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PROBABILITY: SEX AND GRADE LEVEL DIFFERENCES AND
THE EFFECT OF INSTRUCTION ON ThE PERFORMANCE
AND ATTITUDES OF MIDDLE SCHOOL
BOYS AND GIRLS
BY

Zacchaeus Kunle Oguntebi

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment or the requirements
for the degree of

DOCTOR OF PHILOSOPHY

College of Education

1983

r¥1'3985‘

ABSTRACT

PROBABILITY: SEX AND GRADE LEVEL DIFFERENCES AND
THE EFFECT OF INSTRUCTION ON THE PERFORMANCE
AND ATTITUDES OF MIDDLE SCHOOL
BOYS AND GIRLS

BY

Zacchaeus Kunle Uguntebi

Purpose

This study had four related purposes. The first
purpose was to determine existing differences in probability
knowledge and in attitudes toward mathematics of grades six
through eight students by sex and grade prior to probability
intervention. The other purposes were to analyze the
effects of instruction on probability skill develOpment, on
attitudes toward mathematics, and toward probability, by sex

and grade.

Methodology
The probability intervention and data collection took
Place during Fall 1982 and Winter 1983. About 1460 sixth
through eighth graders, from three sites (urban, suburban
and rural) in and around Lansing and Pontiac, Michigan,

participated in the entire study.

Zacchaeus kunle Oguntebi

The instruments used included the Mathematics Attitude
Scale (MAS), Probability Attitude Scale (PAS) and a
Probability Tesc (PT). MAS and PT were pre and posttest
measures while PAS was posttest only. PT was a 25-item test
while MAS.and PAS were similar six-item bipolar semantic
differentials, with high Cronbach a reliability
coefficients. The probability instruction material
contained ten sequenced activities requiring about three
weeks to cover. The statistical analyses included
multivariate and univariate analysis of variance and

repeated measures.

Major Results

Prior to instruction, there were (1) no sex or site
differences in attitudes toward mathematics, but boys
outperformed girls in probability performance. (2) grade
differences in probability performance (increasing with age)
and in mathematics attitudes (decreasing with age), with
slight variations.

After instruction: (1) In all grade levels and sites,
boys and girls benefited significantly from the interven-
tion. (2) While seventh graders tepped the grades, there
were no site or sex differences in probability knowledge
gains, (in spite of boys' slight superiority in both pretest
and posttest scores). In the suburban site, girls slightly

but consistently outgained boys. (3) Attitudes to

 

Zacchaeus Kunle Uguntebi
mathematics declined slightly over the period but these were
not meaningfully significant. (4) There were no site, sex,
or grade differences in attitude change toward mathematics.
(5) Boys and girls did not disagree in attitudes toward
probability and mathematics. (6) Seventh graders had more

favorable attitudes to probability than the other grades.

This Thesis is Dedicated
to
Lawrence Tayo Oguntebi
(my late brother)
to
Rachel Jibike Oguntebi
(my beloved wife)
to
All my Children
to

All my Friends

ii

ACKNOWLEDGEMENTS

My heartfelt appreciation goes to my doctoral committee
members: to Professbr Perry Lanier, my chairman, whose
thoughtful guadance and cooperation helped me throughout my
academic experience; to Professor Glenda Lappan, my
dissertation director, for her profound help and
understanding. Without her relentless readiness to
cooperate, the successful completion of this dissertation
would have remained a problem for a much longer period; to
Prefessor William Fitzgerald whose interest and efforts in
seeing me through were very consistent; to Professor James
Buschman, who consistently enhanced my other ways of seeing
and practicing disciplined inquiry.

My acknowledgement also goes to Dr. David Ben-haim and
Alex Friedlander whose work and advice helped in various
ways. My typist Paula Moan also deserves mention for her
full c00peration.

My special gratitude goes to my wife Rachel and to my
children Blessing and Joy for their assistance and

perseverance throughout my graduate studies.

iii

TABLE OF CONTENTS

L181. UF TABLES O O O O O O O O O O O O O O O 0

LIST OF E‘IGURES O O O O O O O O O O O O O O 0

Chapter

I.

II.

III.

l‘hE PROBLbM O O O . I O O O O O O O O O 0
Introduction and Rationale

Purpose of the Study . . . . . . . . . .
Research-Questions . . . . . . . . . . .
Research Hypotheses . . . . . . . . . .
Assumptions of the Study . . . . . . . .
Scope and Delimitations of the Study . .

REVIEW OF RELATED LITERATURE . . . . . .

Introduction . . . . . . . . . . . .

Development Of Probabilistic Thinking in
Children Prior and Adolescents . . . .

Studies on Children's Understanding of

Probability Concepts Prior to Instruction

Curriculum Innovations in Probability .

Studies in Achievement and Attitudes Toward

Mathematics and Probability . . . . .
Achievement in and Attitudes Toward
Probability . . . . . . . .
Sex Differences in Probability . . .
Achievement in and Attitudes Toward
Mathematics . . . . . . . . . . . .
Sex Differences in Attitudes Toward
Mathematics . . . . . . . . . . . . .
Contemporary Controversy Regarding Sex
Differences . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . .

METHODOLOGY 0 O O O I O O O O I

Introduction . . . . . . .
The Philosophy of MGMP Materials
The Probability Unit Activities
Population and Sample . . . .
Instrumentation . . . . . . .
Procedure and Data Collection
The Design of the Study . . .
Summary . . . . . . . . . . .
iv

Page
vii

xii

d—d—O
UNCGU“ -‘

15
15

16

20
25

33

3b
38

4O
46

47
49
52
52
DZ
55
58
62
b7

68
74

Chapter

IV.

PRESENTATION AND ANALYSIS OF DATA . . . . . .

Sex and Grade Level Differences in Probability
Knowledge and in Attitudes Toward
Mathematics . . . . . . . . . .

Site 1: The Urban Site . . .
Site 2: The Suburban Site .
Site 3: The Rural Site . . .

Comparison of Pretest Results Among S
2, and 3 o o o o o o o o o o o o o

The Effects of Instruction . . . . .

Site 1: The Urban Site . . .

O 0 O O H

Site 2: The Suburban Site. .

r:
O o o o O m 0 o o o
m

1
Site 3: The Rural Site . . . . . .
Comparison of the Effects of Instruction Am
Sites 1, 2 and 3 . . . . . . . . . . . .
Comparison of Attitudes Toward Mathematics
With Attitudes Toward Probability . . .
Site 1: The Urban Site . . . .
Site 2: The Suburban Site. . .
Site 3: The Rural Site . . . .
Sex and Grade Level Differences in Attitude

Toward Probability . . . . . . .
Site 1: The Urban Site . .

Site 2: The Suburban Site.

Site 3: The Rural Site . .
Summary ... . . . . . . . . . .

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

Summary and bindings .
Purpose of the Study .
Research Questions . .
Related Literature .
Methodology . . . .
Hypotheses and Design .
Findings and Conclusions
Discussion . . . . . . . . . . . . .
Implications for Mathematics Education
Recommendations for Future Research .

APPENDICES

Appendix

A.

Brochure of Middle Grades Mathematics Project
(MGMP) Department of Mathematics Michigan
State univerSity O O O O O O I O O O O I O O

Page
77

77
78
85
93

99
103
104
112
124

129

143
144
147
147

153
153
159
160
161

160

166
167
168
169
171
173
176
186
190
191

194

Appendix Page
B. MGMP Probability Test Mathematics Attitude
Scale (MAS) Probability Attitude Scale
(PAS) Now It's your turn . . . . . . . . . . . 198

C. Pearson Correlation Matrices and Reliability
Coefficients . . . . . . . . . . . . . . . . . 209

D. Mean, Standard Deviations and ANOVA Tables . . . 216
E. Scheffé's Post Hoc Comparisons . . . . . . . . . 223

BIBLIOGRAPHY O O O O O O O O O O O O O O O O O O O O O 237

vi

LIST OF TABLES

Table Page

3.1 Distribution of the Whole Sample by Grade and
by sex in EaCh Site 0 O O O O O O O O O O O O bu

3.2 Distribution of the Whole Sample by Subject by
Class by Teacher by Subject and Class . . . . b1

3.3 Reliability Coefficients-Cronbach a for MGMP
PT, MAS and PAS by Site by Time by Grade by
sex 0 O O O O O O O O O O O O O O O O O O 0 O 65

3.4 The 3 x 2 Multivariate Crossed Design Data
matrix C O O O O O O O O O O O O O O O I O O O 70

3.5 The Multivariate Analysis of Repeated Measures
Design for the Subsample for Each Site . . . . 73

4.1 Means and Standard Deviations of MGMP PT and
MAS Pretest Scores for Site 1 by Grade and
by sex 0 O O O O O O O O O O O I O O O O O 0 O 7 9

4.2 A Summary of Multivariate and Univariate
Analysis of Variance for the 3 x 2 Design
for Site 1 O O O O O 0 O O O O O O O O O O O O 83

4.3 Means and Standard Deviations of MCMP PT and
MAS Pretest Scores for Site 2 by Grade and
by Sex 0 O O O O O O O O O O O O O O O O O O O 86

4.4 Summary of Multivariate and Univariate
Analysis of Variance for the 3 x 2 Design
for Site 2 O O O O O O O O O O O O O O O O O O 90

4.5 Means and Standard Deviations of MGMP PT and'
MAS Pretest Scores for Site 3 by Grade and
by sex 0 O O O O O O O O O O O O O O O O O O O 94

4.6 Summary of Multivariate and Univariate Analysis
of Variance for the 3 x 2 Design for Site 3. . 97

4.7 Means and Standard Deviations of MGMP PT, MAS,
and PAS Scores for the Entire Sample by Time
by Grade and by Sex . . . . . . . . . . . . . 101

vii

Table
4.8

4.9

4.12

4.13

4.14

4.21

4.22

Pre and Posttest Means of MGMP PT and MAS Scores
for Site 1 Grade and by Sex . . . . . . . . .

A Summary of Multivariate and Univariate
Analysis of Repeated Measures for Data from
Site 1 O O O O O O O O O O O O O O O O O O O 0

Pre and Posttest Means of MGMP PT and MAS Scores
For Site 2 by Grade and by Sex . . . . . . . .

A Summary of Multivariate and Univariate
Analysis of Repeated Measures for Data from
Site 2 O O O O O O O O O O O O O I O O O O I 0

Pre and Posttest Means of MGMP PT and MAS Scores
for Site 3 by Grade and by Sex . . . . . . . .

A Summary of Multivariate and Univariate
Analysis of Repeated Measures for Data from
Site 3 O O O O O O O O O O 0 O O O O O O O O 0

Means and Standard Deviations of MGMP PT, MAS,
and PAS Scores for the Entire Sample by Time
by Grade by Sex . . . . . . . . . . . . . . .

AVGPTOT and DIPPTUT Means of the MGMP PT
Scores by Grade by Sex per Site . . . . . . .

Pre-Posttest Mean Differences and Averages
of the MGMP PT Scores by Grade by Sex per

Site 0 O O O O O O O O O O O O O O O O O O O O

POSTMAS and
and by Sex . .

Means and Standard Deviations of
PAS Scores for Site 1 by Grade

Analysis of Variance Summary for Mean
Difference between the POSTMAS and PAS
scores for Site 1 O O O O O O O O O O I O 0 O

POSTMAS and
and by Sex . .

Means and Standard Deviations of
PAS Scores for Site 2 by Grade

Analysis of Variance Summary for Mean
Difference Between the POSTMAS and PAS
Scores for Site 2 . . . . . . . . . . . . . .

POSTMAS and
and by Sex . .

Means and Standard Deviations of
PAS Scores for Site 3 by Grade

Analysis of Variance Summary for Mean
Difference Between the POSTMAS and PAS
Scores for Site 3 . . . . . . . . . . . . . .

viii

Page

104

108

115

119

124

130

131

132

145

146

148

149

150

151

Table
4.23

4.24

4.25

4.26

C.1

C02

C.3

C.4

C.5

C.6

C.7

D.1

D02

D.3

D.4

D.5

Means and Standard Deviations of PAS Scores
by Site by Sex by Grade . . . . . . . . . . .

Analysis of Variance Summary Table for PAS
Scores in Site 1 . . . . . . . . . . . . . . .

Analysis of Variance Summary Table for PAS
Scores in Site 2 . . . . . . . . . . . . . . .

Analysis of Variance Summary Table for PAS
Scores in Site 3 . . . . . . . . . . . . . . .

Pearson Correlation Matrix for Site 1 by
Grade 0 O O O O O O O O O O O O O O O O O O 0

Pearson Correlation Matrix for Site 2 by
Grade . . . . . . . . . . . . . . . . . . . .

Pearson Correlation Matrix for Site 3 by
Grade 0 O O O O O O O O O I O O O O O O O O 0

Pearson Correlation Matrix for Site 1 by Sex . .

- Pearson Correlation Matrix for Site 2 by Sex . .

Pearson Correlation Matrix for Site 3 by Sex . .

Reliability Coefficients-Cronbach a for MGMP
PT, MAS and PAS by Site by Time by Grade
by sex 0 O O O O O O O O O O O O O 0 I O O O 0

Means and Standard Deviations of MGMP PT, MAS,
and PAS Scores for the Entire Sample by
Time by Grade by Sex . . . . . . . . . . . . .

Means and Standard Deviations of MGMP PT, MAS
and PAS Scores for Site 1 by Time by Grade
by sex 0 O O O O O O O O O 0 O O O O O O O O 0

Means and Standard Deviations of MGMP PT, MAS
and PAS Scores for Site 2 by Time by Grade
by Sex . . . . . . . . . . . . . . . . . . . .

Means and Standard Deviations of MGMP PT, MAS,
and PAS Scores for Site 3 by Time by Grade
by sex 0 O O O O O I O O O O O O O O O O O O 0

Analysis of Variance Summary for Mean Difference

Between the POSTMAS and PAS Scores for the
Entire smple O O C O O O O O O O O O O I O 0

ix

Page
154
155
157
158
209
210
211
212
213
214
215
216

217

218

Table

0.6
D.7

D.8

D.9
E.1

E.2A

E.3

E.4
E.5
E.6
E.7

E.8

Analysis of Variance Summary for Mean

Difference Between the POSTMAS and PAS Scores

for Site 2 O O O O O O O O O O O O O O O O 0

Analysis of Variance Summary for Mean

Difference Between the POSTMAS and PAS Scores

tor Site 3 C O O O O O O O O O O O O I O O 0

Analysis of Variance Summary for PAS Scores for

Site 3 O O O O O O O O O O O O O O O O O O 0

Analysis of Variance Summary for Mean Averages
Between the POSTMAS and PAS Scores for
EaCh Site I O O O O O O O O O O O O O O O 0

Summary of Scheffé's Posteriori Comparisons
on the Probability Pretest of Grade Level
Boys and Girls . . . . . . . . . . . . . . .

Summary of Scheffé's Posteriori Comparisons
of MGMP Probability Pretest Means for boys
and for Girls in Site 2 . . . . . . . . . .

Summary of Scheffé's Posteriori Comparisons
of Mathematics Attitude Pretest Means of
Grade Level Boys and Girls in Site 2 . . . .

Summary of Scheffé's Posteriori Comparisons of
Both Probability and Mathematics Attitude
Scale Pretest Means of Sex and Grade Level
Effects in Site 3 . . . . . . . . . . . . .

Summary of Scheffé's Posteriori Comparisons of
PT and MAS Mean Differences and Averages
from Pretest to Posttest in Site 1 . . . . .

Summary of Scheffé's Posteriori Comparisons of
Probability Test Mean Gains and Averages
from Pretest to Posttest in Site 2 . . . . .

Summary of Scheffé's Posteriori Comparisons of
Probability Test Mean Gains from Pretest
to Posttest in Site 3 . . . . . . . . . . .

Summary of Scheffé's Posteriori Comparisons for

Probability Attitude Scale by Grade Level
for Site 1 O O O O O I I O O O O O O O O O 0

Summary of Scheffé's Posteriori Comparisons
for Probability Attitudes Scale Means of
Grade Levels for Site 2 . . . . . . . . . .

X

Page

121

122

123

124

128

129

130

133

134

135

Table ’ Page
E.9 Summary of Scheffé's Posteriori Comparisons

for Probability Attitudes Scale Means of
Grade Levels for Site 3 . . . . . . . . . . . 136

xi

LIST OF FIGURES

Figure Page

4.1 PREPTOT Means--Profiles of Sex by Grade Level
at Site 1 O O O O O O O O O O O O O O O O O 0 8]

4.2 PREMAS Means--Profiles of Sex by Grade Levels
at Site 1 O O ‘0 O O 0 O O O O O O O O O O O I 82

4.3 PREPTUT Means--Profiles or Grade Levels by Sex
at Site 2 O O O O O O O O O O O .0 O O O O O O 87

4.4 PREMAS Means--Profiles of Grade Levels by Sex
at Site 2 O O O O O O O O I O O O O O O O O 0 58

4.5 PREPTOT Means-~Profiles of Grade Levels by Sex
at Site 2 O O O O O O O O O O O O O O O O O O 69

4.6 PREPTOT Means--Profiles of Sex by Grade Levels
at Site 3 O O O O O O O O O O O O O O O O O O 95

4.7 PREMAS Means--Profiles of Sex by Grade Level
at Site 3 O O O O O O O O O O O O O O O O O 0 96

4.8 Pretest Probability Means--Profiles of All
Sites by Grade Level . . . . . . . . . . . . . 102

4.9 Profiles of Means on the MGMPPT by Grade by Sex
by Time in Site 1 . . . . . . . . . . . . . . 107

4.10 Profiles of MGMP PT Means by Grade by Sex
by Time in Site 2 . . . . . . . . . . . . . . 114

4.11 Profiles of MGMP PT Means by Grade by Sex
by Time in Site 3 I O O O O O O O O O O O O 0 120

4.12 Profiles of MGMP PT Mean Gains and Averages
by Time in Each Site by Grade Level . . . . . 133

4.13 Profiles of Mean Gains and Averages by Time
by sex 0 O O O O O O O O O O O O O O O I O O I 134

4.14 Profiles of Mean Gains and Averages by Site by
Time for the Entire Sample . . . . . . . . . . 135

xii

Figure

4.15

5.1

5.2

503

Profiles of Mean Gains and Averages by Grade
Level by Time for Entire Sample . . . . .

Profiles
Sample

Profiles
Sample

Profiles
Sample

Profiles
Sample

Profiles
by Sex

of
by
of
by
of
by

of
by

of
in

Pre-Post MGMP PT Means of Entire
Grade . . . . . . . . . . . . .

Pre-Post MGMP PT Means of Entire
sex by Site 0 O O O O O O O O O

Pre-Post MAS Means of Entire
Grade . . . . . . . . . . . . .

Pre-Post MAS Means of Entire
Sex and by Site . . . . . . . .

Means of PAS Scores by Grade and
Site 1 I O O O O O O O O O O O O

Pretest Probability Means--Profiles of All
Sites by Grade Level by Sex . . . . . . .

Profiles of PREMAS Means by Site by Sex and
by Grade Level . . . . . . . . . . . . . .

Profiles of MGMP PT Mean Gains and Averages
Over Time in Each Site by Grade Level and

by Sex

xiii

Page

136

137

‘13::

139

140

156

177

180

182

CHAPTER I

THE PROBLEM

Introduction and Rationale

Probability has enormous importance in modern society.
In a world of uncertainty, we must make choices, take
chances and live by the consequences of our judgments.
Probabilistic thinking is frequently involved, directly or
indirectly, when choosing between alternative courses of
action. Many and diverse daily activities and realities
depend heavily on probabilistic thinking. Decision making
in scientific and educational research, weather forecasts,
military Operations, business predictions, insurance
calculations, design and quality control of consumer
products, genetics, politics, computer technology and social
science, are a few areas of application of aspects of
probability.

Many researchers, scholars and organizations have
emphasized the importance of probability and statistics.
Shulte (1981) points out that statistics and probability

provide methods for dealing with uncertainty and are

Z
inherently interesting, exciting and motivating tOpics for
students. The National Council for Teachers of Mathematics
(NCTM) has long recognized the importance of statistics and
probability in school mathematics. In "An Agenda for
Action:. Recommendations for School Mathematics of the
1980's," deve10ped by the NCTM, probability and statistics
were emphasized as topics deserving attention in school
mathematics (1980). Over eighty percent of the scientific
(mathematics) community surveyed by the NCTM in a
"Priorities in School Mathematics" (PRISM) project (1981,
11-12) strongly support the inclusion of statistics and
probability tOpics in school mathematics for all secondary
school students.

In writing the preface of the 1981 Yearbook of the
National Council of Teachers of Mathematics (NCTM), the
editor, Shulte (1981) observes that the 1981 Yearbook theme
of statistics and probability was selected cognizant of the
importance and appropriateness or probability and statistics
in the school mathematics curriculum. Shulte (1981, ix)
asserts:

All major curriculum groups in this century --

including the NGTM in its recommendations for the

curriculum of the 1980's -- have stressed the
importance of statistics and probability... We

hope the material in this yearbook will capture

your interest and give you a springboard for

beginning the teaching of statistics and

probability.

In an overview and analysis of school mathematics in the

secondary school, the Conference board of the Mathematical

Sciences - National Advisory Committee on Mathematical

3

Education (NACOME) submits that probability and statistics
are "indispensable for the solution of policy questions" and
other facets of life. The NACOME report laments the little
understanding and interest that teachers in general show in
probability and statistics, as revealed in an NCTM
exploratory survey. Shulte (1981) also comments on the
relatively little instructional time that teachers and most
school systems give to these topics. Both Shulte (1981) and
NACOME (1975) advocate the provision of curriculum materials
for teachers in order to encourage teachers to teach
probability and statistics.* Even though the NCTM considers
these tOpics important in upper elementary grades and junior
high school (NCTM, 1983), not a single topic in the NCTM
(1982) YearbOok titled "Mathematics for the Middle Grades
(5-9)", is devoted to statistics and probability.

Other writers or groups who stress the importance of
probability include Shaughnessy (1976), Wilks (1958), the
Cambridge Conference on School Mathematics - Goals for
School Mathematics (1963), Johnson (1980), Kass (1964), Lee
and hoban (1975), White (1980) and huff (1954). Huff and
Geis (1959) powerfully sum up the importance of probability
this way:

Probability theory is the underpinning of the

modern world. Current research in both the

physical and social sciences cannot be understood

without it. Today's politics, tomorrow's weather,
and next weeks' satellite all depend on it.

 

. * The present study includes an evaluation of one such
set of curriculum materials. This material was developed by
the Middle Grade Mathematics Project (MGMP) and will be
described in detail later in the study.

4

If probability is so important and useful in our modern
society, it is worth treating as such in the school
mathematics curriculum.

The literature, however, shows that schools and
teachers for the most part do not teach probability.

Causes include the teachers' lack of knowledge in the
subject and the nonavailability of well organized materials
to help teachers manage the teaching of the subject.

The objective of this study is to consider questions
that will have implications on teaching probability at the
middle grades level. For example, what is the level of
understanding of middle grade students in probablility prior
to any curriculum intervention? A similar question has been
investigated by a number of researchers. Among them are
Jones (1974), Leake, Jr. (1965), Doherty (1965) and Mcleod
(1971), who conducted their studies respectively on grade
levels (1-3), (7-9), (4-6), and on selected elementary
grades. These researchers conclude that elementary and
junior high school boys and girls in general possess
considerable knowledge of some probability concepts prior to
formal instruction. In particular, Jones reports that
grades one through three pupils already have some concept of
outcomes of a sample space. Doherty concludes that by
grades tour through six, children have already acquired some
familiarity with the probabilities of a sample Space, sample
events and the union of two or more mutually exclusive

events. Leake concludes that in grades seven through nine,

5

students already possess considerable knowledge of the same
probability concepts.

Of interest also are the questions of sex differences
in probabilistic thinking prior to any instruction, and how
these sex differences change with grade level, In other
words, do boys and girls develop probabilistic concepts
differently or equally without any systematic probability
curriculum?

The Comprehensive School Mathematics Project (CSMP) has
developed a curriculum which introduces considerable
probability in grades 1-6. Evidence obtained during the
national evaluation of this project indicates that sex
differences which seemed apparent prior to instructibn
vanished as a result of instruction.

The issue of sex differences in mathematics, of which
probability is a part, is widely addressed in the litera-
ture. However, there seems to be little consensus on sex
differences in mathematics in research studies. Investiga-
tions and findings including those of Benbow and Stanley
(1980) tend to conclude that boys naturally have a higher
mathematical ability than girls. Wilson (1972), Flanagan et
a1. (1964) and others claim evidence to support this
position. 0n the other hand, others, especially Fennema
(1977) and Senk and Usiskin (1982), claim that when one
controls for experiences both in course work and informally
outside school, there are no sex differences in mathematics

achievement. They therefore conclude that differences are

b

largely environmental. More research is therefore desirable
on this issue.

In a study involving grades five through eight boys and
girls on the concept of spatial visualization, ben-haim
(1982) reports significant sex differences in the concept
prior to instruction but no sex differences in gains were
observed from pretest to posttest. This raises the related
questions of any sex differences in achievement as a result
of probability instruction. Even if boys and girls differ
in their knowledge of probability prior to instruction,
another important question is whether they gain differently
or equally from probability instruction. At what grade
level are any differences minimal or maximal? These are
questions that have important curriculum implications in
mathematics education.

Leake (1965) and Armstrong (1972) all report
achievement gains resulting from probability interventions.
More research is needed to determine the nature and
magnitude of these gains and in which grade levels
intervention has the best chances of success.

Attitude is another issue frequently studied in
mathematics education. Of particular interest in this study
is both an investigation of students' attitudes toward
mathematics prior to and after studying a unit on
probability and the relationship between their attitudes
toward mathematics in general and toward probability in

particular after a given probability intervention. Do boys

and girls differ in their attitudes toward mathematics and
toward probability? How do any differences change after
instruction, and from grade to grade?

Many studies involving attitudes to mathematics and
probability activities are reported in the literature.
Shulte (1967), Clemente (1982), Moliver (1977), Lee (1975)
and Moyer (1974) all report little or no gain in attitude
toward mathematics as a result of probability instruction.
Clemente (1982) and Fennema (1977) report that generally
middle grade boys tend to have more positive attitude than
girls. On attitude in general, Fennema (1977) gives what
seems to be representative of most literature:

1. There is a positive relationship between

attitude and mathematics achievement which
seems to increase as learners progress in
school.

2. Attitudes towards mathematics are fairly
stable particularly above the sixth grade,
although one longitudinal study showed a
marked decrease from sixth grade to twelfth
grade (Anttonen, 1969).

3. Grades six through eight seem to be critical
in the development of attitudes.

4. Extremely positive or negative attitudes
appear to be better predictors of achievement
than more neutral feelings.

5. There are sex-related differences in atti-

tudes toward mathematics (p. 104).
Although the above seems to be the general belief, some
reports on attitudes toward mathematics still leave us
questions about the magnitude and nature of attitudes to
mathematics, sex differences, grade level differences and

site differences.

8

Purpose of the Study

There are four purposes of this study. The first is to
determine any existing differences in probability knowledge
and attitudes toward mathematics of grades six through eight
students by sex, by grade level and by school setting, prior
to formal instruction.

The second purpose is to examine the effect of
instruction on the probability achievement and attitudes
towards mathematics of the students by sex, by grade level
and by school setting.

The third purpose is to compare attitudes towards
mathematics with attitudes toward probability by sex across
these grade levels.

The fourth purpose of the study is to compare attitudes

toward probability by grade and by sex.

Research Questions

 

There are two types of questions for consideration in
this study. The first set of questions deals with the
existing differences in probability skills and attitudes
toward mathematics of grades six, seven and eight students
by sex, by grade level and by school setting, prior to
instruction. These will be called type A questions. The
second set of questions, type B, focuses on the effects of
instruction on the probability skills of the same students.

These questions also concern the effects of instruction on

differences in attitudes toward mathematics and probability

by sex, by grade level and by school setting after

instruction.

Type A Questions

Prior to instructional intervention:
What effect, if any, does grade level have on knowledge
of probability and/or on attitudes toward mathematics?
What effect, if any, does sex have on knowledge of
probability and/or on attitudes toward mathematics?

Do differences between boys and girls in knowledge of
probability skills and/or in attitudes toward
mathematics change with grade level?

What effect, if any, does school setting have on
knowledge of probability and/or on attitudes toward

mathematics?

Type B Questions

After instructional intervention:

What effect, if any, will probability instructional
intervention have on achievement in probability tasks
and/or on attitudes toward mathematics of sixth, seventh
and eight grade students? Will these effects be
different for boys and girls? Will these effects differ
by grade level? Will the effects differ by school

setting?

10

Do differences exist between students' attitudes toward
mathematics in general and the probability activities in
particular? Will these differences exist for both
sexes? Will these differences exist for each grade
level in the study? Will these differences exist for
each of the sites 1, 2, and 3?

Do differences exist between the sexes in attitudes
toward probability accivities? Will these differences
exist for each grade level? For each site?

Do differences exist among the three school settings
(site 1, site 2, and site 3) in their attitudes toward
probability activities? Will these differences exist

for each grade level? Will they exist for each sex?

Research Hypotheses

 

The hypotheses that will be tested in the investigation

of the research questions will be in two parts: Type A, and

Type E.

Type A hypotheses:

These hypotheses are designed to test for differences

among a sample of sixth through eight grade boys and girls

in their knowledge prior to instruction in probability

activities and their attitudes toward mathematics by sex,

grade level and school setting.

11

H01: There will be no difference among the mean scores
for each of the three grade levels (six, seven and
eight) tested, on both the Middle Grades Mathematics
Project Probability Test (MGMPPT) and on the
Mathematics Attitude Scale (MAS).

H02: There will be no difference between the mean
scores for boys and for girls in grades 6 through 8 on
both the MGMPPT and MAS. -
H03: There will be no interaction of grade by sex among

the mean scores for 6th through 8th graders on both
the MGMPPT and MAS.

Type E hypotheses:

These research hypotheses are designed to test for
differences in two major areas among the sample of students
after instructional intervention. Some of these hypotheses
are to test for any effects of the instruction on the
probability skills and on attitudes toward mathematics of
the middle grades students by sex, grade level and school
setting. Others are to compare the same students' attitudes
toward probability, as well as examine sex and grade level
differences in attitudes toward probability.

H04: There will be no difference between the posttest
means and pretest means of the sixth, seventh, and
eighth grade students on both the MGMP probability
test and Mathematics Attitudes Scale (MAS).

H05: There will be no difference between the mean gain
scores for each of the three grades levels tested in
both the MGMPPT and MAS.

H06: There will be no difference between the mean gain
scores (posttest minus pretest) for boys and for girls
in grades six, seven, and eight on both the MCMPPT and
MAS.

H07: There will be no difference between students' mean
scores on the Mathematics Attitude Scale (MAS) and

H03:
H09:

H10:

12

students' mean scores on the Probability Attitude
Scale (PAS).

There will be no interaction of grade by sex among
othe mean difference scores--MAS score minus PAS
score. ~

There will be no significant difference between

the mean scores for boys and for girls in grades six,
seven, and eight on the Probability Attitude Scale.
There will be no difference between the mean

scores for each of the three grade levels (six, seven
and eight) on the Probability Attitude Scale.

Assumptions of the Study

For the purpose of this study, the following

assumptions are made:

1.

It is assumed that a paper-pencil, multiple-choice
response instrument is a valid means of assessing
student's ability in probability skill.

It is assumed that the sample does not differ
significantly from the p0pulation with reSpect to the
variables being measured in the study.

It is assumed that all the testing conditions (pretest
and posttest) do not differ significantly from school
setting to school setting, and from class to class
within a setting. Examples of such testing conditions
are place, timing, length of testing, the explanation of
testing instructions and other administration

conditions.

13

4. It is assumed that teacher effect will not differ
significantly from setting to setting and from grade to

grade.

Scope and Delimitations of the Study

This study concerns itself with sex and grade level
differences in attitudes toward and achievement in probabil-
ity as contained in the Middle Grades Mathematics Project
Probability Unit (MGMPPU). The MGMPPU is implemented by
teachers who are most probably of varying mathematical
backgrounds and teaching experiences. This study cannot
control effects due to these. However, the teachers use the
same specified probability unit, activity by activity. The
teacher is expected to follow these daily activities as
closely as possible, including materials to use, questions
to ask and assignments to give. The teachers attended a
workshOp before teaching the unit. The study does not
attempt to compare teachers' attitudes and their students'
achievements to others', neither does it attempt to examine
the effect of the MGMPPU on the teachers' attitudes toward
mathematics or probability or on their knowledge in
probability.

1 The generalizability of the findings of this study is
limited to the participating school sites and students
during the period of data collection. However, with a large

sample, over 1440 students, a case can be made that the

14

sample is representative of grades six through eight
students. Moreover, these students were drawn from a wide

variety of schools and were instructed by a diverse group of

teachers.

CHAPTER II

REVIEW OF RELATED LITERATURE

Introduction

 

Literature and arguments for the importance and
inclusion of probability in any contemporary school
mathematics curriculum were briefly presented in the
previous chapter. Also discussed briefly were the issues of
sex and grade level differences in mathematics and
probability. In this chapter, the same aspects of
probability and mathematics will be reviewed in detail.
Precisely, the following will constitute the focus of the
review of the related literature in this study.

1. DeveIOpment of Probabilistic Thinking in Children and
Adolescents.

2. Studies on Children's Understanding of Probability
concepts prior to instruction.

3. Curriculum Innovations in Probability.

4. Studies on Achievement and Attitudes Toward Mathematics
and Probability.

5. Sex Differences in Achievement and Attitudes Toward

Mathematics and Probability.

15

16

6. Summary of the Literature Review.

Development of Probabilistic Thinking in Children and
Adolescents

We will now turn to the fields of deveIOpmental
psychology and mathematics education to cite literature
concerning the deveIOpment of probabilistic thinking in
children and adolescents.

The work Piaget and Inhelder reported in their book
(Piaget and Inhelder, 1951) is the source of much of the
research in the deveIOpment of the probability concept in
young children. Piaget presents clinical evidence from
interviews with children and concludes that the learning of
probability concepts proceeds in stages, in accord with his
theory of the development of thought in children. There are
three stages in Piaget's theory of the development of the
probability concept in children.

In the first stage, generally characteristic of
children under seven years of age, the child is unable to
distinguish betwen the necessary and the possible. In this
stage, uncertainty means only unpredictability of events in
the near future. The child does not possess a concept of
logical uncertainty, and so does not understand the true
nature of a random mixture. Piaget found that children in
this first stage of development tried to superimpose an
order or discover a pattern amid the chaos of a random

mixture.

17

Two behaviors that Piaget observed in children in the
first stage are worth noting in connection with the present
study.

In the first place, if a subject was shown instances of
events A and B, and if A appeared more frequently than B,
the subject would tend to bet on B because it had been
skipped too often. This type of behaviour, sometimes re-
ferred to as the gambler's fallacy, exemplifies a subject's
use of the representativeness heuristic, in the language of
Kahneman and Tversky (1972). A truly representative
sequence of instances of A's and 8's should not favor one or
the other (provided, of course, that the probability of
instance A is the same as that of instance B.

In the second place, Piaget's subjects tended to
predict those events which had been observed most
frequently, with total disregard for the population
distribution. This type of behaviour is characteristic of
the availability heuristic (Tversky and Rahneman, 1973),
wherein events are predicted based upon constructible
instances.

In the second stage of the deveIOpment of the
probability concept (up to about 14 years), Piaget claims
that a child recognizes the distinction between the
necessary and the possible, but has no systematic approach
to generating a list of the possibles. The present study is
concerned with pupils in this stage of probability

development. It therefore suggests, if Piaget is correct,

18

that a pupil in the middle grades has no systematic approach
to combinational analysis, thus lacking the ability to list
the sample space for a probability experiment. For example,
it may be too much for a grade six pupil to be required to
understand that there are 36 sample points in a single throw
of two dice. This child, to Piaget, does not as yet possess
the formal Operations needed to perform such tasks.

In the third stage, a child begins to develop a
combinatorial analysis, understands probability as the limit
or relative frequency (law of large numbers), and can deal
with the probability of isolated instances as a function of
the whole distribution.

Piaget's interview technique requires a high degree of
verbalization for the subjects. Some studies have been
conducted to see if very young children indicate an
understanding of some pr0bability concepts when their
decisions are made in a nonverbal format. Davis (1965), and
Yost, Siegal, and Andrews (1962) present evidence for the
existence of some concepts of probability in children age 3
and 4. The children were permitted to determine probability
or frequency by utilizing a non-verbal decision process.
Yost et al. claim that the amount of reinforcement in a
probability learning experiment with four-year-olds had a
significant effect upon the accuracy of the children's
predictions.

Smock and Belovicz (1968) claim that the children in

Yost's experiment really learned about reinforcement, and

19

not about probability. They present substantial evidence
that subjects of junior high school age have a very poor
conception of the laws of probability. Smock's subjects
could not consistently generate correct sample spaces, and
did not recognize or utilize the concept of independence
when predicting outcomes.

Cohen and Hansel (1956) identify four stages that
children go through in the develoPment of the idea of a
probability distribution. At first there is just a
"glimmering belief" that the numbers in a distribution will
really vary. This corresponds somewhat to recognizing the
distinction between the necessary and the possible in
Piaget's theory. Secondly, a child feels that the category
of exactly equal proportions will occur most often, that is,
that every probability distribution is a uniform
distribution. In the third stage, likelihoods are assigned
to outcomes based upon their similar structure. For
example, the outcome one blue and four yellow beads is
judged as likely to occur as the outcome one yellow and four
blue beads, regardless of the population composition. In
this stage, the child applies the principle of symmetry
universally. Finally, Cohen and Hansel claim, a child is
able to assign a greater probability to the event "one blue
and three yellow beads" than the event "four blue beads" in
a 50-50 distribution. Cohen and Hansel attribute the stages
of mental development both to maturation and physical

experience, and say that a child is ordinarily in the fourth

20
stage of development around the age of 15. This theory is

very much in accord with that of Piaget.

Studies on Children's Understanding of Probability Concepts
Prior to Instruction

In the problem statement in Chapter One, it was shown
that mathematics associations (the NCTM being foremost) and
mathematics educators have strongly advocated the inclusion
of probability curriculum in elementary and secondary school
mathematics (NCTM; 1973, 1960, 1975, 1980-1982). Several
mathematics educators came to such a conviction following
the challenge that resulted from Piaget's theory of
probability deveIOpment and the controversy surrounding the
level of probability concept attainment in children at
various ages. The College Entrance Examination Board (1959)
and the Cambridge Conference on School Mathematics (1963)
were motivated by these psychological findings to advocate
the teaching of probability and statistics in school
mathematics. Until recently (NCTM, 1981 Yearbook), most
efforts undertaken in mathematics education were in the
words of Shaughnessy:

...either feasibility studies undertaken to deter-

mine the teachability of probability and statistics

in the elementary or secondary schools, or experi-

mental and correlational studies which attempted to

measure the effects of teaching a unit of probability

(1977).
Next is a review of some recent studies or reports on

children's understanding of probability concepts prior to

any formal instruction.

21

Jones (1974) used taped interviews with first, second,
and third graders, and embodiments of set and measure to
investigate the status of five concepts of probability among
early elementary school children. The embodiments were
spinners with equal and unequal area divisions, and jars
containing discrete objects. Interviews were taped in order
to gain insight into the errors made by the subjects. The
concepts were sample space; comparison (P1) of the pro-
bability of two events within a fixed sample Space; compar-
ison (P2) of the probability of a given event across three
sample spaces with the number of total outcomes held
constant, %, g, %; identification of (P3) uniform proba-
bility distribution; and comparison (P4) of one event
across three sample spaces in which the frequency of that
event was constant but the total number of outcomes was
varied, %, 3, %. Jones found evidence in support of the
children's understanding of P2, P4, and of sample space.
He suggests that for primary children, an apparent under-
standing of probability in one situation does not guarantee
understanding will be evidenced in another situation. There
is also further evidence in Jones' study that 1.0. predicts
the extent of the development of probabilistic thinking in
young children, in accord with the findings of Leake,
Doherty, and Leffin (discussed later). The use of embodi-
ments seemed to help the children understand probability

although Jones reports that the use of manipulatives to

perform an experiment sometimes interferred with the

22
children's ability to list the outcomes of a sample space.
Color biases and individual preferences prevented some
children from making accurate responses to questions
involving the spinners.

Mullenex (1969) investigated the relationships between
understanding of probability in grades 3 - 6, and the
variables of sex, age, grade level, and skill in other
school subjects. His test was based upon the questions that
Piaget asked children in interviews. Multiple linear
regression techniques indicated a tendency for arithmetic
computational skills and reading skills to be relevant
predictors of performance on probability measures. Mullenex
found sufficient evidence for the understanding of
probability in children to warrant inclusion of probability
topics in grades 3 - 6.

Doherty (1965) carried out a similar study with fourth,
fifth, and sixth graders. An investigation of children's
understanding of independent events was added to the three
concepts of sample space, simple probability, and mutually
exclusive events of Leake's study. Doherty found that
children in grades 4 - 6 possess considerable familiarity
with these concepts prior to formal instruction. Age,
mental age, and achievement were found to be significantly
related to the level of understanding of probability
concepts. Doherty interprets her results as indicative of
the feasibility of teaching probability in the elementary

school. She recommends that tapics from probability be

23

included in elementary school curricula, and that teacher
training programs make provisions for informing prospective
elementary teachers about probability topics that would be
suitable for elementary school children.

In a study of probability concepts possessed by
children in grades 4 - 7 prior tO formal instruction, Leffin
(1971) reports that children have considerable knowledge of
the concepts of finite sample space, probability of a simple
event, and quantification of probability. I.Q., sex, and
grade level were all found to be significantly related to
the understanding Of probability. I.Q. was found to be the
most accurate predictor of performance on probability tests.
In analyzing the children's errors, Leffin mentions that the
concept of combinations was very difficult for them to
comprehend or to use. When Leffin's subjects could list all:
the outcomes in a sample space that counted combinations,
92% of them could not use the information from the sample
space to calculate a probability. This evidence appears to
support Piaget's position that children of this age are in
the stage Of concrete Operations. Leffin's subjects could
successfully handle probability in simple situations like
drawing balls out Of a box given the number of balls Of each
color that are in the box. However, the more complicated
combinatorially-generated sample spaces were not understood
by these children. This finding caused Leffin to speculate
on how early children can be taught a systematic method of

counting. He recommends taped interviews and the use of

 

24
manipulatives with children in order to Obtain more
information about children's readiness to learn counting
principles.

In an investigation of the develOpment of the notions
of chance and probability, Relsey (1980) concluded that
adolescentshad a poor understanding of these notions.
Thus, Kelsey's findings agreed with those of Beyth-Maron and
Shaughnessy.

Leake (1962) found that seventh, eighth, and ninth
graders had some understanding of sample space, probability
of a simple event, and probability Of the union of two
disjoint (mutually exclusive) events. As in Doherty's
study, mental age and achievement both correlated
significantly with understanding of probability. Leake
recommends the inclusion of probability tOpics, in grade
levels seven to nine, based on the results of his
investigation.

The above literature reviews refer chronologically to
elementary and middle grades probability concepts prior to
instruction. The studies all recommend that topics in
probability be included in elementary through middle grades
mathematics curricula. Quite a few probability studies or
curriculum develOpments have been carried out - involving
all levels of learners - elementary, intermediate, middle
grade, high school and college levels. Some of these
studies will now be reviewed in turn. The extent of the

review depends on the relevance or relationship of the

25

particular study or intervention to the present

investigation.

Curriculum Innovations in Probability

Attempts at develOping curriculum materials in
probability have been made by a number of innovators. Some
of these attempts will next be reviewed.

Studies were carried out by Gipson (1972) to determine
what materials would be appropriate for introducing
probability concepts to third graders. In one study,
children received instruction in small groups and in another
instruction was individualized. The instructional sequence
dealt with the concept of sample space and the probability
of a simple event. Audio and video tapes of the subjects
were made to gain deeper insight into the process through
which children learn about probability concepts. Gipson,
like Shepler, reports that the children had difficulty
specifying estimated probability from an experiment. Gipson
also adds that the use of the interviewbtype procedure
(clinical interview) would give a deeper insight into how
children develOp probabilistic concepts. Gipson concludes
with a recommendation that the third grade level is an
appropriate school stage to introduce selected concepts of
probability.

Armstrong (1981) describes the probability included as
an integral part of the elementary mathematics curriculum

developed by the Comprehensive School Mathematics Program

26

(CSMP). The CSMP developed stories and games for the second
and third grades that introduce such concepts as eXpected
frequency, equally likely events, and prediction. Armstrong
reports that third grade CSMP students "considered the
thirty-six equally likely outcomes when two dice are thrown
and determined that there are six ways for a sum Of seven to
occur". Armstrong further describes the area model*
technique for solving probability problems. In this model, a
unit square is divided into regions so that the areas Of the
regions are proportional to the probabilities involved in

the situation. The area model, Armstrong continued, 13 a
geometric model that satisfies the CSMP criteria for solving
probability problems". To be appropriate for students in
the intermediate grades, Armstrong claims the model should:

- be sufficiently powerful to handle fairly

sophisitcated probability problems;

- rely primarily on mathematical skills that

students already have acquired;

- be consistent with the students' current

understanding of probabilistic concepts;

— support the eventual development of more

advanced solution techniques (1981).

Shepler (1970) developed a unit on probability dealing
with sample spaces of one, two, and three dimensions, and
necessary counting techniques. The unit was taught to a
class of 25 specially selected sixth graders of above
average ability. The unit was taught using a mastery

learning model that incorporated self-correcting exercises,

 

* The area model technique was also used in the probability
instruction implemented for the present study.

27

specific prescriptions to diagnose and remedy errors, extra
help sessions, and extra group instruction when mastery was
not satisfactorily attained by a large majority of the
class. Objectives included counting outcomes, probability
of a simple event, probability of a compound event, equally
likely versus unequally likely probability models, and
estimating the probability of an event from data in an
experiment. A criterion level of 90% correct by 90% Of the
students was set for mastery of the Objectives. All the
behavioral objectives were mastered at this level by the
students except those dealing with counting the number Of
outcomes and estimating prObability from data. Shepler's
results agree with those of Leffin (1971), and suggest that
sixth graders do not yet possess the formal operations that
Piaget claims are necessary to count all the outcomes
systematically. A follow up study (Shepler and Romberg,
1973) indicated that after four weeks the subjects were able
to retain most of what they had acquired at the mastery
level.

Beyth-Maron (1980) innovated a probability curriculum
entitled "Thinking under Uncertainty: A Curriculum".
Beyth-Maron's work has a lot in common with the present
study. Hence her work will be reviewed in some detail.

Prompted with concerns similar to those already
expressed in this study, Beyth-Maron conducted a five-year
study which culminated in a workable curriculum in "Thinking

under Uncertainty". Beyth-Maron, in her study, reviewed

28
several scholarly studies on thought processes (Miller,
1956; Bruner's concept formation, 1956; Slovic et al., 1977;
and Tversky b Kahneman's 'thinking and uncertainty,‘ 1974).

"These studies, claims Beyth-Maron, "have demonstrated
cognitive limitations in perceiving, memorizing and
processing information." How then do people perceive
uncertainty, assess probabilities, evaluate risks and judge
the quality of their own and others' decisions? Noting
several limitations associated with probabilistic thinking,
Beyth-Maron sought in her curriculum to help correct some of
these limitations. She asked these leading questions. "Can
corrective procedures be devised? Can we show people when
and how their judgments are wrong and how they can be
improved?" She also remarked on the usual difficult and
inapplicable way in which statistical and probability
concepts were taught to students and made these
observations:

In teaching, it can be difficult to convince

students that probability is relevant to life

events and not just the science of coins and

playing cards. Even experts who appreciate the

relevance of probability to daily matters are

prone to the same mistakes as lay peOple. This

may occur because most daily problems are not

formulated as neat, textbook probability problems

and experts Often fail to make the reformulation

intuitively. In addition, most courses in

statistics, probability are taught without taking

account Of cognitive processes (Beyth-Maron, 1980).
In her probability curriculum, Beyth-Maron demonstrated five
stages of teaching that she considered appropriate for
teaching probability to middle grades students. These

stages were: (1) Demonstrating by example(s); (2) Analyzing

29

thought processes (introspection); (3) Strengthening good
intuitions and showing absurdities by considering
alternative thinking and nonexamples; (4) Analyzing the
pupils' answers and arguments with them, making them
understand how similarity rules do not obey probability
rules; and (5) Deciding what the pupil should learn and use
specifically.

Junior high school was chosen for Beyth-Maron's
curriculum development. Three reasons were given for this
choice. First, junior high school students already have a
grasp Of the minimal mathematics demanded by the probability
activities. Second, these students are mature enough for
introspection ability, and may even enjoy doing so; and
third, they have the time and willingness to accept new
experimental areas.

The proability unit included:

(1) General framework for thinking under uncertainty,

(2) Some tools for judgment, and

(3) Probability Instruction.

In the questionnaire evaluation that followed, curriculum
participants and non-curriculum participants were compared
in 20 items. Beyth-Maron concluded that the program
recipients did significantly better than the control group.
It was also found that children from high academic schools
gained more from the program than those from low academic
schools. However, every participant gained significantly

from the program.

30

In another probability curriculum, White (1974)
developed and taught some concepts to seventh and eighth
grade students. On comparing pretest and posttest results,
White found that the subjects benefited significantly from
the program. Achievement in probability was correlated
significantly with concept attainment, computational
ability, and reading ability. McKinley (1960) develOped a
probability unit for twelfth grade students. McKinley
reports that intelligence, language skills, reading
comprehension, and mathematics achievement, all correlate
significantly with achievement in the unit.

An experimental probability curriculum was developed
and implemented by Shaughnessy (1977). Using college
students as his subjects Shaughnessy, like Beyth-Maron,
attempted to correct certain probabilistic errors in young
people. His Objective was to provide a probability inter-
vention that would maximize the students' chances of over—
coming certain misconceptions of probability and statistics.
He argued, like Beyth-Maron, that a conventional lecture
approach to the teaching of probability may not be the best
way to overcome students' misconceptions about probability.
He therefore developed an experimental approach that used
small-group, activity based strategies in teaching
probability.

The misconceptions that were investigated were those
that arise from reliance upon heuristics of representa-

tiveness and availability. These heuristics "enable human

31
beings to decode complex probabilistic situations"
(Shaughnessy, 1977).

According to the representativeness heuristic, peOple
tend to make decisions about the likelihood of an event
based upon how similar the event is to the distribution from
which it was drawn. For example, a nursing mother whose six
children are boys would see this as not being representative
of the random process of child bearing, and would tend to
expect the seventh child (if any) to be female, even though
she might know that these are independent outcomes. Accord-
ing to the heuristic of availability, subjects tend to base
their judgments upon the relative likelihood of the events
based upon the ease with which instances Of that event can
be constructed or called to mind (Tversky b kahneman, 1973).
For example, subjects employing the availablity heuristic
tend to favour the misconception that out of a group Of 11
people, these are more distinct 4-person committees than
there are distinct 7-person committees. It is easier to
call to mind more examples of 4-person committees than 7,
even though the number of distinct committees is the same
(330) in each case. It was found that the experimental
course was more effective in overcoming some misconceptions
that are attributable to the use of representativeness and
availability than the control course.

The experimental activity-based course was constructed
as an alternative to the lecture method for an undergraduate

course in finite mathematics. A series Of nine activities

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIllllllllllll----—'—‘

32

in probability, combinatories, game theory, expected values,
and elementary statistics were developed by Shaughnessy.
Students in the experimental course worked together in class
on the activities in small groups of four or five members.
Each activity required the groups to perform experiments,
gather data, organize and analyze the data, and finally
reach some conclusions which could be stated in the form of
a mathematical principle or mathematical model. The
students were strongly encouraged to cooperate with one
another, to solve problems as a group rather than
individually, and to help all the members Of their group to
understand the concepts and problems of each activity. The
groups were changed often so that everyone had a chance to
work with everyone else during the course.

Shaughnessy also remarks that the manner in which
college students learn probability makes a difference in
their ability to overcome misconceptions that arise from
availability and representativeness. He concluded his study
with this implication for the mathematics teacher (1977,

p. 314):

Peoples' intuition of probabilistic thinking is

distorted by science education's emphasis on the

necessary, and neglect of the possible. This
experiment suggests that the course methodology

and the teaching model used in an elementary

probability course can help develOp peOples' in-

tuition for probabilistic thinking. A course in
which students carry out experiments, work through
activities to build their own probability models,

and discover counting principles for themselves

canhelp students to overcome their misconceptions

about probability, and can help restore the syn-

thesis between the necessary the the possible
which is essential to probabilistic thinking.

33

From Beyth-Maron's, Shaughnessy's and other curriculum
studies, one sees a lot of similar concerns and experimental
results regarding the use of probabilistic thinking and

learning that will be examined in the present study.

Studies in Achievement and Attitudes Toward Mathematics and
PrObability

Studies involving both mathematics and probability at
the elementary grades level, with respect to achievement and
attitudes, are not common in the research literature.
However, several studies on these tOpics, involving
adolescents and adults, are reported.

Shulte (1968) investigated the effects of a probability
and statistics unit on the achievements and attitudes of
ninth grade general mathematics students. Shulte concluded
that the probability and statistics presented in his unit,
"The Mathematics of Uncertainty", did not effectively
promote student attitude or achievement in computational
skills. The intervention however effectively increased
proficiency in other mathematics areas.

Moyer (1975) designed and conducted a study to "test
the claim that probability has the potential to improve
arithmetic computation skill, arithmetic reasoning, and
attitudes toward mathematics". Moyer whose subjects were
ninth grade general mathematics students, did not find any
significant difference in attitudes toward mathematics, but

the experimental group outperformed (P < .05) the comparison

34

group in knowledge about probability. However, like Shulte,

Moyer's study:

does not support the contention that probability,
at least that part of probability contained in

the unit taught in this study, can be used in

the ninth grade general mathematics classes to
improve arithmetic computational skill, arithmetic
reasoning, or attitude toward mathematics. However,
the study indicated that while gaining knowledge
about probability, the experimental group showed
equivalent improvement with the comparison group
in the ordinary general mathematics areas of
arithmetic computation skill and arithmetic
reasoning.

Lee (1975) worked with low-achieving junior college

students in a study on developing basic mathematics skills

through elementary probability and statistics. Lee had

three goals.

1.

3.

To improve students' mastery of some basic mathematics
skills.

To help obtain some understanding Of probability and
statistics and their uses in real world situations.

To improve students' attitudes toward mathematics.

In the summative evaluation of the 191 subjects involved,

Lee reported that students' attitudes toward mathematics

remained unchanged, but over 80 percent of the subjects

claimed to have acquired a better understanding of

probability and statistics and their applications.

Shevokas (1974) carried out a study using, as subjects,

the students taking the general mathematics course in a

community college. In the study in which computer oriented

and manual Monte Carlo approaches were employed, Shevokas

had two purposes. The first was to investigate the effects

35

of the Monte Carlo approach on achievement in and attitude
toward mathematics. The second purpose was to examine
similar effects in probability and statistics. No
significant difference was found in the measures of
attitudes toward mathematics. However, both experimental
groups (one with computer and the other with manual Monte
Carlo procedures) achieved higher (P < .01) than the control
group which used analytic methods only. Shevokas concluded
with the assertation that the non-computer Monte Carlo
approach was an Optimal method for introoucing a probability
unit to community college students.

Kipp (1975) investigated the effects of integrating
topics from probability with those of elementary algebra in
an experiment with college students. She compared
experimental and control groups on achievement, retention,
and attitude. Greater retention and improved attitude
towards mathematics were found in the groups receiving the
algebra integrated with probability. Kipp recommends that
experimentation be introduced before college students are
taught probability formally. She suggests that college
students should encounter physical models of both uniform
and nonuniform probability distributions.

In an experimental study involving graduate students in
the behavioral sciences, Monroe (1980) developed, taught,
and evaluated two probability concepts. Monroe reached the

following conclusions:

 

36

the relationship between age and performance on

the probability test was strong and negative among

students with a poor mathematics background; stu-

dent attitude toward mathematics was not related to

performance on the probability test; and, important

probability concepts can be taught using a non-

traditional curriculum (Monroe, 1980).

Crouse (1977) in his study investigated the effect of
the teacher's probability knowledge and mathematics
attitudes on student probability achievement. Crouse found
that higher student achievement in the selected probability
tasks taught was significantly associated with higher
teacher knowledge in these tasks. Crouse concluded that

teacher attitude toward mathematics had no effect on student

achievement in the probability tasks taught.

Achievement in and Attitudes Toward Probability

The studies reviewed above each dealt with probability
and mathematics. Quite a few investigations have addressed
probability alone, with respect to student achievements and
attitudes. Some of these studies are now reviewed.

In a study on first grade children's understanding of
probability, Dunlap (1980) indicates that even children with
limited or no understanding of probability can be trained to
evidence such understanding.' He reached this conclusion
from the result of pretest-posttest data involving seven
groups. Dunlap however suggested that the "rule training"
(tutorial) methOd was more successful with first graders

than the "self-discovery training" method.

37

Armstrong (1972) investigated the ability of fifth and
sixth graders to learn selected topics in probability.
Armstrong concluded that while sixth graders possessed the
ability to learn all the concepts of probability taught,
fifth graders were unable to learn the concept of outcome
space. The concepts taught in the study were outcome,
outcome space, even, probability of a finite event, and
mutually exclusive events. Sixth graders gained
significantly on all of these concepts and on the total
probability test.

In another study, Smith (1966) develOped and taught a
unit on probability and statistics for seventeen days to
three groups of seventh graders. The three groups were low,
middle, and high experimental groups. Smith found that all
three groups learned significantly (P < .01) from the
instruction. Smith cOncluded that the seventh grade was an
apprOpriate level to introduce "at least some tOpics in
probability and statistics."

V McClenahan (1974) carried out a study involving an
application of Piagetian research to the growth of chance
and probability concepts. McClenahan's subjects were low
achievers in secondary school mathematics. His conclusions
included

There is a strong indication that the low achiever

in mathematics may not have attained the formal

Operational stage, at least as far as the topic of

probability is concerned (McClenahan, 1974).

This study tends to suggest that probability is a relatively

abstract topic in mathematics. As such, children's level of

38
development ought to be taken into consideration when
introducing probability in school mathematics programs.
Arehart (1978) explored the relationship between ninth
and tenth grade student achievement on a probability unit
and student opportunity to learn the unit objectives.
Twenty-three teachers taught the unit to twenty—six classes.
In the analysis of the pretest-posttest scores, Arehart
found the following:
1. Student achievement in probability is related to the
amount of exposure or Opportunity he has to learn that

objective.

N
o

The study also supports the tenet that amount of student

work is related to student achievement. 5

3. Teacher information turns out to be as important as
teacher questioning behaviour.

4. The amount of teacher information and teacher

questioning about objectives of a lesson relate

positively to the achievement of them.

Sex Differences in Probability

Research findings with respect to sex differences in
probability seem to concur, irrespective of grade level,
that little or no sex differences exist. Studies by
Mullenex (1968), Doherty (1965), Smith (1966), and Wavering
(1979) were conducted respectively with grades levels 3-6,
4-6, 7, and (8, 10, and 12); and in varied settings. Yet

all conclusions were unanimously in favour of no significant

39
sex differences in probability. Doherty's study, reviewed
much earlier in this chapter, was carried out prior to
instruction. Also McLeod (1972) found no sex differences in
his own study. Three treatments in a unit on probability
were administered to second and fourth grade children. The
treatments were laboratory experience, a teacher
demonstration, and a control in which no probability was
taught. The unit on probability covered the law of large
numbers, prediction of a set of outcomes from an experiment
involving repeated trials, and uses or probabilistic terms

such as "certain, impossible", "likely", and "unlikely".
McLeod also found-no differences among the three treatments
in probability achievement.

In an evaluation of the Comprehensive School
Mathematics Project (CSMP) probability curriculum, sex
differences were reported prior to instruction in both CSMP
and non-CSMP students. However, after instruction,
according to Dougherty (1981),* sex differences were not
found with CSMP students, but sex differences persisted with
non-CSMP students.

However, Kass (1964) reported sex differences in
probability achievement in favour of boys. Kass, in his

study, found that boys outperformed girls in binomial

probability tasks. Kass' study is one of the very few

 

* This report was given by Dougherty of the CSMP at the NCTM
(1981) Annual Conference, at St. Louis, Missouri, U.S.A.

40

studies that found any sex differences in probability. More
research is needed with respect to sex differences in

probability.

Achievement in and Attitudes Toward Mathematics

So far, reviewed in this chapter are studies in which
probability and mathematics are the focus, or in which
probability alone is the concern, with respect to
achievement and attitudes. Other studies have addressed
achievements in and attitudes toward mathematics alone.
This section contains a review of attitudes and achievement,
and the relation between them, with respect to mathematics.

The general question asked by current researchers is
"What is the strength of relationship between attitudes
toward mathematics and achievement in mathematics?"
Affective variables are believed by many educators to be as
important contributors to the learning of mathematics as
cognitive variables. Evidently, research is needed to
verify or nullify the common sense feeling of heavy
dependence, or even causality, between attitudes and
achievement with respect to mathematics.

Malcolm (1971) reviewed the question of attitude
formation through a ten-month longitudinal study. Malcolm
used a sample of 858 students from a large suburban school
district, in grades three to four, five to six, and six to
seven. The purpose of the study, among other concerns, was

to determine if attitudes do decline with age, and if any

41

grade level would emerge as producing the greatest amount of

attitude change. Two arithmetic attitude scales were

employed. The first scale was the Hoyt Minnesota Pupil

Opinion, a 28-item yes-no instrument. The second scale was

a semantic differential with fifteen bipolar adjective

pairs. Both scales were proved reliable and found to have

acceptable internal reliability, with the Hoyt instrument
yielding the highest correlations. Like in the present
study, sex and grade were the independent variables.

Inconsistent results were obtained. With the Hoyt posttest

scores, fourth graders had the highest attitudes toward

arithmetic and the sixth graders had the lowest. Sex
differences were found on the semantic differential scale
only. Malcolm submitted these conclusions:

1. Attitudes do decline as one proceeds through school.

2. The later (elementary) grades; i.e., grades five to six
and grades six to seven, appear to be important in
attitude formation.

3. Girls tend to register more negative attitude change
than boys across the grades.

Malcolm concluded his study with a recommendation for

longitudinal studies dealing with the identification of

factors influencing attitude formation.
As if in response to Malcolm's recommendation,

Shaughnessy, Haladyna and Shaughnessy (1983) conducted a

study on factors that influence attitude toward mathematics.

Admitting that "poor attitudes may be behind a decreased

42

enrollment in advanced mathematics classes in high school,

especially on the part of females", Shaughnessy et al.

examined the relations of student, teacher, and learning
environment variables to attitude toward mathematics. They
argued that attitude studies need not be designed in
relationship with achievement all the time. They asserted,

"Improvement of student attitude has been regarded as a

valuable end product in and of itself".

In the study, the research questions examined were:

1. To what extent do student, teacher, and learning
environment variables of both types (exogenous and
endogenous) account for the variance of a measure of
students' attitude toward mathematics?

2. Are these patterns consistent across three different
grade levels?-

3. Is gender a significant variable in the study of these
relationships?

Grades four, seven, and nine students participated in this

study. The aspects of affective components measured

included student motivation, teacher quality, social-psycho-
logical aspects, management and organization, and attitudes
toward mathematics. In the attitude toward mathematics
questionnaire, items included the composite question:

"How do you feel...

1. when it is time for mathematics?

2. during mathematics?

3. when mathematics is over?

43

4. if you knew you would never go to mathematics again?

Shaughnessy et al. submitted the following conclusions:

1. Exogenous student variables (e.g. gender and
socioeconomic status) showed little direct relationship

to attitude.

N
o

Endogenous student variables (e.g. teacher quality and
class cohesiveness) showed consistently notable
correlations with attitudes toward mathematics.

3. Fatalism (students' perception of their ability to
affect school success), and teacher quality indicated
the strongest relationships toward attitudes across all
three goal levels.

4. The strength of fatalism grows steadily with grade
level, and it is higher for girls than for boys.

5. The teacher quality effect is higher for girls in grade
level seven, but reversely true in grade level nine.

Shaughnessy et al. concluded their study with some

implications for mathematics education. First, there is a

need for good teacher quality in order to enhance more

positive attitudes toward mathematics. Thus, attention is
called to more comprehensive mathematics teacher education
programs. Second, student, teacher, and learning
environment variables are importantly related to mathematics
attitude. These variables must be adequately recognized and
taken into consideration in mathematics staff develOpment.

Third, more investigation on student fatalism is needed.

There seems to be a significant correlation between a

44

student's perception of his ability to affect his school
success and his attitude toward mathematics. While this
relationship does not necessarily imply causality, more
knowledge about its strength is desirable.

Thus, from studies by Malcolm (1971), Shaughnessy et
al. (1983), and by others, Knaupp (1973), Epstein (1981),
Shaughnessy et al. (1982), and Suydam and Weaver (1975),
research evidence abounds that tend to suggest that
achievement is not the only variable positively related to
attitude toward mathematics. Suydam and Weaver (1975) made
the following Observation with respect to elementary school
studies:

There is no consistent body of research evidence

to support the popular believ that there is a sig-

nificant positive relationship between pupil atti-

tudes toward mathematics and pupil achievement in
mathematics...We have little research basis for
believing that these two things are causally

related (p. 1-3).

Callahan and Glennon (1975) are also in agreement with
Suydam and Weaver. Also reviewing elementary school studies
for the same age, they conclude that the state of the art
"makes it difficult to present compelling research
evidence...that positive attitudes play an important role in
contributing to mathematics achievement" (p. 80). Aiken
(1976) argues that "when attitudes scores are used as
predictors of achievement in mathematics, a low but
significant positive correlation is usually found" (p. 295)

at the elementary, secondary, college undergraduate and

postgraduate levels.

45

The above studies seem to deemphasize positive
relationships between attitude and achievement. Other
studies by Anttonen (1969), Malcolm (1971) and Norman (1977)
report a decline in attitudes occurring with grade level.
However, other equally valid studies, Fennema (1981) in
particular, have reported Opposite findings.

Fennema (1977) suggests that part of the contradictory
conclusions can be explained by the age of the subjects
being considered in the reviews. Two reviews, Suydam and
Weaver (1975), and Callahan and Glennon (1975), were
concerned basically with children in grades one through six.
Problems of assessing attitude in these grades have not been
addressed adequately and lack of carefully designed
measuring instruments may have caused reviewers to seriously
question any significant differences reported. .Aiken, in
his 1976 review, was concerned with a much broader age
spectrum. Even while recognizing the serious problems
connected with the studies_of young children, he was willing
to accept the evidence as having some validity because the
results coincided with studies having older subjects.

Fennema (1977) summarized the conclusions most often
reported in the literature, but which are now being
contended:

1. There is a positive relationship between

attitude and achievement which seems to in-
crease as learners progress in school.

2. Attitudes toward mathematics are fairly

stable - particularly after about the sixth
grade, although one longitudinal study showed

a marked decrease from 6th to 12th grade
(Antonnen, 1969).

46

3. Grades 6-8 seem to be critical in the develop-
ment of attitudes.
4. Extremely positive or negative attitudes appear
to be better predictors of achievement than more
neutral feelings (p. 104).
Fennema indicates there is a fifth conclusion related
to sex differences in attitudes toward mathematics which

will be discussed in the succeeding paragraphs.

Sex Differences in Attitudes Toward Mathematics

 

Although it was not explicitly emphasized in the works
of Malcolm (1971), and Shaughnessy et al. (1983) reviewed
earlier, sex differenCes were indicated in attitudes toward
mathematics.

On the whole Fennema (1977) concludes that "there are
sex-related differences in attitudes toward mathematics
(p. 104)." But, even though there is consensus that
sex-related differences in mathematics attitude exist, the
magnitude and specific dimensions of these differences are
unclear. Although denoting some studies which failed to
find significant sex differences in attitudes and
achievement in mathematics, Aiken (1976) indicates that
"differences in both attitudes and achievement in
mathematics are frequently found to favor boys over girls at
junior-high level and beyond" (p. 296). With regard to sex
differences in attitudes, Suydam and Weaver (1975) quote
studies with contradictory results and say that in Other
studies no significant sex-related differences were found.

Aiken (1976) states that the correlation between attitude

47

and achievement varies not only with grade level but also
with the sex of the student and is generally someWhat higher
for girls than for boys.

Basic agreement with the conclusion that significant
differences in attitudes are frequently found to favor males
over females, was reported in the Fennema and Sherman study
(1977, 1978) with learners in grades six through eleven. It
has also been reported that mathematics test anxiety is
significantly higher for eight grade girls than for eighth
grade boys (Szetela, 1973). Finally, Ben-Haim (1982) sums
up these invescigations with this quote from Aiken (1976).
This is a summary of some tentative findings of these kinds
Of investigations in mathematics education:

1. Modern mathematics programs do not improve
attitudes more than traditional programs.

2. Compared to regular classes, "continuous
progress" classes do not have a different
effect on attitudes toward mathematics.

3. Discovery methods are not superior to
expository methods in their effects on
attitudes toward mathematics.

4. Neither follow-up instruction nor flexible
scheduling improves attitudes more than
traditional instruction.

5. An individual approach to instruction in
elementary and junior high mathematics
sometimes has a more positive effect on
attitude than a traditional approach; other
times no difference in the effects of the
two types of programs is found.

6. Certain units or topics in mathematics have a
more positive or a more negative effect on at-
titudes than other units or tOpics (p. 300-301).

‘Contemporary_Controversy Regarding Sex Differences

A number of studies have identified sex differences in

mathematics achievement (Flanagan et al., 1964; NAEP, 1975;

48

Wilson, 1972, Clemente, 1982; Benbow and Stanley, 1980).
Other studies challenge the notion and argue that recent
studies tend to prove otherwise (Senk a Usiskin, 1982;
Fennema, 1982; Armstrong, 1981; and Becker, 1981). Salient
among the prOponents of the existence of sex differences in
mathematics achievement are Benbow and Stanley (1980) who
became strong prOponents as a result of a controlled
longitudinal study involving high achieving boys and girls.
They conclude the following on finding significant
differences in favour of boys:

It is therefore obvious that differential course-

taking in mathematics cannot alone explain the

sex differences we observed...Sex differences in

achievement in and attitude toward mathematics

result from superior male mathematical ability.

Thus Benbow and Stanley tend to advocate that boys naturally
do better in mathematics than girls. The above study was
conducted to investigate Fennema's assertion (Fennema, 1972)
that any sex differences in mathematics achievement are due
to differential course-taking, especially at high school
level, since sex differences are not apparent prior to high
school.

In another study on mathematics achievement involving
general and high achieving boys and girls, Senk and Usiskin
(1982) report findings quite contradictory to those of
benbow and Stanley. In their extensive investigation of sex
differences in achievement in geometry proof, Senk and

Usiskin report that the more an instrument directly measures

a student's formal educational experiences in mathematics,

49

the less the likelihOOd of sex differences. They concede
that boys perform better than girls in tests or problem
solving, consumer applications and the Scholastic Aptitude
Test-Measure (SAT-M). however, they insist that these are
not a measure of students' formal educational experiences in
which mathematical ability should be tested, but a measure
of students' experiences outside classroom mathematics.
Hence, boys tend to out-perform girls in those tasks because
they tend to have more experiences than girls in those
tasks. Thus, Senk and Usiskin continue their argument that
achievement in geometry proof is achievement in complex and
high level cognitive reasoning. Senk and Usiskin (1982)
conclude:

Our results with proof, together with our analysis

of other studies, lead us to believe that boys and

girls are of equal mathematical ability...We have

found that when male and female students are tested

on geometry proof, a high level cognitive task with

spatial requirements that is encountered almost

exclusively in the classroom, no sex differences

in performance exist. Our results hold for both our

national sample of mixed ability students and for

select high-scoring samples...Girls and boys perform

equally well.

There is therefore some controversy as to the existence

of sex differences in mathematics achievement.

My

There appears to be a good deal of support and
agreement in the literature that the develOpment of the
probability concept in children does proceed in stages in

accord with the theory of Piaget. However, there is

50

considerable disagreement among investigators as to which
probability concepts are actually known by children, and at
what age levels. However, most of them are in favour of the
introduction of probability in elementary school
mathematics. Curriculum innovators in probability tend to
suggest that the child's level of probability develOpment,
teacher quality, and certain probabilistic errors are the
major concerns in any probability instruction. Grade level
differences exist in probability achievement but sex differ-
ences are rare. There is no consensus in the literature
with respect to the nature of sex differences in mathematics
achievement and attitudes. Results are conflicting. While.
some researchers argue that achievement differences are
innate and unchangeable, others insist they are environ-
mental and correctable with apprOpriate instructional pro-
cedures. There is also some disagreement in the literature
with respect to mathematics attitude change with grade
level. While many assert these attitudes are developed
early and decline with grade level, others submit that
mathematics attitudes increase with grade level. It is thus
apparent that more research is needed on these issues that
have so much implication on mathematics education. Even
when agreement exists with respect to existence of
differences, it is desirable to know the extent of these
differences.

Finally, the present study will compare achievements in

probabilistic skills and attitudes to mathematics and

51

probability activities across settings. An underlying
assumption is that students from urban, suburban and rural
areas differ in socio-economic status and background. One
objective is to investigate how differences in setting
affect achievement in and attitudes toward mathematics and
probability activities. Studies on attitudes toward and
achievement in mathematics, in which setting is one of the
independent variables, are hard to come by. In the only
similar study available to this investigator, the effects of
race, sex, and grade level on change in mathematics
achievement were investigated. In that study, Rule (1981)
concluded that grade level, when used with sex, race and
teaching method, significantly contributed to the prediction
of change in student attitudes toward and achievement in
mathematics. The next chapter gives a detailed description

of the procedures followed in this investigation.

CHAPTER III

METHODOLOGY

Introduction

 

In this chapter a detailed description of the
probability intervention is presented. Also included are
descriptions of the pOpulation and sample, the procedure and
data collection, the instrumentation of the study, the
hypotheses to be tested and the statistical design of the
study. A summary of these aspects of the study concludes

the chapter.

The Philosophy of MGMP Materials

 

With the major goal of developing units of high quality
mathematics curriculum for middle grades students, the MGMP
staff developed four mathematics units. Among these is the
unit entitled Probability, which is the focus of this study.
The other three MGMP units are called Factors and Multiples,
Spatial Visualization and Similarity.

Utilizing an instructional mooel developed by Shroyer
and Fitzgerald (1979), MGMP attempts to help students

52

53

develOp a deep, lasting understanding of the mathematical
concepts and strategies studied. The model consists of
three phases: launching, exploring, and summarizing
(Appendix A), and clearly describes what is expected from
the teacher and students during each instructional phase in
this way:

During the Launching the teacher follows the
script very closely posing the questions and
challenges in the sequence they are intended and
presented. This sequence allows each student to
be engaged in the task at his/her appropriate
level with some degree of success.

After the major challenge has been posed,
the class can begin workin individually or in
small groups. The teacher can float around the
class to keep abreast of developments. Some
children will need additional help beginning the
task as one presentation of the challenge is often
not sufficient. Other children will need help
maintaining progress toward the challenge. The
teacher may spot errors the students have made and
help the children correct their error. Still
other children will finish the task and will need
to be presented with an extra challenge to keep
them working productively.

Such a work period will result in the child-
ren being more different from each other than
before. While all children have made progress,
some have made much more than others. This is as
it should be.

However, it is desirable to bring the class
together again to summarize the results of the
activity. The orderly tabulation of results will
allow children to recognize patterns and generate
rules. Again, one should expect great differences
among the children, but all can profit from a dis-
cussion of the generalizations which might surface
from the group (Fitzgerald and Shroyer, 1979).

 

 

 

 

 

Simply put, the model is designed to present important,
related mathematical concepts to children, using activity-
oriented lessons. Children are provided with manipulative

experiences and multiple embodiments.

54
A detailed instructional guide is provided to enhance
easy implementation of the teaching model described. It was
developed to provide Specific suggestions for important
questions to be asked at appropriate stages of the
activities. Additional questions which involve
generalizations and further challenges for high ability

students are also included.

The Probability Unit

The probability instructional material used in this
investigation was first developed during the 1981/82 school
year by the staff of the Middle Grades Mathematics Project
(MGMP), Department of Mathematics, Michigan State
University, East Lansing, Michigan.

The MGMP is a curriculum develOpment project jointly
funded by the National Science Foundation-DeveIOpment in
Science Education (NSF-OISE) and Michigan State University

(MSU).

Pilot Testing

 

Before the Probability Unit was implemented for the
purpose of this study, it had been through several phases of
pilot testing and modification. One of the later stages of
pilot testing took place in Summer 1982 when the MGMP staff
taught the unit to forty middle grade students. Eight

affiliated middle school teachers participated in this

55

summer teaching institute. The affiliated teachers observed
the classes taught by the staff. With suggestions and.
criticisms from these teachers, the MGMP probability unit
was modified. This modified version was retested as schools
reopened in September 1982. At this time, it was taught in
one school (which later did not participate in this study)
by one of the teachers who had watched the summer
demonstration. Minor changes were made in the unit in

preparation for the present study.

The Probability Unit Activities

The Probability Unit includes ten sequentially
developed activities requiring about three weeks of
instructional time. The activities of the unit include the
following:‘ State lottery, three activities on fair and
unfair games, surveys, area models, expected value,
newspapers pay, Jonesville families and Pascal's Triangle.

The first five of these activities strictly involve
determining probabilities of independent events. These are
probabilistic conditions in which the outcome of one event
does not depend on another. Simple examples of independent
events are observed in the repeated tossing of a die or
coins. The remaining five acitivites deal with compound
events (for example the probability of a 60% free throw
shooter in basketball hitting two in a row) and binomial

probabilities.

56

The unit assumes that the students are being exposed to
probability instructions for the first time. Hence in the
first activity, the definition of probability as a fraction
or ratio is given as follows:

The probability that an event A will happen is the
number of times A occurs divided by the total number of
possible events. That is,

P(A) = number of A

 

total number of outcomes

Activities two to four introduce the probabilistic
thinking involved in deciding if a game is fair or unfair.
Playing fair and unfair games in pairs, the students are
introduced to experimental and theoretical probability.
Simple tree diagrams are also used to explain theoretical
probability. Students are introduced to various ways of
conceptualizing probabilities rather than given an abstract
definition. Through such experimental approaches as coin
and die tossing, and spinner activities, students have
experiences both with fractions and decimals, and with
identifying a relationship between geometry and probability.
For example through the use of spinners as an experimental
tool (activity 4), area models (activity 6, 7) students are
exposed to such concepts as circles and angles, rectangles
and squares.

Activity 5 exposes the students to experimental

probability through useful and practical survey activities.

57

Examples of surveys introduced are traffic patterns, weather
predictions, political voting and rating.

Dividing geometrical shapes, especially squares and
circles, into equal units of area, and calculating
probabilities from these areas, are the focus of Activity 6.

In Activities 7 and 8, the use of probabilities to make
predictions, and calculate expected values are intrOduced.
Area models are also used to analyze compound situations.

In Activity 8, students are given an Opportunity to
plan a simulation of a problem, to carry out the simulation,
to analyze the problem theoretically and to compare the
results. .

The last two activities, 9 and 10, deal with binomial
probabilities. Students are introduced to the calculation
of probabilities involving dichotomous situations in which
two, and only two, possible responses exist at a time.
Examples are yes or no, boy or girl, true or false, heads or
tails events.

Activity 9 introduces these concepts through a "boy or
girl" activity, entitled Jonesville families. Activity 10,
with the introduction of the Pascal's Triangle, leads the
students to understand and appreciate the theoretical basis
of dichotomous probabilities.

Each activity is followed by practice questions. Also
at the end of the probability unit are rOurteen compre-
hensive review problems on all the activities in the entire

package.

58
Appendix B contains the test used to evaluate student

performance on the unit.

Population and Sample

 

The subjects of the study are grade six, seven and
eight students from six different schools, situated in three
distinct sites, two schools per site. One of these schools
is an elementary school, three are middle schools and two
are junior high schools. The three sites are categorized
into urban, suburban and rural settings and are respectively
referred to as site 1, site 2 and site 3 throughout this
study.

Site 1 comprises two inner city schools. One of these
is a junior high school situated in Pontiac, Michigan. The
other is an elementary school in the inner city of Lansing,
the state capital of Michigan. Their distance apart not
withstanding (about 80 miles), these two schools are similar
in socioeconomic, racial and demographic distributions. The
Lansing district demographic data for 1980-81 shows 65
percent White, 23 percent Black, and 10 percent Latino. The
site 1 Lansing inner city school demographic data for the
same year shows 56 percent White, 17 percent Black, and 23
percent Latino. These distributions are presumed stable
till the time of this study. Site 2, the suburban site,
comprises one middle school and one junior high school.
Though about fifteen miles apart, both are schools situated

in metropolitan Lansing, Michigan, and are also similar in

59

social, economic and racial characteristics. Children
attending these schools come from upper-middle class
populations. These site 2 schools are situated in a
predominantly white domain. Site 3, the urban site, also
comprises two schools, a middle school and a junior high
school. These schools are situated in rural areas in the
suburbs of Lansing and serve middle class, predominantly
White communities. In each of these three sites, several
sixth, seventh and eight grade classes participated in the
study. However, not all sixth, seventh, or eighth grade
classes in each site participated in the study.

The entire sample comprised about 1460 boys and girls.
These students were from 66 classes taught by 30 different
teachers. Some teachers taught more than one class, and a
few taught classes in more than one grade level.

Tables 3.1 and 3.2 show descriptive information on the
subjects. Table 3.1 shows the distribution of the entire
sample by grade level (six through eight) and by sex in each
of the three sites. Table 3.2 shows the distributions of
each site by the number of students, number of classes,
number of teachers and by the average number Of students per
class.

From the two tables presented, the information shows
that a large number of subjects were involved in each site
in the study. Any differences observed were therefore
pressumably due to factors other than the size of the

sample.

 

 

 

 

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61

TABLE 3.2

DISTRIBUTION OF THE WHOLE SAMPLE BY SUBJECT
BY CLASS BY TEACHER BY SUBJECT AND CLASS

 

 

 

 

 

 

N33 Nab NTC Nsycd
Site 1 444 20 12 22
Grade 6 149 b 6 25
Grade 7 138 7 3 20
Grade 8 157 7 3 22
Site 2 621 30 11 21
Grade 6 340 15 6 23
Grade 7 101 6 2 17
Grade 8 180 9 3 20
Site 3 381 16 13 25
Grade 6 133 5 5 27
Grade 7 '140 6 6 25
Grade 8 108 5 2 22
All 1446 66 36* 24
Grade 6 622 26 17 20
Grade 7 379 19 11 21
Grade 8 445 21 8 22
a NS-Number of subjects
bNC-Number of classes
3 NT-Numkber of Teachers

3(-

NS/c-Average number of subjects per class
to the nearest whole N.
Teachers who taught two grade levels were

counted twice.
teachers.

A total of 30 distinct

62

Instrumentation

 

The instruments used in this study were two semantic
differential attitude scales and a performance test on
probability ability. The two attitude scales were called
the Mathematics Attitude Scale (MAS) and the Probability
Attitude Scale (PAS). Throughout the study these two
attitude scales are referred to as MAS and PAS, while the
performance test instrument is called the Middle Grades

Mathematics Project Probability Test (MGMPPT).

Attitudes

 

The instrument to measure student attitudes toward
mathematics, MAS, was developed by Shumway and White (1981).
The same instrument was used in this study to measure the
subjects' attitudes toward the probability activities
treated in the instruction. The MAS was administered as a
pretest while both the MAS and PAS were administered as
posttest attitude measures. The Mathematics Attitudes Scale
and the Probability Attitude Scales appear in Apendix C.
both scales are a six-item semantic differential with five
Options each. The scales were scored by assigning scores of
5 through 1 to the subject responses. 5 and 1 signified the
most favorable and least favorable response respectively,
while the in between responses were accordingly scored 4, 3

and 2. The average, which marked the subject's attitude,

63

the subject's attitude, was then calculated on an attitude
score of a maximum of 5 and a minimum of 1.

According to Shumway and White (1981), the internal
consistency reliability estimates for the MAS ranged from
0.82 to 0.92. In this study, this MAS will be used with all
the subjects and the Cronbach a reliability estimates will
be computed by Sex, by grade level and by site.

In a study in which these attitude scales were used,
Ben-Haim (1982) confirmed the reliability of the
instruments. Ben-Haim computed Cronbach a reliability
estimates for the MAS and these estimates ranged from 0.79
to 0.91 in each Site, by time, by grade and by sex. For the
PearSon Correlation coefficients between the pretest and
posttest scores on the MAS, Ben—Haim also found these to
range from 0.64 to 0.82 which was significantly high
(P < 0.01).

The stability of the Mathematics Attitude Scale
instrument was supported by the high correlation between the
pre- and post-administration of the instrument. In this
study, the Pearson correlation coefficients between the math
attitudes pretest and posttest (a time period of about three
to four weeks) ranged mostly from 0.60 to 0.76. Tables C.1
to 0.6 Appendix C contain the Pearson correlation
coefficients between the pretest and posttest scores on the
Math Attitude Scale, for the entire subsample per Site by
grade by sex. Most of the correlations were Significant

(P < .01).

64

The Cronbach a reliability coefficients for the
Probability Attitude Scale in this investigation, ranged
from 0.80 to 0.89. Table 3.3 contains the reliability
coefficients of the Probability Attitudes Scale for the

post testing by Site by grade by sex.

Measuring Probability knowledge

\

 

The subject performance in probability was measured by
the Middle Grades Mathematics Project Probability Test
(MGMPPT). This test was developed by the MGMP staff,
including this investigator, in the Mathematics Department
of Michigan State University. The test was pilot tested
with hundreds of middle grades boys and girls in the Lansing
area during development. Its final form contains 25
multiple choice items with 5 options each. Before
administering the test, the teacher was instructed to
discuss the two sample items on the sample sheet that
preceded the test. This writer and members of the prOject
staff supervised the teachers to ensure that each teacher
carefully follow the instructions provided for the
administration of the test. The same probability test was
given for both the pretest and posttest measures, keeping
all the testing conditions as much the same as possible. The
test was not timed but the average time taken during the
pilot trials was about twenty-five minutes. The time lapse
between the pretest and posttest was three to four weeks.

This varied slightly from school to school. The items were

RELIABILITY CUEFFICIENTS-CRONBACH 6 FOR MGMP PT,
MAS AND PAS BY SITE BY TIME BY GRADE BY SEX

65

TABLE 3.3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PTa MASb PASC
PRETEST POSTTEST PRETEST POSTTEST
N
Site 1
Grade 6 149 .68 .67 .84 .83 .80
Grade 7 139 .61 .81 .83 .82 .82
Grade 8 157 .75 .82 .77 .84 .84
Total 435 .70 .79 .81 .83 .82
Boys 217 .73 .80 .so .30 .82
Girls 228 .66 .77 4.83 .85 .83
Site 2
Grade 6 340 .58 .74 .85 .86 .86
Grade 7 101 .73 .83 .88 .88 .89
Grade 8 180 .67 .73 .84 .85 .86
Total 621 .69 .78 .88 .88 .87
Boys 317 .74 .81 .86 .89 .88
Girls '306 .59 .73 .89 .87 .87
Site 3
Grade 6 133 .45 .58 .77 .83 ‘.84
Grade 7 150 .59 .79 .84 .86 .86
Grade 8 109 .58 .73 .82 .83 .82
Total 392 .56 .75 .82 .86 .85
Boys 203 .54 .77 .82 .86 .86
Girls 189 .56 .72 .82 .86 .84
All
Grade 6 622 .61 .71 .83 .85 .85
Grade 7 380 .70 .83 .84 .85 .87
Grade 8 444 .70 .77 .83 .86 .88
Total 1446 .68 .79 .84 .86 .87
boys 731 .72 .80 .84 .86 .86
Girls 717 .75 .62 .85 .86 .87

 

a PT - Probability TeSt (Range 0-25).

b MAS - Mathematics Attitudes Scale (Range 1-5).
C PAS - Probability Attitudes Scale (Range 1-5).

66

scored by assigning 1 point for each correct item. Thus it
was possible for each subject to score a total between 0 and
25 (inclusive). Appendix 8 contains the 25 items that
constitute the MGMP PT.

The MGMP Probability Test served two purposes. _First,
it served as a measure of probability knowledge in order to
assess existing differences by grade level and by sex prior
to the intervention. Second, it served as a pretest-
posttest on probability in order to evaluate the effects of
instruction in activities involving probability tasks.

The Cronbach a reliability coefficients calculated for
the MGMP PT ranged largely between 0.65 and 0.83. With the
exception of the value of 0.58 for site 3 grade 6 pretest,
each of the 18 Cronbach a reliability coefficients
calculated for the MGMP PT posttest was well above 0.60.
Table 3.3 includes the reliability coefficients for the MGMP
PT for each site, by time by grade and by sex.

Test validity and reliability were based upon scholarly
analyses of test items by researchers, mathematics educators
and mathematicians. Another indicator of the quality or the
instrument was the Significantly high correlations between
the pre-post test scores. In this Study, the Pearson
correlation coefficients ranged from 0.53 to 0.76. Tables
C.1 to 0.6 in Appendix C contain the Pearson correlation
coefficients on the MGMP PT scores for each Site, by grade

and by sex. Almost all the correlations were significant at

P < .01.

67

Procedure and Data Collection

 

The study was conducted during Fall 1982 and Winter
1983 and the duration of the data collection was between
October 1982 and January 1983. Each teacher gave the
Mathematics Attitude Scale (MAS) and MGMP Probability Test
on the day prior to the beginning of the probability
intervention.1

The administration or the tests was restricted to the
regular mathematics hour. The unit had not been taught
previously by any of the teachers involved with the unit.
The probability instruction continued through every
mathematics lesson until the completion and administration
of the two posttests and the Probability Attitude Scale

(PAS).

Data Collection

 

The following data were collected:
1. General Information Such information on the subjects
and the schools participating in the study as the size,

type, and setting.

 

1 Due to factors beyond the control of the researcher,
each teacher started the instruction from any Monday of
his or her convenience and completed it in three to four
weeks.

68

Pretesting Pretest scores on the Mathematics Attitude

 

Scale and MGMP Probability Test from the whole sample in

all three Sites.

Observations During the instruction in the probability

unit which required about three weeks of instructional
time, the investigator made some observations in many of
the participating classrooms. The purpose of these
observations was to supervise as well as to document the
progress of the activities.

Posttesting At the end of the instructional

 

intervention all subjects were given the Mathematics

Attitude Scale, the Probability Attitude Scale (PAS) and

‘the MGMP Probability Test.

The Design of the Study

 

Several statistical procedures were selected to analyze

the data collected during the study. Particular statistics

were chosen to test the hypotheses given in the following

paragraphs. The analyses included the following: means,

standard deviations, correlations, multivariate and

univariate analyses of variance and repeated measures.

Planned comparisons and Scheffe's Post Hoc comparison were

also selected. All these analyses were carried out on the

3600 Computer at the Michigan State University Computer

Center, using the Statistical Package for the Social

Sciences (SPSS).

69

The multivariate model with a two-way fixed effects
analysis of variance (MANOVA) was used to assess the
differences in probability knowledge and attitudes toward
mathematics by grade level and sex prior to the
instruction.

As described earlier in this chapter, the populations
of the three sites (1, 2, and 3) were different in their
characteristics. Hence the data from each site was analyzed
separately. For each Site, the design was 3 x 2 completely
crossed with two criterion measures and with unequal numbers
of subjects per cell. The multivariate design for each site
appears in Table 3.4. The table Shows that the data matrix
is completely crossed Since in each site there were three.
grade levels (6, 7 and 8) with boys and girls in each grade
level. The grade level and sex main effects constituted the
independent variables, while the MGMP probability test and
attitude scores constituted the dependent variables. A
total of five measures were taken on each subject. Two of
these were administered prior to instruction. These two
were the pretest in probability and pretest in mathematics
attitude. The other three were given immediately after the
instruction. These were the Posttest in Probability
(POSTPTOT), Posttest Mathematics Attitude Scale (POSTMAS)

and the Probability Attitude Scale (PAS).

70

TABLE 3.4

THE 3 X 2 MULTIVARIATE CROSSED DESIGN DATA MATRIX

 

 

 

 

 

 

 

 

Indepedent Dependent
Variables Variables
Grade Sex 4_ MGMP PTa Scores MASb Scores

6 Boys X1 X2
Girls X1 Mg

7 Boys X1 X2
Girls x1 My,

8 boys x1 x2
Girls x1 x2

 

a MGMP PT - Middle Grades Mathematics Project

Probability Test.

b MAS - Mathematics Attitude Scale.

71

The hypotheses tested within this design, given in null

form, were:

H01: There will be no difference among the mean scores
for each of the three grade levels (six, seven, and
eight) tested, on both the Middle Grades Mathematics
Project Probability Test and on the Mathematics
Attitudes Scale.

H02: There will be no difference between the mean
scores for boys and for girls in grades six through
eight on both the Middle Grades Mathematics Project
Probability Test and on the Mathematics Attitude
Scale.

H03: There will be no interaction of grade by sex among
the mean scores for sixth through eighth grade
students on both the Middle Grades Mathematics Project
Probability Test and on the Mathematics Attitude
Scale.

For the tests of significance, the assumptions or
independence and normality were met. In other words, the
characteristics of the measures of all the dependent
variables in the study included the fact that subjects
responded independently of each other, and that any
measurement errors within each group were normally
distributed with mean zero and general population variance.
Another assumption met was that measures of the dependent
variable must not be linear functions of one another for
each subject. For example, in the measure of each pair of
dependent variables in this Study, not every subject had the
same subset scores or total score for both dependent
variables. Also, Since the model had only six (3 x 2)
subclass means, and Since assumptions of independence of

these subclass means was met, it was permissible to test or

estimate the significance of these six means. however,

72

because there were only Six degrees of freedom among means,

not more than Six independent Significance estimations was

allowed.

To analyze the effects of instruction on the
probability skills, differences in probability and attitudes
toward mathematics, by sex and grade level, the Multivariate
Analysis of Repeated Measures Model was selected.

The design for the subsample from each site was a
Two-Way Six-group design, with four measures per subject.
The design appears in Table 3.5.

The hypotheses tested within this design, given in null
form were:
h04: There will be no difference between the posttest

means and the pretest means of sixth, seventh, and
eighth grade Students on both the Middle Grade
Mathematics Project Probability Test and on the
Mathematics Attitudes Scale.

H05: There will be no difference between the mean gain
scores (posttest minus pretest) for each of the three
grade levels tested, Six, seven, and eight on both the
Middle Grades Mathematics Project Probability Test and
on the Mathematics Attitude Scale.

H06: There will be no difference between the mean gain
scores (posttest minus pretest) for boys and for girls
in grades six, seven, and eight on the Middle Grades
Mathematics Project Probability Test and on the
Mathematics Attitude Scale

To analyze the effects of instruction on probability, the

Multivariate Analysis of Repeated Measures was used. Both

the Finn and the SPSS methods were available for the

repeated measures analysis but the latter was used to keep

uniformity with the other statistical analyses tested in the

study.

73

TABLE 3.5

THE MULTIVARIATE ANALYSIS OF REPEATED MEASURES
DESIGN FOR THE SUBSAMPLE FOR EACH SITE

 

Grade Sex MGMPPT Scoresa MAS Scoresb

Pretest Posttest Pretest Posttest

 

 

 

 

 

6 Boys X1 X2 Y1 Y2
Girls X1 X2 Y1 Y2
7 Boys X1 X2 Y1 Y2
Girls X1 X2 Y1 'Y2
8 Boys x1 x2 Y1 Yxr
Girls X1. X2 Y1 Y2

 

a MGMP PT - Middle Grades Mathematics Project Probability
Test.
b MAS - Mathematics Attitude Scale.

74

To compare the attitudes toward mathematics and
probability by sex and by grade after instruction, a Two-Way
ANOVA design was used. The hypotheses tested within this
design for each Site were:

H07: There will be no difference between students' mean

' scores on the Mathematics Attitude Scale (MAS) and
students' mean scores on the Probability Attitude
Scale (PAS).

Hog: There will be no interaction of grade by sex among
the mean difference scores--MAS score minus PAS
score.

A Two-Way ANOVA design was also applied in order to
examine sex and grade level differences in attitudes toward
probability. The hypotheses tested within this design in
each site were:

H09: There will be no significant difference between
the mean scores for boys and for girls in grades six,
seven, and eight on the Probability Attitude Scale.

H10: There will be no difference between the mean
scores for each of the three grade levels (six, seven
and eight) on the Probability Attitude Scale.

Planned comparisons for the grade effect and Scheffe's
Post Hoc comparisons were used to identify the sources of
significant main effects or interactions. The .05 level of

significance was the limit accepted in testing all the

hypotheses in the study.

Summary

This chapter described in detail the entire research
methodology of this study. Specifically, the procedures

followed in the investigation of probability knowledge and

75

attitudes toward mathematics per Site by sex and by grade
level were presented.

This chapter also described in detail the pOpulation
and sample of the Study including the distribution by Site
(1, 2, and 3), by grade level (6, 7, and 8). The unit of
analysis chosen for the Study was the Student. From Site 1,
the urban setting, 444 subjects took part. From the
suburban setting, Site 2, 621 subjects participated while
381 subjects took part in the study from Site 3, the rural
setting. Of the 1460 subjects involved in the study, 1446
took part consistently in the data analyses. The difference
of 14, which was highly insignificant, was due to random
omission. There were 622 Sixth graders, 379 seventh
graders, 445 eighth graders, 725 boys and 721 girls who took
part in the entire study.

Also described in this chapter was the Middle Grades
Mathematics Project (MGMP) Probability Unit which consti:
tuted the instructional intervention of the Study. The
sequence of activities and the instructional mOdel were
described. The instrumentation included two semantic dif-
ferential scales (MAS and PAS) for measuring attitudes to-
ward mathematics and toward probability. It also included
the MGMP probability test for measuring the probability per-
formance of the subjects. Reported reliability coefficients
for the two attitude scales ranged from .80 to .89, while
those of the probability test ranged from .54 to .83 except

for the .45 for site 3 grade 6. The Pearson correlation

76

coefficients between successive administration of the
instruments were reported to be mostly Significant at
P < .01.

The statistical design, hypotheses and methoos of
analyses were presented. These included means, standard
deviations, correlations, univariate and multivariate
analysis of variance, and repeated measures, planned
comparisons, and Sheffe's Post Hoc comparisons for
significance. The Significance level chosen was at least
.05 for all hypotheses.

Results obtained from the different analyses and their
interpretation are presented and discussed in the next

chapter.

CHAPTER IV

PRESENTATION AND ANALYSIS OF DATA

This chapter presents a summary of the data collected

during this investigation, the analysis of data, and find-

ings based on this analysis. It consists of seven sections:

1.

Sex and Grade level differences in probability
performance and in attitudes toward mathematics
Comparison of results from the three sites.

The effects of instruction.

Comparison of effects of instruction among the three
Sites.

Comparison of attitudes toward mathematics with
attitudes toward probability.

Sex and grade level differences in attitudes toward
probability.

The chapter concludes with a summary of results.

Sex and Grade Level Differences in Probability Knowledge and
in Attitudes Toward Mathematics

For the data for Site 1, Site 2 and Site 3, the

Multivariate model with a Two-Way fixed effect analysis of

variance (MANOVA) was used to determine differences in

77

78

probability knowledge and in attitudes toward mathematics by
grade level and by sex prior to the intervention. As
explained earlier (Chapter 3), the data from each Site was
analyzed separately since the pOpulations from the three
Sites were different in their characteristics.

The analysis for each site includes the hypotheses to
be tested, means and standard deviations, profiles by grade
and by sex for each measure, a summary table of multivariate
and univariate analysis of variance, results of post hoc
comparisons, and the results of the significance tests for
each null hypothesis. The level of Significance for each
test was (P < .05) and the design was 3 x 2 crossed with two

criterion measures.

Site 1: The Urban Site

The multivariate null hypotheses tested within this
design were:

H01: There will be no difference between the mean
scores of the three grade levels (6, 7 and 8) tested,
on both the MGMP Probability Test and on the
Mathematics Attitude Scale.

H02: There will be no difference between the mean
scores for boys and for girls in grades Six, seven and
eight on both the MGMP Probability Test and on the
Mathematics Attitude Scale.

H03: There will be no interaction of grade by sex
between the mean scores for Sixth, seventh and eighth
grade students, on both the MGMP Probability Test and
on the Mathematics Attitude Scale.
Table 4.1 provides means and standard deviations for
the Middle Grades Mathematics Project Probability Test

(MGMPPT) and the Mathematics Attitude Scale (MAS) scores for

79

TABLE 4.1

MEANS AND STANDARD DEVIATIONS OF MGMP PT AND MAS PRETEST
SCORES FOR SITE 1 BY GRADE AND BY SEX

 

 

 

 

 

MGMP PTa MASb
Grade u m 3.0. M 5.0.
Grade 6 149 6.63 3.44 3.607 .947
Boys 76 6.97 4.04 3.568 .964
Girls 73 6.27 2.65 3.648 .934
Grade 7 138 7.62 3.30 3.457 .920
Boys 66 7.45 3.25 3.548 .371
Girls 72 7.78 3.35 3.373 .962
Grade 8 157 7.92 4.09 3.462 .818
Boys 74 8.10 4.20 3.532 .807
Girls 83 7.70 3.99 3.400 .828
Total 444 7.39 3.67 3.509 .896
Boys 216 7.53 3.89 3.549 .880
Girls 228 7.27 3.46 3.471 .911

 

a MGMP PT - Middle Grade Mathematics Project Probability
Test (Range 0-25).
b MAS - Mathematics Attitude Scale (Range 1-5).

80

site 1 by grade and by sex. The mean scores from this table
were used in the proriles Shown in Figures 4.1 and 4.2 for
grade levels by sex for both the MGMP PT and MAS measures.
The same means in Table 4.1 were also used in Figures 4.1
and 4.2 to display the sex by grade level profiles for the
same two measures. 4

A summary of the Multivariate and Univariate analysis
of variance for the 3 x 2 crossed design for this Site is
presented in Table 4.2. The multivariate test indicated
that the grade by sex interaction and sex main effects were
not significant but were significant for the grade level
main effect (P < .05). The univariate tests showed the
Significance to be confined to the pretest probability test
(PREPTOT) scores, and not to the pretest attitude scale
(PREMAS) scores. This means that in Site 1, there was no
significant difference among grades 6, 7 and 8 in attitudes
toward mathematics prior to instruction.

In order to determine in which of the three grade
levels significant differences occured in the MGMP
Probability test scores prior to instruction, Scheffe's Post ‘
Hoc tests were used. AS can be observed in Table 4.2 and
confirmed statistically with the Post Hoc tests, the
significant grade level differences in the PREPTUT was'
confined to between sixth and seventh graders and not
between seventh and eighth graders. both the profiles of
grade levels in Figure 4.1 and the mean PREPTOT scores in

Table 4.1 indicate a difference on the Side of seventh

81

Grade 6: N = 149
8.5 Grade 7: N = 139
. Grade 8: N = 157

8.0

7.5

 

7.0

3&9

6. 5 99‘6“

PREPTOT Means

6.0

Girls Boys Sex

 

Fig. 4.1 PREPTOT MeanS--Profiles of Sex by Grade

. Level at Site 1.

S2

 
   
  

Grade 6: N 8 149
Grade 7: N r 139
Grade 8: N - 157

PREMAS Means

Girls Boys Sex

 

Fig. 4.2 PREMAS Means--Profiles of Sex by Grade Levels
at Site 1.

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84
graders. Appendix E (Table E.1) includes the summary table
of Scheffe's posteriori comparisons of the probability
pretest means for boys and for girls in grade levels six and
seven. These comparisons were found to be statistically
different in favour of grade level seven girls over grade
level Six girls at .05 level. However, an examination of
the two means showed a difference of only 1.50. Hence this
statistical difference was detected due to the high
precision level of the test. In other wOrds, because of the
large sample size used, the test was powerful enough to
detect small differences which were not necessarily meaning-
ful. Thus, the statistical difference identified not-
withstanding, the grade seven girls were not necessarily
superior to grade six girls in MGMP probability knowledge
prior to instruction.

To summarize, based on the statistical analySiS of the
data from Site 1, the following decisions were made with
respect to the hypotheses H01-H03 (starting with the
interaction):

1. The multivariate null hypothesis (H03) of no
interaction of grade by sex was retained.

2. The multivariate null hypothesis (H02) of no
difference between boys and girls in grades six through
eight was also retained.

'3. The multivariate null hypothesis (H01) of no
difference between grade levels six through eight was

rejected.

85
The ensuing post hoc contrast between grade levels eight
versus six and seven was not significant. However, the
contrast between grade level six versus seven was
significant (P < .05). The univariate analyses for these
two grades for the MGMP PT and MAS scores showed that the
statistical significance was limited to the MGMP PT scores
only. Lastly, the Scheffe's Post Hoc comparison of the
probability test scores between grade levels six and seven
was statistically but not meaningfully Significant between

grade Six girls versus grade seven girls.

Site 2: The Suburban Site

 

The same three multivariate null hypotheses
(H01-H03) were tested in the analysis of the data
from this Site.

Table 4.3 presents the means and standard deviations of
both the MGMP PT and the MAS criterion measures by grade and
by sex. These means were used in Figures 4.3 to 4.5 to
illustrate various profiles for both criterion measures by
grade level and by sex.

The summary of the Multivariate and Univariate analysis
of variance for the 3 x 2 design of Site 2 is presented in
Table 4.4 No interaction was found of grade by sex but the
multivariate analysis of sex and grade main effects were
Significant. The test for sex main effects was significant
(P < .001) and the corresponding univariate test Showed only

the probability test scores to be significant (P < .001).

86

TABLE 4.3

MEANS AND STANDARD DEVIATIONS OF MGMP PT AND MAS PRETEST
SCORES FOR SITE 2 BY GRADE AND BY SEX

 

 

 

 

 

MGMP PTa MASb
Grade N M S.D. .M S.D.
Grade 6 340 7.80 3.11 3.510 .908
Boys 162 8.14 3.41 3.422 .930
Girls 178 7.50 2.78 3.590 .883
Grade 7 101 11.71 3.86 3.406 .856
Boys 53 12.68 4.09 3.582 .947
Girls 48 10.65 3.31 3.212 .702
Grade 8 180 9.66 3.65 2.622 .866
Boys 102 10.13 4.02 2.613 .850
Girls 78 9.05 3.03 2.635 .892
Total 621 8.98 3.70 3.236 .970
Boys 317 9.54 4.08 3.188 .990
Girls 304 8.40 3.15 3.285 .948

 

a MGMP PT - Middle Grade Mathematics Project Probability
Test (Range 0-25).
b MAS - Mathematics Attitude Scale (1-5).

13.0 ‘ Boys: u - 317

Girls: N = 304
12.5 I
12.0
11.5

11.0

PREPTOT Means
\0
U1

 

6 7 ' 3 Grade

Fig. 4.3 PREPTOT Means--Profiles of Grade Levels by Sex
at Site 2.

66

4.20 ‘ Boys: N = 317
8 Girls: N 3 304

503

PREMAS Means

6 7 ‘ 8 Gra-e

Fig. 4.4 PREMAS Means--Profiles of Grade Levels by Sex
at Site 2.

-A_- -.-.- -_. ...___..._.-

89

Grade 6: N = 340
13.0 . Grade 7: N - 101
Grade 8: N = 180

12.5

12.0

11.5

11.0

10.5

10.0

 

PREPTOT Means
\0
Ln

Gr ade

sixth

Girls 5 Boys Sex

Fig. 4.5 PREPTOT-Means--Profiles of Grade Levels
by Sex at Site 2.

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91

Scheffe's posteriori sex comparisons, presented in
Appendix E (Table E.2) showed no Significant difference
between boys and girls in grade six. However, these
comparisons Showed seventh and eighth grade boys to be.
significantly higher (P < .05) than the seventh and eighth
'grade girls. The profiles of grade levels by sex in Fig.
4.3 also verify this. No statistically Significant sex
differences were Observed in MAS scores.

For the grade main effects, both planned comparisons,
grades seven versus grade six and grade eight versus grades
six and seven, were Significant (P < .0001 and P < .001
respectively). The corresponding univariate tests showed
grade seven to be significantly higher than grade Six in the
probability test (P < .0001) but no such difference was
'observed when grade eight was compared to grades six and
seven. However, both Fig. 4.5 and the Table E.2A in
Appendix E, Showing the summary table of the Scheffe's
poSteriori comparisons of grade levels for boys and for
girls in the MGMP PT, Show the mean of grade seven to be
significantly higher than that of grade eight. The Post Hoc
tests, and the profiles of grade levels by sex in Fig. 4.3,
Show the superiority of grade seven boys and girls over
“their corresponding sex in grades six and eight in the MGMP
PT. The Post Hoc posteriori comparison of grade levels for
boys and for girls on the MAS is presented in Table E.ZB of
Appendix E. Mathematics attitude differences were not

observed between grade levels Six and seven but eighth

92

graders Significantly (P < .05) demonstrated lower

mathematics attitude on the MAS than either the Sixth or

seventh graders.

To summarize, based on the statistical analysis of the

data from Site 2, the following decisions were made with

respect to hypothesis H03, H02 and H01 in that order:

1.1

The multivariate null hypothesis (h03) of no

interaction of grade by sex was retained.

The multivariate null hypothesis (H02), of no sex
differences in each of grades six, seven and eight, was
rejected (P < .001). The correSponding univariate test
for the prODability test mean scores was also rejected
(P < .001) while that of the MAS mean scores was
retained.

The multivariate null hypothesis (H01), of no
difference among mean scores for each of the three grade
levels six through eight, was rejected (P < .001)._ The
two planned comparisons, between grade level seven
versus six, and grade levels eight versus Six and seven,
were both Significant (P < .0001). The corresponding
univariate null hypotheses for the MGMP PT mean scores
was rejected for the contrast between grade seven and
Six (P < .0001), but retained for that between grade
eight versus grades Six and seven. The univariate null
hypotheses for the MAS mean scores was also rejected

(P < .0001).

Site 3: The Rural Site

Again, the three multivariate null hypotheses
(H01-H03) were tested in the analysis of the data
from this site.

The means and Standard deviations of both the MGMP PT
and the MAS criterion measures are presented in Table 4.5 by
grade and by sex. These means were used in the profiles of
grade level and of sex in Figures-4.6 and 4.7.

A summary of the Multivariate and Univariate analysis
of variance for the 3 x 2 design of site 3 is presented in
Table 4.6. The multivariate analysis showed no interaction
of grade by sex, but showed the sex main effects to be
Significant (P < .01), and the two planned comparisons for
the grade main effects to be significant; (P < .0001) for
G2, the contrast of grade eight versus grades Six and
seven, and (P < .01) for G1, the contrast of grade seven
versus grade Six.

The univariate test for the sex main effects was
significant (P < .01) for the MGMP PT mean scores but not
significant for the MAS mean scores. Employing Scheffe's
Post Hoc posteriori comparisons to determine in which grade
levels these sex differences existed, the differences were
observed to be Significant (P < .05) only at grade level
six, in favour of boys. Table E.3 in Appendix E includes a
summary of these comparisons.

The univariate tests for the contrast G2 (grade 8

versus grades 6 and 7) were Significant for both the MGMP PT

94

TABLE 4.5

MEANS AND STANDARD DEVIATIONS OF MGMP PT AND MAS PRETEST
SCORES FOR SITE 3 BY GRADE AND BY SEX

 

 

 

 

 

MGMP PTa MASb
Grade N M S.D. M S.D.
Grade 6 133 8.64 2.69 3.450 .752
boys 70 9.26 2.83 3.419 .756
Girls 63 7.95 2.36 3.484 .752
Grade 7 140 9.13 3.20 3.207 .819
Boys 65 9.40 3.12 3.177 .706
Girls 75 8.89 3.27 3.233 .910
Grade 8 108 9.99 3.16 _ 3.005 .804
Boys 57 10.27 3.16 2.842 .902
Girls 51 9.69 3.15 2.186 .640
Total 381 9.20 3.03 3.236 .810
Boys 192 9.60 3.04 3.166 .817
Girls 189 8.79 3.03 3.304 .797

 

a MGMP PT - Middle Grade Mathematics Project Probability
Test (Range 0-25).
b MAS - Mathematics Attitude Scale (Range 1-5).

 

 

95

Grade 6: N = 133
Grade 7: N 3 140
10.5 . Grade 8: N = 108

10.0

9.5

9.0

PREPTOT Means

8.5

8.0

 

Girls Boys Sex

Fig. 4.6, PREPTOT Means--Profiles of Sex by Grade
Level at Site 3.

96

Grade 6: N - 133
Grade 7: N - 140
Grade 8: N = 108

 

3.5 —_ Sixth Grade

3.4 4'
3.3

3.2 ¥ Seventh Grade

 

PREMAS Means

      

Girls Boys Sex

Fig 4.7 PREMAS Means--Profi1es of Sex by Grade Level
at Site 3.

97

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98

mean scores and the MAS mean scores (P < .01 and P < .001
respectively). For the contrast G1 (grade 7 versus grade
6), these tests were not significant tor the MGMP PT mean
scores but were significant (P < .01) for the MAS mean
scores as Observed trom Table 4.6. Using the Scheffe's Post
Hoc comparison, significant grade level ditferences were
observed only between the boys in grades eight and six tor
the MAS mean scores, and only between the girls in the same
grades (eight and six) for the MGMP PT mean scores. The
profiles or sex by grade level in Fig. 4.7 shows a drOp in
the MAS mean scores from grade level six to eight. however,
as mentioned already, the post hoc tests showed the
significant difterence on the MAS scores (P < .05) was
limited to between eight grade boys versus sixth grade boys,
and in tavour of the latter. No signiticant contrasts were
found in the other grade levels.

As a summary, based on the statistical analysis of the
data from Site 3, the following decisions were made
concerning the hypotheses (H03, hog, h01) in
that order:

1. The multivariate null hypothesis (h03), or no
interaction of grade by sex among the mean scores for
grades six through eight, was retained.

2. The multivariate null hypothesis (hug), or no
difterence between the mean scores tor boys and for
girls six, seven and eight, was rejected (P < .01). The

univariate test for the MGMP PT mean scores was also

99

rejected (P < .01), but retained for the MAS scores.
Scheffe's Post hoc posteriori comparisons showed that
these significant sex differences, observed in the MGMP
PT mean scores, were found between boys and girls in the
sixth grade, but in no other grade level.

3. The multivariate null hypothesis (H01) of no
difference among mean scores for each of the grade
levels six to eight was rejected (P < .01). The two
planned comparisons, between grade levels eight versus
six and seven, and between grade levels seven versus
six, were significant (P < .0001 and P < .01
respectively). Hith‘the exception of the MGMP PT mean
scores for the contrast of grades seven and six, the
corresponding univariate hypotheses for both criterion

measures were each significant for each contrast.

Comparison of Pretest Results Among Sites 1, 2, and 3

 

Although the data from each site was analysed
separately on the assumption that the three subsamples were
systematically different and representative of three
different populations, it was desirable and statistically
permissible to compare differences and similarities of the
results across the three sites.

The multivariate null hypothesis (h03) of no
interaction of grade by sex was retained for the data in
each site. The multivariate null hypothesis (H02) of no

sex effects was retained for the data in Site 1, but

100

rejected in Sites 2 and 3, and similar conclusions were
reached with respect to the univariate analyses.
Significant grade level differences were concluded for both
criterion measures in all three sites with respect to the
multivariate analyses, but results differed from site to
site in the corresponding univariate analyses. In Site 1,
no significant grade differences were found in the MAS mean
scores and the only MGMP probability test significant
differences were found between girls in grades seven and
six. No significant sex differences were found in any site
with respect to the MAS mean scores. In each of the sites,
graph profiles showed that grade six performed lower than
grades seven and eight on the MGMP PT mean scores. In Site
2, grade seven significantly outperformed grades six and
eight in the MGMP PT mean scores. Also, a comparison of the
grade seven MGMP PT mean scores for all sites from Tables
4.1, 4.3, and 4.5 snowed that Site 2 grade seven recorded
the highest mean scores of all grades of all sites. Fig.
4.8 includes the MGMP PT profiles of all sites by grade
levels. Table 4.7 includes the means and standard
deviations of totals for each site, for each grade level and
for each sex for all the five measures of the dependent

variables for the entire study.

MEANS AND STANDARD DEVIATIONS OF MGMP PT,

101

TABLE 4.7

MAS, AND PAS
SCORES FOR THE ENTIRE SAMPLE bY TIME BY GRADE bY SEX

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PRLPTOTa POSTPTUTb PREMASC POSTMASG PASe
M M M M M
(S.D.) (S.D.) (S.D.) (5.0;) (S.D.)
Site 1 444 7.394 10.712 3.509 3.351 3.315
(3.074)) (4.432) (.890) (.811) (.840)
Site 2 021 8.977 12.034 3.230 3.101 2.654
(3.095) (4.309) (.970) (.954) )(.939)
Site 3 381 9.202 12.405 3.230 3.104 3.081
(3.058) (3.935) (.810) (.841) (.889)
Grade 6 022 7.701 10.963 3.520 3.389 3.027
(3.177) -(3.750) (.887) (.878) (.043)
Grade 7 379 9.209 13.193 3.351 3.258 3.048
(3.775) (4.838) (.872) (.855) (.897)
Grade 8 445 9.120 11.593 3.011 2.952 2.823
(3.802) (4.259) (.909) (.802) (.900)
Boys 737 9.959 12.104 2.292 3.209 2.965
(3.871) (4.55[) (.920) (.910) (.901)
Girls 721 9.143 11.447 3.349 3.230 2.909
(3.274) (4.055) (.902) (.800) (.919)
All 1446 8.551 11.741 3.319 3.220 2.070
, (3.613) (4.309) (.015) (.886) (.941)
a Probability Pretest (Range 0-25).
b Probability Posttest (Range 0-25).
c Pretest Mathematics Attitude Scale (Range 1-5).
2 Posttest Mathematics Attitude Scale (Range 1-5).

Probability Attitude Scale (Range 1-5).

PREPTOT Means

12.

11.

11.

10.

10.

102

Grade 6: N
Grade 7: N
Grade 8: N

622
379
445

 

 

1 2 3 Site

Fig 4.8 Pretest Probability Means--Profiles of All Sites
Grade Level.

103
The Effects of Instruction

The analysis of the effects of instruction on
probability skills and on differences in probability and
attitudes towards mathematics, by sex and by grade level,
was conducted separately for each site. The design for the
subsample from each site was a Two-Nay 3 x 2 group design,
with four measures on each subject. The Multivariate and
Univariate Analysis of Repeated Measures was used for the
data from each site. The following three multivariate null
hypotheses were tested for each site:

H04: There will be no difference between the posttest
means and the pretest means or sixth, seventh, and
eighth grade students, on both the Middle Grades
mathematics Project Probability Test and on the
Mathematics Attitude Scale.

H05: There will be no difference between the mean gain
scores (posttest minus pretest) for each of the three
grade levels tested, six, seven, and eight, on both the
Middle Grades Mathematics Project Test and on the
Mathematics Attitude Scale.

H00: There will be no difference between the mean gain
scores (posttest minus pretest) for boys and for girls
in grades six, seven, and eight, on the Middle Grades
Mathematics Project Probability Test and on the
Mathematics Attitude Scale.

Site 1: The Urban Site

 

The pretest and posttest means and standard deviations
of the MGMP Probability Test (PT) scores and the Mathematics
Attitudes Scale (MAS) scores for the data from Site 1 are
presented by grade and by sex in Table 4.8.

A Summary of Multivariate and Univariate Analysis of

Repeated Measures for the data from Site 1 is presented in

104

TABLE 4.8

PRE AND POSTTEST MEANS* OF MGMPT PT AND MAS SCORES
FOR SITE 1 BY GRADE AND BY SEX

 

 

 

 

 

MGMP PTa MASb
Grade Pretest Posttest Pretest Posttest
N M M M M
Grade 6 149 6.62 9.79 3.607 3.259
Boys 76 6.97 10.16 3.568 3.241
Girls 73 6.27 9.41 3.648 3.276
Grade 7 138 7.62 11.07 3.450 3.379
Boys 66 7.45 10.38 3.548 3.407
Girls ' 72 7.78 11.75 3.373 3.350
Grade 8 157 7.93 11.29 3.466 3.419
Boys 74 8.16 11.91 3.532 3.453
Girls 83 7.70 10.66 3.400 3.384
Total 444 7.39 10.71 3.509 3.351
boys 216 7.53 10.82 3.549 3.364
Girls 228 7.27 10.61 3.471 3.339

 

31-0”

MGMP PT - Middle Grade Mathematics Project Probability
Test (Range 0-25).
MAS - Mathematics Attitude Scale (Range 1-5).
The corresponding standard deviations are included in
Appendix D, Table 0.2.

105

Table 4.9, following the format used by Winer (1962). The
table is in two parts. The first part shows averages of PT
and MAS mean scores over time, respectively abbreviated as
AVGPTOT and AVGMAS in Table 4.9. AVGPTOT can be taken as a
measure of the average amount of MGMP probability knowledge
possessed by subjects over time from pretest through
posttest. AVGMAS is the corresponding average of the
Mathematics Attitude Scale scores. Although AVGPTOT and
AVGMAS indicate some measure of the effect of instruction
over time, they were not the major concern of the null
hypotheses tested on the effect of instruction in this
study. hence these two measures were of little interest in
these analyses.

The second part of Table 4.9 shows differences of PT
and MAS mean scores over time, respectively abbreviated as
DIFPTOT and DIFMAS. DIFPTOT and DIEMAS are a measure of
time effect (differences or changes) over subjects from
pretest to posttest. Since the multivariate hypotheses
(H04-H06) tested in this study were concerned with
differences between posttest means and pretest means,
DIFPTOT and DIFMAS were the major focus of the analyses of
these hypotheses.

In all these analyses of repeated measures, each
DIFPTOT turned out to be a gain regardless of the
independent variable considered over time. DIFMAS generally
stayed the same over the given time interaction. The MGMP

PT pre-posttest means from Table 4.8 were used to draw the

106

profiles for pre-post test scores or gains by grade and by
sex in Figures 4.9, 4.12 and 4.13 (pages 57 and 58).

The Multivariate Analysis of Repeated Measures for Site
1 showed no significant interaction of grade by sex by time,
nor of sex by time. The test however snowed significant
interaction of one of the contrasts (G1) by time (P < .05).
The univariate test was not significant for UIPPTUT but was
significant (P < .01) for DIPMAS. This means that in the
G1 contrast (of grade 7 versus grade 6), grade 7 was
significantly different from grade 6 in attitude change to
mathematics over the period of instruction, but not
significantly different in probability knowledge gain from
pretest to posttest as measured by the MGMP PT.

Indeed, from the table of pre-posttest means, averages
and differences (Table 4.16, page 56), the attitude change
was negative for both grades six and seven. Scheffe's Post
hoc posteriori contrasts (Appendix 3.4) showed grade six
girls to have significantly changed more in mathematics
attitude than grade seven girls at the .05 level. however,
a comparison of their actual mean losses in the MAS scores
showed that the difference was -.35. This difference was
therefore concluded to be non meaningfully significant,
given the high power level of the statistical test as a
result of the large sample size characterizing the study.
The other contrast G2 by time (grade 8 versus grades 6 and
7 over time) was confounded in the G] by time test and

hence could not be tested. however, from a close look at

N

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63 76
AL 6G 73
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Fig. 4.9 Profile of Means on the MGMP PT by Grade by
Sex by Time in Site 1.

 

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110

the table and profiles of mean differences, Tables 4.16 and
4.17 and Fig. 4.9, it was apparent that boys and girls did
not differ significantly within and between grade levels in
MGMP probability knowledge gains over time. Table E.4 in
Appendix E also shows the post hoc contrast of a significant
.36 MAS mean difference between girls in grades six and
eight. The test of the overall time effect over the
subjects was highly statistically significant (P < .0001).
Although this test was also confounded in the G1 by Time
interaction (Table 4.9), the prOLiles (Fig. 4.9) and Table
(4.16) of mean gains show systematic mean gain scores in the
MGMP PT, ranging from 3.97 (by grade 7 girls) to 1.92 (by
grade 8 boys), from pretest to posttest, in each of the six
mean (difference) cells (Table 4.15). Thus, the significant
difference of overall time effect on the subjects in the
MGMP PT scores, showed in Table 4.9, was retained. However,
since the DIFMAS changes showed a range of magnitiude of .35
across the six mean (difference) cells, Table E.4, Appendix
B, it was concluded that there was no meaningfully
significant change of attitudes toward mathematics from
pretest to posttest, the post hoc result between girls in
grades eight and six notwithstanding.

To summarize, based on the statistical analysis of the
pretest-posttest data from Site 1, the following decisions

were made with respect to hypotheses (HOQ-HOO):

111

1. The multivariate null hypothesis (504) of no
difference between the posttest means and pretest means
was rejected (P < .0001).

2. The only statistically significant difference in grade
levels (six through eight) in mean gain scores was in
the mathematics attitude change. Further tests showed
this difference to be between grades six and eight
girls. This was considered logically small and
non-meaningfully significant. The multivariate null
hypothesis (H95) of no difference between the mean
gain scores for each of the three grade levels was
however rejected (P < .05) to avoid the risk of a Type
11 error.

3. The hypothesis (H06) was retained, and the
conclusion was in favour of no significant difference
between the mean gain scores for boys and for girls in
each of the grade levels six, seven and eight, on the
MGMP PT and on the MAS.

The summary of the Multivariate Analysis of Repeated
Measures for Site 1 (Table 4.9) and the graphical represen-
tation in Figs. 4.13 and 4.17 (pages 58, 62) show that
results with respect to AVGPTUT were similar to those or
DIFPTOT. There was no significant grade by sex interaction,
neither was the sex main effect significant. Boys and girls
in Site 1 therefore did not differ in their overall
knowledge (averages) in the probability measured by the MGMP

Probability Test. The grade levels seven and eight did not

112

differ statistically from each other in their knowledge of
MGMP probability, but each differed statistially from grade
level six as showed by Scheffe's Post Hoc posteriori compar-
ison. There was a mean average difference of only 1.37
between grades eight and six, and only 1.3 between grades
seven and six out of a maximum mean difference of 25.

Hence, these statistically significant differences were not
considered meaningful. Finally although grades seven and
eight knew more MGMP Probability than grade six (statisti-
cally higher AVGPTUT) they did not gain more (DIFPTOT). One
might conclude that, in Site 1, the MGMP instruction was

more effective in grade six than in grades 7 and 8.

Site 2: The Suburban Site

The pretest and posttest means and standard deviations
of the MGMP PT and MAS scores for the data from Site 2 are
presented by grade and by sex in Table 4.10.

A summary of Multivariate and Univariate Analysis or
. Repeated Measures for the data from Site 2 is included in
Table 4.11. The layout and the interpretation format of the
table are similar to the corresponding Table (4.8) in Site
1. AVGPTOT, AVGMAS, DIFPTOT and DIFMAS all refer to the
same measures in each site.

The results of the Multivariate Analysis of Repeated
Measures for Site 2 (Table 4.11) showed no significant
interactions of grade by sex by time, nor of sex by time.

The grade by time interaction was significant. however, in

113

TABLE 4.10

PRE AND POSTTEST MEANS* OF MGMP PT AND MAS SCORES
bOK SITE 2 BY GRADE AND bY SEX

 

 

 

 

 

MGMP PTa MASb
Pretest Posttest Pretest Posttest
Grade N M M M M
Grade 6 340 7.80 11.27 3.510 3.404
Boys 102 8.14 11.40 3.422 3.300
Girls 178 7.50 11.15 3.590 3.443
Grade 7 101 11.71 14.97 3.406 3.320
Boys 53 12.60 15.91 3.582 3.547
Girls 48 10.05 13.94 3.212 3.009
Grade 8 100 9.00 11.03 2.022 2.013
Boys 102 10.13 12.05 2.013 2.001
Girls I 73 9.05 11.55 2.035 2.028
Total 021 8.98 12.03 3.230 3.101
Boys 317 9.54 12.30 3.100 3.147
Girls 304 8.40 11.09 3.285 3.175

 

3&0"

MGMP PT - Middle Grade Mathematics Project Probability
Test (Range 0-25).
MAS - Mathematics Attitude Scale (Range 1-5).

The corresponding standard deviations are given in

Appendix D. Table D.3.

114

 

16.0
15.0
14.0
13.0
03
6 12.0
6
z.
‘5'. _ 11.0
95
o
:3 10.0
=- 162
9.0 = 178
=- 53
= 48
8.0 = 102
- 78
7.0
Pre Post Time.

Fig. 4.10 Profiles of MGMP PT Means by Grades by Sex
by Time in Site 2.

115

 

 

 

 

 

 

 

 

 

 

 

 

 

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117

the planned comparison of the contrasts 6} (grade 7 versus
grade 6), and G2 (grade 8 versus grades 6 and 7), the

former (6]) was not significant. This means that there

were no significant differences in MGMP PT and MAS gains due
to interaction between grade and sex over the time. Also,
there were no significant differences between boys and girls
in each grade level in MGMP Probability mean gains or mean
attitude changes. Grade 7 also did not differ significantly
from grade 6 in these criterion measures. however, the
contrast of grade 5 versus grades 6 and 7 was significant

(P < .0001). The corresponding univariate analysis confined
these differences to (P < .0001) to DIFPTOT and not DIFMAS
(Table 4.11). That is, differences between grade 8 versus
grades 6 and 7 were found in mean probability achievement
gains (DIFPTOT), but not in change in attitude to mathema-
tics (DIFMAS). In short, in the entire analysis of the data
from Site 2, no significant differences were observed among
both independent variables (sex and grade levels) in atti-
tude change toward mathematics (Table 4.11), as measured by
the Mathematics Attitude Scale. however, Table 4.10 of pre-
test and posttest means showed that boys kept more steadily,
on the positive side than girls, and grades eight and seven
than grade six. 'however, all differences in attitude change
toward mathematics over time were small and non statistical-
ly significant. Hence, it was considered unecessary to
display tables and profiles of attitude change towards

mathematics.

118

To determine which grade level and sex cells were
responsible for the MGMP PT gain differences observed in the
contrast of grade 8 versus grades 6 and 7, Scheffe's Post
hoc posteriori comparisons were employed. These pairwise
comparisons showed that significant probability gain
differences occurred between boys in grades 7 and 8, between
boys in grades 0 and 8, and between girls in grades 0 and 0
(Table E.5, Appendix E). These probability gain differences
were unexpectedly always in favour of the lower grade!
Similar conclusions are also observable from the table of
MGMP Probability gains and averages (Table 4.15), and from
the profiles of Site 2 mean gains and averages in figure
4.12. The same protiles show an ordinal interaction between
boys and girls across all the grade level factors in both
the gains and averages in the MGMPPT. In other words, in
each grade level in Site 2, girls gained more, but knew
less, probability than boys, as measured by the MGMP
Probability Test. Similar conclusions followed a close
examination of the profiles by grade, by sex, and by time
(Fig. 4.10), of the MGMP PT pre-posttest means.

The multivariate test showed the over all time effect
to be significant (P < .0001, Table 4.11). This test was
however confounded in the 62 by time contrast and the
result cannot be reported as significant with absolute
certainty. however, the table of probability achievement
gains (DIFPTOT), in Table 4.15, show some substantial gains

by each grade and sex with the exception of boys in grade 8

119

TABLE 4.12

PRE AND POSTTEST MLANS* 0k MGMP PT AND MAS SCORES
FOR SITE 3 bY GRADE AhD BY Shh

 

 

 

 

 

111cm PTa MASb
Pretest Posttest Pretest Posttest
Grade h M h h M
Grade 6 133 8.64 11.50 3.450 3.496
Boys 70 9.26 12.06 3.419 3.502
Girls 63 7.95 10.87 3.484 3.489
Grade 7 140 9.13 13.98 3.207 3.096
Boys 65 9.40 14.43 3.177 3.085
Girls 75 8.89 13.59 3.233 3.107
Grade 8 108 9.99 11.69 3.005 2.843
Boys 57 10.26 12.04 2.842 2.766
Girls 51 9.69 11.32 3.186 2.928
Total 381 9.20 12.47 3.236 3.164
Boys 192 9.66 12.85 3.166 3.142
Girls 189 8.79 12.07 3.304 3.186

 

a MGMP PT - Middle Grade Mathematics Project Probability
Test (Range 0-25).
b MAS - Mathematics Attitude Scale (Range 1-5).
* The corresponding Standard Deviations are given in
Appendix D, Table D.4.

MGMP PT Means

120

N of Grade 63 = 70
6G 3 63
78 = 65
7G - 75
BB - 57
86 - 51

 

Pre Post Time

Fig. 4.11 Profiles of MGMP PT Means by Grade by
Sex by Time in Site 3.

121

(with only a gain of 1.92). Hence the overall time effect
hypothesis was not considered tenable. Moreover, the risk
of a Type I error involved was small since both the
multivariate and univariate tests were highly significant
(P < .0001) in favour of MGMP Probability gains. Hence, the
conclusion was in favour of significant overall gains in the
MGMP PT scores by Site 2 subjects.
In summary, based on the Multivariate Analysis
of Repeated Measures of the data from Site 2, the following
decisions were made with respect to the null hypotheses
(H04-H06>=
1. The null hypothesis (H04) of no difference between
the posttest means and pretest means of the three grade
levels studied was rejected. However, no meaningfully
significant difference was found between the MAS mean
gain scores. This means that there was an overall
significant time effect on the subjects due to the MGMP
Probability instruction.

2. The null hypothesis (H05) of no difference between
the mean gain scores for the three grade levels was
rejected. The tests showed evidence of significantly
different mean gain scores among the three grade levels
studied on the MGMP PT scores, but not on the MAS
scores. Gain differences were found between grades 6
and 8, between grades 7 and 8, always in favour or the

lower grade; but not between grades 6 and 7.

122
3. The null hypothesis (H00) was tenable and retained.
The conclusion was that there was no significant
difference in the mean gain scores for boys and for
girls in grades six, seven, and eight on both the MGMP
PT and MAS.

Although the hypotheses tested did not involve averages
of probability achievement means and mathematics attitude
means, AVGPTUT and AVGMAS, it was deemed desirable to
comment on the Multivariate and Univariate results involving
these measures. The multivariate grade by sex interaction
test was significant (P < .05), and the univariate test
showed this significance to be in AVGMAS only, (P < .05,
Table 4.11). This means that there was a grade by sex
interaction in the subjects' overall attitudes to
mathematics as measured by the MAS. Because of this
significant grade by sex interaction the sex and grade
level, main effects could not be tested separately. Hence,
no statistical conclusions were feasible. However, the
profiles of mean averages by time by sex (Fig. 4.13) and by
grade level (Fig. 4.12) showed an ordinal interaction
between boys and girls across the three levels (6, 7 and 8)
of the grade factor. From the same two profiles and from
the Table 4.15, it was observed that boys consistently
scored higher averages than girls in the MGMP PT, most
conspicously in the seventh grade. A conclusion therefore
was that while girls consistently gained more probability

than boys across the three grade levels (as measured by the

123

MGMP PT), boys consistently knew more (Figures 4.12 and

4.13).

Site 3: The Rural Site

The pretest and posttest means and standard deviations
of the MGMPPT and MAS scores for the data from Site 3 are
presented by grade and by sex in Table 4.12.

Table 4.13 presents a summary of Multivariate and
Univariate Analysis of Repeated measures for the data from
Site 3. There was no significant interaction of grade by
sex by time, nor of sex by time. The profiles of mean gains
by sex in Fig. 4.13 and by grade level in Fig. 4.12 both
attest to the fact that boys and girls gained statistically
equally from pretest to posttest. Both the G1 by Time and
,G2 by Time planned comparisons were significant
(P < .0001) according to the multivariate tests, even when
the grade and sex main effects were reordered (Table 4.13).
The correSponding univariate results showed that DIPMAS was
not significant in each case. hence, in the Multivariate
and Univariate Analysis of the data from Site 3, there were
no significant differences in attitude change to mathematics
(as measured by the MAS) by sex, or by grade, or by the
interaction of both. Hence as before, tables and profiles
of mathematics attitude changes from pretest to posttest
were not necessary. However, from a survey of the

pretest-posttest means in Table 4.12, it was observed that

124

 

 

 

 

 

 

 

 

 

 

 

 

 

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126
the sixth graders had and gained more positive attitudes
toward mathematics than seventh and eighth graders.

The univariate analysis did show the contrasts (G1)
of grades seven versus six, and (G2) of grade eight versus
grades seven and six to be significant (P < .0001) with
respect to DIFPTOT. That is, the three grade levels
differed significantly in their gains in the MGMP PT scores
from pretest to posttest.

Scheffe's Post Hoc posteriori comparisons were con-
ducted in order to determine details of these differences.
Table E.b in Appendix E includes a summary of these pairwise
contrasts. Significant grade level differences in MGMP
Probability gains were found (P < .05) between boys in grade
six versus grade seven, and between girls in grade six
versus grade seven; always in favor of grade seven. Similar
results, also in favour of grade seven, were found between
grades seven and eight. Differences were also found (P <
.05) between boys in grades six and eight, and between girls
in these two grade levels - both in favour of grade six.

The contrast differences between grades six and eight were
however considered small and non meaningful - judging from
the high precision of the tests, due to the large sample
size used in the study. From these post hoc results and
from the profiles of mean gains in Fig. 4.12, the conclusion
was that grade seven gained most while grade eight gained
least by time in the MGMP Probability Test scores. Similar

conclusions were observed from the profiles of means by

127
grade, by sex, and by time in Fig. 4.1] and from the AVGPTOT
and DlFPTUT of MGMP PT mean scores in Table 4.15.

With respect to the overall time effect over the
subjects, the multivariate and univariate analyses were each
highly significant (P < .0001) in favour or gains in the
MGMP PT scores. hence, with arguments congruent to those
used previously with respect to time effect in Site 2, it
was concluded that there was significant overall gain in the
MGMP Probability by the subjects.

In summary, based on the Multivariate Analysis of
Repeated Measures of the data from Site 3, the following
decisions were made with respect to the null hypotheses
(Boa-H06):

1. The null hypothesis (H04), of no differene between
the posttest means and pretest means of the three grade
levels studied was rejected. however, no meaningfully
significant difference was found between the MAS mean
gain scores. This means that there was an overall
significant time effect on the subjects due to the MGMP
Probability instruction.

2. The null hypothesis (h05), of no difference between
the mean gain scores for the three grade levels, was
rejected. The tests showed evidence or significantly
different mean gain scores among the three grade levels
studied in the MGMP PT scores, but not on the MAS
scores. Gain differences were found between grades six

and eight, between grades seven and eight, always in

.128

favour of the lower grade, and between grades six and
seven, in favour of grade seven.

3. The null hypothesis (hog) was retained. The
conclusion was that there were no significant
differences in the mean gain scores for boys and for
girls in grades six, seven, and eight, on both the MGMP
PT and MAS.

With respect to results involving MGMP PT mean
averages, AVGPTUT, the Multivariate and Univariate tests
were significant with reSpect to sex main effects, to the
G1 contrast, and to the G2 contrast (P < .01, P < .0001
repectively). The profiles of mean averages by sex (Fig.
4.13) and by grade level (Fig. 4.12) indicate that boys
scored significantly higher averages than girls across the
three grade levels - a highly ordinal interaction, as
opposed to the disordinal (crossing) interaction of sex with
grade levels in the profiles of gains (Fig. 4.12). As in
Site 2, grade seven recorded the highest averages in the
probability means from pretest to posttest. Thus in Site 3,
seventh graders significantly gained and knew more of the
MGMP Probability than sixth or eighth graders. While boys
significantly knew more of the MGMP Probability than girls,

they did not gain more.

129

Comparison of the Effect of Instruction Among Sites
1, 2, and 3

 

The same three null hypotheses were tested separately
for each site using Multivariate and Univariate Analysis of
Repeated Measures.

The null hypothesis (h04), of no difference between
the posttest means and pretest means of the three grade
levels (6, 7, and 8) studied, was rejected, (P < .0001) for
the data in each site. This means that, in the entire
study, there was evidence of a significant overall time
effect on the subjects. In each site, the effect of
instruction was found to cause significant gains from
pretest to posttest in the MGMP PT scores. All attitude
changes were small and non-meaningfully significant. The
profiles of pre-posttest means of the MAS of the entire
sample by grade (Fig. 4.18), and, by sex and by site'(Fig .
4.19), all show slight mathematics attitude changes from
pretest to posttest, none of which was meaningfully
significant. Fig 4.19 shows that on the whole, boys were
more steady than girls in their attitudes toward mathematics
as measured by the MAS. Although girls in general drOpped
more in attitudes than boys, they still scored higher on
average attitude (AVGMAS) in the pretest and posttest
measures (Tables 4.14 and 4.15).

The null hypothesis (hos) was rejected (P < .05)
for the data from Site 1. The rejection was due to change

differences found between girls in grades six and seven

130

TAbLE 4.14

MEANS AND STANDARD DEVIATIONS 0F MGMP PT, MAS, AND PAS
SCORES FOR THE ENTIRE SAMPLE BY TIME BY GRADE BY SEX

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PREPTOTa POS'I‘PTO'I'b PREMASC POSTMASd PASe
M M 'M M M
(S.D.) (S.D.) (S.D.) (S.D.) (8.0.)
Site 1 444 7.394 10.712 3.509 3.351 3.315
(3.674) (4.432) (.896) ((811) ((840)
Site 2 621 8.977 12.034 3.236 3.161 2.654
(3.695) (4.309) (.970) (.954) (.939)
Site 3 381 9.202 12.465 3.236 3.164 3.081
(3.058) (3.935) (.810) (.841) (.889)
Grade 6 622 7.701 10.963 3.520 3.389 3.027
(3.177) (3.750) (.887) (.878) (.043)
Grade 7 379 9.269 13.193 3.351 3.258 3.048
(3.775) (4.838) (.872) (.855) (.897)
Grade 8 445 9.126 11.593 3.011 2.952 2.823
(3.802) (4.259) (.909) (.862) (.960)
Boys 737 9.959 12.104 2.292 3.209 2.965
(3.871) (4.551) (.926) (.910) (.961)
Girls 721 9.143 11.447 3.349 3.230 2.969
(3.274) (4.055) (.902) (.860) (.919)
All 1446 8.551 11.741 3.319 3.220 2.070
(3.613), (4.309) (.015) (.886) (.941)
a Probability Pretest (Range 0-25).
h Probability Posttest (Range 0-25).
C Pretest Mathematics Attitude Scale (Range 1-5).
2 Posttest Mathematics Attitude Scale (Range 1-5).

Probability Attitude Scale (Range 1-5).

131

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132

TABLE 4.16

PRE-POSTTEST MEAN DIFFERENCES AND AVERAGES OF
THE MGMP PT SCORES BY GRADE BY SEX PER SITE

 

 

 

 

 

N AVGPTOTa DIFPTUTb
Site 1 444 9.05 3.32
Boys 216 9.18 3.30
Girls 228 8.94 3.34
6 149 8.21 3.16
7 138 - 9.36 3.47
8 157 9.58 3.33
Site 2 621 10.51 3.06
Boys 317 10.95 '2.83
Girls 304 10.04 3.30
6 340 9.54 3.46
7 101 13.34 3.26
8 180 10.75 2.17
Site 3 381 10.83 3.26
Boys 192 11.23 3.25
Girls 189 10.43 3.28
6 133 10.07 2.86
7 140 11.55 4.85
8 108 10.84 1.70
All Sites 1446 10.15 3.19
Boys 725 10.50 3.08
Girls 721 9.79 3.30
6 622 9.33 3.26
7 389 11.23 3.92
8 445 10.36 2.48

 

a AVGPTOT - Averaging Probability Test Mean
Scores (Pre + Posttest).

b DIFPTOT - Differences in Probability Test
Mean Scores (time effect) over
subjects.

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6th Grade: N
7th Grade: N
8th Grade: N

MGMP PT Means

Pre Post Time

Fig. 4.16 Profiles of Pre—Post MGMP PT Means of
Entire Sample by Grade.

MGMP PT Means

138

By Sex By Site
12.5 '

12.0
11.5
11.0
10.5

10.0

 

Pre Post 'Pre Post Time

Fig. 4.17 Profiles of Pre-Post MGMP PT Means of Entire
Sample by Sex by Site. ’

MAS Means

Fig.
Sample by

139

 

 

6
5
4. 6th Grade- M = 622
\ 3 . ‘-
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7th Grade: N = 379
2 .
l
0 \
“M
9 8th Grade: N = 445

 

Pre Post Time

4.18 Profiles oprre-Post Mas Means of Entire
Grade.

140

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141

in attitudes toward mathematics. That is, in Site 1, no
significant grade level differences were found in the MGMP
PT mean gain scores, but were in the MAS mean scores. In
Sites 2 and 3, significant grade level differences were
found but in the MGMP PT mean gain scores only. In Site 2,
these differences were detected between boys in grade levels
seven and eight, between girls in grade levels six and
eight, and between boys in grade levels six and eight,
always in favour of the lower grade. In Site 3, these same
grade level pairwise differences were also found to be
statistically significant (P < .05), but gain differences
between grades six and eight were considered not
meaningfully significant. In addition, significant grade
level MGMP Probability gain differences were found between
boys and girls in grades six and seven, and in grades seven
and eight always in favour of grade seven. Thus, in Site 3,
the seventh graders gained more in MGMP PT scores than
grades six and eight. In Sites 2 and 3, the least MGMP mean
gain scores were recorded by eighth graders (Fig. 4.12).
Although average scores by time were not the focus of
the null hypotheses tested in the study, interesting results
were found in Sites 2 and 3. In both sites, seventh graders
averaged significantly higher by time (P < .05) than sixth
and eight graders in the MGMP Probability Test. In Site 3,
seventh graders gained more from pretest to posttest and had
higher average scores on the MGMP PT than sixth and eighth

graders.

142

These results can be observed from the profiles in
Figures 4.16, and 4.12, and also from Table 4.15.

From the profiles for the entire sample of mean gains
and averages (Figs. 4.14, 4.15) in MGMP PT scores, and of
pre-posttest means (Figs. 4.16 and 4.17), more comparisons
can be made among the three sites, with respect to grade
level differences in mean gains and averages. For the
entire study, there were no site by time interactions in the
MGMP PT mean scores (Fig. 4.17). In other words, the three
sites differed in the same order in both the pretest and the
posttest. however, Site 1 gained slightly more than Sites 2
and 3 (Fig. 4.14 and Table 4.12). These gain differences
were small and therefore considered nonmeaningful. 0n the
whole, grade seven gained more and averaged higher than
grades six and eight (Figs. 4.15, 4.16, and Table 4.12).

The null hypothesis (H06) of no difference between
the mean gain scores for boys and for girls in grades six,
seven and eight in both criterion measures was not rejected
for the data from any site. This means that no significant
sex differences per grade level were found in any site, with
respect to gains in the MGMP PT scores and in Mathematics
Attitude Scale. In every site, girls gained slightly higher
than boys in the MGMP PT scores, but none of these was found
to be significant (Table 4.12 and Fig. 4.13). In MGMP PT
knowledge (averages) by time, boys significantly

outperformed girls in Sites 2 and 3 but not in Site 1.

143

In conclusion, no significant differences were found
between boys and girls, on the whole, with respect to gains
in the MGMP PT and MAS mean scores. No meaningfully
significant differences were found in the entire study in
change in MAS scores by sex, by grade, or by site. Girls
changed (drOpped) more than boys in the MAS scores (Fig.
4.19), but these were not statistically significant.

Grade level differences in MGMP mean gains were found
in Sites 2 and 3, but not in Site 1. In Sites 2 and 3,
grade seven averaged significantly more than grades 6 and 8
in the MGMP PT scores. In Sites 2 and 3, grades six and
seven gained more than grade eight, while in Site 3, grade
seven gained significantly more than grades six and eight.

Finally, in the entire study, no meaningfully signifi-
cant differences were found by sex, by grade, and by site,
in attitude change toward mathematics. With respect to
knowledge gain in the MGMP PT, no overall differences were
found between boys and girls, or among the three sites, but

grade seven significantly outperformed the other grades.

Comparison of Attitudes Toward Mathematics With Attitudes
Toward Probability

 

 

Attitudes toward mathematics were compared by sex and
by grade level with attitudes toward probability, after the
instruction. A Two-Way Analysis of Variance (ANOVA) design
was employed in these comparisons and the data from each

site was analysed separately for reasons earlier explained.

144

The null hypotheses tested within this design were:

H07: There will be no difference between students' mean
scores on the Mathematics Attitude Scale (MAS) and
students' mean scores on the Probability Attitude
Scale (PAS).

hog: There will be no interaction of grade by sex among

the mean difference scores--MAS score minus PAS
score. ,

Site 1: The Urban Site

 

The means and standard deviations of posttest scores in
Mathematics Attitude Scale (POSTMAS) and scores in the
Probability Attitude Scale (PAS)* for Site 1 are included in
Table 4.17. These statistics are tabulated by grade and by
sex. Table 4.18 presents a summary of the Analysis of
Variance (ANOVA) for the results of the test on the mean
difference between POSTMAS and PAS scores.

From the ANOVA results, it was found that the
interaction of grade by sex was not significant. The F
values of the sex and grade main effects were also not
significant.

Hence, both null hypotheses H07 and H03 were
retained. This means that neither boys nor girls, nor the
grade levels six, seven, and eight studied in Site 1
significantly differed in their attitudes toward
mathematics (MAS) versus attitudes toward probability (PAS)

after the probability instruction.

 

* PAS - The Probability Attitude Scale test had no preteSt.

145

TABLE 4.17

MEANS AND STANDARD DEVIATIONS OF POSTMAS AND PAS
SCORES FOR SITE 1 bY GRADE AND bY SEX

 

 

 

 

 

POSTMASa PASD
Grade N M S.D. M S.D.
Grade 6 149 3.258 .816 3.156 .869
Boys 76 3.241 .826 3.118 .937
Girls 73 3.276 .810 3.194 .796
Grade 7 138 3.377 .858 3.460 .797
Boys 66 3.407 .744 3.288 .787
“Girls 72 3.350 .933 3.818 .778
Grade 8 157 3.416 .760 3.339 .828
Boys 74 3.453 .776 3.426 .735
Girls 83 3.384 .749 3.261 .901
Total 444 3.351 .811 3.315 .840
Boys 216 3.364 .791 3.276 .833
Girls 228 3.339 .828 3.352 .847

 

a POSTMAS - Posttest Mathematics Attitude Scale

(Range 1-

5).

b PAS - Probability Attitude Scale - no pretest for PAS

(Range 1-5).

146

TABLE 4.18

ANALYSIS OF VARIANCE SUMMARY FOR MEAN DIFFERENCE BETWEEN
THE POSTMASa AND PASb SCORES EOR SITE 1

 

 

 

Source of Variation D.F. MS F P
Grade 2 1.424 1.489 .227
Sex 1 1.114 1.165 .281
Grade x sex 2 2.233 2.335 .098

Between Groups 5

Within Groups 438 .956

 

a POSTMAS - Posttest Mathematics Attitude Scale.
9 PAS - Probability Attitude Scale.

147

Site 2: The Suburban Site

Table 4.19 includes the means and standard deviations
or the POSTMAS and PAS scores for the data from Site 2 by
grade and by sex. Table 4.20 presents a summary of the
ANOVA results tor the mean ditterence between the POSTMAS
and PAS scores. These (ANOVA) results indicated that both
the interaction of grade by sex and the sex main effects
were not significant. Hence, the null hypothesis (Hog)
was not rejected. However, the grade level main effect was
significant. The results or the planned comparisons, G2
(grade 8 versus grades 6 and 7), and G1 (grade 7 versus
grade 6), were signiticant (P < .05) in tavour or Gz. The
null hypothesis (H07) was thereiore rejected. The
conclusion was that grades 8 and 7 did not differ
signiticantly from each other in their attitudes to
mathematics and prooability but together ditfered trom grade
8 in these attitudes. Further, Table 4.19 indicates that,
in Site 2, Sixth and seventh graders preferred mathematics

to probability, but eighth graders did not.

Site 3: The Rural Site

 

The means and standard deviations of the POSTMAS and
PAS scores tor Site 3 are presented by grade and by sex in
Table 4.21. A summary of the ANOVA results tor the mean
difterence between the POSTMAS and PAS scores is contained

in Table 4.22. The results of the ANOVA were similar to

TABLE 4.19

MEANS AND STANDARD DEVIATIONS OF POSTMAS AND PAS
SCORES FOR SITE 2 BY GRADE AND BY SEX

 

 

 

 

 

POSTMASa PASb
Grade .N M S.D. M S.D.
Grade 6 340 3.404 .930 2.808 .943
Boys 162 3.360 .995 2.840 1.004
Girls 178 3.444 .869 2.780 .886
Grade 7 101 3.320 .850 2.790 .923
boys 53 3.547 .945 2.937 1.011
Girls 48 3.069 .653 2.629 .793
Grade 8 180 2.613 .824 2.285 .839
Boys 102 2.601 .819 2.335 .849
Girls 78 2.628 .836 2.220 . .827
Total 621 3.161 .954 2.654 .939
Boys 317 3.147 1.006 2.694 .987
GirIS‘ 304 3.175 .898 2.612 .887

 

5).

a POSTMAS - Posttest Mathematics Attitude Scale
(Range 1-

b PAS - Probability Attitude Scale - no pretest for PAS

149

TABLE 4.20

ANALYSIS OF VARIANCE SUMMARY FOR MEAN DIFFERENCLa EETNEEN
THE POSTMASb AND PASc SGGRES FOR SITE 2

 

 

 

Source of Variation D.P. MS F P
Grade** 2
sz (1) 3.15 5.83* .016
Gle (1) 1.10 2.05 .153
Sex 1 1.30 1.20 .273
Grade x sex 2 1.03 .95 .388
Between Groups 5
Within Groups . 615 .540

 

a The corresponding ANOVA Table of Averages is in
Appendix D.9.
POSTMAS - Posttest Mathematics Attitude Scale.
PAS - Probability Attitude Scale.
G2 - Contrast of Grade 8 versus Grades 6 and 7.
G1 - Contrast of Grade 7 versus Grade 6.
Significant P < .05.
* Table 0.6 in Appendix D includes the alternative ANOVA
test without grade level contrasts.

*X’beOU'

TABLE 4.21

MEANS AND STANDARD DEVIATIONS OF POSTMAS AND PAS

SCORES FOR SITE 3 BY GRADE AND BY SEX

 

 

 

 

 

POSTMASa PASb
Grade N M S.D. M S.D.
Grade 6 133 3.496 .792 3.440 .863
Boys 70 3.507 .805 3.469 .885
Girls 63 3.489 .783 3.407 .843
Grade 7 140 3.096 .837 2.828 .826
Boys 65 3.085 .782 2.805 .825
Girls 75 33.107 .886 2.849 .831
Grade 8 108 2.843 .761 2.968 . .866
Boys 57 2.766 .827 2.927 .977
Girls 51 2.928 .678 3.013 .729
Total 381 3.164 .841 3.081 .889
boys 192 3.142 .855 3.083 .938
Girls 189 3.186 .828 3.079 .840

 

a POSTMAS - Posttest Mathematics Attitude Scale

(Range 1-

5).

b PAS - Probability Attitude Scale - no pretest tor PAS

(Range 1-5).

151

TABLE 4.22

ANALYSIS OF VARIANCE SUMMARY FOR MEAN DIFFERENCEa BETWEEN
ThE POSTMASb AND PASC SCORES FOR SITE 3

 

 

 

 

Source oi Variation D.P. MS F P
Grade** 2
sz (1) 3.25 6.91* .009
G19 (1) 1.53 3.24 .073
Sex 1 .09 .09 .759
Grade x sex 2 .08 .09 .918
Between Groups 5
Within Groups 375 .470
a The corresponding ANOVA Table or Averages is in
Appendix D.9.
b POSTMAS - Posttest Mathematics Attitude Scale.
9 PAS - Probability Attitude Scale.
9 G2 - Contrast oi Grade 8 versus Grades 6 and 7.
e G1 - Contrast of Grade 7 versus Grade 6.
* Signiticant P < .01.
*

* Table D.7 in Appendix D includes the alternative ANOVA
test without grade level contrasts.

152

those of Site 2. No significant interaction of grade by sex
was found. hence the null hypothesis (Hug), of no
interaction of grade by sex, was retained. However, grade
level differences were found (P < .01) between grade 8
versus grades 6 and 7 (contrast G2). Thus, the null
hypothesis (H07) was rejected. hence, statistically,
attitudes to mathematics differed from attitudes to
probability among the three grade levels. however, from a
close comparison of the PASTMAS and PAS means (Table 4.21),
one might conclude that these grade level differences in
attitudes toward mathematics versus attitudes toward
probability were not meaningful.

In conclusion, the null hypothesis (808) was re-
tained for the data in Site 2 and 3. There was no evidence
of significant interactions of grade by sex in attitudes
toward mathematics and probability in Site 2 and 3, but
interactions were present in Site 1. In all sites, boys and
girls did not differ in their attitudes toward mathematics
versus attitudes toward probability. Hence, the hypothesis
(H07) was retained.* In Sites 2 and 3, the three grade
levels studied differed in these comparisons. However, in
Site 3, these differences were considered nonmeaningful. In
Site 2, sixth and seventh graders preferred mathematics to
probability, while eighth graders did not (as measured by

the Mathematics and Probability Attitude Scales.

 

* Table 0.5 in Appendix D contains the ANOVA table for mean
difference between the POSTMAS and PAS for the entire
sample.

153

Sex and Grade Level Differences in Attitudes Toward
Probability

 

 

The attitudes of boys were compared with attitudes of
girls toward probabilty after instruction. The attitudes of
the grade levels (six, seven, and eight) studied were
similarly compared with one another. To examine these
differences in attitudes toward probability after
instruction a Two-Way ANOVA design was used. The following
two null hypotheses were tested separately for the data from
each site:

H09: There will be no difference between the mean
scores for boys and for girls in grades six, seven,
and eight on the Probability Attitude Scale.

H10: There will be no difference between the mean scores
for each of the three grade levels (six, seven and
eight) on the Probability Attitude Scale.

Table 4.23 includes the means and standard deviations
of the PAS scores for Site 1, Site 2, and Site 3, by grade
and by sex. Tables 4.24, 4.25 and 4.26 respectively present
a summary of the ANOVA for Sites 1, 2, and 3, by grade and

by sex, for the results of the tests on attitudes toward

probability.

Site 1: The Urban Site

The results of the ANOVA indicated that the interaction
of grade by sex was significant (P < .05). Hence, the tests
for the grade and sex main effects were confounded in the
interaction between the two effects. However, from the

profiles of PAS mean scores by grade and by sex (Fig. 4.20),

154

TABLE 4.23

MEANS AND STANDARD DEVIATIONS OF PASa SCORES
BY SITE BY SEX BY GRADE

 

 

 

 

 

 

Grade Site 1 Site 2 Site 3
N M S.D. N M S.D. N M. S.D.
Grade 6 149 3.156 .869 340 2.808 .943 133 3.440 .863
Boys 76 3.118 .937 162 2.840 1.004 70 3.469 .885
Girls 73 3.194 .796 178 2.780 .886 - 63 3.407 .843
Grade 7 138 3.460 .797 101 2.790 .923 140 2.828 .826
Boys 66 3.288 .787 53 2.937 1.011 65 2.805 .825
Girls 72 3.618 .778 48 2.629 .793 75 2.849 .831
Grade 8 157 3.339 .828 180 2.285 .839 108 2.968 .866
Boys 74 3.426 .735 102 2.335 .849 57 2.927 .977
Girls 83 3.261 .901 78 2.220 .827 51 3.013 .729
Total 444 3.315 .840 621 2.654 .939 381 3.081 .889
Boys 216 3.276 .833 317 2.694 .987 192 3.083 .938
Girls 228 3.352 .847 304 2.612 .887 189 3.079 .840

 

a PAS - Probability Attitude Scale (Range 1-5).

155

TABLE 4.24

ANALYSIS OF VARIANCE SUMMARY TABLE FOR PASa
SCORES IN SITE 1

 

 

 

Source of Variation D.F. MS F P
GraEe 2 3.336 4.861* .008
Sex 1 .542 .789 .375
Grade x sex 2 2.243 3.268* ".039

Between Groups 5

Within Groups 438 .686

 

a POSTMAS - Posttest Mathematics Attitude Scale.
* Significant P < .01.

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157

TABLE 4.25

ANALYSIS OF VARIANCE SUMMARY TABLE FOR PASa
SCORES IN SITE 2

 

 

 

Source of Variation ' D.F. MS F P
Grade 2 17.760 21.445* .001
Sex 1 2.078 2.510 .114
Grade x sex 2 .603 .728 .484

Between Groups 5

Within Groups 615 .828

 

a POSTMAS - Posttest Mathematics Attitude Scale.
* Significant P < .01.

158

TABLE 4.26

ANALYSIS OF VARIANCE SUMMARY TABLE FOR PASa
SCORES IN SITE 3

 

 

 

Source of Variation D.F. MS F P
Grade 2 13.735 18-883* .001
Sex 1 .034 .047 .829
Grade x sex 2 .179 .246 .782

Between Groups 5

Within Groups 375 .727

 

a POSTMAS - Posttest Mathematics Attitude Scale.
* Significant P < .01.

159
it was observed that sex differences were not significantly
plausible. Hence, the null hypothesis H09 was tenable
and retained. The conclusion was that, in Site 1, boys and
girls did not differ in their attitudes to the MGMP
Probability as measured by the PAS.

Differences were observed among the means for grades
six, seven, and eight (Fig. 4.20), with grade seven
recording the highest mean. Scheffe's Post Hoc posteriori
comparisons (Appendix B, Table E.7) showed the significant
difference to be confined to between grades six and seven
' girls, in favour of the latter. Hence, the null hypothesis
H10 was rejected (P < .01). The conclusion was that in
Site 1, seventh graders differed significantly from-sixth
graders in their attitudes toward the MGMP Probabilty as

measured by the PAS.

Site 2: The Suburban Site

The results of the ANOVA indicated no significant in-
teraction of grade by sex. The sex main effects were also
not significant (Table 4.25). The null hypothesis H09 was
therefore retained. Only the grade main effects were signi-
ficant (P < .01). Hence, the null hypothesis H10 was
rejected. An examination of the PAS means by grade (Table
{E.8, Appendix E, and Table 4.23) indicated grade six to be
significantly higher than grade eight. This means that, in
Site 2, eighth graders had less positive attitudes to the

MGMP Probability than sixth graders, as measured by the PAS.

160

Site 3: The Rural Site

 

The results of the ANOVA for this site were similar to
those of Site 2. No significant interaction of grade by
sex, or of the sex main effects were indicated. Hence, the
null hypothesis H09 was retained. However, grade level
differences in attitudes toward the MGMP Probabilty were
significant (P < .01) among grade levels six, seven, and
eight. Table 4.23 and the post hoc tests (Appendix B, Table
E.9) show that in Site 3, sixth graders were significantly
higher than seventh and eighth graders in attitudes toward
probability. The highest differences were observed between
boys in grades six and seven, in favour of the former.

In conclusion, in each of the three sites studied, the
two null hypotheses (H09 and H10) were retained and
rejected respectively. The interpretation of this was that,
in the entire study, boys and girls did not differ in their
attitudes toward probability, but the grade levels did.
Table D.9 in Appendix D shows the ANOVA table for the three
sites conbined. In Site 1, the urban site, grade seven
recorded the highest attitudes to probability. On the
contrary, the same grade level recorded the lowest attitudes
to probability in Site 3. In general, sites 1 and 3
appeared to have demonstrated more favorable attitudes to

probability than Site 2.

161

Summary of findings

 

The statistical analysis of the data collected in this

investigation was presented in this chapter. The following

were the major findings.

Prior to Instruction: Using MANOVA-on sex and grade level

 

differences in probability knowledge and attitudes toward

mathematics, the following results were found.

1.

The null hypothesis (H01) was rejected in each

site. In Site 1, significant differences were found
between girls in grade levels six and seven in their
existing knowledge in probability prior to instruction.
Attitude differences to mathematics between boys and
girls, or among the grade levels (six, seven, and
eight), were not found in Site 1.

In Site 2, grade seven was found to be significantly
different from grade six in both probability knowledge
and mathematics attitudes. While the probability
knowledge favoured the seventh graders, the reverse was
the case in attitudes toward mathematics. Seventh
graders were also found to be significantly superior to
eighth graders in both prooability knowledge and
attitudes toward mathematics.

In Site 3, grade level differences in probability were
significant between girls in grades Six and eight,

favouring grade eight. However, the differences

162
between the girls were considered nonmeaningful.

4. Sex differences were not found in Site 1 in either
probability knowledge or mathematics attitudes.

5. In Site 2, seventh and eighth grade boys were superior
to the respective grade level girls in probability
knowledge, prior to instruction. No sex differences
were found in Site 2 in attitudes toward
mathematics.

6. Sex differences were not found in Site 3 with respect
to mathematics attitudes. In probability knowledge,
boys in grade six outperformed girls.

7. In all these sites, interactions of grade by sex were
not significant. Also, sex differences were not found

in attitudes toward mathematics.

After Instruction: The Multivariate Analysis of Repeated

 

Measures was used to test the effect of instruction as
stated in the null hypotheses (H04-H06). To compare

attitudes toward mathematics with attitudes toward
probability by grade and by sex, (H07 and H08),

ANOVA was used. Lastly, to compare attitudes to probability

by grade and by sex, (Hog-1Q), ANOVA was also used.

The following were the results found after the instruction.
8. In all sites, time effect was found to be significant

(P < .0001) for all subjects. In other words, a

significant effect of the instruction was found from

10.

11.

12.

163

pretest to posttest, in both probability and attitudes
toward mathematics. Gains over time in the MGMP
Probability Test and change over time in attitudes
toward mathematics (as measured by the MAS) were, on
the whole, significant. However, all attitude changes
were not considered meaningful.

Sex differences were also not found in probability
gains from the instruction and in attitude change
toward mathematics over the time period.

In general, grade level differences were found in
probability knowledge over time (pre + posttest), in
probability gains over time (posttest - pretest), and
in attitude change to mathematics over time.

In Site 1, grade level differences in probability gains
were not found, but significant differences were found
between girls in grades six and seven in favour of
grade seven. Due to inordinal interactions of
probability knowledge (over time) between boys and
girls among the three grade levels, significant grade
level differences were not found in probability
knowledge over time.

In Site 2, sixth and seventh graders gained equally
from the probability instruction, but each outgained
the eighth graders. Boys consistently outscored girls
in probability knowledge (over time) across all grade
levels, but this was significant only in grade seven.

0n the other hand, girls consistently outgained boys

13.

14.

15.

16.

17.

18.

164

in scores on the pre-post probability test, but these
were not statistically significant.

While grade eight gained least from the probability
instruction in Site 3, grade seven gained most. Grade
seven also demonstrated more probability knowledge
(over time) than grades six and eight.

In the entire study, all three sites did not differ
significantly from one another in either probability
gains over time or in attitude change toward mathe-
matics. Boys and girls, and all grade levels, changed
equally in mathematics attitudes. Boys and girls
gained equally, and grade seven outgained the other
grades, on the probability unit test pre-post.

With respect to attitude comparison to mathematics and
to probability, no sex differences were found by site
and by grade level.

Grade level differences were found in Sites 2 and 3 in
attitudes to mathematics versus attitudes to
probability. In Site 2, sixth and seventh graders
preferred mathematics to probability, while the eighth
graders had no preference.

In the comparison of attitudes to probability, boys and
girls did not differ in any grade or Site.

In Site 1, grade seven had significantly higher
attitudes to probability than grades six and eight. 0n
the contrary, the same grade level recorded the lowest

attitude to probability in Site 3. 0n the whole,

165
attitudes to probability were more favourable in Sites
1 and 3 than in Site 2, while grade seven, on the

whole, recorded higher liking of probability than any

other grade level.

CHAPTER V

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

The problem addressed in this investigation, the
rationale, and delimitations of the study, were all
introduced in the first chapter. In the second chapter, the
review of related literature was given. The third chapter
described the methodology of the study. In the fourth
chapter, the analysis of the data and the interpretation of
the findings were reported.

This final chapter, the fifth, contains the following:
1. Summary and Findings.

2. Conclusions and Discussion.
3. Implications for Mathematics Education.

4. Recommendations for Future Research.

Summary and Findings

 

This section contains a summary of the purpose of the
study, research questions, related literature, methodology,

hypotheses, and findings.

106

167

Purpose of the Study

This study had four related purposes. The first pur-
pose was to determine any existing differences in probabil-
ity knowledge and in attitudes toward mathematics of grades
six through eight students by sex and by grade level prior
to a formal probability instruction. The second purpose was
to examine the effect of instruction on the probability
achievement and on attitudes toward mathematics of these
students. The third purpose was to compare attitudes toward
mathematics with attitudes toward probability by sex and by
grade level. The fourth purpose was to compare attitudes

toward probability by sex and by grade level.

Research Questions

 

Two sets of related questions were considered in this
study. The first set of questions, called type A, dealt
with the existing (prior to instruction) differences in
probability knowledge and attitudes toward mathematics of
grades six, seven, and eight students, by sex, by grade
level and by school setting. The second set of questions,
type B, dealt with effects of instruction in three parts.
The first part focussed on the effects of instruction on the
probability knowledge and on attitudes toward mathematics of
the same students. The second part concerned differences,
by grade level and by sex, in attitudes toward mathematics

versus attitudes toward probability, after instruction. The

168

last part was a comparison of attitudes toward probability

after instruction, by grade level and by sex. These

questions were all examined for each of three school

settings (Sites 1, 2, and 3). Type A questions included the

following:

1.

What effect, if any, does grade level have on knowledge
of probability and/or on attitudes toward mathematics?
What effect, if any, does sex have on knowledge of
probability and/or on attitudes toward-mathematics?

Do differences between boys and girls in knowledge of
probability skills and/or in attitudes toward
mathematics change with grade level?

What effect, if any, does school setting have on
knowledge of probability and/or on attiudes toward

mathmatics?

Type B Questions

 

After instructional intervention:

What effeCt, if any, will probability instructional
intervention have on achievement in probability tasks
and/or on attitudes toward mathematics of sixth, seventh
and eighth grade students? Will these effects be
different for boys and girls? Will these effects differ
by grade level? Will the effects differ by school
setting?
Do differences exist between students' attitudes toward

mathematics in general and the probability activities in

169

particular? Will these differences exist for both
sexes? Will these differences exist for each grade
level in the study? Will these differences exist for
each of the sites 1, 2, and 3?

3. Do differences exist between the sexes in attitudes
toward probability activities? Will these differences
exist for each grade level? For each site?

4. Do differences exist among the three school settings in
their attitudes toward probability activities? Will
these differences exist for each grade level? Will they

exist for each sex?

Related Literature

 

The review of the literature revealed how a number of
inveStigators have concluded scientifically that elementary
and middle grades boys and girls possess some knowledge of
probability concepts prior to formal instruction. Further
review indicated that Piaget prOpounded the existence of
three stages of probability development just as his widely
known four stages of mental maturation. Other researchers
have also concurred with Piaget's assertion. Thus, there is
a popular suggestion in the literature that the amount of
probability concepts possessed by a pupil, and the level of
probability sophistication the pupil is ready to learn, are
a function of the child's stage of probability development.
A corrolary to this assertion is the expectation of grade

level differences in probability gains, in favour of the

170

higher grade level. Also, probability achievement in favour
of higher I.Q., is another pOpular assertion made frequently
in the literature.

Only a relatively few investigations on sex differences
in probability were detected by the investigator. In
general, the literature reviewed tended to conclude that
there are no significant sex differences in existing
probability ability or gains from instruction. However, a
few studies have suggested sex differences in favour of
boys, prior to instruction. Further, girls tended to gain
more than boys from probability interventions.

Consequently, only one study (reviewed) reported the
persistence of any sex differences from posttest measures.

Quite a few probability curriculum innovations were'
identified in the literature. Of these, many claimed that
peOple of all ages tend to commit certain errors in
probabilistic thinking. These errors, according to these
investigators, tend to interact with the learning of
probabilistic concepts. However, research evidence abounds
that boys and girls do benefit significantly from
probability intervention.

The issue of sex differences in mathematics is widely
addressed in the literature. However, although most of
these studies tend to conclude in favour of boys, there is
disagreement as to the nature or extent of these differ-
ences. Renown researchers have explained mathematics

achievement differences in terms of environmental

171

conditions, while equally disciplined investigators have
adduced heredity for their explanation. Thus, the issue of
the cause of sex differences in mathematics achievements is
essentially the issue of nurture versus nature.

The review of literature related to attitude studies
suggested that earlier investigators tended to conclude in
favour of a positive relatibnship between attitude and
mathematics achievement. However, more contemporary studies
have challenged this assertion. They have suggested that
such other factors as teacher quality, classroom
cohesiveness and student fatalism correlate as strongly as

attitude, with mathematics achievement.

Methodology

 

The probability intervention employed in the study took
place during Fall 1982 and Winter 1983; while data
collection took place between October 1982 and January 1983.
The information collected included pretest and posttest
measures in probability performance and in attitudes toward
mathematics, and attitudes toward probability (posttest
measures only). All posttest measures were the same as the
corresponding pretest measures.

The subjects of the study were grade six, seven and
eight students from six different schools, Situated in three
distinct sites; two schools per site. One of these schools
was an elementary school, three were middle schools and two

were junior high schools. The three sites were categorized

172

into urban, suburban and rural settings and were respective-
ly referred to as Site 1, Site 2 and Site 3 throughout this
study. From Site 1, the urban setting, 444 subjects took
part. From the suburban setting, Site 2, 621 subjects
participated while 381 subjects took part in the study from
Site 3, the rural setting. Of the 1460 subjects involved in
the study, 1446 took part consistently in the data analyses.
The difference of 14, which was highly insignificant, was
due to random omission. There were 622 sixth graders, 379
seventh graders, 445 eighth graders, 725 boys, and 721 girls

who took part in the entire study.

Instrumentation

 

The instruments used in this study were two semantic
differential attitude scales and a performance test on
probability ability. The two attitude scales were called
the Mathematics Attitude Scale (MAS) and the Probability
Attitude Scale (PAS). Throughout the study these two
attitude scales were referred to as MAS and PAS, while the
performance test instrument was called the Middle Grades
Mathematics Project Probability Test (MGMP PT). Both
attitude scales were simple six-item bipolar semantic
differentials, with five response options. The MGMP PT
consisted of 25 multiple choice items. ApprOpriate test
instructions and conditions were provided and the teachers
involved invigilated all tests. Appendix B includes these

tests and test instructions.

173

The Probability Unit included ten sequentially
developed activities requiring about three weeks of
instructional time. The activities of the unit included the
following: State lottery, three activities of fair and
unfair games, surveys, area models, expected value,
newspapers pay, Jonesville families and Pascal's Triangle.
The Probability Unit employed an instructional model
consisting of three phases: launching, exploring, and
summarizing. Each teacher was provided with a detailed
instructional guide which was presumably followed throughout
each probability activity.

The first five activities strictly involved determining
probabilities of independent events. The last five
activities treated a variety of probabilty concepts. These
activities included the.calculation of probabilities from
geometric shapes, the use of probabilities to make
predictions, the calculation of eXpected values, and
simulation and analysis of theoretical problems. Binomial

probabilities were also introduced.

Hypotheses and Design

 

Several statistical procedures were selected to analyze
the data collected during the study. Particular statistics
were chosen to test the hypotheses given in the following
paragraphs. The analyses included the following: means,
standard deviations, correlations, multivariate and

univariate analyses of variance and repeated measures.

174
Planned comparisons and Scheffe's Post Hoc comparison were
also selected. All these analyses were carried out on the
3600 Computer at the Michigan State University Computer
Center, using the Statistical Package for the Social
Sciences (SPSS).

The multivariate model with a two-way fixed effects
analysis of variance (MANOVA) was used to assess the
differences in probability knowledge and attitudes toward
mathematics, by grade level and sex, prior to the
instruction.

The hypotheses tested within this design, given in null
form, were:

H01: There will be no difference among the mean scores
for each of the three grade levels (six, seven, and
eight) tested, on both the Middle Grades Mathematics
Project Probability Test and on the Mathematics
Attitudes Scale.

H02: There will be no difference between the mean
scores for boys and for girls in grades six through
eight on both the Middle Grades Mathematics Project
Probability Test and on the Mathematics Attitude
Scale.

H03: There will be no interaction of grade by sex among
the mean scores for sixth through eighth grade
students on both the Middle Grades Mathematics Project
Probability Test and on the Mathematics Attitude
Scale.

To analyze the effects of instruction on the
probability skills, differences in probability and attitudes
toward mathematics, by sex and grade level, the Multivariate
Analysis of Repeated Measures Model was selected.

The hypotheses tested with this design, given in null

form were:

175

H04: There will be no difference between the posttest
means and the pretest means of sixth, seventh, and
eighth grade students on both the Middle Grade
Mathematics Project Probability Test and on the
Mathematics Attitudes Scale.

H05: There will be no difference between the mean gain
scores (posttest minus pretest) for each of the three
grade levels tested, six, seven, and eight on both the
Middle Grades Mathematics Project Probability Test and
on the Mathematics Attitude Scale.

“06" There will be no difference between the mean gain
scores (posttest minus pretest) for boys and for girls
in grades six, seven, and eight on the Middle Grades
Mathematics Project Probability Test and on the
Mathematics Attitude Scale.

To compare the attitudes toward mathematics and
probability by sex and by grade after instruction, a Two-Way
ANOVA design was used. The hypotheses tested within this
design for each site were:

H07: There will be no difference between students' mean
scores on the Mathematics Attitude Scale (MAS) and
students' mean scores on the Probability Attitude
Scale (PAS).

Hog: There will be no interaction of grade by sex among
the mean difference scores--Mas score munus PAS
score.

A Two-Way ANOVA design was also applied in order to
examine sex and grade level differences in attitudes toward
probability. The hypotheses tested within this design in
each site were:

H09: There will be no significant difference between

the mean scores for boys and for girls in grades six,
seven, and eight on the Probability Attitude Scale.

H10: There will no difference between the mean scores
for each of the three grade levels (six, seven and
eight) on the Probability Attitude Scale.
Planned comparisons for the grade effect and Scheffe's

Post Hoc compariSons were used to identify the sources or

176

significant main effects or interactions. The .05 level of
significant was the limit accepted in testing all the

hypotheses in the study.

Findings and Conclusions

The statistical analysis of the data collected in this
investigation was presented in Chapter IV. The following

were the major findings.

Prior to Instruction: Using MANOVA on sex and grade level

 

differences in probability knowledge and attitudes toward

mathematics, the following results were found.

1. The null hypothesis (H01) was rejected in each site.
In Site 1, significant differences were found between
girls in grade levels six and seven in their existing
knowledge in probability prior to instruction. Attitude
differences to mathematics between boys and girls, or
among the grade levels (six, seven, and eight) were not
found in Site 1. The MGMP PT means for grades six and
seven girls were 6.27 and 7.78 respectively. (All
pairwise mean differences can be estimated from the
profiles of probability pretest means of all sites by
grade level by sex in Fig. 5.1).

2. In Site 2, grade seven was found to be significantly
different from grade six in both probability knowledge
and mathematics attitudes. While the probability

knowledge favoured the seventh graders, the reverse was

7B

622
379
445

Grade 6:
12.5 Grade 7:
Grade 8:

222
111111

12.0

11.5

11.0

10.5

 

 

10.0

Probability Test Means

 

1 2 3 Site

Fig. 5.1 Pretest Probability Means-Profiles of All
Sites by Grade Level by Sex.

178

the case in attitudes toward mathematics. Seventh
graders were also found to be significantly superior to
eight graders in both probability knowledge and atti-
tudes toward mathematics. The means for grades six,
seven, and eight were respectively 7.80, 11.71, and 9.66
for the PT; and 3.51, 3.41, and 2.62 for the PREMAS
scores. The post hoc comparison restricted the attitude
differences to differences between girls in grades six
and seven, favouring grade six. As measured by the MAS,
these attitude means were 3.59 and 3.21. Also in this
site, the suburban setting, boys and girls in grade
seven outperformed boys and girls in grade eight in both
probability knowledge and mathematics attitude. Hence,
in the suburban site, attitudes toward mathematics
drOpped with grade level or with age. With respect to
probability knowledge prior to instruction, seventh
graders unexpectedly outscored eighth graders.

In Site 3, the rural site, grade level differences in
probability were significant between girls in grades six
and eight, favouring grade eight. The mean scores were
7.95 and 9.69. The differenCe was, however, only 1.73,
and was considered nonmeaningful. With respect to
mathematics attitudes in Site 3, the means for grades 6,
7, and 8 were respectively 3.45, 3.21, and 3.01. Hence,
although these differences were not significant, a

gradual drop in mathematics attitudes with grade level

179

was observed like in Site 2. Fig. 5.2 Shows an ordinal
interaction by grade level.

4. No significant sex differences were found in all Sites
in attitudes toward mathematics, as measured by the MAS,
prior to instruction. However, sex differences in
probability knowledge were found in Sites 2 and 3, but
not in Site 1. From further tests, all these
significant differences favoured boys. In Site 2, the
suburban site, these differences occurred in grades
seven and eight. In Site 3, the rural site, they
occurred in grade six only. These findings are
observable from the profiles of probability means by
grade level and by sex in Fig. 5.1.

5. In all the sites, interactions of grade by sex were not
significant. Thus, the sex and grade level main effect

tests were plausible.

After Instruction: The Multivariate Analysis of Repeated
Measures was used to test the effect of instruction as
stated in the null hypotheses (HO4'H06)- To compare
attitudes toward mathematics with attitudes toward
probability by grade and by sex, (H07 and H08), ANOVA

was used. Lastly, to compare attitudes to probability by
grade and by sex, (Hog-H10), ANOVA was also used.

The following were the results found to be significant.

18L)

 

 

Pretest MAS Means

 

I 2 3 Site

Fig. 5.2 Profiles of PREMAS Means by Site by Sex and by
Grade Level.

181
In all sites, time effect was found to be significant (P
< .0001) for all subjects. In other words, a signifi-
cant effect of the instruction was found from pretest to
posttest, in both probability and attitudes toward
mathematics. Gains over time in the MGMP Probability
Test and change over time in attitudes toward mathe-
matics (as measured by the MAS) were, on the whole,
significant. Fig. 5.3 summarizes all probability gains
and averages. However, all attitude changes were not
considered meaningful. The probability mean gains over
time were 3.32, 3.06, and 3.26, in Sites 1, 2, and 3
respectively, while the averages over time were re-
spectively 9.05, 10.51, and 10.83. Hence, although Site
1 averaged (knew) less probability than Sites 2 and 3,
it outgained them. Attitude change to mathematics was
also significant over time. However, as discussed in
Chapter IV, the differences between pretest and posttest
attitudes were small. The significant difference was
most likely due to the high precision level of the test.
In other words, because the sample used was large, the
degrees of freedom used were large. Hence, the test was
powerful enough to detect differences which were not
necessarily meaningful. The pretest and posttest
attitude means were 3.51 and 3.35 for Site 1, 3.24 and
3.16 for Site 2, and 3.24 and 3.16 for Site 3. because
of these small differences, the conclusion was that, on

the whole, middle grades (six, seven, and eight) boys

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183
and girls did not change their attitudes appreciably
toward mathematics during the study.
Sex differences were also not found in probability gains
from the instruction and in attitude changes toward
mathematics over the time period. Thus, in the entire
study, boys and girls gained equally from the probabil-
ity intervention. In Site 2, there was an ordinal
interaction of sex by grade level in probability gains
over time (Fig. 5.3). This means that girls gained more
than boys in each grade level, but these differences
were not statistically significant. Although attitude
changes by sex were not significant, girls in general
dropped slightly more than boys from pretest to posttest
in attitudes toward mathematics. The pretest and
posttest means were 3.29 and 3.21 for boys, while these
means were 3.35 and 3.23 for girls. These attitude
changes by sex are observable graphically in Fig. 4.19.
In general, grade level differences were found in
probability knowledge over time (pre + posttest), in
probability gains over time (posttest-pretest), and in
attitude change to mathematics over time. Both
probability knowledge and gains over time were found to
be in favour of the seventh graders (Fig. 4.15). The
overall pretest and posttest probability means were 7.70
and 10.96 for grade six, 9.27 and 13.19 for grade seven,
and 9.13 and 11.59 for grade eight. Although attitude

changes by grade were not significant as measured by the

10.

11.

184

MAS, attitude decline by grade level was observed in

both the pretest and posttest scores (Fig. 4.18, Table
4.14). That is, the higher the grade level, the lower
the attitudes toward mathematics.

In Site 1, grade level differences in probability gains

. were not found, but significant differences were found

between girls in grades six and seven in favour of
grade seven. Due to inordinal interactions of
probability knowledge (over time) between boys and
girls among the three grade levels, significant grade
level differences were not found in probability
knowledge over time. In Site 1, the mean gains were
3.16, 3.47, and 3.33 for grades 6, 7, and 8
respectively.

In Site 2, sixth and seventh graders gained equally
from the probability instruction, but each outgained
the eighth graders. boys consistently outscored girls
in probability knowledge (over time) across all grade
levels, but this was significant only in grade seven.
On the other hand, girls consistently outgained boys in
scores on the pre-post probability test, but these were
not statistically significant.

While grade eight gained least from the probability
instruction in Site 3, grade seven gained most. Grade
seven also demonstrated more probability knowledge

(over time) than grades six and eight.

12.

13.

14.

15.

16.

185
In the entire study, all three sites did not differ
significantly from one another in either probability
gains over time or in attitude change toward
mathematics.’ Boys, and girls, and all grade levels,
changed equally in mathematics attitudes. Boys and
girls gained equally, and grade seven outgained the
other grades, on the probability unit test pre-post.
Fig. 5.3 presents all gains and averages by site, by
grade, and by sex.
With respect to attitude comparison to mathematics and
to probability, no sex differences were found by site
and by grade level. This means boys and girls did not
disagree in their attitudes toward probability and
mathematics.
Grade level differences were found in Sites 2 and 3 in
attitudes toward mathematics versus attitudes toward
probability. In Site 2, sixth and seventh graders
preferred mathematics to probability, while the eighth
graders had no preference.
In the comparison of attitudes to probability, boys and
girls did not differ in any grade or site.
In Site 1, grade seven had significantly higher
attitude to probability than grades six and eight. On
the contrary, the same grade level recorded the lowest
attitude to probability in Site 3. On the whole,
attitudes to probability were more favourable in Sites

1 and 3 than in Site 2, while grade seven, on the

186

whole, recorded more liking of probability than any

other grade level.

Discussion

 

The findings and conclusions reached in this study were
presented in the previous section. A number of observations
are made about middle grades (six, seven, and eight) boys
and girls, with respect to their probability knowledge and
attitudes toward mathematics prior to a probability
intervention.

Perhaps a major observation from this study is the
presence of sex differences in probability prior to
instruction among middle school pupils. In general, boys
seem to outperform girls in probabilistic concepts in the
absence of instruction. However, although boys may still
outperform girls after instruction, differences may no
longer be tangible. In fact, in the present study, girls
benefited more than boys from the probability intervention,
as demonstrated in the pretest-posttest scores. Evidence of
such sex differences in research studies exists. Kass
(1964) found sex differences in favour of boys in binomial
probability tasks. The explanation Kass adduced for this
was that boys have a natural tendency to interact with
out-of-class probabilistic events that involve dichotomous
choices. For example, boys tend to be associated more than
girls in gambling activities involving head or tail, win or

lose situations.

187

Similar to the present study, in the evaluation of the
Comprehensive School Mathematics Project (CSMP), sex
differences which were found in the pretest measures,
vanished after the program, for CSMP participants, but sex
differences persisted among non-CSMP participants. On the
whole, the present finding is in agreement with most studies
in the literature. For example, in studies by Mullenex
(1968), Doherty (1965), Wavering (1979), Smith (1966) and
McLeod (1972), sex differences were not found in probability
achievement.

.That sex differences in attitudes to mathematics were
not found in this study either prior to, or sequel to
instruction, may perhaps surprise some. Notably,
investigators such as Fennema (1977), Fennema and Sherman
(1978), Malcolm (1971), and Shaughnessy et al. (1983), all
found sex differences in attitudes toward mathematics.
However, this finding agrees with observations by Suydam and
Weaver (1975) that sex differences were not found in
attitudes to mathematics in some studies. It is, however,
worth remarking that although sex differences were not
significant in the present study, boys had a tendency to
remain more steady than girls. For example, while girls on
the whole had more positive attitudes toward mathematics,
they dropped more than boys from pretest to posttest, but
again, these were negligible differences.

The findings from the present study suggest that grades

six, seven, and eight boys and girls do differ in

188

probability knowledge prior to instruction. Although these
three grade levels are all within Piaget's second stage of
probability development, from seven to fourteen years of
age, higher grades would be expected to show more maturity
in mathematical ability, of which prooability is a part.
Hence, it was not a surprise that grade six performed less
well than the other grades in both pretest and posttest
probability scores. However, the most important grade level
question is which grade benefited (gained) most from the
probability intervention. Interestingly, the seventh
graders gained more probability knowledge than sixth and
eighth graders, in the present study. This important result
is similar to the conclusion reached by Smith (1966) in a
study in which grade seven constituted the subjects used.
‘Smith reported that seventh graders gained significantly
from a 17-day probability intervention. Several questions
arise as to why grade seven should do better in probability
than grades six and eight, especially grade eight. First,
this superiority might be due, by chance, to higher teacher
and/or student quality. Or the reverse might be true of
grade eight, especially in the suburban setting. But, the
seventh graders outperformed the others in the rural
setting, and did not significantly differ from the eighth
graders in the urban setting. Hence, grade seven may
possess some interesting characteristics with respect to
probability. Another explanation might be found in the

nature of mathematics topics that were covered just before

189

the probability intervention. These tOpics may tend to
promote the learning 0f the probability activities in this
study.

According to the present study, grades six, seven, and
eight do not differ appreciably in their attitude change as
a result of probability instruction. This result agrees
with Fennema (1977) who reported that attitudes toward
mathematics remain fairly stable between grades six and
twelve. Although attitude differences were not significant
in the present study, distribution showed that attitudes
tended to decrease with grade level, with girls decreasing
more than boys. Malcolm (1974) reached very similar
conclusions.

Finally, a major result of this study was that
irrespective of sex, grade level or site, middle school
students benefited significantly from the training program
in probability tasks. Similar conclusions were arrived by
such investigators as Beyth-Maron (1980), Shaughnessy
(1977), and White (1974). Perhaps a partial explanation for
overall significant student gains from the Middle Grades
Probability Project Unit is the experimental nature of the
activities. The activities employed the strategies of
launching, exploration, and summarization. Moreover,
concrete operations and multiple embodiments, proved to be
effective by Piaget and Inhelder (1951) and Jones (1974),
were utilized in all the probability activities in this

study.

190

Implications for Mathematics Education

In the present investigation, it was demonstrated that
middle school Students can respond very well to probability
insruction. Although grade seven appears optimal for the
introduction of the (MGMP) Probability Unit, it has worked
very well for grades six and eight too.

Not only do grade levels six, seven, and eight respond
well to probability instruction, it was found, in this
study, that boys and girls respond equally and favorable
well to the probability instruction.

Also, probability instruction among middle school
students was demonstrated in this investigation to work
equally well in urban, suburban, and rural areas despite
socioeconomic and other background differences. Although
pretest and posttest measures indicated that students in
urban settings performed less well, findings revealed that
they benefited equally from the probability training
program as students from the other settings.

Another implication is for mathematics teachers.
Regardless of grade level, sex, or school setting, all
teachers, when supplied with well-sequenced instructional
activities, successfully taught a unit on probability. The
test and unit materials are easy and handy, and almost all
the manipulatives can be improvised locally.

Another implication is for mathematics teacher
education. The importance and use of probability knowldege

are being emphasized by contemporary mathematics educators

191

(Shulte 1981). However, for teachers to be encouraged

to take the topic more seriously, adequate staff development

is necessary for preservice and inservice teachers.

Undergraduate and graduate mathematics teacher education

programs should include the teaching of probability.

Recommendation for Future Research

The following recommendations are based on the

investigator's findings and conclusions in the present

study.

1.

It is recommended that this study be replicated with
similar subjects and extended to include a test of their
retention span in probability knowledge.

It is recommended that the study be replicated among
grade levels nine through twelve students to complete
the investigation through all postprimary grades. In
this case, efforts should be made to reduce teacher
differences as much as possible. For example, design
the study so that teacher participants have a
mathematics teaching certificate. This should limit
variability in teacher content knowledge.

It is recommended that the study be replicated in same
grade levels using identified high, middle, and low
ability students. This would afford the information on
how various ability levels respond to probability, and
to the Middle Grades Mathematics Project materials in

particular.

192

The initial question of interest to the investigator was
an analysis of students' patterns of errors in proba-
bilistic thinking. An appropriate question to investi-
gate might be an examination of the heuristics of avail-
ability and representativeness (reviewed in Chapter II).
In addition, analysis of students' errors could help
identify at what grade level certain probability con-
cepts are not amenable to instruction. This would have
implications for mathematics curriculum develOpment. A
corollary advantage would be availability or findings
that speak critically to Piaget's three stages of
probability development.

With the use of the same Probability Unit, it is
recommended that the teaching of probability be studied
by qualitative methods, in particular those associated
with ethnographic research. Especially in conjunction
with the statistical methods employed in the present
study, ethnographic perspectives could provide theoreti-
cal explanations of pretest-posttest results which would
not be feasible otherwise. The researcher could
document, through systematic participant observations of
the teacher-student class interactions, behavioral
patterns that lead to certain results. Had such methods
been available to complement the present study, such
perplexing questions as why seventh grade girls tended
to gain more than boys in the probability instruction

even though they knew less could have been investigated.

193

Other questions are to what extent the teacher followed
the curriculum materials, felt comfortable in responding
to student questions, or treated boys and girls
differentially. These could also be observed and
analyzed through the use of ethnographic methoas. Such
methods can aid in providing a more complete description
of what took place (or did not take place) during the
teaching-learning process. They also make it possible
for more relevant and useful questions to be raised as
the study progresses; questions which can have far
reaching implications for curriculum develOpment and

research in mathematics education.

APPENDICES

APPENDIX A

Brochure of Middle Grades Mathematics PrOJect (MGMP)
Department of Mathematics
Michigan State University

194

-MIDDLE GRADES
MATHEMATICS PROJECT
DEPARTMENT OF MATHEMATICS
MICHIGAN STATE UNIVERSITY

The MGMP is a curriculum develOpment project funded by

NSF - DISE, to develop units of high quality mathematics

instruction for grades 5 through 8. Each unit

* is based on_a related collection of important

*

*

mathematical ideas,

provides a carefully sequenced set of activities

which lead to an understanding of the mathematical
challenges,

helps the teacher foster a problem-solving atmosphere
in the classroom,

uses concrete manipulatives where appropriate to help
provide the transition from concrete to abstract
thinking,

utilizes an instructional model which consists of
three phases...launching, exploring, and
summarizing,

provides a carefully developed instructional guide
for the teacher,
requires two to three weeks of instructional time.

The goal of the MGMP materials is to help Students

develop a deep, lasting understanding of the mathematical

concepts and strategies studied. Rather than attempting to

break the curriculum into small bits to be learned in

isolation from each other, MGMP materials concentrate on a

195

cluster of important ideas and the relationships which exist
among these ideas. Where possible the ideas are embedded in
concrete models to assist the students in moving from this
concrete stage to more abstract reasoning.

Many of the activities are built around a specific
mathematical-challenge. The instructional model used in the
units focuses on helping the students solve the mathematical
challenge. The instruction is divided into three phases.
During the first phase the teacher launches the challenge.
The launching consists of introducing new concepts,
clarifying definitions, reviewing old concepts, and issuing
the challenge.

The second phase of instruction is the class
exploration. During the exploration the students work
individually or in small groups. The students may be
gathering data, sharing ideas, looking for patterns, making
conjectures, or develOping other types of problem-solving
strategies. The teacher's role during exploration is to
encourage the students to persevere in seeking a solution to
the challenge. The teacher does this by asking appropriate
questions, encouraging and redirecting where needed. For
the more able students, the teacher provides extra
challenges related to the ideas being studied.

When most of the children have gathered sufficient
data, the class returns to a whole class mode (often
beginning the next day) for the final phase or instruction,

summarizing. Here the teacher has an opportunity to

196

demonstrate ways to organize data so that patterns and
related rules become more obvious. Discussing the
strategies used by the children helps the teacher to guide
the students in refining these strategies into efficient,
effective problem solving techniques.

The teacher plays a central role in this instructional
model. First the teacher provides and motivates the
challenge and then 12323 the students in exploring the
problem. The teacher asks appropriate questions,
encouraging and redirecting where needed. Finally, through
the summary, the teacher helps the students to deepen their
pnderstanding of both the mathematical ideas involved in the
challenge and the strategies used to solve it.

To aid the teacher in using the teaching model de-
scribed, a detailed instructional guide is provided. This
guide was developed as a result of many classroom trials of'
the materials. It provides help with both the mathematics
content and the classroom management of the activities.
Specific suggestions for important questions to be asked at
appropriate stages of the activities are included. Exten-
sion questions and challenges for the more able students are
provided along with suggestions for helping those students
who are having difficulty. The units develOped include:

SPATIAL VISUALIZATION
FACTORS AND MULTIPLES
PROBABILITY
SIMILARITY

197
STAFF
Glenda Lappan, Director
William M. Fitzgerlad
Elizabeth Phillips
Mary Jean Winter
Pat Yarbrough
David Ben-Haim
Alex Friedlander

Zacchaeus Oguntebi

CONSULTANTS
Janet Shroyer (DeveIOpment)

Aquinas College, Grand Rapids, MI

Richard Shumway (Evaluation)

Ohio State University

APPENDIX B

MGMP Probability Test
Mathematics Attitude Scale (MAS)
Probability Attitude Scale (PAS)

Now It's your Turn

PLEASE NOTE:

Copyrighted materia1s in this document
have not been f11med at the request of
the author. They are avai1ab1e for
consu1tation, however, in the author's
university 1ibrary.

These consist of pages:

198-208

 

 

 

 

 

Universg'
Micr 1lms
International

300 N. ZEEB 90.. ANN ARBOR,M| 4810613131 761-4700

198

PROBABILITY TEST

DO NOT WRITE ON THIS TEST BOOKLET.
YOU MAY USE A SHEET OF SCRATCH PAPER.
READ QUESTIONS CAREFULLY.

SELECT THE ANSWER TO THE QUESTION.
MARK YOUR ANSWER ON THE ANSWER SHEET.

D

A B C (E:
EXAMPLE (:) (:) (:) {l’ (:)

 

BE SURE TO FILL THE CIRCLE COMPLETELY.
ERASE COMPLETELY WHEN NECESSARY.

MARK ONLY IN THE RESPONSE CIRCLES PROVIDED.
MAKE NO STRAY MARKS ON THE ANSWER SHEET.
STOP: WAIT UNTIL YOU ARE TOLD TO BEGIN.

199

PROBABILITY
PRETEST

Materials:

A)
B)
C)
D)

Somantic Differential Test about Mathematics
Probability Test Booklet
Answer Sheet

#2 Pencils

Instructions:

A)

B)

C)

Note:

Give Semantic Differential Test first. 5-10 minutes
should be sufficient. Collect this paper before
distributing next test. Students should PRINT their
name and circle girl or boy.

Distribute answer sheets and #2 pencils.
Complete only the name and sex sections.

Distribute Probability Test Booklets.

Provide scrap paper.

Review cover sheet instructions.

Allow as much time as needed for the 25 questions.
(Calculators are not allowed)

Please keep the classes separated and provide a class
list with each class set of Semantic Differential
and Probability Test.

The packages of materials will be collected as soon as the

test

is completed

Thank you very much for your c00peration.

200

PROBABILITY
POSTTEST

Materials:

 

Semantic Differential Test on Attitudes Toward Math.

 

2. Semantic Differential Test on Attitudes Toward
Probability Activities.

3. Now It's Your Turn.

4. Answer Sheets.

5. Probability Test booklets (25 questions).

6. #2 Pencils.

Instructions:

1. Administer attitude test in following order: Have
students print their name and circle their sex.
a) Attitudes Toward Math (about 3 minutes)
6) Attitudes Toward Probability Activities (about 3

minutes)

c) Now it's Your Turn (about 10 minutes)

2. Distribute answer sheets.
Students complete name and sex only.
Use #2 pencils.

3. Distribute scrap paper.

4. Distribute Probability Test Booklets.
Allow as much time as needed for the 25 questions.
About 25 minutes is average. (calculators are not
allowed)

5. Please have marks erased from booklets before passing

them to another teacher, or administering the test to
another class.

Note: Please keep the classes separated and provide a
class list with each class set of Semantic
Differential and Probability Tests.

The packages of materials will be collected as soon as the
testing is completed.

Thank you very much for your c00peration.

N
o

201

A spinner is divided into 15 sections of equal Size.
Five of these sections are red, four are blue, three
are green, and three are yellow. If the spinner is

spun, what is the probability that it will stop on a
blue section?

1

._ _4 11
(A) 3 (B) 15 (G)

J: __

11 (0) 1 (E) 4
.3.

The probability of an event happening is 8. What is

the probability that the event will not happen?

2. .5. .3.
(A) U (h) 6 (b) 8 (D) 4+ (E) 1

A bowl contains 3 red marbles, 5 green marbles, and 4
blue marbles. A blue marble is drawn and not replaced.
Then the contents of the bowl are thoroughly mixed.
After this, you are asked to draw a marble from the
bowl without looking. What is the probability that you
will draw a blue marble?

_3. _3 _4 ' .1-
(A) 12 (B) 11 (C) 12 (D) 11 (E) 3

Which of the following numbers could not be a
probability?

b)

8 5
(A) 1 (B) 7 (c) 9 (D) Z (L) u

A fair coin has been tossed 10 times and has come up
heads each time. Which of the following statements is

true:

(A) The coin will come up heads on the next toss.

(B) The coin will come up tails on the next toss.

(C) There is an equal chance of coming up heads or
tails on the next toss.

(D) The coin is more likely to come up heads on the
next toss than tails.

(B) The coin is more likely to come up tails on the
next toss than heads.

The probability of getting exactly one head and one
tail when two fair coins are tossed is:

1 1 2 1

(A) 4' (B) 3 (C) 1 (D) 3 (E) 2

202

7. If two dice are tossed over and over again, which sum
would you expect to occur most often?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 12

8. The probability of getting a sum of 12 when two dice
are thrown is:

1 1 1 1

1
(A) 2 (B) 3 (C) E (D) 12 (E) '3'?
9. Bill Bailey tossed a thumbtack 50 times. It landed
point up 22 times. If he tossed the same thumbtack 250
times, about how many times would you expect it to land
point up?
(A) 88 (B) 110 (C) 125 (D) 200 (E) 250

—_'1

Questions 10-12 relate to the 5 spinners shown below.

COG) O.

0. Which spinner is the most likely to stOp on red?

 

 

 

(A) I (B) II (C) Ill (0) IV (E) V
11. Kim spun a spinner 100 times and made a record of her B
results.
Outcome Blue Red
Number of times 86 14

 

 

Which spinner is most likely the one Rim used?
(A) I (B) II (C) III (D) IV (E) V

12. If spinner III is to be spun twice, what is the
probability of getting red - red?

1 3
(A) 21‘ (B) '2‘ (C) E (D) 2? (r1) 1

_I
d

 

 

13.

14.

15.

16.

17.

203

A bag contains only red and blue marbles.

3
The probability of drawing a red marble is 5. What is
the probability of drawing a blue marble?

5 i .2. 3
(A) '3' (B) 5 (C) 5 (D) 5 (E) 1

A bag contains 2 yellow, 2 blue, and 4 red marbles.
How many blue marbles must be added to the bag to make

I .
the probability of drawing a blue marble 2.

Three pennies are tossed. What is the probability of
getting 2 heads and 1 tail?

1 1 A. l 2
(A) 8 (B) 3 (G) 3 (0) 2 (E) 8

John is tossing bean bags randomly onto the mat below.
What is the probability of a bean bag landing in an
area marked B?

 

 

 

 

 

 

 

 

A
A
b
B
b .
I .3. .1. 2 2
(A) 4 (B) 8 (U) 2 ~ (U) 6 (h) 3

Sally has a 50% free throw shooting average in
basketball. She goes to the line to take two shots.
What is the probability that she will make both shots?

1 1 1 3
(A) Z (B) 7 (U) 3 (D) K (E) 1

204

18. Hat 1 and Hat 2 contain red and white marbles as Shown
below. A hat is chosen at random and a marble drawn
from it.

1339/ M7

Which area model can be used to find the probability
of drawing a white marble?

 

 

 

 

R W R W
R R
W W R W W R W W
W R R 1———_4
W 1 W W

(A) (b) (C) (D) (E)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

19. Bag 1 and Bag 2 contain blocks as shown below.

__.

 

 

 

 

Which of the following is a tree diagram showing the
possible combined results of drawing a block from bag
1, and then, a block from bag 2?

P-R-G-Y - R P
(A)4w-R-G-Y (B) R G (C) P<
W

(E) P

(u) r<G

20.

21.

22.

23.

205

Two coins are flipped. Player A gets a point if the
coins match and player B gets a point if there is no
match. In this game

(A) A is more likely to win.

(B) B is more likely to win.

(C) A and B have the same chances of winning.
(D) There is not enough information to decide.
(E) B can never win.

Two bills are drawn from a bag containing a five dollar
bill and 3 one dollar bills. If the experiment is
repeated many times, what would you expect the average
amount of money drawn per time to be?

(A) $2 (B) $3 (C) $4 (U) $5 (b) $0
What is the probability that a family of three children
will have 2 girls and 1 boy?

I l 2 l i
(A) 8 (B) 3 (C) 3 (D) 2 (E) 8

How many different ways could you answer a 4 item
true-false test?

(A) 1 (B) 2 (C) 4 (D) 8 (E) 16

 

 

24-25 In order to determine what ice cream flavors to have

24.

25.

in the cafeteria a random poll is taken of 30
Students on their favorite ice cream flaveo. The
results are:

Vanilla 8 Butter Pecan 4
Chocolate 10 Peppermint 3
Strawberry 5

If there are 600 students in the school, about how many
would you expect to prefer chocolate?

(A) 10 (B) 20 (C) 200 (D) 250 (E) 300

If a student is chosen at random, what is the 4
probability that the student favors either chocolate or‘
butter pecan?

10 14 4 6 16

— — _ _
f

 

(A) 3'0 (B) 30 (C) 30 (1)) 3 (E) 30

 

EXAMPLE:

206

MATHEMATICS ATTITUDE SCALE (MAS)

NAME BOY/GIRL

 

FOR EACH PAIR OF WORDS BELOW PLACE A X ON THE
BLANK THAT BEST TELLS HOW YOU FEEL ABOUT--

SNOW
LIKE ___ _ _ _ _I_ _ ‘_ _._ _ _ _‘_ _ ._‘_‘_ _ HATE
UGLY _._ ,_ _._ _ _ ,_ _._ _ ,_ ___ _ _H_ PRETTY
WORK _._ _ '_‘_ _ _' _ _‘_ _ ‘_l_ _ _ ._._._._ PLAY
DIRECTIONS: FOR EACH PAIR OF WORDS BELOW PLACE AN x ON THE

BLANR THAT BEST TELLS HOW YOU FEEL ABOUT--
MATHEMATICS

BAD ____________________ GOOD
SAD ____________________ HAPPY
BORING ____________________ EXCITING
JUMP IN ____________________ HOLD BACK
HARD ____________________ EASY
MORE LESS

207

PROBABILITY ATTITUDE SCALE (PAS)

NAME BOY / G I RL

 

EXAMPLE: FOR EACH PAIR OF WORDS BELOW PLACE A X ON THE

BLANK THAT BEST TELLS HOW YOU PEEL ABOUT--

SNOW
LIKE _M_ _ ‘_ _ _ __ _ _._ _, ___ _ _ ‘_._ _ _ HATE
UGLY _ _‘_ _p ‘_ _ _‘_ _._._ _ ‘_ _q_ _ ‘_ ___ PRETTY
WORK _‘_ _ ‘_._._ _, _j_ _ _ ._._ _ _ ‘_‘_‘_‘_ PLAY
DIRECTIONS: FOR EACH PAIR OF WORDS BELOW PLACE AN X ON THE
BLANK THAT BEST TELLS HOW YOU FEEL ABOUT--
PROBABILITY ACITIVIES

BAD ____________________ GOOD
SAD ____________________ HAPPY
BORING ____________________ EXCITING
JUMP IN ____________________ HOLD bACK
HARD ____________________ EASY
MORE LESS

208

NOW IT'S YOUR TURN

THINK ABOUT THE PROBABILITY ACTIVITIES WE HAVE EXPLORED,
SOME OF THE ACTIVITIES WERE TAKE SURVEYS, AREA MODELS FOR
PROBABILITY, FAIR AND UNFAIR GAMES, EXPECTED VALUE, PASCAL'S
TRIANGLE.

(1)

(3)

(4)

(5)

(b)

DID THESE ACTIVITIES MAKE YOU THINK? EXPLAIN.

WHAT IS THE MOST INTERESTING IDEA THAT YOU LEARNED FROM
THESE ACTIVITIES?

WHAT DID YOU FIND HARD ABOUT THESE ACTIVITIES?

WHAT DID YOU LIKE MOST ABOUT THESE ACTIVITIES? WHY?

WHAT DID YOU LIKE LEAST ABOUT THESE ACTIVITIES? WHY?

OTHER COMMENTS?

APPENDIX C

Pearson Correlation Matrices and
Reliability Coefficients

PEARSON

209

TABLE C.1

CORRELATION MATRIK
FOR SITE 1 BY GRADEa

 

 

Grade g1)_ g2) £3) (4)

PTb 6 1.000
Pretest (1) 7 1.000

8 1.000
PT 6 .374* 1.000
Pretest (2) 7 .631* 1.000

8 .780* 1.000
MASC b .185* .243* 1.000
Pretest (3) 7 .123 .122 1.000

8 .105 .091 1.000
MAS 6 .102 .210* .526* 1.000
Posttest (4) 7 .143 .192* .b39* 1.000

8 .112 .103 .004* 1.000
PAS“ 6 .206* .220* .507* .267*
Posttest (5) 7 .171 .249* .228* .289*

 

a N of 6th Graders = 149, N of 7th Graders = 139, N or 8th
Graders = 157.
PT - Probability Test.
MAS - Mathematics Attitude Scale.
PAS - Probability Attitude Scale.
Signiticant P < .01.

*QOU'

210

TABLE C.2

PEARSON CORRELATION MATRIX

FOR SITE 2 BY GRADEa

 

 

Grade (1) (2) (3) (4)
PTb 6 1.000
Pretest (1) 7 1.000
8 1.000
PT 6 .595* 1.000
Pretest (2) 7 .626* 1.000
8 .708* 1.000
MASC 6 .136* .205* 1.000
Pretest (3) 7 .125 .369* 1.000
B .103* .195* 1.000
MAS o .129* .212*‘ .705* 1.000
Posttest (4) 7 .064 .214* .670* 1.000
8 .171* .154* .764* 1.000
PASd o .149* .168* .184* .250*
Posttest (5) 7 .107 .210* .502* .473*
8 .234 .073* .29B* .428*
a N or 6th Graders = 340, N or 7th Graders a 101, N of 8th

b
c
d
*

Graders - 160.

PT - Probability Test.
MAS - Mathematics Attitude Scale.
PAS - Probability Attitude Scale.
Significant P < .01.

 

211

TABLE C.3

PEARSON CORRELATION MATRIX
FOR SITE 3 BY GRADEa

 

 

 

Grade (1) (2) (3) (4)
PTb 6 1.000
Pretest (1) 7 1.000
8 1.000
PT 6 .303* 1.000
Pretest (2) 7 .565* 1.000
8 .579* 1.000
MASc 6 .233* .174 1.000
Pretest (3) 7 .230* .116 1.000
8 .147* .150* 1.000
MAS 6 .182* .153 .671* 1.000
Posttest (4) 7 .137 .134 .715* 1.000
8 .104 .162* .693* 1.000
PAS“ 6 .086 .089 .174 .382*
Posttest (5) 7 .070 .144 .396* .372*
8 -.106 -.062 .242* .324*
a N of 6th Graders = 133, N or 7th Graders = 109, N of 8th
Graders = 203.
b PT - Probability Test.
C MAS - Mathematics Attitude Scale.
2 PAS - Probability Attitude Scale.

Significant P < .01.

PEARSON

212

TABLE C.4

CORRELATION MATRIX
FOR SITE 1 BY SEXa

 

 

 

(1) (2) (3). (4)
PTb Total 1.000
Pretest (1) Boys 1.000
Girls 1.000
PT Total .686* 1.000
Pretest (2) Boys .712* 1.000
Girls .657* 1.000
MASc Total .123* .130* 1.000
Pretest (3) Boys .134 .125 1.000
Girls .110 .133 1.000
MAS Total .146* .172* .578* 1.000
Posttest (4) Boys .152* .216* .626* 1.000
Girls .141* .129* .535* 1.000
PASd Total .134* .254* .304* .2961*
Posttest (5) Boys .226* .314* .509* .456*
Girls .144* .197* .238* .153*
a Total Sample N - 445, #Boys = 217, #Girls = 228

X-QOU'

PT - Probability Test.

MAS - Mathematics Attitude Scale.
PAS - Probability Attitude Scale.

Significant P < .01.

PEARSON
FOR SITE 2 BY SEXa

213

TABLE C.5

CORRELATION MATRIX

 

 

(I) (2) £3) (4)
PTb Total 1.000
Pretest (1) Boys 1.000
Girls 1.000
PT Total .666* 1.000
Pretest (2) Boys .717* 1.000
Girls .582* 1.000
MASc Total .066 .202* 1.000
Pretest (3) Boys .133* .255* 1.000
Girls -.008 .147* 1.000
MAS Total .057 .176* .755* 1.000
Posttest (4) Boys .119* .244* .728* 1.000
Girls -.030 .087 .790* 1.000
PASd Total .066 .142* .330* .388*
Posttest (5) Boys .100 .202* .286* .339*
Girls .000 .055 .387* .452*

 

g Total Sample N = 621, #Boys = 317, #Girls = 304
PT - Probability Test.

MAS - Mathematics Attitude Scale.
PAS - Probability Attitude Scale.
* Significant P < .01.

(100’

PEARSON

214

TABLE C.6

CORRELATION MATRIX
FOR SITE 3 BY SEXa

 

 

U) (2) (3) (A)
PTb Total 1.000
Pretest (1) Boys 1.000
Girls 1.000
PT Total .546* 1.000
Pretest (2) Boys .579* 1.000
Girls .492* 1.000
MASc Total .148* .114* 1.000
Pretest (3) Boys .147* .150* 1.000
Girls .173* .094 1.000
MAS Total .112* .134* .670* 1.000
Posttest (4) boys .104 .162* .693* 1.000
Girls .130 .109 .709* 1.000
Posttest (5) Boys -.011 -.062 .242* .324*
Girls .014 .011 .287* .422*

 

a Total Sample N - 392, #Boys - 203, #Girls = 189

b PT - Probability Test.

C
(1
*

, MAS

- Mathematics Attitude Scale.

PAS - Probability Attitude Scale.
Significant P < .01.

RELIABILITY COEFFICIENTS-CRONBACH a FOR MGMP PT,
MAS AND PAS BY SITE BY TIME BY GRADE BY SEX

215

TABLE C.7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PTa MASb PASC
PRETEST POSTTEST PRETEST POSTTEST
N
Site 1
Grade 6 149 .68 .67 .84 .83 .80
Grade 7 139 .61 .81 .83 .82 .82
Grade 8 157 .75 .82 .77 .84 .84
Total 435 .70 .79 .81 .83 .82
boys 217 .73 .80 .80 .80 .82
Girls 228 .66 .77 .83 .85 .83
Site 2
Grade 6 340 .58 .74 .85 .86 .86
Grade 7 101 ..73 .83 .88 .88 .89
Grade 8 160 .67 .73 .84 .85 .86
Total 621 .69 .78 .88 .88 .87
Boys 317 .74 .81 .86 .89 .88
Girls 306 .59 .73 .89 .87 .87
Site 3
Grade 6 133 .45 .58 .77 .83 .84
Grade 7 150 .59 .79 .84 .86 .86
Grade 8 109 .58 .73 .82 .83 .82
Total 392 .56 .75 .82 .86 .85
Boys 203 .54 .77 .82 .86 .86
Girls 189 .56 .72 .82 .86 .84
All
Grade 6 622 .61 .71 .83 .85 .85
Grade 7 380 .70 .83 .84 .85 .87
Grade 8 444 .70 .77 .83 .86 .88
Total 1446 .68 .79 .84 .86 .87
Boys 731 .72 .80 .84 .86 .86
Girls 717 .75 .62 .85 .86 .87

 

a PT - Probability Test (Range 0-25).

b MAS - Mathematics Attitudes Scale (Range 1-5).
C PAS - Probability Attitudes Scale (Range 1-5).

APPENDIX D

Mean, Standard Deviations and ANOVA Tables

216

TABLE D.1

MEANS AND STANDARD DEVIATIONS OF MGMP PT, MAS, AND PAS
SCORES FOR THE ENTIRE SAMPLE BY TIME BY GRADE BY SEX

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PREPTUTa POSTPTOTP PREMASC POSTMAS“ PASe -
M M M M M
(S.D.) (S.D.) (S.D.) (S.D.) (S.D.)
Site 1 444 7.394 10.712 3.509 3.351 3.315
(3.674) (4.432) (.896) (.811) (.840)
Site 2 621 8.977 12.034 3.236 3.161 2.654
((3.695) (4.309) (.970) (.954) (.939)
Site 3 381 9.202 12.465 3.236 3.164 3.081
(3.058) (3.935) (.810) (.841) (.889)
Grade 6 622 7.701 10.963 3.520 3.389 3.027
(3.177) (3.750). (.887) (:878) (.043)
Grade 7 379 9.269 13.193 3.351 3.258 3.048
(3.775) (4.838) (.872) (.855) (.897)
Grade 445 9.126 11.593 3.011 2.952 2.823
(3.802) (4.259) (.909) (.862) (.960)
Boys 725 9.959 12.104 3.292 3.209 2.965
(3.871) (4.551) (.926) (.910) (.961)
Girls 721 9.143 11.447 3.349 3.230 2.969
(3.274) (4.055) (.902) (.860) (.919)
All 1446 8.551 11.741 3.319 3.220 2.070
(3.613) (4.309) ((.015) (.886) (.941)
a Probability Pretest (Range 0-25).
b Probability Posttest (Range 0-25).
c Pretest Mathematics Attitude Scale (Range 1-5).
: Posttest Mathematics Attitude Scale (Range 1-5).

Probability Attitude Scale (Range 1-5).

217

TABLE D.2

MEANS AND STANDARD DEVIATIONS OF MGMP PT, MAS, AND PAS

SCORES FOR SITE 1 BY TIME BY GRADE BY SEX

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rTa MAS“ PASC
‘N Pretest Posttest Pretest Posttest
M M M M M

(S.D.) (S.D.) (S.D.) (S.D.) (S.D.)

Grade 149 6.631 9.792 3.607 3.256 3.156
(3.347) (3.548) (.947) (.816) (.869)

Boys 76 6.974 10.16 3.568 3.241 3.118
(4.043) (3.92) (.964) (.826) (.937)

Girls 73 6.274 9.41 3.648 3.276 3.194
(2.647) (3.09) (.934) (.810) (.796)

Grade 138 7.623 11.094 3.456 3.377 3.460
(3.298) (4.698) (.920) (.858) (.797)

Boys 66 7.455 10.379 3.548 3.407 3.288
(3.254) (4.706) (.871) (.774) (.787)

01:13 72 7.778 11.750 3.373‘ 3.350 3.618
(3.354) (4.626) (.962) (.933) (.778)

Grade 157 7.917 11.248 3.462 3.416 3.339
(4.086) (4.821) (.808) (.760) (.828)

Boys 74 8.162 11.905 3.537 3.453 3.426
(4.204) (4.916) (.807) (.776) (.735)

Girls 83 7.700 10.663 3.400 3.384 3.261
(3.990) (4.686) (.828) (.749) (.901)

Total 444 7.394 10.712 3.509 3.351 3.315
(3.674) (4.432) (.896) (.811) (.840)

Boys 216 7.528 10.824 3.549 3.364 3.276
(3.893) (4.570) (.880) (.795) (.833)

Girls 228 7.268 10.605 3.471 3.339 3.352
(3.458) (4.306) (.911) (.828) (.847)

 

a MGMP Probability test (Range 0-25).
b Mathematics Attitude Scale (Range 1-5).
C Probability Attitude Scale (Range 1-5).

218

TABLE D.3

MEANS AND STANDARD DEVIATIUNS 0F MGMP PT, MAS, AND pAs
SCORES FUR SITE 2 BY TIME BY GRADE BY SEX

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PTa MASb PASC
N Pretest Posttest Pretest Posttest
M M M M M

(S.D.) (S.D.), (S.D.) (S.D.) (3.0:)

Grade 340 7.803 11.268 3.510 3.404 2.808
(3.109) (3.974) (.908) (.930) (.943)

Boys 162 8.136 11.401 3.422 3.360 2.840
(3.411) (4.096) (.930) (.995) (1.004)

Girls 178 7.500 11.146 3.590 3.444 2.78
(2.781) (3.867) (.883) (.869) (.886)

Grade 101 11.713 14.970 3.406 3.320 2.937
(3.861) (4.757) (.856) (.850)) (.923)

Boys 53 12.679 15.906 3.583 3.547 2.937
(4.094) (5.208) (.947) (.945) (1.011)

Girls 48 10.646 13.938 3.212 3.069 2.629
(3.310) (4.008) (.702) (.653) (.793)

Grade 180 9.661 11.833 2.622 2.613 2.285
(3.653) (3.969)) (.866) (.824) (.839)

Boys 102 10.128 12.049 2.613 2.601 2.335
(4.019)_ (4.225) (.850) (.819) (.849)

Girls 78 9.051 11.551 2.635 2.628 2.220
(3.028), (3.613) (.892) (.836) (.827)

Total 621 8.978 12.034 3.236 3.161 2.654
(3.695) (4.309) (.970) (.954) (.939)

Boys 317 9.536 12.363 3.188 3.147 2.694
(4.077) (4.618) (.990) (1.006) (,987)

Girls 304 8.395 11.691 3.285 3.175 2.612
(3.151)) (3.940) (.948) (.898)) (.887)

 

a MGMP Probability test (Range 0-25).
b Mathematics Attitude Scale (Range 1-5).
C Probability Attitude Scale (Range 1.5).

219

TABLE D.4

MEANS AND STANDARD DEVIATIONS OF MGMP PT, MAS, AND PAS
SCORES FOR SITE 3 BY TIME BY GRADE BY SEX

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P'l'a MASb PASC
N Pretest Posttest Pretest Posttest
M M M M M

(S.D.) (S.D.) (S.D.) (S.D.) (S.D.)

Grade 133 8.639 11.496 3.450 3.496 3.440
(2.687) (3.071) (.752) (.792) (.863)

boys 70 9.257 12.057 3.419 3.502 3.469
(2.827) (3.261) (.757) (.805) (.885)

Girls 63 7.952 10.873 2.480 3.489 3.407
(2.358) (2.739) (.752) (.783) (.843)

Grade 140 9.129 . 13.979 3.207 3.096 2.828
(3.203) (4.282) (.819) (.837) (.826)

Boys 65 9.400 14.431 3.177 3.085 2.805
(2.122) (4.448) (.706) (.782) (.825)

Girls 75 8.693 13.587 3.233 3.107 2.849
(3.274) (4.123) (.910) (.886) (.831)

Grade 108 9.991 11.694 3.005 2.843 2.968
(3.155) (3.844) (.804) (.761) (.866)

Boys 57 10.263 12.035 2.842 2.766 2.927
(3.160) (4.031) (.902) (.827) (.977)

Girls 51 9.686 11.314 3.186 2.928 3.013
(3.153) (3.625) (.640) (.678) (.729)

Total 381 9.202 12.465 3.236 3.164 3.081

(3.058) (3.935) (.810) (.841) (.889

Boys 192 9.604 12.854 3.166 3.142 3.083
(3.044)), (4.064) (.817) (.855) (.938)

Girls 189 8.794 12.069 3.304 3.186 3.079
(3.026)) (3.770), (.799) (.828) (.840)

 

a MGMP Probability test (Range 0-25).
b Mathematics Attitude Scale (Range 1-5).
C Probability Attitude Scale (Range 1-5).

220

TABLE D.5

ANALYSIS OF VARIANCE SUMMARY FOR MEAN DIFFERENCE BETWEEN
THE PosTMASa AND PASb SCORES FOR THE ENTIRE SAMPLE

 

 

 

Source of Variation D.F. MS F P<
Grade 2 7.427 7.052* .001
Sex . 1 .101 .096 .757
Grade x sex 2 ' 3.256 3.091 .046

Between Groups 5

Within Groups 1440 1.053

 

a POSTMAS - Posttest Mathematics Attitude Scale.
PAS - Probability Attitude Scale.
* Significant P < .001.

221

TABLE 0.6

ANALYSIS OF VARIANCE SUMMARY FOR MAIN DIFFERENCE BETWEEN
THE POSTMASa AND PASb SCORES FOR SITE 2

 

 

 

Source of Variation , D.F. MS F P<
Grade** 2 3.974 3.680* .026
Sex 1 1.300 1.204 .273
Grade x sex 2 1.025 .949 .388

Between Groups 5

Within Groups 615. 1.080

 

a POSTMAS - Posttest Mathematics Attitude Scale.
PAS - Probability Attitude Scale.

* Significant P < .05.

** Alternative to the test in Table 4.20. Grade level
contrasts are omitted from this test.

TABLE D . 7

ANALYSIS OF VARIANCE SUMMARY FOR MAIN DIFFERENCE BETWEEN
THE POSTMASa AND PASP SCORES FOR SITE 2

 

 

 

Source of Variation D.F. MS F P<
Grade** 2 7.711 5.008* .007
Sex 1 .089 .094 .759
Grade x sex 2 .081 .086 .918

Between Groups 5

Within Groups 375 .941

 

a POSTMAS - Posttest Mathematics Attitude Scale.
PAS - Probability Attitude Scale. .

* Significant P < .01.

** Alternative to the test in Table 4.22. Grade level
contrasts are omitted from this test.

TABLE D.8

ANALYSIS OF VARIANCE SUMMARY FOR PASa
FOR SITE 3

 

 

 

 

Source of Variation D.F. MS F P<
Grade 2 7.09 7.979* .001
Sex _ 1 .19 .022 .883
Grade x sex 2 .67 .417 .659

Between Groups 5

Within Groups 1440 .78

a PAS - Probability Attitude Scale.

*

Significant P < .001.

TABLE 0.9

ANALYSIS OF VARIANCE SUMMARY FOR MEAN AVERAGES BETWEEN
THE POSTMAS AND PAS SCORES FOR EACH SITE

 

 

 

 

 

 

 

 

 

 

 

Site 1: Source of Variation D.F. MS P P<
Grade 2 1
G1 (7 vs. 6) (1) 6.52 7.51* .006
02 (8 vs 6 8 7) (1) .85 .98 .323
Sex 1 .09 .10 .752
Grade x sex 2 1.25 1.44 .239
Between Groups 5
Within Groups 438 .868
Site 2: Source of Variation D.F. MS F P<
Grade 2 53.45 49.94* .0001
Sex 1 1.52 1.42 .234
Grade x Sex 2 3.23 3.02 .050
5
615 1.07
Site 3: Source of Variation D.F. MS F P<
Grade 2 24.57 27.28* .0001
Sex 1 .22 .25 .619
Grade x Sex 3 .39 .43 .651
Between Groups 5
Within Groups 375 .90

 

* Significant P < .01.

APPENDIX E

Scheffé's Post Hoc Comparisons

225

Of interest to the investigator was an attempt to find
the exact testmanship (Hays, 1973)* and location of
differences between each pairwise comparison. Thus, after a
significant F was found for a given factor (grade level or
sex), Scheffé's posteriori comparisons were employed to
contrast relevant pairs of means. The method due to
Scheffé (1959) was used for its advantages of simplicity,
applicability to unequal group sizes, and robustness to
Violation of normality and homogeneity of variance.
Moreover, there is no limit, unlike planned comparisons, as
to the number of such pairwise comparisons that can be made.
Finally, Scheffé's Post Hoc contrast permits comparisons for
both within and between group cells.

According to Scheffé's (1959, P.71), a contrast is
significantly different from zero at the .05 level, (and in
favour of the grade level or sex written first), if and only

if:
111 > 83%, where

III = the magnitude of the difference between the two

compared means, and

 

611} 'JMSe (Ir-Tl]: +fi1‘3), where n1, n2 are the cell

sizes compared, and

 

* According to Rays (1973, p. 413), testmanship means "how
big is a difference?"

226
MSe = the mean square error of the relevant test (MAS or

MGMP PT);

 

CD

t/(J'I) F0.95’ J-I, N-J’ where

J = 3 for grade level contrasts
J - 2 for sex contrasts, and
N = the total number of (within group) subjects

involved in the original multivariate and univariate
analysis.*

The following Tables E.1 to E.9, of various kinds of
posteriori contrasts, are constructed, using the formulae
desCribed above. Each significant contrast is in favour of
the grade level or sex factor written first in each

contrast.

 

* More discussion on Scheffé's posteriori contrast was given
by Glass and Stanley (1970, Chapter 16).

227

TABLE E.1

SUMMARY OF SCHEFFE'S POSTERIORI COMYARISONSa
ON THE PROBABILITY PRETEST 0F
GRADE LEVEL BOYS AND GIRLS

 

 

Contrast 30? Tb
Grade 7 Boys versus
Grade 6 Boys 1.50 .48
Grade 7 Girls versus
Grade 6 Girls 1.48 1.50*

 

a A description of these comparisons is found on
page 225.

b 0 indicates the difference between appropriate
means from Table 0.2.

* Significant at the .05 level, in favour of grade 7
girls.

TABLB B.2A
SUMMARY OF SCHEhFE'S POSIERIURI COMPARISONSa
OF MGMP PROBABILITY PBBTBST MBAMS FOR
BOYS AND FOR GIRLS IN SITB 2

 

 

 

Lontrast Effect 38? @b
Grade 6 Boys vs. Grade 6 Girls Sex .73 .64
Grade 7 Boys vs. Grade 7 Girls Sex 1.01 2.03*
Grade 8 Boys vs. Grade 8 Girls Sex 1.01 1.08*
Grade 7 Boys vs. Grade 6 Boys Grade 1.33 4.54*
Grade 7 Girls vs. Grade 6 Girls Grade 1.43 3.15*
Grade 7 Boys vs. Grade 8 Boys Grade 1.42 2.55*
Grade 7 Girls vs. Grade 8 Girls Grade' 1.54 1.00*

 

a A description of these comparisons is found on page 225.
@ indicates the difference between apprOpriate means
from Table D.3.

* Significant at the .05 level, in favour of the sex or
grade level written first in each contrast.

229

TABLE E.2B
SUMMARY OF SUREFEE'S POSTERIORI UOMPARISOMSa 0b
MATHEMATICS ATTITUDE PRETEST MEAMS 0F GRADE
LEVEL BOYS AMD GIRLS IN SITE 2

 

 

Contrast Effect 58; Yb

Grade 6 Boys vs. Grade 7 Boys Grade .34 .16

Grade 6 Girls vs. Grade 7 Girls Grade .35 .38*
Grade 7 Boys vs. Grade 8 Boys Grade .37 .97*
Grade 7 Girls vs. Grade 8 Girls Grade .39 .78*
Grade 6 Boyss vs. Grade 8 Boys Grade .27 .81*
Grade 6 Girls vs. Grade 8 Girls Grade .29 .96*

 

a G description of these comparisons is found on page 225.
W indicates the difference between appropriate means
from Table D.3.

* Significant at the .03 level, in favour of the sex or
grade level written first in each contrast.

230

TABLE E.3

SUMMARY OF SCHEFFE'S PUSTERIORI COMPARISONSa OF BOTH
PROBABILITY AND MAThEMATICS ATTITUDE SCALE PRETEST

MEANS OF SEX AND GRADE LEVEL EFFECTS IN SITE 3

 

 

 

 

 

Contrast Effect 56? @b

Grade 6 Boys vs. Grade 6 Girls Sex 1.01 1.31*
Grade 7 Boys vs. Grade 7 Girls Sex 1.0 .51

Grade 8 Boys vs. Grade 8 Girls Sex 1.13 .58

Grade 7 boys vs. Grade 6 boys Grade 1.26 .14

Grade 7 Girls vs. Grade 6 Girls Grade 1.25 .94

Grade 8 boys vs. Grade 7 Boys Grade 1.33 .86

Grade 8 Girls vs. Grade 7 Girls Grade 1.33 .79

Grade 8 Boys vs. Grade 6 Boys Grade 1.33 1.01

Grade 8 Girls vs. Grade 6 Girls Grade 1.38 1.73*
MAS:

Grade 6 Boys vs. Grade 8 Boys Grade .35 .58*
Grade 6 Girls vs. Grade 8 Girls Grade .36 .30

 

§ A description of these comparisons is found on page 225.
@ indicates the difference between appropriate means

from Table D.4.
* Significant at the .05 level, in favour of the sex or

grade level written first in each contrast.

231

TABLE E.4
SUMMARY OF SChEFFE'S POSTERIOKI COMPARISONSa OP PT
AND MAS MEAN DIFFERENCES AND AVERAGES
FROM PRETEST TO POSTTEST IN SITE 1

 

 

 

Criterion ,
8% @b
Measure Contrast
DIFMASc Grade 6 Boys vs. Grade 7 Boys .23 ;.19
Grade Girls vs. Grade 7 Girls .23 -.35**
AVGPTOTd Grade Boys vs. Grade 6 Boys 1.87 .35

Boys vs. Grade 6 boys .38 .25

b
7

Grade 7 Girls vs. Grade 6 Girls 2.12 1.92
Grade 8
8

Grade Girls vs. Grade 6 Girls .27 .36*

a 3 description of these comparisons is found on page
25.

b @indicates the difference between appropriate means
from Table v.2.

C DIFMAS - Mathematics Attitudes mean differences
(Posttest-Pretest).

d AVGPTOT - Probability Attitudes mean averages (Pretest
+ Posttest).

* Significant at the .05 level, in favour of the sex or
grade level weritten first in each contrast.

** The negative sign indicates that grade 6 girls
‘significantly lost more mathematics attitude than grade 7
girls.

SUMMARY OF SCEEFFE'S POSTERIORI CUMPARISONSa OF

232

TABLE E.5

PROBABILITY TEST MEAN GAINS AND AVERAGES

FROM PRETEST T0 POSTTEST IN SITE 2

 

 

 

 

Criterion .
8°; @b
Measure Contrast
DIFMASc Grade 7 Boys vs. Grade 6 Boys , .90 .04
(Gains in Grade 7 Girls vs. Grade 6 Girls .92 .84
probability Grade 7 boys vs. Grade 8 Boys .96 1.30*
mean scores) Grade 7 Girls vs. Grade 8 Girls 1.04 .79
Grade 6 Boys vs. Grade 8 Boys .72 1.34*
Grade 6 Girls vs. Grade 8 Girls .77 1.15*
AVGP‘I‘OTd Grade 6 Boys vs. Grade 6 Girls 1.01 .45
(Averages in Grade 7 Boys vs. Grade 7 Girls 1.86 2.00*
probability Grade 8 Boys vs. Grade 8 Girls 1.41 .79
mean scores) Grade 7 Boys vs. Grade 6 Boys 1.85 4.52*
Grade 7 Girls vs. Grade 6 Girls 1.90 2.97*

 

a A description of these comparisons is found on page 225.
Y indicates the difference between appropriate means
from Table D.3.

C DIFMAS

(Posttest-Pretest).
d AVGPTOT - Probability Attitudes mean averages (Pretest

+ Posttest).

- Mathematics Attitudes mean differences

* Significant at the .05 level, in favour of the sex or

grade level written first in each contrast.

233

TABLE E.6

SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONSa OF
PROBABILITY TEST MEAN GAINS FROM PRETEST
TO POSTTEST IN SITE 3

 

 

Contrast 86$ Sb
Grade 7 Boys vs. Grade 6 Boys .95 2.23*
Grade 7 Girls vs. Grade 6 Girls .94 1.77*
Grade 7 Boys vs. Grade 8 Boys 1.00 3.26*
Grade 7 Girls vs. Grade 8 Girls 1.00 3.07*
Grade 6 Boys vs. Grade 8 Boys .98 1.03*
Grade 6 Girls vs. Grade 8 Girls . 1.04 1.29*

 

a A description of these comparisons is found on page 225.
Q indicates the difference between appropriate means
from Table D.4.

* Significant at the .05 level, in favour of the sex or
grade level written first in each contrast.

234

TABLE E.7

SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONSa
FOR PROBABILITY ATTITUDE SCALE BY
GRADE LEVEL FOR SITE 1

 

 

Contrast Sag @b P<
Grade 7 to Grade 6 .24 .30* .05
Grade 7 Boys to Grade 6 Boys .34 .17 ns
Grade 7 Girls to Grade 6 Girls .34 . .42* .05
Grade 8 to Grade 6 .23 .18 ns
Grade 7 to Grade 8 .24 .12 ns

 

a A description of these comparisons is found on page 225.
@ indicates the difference between appropriate PAS means
from Table 4.23.

* Significant at the .05 level, in favour of the grade level
written first in each contrast.

235

. TABLE E.8

OF GRADE LEVELS FOR SITE 2

SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONSa
FOR PROBABILITY ATTITUDES SCALE MEANS

 

 

 

 

Contrast 36$ Tb

Grade 6 vs. Grade 7 .25 .02
Grade 6 vs. Grade 8 .21 .52*
Grade 6 Boys vs. Grade 8 Boys .28 .51*
Grade 6 Girls vs. Grade 8 Girls .30 .56*
Grade 7 vs. Grade 8 .28 .51*
Grade 7 Boys vs. Grade 8 Boys .38 .60*
Grade 7 Girls vs. Grade 8 Girls .41 .41
a A description of these comparisons is found on page 225.

@ indicates the difference between appropriate PAS means
from Table 4.23.
* Significant at the .05 level, in favour of the sex or
grade level written first in each contrast.

236

TABLE E.9
SUMMARY OF SCBEFFE'S POSTERIORI COMPARISONSa
FOR PROBABILITY ATTITUUES SCALE MEANS
OF GRADE LEVELS FOR SITE 3

 

A

 

 

 

Contrast so; ' Eb
Grade 6 vs. Grade 7 .25 .61*
Grade 6 Boys vs. Grade 7 Boys .36 .66*
Grade 6 Girls vs. Grade 7 Girls .36 .57*
Grade 6 vs. Grade 8 .27 .47*
Grade 6 Boys vs. Grade 8 Boys .39 .54*
Grade 6 Girls vs. Grade 8 Girls .39 .39
Grade 8 vs. Grade 7 .27 .14
Grade 8 Boys vs. Grade 7 Boys .38 .12
Grade 8 Girls vs. Grade 7 Girls .38 .16

 

a A description of these comparisons is found on page 225.

b ‘ indicates the difference between apprOpriate PAS means
Erom Table 4.23.

* Significant at the .05 level, in favour of the sex or
grade level written first in each contrast.

 

 

B IBLIOGRAPRY

 

 

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