Description of Procedures and Desin
The design of the present study can best be classi
fied as the nonequivalent control group design as described
by Campbell and Stanley (1963). There is not, however, a
control group as such since all groups involved received a
treatment. More specifically, the study is a 3x3x2 factorial
experiment. The three experimental factors are the sequencing
pattern (exploratorydiscovery, verificationapplication,
combination), achievement status (high, average, low) and per
sonality type (sensing, intuitive). The criterion measure is
the error score on a posttest on ratio and similarity with
the error score on a pretest on ratio and similarity as a
covariate.
For the purposes of this study, seven classes of
freshmen students enrolled in a required mathematics course
at Santa Fe Junior College were selected and constituted the
experimental population. These seven classes were selected
on the basis of the willingness of the instructors to parti
cipate in the study and the fact that each instructor had at
least two classes at approximately the same time of day.
There were three instructors involved in the study  two
instructors had two classes apiece while the third had three
classes. Five of the classes were during the day and met
for 95 minutes at each session. The remaining two classes
were at night and met for two hours at a time.
Since it was not possible to assign students randomly
to these seven classes, the classes were randomly assigned to
treatments. The two instructors with two classes each had
two of the three treatments but not the same two. The third
instructor had all three treatments. Diagrammatically, the
design would look something like the following:
TEACHER
A B C
T
S I X X
E
II X X
T
M
III X X X
E
N
T
Each of the seven classes received laboratory exper
iences in conjunction with their study of ratio and
similarity. The variable was in the sequencing of the lab
oratory experience with the class discussion. The one
treatment group received their laboratory experiences before
the classroom instruction, hereafter referred to as the
exploratorydiscovery method. The laboratory experience
consisted of a series of guided experiments on ratio and
similarity. The exploratorydiscovery group received sixty
minutes of laboratory experience followed immediately by
thirty minutes of class presentation on the principles
observed in the experiments.
The second treatment group, hereafter referred to as
the verificationapplication group, received thirty minutes
of class presentation followed immediately by sixty minutes
of laboratory experience.
The third group, hereafter referred to as the com
bination group, received thirty minutes of laboratory exper
iences followed by thirty minutes of discussion, which was
followed by another thirty minutes of laboratory experiences.
In August, 1972, before undertaking their study of
ratio and similarity, all three groups were administered the
MyersBriggs Type Indicator and a pretest on ratio and simi
larity to determine their background knowledge on this topic.
(The pretestposttest was designed by the author and will be
described in the next section.) They were also given the
opportunity to perform some laboratory experiments dealing
with area of a circle and the calculation of pi so that they
would be familiar with this method of instruction. After
the unit on ratio and similarity was completed, each group
was given a brief questionnaire dealing with their personal
reaction to the laboratory experience and a posttest.
The seven classes used had a total enrollment of 129
students. For a student to be included in the study it was
necessary to have four pieces of data on him. They were a
pretest score, a posttest score, a MyersBriggs Type
Indicator classification, and an overall grade point average
for his work at Santa Fe Junior College. Deletion of those
subjects with incomplete data left a sample population of 94
subjects. These 94 subjects were categorized by treatment,
achievement level, and personality type.
The basis for determining a student's achievement
level was his overall grade point average (GPA) at Santa Fe
Junior College. At this junior college only four letter
grades were in use  A, B, C, and W. A grade of A was
worth four points per semester hour of credit earned; a
grade of B was worth three points per semester hour of credit
earned; a grade of C was worth two points per semester hour
of credit earned; and a grade of W, which normally is not
used in the calculation of the GPA, was assigned one point
per semester hour of credit attempted. If a student's GPA
was greater than or equal to 3.35, he was termed a high
achiever. If the GPA was between 2.65 and 3.35, he was
termed an averageachiever. If the GPA was less than or
equal to 2.65, he was classified a lowachiever. The dis
tribution of the subjects taking into account treatment and
achievement level is shown in Table I.
TABLE I: Subject Distribution by Treatment and
Achievement Level
High Average Low Total
Achievers Achievers Achievers
Exploratory 14 10 5 29
Discovery
Verification 8 4 9 21
Application
Combination 31 9 4 44
Total 53 23 18 94
In order to divide the students into two broad per
sonality types, all subjects were administered the Myers
Briggs Type Indicator. This test measures four dichotomous
dimensions of the personality. They are: judgment
perception, thinkingfeeling, sensingintuition, and
extraversionintroversion. For the purposes of this study,
only the sensingintuition dimension was used. (This test
will be described in detail in the next section.) On the
basis of this test, students were classified as sensing,
that is, using data perceived through the senses to draw
conclusions or make decisions; or as intuitive, that is,
tending to rely upon imagination and inspiration for
decisions. The distribution of the subjects taking into
account treatment and personality type is shown in Table II,
while Table III gives the distribution using the factors of
achievement level and personality type.
TABLE II: Subject Distribution
Personality Type
by Treatment and
Sensing Intuitive Total
Exploratory 18 11 29
Discovery
Verification
Application 13 8 21
Combination 19 25 44
Total 50 44 94
TABLE III: Subject Distribution by Achievement Level
and Personality Type
Sensing Intuitive Total
HighAchiever 26 27 53
AverageAchiever 12 11 23
LowAchiever 12 6 18
Total 50 44 94
Instrumentation
This section is devoted to an examination of the
experimental materials and two test instruments utilized in
the present study.
MyersBriggs Type Indicator
As indicated in the preceding section, the subjects
were classified by personality types by means of the Myers
Briggs Type Indicator, Form F (grades 916). This is a
forced choice, selfreport inventory consisting of 166 ques
tions and is designed to be used with normal subjects. It
is administered in a group setting and requires approximately
fiftyfive minutes to complete.
The test purports to measure the following four
dichotomous dimensions: judgmentperception, thinking
feeling, sensingintuition, and extraversionintroversion.
Each student's answer sheet must be graded eight times to
obtain a preference for each of these dimensions. An
adjusted score is determined through the use of prepared
tables found in the MyersBri=gs Type Indicator Manual.
This adjusted score gives not only a preference but also
an indication of the strength of that preference. Since the
present study dealt principally with a student's reasoning
ability, it was decided to use only the sensingintuition
dimension. This dimension has been characterized in the
following way. "When people prefer sensing, they find too
much of interest in the actuality around them to spend much
energy listening for ideas out of nowhere. When people pre
fer intuition, they are too much interested in all the
possibilities that occur to them to give a whole lot of
notice to the actualities" (Wyers 1962, p. 51).
The MyersBriggs Type Indicator has been developed
over a twenty year period. The developers assert that it is'
based on the Jungian theory of type, but the true dichotomy
of the dimensions has been questioned by a number of
psychologists. To determine content validity, splithalf
reliability coefficients, corrected by the SpearmanBrown
prophecy formula, were calculated for each dimension at dif
ferent grade levels. The sensingintuition scale had a
reliability coefficient of 0.87 for college students.
PretestPosttest
In June 1972, twenty behavioral objectives on the
topic of ratio and similarity were developed. These were
submitted to a panel of judges consisting of three junior
college mathematics teachers. The panel assessed the objec
tives and found them to be appropriate for both the topic of
study and the grade level. From these objectives a pretest
posttest designed to assess the subjects' knowledge of ratio
and similarity was developed. One question was prepared for
each objective. The test was submitted to the same panel
and adjudged to be appropriate for the stated objectives.
The test was administered to the students in a sec
tion of the required mathematics course which was not to be
involved in the study. There were twenty students in the
class. The results of this trial run were subjected to a
difficulty test using the following criterion If X repre
sents the number of correct responses to a particular ques
tion, then the question is judged to be acceptable only if
.10N < X < .90N, where N represents the total number of
students taking the test. According to this formula, all
questions were acceptable.
The test was then administered to all subjects in
the seven experimental classes at the beginning and end of
the unit on ratio and similarity. Copies of the performance
objectives and pretestposttest may be found in Appendix A.
Experiment Materials
All of the experiments used in this study were taken
from The Laboratory Approach to Mathematics by Kidd, Myers
and Cilley or from unpublished materials developed by
Kenneth P. Kidd. Some modifications in the materials used
were made. Copies of the experiments may be found in
Appendix B.
Statistical Treatment
The data gathered in the present study were analyzed
using the system of multiple linear regression. A computer
program called MANOVA was employed to compute the error sum
of squares and Fstatistics for all main effects and
interactions. The criterion variable was the posttest error
scores while the pretest error scores were used as a
covariate. The calculated Fvalues were used to determine
whether to accept or reject the null hypotheses at a prede
termined level of confidence. Scheffe's Method was also
used to determine whether reductions in error scores were
significant.
CHAPTER IV
ANALYSIS OF DATA
The first three hypotheses involve a comparison of
the exploratorydiscovery method, the verification
application method and the combination method without regard
to achievement level or personality type. The mean error
scores on the pretest and posttest for these hypotheses are
found in Table IV. Table V is the analysis of covariance
table for the entire study.
TABLE IV: Mean Error Scores for Subjects in the
ExploratoryDiscovery Group, the Verification
Application Group and Combination Group on the
Pretest and Posttest
Pretest Posttest Difference
Exploratory
ExDsovry 9.897 5.379 4.518
Discovery
Ve ication 8.048 5.190 2.858
Application
Combination 7.636 4.341 3.295
H1. There is no significant difference between the mean
posttest score of students in the exploratory
discovery group and the mean posttest score of
students in the verificationapplication group.
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H2. There is no significant difference between the mean
posttest score of students in the exploratory
discovery group and the mean posttest score of
students in the combination group.
H3. There is no significant difference between the mean
posttest score of students in the verification
application group and the mean posttest score of
students in the combination group.
Hypotheses H1H3 state that there are no differences
among the mean error scores of subjects in the exploratory
discovery group, the verificationapplication group and the
combination group. If these hypotheses are in fact true,
then differences as large or larger than those observed
could occur by chance 14.1 percent of the time. The Fratio
for method in Table V is less than that required for signif
icance at the 0.05 confidence level, and hence none of the
null hypotheses H1H3 can be rejected. This indicates that
there is no significant difference among the mean error
scores for the three methods. However, use of Scheffe's
Method to compare the differences between the posttest and
pretest mean error scores indicates that the exploratory
discovery group achieved a significantly greater reduction
in mean error score than either of the other methods. This
is shown by the 95 percent confidence intervals found in
Table VI.
TABLE VI: 95 Percent Confidence Intervals for Comparisons
Among the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group
Method Contrasts Confidence
Interval
ExploratoryDiscovery minus 1.660 0.632 to 2.688
VerificationApplication
ExploratoryDiscovery
minus Combination 1.223 0.257 to 2.189
VerificationApplication 0437 1.508 to 0.634
minus Combination 0 1508 to
Hypothesis H4 involves a comparison of sensing sub
jects and intuitive subjects. The mean error scores for this
hypothesis are given in Table VII.
TABLE VII: Mean Error Scores of Sensing Subjects and
Intuitive Subjects on the Pretest and Posttest
Pretest Posttest Difference
Sensing Subjects 8.700 5.420 3.280
Intuitive Subjects 8.114 4.205 3.909
H4. There is no significant difference between the mean
posttest score of sensing students and the mean
posttest score of intuitive students.
Hypothesis H4 states that there are no differences
between the mean error scores of subjects who have been
categorized as sensing and those who have been categorized
as intuitive. If this hypothesis is in fact true, then
differences as large or larger than those observed could
occur by chance 2.4 percent of the time. The Fratio for
personality in Table V exceeds that required for significance
at the 0.05 confidence level, and hence the null hypothesis
H4 can be rejected. This means that the sensing students did
significantly better on the posttest than the intuitive
students. This would imply that laboratory experiences are
more meaningful for those students who rely upon their senses
than for those who rely upon their feelings and imagination.
Hypotheses H5H13 involve the comparison of the three
methods of instruction within achievement levels. Table VIII
shows the mean pretest scores and mean posttest scores for
highachievers in the exploratorydiscovery group, the
verificationapplication group and the combination group.
TABLE VIII: Mean Error Scores for HighAchievers in the
ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory 8.214 5.000 3.214
Discovery
Ve ication 6.000 3.750 2.250
ApplCombination 7.000 3.774 3.226
Combination 7.000 3.774 3.226
The mean pretest scores and mean posttest scores for average
achievers in the exploratorydiscovery group, the verification
application group and the combination group may be found in
Table IX.
TABLE IX: Mean Error Scores for AverageAchievers in the
ExploratoryDiscovery Group, the Verification
Application Group and the Combination Group on
the Pretest and Posttest
Pretest Posttest Difference
Exploratory 12.600 5.800 6.800
Discovery
Verification
pication 8.250 6.250 2.000
Application
Combination 8.556 6.000 2.556
Table X displays the mean pretest scores and mean posttest
scores for lowachievers in the exploratorydiscovery group,
the verificationapplication group and the combination group.
TABLE X: Mean Error Scores for LowAchievers in the
ExploratoryDiscovery Group, the Verification
Application Group and the Combination Group on
the Pretest and Posttest
Pretest Posttest Difference
exploratory 9.200 5.600 2.600
Discovery
Verification
Application 9.778 6.000 3.778
Combination 10.500 5.000 5.500
H5. There is no significant difference between the mean
posttest score of highachieving students in the
exploratorydiscovery group and the mean posttest
score of highachieving students in the verification
application group.
H6. There is no significant difference between the mean
posttest score of highachieving students in the
exploratorydiscovery group and the mean posttest
score of highachieving students in the combination
group.
H7. There is no significant difference between the mean
posttest score of highachieving students in the
verificationapplication group and the mean posttest
score of highachieving students in the combination
group.
H8. There is no significant difference between the mean
posttest score of averageachieving students in the
exploratorydiscovery group and the mean posttest
score of averageachieving students in the
verificationapplication group.
H9. There is no significant difference between the mean
posttest score of averageachieving students in the
exploratorydiscovery group and the mean posttest
score of averageachieving students in the
combination group.
H10. There is no significant difference between the mean
posttest score of averageachieving students in the
verificationapplication group and the mean posttest
score of averageachieving students in the
combination group.
H11. There is no significant difference between the mean
posttest score of lowachieving students in the
exploratorydiscovery group and the mean posttest
score of lowachieving students in the verification
application group.
H12. There is no significant difference between the mean
posttest score of lowachieving students in the
exploratorydiscovery group and the mean posttest
score of lowachieving students in the combination
group.
H13. There is no significant difference between the mean
posttest score of lowachieving students in the
verificationapplication group and the mean posttest
score of lowachieving students in the combination
group.
Hypotheses H5H7 state that there are no significant
differences among the mean error scores on the posttest of
highachievers in the three laboratory sequencing treatments.
Hypotheses H8H10 state that there are no significant
differences among the mean error scores on the posttest of
averageachievers in the three treatment groups. Hypotheses
H11H13 state that there are no significant differences
among the mean error scores on the posttest of lowachievers
in the three treatment groups. The Fvalues found in
Table V indicate that both main effects are not significant
at the 0.05 level of confidence. The Fratio of 1.501 for
methodachievement interaction also is less than that required
for significance at the 0.05 confidence level. Therefore, we
cannot reject the hypothesis of no interaction. This also
means that we can reject none of the hypotheses H5H13.
Use of Scheffe's Method to compare the differences
between the posttest and pretest mean error scores for
averageachievers indicates that the exploratorydiscovery
group achieved a significantly greater reduction in mean
error scores than either of the other methods. This is
shown by the 95 percent confidence intervals found in
Table XI.
TABLE XI: 95 Percent Confidence Intervals for Comparisons
Among the AverageAchievers in the Exploratory
Discovery Group, the VerificationApplication
Group and the Combination Group
Method Contrasts Confidence
Interval
ExploratoryDiscovery minus 4.800 2.411 to 7.189
VerificationApplication
ExploratoryDiscovery 4.224 2.369 to 6.079
minus Combination
VerificationApplication 0.556 2.982 to 1.870
minus Combination
Use of Scheffe's Method to compare the differences
between the posttest and pretest mean error scores for low
achievers indicates that the combination group achieved a
significantly greater reduction in mean error scores than
the exploratorydiscovery group. This is shown by the 95
percent confidence intervals found in Table XII.
TABLE XII:
95 Percent Confidence Intervals for
Comparisons Among the LowAchievers in the
ExploratoryDiscovery Group, the Verification
Application Group and the Combination Group
Method Contrasts Confidence
Interval
ExploratoryDiscovery minus 1.178 3.431 to 1.074
VerificationApplication
ExploratoryDiscovery 2.900 5.609 to 0.191
minus Combination
VerificationApplication 2.222 4.648 to 0.204
minus Combination
Hypotheses H14H19 involve the comparison of the
three methods of instruction within personality types.
Table XIII shows the mean pretest scores and mean posttest
scores for sensing students in the exploratorydiscovery
group, the verificationapplication group and the combination
group.
TABLE XIII: Mean Error Scores for Sensing Subjects in
the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory 10.611 6.167 4.444
Discovery
Verification
Apication 7.154 5.077 2.077
Combination 7.947 4.947 3.000
The mean pretest scores and mean posttest scores for intuitive
students in the exploratorydiscovery group, the
verificationapplication group and the combination group may
be found in Table XIV.
~
TABLE XIV: Mean Error Scores for Intuitive Subjects in
the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exslotory 8.727 4.091 4.636
Discovery
Verification
Application 9.500 5.375 4.125
Combination 7.400 3.880 3.520
H14. There is no significant difference between the mean
posttest score of sensing students in the
exploratorydiscovery group and the mean posttest
score of sensing students in the verification
application group.
H15. There is no significant difference between the mean
posttest score of sensing students in the
exploratorydiscovery group and the mean posttest
score of sensing students in the combination group.
H16. There is no significant difference between the mean
posttest score of sensing students in the
verificationapplication group and the mean posttest
score of sensing students in the combination group.
H17. There is no significant difference between the mean
posttest score of intuitive students in the
exploratorydiscovery group and the mean posttest
score of intuitive students in the verification
application group.
H18. There is no significant difference between the mean
posttest score of intuitive students in the
exploratorydiscovery group and the mean posttest
score of intuitive students in the combination
group.
H19. There is no significant difference between the mean
posttest score of intuitive students in the
verificationapplication group and the mean posttest
score of intuitive students in the combination
group.
Hypotheses H14HI6 state that there are no signifi
cant differences among the mean error scores on the posttest
of sensing subjects in the three laboratory sequencing
treatments. Hypotheses H1117H19 state that there are no sig
nificant differences among the mean error scores on the
posttest of intuitive subjects in the three treatment groups.
The Fratios found in Table V indicate that only the main
effect of personality is significant at the 0.05 level of
confidence. The Fratio of 0.578 for methodpersonality
interaction is less than that needed for significance at the
0.05 confidence level. Therefore, we cannot reject the
hypothesis of no interaction. This also means that we can
reject none of the hypotheses H14H19.
Use of Scheffe's Method to compare the difference
between the posttest and pretest mean error scores for
sensing students indicates that the exploratorydiscovery
group achieved a significantly greater reduction in mean
error scores than either of the other methods. This is
shown by the 95 percent confidence intervals given in
Table XV.
TABLE XV: 95 Percent Confidence Intervals for Comparisons
Among the Sensing Subjects in the Exploratory
Discovery Group, the VerificationApplication
Group and the Combination Group
Method Contrasts Confidence
Interval
ExploratoryDiscovery minus 2.367 0.660 to 4.074
VerificationApplication
ExploratoryDiscovery 1.444 0.116 to 2.772
minus Combination
VerificationApplication 0.923 2.376 to 0.530
minus Combination
Hypotheses H20H37 involve the comparison of the
three methods of instruction within achievement levels taking
personality type into account. Table XVI shows the mean pre
test scores and mean posttest scores for sensing high
achievers in the exploratorydiscovery group, the
verificationapplication group and the combination group.
TABLE XVI:
Mean Error Scores for Sensing HighAchievers
in the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory
Dlortory 7.143 5.571 1.572
Discovery
Verification 4.600 4.000 0.600
Application
Combination 6.857 4.071 2.786
The mean pretest scores and mean posttest scores for sensing
averageachievers in the exploratorydiscovery group, the
verificationapplication group and the combination group may
be found in Table XVII.
TABLE XVII:
Mean Error Scores for Sensing Average
Achievers in the ExploratoryDiscovery Group,
the VerificationApplication Group, and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory 14.143 6.571 7.572
Discovery
Verification
Application 3.500 5.000 1.500
Combination 10.333 8.667 1.666
Table XVIII displays the mean pretest scores and mean post
test scores of sensing lowachievers in the exploratory
discovery group, the verificationapplication group and the
combination group.
TABLE XVIII: Mean Error Scores for Sensing LowAchievers
in the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory 10.500 6.500
Discovery000 6. 4
Verification
Appication 10.500 6.000 4.500
Combination 12.000 5.500 6.500
Table XIX shows the mean pretest scores and mean posttest
scores for intuitive highachievers in the exploratory
discovery group, the verificationapplication group and the
combination group.
TABLE XIX:
Mean Error Scores for Intuitive highAchievers
in the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory
Exploratory 9.286 4.429 4.857
Discovery
Verification
VerA ication 8.333 3.333 5.000
Application
Combination 7.118 3.529 3.589
The mean pretest scores and mean posttest scores for intui
tive averageachievers in the exploratorydiscovery group,
the verificationapplication group and the combination group
may be found in Table XX.
TABLE XX: Mean Error Scores for Intuitive Average
Achievers in the ExploratoryDiscovery Group,
the VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory
ED overry 9.000 4.000 5.000
Discovery
Verification
Application 13.000 7.500 5.500
Combination 7.667 4.667 3.000
Table XXI displays the mean pretest scores and mean posttest
scores of intuitive lowachievers in the exploratory
discovery group, the verificationapplication group and the
combination group.
TABLE XXI:
Mean Error Scores for Intuitive LowAchievers
in the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group on the Pretest and Posttest
Pretest Posttest Difference
Exploratory 4.000 2.000 2.000
Discovery
erification 8.333 6.000 2.333
Application
Combination 9.000 4.500 4.500
H20. There is no significant difference between the mean
posttest score of highachieving sensing students
in the exploratorydiscovery group and the mean
posttest score of highachieving sensing students
in the verificationapplication group.
H21. There is no significant difference between the mean
posttest score of highachieving sensing students
in the exploratorydiscovery group and the mean
posttest score of highachieving sensing students
in the combination group.
H22. There is no significant difference between the mean
posttest score of highachieving sensing students
in the verificationapplication group and the mean
posttest score of highachieving sensing students
in the combination group.
H23. There is no significant difference between the mean
posttest score of averageachieving sensing students
in the exploratorydiscovery group and the mean
posttest score of averageachieving sensing students
in the verificationapplication group.
H24. There is no significant difference between the mean
posttest score of averageachieving sensing students
in the exploratorydiscovery group and the mean
posttest score of averageachieving sensing students
in the combination group.
H25. There is no significant difference between the mean
posttest score of averageachieving sensing students
in the verificationapplication group and the mean
posttest score of averageachieving sensing students
in the combination group.
H26. There is no significant difference between the mean
posttest score of lowachieving sensing students in
the exploratorydiscovery group and the mean post
test score of lowachieving sensing students in the
verificationapplication group.
H27. There is no significant difference between the mean
posttest score of lowachieving sensing students in
the exploratorydiscovery group and the mean post
test score of lowachieving sensing students in the
combination group.
H28. There is no significant difference between the mean
posttest score of lowachieving sensing students in
the verificationapplication group and the mean
posttest score of lowachieving sensing students in
the combination group.
H29. There is no significant difference between the mean
posttest score of highachieving intuitive students
in the exploratorydiscovery group and the mean
posttest score of highachieving intuitive students
in the verificationapplication group.
H30. There is no significant difference between the mean
posttest score of highachieving intuitive students
in the exploratorydiscovery group and the mean
posttest score of highachieving intuitive students
in the combination group.
H31. There is no significant difference between the mean
posttest score of highachieving intuitive students
in the verificationapplication group and the mean
posttest score of highachieving intuitive students
in the combination group.
H32. There is no significant difference between the mean
posttest score of averageachieving intuitive stu
dents in the exploratorydiscovery group and the
mean posttest score of averageachieving intuitive
students in the verificationapplication group.
H33. There is no significant difference between the mean
posttest score of averageachieving intuitive stu
dents in the exploratorydiscovery group and the
mean posttest score of averageachieving intuitive
students in the combination group.
H34. There is no significant difference between the mean
posttest score of averageachieving intuitive stu
dents in the verificationapplication group and the
mean posttest score of averageachieving intuitive
students in the combination group.
H35. There is no significant difference between the mean
posttest score of lowachieving intuitive students
in the exploratorydiscovery group and the mean
posttest score of lowachieving intuitive students
in the verificationapplication group.
H36. There is no significant difference between the mean
posttest score of lowachieving intuitive students
in the exploratorydiscovery group and the mean
posttest score of lowachieving intuitive students
in the combination group.
H37. There is no significant difference between the mean
posttest score of lowachieving intuitive students
in the verificationapplication group and the mean
posttest score of lowachieving intuitive students
in the combination group.
Hypotheses H20H22 state that there are no signifi
cant differences among the mean error scores on the posttest
of highachieving sensing subjects in the three laboratory
sequencing treatments. Hypotheses H23H25 assert that there
are no significant differences among the mean error scores
on the posttest of averageachieving sensing subjects in the
three treatment groups. Hypotheses H26H28 state that there
are no significant differences among the mean error scores
on the posttest of lowachieving sensing subjects in the
three treatment groups.
Hypotheses H29H31 state that there are no signifi
cant differences among the mean error scores on the posttest
of highachieving intuitive subjects in the three laboratory
sequencing treatments. Hypotheses H32H34 assert that there
are no significant differences among the mean error scores
on the posttest of averageachieving intuitive subjects in
the three treatment groups. Hypotheses H35H37 state that
there are no significant differences among the mean error
scores on the posttest of lowachieving intuitive subjects
in the three treatment groups.
The Fratios found in Table V indicate that the main
effect of personality is the only one which is significant
at the 0.05 level of confidence. The Fratio of 0.982 for
methodachievementpersonality interaction is less than that
required for significance at the 0.05 level of confidence.
Therefore, we cannot reject the hypothesis of no interaction.
In addition, none of the hypotheses H20H37 can be rejected.
Use of Scheffe's Method to compare the differences
between the posttest and pretest mean error scores for
sensing highachievers, indicates that the combination group
achieved a significantly greater reduction in rrean error
scores than the verificationapplication group. This is
shown by the 95 percent confidence intervals found in
Table XXII.
TABLE XXII: 95 Percent Confidence Intervals for
Comparisons Among the Sensing HighAchievers
in the ExploratoryDiscovery Group, the
VerificationApplication Group and the
Combination Group
Method Contrasts Confidence
Interval
ExploratoryDiscovery minus 0.972 1.392 to 3.336
VerificationApplication
ExploratoryDiscovery 1.214 3.083 to 0.655
minus Combination
VerificationApplication 2.186 4.290 to 0.082
minus Combination
Use of Scheffe's Method to compare the differences
between the posttest and pretest mean error scores for sensing
averageachievers indicates that the exploratorydiscovery
group achieved a significantly greater reduction in mean
error scores than either of the other groups. This is shown
by the 95 percent confidence intervals found in Table XXIII.
TABLE XXIII:
95 Percent Confidence Intervals for
Comparisons Among the Sensing Average
Achievers in the ExploratoryDiscovery
Group, the VerificationApplication Group
and the Combination Group
Method Contrasts Confidence
Interval
ExploratoryDiscovery minus
VerificationApplication
ExploratoryDiscovery
minus Combination
VerificationApplication
minus Combination
9.072
5.575 to 12.569
5.906 3.120 to 8.692
3.166 6.852 to 0.520
At the conclusion of the study each student was
asked to complete a questionnaire designed to measure his
reactions to the laboratory experience. These questionnaires
have been tabulated according to instructional treatment. In
Table XXIV is the tabulation for the exploratorydiscovery
group. In Table XXV, is the tabulation for the verification
application group while the tabulation for the combination
group is given in Table XXVI. A few selected comments by
students in each of these groups will be found in the next
three sections. The comments were also tabulated by instruc
tional treatment to give an indication of the frequency of
the various comments. These will be found in Tables XXVII,
XXVIII and XXIX. All three methods received both favorable
and unfavorable comments but the combination group seemed to
be the most popular.
Comments from the ExploratoryDiscoverv Group
1. "I really enjoyed the experiment because you see things
different after you learn it especially the ratios and
things like that ...."
2. "Excellent for students who have difficulty with theory."
3. "If someone is slow to grasp concepts, this method is
really hard to grasp. This method is fine for someone
who has a good background in something similar."
4. "Not enough time to complete all experiments."
5. "The experiments would have been good for a fourth grade
class. As a college course they were terribly BORING."
6. "When I don't understand I quit."
7. "I think I could have learnt more with the aid of an
instructor previewing the work."
Comments from the VerificationApplication Group
1. "The experiments were fun. It was like a learning game.
Took the boredom out of the classroom."
2. "I found doing the experiments fun, and learned a great
deal from doing them."
3. "It seems to be a pretty good method for teaching this
subject matter. It still could use some refinement."
4. "Experiments were well thoughtout. They seemed a little
lengthy, though and I feel like there didn't need to be
so many of them."
5. "I felt the experiments were too easy and the same thing
could have been taught quicker in a classroom lecture."
Comments for the Combination Group
1. "It seems to visualize math and make it more understanding.
Easier to handle and appreciate. It also gives the
student a chance to do work and exercise without fear
of failing a test, trying. We need this approach more."
2. "Very good. I wish I could of done this type learning
all the way through the math course."
3. "It beats listening to lectures type classes all period
and its easy to figure somethings out better on your
own."
4. "It all seemed too easy, more like a game than like math,
although it did get the point across rather well."
5. "I feel being able to 'stick your hands into it' teaches
you more than watching someone else 'have all the fun'
by being shown. I was very enthusiastic about it!!"
6. "Try something else."
7. "Most people were confused and turned off by the
experiments."
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TABLE XXVII: Summary of Comments
Discovery Group
for the Exploratory
Frequency Percent
Too easy and/or boring 4 12.9
Too difficult and/or not clear 7 22.6
Interesting and valuable 5 16.1
No comment 15 48.4
TABLE XXVIII: Summary of Comments for the Verification
Application Group
Frequency Percent
Too easy and/or boring 5 17.2
Too difficult and/or not clear 0 0.0
Interesting and valuable 5 17.2
No comment 19 65.6
TABLE XXIX: Summary of Comments for the Combination Group
Frequency Percent
Too easy and/or boring 3 7.1
Too difficult and/or not clear 4 9.5
Interesting and valuable 11 26.2
No comment 24 57.2
~
CHAPTER V
SUMItARY, CONCLUSIONS, LIMITATIONS,
AND IMPLICATIONS
Summary
The purpose of this study was to investigate the
relative effectiveness of the mathematics laboratory when
used in three different ways in conjunction with a tradi
tional lecturediscussion approach to teach community
college freshmen enrolled in a required mathematics course.
The laboratory experience was used as an introduction to a
topic, as a reinforcer, or as both. All subjects were
classified by achievement level and personality type, as
determined by the MyersBriggs Type Indicator, so that the
interaction of these factors with the various laboratory
approaches could be assessed. Previous research in scien
tific fields other than mathematics had studied the question
of sequencing the laboratory experience with traditional
teaching techniques but had found no significant results.
These studies had not, however, taken achievement and/or
personality into account. Because of the current popularity
of the laboratory approach to teaching mathematics at all
grade levels, it was decided to examine the issue of
sequencing at the college level.
Ratio and similarity was selected as the unit to be
taught using the combination of laboratory experiments and
class discussion. This topic was selected for two reasons.
First, several years of teaching had shown it to be a topic
which was not familiar to the vast majority of college
freshmen. Second, twenty tested and refined experiments on
this topic were available.
A pretestposttest was developed to determine a
subject's knowledge of ratio and similarity. This test was
submitted to a panel of judges for evaluation as to content
validity and appropriateness for the grade level. It was
also pilot tested on a class of freshmen mathematics stu
dents at Santa Fe Junior College. Subsequently, the pretest
and MyersBriggs Type Indicator were administered to seven
mathematics classes of college freshmen who had been
selected to participate in the study. These seven classes
were randomly assigned to one of three laboratory sequencing
patterns. Some groups received their laboratory experiences
before the class discussion; some received their laboratory
experience after the class discussion; and some received
laboratory experiences both before and after the class
discussion. All groups had the same length of time in the
laboratory. At the end of the unit, the test on ratio and
similarity was again administered. The error scores on the
pretest were used as a covariate while the posttest scores
served as the criterion variable. Prior to analyzing the
data, a college grade point average was determined for each
~
subject and used to classify him as a highachiever,
averageachiever, or lowachiever. Only those students who
had taken the pretest, posttest and MyersBriggs Type
Indicator were actually used in the study. There were 94
such students and these formed the sample for the study.
Each student was classified according to the method of in
struction he received, his achievement level and his per
sonality type as determined by the sensingintuition scale
of the MyersBriggs Type Indicator. This divided the sample
population into eighteen subcategories varying in size from
one to seventeen members.
The mean error scores for the eighteen groups were
compared in a 3x3x2 factorial design using multiple linear
regression techniques. A computer program called MANOVA was
used to perform an analysis of covariance. The calculated
Fratios were used in testing the 37 null hypotheses.
Scheffe's Method was used to determine the significance of
reductions in mean error scores.
Conclusions
The following conclusions may be drawn from the
study:
1. In a comparison of the exploratorydiscovery group, the
verificationapplication group and the combination group
there were no significant differences among the mean
posttest error scores. However, the exploratorydiscovery
group achieved a significantly greater reduction in mean
error score than either of the other groups.
2. In a comparison of sensing subjects to intuitive sub
jects, without regard to achievement level or method,
the sensing subjects did significantly better than the
intuitive students. This would imply that laboratory
experiences are more meaningful to sensing students than
to intuitive students.
3. In a comparison of the three methods of instruction
within the highachiever category without regard to
personality type, there were no significant differences
among the mean posttest error scores.
4. In a comparison of the three methods of instruction
within the averageachiever category without regard to
personality type, there were no significant differences
among the mean posttest error scores. The subjects in
the exploratorydiscovery group did, however, achieve
a significantly greater reduction in mean error score
than either of the other groups.
5. In a comparison of the three methods of instruction
within the lowachiever category without regard to per
sonality type, there were no significant differences
among the mean posttest error scores but the combination
group did attain a significantly greater reduction in
mean error score than the exploratorydiscovery group.
6. In a comparison of the three methods of instruction for
sensing subjects, there were no significant differences
~ ~
among the mean posttest error scores. However, subjects
in the exploratorydiscovery group achieved a signifi
cantly greater reduction in mean error score than either
of the other groups.
7. In a comparison of the three methods of instruction for
intuitive subjects, there were no significant
differences among the mean posttest error scores.
9. The interaction of method, achievement level and person
ality type did not make a statistically significant dif
ference in error scores on the posttest. The sensing
highachievers in the combination group achieved a sig
nificantly greater reduction in mean error score than
the verificationapplication group. The sensing average
achievers in the exploratorydiscovery group achieved
a significantly greater reduction in mean error score
than either of the other groups.
In general, the results of the present study support
the findings of Reuss (1970) and Emslie (1971). Although
there were no significant differences among the posttest
error scores for the three sequencing patterns, the signifi
cant reductions in mean error scores do have some educational
implications. Significant reductions in the mean error score
were most frequently attained by the exploratorydiscovery
group with the combination group a near second. The higher
percentage of favorable comments for the combination group
would suggest it is the better method in most situations.
In addition, this method allows for the greatest diversity
Among the students in the class. Those students who learn
well on their own have an opportunity to discover new con
cepts for themselves and to receive almost immediate confir
mation from the teacher. The additional laboratory exper
iences give further support to their findings. For those
students who have difficulty abstracting or generalizing, the
first laboratory experience may be frustrating and of limited
value. The laboratory experience after the class discussion
does, however, provide the opportunity to physically verify
what has been taught in the classroom.
Limitations
Some of the limitations of the study are as follows:
1. Each teacher did not use all three methods of
instruction. This means that teachermethod interaction
could not be checked. It was assumed in the present
study that the teacher effect would be nonsignificant.
2. The unit on ratio and similarity lasted for only two and
a half weeks. This may have effected the mean error
scores of lowachievers who possibly were slower in
adjusting to the new technique and in performing the
experiments.
3. The concentration of the research on a single unit of
study may mean that the results are not valid for a
different topic and/or longer periods of study.
4. The unusual grading system of the junior college used in
the study may have skewed the achievement level cate
gories so that some lowachievers were called average
achievers and some averageachievers were called high
achievers. This skewing may have affected the results
of the study within achievement levels.
5. The size of the sample population was smaller than
originally anticipated. This resulted in some mean
error scores being based on as few as one observation.
The results obtained cannot be safely generalized to
larger populations.
Implications
The present study contributes to the growing body of
information about the effectiveness of the mathematics labo
ratory as a mode of instruction. Although none of the
sequencing patterns was found to be more effective than the
others, the significant difference between the two person
ality types does have some implications for further research.
A study of personality interaction with a laboratory exper
ience using all sixteen categories obtained from the Myers
Briggs Type Indicator might prove informative. Further
research into the laboratory effectiveness within achieve
ment levels also should be undertaken with greater rigor,
than was possible in this study, on the definitions of high
achiever, averageachiever and lowachiever. As noted
earlier the combination method would seem to be the best
68
approach at present for all achievement levels and all
personality types.
Finally, it is hoped that this study will encourage
other studies dealing with the mathematics laboratory as an
adjunct to traditional modes of instruction at all levels of
education.
APPENDIX A
PERFORMANCE OBJECTIVES AND
PRETESTPOSTTEST
PERFORMANCE OBJECTIVES FOR RATIO AND SIMILARITY
1. The student should be able to write a ratio in the form
(a:b) to compare the cardinalities of two sets.
2. The student should be able to write a ratio in the form
(a:b) to compare the lengths of two line segments.
3. The student should be able to write an extended ratio
in the form (a:b:c) to compare the cardinality of three
sets.
4. The student should be able to write an extended ratio
in the form (a:b:c) to compare the lengths of three
line segments.
5. The student should be able to give two sets to illustrate
a particular ratio.
6. The student should be able to give two line segments to
illustrate a particular ratio.
7. The student should be able to give three sets to illus
trate a particular extended ratio.
8. The student should be able to give three line segments
to illustrate a particular extended ratio.
9. The student should be able to partition two sets and
write the resulting equivalent ratio.
10. The student should be able to divide two line segments
into a specified number of congruent pieces and write
the resulting equivalent ratio.
11. The student should be able to translate verbal ratios
into symbolic ratios of the form (a:b).
12. The student should be able to determine when two ratios
are equivalent.
13. The student should be able to write (a:b) = (c:d) in the
product form a x d = b x c.
14. The student should be able to supply the missing part
of a ratio needed to make two ratios equal.
Example: (a:_) = (c:d)
15. The student should be able to translate word problems
involving ratios into equations and solve them.
16. The student should be able to determine whether or not
two triangles are similar.
17. The student should be able to determine whether or not
two rectangles are similar.
18. The student should be able to construct a triangle which
is similar to a given triangle.
19. Given two similar figures, the student should be able to
find the constant of proportionality.
20. The student should be able to determine whether polygons
of more than four sides are similar or not.
PRETESTPOSTTEST
1. Write a ratio comparing the cardinality of
A = (1,4,9,103 to the cardinality of set B
in that order. Answer
2. Write a ratio comparing the length of line
line segment CD, in that order.
A 1 3/4 B
the set
= e2,5,6,7,i11
segment AB to
C 1 D
Answer
3. Write a ratio comparing the cardinality of
B = (1,4,6,9,10) and C = [1,3,4,5,8,9 in
Answer
A = f2,3,5.7
that order.
4. Write a ratio comparing the lengths of line segments AB,
CD and EF, in that order.
C 1 1/2 D
E 2 F
Answer
5. Give an example of two sets whose cardinalities are in
the ratio 7:5. Answer
6. Draw two line segments whose lengths are in the ratio
2:3. Answer
7. Give an example of three sets whose cardinalities are in
the ratio 2:3:5.
Answer A = B = C =
8. Draw three line segments whose lengths are in the ratio
1:2:3. Answer
9. Divide each of the following sets into equal subsets and
express the resulting equivalent ratio.
*
Answer
10. Divide each of the line segments in half and express the
resulting ratio.
Answer
A 1 B C 2 D
A 1 B
73
11. Write a ratio to represent "for every two sweaters there
are three skirts." Answer
12. Which of the following ratios is equivalent to 3:4?
635 4:3 6:8 6:7 Answer
13. Express the statement (5:7) = (10:14) as a product.
Answer
14. Fill in the blank to make the statement (3:5) = (_:15)
true. Answer
15. If a bicycle wheel makes 10 revolutions in going 33 feet
how far does it travel in making 13 1/3 revolutions?
Answer
16. Which of the following triangles is similar to
triangle A?
5 6
3 2
4 3 6 8
A B C D
Answer
17. Which of the following rectangles is similar to
rectangle A?
9
7
3 2
5 15 9 4
A B C D
Answer
18. Construct a triangle similar to the one at the left
with the indicated base line.
1 1 1/2
19. Given that the following two figures are similar, what
is the constant of proportionality when comparing the
left figure to the right figure?
9
6 6 6
3 2
Answer
20. Determine which of the
figure A?
6
3 2
1 8
2
following figures is similar to
E
Answer
APPENDIX B
EXPERIMENTS
EXPERIMENT 1
Problem: How do you use ratios to compare the numbers of
objects in two sets?
Materials: Envelope containing 15 paper clips, 18 nails,
24 match sticks, 6 beans, 9 triangles, 24 squares
Procedures:
Example: Refer to Figure 1. The two sets are compared
element to element. You can see that there are
more hexagons than squares.
00
LI 00
Fig. 1
Refer to Figure 2. The two sets are compared set
to set. There are squares and
hexagons. We can write the comparison in the form
(4:6). We call this comparison a ratio.
Fig. 2
Refer to Figure 3. Each set is separated into two
equivalent subsets. They are compared subset to
subset. The ratio (4:6) describes the comparison
by sets. The ratio (2:__) describes the
comparison by subsets.
Fig. 3
1. Complete each ratio. Watch the direction of the arrow
to see which set comes first.
a) (3 : )
A El 11
An
b) (__ :5)
OO => o
d)
2. In exercise 1 we compared (circle one)
a) the shapes of objects in two sets;
b) the sizes of objects in two sets;
c) the colors of objects in two sets;
d)
d0D
2. In exercise 1 we compared (circle one)
d) the shapnumbers of objects in two sets.
b) the sizes of objects in two sets;
c) the colors of objects in two sets;
d) the numbers of objects in two sets.
3. Write three ratios to compare the number of triangles
with the number of squares in three different ways.
Aaan
aAA
naA
aAA
EJ Ul0
El El El [D E
r3 ~ CC COOOU
C El
(__ :10)
(12: _)
The first ratio means there are twelve triangles for
every squares.
The second ratio means there are triangles for
every ten squares.
This ratio can also be written (3:5), and it means
4. Refer to the envelope of paper clips, nails, match
sticks, and beans. Write a ratio for each of the
following:
a) The number of paper clips to the number of nails
b) The number of match sticks to the number of beans
c) The number of squares to the number of triangles
5. Here is how one group of students worked exercise 4.
a) Tom wrote (15:18) to describe the ratio of the number
of paper clips to the number of nails.
Marcia wrote the ratio (5:6).
Paul wrote (18:15).
Which were right?
b) David wrote (6:24) to describe the ratio of the
number of match sticks to the number of beans.
Dana wrote the ratio (12:3) to describe the same
ratio.
Diane wrote (2416).
Which were right?
c) Three students wrote (3:8) to show the ratio of the
number of squares to the number of triangles. Were
they right?
Write two more ratios that show this comparison.
and
6. Write a ratio to compare the number of boys with the
number of girls in your class.
7. Write a ratio to represent each of the following
a) There are eighteen bicycles and thirty students.
b) There are two bicycles for every three students.
c) There are twice as many students as there are
bicycles.
From THE LABORATORY APPROACH TO MATHEMATICS by
Kenneth P. Kidd, Shirley Myers, and David M. Cilley.
copyright 1970, Science Research Associates, Inc.
Reproduced by permission of the publisher.
EXPERIMENT 2
Problem: How do you use ratios to compare lengths of
objects?
Materials: 12inch ruler, 1 red, 1 green, 1 yellow and
1 black stick, nail, clothespin, centimeter ruler
Procedures:
Example: Measure the red stick and the green stick in
inches. The red stick is inches long. The
green stick is inches long.
The ratio (_ :6) can be used to compare the
length of the red stick with that of the green
stick. The ratio (1:__) could also be used.
This means "one for every two." The red stick
has a length of 1 inch for every __ inches of
the green stick.
1. Measure each object and record the length.
a) red stick in.
b) green stick in.
c) yellow stick in.
d) black stick in.
e) nail ___ in.
f) clothespin in.
2. Write a ratio for each of the following:
a) The length of the yellow stick to the length of the
black stick.
b) The length of the nail to the length of the
clothespin.
c) The length of a table to its width.
d) Your height (inches) to your weight (pounds).
e) The length of the red stick to the length of the
yellow stick.
f) The length of the green stick to the length of the
black stick.
g) The length of the yellow stick to the length of the
green stick.
h) The length of the classroom to the width of the
classroom.
3. Pat's desk is 12 icecream sticks long and 8 icecream
sticks wide. Circle the ratios that can be used to
compare its length with its width.
a) (8:12)
b) (12:8)
c) (6:4)
d) (3:2)
e) (2:3)
f) (4:1)
4. Write a ratio for each statement.
a) A wall had 5 feet of width for every 2 feet of
height.
b) A dog was winning a tugofwar with a boy, since
there were 10 pounds of dog for every 8 pounds of
boy.
c) A very thin man weighs 130 pounds and is 74 inches
tall.
5. Refer to the line segments below. Measure each segment
using the centimeter ruler. Let each letter represent
the measure of the line segment. Complete the following
to indicate the ratios of these measures.
(2:3) = (f:
(5:2) = ( : )
(1:9) = (
(7:5) =
a
0c
f
From THE LABORATORY APPROACH TO MATHEMATICS by
Kenneth P. Kidd, Shirley iyers, and David M. Cilley.
copyright 1970, Science Research Associates, Inc.
Reproduced by permission of the publisher.
~I~
EXPERIMENT 3
Problem: How do you illustrate a given ratio?
Materials: 10 red cubes, 20 blue cubes, 5 beans, 25 match
sticks, and 12inch ruler
Procedures:
1. Suppose we wish to illustrate a matching of a set of
triangles with a set of squares that are in the ratio
(3:4). To do this, we might draw the following:
A n
Complete the following drawing to illustrate the same
ratio:
AAA
2. A ratio for the number of tables to the number of
students is (1:4). How many tables are there for 28
students? Therefore (1:4) = ( :28).
3. Place three red cubes and five blue cubes on the table.
A ratio for the number of red cubes to the number of
blue cubes is (__ ).
Add six red cubes for a total of nine. How many blue
cubes must you have so that there are three red cubes for
every five blue cubes? A ratio for the number
of red cubes to the number of blue cubes is (9:_ ).
We can say that (3:5) = (9: ).
4. Place four beans on the table. Place enough match
sticks on the table so that the ratio of the number of
beans to the number of match sticks is (1:6) =
5. Draw line segments AB and CD so that a ratio for the
length of segment AB to the length of segment CD is
(2:3). If segment AB is four inches long, how long
would CD be?
6. Draw segment GH so that a ratio of the length of
segment EF (below) to the length of segment GH is (1:2).
E F
Segment EF must have one unit of length for every
units of length of segment GH.
7. Draw segment JK so that a ratio for the length of
segment JK to the length of segment LM (below) is (2:3).
From THE LABORATORY APPROACH TO MATHEMATICS by
Kenneth P. Kidd, Shirley Myers, and David M. Cilley.
copyright 1970, Science Research Associates, Inc.
Reproduced by permission of the publisher.
EXPERIMENT 4
Problem: How are ratios used to make comparisons?
Materials: Envelope containing 8 red squares and 12 blue
squares, egg beater
Procedures:
1. In the envelope there are eight red squares and
blue squares. The ratio of the number of red squares to
the number of blue squares can be written as (8:12).
Red El El E Blue
El O E O El
One student shows that there are four red squares for
every six blue squares. He uses the ratio (4:6). A
second student prefers to match two red squares with
blue squares. His ratio is (2:_ ). The set
of red squares can be compared with the set of blue
squares by any of these ratios: (8:12), (4:_),
(__' 3).
First student's comparison
El .D DD L07
Second student's comparison
The ratio (2:3) means 
a) There are __ red squares for every __ blue
squares.
b) There are twothirds as many red squares as
squares.
c) For every __ red squares there are blue
squares.
2. Turn the handle of the egg beater. While the handle
makes one turn, the beater makes turns. While
the handle makes two turns, the beater makes
turns. We can use the ratio to show that for
every turn of the handle the beater makes
turns. The ratio (2:__ ) could also be used.
From THE LABORATORY APPROACH TO MATHEMATICS by
Kenneth P. Kidd, Shirley Myers, and David M. Cilley.
copyright 1970, Science Research Associates, Inc.
Reproduced by permission of the publisher.
EXPERIMENT 5
Problem: How do you use extended ratios to compare the
numbers of objects in three sets or to compare
three measures?
Materials:
3 green sticks, 3 blue sticks, 9 red cubes,
15 green cubes, 21 blue cubes, 12inch ruler,
and masking tape
Procedures:
1. An extended ratio for the number of triangles to the
number of squares to the number of circles, as pictured
below, is (2:3:5).
A A > 000
AAO o00
Finish this drawing so that the number of triangles in
the first rectangle is to the number of squares in the
second rectangle is to the number of circles in the
third rectangle as (2:3:5).
A A n
AA
Finish this drawing so that it also shows the extended
ratio (2:3:5).
In each of the drawings above, for every two triangles
there are three squares and five circles. Make another
drawing, different from the first three, in which the
comparison of triangles with squares with circles is
(2:3:5).
>L
2. Complete the following extended ratio to show the
comparison of the number of red, green and blue cubes:
(3: ).
3. The three green sticks have lengths of 3 inches, 4 inches
and 5 inches. The extended ratio that compares their
lengths (from shortest to longest) is (3: : ). The
lengths of the three blue sticks are __ inches,
inches and inches. The extended ratio that
compares their lengths (from shortest to longest) is
(6: : ).
4. Tape the green sticks together
(as in Figure 4) to form a
triangle. Do the same with the
blue sticks. Do you notice
anything about the two triangles?
Figure 4
5. Dave plans to cut three red sticks so that their lengths
can be compared by the extended ratio (3:4:5). If he
cuts the shortest stick 9 inches long, how long should
he make the other two? and
6. Sam Gravelcement makes concrete by mixing cement, sand,
and gravel in the extended ratio (1:2:3). This means
that for every measure of cement he uses measures
of sand and measures of gravel. He has placed
four buckets of cement in a cement mixer. How many
buckets of sand must he add? How many buckets
of gravel? Complete this proportionality so that
it describes Sam's mixture: (1:2:3) = (4: __).
From THE LABORATORY APPROACH TO MATHEMATICS by
Kenneth P. Kidd, Shirley Myers, and David M. Cilley.
copyright 1970, Science Research Associates, Inc.
Reproduced by permission of the publisher.
EXPERIMENT 6
Problem: How do you write equal ratios and extended ratios?
Materials: 18 triangles, 12 small squares, 16 large squares,
4 beans, and 24 match sticks
Procedure:
1. Place eight triangles and twelve small squares on the
table. What is the ratio of the number of triangles to
the number of squares?
2. Place the eight triangles and twelve squares in two
equal piles on the table. (One pile should contain the
same number of squares and the same number of triangles
as the other pile.) For every four triangles there are
__ squares. This ratio can be written
3. Next make four equal piles of the eight triangles and
twelve squares. In each pile place two triangles and
Squares. For every two triangles there are
Squares. This ratio can be written
4. You should have written three different ratios. Each
ratio represents the same comparison of the number of
triangles with the number of squares. Therefore these
ratios are equal. That is, (8:_) = (_;6) = _
An equation relating two ratios is called a proportion.
5. Place sixteen large squares on the table. Pretend they
are sandwiches to be divided among four hungry students.
A ratio for the number of sandwiches to the number of
students is Divide the sandwiches evenly
among the four students. There are ____ sandwiches
for each student. This ratio can be written ( l1).
So, (16:4) = ( :1).
6. Place one bean and six match sticks on the table. Six
match sticks are needed for each bean. The ratio of the
number of match sticks to the number of beans must be
(6:_). Place three more beans on the table with as
many match sticks as are needed. Since we will need
Match sticks for the four beans, (6:1) = ( :4).
7. Place eight triangles, twelve small squares and sixteen
large squares in a pile on the table. The extended
ratio that compares triangles with small squares with
large squares is (_:12:_).
Now make two equal piles. Place four triangles,
small squares, and eight large squares in each pile.
