Table Of ContentTHE ROLE OF ALGEBRA IN THE DEVELOPMENT
OF1 RELATIONAL THINKING
Dissertation
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
William Nichols Jackson, B.S., M.S.
The Ohio State University
1952
Approved by:
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ACKNOWLEDGMENTS
It Is with gratitude that I acknowledge the guidance
given by my adviser, Dr. Harold P. Fawcett, in the planning
and reporting of this study. The.type of thinking which he
stimulated was a frequent source of inspiration to me.
The suggestions of Dr. Earl W. Anderson and Dr. G. P. Cahoon
were unusually helpful to me in the preparation of the
dissertation.
The untiring efforts of my dear wife, Dorwatha, in the
typing of the first and final drafts of the dissertation
merit ray continuous thankfulness.
William N. Jackson
£02420
ill
TABLE OF CONTENTS
CHAPTER PAGE
ACKNOWLEDGEMENTS ............. 11
I THE NATURE OF RELATIONAL THINKING AND ITS
RELATIONSHIP TO THE INTERPRETATION OF DATA ___ 1
Statement of the Problem ....... 1
Plan of the Study ........................... 1
Need for and Importance of the Study ....... 2
The Nature of Relational Thinking .......... 10
Relational Thinking and Interpretation of Data 12
II PROBLEMS IN THE INTERPRETATION OF DATA ....... 16
Introduction ......................... 16
Definition and,Analysis of the Ability ..... 19
Studies Related to Interpretation of Data ... 30
III SUGGESTED ALGEBRAIC CONTENT AND TEACHING METHOD
FOR DEVELOPING ABILITY TO INTERPRET DATA ..... 47
Introduction . . .......................-..... 47
Algebraic Content ........................... 53
Relationship ........................... 53
Number ................................. 62
Measurement ............................ 67
Symbolism .............................. 71
Operation .............................. 75
Proof .................................. 78
Basic Assumptions ........................... 81
Outcomes ........................... 85
Teaching Method ............................. 87
IV GENERAL PROCEDURES AND EXPERIMENTAL CONDITIONS . 90
General Procedure ........................... 90
The School .................................. 91
The Students .................. 92
V EXPERIMENTAL MATERIALS AND CLASSROOM PROCEDURES 101
Summary of Instructional Units .............. 101
The Study Guide ............................. 107
Unit One - The Language of Algebra ......... 110
Unit Two - Extending the Number System ..... 126
Unit Three - Approximate and Functional
Relationships ............... 151
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CHAPTER,-, PAGE
Unit Four - Extending Operations
in the Number System ....... 189
VI EVALUATING- STUDENT ABILITIES TO INTERPRET,
DATA ....................................... 209
Significance of Gains in Ability to
Interpret Data .......................... 209
Apparent Influence of Intelligence
on Interpretation of Data .............. 225
Apparent Influence of Reading Ability
on Interpretation of Data .............. 226
Apparent Influence of Sex on
Interpretation of Data ............,.. . 233
Conclusions ........................... 235.;.
Discussion of Individual Students ........ 237
Inventory of Behaviors Evaluated .... ; .. . 241
.VII IMPLICATIONS FOR GENERAL EDUCATION ....... 246
General Summary ........................... 246
Recommendations ........................... 247
New Studies ............................ 249
BIBLIOGRAPHY .............................. 255
APPENDIX .................................. 259
AUTOBIOGRAPHY 369
V.
LIST OF TABLES
TABLE PAGE
1 SPEED AND REACTION DISTANCE.....J............... 27
2 GRADUATES ATTENDING COLLEGE .................... 92,
3 PARENTAL STATUS OF STUDENTS .................... 93
4 RESIDENTIAL STATUS OF STUDENTS ..... 93
5 OCCUPATIONS OF PARENTS '......................... 94
6 AGES OF STUDENTS ............................... 94
7 INTELLIGENCE QUOTIENTS OF STUDENTS IN SECTION 1 . 9&
8 INTELLIGENCE QUOTIENTS OF STUDENTS IN SECTION 2 . 97
9 READING PERCENTILES OF STUDENTS ................ 98
10 CLASS SCHEDULES ................ 99
11 NEGATIVE RESPONSES IN LISTING OF ANTONYMS ...... 127
12 FREQUENCY OF OPPOSITES SUGGESTED BY STUDENTS .... 128
13 FREQUENCY OF NUMBER OPPOSITES SUGGESTED BY
STUDENTS ........................................ 129
14 STUDENT RESPONSES TO PROBLEM 1 131
15 STUDENT RESPONSES TO PROBLEM 3 132
16 RESPONSES OF STUDENTS TO PROBLEM 4 132
17 STUDENT RATINGS OF CONVERSE STATEMENTS ......... 177
18 STUDENT RATINGS OF STATEMENTS ILLUSTRATING
TRANSITIVITY .................................... 180
19 STUDENT RATINGS OF STATEMENTS IN STUDY GUIDE
NO. 42 182
20 STATEMENTS CHECKED BY STUDENTS ON TEST_5.4 187
21 STUDENT EVALUATION OF INTERPRETATIVE STATEMENTS . 204
22 MEDIANS OF STUDENT RESPONSES TO TEST.5.5 208
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TABLE ■. PAGE,
23 SCORES ON INTERPRETATION OF DATA TEST.,
FOR SECTION 1, December, 1950 213
24 SCORES ON INTERPRETATION OF DATA TESTl
FOR,SECT ION 2, December, 1950 214
25 SCORES ON INTERPRETATION OF DATA TESTi
FOR SECTION. 1, June, 1951 215:
26 SCORES ON INTERPRETATION OF DATA TEST.
FOR SECTION 2, June, 1951 216
27 MEAN GAINS ON INTERPRETATION OF DATA TEST ...... 217.
28 SIGNIFICANCE OF DIFFERENCE IN MEANS ............ 219
29 SCORES ON INTERPRETATION OF DATA TEST.BY
STUDENTS HAVING HIGHER INTELLIGENCE QUOTIENTS ... 223;
30 SCORES ON INTERPRETATION OF DATA TEST,BY.
STUDENTS HAVING LOWER INTELLIGENCE QUOTIENTS .... 224
31 MEAN GAINS ON INTERPRETATION OF DATA TEST .... 225
32 SIGNIFICANCE OF DIFFERENCE IN MEANS ............ 226
. 33 SCORES ON INTERPRETATION OF DATA TEST BY
STUDENTS HAVING BETTER READING ABILITY .......... 227,
34 SCORES ON INTERPRETATION OF DATA TEST BY
STUDENTS HAVING POOR READING ABILITY ............ 228
35:: MEAN GAINS ON INTERPRETATION OF DATA TEST ...... 229
36 SIGNIFICANCE OF DIFFERENCE IN MEANS ............ 230
37 SCORES ON INTERPRETATION OF DATA TEST BY BOYS ... 231
38 SCORES ON INTERPRETATION OF DATA TEST BY GIRLS .. 232
39 MEAN GAINS ON INTERPRETATION OF DATA TEST ...... 233
40 SIGNIFICANCE OF DIFFERENCE IN MEANS ........ . ... 234
41 GAINS BY STUDENT F IN ABILITY TO INTERPRET DATA .. 23.8
42.. GAINS BY STUDENT A' IN ABILITY TO INTERPRET DATA . 240
I
vii
LIST OF CHARTS.
CHARTj.' PAGE']
1 WEIGHT-HEIGHT RELATION FOR BABYj.BOY .............. 15.8
2. WEIGHT «*HEIGHT iRELAT ION. .FOR AVERAGE TYPE
GIRLS ............................................ 166
3: AREAS OF SQUARES . .................... 194
1
THE: ROLE': OP ALGEBRA’ . IN THE DEVELOPMENT.
OF RELATIONAL THINKING
Chapter 1
The Nature of Relational Thinking and its Relationship to
the Interpretation of Data-
Statement of the Problem
The problem investigated was; concerned with the
teaching of elementary algebra. The purpose of the study
was; to determine the extent to which the students in two
ninth grade classes in the William Grant High School lo
cated at Cbvington, Kentucky,, can develop understandings:
and abilities associated with the interpretation of data^
through the study and use of algebraic concepts. Since
only those concepts are considered which, in the judgment
of the writer seem to contribute to student facilities for
interpreting data,, this Involves the selection of appro
priate content and teaching methods:. Ah important aspect
of the problem is the generalization of these under
standings and abilities for use beyond the narrow confines:
of elementary algebra. It is the writer*s belief that the
ability to interpret data is an integral part of one of
the great: fields of mathematics, relational thinking.
Plan of the Study
The plan for the discussion of this study follows.
In the first chapter the characteristics of relational
thinking have heen discussed, and its intimate relation
ship with the interpretation of data have "been shown.
In chapter two, a detailed analysis has been made of the
behaviors characteristic of the Interpreter of data.
Chapter three is concerned with the theory for planning
a course in algebra designed to Improve ability to inter
pret data.
Ih chapter four, a description has been given of the
students and facilities Involved in the experiment. A
detailed description of the experimental course has been
presented in chapter five. This description Includes
teaching techniques, classroom materials, and summaries
of the accomplishments of students.
The sixth chapter consists of an evaluation of
student growth in the abilities Involved in the Interpre
tation of data. The final chapter is a discussion of the
implications for general education as suggested by the
results of the study.
Need for and Importance of the Study
Our environment is: composed of many elements which
are forever changing, but the changes in these elements
are determined by changes in related elements. To live
happily a person must be able to perceive these relations
as they apply to his welfare and the welfare of his
neighbors. Many of these elements are presented falsely
through numerous communication devices in use today.
3
Frequently the elements are so presented as to hide or
distort the relationships between them. It Is the common
practice of some persons to emphasize the relations which
favor an individual or group and to ignore or belittle
those which are not favorable to them. In this world of
advertising, political platforms, accidental deaths., and
propaganda, young people need to become aware of the
devious, as well as proper, ways to interpret relations
between significant elements in any situation.
To think at all is to think in terms of relations.
For example,, relational thinking is used no less by the
carpenter as he decides to use white pine lumber because
of such elements as humidity, wear, and type of finish
desired, than by the physician who considers such ele
ments as temperature, blood pressure, skin condition and
other variables in the human body as he concludes that
his patient suffers from a particular malady. Such
thinking is used by the cook who knows that a "pinch of
salt" along with varying amounts of other ingredients
with proper baking will produce a certain type of deli
cacy j- and by the truck driver who knows the relations
between speed, the condition of the road surface and the
braking system such that he can negotiate a highway
curve safely at a speed which makes him neither an over
cautious nor a reckless driver. Even a small child who
has used data derived from the actions of his parents to
