Group Title: investigation of the relative effectiveness of three methods of utilizing laboratory activities in selected topics of junior college mathematics
Title: An investigation of the relative effectiveness of three methods of utilizing laboratory activities in selected topics of junior college mathematics
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Title: An investigation of the relative effectiveness of three methods of utilizing laboratory activities in selected topics of junior college mathematics
Physical Description: xii, 122 leaves. : illus. ; 28 cm.
Language: English
Creator: Golliday, Joan Maries, 1942-
Publication Date: 1974
Copyright Date: 1974
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Subject: Mathematics -- Study and teaching   ( lcsh )
Computation laboratories   ( lcsh )
Curriculum and Instruction thesis Ph. D
Dissertations, Academic -- Curriculum and Instruction -- UF
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non-fiction   ( marcgt )
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Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 118-120.
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
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Resource Identifier: alephbibnum - 000869068
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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

Inductive-Deductive Studies ............... 11
Laboratory Studies ........................ 15
Summary .................................. 19

III. THE EXPERIMENTAL DESIGN ....................... 21

Statement of Hypotheses ................... 21
Description of Procedures and Design ...... 26
Instrumentation ......................... ... 31
Myers-Briggs Type Indicator .......... 31
Pretest-Posttest ..................... 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
Pretest-Posttest .................... 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
Exploratory-Discovery Group, the
Verification-Application Group and
Combination Group on the Pretest and
Posttest 35

V Analysis of Covariance 36

VI 95 Percent Confidence Intervals for
Comparisons Among the Exploratory-Discovery
Group, the Verification-Application 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 High-Achievers in the
Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group on the Pretest and
Posttest 39

IX Mean Error Scores for Average-Achievers in
the Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group on the Pretest and
Posttest 40









LIST OF TABLES -- (Continued)


Ta ble Page

X Mean Error Scores for Low-Achievers in the
Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group on the Pretest and
Posttest 40

XI 95 Percent Confidence Intervals for
Comparisons Among the Average-Achievers in
the Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group 43

XII 95 Percent Confidence Intervals for
Comparisons Among the Low-Achievers in the
Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group 43

XIII Mean Error Scores for Sensing Subjects in
the Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group on the Pretest and
Posttest 44

XIV Mean Error Scores for Intuitive Subjects
in the Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group on the Pretest and
Posttest 45

XV 95 Percent Confidence Intervals for
Comparisons Among the Sensing Subjects in
the Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group 47

XVI Mean Error Scores for Sensing High-
Achievers in the Exploratory-Discovery
Group, the Verification-Application Group
and the Combination Group on the Pretest
and Posttest 47

XVII Mean Error Scores for Sensing Average-
Achievers in the Exploratory-Discovery
Group, the Verification-Application Group
and the Combination Group on the Pretest
and Posttest 48









LIST OF TABLES -- (Continued)


Table Page

XVIII Mean Error Scores for Sensing Low-Achievers
in the Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group on the Pretest and
Posttest 48

XIX Mean Error Scores for Intuitive High-
Achievers in the Exploratory-Discovery
Group, the Verification-Application Group
and the Combination Group on the Pretest
and Posttest 49

XX Mean Error Scores for Intuitive Average-
Achievers in the Exploratory-Discovery
Group, the Verification-Application Group
and the Combination Group on the Pretest
and Posttest 49

XXI Mean Error Scores for Intuitive Low-
Achievers in the ExpLoratory-Discovery
Group, the Verification-Application 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 Exploratory-Discovery
Group, the Verification-Application Group
and the Combination Group 54

XXIII 95 Percent Confidence Intervals for
Comparisons Among the Sensing Average-
Achievers in the Exploratory-Discovery
Group, the Verification-Application 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 lecture-discussion 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 Myers-Briggs 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









exploratory-discovery group, received sixty minutes of labo-

ratory experiences followed by thirty minutes of discussion.

The second group, known as the verification-application

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 exploratory-discovery group attained a

significantly greater reduction than the other two groups.

Within the categories of average-achievers and sensing

students, the exploratory-discovery group also achieved a


~ ~









significantly greater reduction than the other two groups.

Low-achievers in the combination group achieved a signifi-

cantly greater reduction in mean error score than those in

the exploratory-discovery group. Finally, sensing high-

achievers in the combination group achieved a significantly

greater reduction in mean error score than those in the

verification-application 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 laboratory-type 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 multi-sensory aids were used by the

teachers to demonstrate principles which they expounded

rather than as a means of hands-on 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 lecture-discussion 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 lecture-discussion 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

lecture-discussion 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 Myers-Briggs 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 high-achieving college freshmen studying ratio and

similarity perform significantly better under any one of

the three sequencing arrangements?

3. Do average-achieving college freshmen studying ratio and

similarity perform significantly better under any one of

the three sequencing arrangements?

4. Do low-achieving 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 Myers-Briggs 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 Pyers-Briggs






5


Type Indicator perform significantly better under any

one of the three sequencing arrangements when studying

ratio and similarity?

7. Do high-achieving 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 high-achieving 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 average-achieving 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 average-achieving 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 low-achieving 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 low-achieving 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 high-achievers,

sensing average-achievers, sensing low-achievers, intuitive

high-achievers, intuitive average-achievers, or intuitive

low-achievers on the basis of their college grade point

average and the Myers-Briggs 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 Myers-Briggs Type
Indicator score.

Intuitive Subject: a subject who has been classified as an
intuitive personality on the basis of his Myers-Briggs Type
Indicator score.

High-Achiever: a subject whose grade point average at his
current community college is greater than or equal to 3.35.

Average-Achiever: a subject whose grade point average at
his current community college is between 2.65 and 3.35.

Low-Achiever: a subject whose grade point average at his
current community college is less than or equal to 2.65.

Exploratory-Discovery 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.

Verification-Application 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 team-teaching, 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 lecture-discussion 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 lecture-discussion?

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

lecture-discussion 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 Myers-Briggs 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 inductive-deductive 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.



Inductive-Deductive 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 ninth-grade 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 problem-solving and rule-stating 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 high-intelligence group completed their work in

significantly less time than the low-intelligence 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 ninth-grade students enrolled in a general

science course. These subjects were randomly assigned to

three groups. One group received materials programmed in

an expository-deductive format. The second group received

materials programmed in a discovery-inductive 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

teacher-textbook 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 above-average ability, but

with below-average achievement were used. One group was

taught fraction concepts and computation with fractions

using the traditional textbook-discussion approach. The









second group was taught the same material in a laboratory

setting using manipulative devices and multi-sensory

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 low-achiever 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 re-encounter 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 college-level 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 either-or 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 laboratory-theory sequence while Method II

was a theory-laboratory 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 laboratory-theory 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 lecture-discussion 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 lecture-discussion 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
exploratory-discovery group and the mean posttest
score of students in the verification-application
group.









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 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 high-achieving students in
the exploratory-discovery group and the mean post-
test score of high-achieving students in the
verification-application group.

H6. There is no significant difference between the
mean posttest score of high-achieving students in
the exploratory-discovery group and the mean post-
test score of high-achieving students in the
combination group.

H7. There is no significant difference between the
mean posttest score of high-achieving students in
the verification-application group and the mean
posttest score of high-achieving students in the
combination group.

H8. There is no significant difference between the
mean posttest score of average-achieving students
in the exploratory-discovery group and the mean
posttest score of average-achieving students in
the verification-application group.

H9. There is no significant difference between the
mean posttest score of average-achieving students
in the exploratory-discovery group and the mean
posttest score of average-achieving students in
the combination group.

H10. There is no significant difference between the
mean posttest score of average-achieving students
in the verification-application group and the
mean posttest score of average-achieving students
in the combination group.

H11. There is no significant difference between the
mean posttest score of low-achieving students in
the exploratory-discovery group and the mean









posttest score of low-achieving students in the
verification-application group.

H12. There is no significant difference between the
mean posttest score of low-achieving students in
the exploratory-discovery group and the mean
posttest score of low-achieving students in the
combination group.

H13. There is no significant difference between the
mean posttest score of low-achieving students in
the verification-application group and the mean
posttest score of low-achieving students in the
combination group.

H14. There is no significant difference between the
mean posttest score of sensing students in the
exploratory-discovery 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
exploratory-discovery 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
verification-application 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
exploratory-discovery 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
exploratory-discovery 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
verification-application 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 high-achieving sensing









students in the exploratory-discovery group and
the mean posttest score of high-achieving sensing
students in the verification-application group.

H21. There is no significant difference between the
mean posttest score of high-achieving sensing
students in the exploratory-discovery group and
the mean posttest score of high-achieving sensing
students in the combination group.

H22. There is no significant difference between the
mean posttest score of high-achieving sensing
students in the verification-application group
and the mean posttest score of high-achieving
sensing students in the combination group.

H23. There is no significant difference between the
mean posttest score of average-achieving sensing
students in the exploratory-discovery group and
the mean posttest score of average-achieving
sensing students in the verification-application
group.

H24. There is no significant difference between the
mean posttest score of average-achieving sensing
students in the exploratory-discovery group and
the mean posttest score of average-achieving
sensing students in the combination group.

H25. There is no significant difference between the
mean posttest score of average-achieving sensing
students in the verification-application group
and the mean posttest score of average-achieving
sensing students in the combination group.

H26. There is no significant difference between the
mean posttest score of low-achieving sensing
students in the exploratory-discovery group and
the mean posttest score of low-achieving sensing
students in the verification-application group.

H27. There is no significant difference between the
mean posttest score of low-achieving sensing
students in the exploratory-discovery group and
the mean posttest score of low-achieving sensing
students in the combination group.

H28. There is no significant difference between the
mean posttest score of low-achieving sensing
students in the verification-application group
and the mean posttest score of low-achieving
sensing students in the combination group.









H29. There is no significant difference between the
mean posttest score of high-achieving intuitive
students in the exploratory-discovery group and
the mean posttest score of high-achieving intui-
tive students in the verification-application
group.

H30. There is no significant difference between the
mean posttest score of high-achieving intuitive
students in the exploratory-discovery group and
the mean posttest score of high-achieving
intuitive students in the combination group.

H31. There is no significant difference between the
mean posttest score of high-achieving intuitive
students in the verification-application group
and the mean posttest score of high-achieving
intuitive students in the combination group.

H32. There is no significant difference between the
mean posttest score of average-achieving intui-
tive students in the exploratory-discovery group
and the mean posttest score of average-achieving
intuitive students in the verification-application
group.

H33. There is no significant difference between the
mean posttest score of average-achieving intui-
tive students in the exploratory-discovery group
and the mean posttest score of average-achieving
intuitive students in the combination group.

H34. There is no significant difference between the
mean posttest score of average-achieving intui-
tive students in the verification-application
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 low-achieving intuitive
students in the exploratory-discovery group and
the mean posttest score of low-achieving intuitive
students in the verification-application group.

H36. There is no significant difference between the
mean posttest score of low-achieving intuitive
students in the exploratory-discovery group and
the mean posttest score of low-achieving intuitive
students in the combination group.

H37. There is no significant difference between the
mean posttest score of low-achieving intuitive









students in the verification-application group
and the mean posttest score of low-achieving
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 (exploratory-discovery, verification-application,

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

exploratory-discovery method. The laboratory experience

consisted of a series of guided experiments on ratio and

similarity. The exploratory-discovery 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 verification-application 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

Myers-Briggs Type Indicator and a pretest on ratio and simi-

larity to determine their background knowledge on this topic.

(The pretest-posttest 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 Myers-Briggs 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 average-achiever. If the GPA was less than or

equal to 2.65, he was classified a low-achiever. 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, thinking-feeling, sensing-intuition, and

extraversion-introversion. For the purposes of this study,

only the sensing-intuition 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

High-Achiever 26 27 53

Average-Achiever 12 11 23

Low-Achiever 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.

Myers-Briggs 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 9-16). This is a









forced choice, self-report 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

fifty-five minutes to complete.

The test purports to measure the following four

dichotomous dimensions: judgment-perception, thinking-

feeling, sensing-intuition, and extraversion-introversion.

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 Myers-Bri=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 sensing-intuition

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 Myers-Briggs 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, split-half









reliability coefficients, corrected by the Spearman-Brown

prophecy formula, were calculated for each dimension at dif-

ferent grade levels. The sensing-intuition scale had a

reliability coefficient of 0.87 for college students.


Pretest-Posttest

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 pretest-posttest 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 F-statistics 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 F-values 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 exploratory-discovery 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
Exploratory-Discovery 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 verification-application 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 H1-H3 state that there are no differences

among the mean error scores of subjects in the exploratory-

discovery group, the verification-application 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 F-ratio

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 H1-H3 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 Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group


Method Contrasts Confidence
Interval

Exploratory-Discovery minus 1.660 0.632 to 2.688
Verification-Application

Exploratory-Discovery
minus Combination 1.223 0.257 to 2.189

Verification-Application 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 F-ratio 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 H5-H13 involve the comparison of the three

methods of instruction within achievement levels. Table VIII

shows the mean pretest scores and mean posttest scores for

high-achievers in the exploratory-discovery group, the

verification-application group and the combination group.


TABLE VIII: Mean Error Scores for High-Achievers in the
Exploratory-Discovery Group, the
Verification-Application 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 exploratory-discovery group, the verification-

application group and the combination group may be found in

Table IX.









TABLE IX: Mean Error Scores for Average-Achievers in the
Exploratory-Discovery 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 low-achievers in the exploratory-discovery group,

the verification-application group and the combination group.


TABLE X: Mean Error Scores for Low-Achievers in the
Exploratory-Discovery 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 high-achieving students in the
exploratory-discovery group and the mean posttest
score of high-achieving students in the verification-
application group.

H6. There is no significant difference between the mean
posttest score of high-achieving students in the









exploratory-discovery group and the mean posttest
score of high-achieving students in the combination
group.

H7. There is no significant difference between the mean
posttest score of high-achieving students in the
verification-application group and the mean posttest
score of high-achieving students in the combination
group.

H8. There is no significant difference between the mean
posttest score of average-achieving students in the
exploratory-discovery group and the mean posttest
score of average-achieving students in the
verification-application group.

H9. There is no significant difference between the mean
posttest score of average-achieving students in the
exploratory-discovery group and the mean posttest
score of average-achieving students in the
combination group.

H10. There is no significant difference between the mean
posttest score of average-achieving students in the
verification-application group and the mean posttest
score of average-achieving students in the
combination group.

H11. There is no significant difference between the mean
posttest score of low-achieving students in the
exploratory-discovery group and the mean posttest
score of low-achieving students in the verification-
application group.

H12. There is no significant difference between the mean
posttest score of low-achieving students in the
exploratory-discovery group and the mean posttest
score of low-achieving students in the combination
group.

H13. There is no significant difference between the mean
posttest score of low-achieving students in the
verification-application group and the mean posttest
score of low-achieving students in the combination
group.

Hypotheses H5-H7 state that there are no significant

differences among the mean error scores on the posttest of

high-achievers in the three laboratory sequencing treatments.

Hypotheses H8-H10 state that there are no significant









differences among the mean error scores on the posttest of

average-achievers in the three treatment groups. Hypotheses

H11-H13 state that there are no significant differences

among the mean error scores on the posttest of low-achievers

in the three treatment groups. The F-values found in

Table V indicate that both main effects are not significant

at the 0.05 level of confidence. The F-ratio of 1.501 for

method-achievement 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 H5-H13.

Use of Scheffe's Method to compare the differences

between the posttest and pretest mean error scores for

average-achievers indicates that the exploratory-discovery

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 Average-Achievers in the Exploratory-
Discovery Group, the Verification-Application
Group and the Combination Group


Method Contrasts Confidence
Interval

Exploratory-Discovery minus 4.800 2.411 to 7.189
Verification-Application

Exploratory-Discovery 4.224 2.369 to 6.079
minus Combination

Verification-Application -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 exploratory-discovery group. This is shown by the 95

percent confidence intervals found in Table XII.


TABLE XII:


95 Percent Confidence Intervals for
Comparisons Among the Low-Achievers in the
Exploratory-Discovery Group, the Verification-
Application Group and the Combination Group


Method Contrasts Confidence
Interval

Exploratory-Discovery minus -1.178 -3.431 to 1.074
Verification-Application

Exploratory-Discovery -2.900 -5.609 to -0.191
minus Combination

Verification-Application -2.222 -4.648 to 0.204
minus Combination









Hypotheses H14-H19 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 exploratory-discovery

group, the verification-application group and the combination

group.


TABLE XIII: Mean Error Scores for Sensing Subjects in
the Exploratory-Discovery Group, the
Verification-Application 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 exploratory-discovery group, the

verification-application group and the combination group may

be found in Table XIV.


~









TABLE XIV: Mean Error Scores for Intuitive Subjects in
the Exploratory-Discovery Group, the
Verification-Application 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
exploratory-discovery 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
exploratory-discovery 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
verification-application 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
exploratory-discovery 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
exploratory-discovery 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
verification-application group and the mean posttest
score of intuitive students in the combination
group.









Hypotheses H14-HI6 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 H1117-H19 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 F-ratios found in Table V indicate that only the main

effect of personality is significant at the 0.05 level of

confidence. The F-ratio of 0.578 for method-personality

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 H14-H19.

Use of Scheffe's Method to compare the difference

between the posttest and pretest mean error scores for

sensing students indicates that the exploratory-discovery

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 Verification-Application
Group and the Combination Group


Method Contrasts Confidence
Interval

Exploratory-Discovery minus 2.367 0.660 to 4.074
Verification-Application

Exploratory-Discovery 1.444 0.116 to 2.772
minus Combination

Verification-Application -0.923 -2.376 to 0.530
minus Combination



Hypotheses H20-H37 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 exploratory-discovery group, the

verification-application group and the combination group.


TABLE XVI:


Mean Error Scores for Sensing High-Achievers
in the Exploratory-Discovery Group, the
Verification-Application 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

average-achievers in the exploratory-discovery group, the

verification-application group and the combination group may

be found in Table XVII.


TABLE XVII:


Mean Error Scores for Sensing Average-
Achievers in the Exploratory-Discovery Group,
the Verification-Application 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 low-achievers in the exploratory-

discovery group, the verification-application group and the

combination group.


TABLE XVIII: Mean Error Scores for Sensing Low-Achievers
in the Exploratory-Discovery Group, the
Verification-Application 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 high-achievers in the exploratory-

discovery group, the verification-application group and the

combination group.


TABLE XIX:


Mean Error Scores for Intuitive high-Achievers
in the Exploratory-Discovery Group, the
Verification-Application 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 average-achievers in the exploratory-discovery group,

the verification-application group and the combination group

may be found in Table XX.


TABLE XX: Mean Error Scores for Intuitive Average-
Achievers in the Exploratory-Discovery Group,
the Verification-Application 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 low-achievers in the exploratory-

discovery group, the verification-application group and the

combination group.


TABLE XXI:


Mean Error Scores for Intuitive Low-Achievers
in the Exploratory-Discovery Group, the
Verification-Application 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 high-achieving sensing students
in the exploratory-discovery group and the mean
posttest score of high-achieving sensing students
in the verification-application group.

H21. There is no significant difference between the mean
posttest score of high-achieving sensing students
in the exploratory-discovery group and the mean
posttest score of high-achieving sensing students
in the combination group.

H22. There is no significant difference between the mean
posttest score of high-achieving sensing students
in the verification-application group and the mean
posttest score of high-achieving sensing students
in the combination group.

H23. There is no significant difference between the mean
posttest score of average-achieving sensing students
in the exploratory-discovery group and the mean
posttest score of average-achieving sensing students
in the verification-application group.

H24. There is no significant difference between the mean









posttest score of average-achieving sensing students
in the exploratory-discovery group and the mean
posttest score of average-achieving sensing students
in the combination group.

H25. There is no significant difference between the mean
posttest score of average-achieving sensing students
in the verification-application group and the mean
posttest score of average-achieving sensing students
in the combination group.

H26. There is no significant difference between the mean
posttest score of low-achieving sensing students in
the exploratory-discovery group and the mean post-
test score of low-achieving sensing students in the
verification-application group.

H27. There is no significant difference between the mean
posttest score of low-achieving sensing students in
the exploratory-discovery group and the mean post-
test score of low-achieving sensing students in the
combination group.

H28. There is no significant difference between the mean
posttest score of low-achieving sensing students in
the verification-application group and the mean
posttest score of low-achieving sensing students in
the combination group.

H29. There is no significant difference between the mean
posttest score of high-achieving intuitive students
in the exploratory-discovery group and the mean
posttest score of high-achieving intuitive students
in the verification-application group.

H30. There is no significant difference between the mean
posttest score of high-achieving intuitive students
in the exploratory-discovery group and the mean
posttest score of high-achieving intuitive students
in the combination group.

H31. There is no significant difference between the mean
posttest score of high-achieving intuitive students
in the verification-application group and the mean
posttest score of high-achieving intuitive students
in the combination group.

H32. There is no significant difference between the mean
posttest score of average-achieving intuitive stu-
dents in the exploratory-discovery group and the
mean posttest score of average-achieving intuitive
students in the verification-application group.









H33. There is no significant difference between the mean
posttest score of average-achieving intuitive stu-
dents in the exploratory-discovery group and the
mean posttest score of average-achieving intuitive
students in the combination group.

H34. There is no significant difference between the mean
posttest score of average-achieving intuitive stu-
dents in the verification-application 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 low-achieving intuitive students
in the exploratory-discovery group and the mean
posttest score of low-achieving intuitive students
in the verification-application group.

H36. There is no significant difference between the mean
posttest score of low-achieving intuitive students
in the exploratory-discovery group and the mean
posttest score of low-achieving intuitive students
in the combination group.

H37. There is no significant difference between the mean
posttest score of low-achieving intuitive students
in the verification-application group and the mean
posttest score of low-achieving intuitive students
in the combination group.

Hypotheses H20-H22 state that there are no signifi-

cant differences among the mean error scores on the posttest

of high-achieving sensing subjects in the three laboratory

sequencing treatments. Hypotheses H23-H25 assert that there

are no significant differences among the mean error scores

on the posttest of average-achieving sensing subjects in the

three treatment groups. Hypotheses H26-H28 state that there

are no significant differences among the mean error scores

on the posttest of low-achieving sensing subjects in the

three treatment groups.

Hypotheses H29-H31 state that there are no signifi-

cant differences among the mean error scores on the posttest









of high-achieving intuitive subjects in the three laboratory

sequencing treatments. Hypotheses H32-H34 assert that there

are no significant differences among the mean error scores

on the posttest of average-achieving intuitive subjects in

the three treatment groups. Hypotheses H35-H37 state that

there are no significant differences among the mean error

scores on the posttest of low-achieving intuitive subjects

in the three treatment groups.

The F-ratios 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 F-ratio of 0.982 for

method-achievement-personality 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 H20-H37 can be rejected.

Use of Scheffe's Method to compare the differences

between the posttest and pretest mean error scores for

sensing high-achievers, indicates that the combination group

achieved a significantly greater reduction in rrean error

scores than the verification-application 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 High-Achievers
in the Exploratory-Discovery Group, the
Verification-Application Group and the
Combination Group


Method Contrasts Confidence
Interval

Exploratory-Discovery minus 0.972 -1.392 to 3.336
Verification-Application

Exploratory-Discovery -1.214 -3.083 to 0.655
minus Combination

Verification-Application -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

average-achievers indicates that the exploratory-discovery

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 Exploratory-Discovery
Group, the Verification-Application Group
and the Combination Group


Method Contrasts Confidence
Interval


Exploratory-Discovery minus
Verification-Application

Exploratory-Discovery
minus Combination

Verification-Application
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 exploratory-discovery

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 Exploratory-Discoverv 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 Verification-Application 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 thought-out. 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 lecture-discussion 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 Myers-Briggs 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 pretest-posttest 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 Myers-Briggs 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 high-achiever,

average-achiever, or low-achiever. Only those students who

had taken the pretest, posttest and Myers-Briggs 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 sensing-intuition scale

of the Myers-Briggs 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

F-ratios 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 exploratory-discovery group, the

verification-application group and the combination group

there were no significant differences among the mean

posttest error scores. However, the exploratory-discovery










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 high-achiever 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 average-achiever category without regard to

personality type, there were no significant differences

among the mean posttest error scores. The subjects in

the exploratory-discovery 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 low-achiever 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 exploratory-discovery 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 exploratory-discovery 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

high-achievers in the combination group achieved a sig-

nificantly greater reduction in mean error score than

the verification-application group. The sensing average-

achievers in the exploratory-discovery 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 exploratory-discovery

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 teacher-method 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 low-achievers 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 low-achievers were called average-

achievers and some average-achievers 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, average-achiever and low-achiever. 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
PRETEST-POSTTEST









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.









PRETEST-POSTTEST


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: 12-inch 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 ice-cream sticks long and 8 ice-cream
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 tug-of-war 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 12-inch 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 two-thirds 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, 12-inch 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.




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