A COMPARISON OF THE ABILITIES 0F LATE ’ ,
ELEMENTARY SCHOOL CHILDREN TO LEARN TASKS r =

ON THE OPERATIONS 0F SIGNED NUMBERS- - .

Thesis for the Degree of Ph. D.

MECHIGAN STATE UNIVERSm'

JAMES EDWARD mm ’
1970

  
      

  

LIBRARY
Michigan State
' ' University

'w'

THgg‘C

This is to certify that the

thesis entitled

A COMPARISON OF THE ABILITIES OF LATE
ELEMENTARY SCHOOL CHILDREN TO LEARN
TASKS ON THE OPERATIONS OF SIGNED
NUMBERS

presented by

James Edward Riley

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Education

Z/ 1
M 4-; / “‘7
Major profes(or

Date 7/} Y/70

0-169

 

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x51. 317'”! \ t l. ' * .‘ 1 . ‘ ’3‘" - 17‘ r 5 (Eur riti". OPERATIONS
L r‘ N l Y {:1
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Elementary sch;fil mathemat;cs has undergone dramatic

changes in both cornenfi and proueitres within the past ten

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with recommendations for the ineiusion of still newer ideas
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signed numbers.

The numbersa in fact integersa were represented on the
number line as biwdireCtienal vectors. The number line was
coordinated by indicating tne direcrion and distance a point
was located from zero, :he onerattcn of addition was defined
as vector additiono The operation of subtraction was motiva-

ted by yresentin- that cperatien as the inverse operation

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of addition. The rti:s for mglt1p.ication weme developed
as a conseQUence cf the distributive and additive inverse
properties° fhe skills needed t3 effectively work with these

operations were organized ints seve.:etn obgectively scored

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taskso fines tasks were Egrthsr group,i into six lessons.

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retentiono

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ed than the iyuith grassrso Oxeiailg a claSs was said to have

at achievement 'or a particular
lessen if 5(% cr more :5 the class sacred 50% or more on the
test follewing teat lessens It was reund that sixth graders
could be expected t; achieve at this level 90% of the time,

whereas fourth ind 31fth graders could fig so only 80% of the

n

timeo

A COMPARISON OF THE ABILITIES OF LATE ELEMENTARY
SCHOOL CHILDREN TO LEARN TASKS ON THE OPERATIONS
OF SIGNED NUMBERS
by

James Edward Riley

A THESIS

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

DOCTOR OF PHILOSOPHY
College of Education

1970

To my three girls:

Netty,

Lisalu,

and Jennifer

ACKNOWLEDGMENTS

The Author wishes to acknowledge and thank, first and
foremost, Dr. Wayne Taylor for his help and guidance over
the years that have lead to the completion of this work.
A debt of gratitude is due Dr. Isobel Blyth, Dr. George Meyers,
and Dr. William Welsh for their help in formulating the prob-
lem and the design as well as performing the lengthy task of
critically reading the thesis. Special expressions of thanks
are given to two of my friends. They are Dr. Michael Stoline,
for sharing his time and his knowledge of statistics and de-
sign, and Miss Bernice Gan, for sharing her time and her
knowledge of FORTRAN and the IBM 1620. Also, the author
wishes to thank Mrs. Judith Warriner for the long hours she
spent at the typewriter preparing the manuscript. Finally,
the outstanding cooperation and enthusiasm of the many teach~
ers who participated in the study with their classes have
made this a truly memorable experience.

iii

TABLE OF CONTENTS

LIST OF TABLES o o o o o o o o o o o o o o o o o o o
CI‘H’LPTER l: TI'IE PROBLEM o o o o o o o o o o o o o O

The Need

The Purpose
The Assumption
Definitions
The Overview

CHAPTER 2: REVIEW OF LITERATURE o o o o o o o o o o

The Integers: Their Development and Pedagogy

Teaching Mathematical Structure in the
Elementary School

Theories of Instruction in Mathematics

Summary

CHAPTER 3: DESIGN OF THE STUDY 0 o o o o o o o o o

The Curriculum.And Its Presentation
Sample

Measures

The Design of the Study

Treatment Procedures

The Hypotheses

Analysis

Summary

CI'EPTER 4' 3 ANALYSIS OF DATA 0 o o o o o o o o o o 0
Summary

CHAPTER 5: SUMMARY AND CONCLUSIONS . . . . . . . .
Summary
Conclusions
Discussion

Recommendations

BIBLIOGRAPHY O O O O O O O O O O O O O O O O O O O 0

iv

APPENDIX.A:

APPENDIX B:

PROBLEM SETS AND POST TEST . . .

PLANNED COMPARISONS COMPUTATIONAL

PROCEURES O O O U 0 O O O O O O O O 0

APPENDIX C: TABLES . . . . . . . . . . . . .

LIST OF TABLES

Table Page

3 O l RELIABILITY MEASURES O O O O O O O O O C O O O O 34

4.1 SUMMARY OF MEAN SCORES FOR TASKS AND
POST TEST . O O O O C O O C O C O O O C 0 C O 44

4.2 THE TESTS OF HYPOTHESIS lA ON THE
COMPARISON OF LEARNING BETWEEN THE
SEXES FOR 17 LEARNING TASKS . . . . . . . . . 45

4.3 THE TESTS OF HYPOTHESIS 2Al ON THE

COMPARISON OF LEARNING BETWEEN
FIFTH.AND SIXTH GRADERS FOR 17
LMRNING TASKS C O C O O O C O O O O O O O O O 47

4.4 THE TESTS OF HYPOTHESIS 2A2 ON THE

COMPARISON OF LEARNING BETWEEN
FOURTH.AND FIFTH GRADERS FOR 17
LEWING TASKS O O O C O O C O O O O O O O O O 48

4.5 THE TESTS OF HYPOTHESIS 3A ON THE
COMPARISON OF LEARNING BETWEEN
HIGH AND LOW IoQ. GROUPS FOR 17
LEARNING TASKS . . . . . . . . . . . . . . . . 50

4.6 THE TESTS OF HYPOTHESIS 4A ON THE COMPARISON
OF LEARNING BETWEEN DISCOVERY-INSTRUCTION
GROUPS FOR 17 LEARNING TASKS . . . . . . . . . 51

4.7 THE TESTS OF HYPOTHESIS 1B ON THE COMPARISON
OF RETENTION BETWEEN THE SEXES IN GRADES
FOUR THROUGH S IX 0 O O O I O O O C O C O O O O 5 3

4.8 THE TESTS OF HYPOTHESIS BB ON THE COMPARISON

OF RETENTION BETWEEN HIGH AND LOW I.Q. GROUPS
IN GRADES FOUR THROUGH SIX . . . . . . . . . . 55

vi

Table

THE TESTS OF HYPOTHESIS 48 ON THE COMPARISON
OF RETENTION BETWEEN DISCOVERY-INSTRUCTION
GROUPS IN GRADES FOUR THROUGH SIX . . . .

NUMBERS OF TESTS THAT SUPPORT THE HYPOTHESES
COMPARING MEANS (H) WITHIN THE CLASSIFICA-

TIONS OF SEX, GRADE,
TECHING C O O O O O C O O

I.Q., AND METHOD OF

SUMMARY OF TESTS OF HYPOTHESES COMPARING
POST TEST SCORE MEANS (H) WITHIN THE

CLASSIFICATIONS OF SEX,
METHODS, AND GRADE . . . .

PLANNED COMPARISONS ANALYSIS
FOR TASK l O O O O O O O O

PLANNED COMPARISONS ANALYSIS
FOR TASK 2 . . . . . . . .

PLANNED COMPARISONS ANALYSIS
FOR TASK 3 O O O O O O C O

PLANNED COMPARISONS ANALYSIS
FOR TASK 4 . . . . . . . .

PLANNED COMPARISONS ANALYSIS
FOR TASK 5 O O Q C O O O O

PLANNED COMPARISONS ANALYSIS
FOR TASK 6 . . . . . . . .

PLANNED COMPARISONS ANALYSIS
FOR TASK 7 O O O O O O O O

PLANNED COMPARISONS ANALYSIS
FOR TASK 8 . . . . . . . .

PLANNED COMPARISONS ANALYSIS
FOR TASK 9 O O O O O O O O

PLANNED COMPARISONS ANALYSIS
FOR TASK lO . . . . a . .

PLANNED COMPARISONS ANALYSIS
FOR TASK 11 . . . . . . .

vii

I.Q., TEACHING

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

OF VARIANCE

Page

56

58

59

107

107

108

108

109

109

110

110

111

111

112

PLANNED COMPARISONS ANALYSIS OF VARIANCE
FOR TASK l3 0 O C O O O O O O O O O O 0

PLANNED COMPARISONS ANALYSIS OF VARIANCE
FOR TASK 14 C C O O O . O O O O O O O .

PLANNED COMPARISONS ANALYSIS OF VARIANCE
FOR TASK 15 O O O O O O C O O C C O O O

PLANNED COMPARISONS ANALYSIS OF VARIANCE
FOR TASK l6 O O O O O O O O O O C C C .

PLANNED COMPARISONS ANALYSIS OF VARIANCE

FOR TASK l7 0 O O O 0 0 O O O O O O O 0

ANALYSIS
ON THE

ANALYSIS
ON THE

ANALYSIS
ON THE

ANALYSIS
ON THE

ANALYSIS
ON THE

ANALYSIS
ON THE

ANALYSIS
ON THE
FOURTH

ANALYSIS
ON THE

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF SEX IN THE FOURTH GRADE

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF SEX IN THE FIFTH GRADE

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF SEX IN THE SIXTH GRADE

OF VARIANCE FOR REPEATED MEASURES

FACTOR OF I.Q. IN THE FOURTH GRADE.

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF 1.0. IN THE FIFTH GRADE

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF I.Q. IN THE SIXTH GRADE

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF TEACHING METHOD IN THE
GRADE 0 o o o o o o o o o o o o 6

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF TEACHING METHOD IN THE

FIFTH GRADE 0 O '0 O O O O I I O C O O 0

ANALYSIS
ON THE

OF VARIANCE FOR REPEATED MEASURES
FACTOR OF TEACHING METHOD IN THE

SIXTH GMDE O O O O C O 0 O O O O O C 0

viii

page

113

113

114

114

115

115

116

116

117

117

118

118

119

119

Table Page

C.27 ANALYSIS OF VARIANCE FOR REPEATED MEASURES
ON THE FACTOR OF GRADE . . . . . . . . . . . 120

C.28 CUMULATIVE PERCENTAGE OF STUDENTS WITHIN
FOURTH GRADE CLASSES TO REACH ACHIEVEMENT
LEVELS O C O O C C C O C 0 O O O O O . O . O 121

C.29 CUMULATIVE PERCENTAGE OF STUDENTS WITHIN
FIFTH GRADE CLASSES TO REACH ACHIEVEMENT
LEVELS O 9 O O O O O O O C O C O C C C C O O 1 2 2

C.30 CUMULATIVE PERCENTAGE OF STUDENTS WITHIN

SIXTH GRADE CLASSES TO REACH ACHIEVEMENT
LEVELS I O C O O O O O O O O O O O C C O O . 12 3

ix

CI-HXPTER 1: THE PROBLEM

In 1963, a group of 25 mathematicians and scientists
were brought together by Professor A.M. Gleason of Harvard
and Professor W.T. Martin of the Massachusetts Institute
of Technology for the purpose of conjecturing the content
of the mathematics curriculum in the year 1990. The conclus-
ions of the conference were published in a report1 generally
known as the Cambridge Conference report. Essentially the
conference foresaw the mathematics content of the first
sixteen years telescoped into a period of thirteen years.

A number of topics normally introduced in the secondary
school will necessarily be introduced in the elementary
school. Operations on signed (positive and negative) numbers,
a topic considered appropriate for seventh or eight grade

by some present-day writers as Kingston2 and Butler,3 is
proposed to be introduced at the third grade level. This

curricular innovation is the motivation of this study to

 

lCambridge Conference on School Mathematics, Goals
for School Mathematics (Boston: Published for Education-
al Services, Inc. by Houghton Mifflin, 1963).

2Kingston, J. Maurice, Mathematics for Teachers of
the Middle Grades (New York: John Wiley & Sons, Inc.,
19665, p. 59.

3Butler, Charles H. and Wren, Lynwood F., The Teach-
ing of Secondary Mathematics (New York: McGraw-Hill Book
Company, 1965), pp. 340-349.

 

1

2
compare the achievements of elementary school children in

learning tasks involving signed numbers.

The Need

 

Since the ideas presented in the Cambridge Conference
report reflect the thoughts of respected men active in the
development of mathematics pedagogy, they need and deserve to
be tested. Mathematics educators, like Irving Adler,l have
strongly urged experimentation with topics found in the report.
The reasons that justify this study then parallel those that
motivate efforts as the Cambridge report.

A listing of writers that have outlined the causes and
rationalizations of curricular changes in elementary school
mathematics would be extensive. However, they all have com-
mon themes as cited in the following samples. Willoughby2
states that the changes have been affected by acceleration
in mathematics research, the reorganization and restructur—
3

ing of mathematics, and new pedagogical methods. Folsom

and Butler4 attribute the changes to the rapid development

 

IAdler, Irving, "The Cambridge Conference Report:

Blueprint or Fantasy?," The Arithmetic Teacher, Vol. 13
(March, 1966), pp. 179-187.

2Willoughby, Stephen 8., Contemporary Teaching of Sec-
ondary School Mathematics (New York: John Wiley and Sons,
Inc., 1967), pp. 29-35.

3Folsom, Mary, "Why the New Mathematics?." The Instruc-
tor, Vol. 73 (December, 1963), pp. 6-7.

4Butler,_gp_. cit., pp. 4, 56-57.

3
of new mathematics and the changing needs of society for
mathematics. The Cambridge Conference reportl cites changing
social needs, new developments in mathematics, and new teach—
ing methods as reasons for change. In summary, four reasons
are given as justification for mathematics curricular innova-
tion, namely, (1) the increasing rate of the discovery of new
mathematics, (2) the reorganization of mathematical struc-
tures, (3) the development of new educational methods, and
(4) the changing need of society.

Consider the argument that the increasing volume of new-
ly discovered mathematics justifies changes in the mathematics
curriculum. Evenson2 argues that since more mathematics is
being created and used, there is a need for more mathematics
to be learned. Frequently, the number of pages in the.fl§£22f
matical Review3 is cited as evidence of the expanding world
of mathematical knowledge. However, a drive to learn more
mathematics because there is more mathematics to learn, in
some remote hope to close the gap, is indeed futile. Rather,
the mathematics student must develop the skills of how to
learn on his own the mathematics he will need in his life-

time.

 

lCambridge Conference on School Mathematics, op. cit.,
pp. 7-120

2Evenson, A.B., Modern Mathematics (Chicago: Scott,
Foresman and Company, 1962), p. 8.

 

3Volume 35, 1968 contains 1,437 pages.

4

The reorganization of mathematical structures may be
a more powerful force in the revamping of the mathematics
curriculum. It is this reorganization that delineates "Mod—
ern" mathematics from "old” mathematics. Butlerl describes
the difference in this way:

The origin of what might be called the modern point

of view in mathematics can be traced to the pioneer-

ing efforts of Gauss, Bolyai, Lobachevski, and

Riemann in the creation of non-Euclidean geometries.

By daring to challenge that which for two millenniums

had been accepted as absolute, they freed the intellect

to reject the evidence of the senses for the sake

of what the mind might produce . . . . This new meth-

od no longer recognizes postulates (axioms) as 'self

evident truths,‘ but merely as 'acceptable assumptions.‘
The "modern" mathematics growing out of this realization has
resulted, according to Allendorfer,2 in two trends. First,
mathematical systems have been developed which exist only of
and for themselves with no obligation to relate to the real
world and, secondly, theories that may have grown fldm differ-
ent models in nature are combined into a single abstract
system that gives greater insight into the original systems
as well as producing greater economy of thought. This struc-
turing is in a sense the essence of mathematics, and, since
an aim of mathematics education is to convey the nature of

mathematics, it follows that this structuring should be a

U I I 1 O 2
factor in determining the mathematics curriculum. Bruner

 

1Butler, 92, cit., pp. 55m56.

2Allendorfer, Carl B., Naihena+1rs fox Parents (New

York: The Macmillan Company" 196)) pp. 8—9.

 

3Bruner, J. 8., The Proce 0:5 of Eiuc cation (Cambridge,
Massachusetts: Har*vard Unixersity :i: 53, 1962), p. 31.

states the case this way:
. . . the curriculum of a subject should be deter-
mined by the most fundamental understanding that can
be achieved of the underlying principles that give
structure to the subject. Teaching specific topics
or skills without making clear their context in the
broader fundamental structure of a field of know-
ledge is uneconomical . . . .
Next, contributions in educational psychology by men
like Skinner,1 Bruner,2 Piaget,3 and Gagné,4 have given
rise to new theories of instruction. The new theories, while
they do not suggest that revolutionary curricular changes as
advocated in the Cambridge Conference report need to be under-
taken, do indicate methods by which changes may be made.
They give the curricular innovator a hope to succeed.
Finally, the changing ways in which we live have strong
effects upon changes in the mathematics curriculum. People
over thirty, remembering the neighborhood store, can probab-
ly recall a store clerk totaling the costs of groceries on

a grocery list. Today, the supermarket check-out girl uses

a very efficient machine, that not only totals the costs, but

 

1Skinner, B.F., "Teaching Machines," Science, Vol. 128
(October, 1958), pp. 969u977.

2Bruner, J.S. Toward a Theory of Instruction (Cambridge,
Massachusetts: The Belknap Press of Harvard University Press,
1966).

 

3Piaget, Jean, "How Children Form Mathematical Concepts,"
Scientific American, Vol. 189 (November, 1953), pp. 74-79.

4Gagné, Robert M., The Conditions of Learning (New York:
Holt, Rinehart and Winston, 1965).

 

6
also calculates the change the customer is to receive. The
world is in a computer revolution. Kemenyl has stated that
one cent will buy about 2,000 arithmetical computations to-
day and, therefore, no man can earn a living doing arithme-
tic. This is not to say that there is no longer a need for
one to learn to compute. There is general need for numerous
skills associated with the study of arithmetic, ranging from
telling time to balancing a checkbook. However, the society
no longer needs a large number of peOple, highly competent
in arithmetic, to serve as accountants, bookkeepers, time-
keepers, and stockmen. The computational aspects of their
work is being increasingly handled by machines. Further,
the growth of the use of computers is placing on our age a
need for a new set of skills requiring more, not less, mathe-
matics.

The introduction of signed numbers into the elementary
school curriculum is justified on at least three of the four
stated reasons. The rules of operations (addition and multi-
plication) on signed numbers provide an excellent illustration
of the consequence of a mathematical structure. Also, an
understanding of the operations of signed numbers is prerequis-
ite to an understanding of the real number system. Knowledge
of the real number system provides a foundation for a great

deal of new mathematics. Finally, the real number system is

 

lKemeny, John, "The Impact of the Computer on Teach-
ing," an address given at the Cleveland Meeting of the Nation—
al Council of Teachers of Mathematics, Cleveland, Ohio,
November 13, 1969.

7
probably the best model for application in the real world
through disciplines as calculus and statistics. For these

reasons the study is justified.

The Purpose

The purpose of this study is to investigate the abili-
ties of elementary school children in learning tasks involv-
ing the operations of addition, subtraction, and multiplica-
tion of signed numbers. The effects of grade level, 1.0.,
sex, and different teaching methods upon the learning of
tasks as measured by test scores are analyzed. An objective
test for satisfactory achievement is defined and applied to

the tasks.

The Assumptions

The crucial issue of the study is to consider the feasi-
bility of introducing Operations on signed numbers in the
elementary school course of study. The assumptions used as
the basis for the hypotheses are conservative.

(1) It is assumed that the general mathematical ability
of boys and girls is the same. The results of research test-
ing the mathematical abilities of boys and girls are mixed.
Studies indicating that boys achieve better than girls in
tasks dealing with mathematical concepts, where as girls

achieve better on tasks involving computation are reported by

Jarvis1 and Parsleyz.

(2) It is assumed that children, as they grow older
and gain learning experience, can learn new tasks more readi-
ly and remember them longer.

(3) It is assumed that children with greater intellec-
tual ability can learn new tasks more easily and remember
them longer than children with less intellectual ability.

(4) It is assumed that teaching is an art. Theories
of instruction may be constructed compatible with various
theories of learning, but the success of the "average" teach-
er in the "average" classroom is due more to the personality
of the teacher and her ability to adopt a teaching style that
works for her.

(5) Bruner's3 famous axiom that "any subject can be
taught effectively in some intellectually honest form to any

child at any stage of development" is accepted.

The Hypotheses

The following hypotheses, based upon the assumptions,

 

1Jarvis, 0.T., "Boy-Girl Ability Differences in Elemen-
tary School Arithmetic," School Science and Mathematics,
Vol. 64 (November, 1964), pp. 657-659.

2Parsley, Kenneth M., "Further Investigation of Sex
Differences in Achievement of Under-Average and OvereAverage
Achieving Students Within Five 1.0. Groups in Grades Four
through Eight," Journal of Educational Research, Vol. 57
(January, 1964), pp. §6§-§70.

3Bruner, J,S., The Process of Education, p. 33.

are tested in the study:

(1) Hypothesis 1A: There will be no difference in the
scores on the tasks between boys and girls.

Hypothesis 18: There will be no difference in the
retention of task skills between boys and girls

(2) Hypothesis 2A: The mean score of children on the
tasks at any grade level will be higher than the mean score
of children at a lower grade level on the same tasks.

Hypothesis 2B: The retention of task skills by
children at any grade level will be greater than the reten-
tion of task skills by children at a lower grade level on the
same tasks.

(3) Hypothesis 3A: The mean score on tasks of children
with higher intellectual ability will be higher than the mean
score on tasks of children with lower intellectual ability.

Hypothesis 3B: The retention of task skills by
children with higher intellectual ability will be greater
than the retention of task skills by children with lower in-
tellectual ability.

(4) Hypothesis 4A: There will be no difference in the
mean scores on the tasks between groups receiving different
instructional methods.

Hypothesis 4B: There will be no difference in the
retention of skills between groups receiving different instruc—
tional methods.

(5) Hypothesis 5: Fourth, fifth, and sixth grade classes
will attain satisfactory achievement in learning tasks involv-

ing signed numbers.

lO

Qefinitions

The following terms, unique in this study, are defined.

(1) Direction number: An integer represented on a numb—

 

er line as a vector.

(2) Discovery learning: A learning experience, as
described on pages 35-36, where the responsibility for learning
remains with the student.

(3) Instructional (Didactic) learning: A learning ex-
perience, as described on pages 35-36, where the responsibility

for learning remains with the teacher.

The Overview

 

This chapter, the first, contains the statement of the prob-
lem and a justification for the study. In Chapter 2, the rele-
vant literature is reviewed. The emphasis is placed on three
areas; namely, the development of signed numbers in mathematics
education, the use of mathematical structure in the elementary
school, and the psychological foundations underlining the
teaching methods used in the study. Those aspects of the
study dealing with the design are found in Chapter 3. The
selection of subjects, measures, and experimental design are
reviewed, as well as the development of the curricular mater-
ial. In Chapter 3 the hypotheses are restated in testable
form and the statistical procedures for testing them are
listed. Chapter 4 contains an analysis of the data and Chapter

5 ends the report with some conclusions and a summary.

CHAPTER 2: REVIEW OF LITERATURE
The Integers: Their Development and Pedagogy

The slow acceptance of the concept of negative numbers
by mathematicians is remarkable. A survey of the development
of integers by Gorzal states that not until 1637 were signed
numbers firmly established as a number system through the work
of Descartes, who referred to positive and negative numbers
as true and false numbers. Prior to this, medieval mathema-
ticians thought expressions as 2-5 to be "meaningless" and,
even earlier, Diophantus (Ca. 275) called the equation
4x + 20 = 4 absurd. However, the survey continues, not all
mathematicians denied the existence of negative numbers. The
Arabian al-Khowarizmi (Ca. 825) is known to have stated the
rules of signed numbers, placing a "dot" over the numeral to
indicate a negative number. At about the same time the Hindus
denoted negative numbers by enclosing the numeral in a circle.
But, according to Miller,2 a refusal by some mathematicians
to accept negatives persisted until the 19th century.

After the acceptance of signed numbers into the domain

 

lGorza, Vivian S., A Surveyyof Mathematics: fiEarly
Concepts and their Historical Development (New York: Holt,
Rinehart and Winston, 1968), pp. 244-247.

2Miller, G.A., "Crusade Against the Use of Negative
Numbers," School Science and Mathematics, Vol. 33 (December,
1933), pp. 959-964. 7 ’

 

ll

12

of mathematics by mathematicians, the study of Operations
on signed numbers became an integral part of the study of
algebra. The teaching of signed numbers has evolved from a
period When the study was developed by seemingly arbitrary
rules of operation to the present attempt to show signed
numbers as rational, necessary, functionaries in the struc-
ture of a number system. A survey of Older algebra texts,
as those by Wentworth,l Beman,2 and Milne,3 show the rules
Of Operations on signed numbers to be based upon the "like-
ness" or "unlikeness" of the signs. In text books used to-
day, as those by Beberman4 and Price,5 the rules are pre-
sented as the consequence of the algebraic structures of
the number system.

The introduction Of advanced mathematics topics into

the elementary school curriculum brings with it problems

 

lWentworth, G.A., School Algebra (Boston: Ginn and

2Beman, Woster W., Elements of Algebra (Boston: Ginn
and Company, 1900), pp. 27-28.

3Milne, William J., High School Algebra (new York:
American BOOk Company, 1892), pp. 20, 29, 43.

4Beberman, Max, and Vaughn, Herbert B., High School
Mathematics (Boston: D.C. ieath and Company, 1966),
pp. 20'590

5Price, H.V., Peak, P., and Jones, P.S., Mathematics:
An Integrated Series, Book One (New York: Harcourt, Brace
and WOrld, Inc., 1965), pp. 135~154.

13
not found in the secondary school. Rombergl has described
this difficulty simply and adequately:
The means of embouying advanced concepts in simple
forms and the techniques of implementing such forms
in successful instructional sequences remain to be
found.
The literature provides some hints as to how this may be
done in the case Of signed numbers. Patterson2 suggests
using pictures on the number line to indicate positive and
negative direction at the first grade level. For the fourth
grade, Davis3 suggests motivating the concepts Of "plus"
and "minus" numbers by "real life" credit and debit situa-

tions. The students then continue on tO more abstract Prob-

lems involving frames as

+5 + '5] = 43 .

Havenhill4 proposes the use of arrows to indicate positive

 

lRomberg, T.A. and DeVault,.M.V., "Mathematics Curricu-
lum: Needed Research," Journal of Research and Development
in Education, Vol. 1 (Fall, 1967), pp. 95-110.

2Patterson, Katherine, "A picture line can be fun!,"
The Arithmetic Teacher, Vol. 16 (December, 1969), pp. 603-
605.

3Davis, Robert B., The Madison Projects Approach to
A Theory Of Instruction, a report of the Madison Project,
Webster College, St. Louis, Missouri, p. 12.

4Havenhill, Wallace P., "Though This Be Madness...,"
The Arithmetic Teacher, Vol. 16 (December, 1969) pp. 606-
608.

l4
and negative dilection as well as magnitude. D'Z3iucjustine“i
recommends that the pOlnts on the number line be identified
by numerals with arrows over them (3). The arrows indicate
the direction of the point from zero and the numeral indica-
tes the distance of the point from zero. After a few exer-
cises in addition using this representation, the arrows would
be replaced by the traditional + and — signs. Further work
would involve problems using frames similar to those previous-
ly attributed to Davis. Riedesel2 and the School Mathematics
Study Group3 advocate using the thermometer for introducing
signed numbers and then proceeding tO addition by using arrows
to find vector sums on the number line.

The suggestions Offered thus far deal only with the
representation, addition, and subtraction of signed numbers
while ignoring the problems of multiplication. There is good
reason for this. The teaching of the multiplication of signed

numbers presents some imposing problems. The Cambridge report4

 

lD'Augustine, C.H., Multiple Methods Of Teaching Mathe-
matics in the Elementary School (New York: Harper and Row,
196 , PP. 260-270.

2Riedesel, C. Alan, Guiding Discovery in Elementary
School Mathematics (New York: Appleton-Century-Crofts, 1967),
pp. 00-1010

3School Mathematics Study Group, Mathematics for the
Elementary School, Teacher CommentarYJ Part 14(New riven,
Yale University Press, 1963), pp. 349-376.

4The Cambridge Conference on School Mathematics,
op, cit., p. 37.

15
states:

Perhaps no area of discussion brought more view-
points than the question of how the multiplication
of signed numbers should be introduced. The simple
route via the distributive law was considered, but
a closely related approach was more popular. One
Observes that the definition of multiplication is
ours to make but only one definition will have the
desireable properties. Others favored an experi-
mental approach involving negative weights on
balance boards, etc. Still others favored the
"negative" debt approach. Even the immediate in-
troduction Of signed area was proposed . . . . The
question is evidently not mathematical, it is purely
pedagogic. The problem is tO convey the "inner

reasonableness Of (-1) x (-l) = +1."
Havenhill1 suggests that the rules for multiplication be
developed in the following way:
By utilizing the two interpretations of the + and -
signs, the multiplication sentence, a x b = c, may
be interpreted as follows. The magnitude Of the
multiplicand (b) is the length of each arrow. Its
sign points the arrows to the right (+) or left
(-). The magnitude of the multiplier (a) tells
how many arrows tO lay end to end beginning at the
origin. Its sign tells whether to reverse their
direction (-) or not (+).
This may seem tO be confusing. The fault is not Havenhill's.
The pedagogical problem is real. Havenhill's procedure
underlines the difficulty. The rules for multiplication
can be justified many ways. But most, like the use of equiva-
lence classes Of ordered pairs Of natural numbers as described

by Banks,2 the geometrical approach Of using projections on

the real line with the ratios Of similar triangles suggested

 

lHavenhill, loc. cit.

 

2Banks, J. Houston, Elements Of MathematicsL_Second
Edition (Boston: Allyn and Bacon, Inc., 1960), pp. 136-148.

16
by Petro,l and the product line method, can be rejected,
‘a priori, as unsuitable for the elementary school. The
search for an adequate way to teach the multiplication Of
signed numbers continues.

Research specifically attending to the problems of
developing the concepts of signed numbers is exceedingly
rare. Parsons,2 working in the Madison Project, reports
trials with fourth grade children have been determined a
”success" though a criteria for "success" is not reported.
Carlton3 reports that instruction in the elementary school
on Operations of positive and negative integers is under
evaluation in the Soviet Union. NO results are available
at the present time.

A review of the current elementary texts used in the
United States reveals that the study of signed numbers is
being slowly introduced to sixth grade children. Most pro-
grams on this topic deal only with addition and, in some

cases, with subtraction as exemplified in texts by Duncan,4

 

1Interview with John Petro, Associate Professor of
Mathematics, Western Michigan University, March 16, 1970.

2Parsons, Cynthia, "Algebra as Presented to Fourth
Graders is Grasped with Enthusiasm," Christian Science
Monitor, January 9, 1960, p. 11.

 

3Carlton, Virginia, "Mathematics Education in the
Elementary Schools of the Soviet Union," The Arithmetic
Teacher, VOl. 15 (February, 1968), pp. 108—114.

4Duncan, Ernest R., Modern School Mathematics: Struc-
ture and Use (Boston: Houghton Mifflin Company, 1970),
pp. 332-339.

 

l7
1 2 3 4 . 5
Fouch, Hartung, Keedy, Glennon, and Spitzer. In every
case the subtraction is considered, it is motivated as the
inverse of addition. Only one author, Eicholz,6 also in—
cludes the Operation of multiplication. The justification
of the rules of signed numbers is handled by the distribu-

tive law and the additive inverse prOperty.

Teaching Mathematical Structure in the Elementary School

As previously stated, the teaching of signed numbers
has evolved from a time when the study was developed from
apparently arbitrary rules to the present procedure of

developing the operations on the numbers as consequences

 

lFouch, Robert S., and Haas, Raymond, SRA Elementary
Mathematics ProgramL_BOOk 6 (Chicago: Science Research
Associates, 1968), pp. 143-150.

2Hartung, Maurice L., et al., Seeing Through Arith-
meticz 6 (Glenview, Illinois: Scott Foresman and Company,

_ploring Elementary Mathe-

3Keedy, Mervin J., et
nehart, and Winston, I970),

maticsLié (New York: HdIE,
pp. 224-231. 234-235.

 

.1-
Ri

 

4Glennon, Vincent J., Short, Roy P., and Brownell,
M.A., Mathematics We Need (Boston: Ginn and Company, 1966),
pp. 312-313.

5Spitzer, Herbert F., et al., Elementary_Mathematics
(St. Louis, Missouri: McGraw-Hill Book Company, 1967),
pp. 23-24, 300

6Eicholz, Robert E. and O'Daffer, Phares C., Elemen-
tary_SchOOl Mathematics, second editiohi_BOOk 6 (Menlo
Park, California: Addison Wesley Publishing Company, 1968),
pp. 291-293.

18
of the structure of the number system. This follows the
generally accepted belief that mathematics that is learned
through understanding is learned with greater retention
and greater facility for transfer than mathematics learned

by rote. Studies by Brownell,l Dawson,2 Greathouse,3 Krich,4

Miller,5 and Rappaport6'7

confirm this belief. It is argued
that meaning in arithmetic is attained through the laws

that give the subject structure by mathematics educators as

 

1Brownell, William A. and Moser, Harold B., "Mean-
ingful vs. Mechanical Learning: A Study in Grade 3
Subtraction," Duke University §tudies in Education, VOl.
8 (1949): PP. 1-207.

2Dawson, Dan T., "The Case for the Meaning Theory in
Teaching Arithmetic," Elementary School Journal, Vol. 55
(March, 1955), pp. 393-399.

3Greathouse, Jimmie Joe, "An Experimental Investi-
gation Of Relative Effectiveness Among Three Different
Arithmetic Teaching Methods," unpublished Ph.D. Thesis,
The University Of New Mexico, 1965.

4Krich, Percy, "Grade Placement and Meaningful Learn-
ing," School Science andflMathematics, Vol. 64 (February,
1964), Pp. Iil-I57e

5Miller, G.H., "How Effective is the Meaning Method7,"
The Arithmetic Teacher, Vol. 4 (March, 1957), pp. 45-49.

6Rappaport, David, "Understanding Meanings in Arith-
metic," The Arithmetic Teacher, Vol. 5 (March, 1958),
pp. 96-99.

7 "The Meaning Approach in Teaching Arith-
metic," ChicagoSchool Journal, Vol. 44 (January, 1963),
pp. 172-174.

 

l9
Flournoy,l Gordon,2 and Schraf.3 They reason that, since
computational algorithms are governed by algebraic struc-
tural laws, an understanding of these laws by students and
the use of these laws by teachers in justifying the algor-
ithms will result in more meaningful learning.

The research investigating the ability Of elementary
school children to learn and apply structural laws is fairly
extensive. Studies by Schmidt4 and Hall5 indicate that child-
ren who have developed an understanding of the commutative
and associative laws show an improvement in fundamental ad-

dition and multiplication skills. Research reports by Gray6

 

lFlournoy, Frances, "Understanding Relationships:
An Essential for Solving Equations," The Elementary School
Journal, Vol. 64 (January, 1964), pp. 214-217.

2Gordon, David X., "Clarifying Arithmetic Through
Algebra," School Science and Mathematics, Vol. 42 (March,
1942), pp. 585- 289.

BSchraf, William L., "Arithmetic Taught as a Basis
for Later Mathematics," school Science and Mathematics,
Vol. 46 (May, 1946), pp. 413-423.

4Schmidt,.Mary M., "Effects Of Teaching the Commuta-
tive Laws, Associative Laws on Fundamental Skills Of Fourth
Grade Pupils," Dissertation Abstracts, VOl. 26 (February,
1966)! P0 4510.

5Hall, Kenneth Dwight, "An Experimental Study Of Two
Methods Of Instruction for Mastering Multiplication Facts
at the Third Grade Level," unpublished Ph.D. Thesis, Duke
University, 1967.

6Gray, Roland F., "An Experiment in the Teaching Of
Introductory.Multiplication," The Arithmetic Teacher, Vol.7
(March, 1965), pp. 199-203.

20
and Schelll'2 show that children with an understanding of
the distributive law develop a better understanding of multi-
plication than children motivated by "repeated addition" or
"rectangular array" methods.

While knowledge Of mathematical structure may help
children learn arithmetical Operations, the teaching of
mathematical structure, itself, presents some problems.
Baumann3 found that the attainment Of the concepts Of com-
mutativity, closure, and identity were quite difficult for
second and fourth grade children. Flournoy4 and Gray5 have
demonstrated that elementary school children could not
apply the structural laws without specific instruction
into the nature of the laws. The order of difficulty in

learning the structural laws is reported by Crawford6 tO

 

lSchell, Leo M., "Two Aspects of Introductory Multi-
plication: The Array and the Distributive PrOperty,"
Dissertation Abstracts, Vol. 25 (April, 1965), p. 5161.

2Schell, Leo M., "Learning the Distributive Property
by Third Graders," School Science and Mathematics, VOl.
68 (January, 1968), pp. 28-32.

 

3Baumann, Raemt R., "Childrens Understanding of Select-
ed Mathematical Concepts in Grades Two and Four," Disserta-
tion Abstracts, Vol. 26 (March, 1966), p. 5219.

 

4Flournoy, Frances, "Applying Basic Mathematical
Ideas in Arithmetic," The Arithmetic Teacher, Vol. 11
(February, 1964), pp. 104-108.

 

5Gray, pp, cit.

6Crawford, Douglas H., "An Inventory of Age-Grade
Trends in Understanding the Field Axioms,“ Dissertation
Abstracts, Vol. 25 (April, 1965), pp. 5728-5729.

 

 

21

be commutativity (easiest), inverse, closure, identity,
associativity, and distributivity (most difficult). In at
least one case, the structural development has proved less
reliable than the traditional approach. Hervy,l comparing
the equal additions approach with the use of Cartesian pro-
ducts, reported that equal-additions multiplications prob-
lems were less difficult to solve and conceptualize, and
that cartesian-product problems were more readily solved

by high achievers than by low achievers.

Theories of Instruction in Mathematics

Developments in learning theory have lead to the estab-
lishment of theories Of instruction in mathematics. A
spectrum Of ideas on teaching procedures range from rigid-
ly guided learning experiences to those encouraging student
experimentation and discovery. The two essential views
that are being proposed have been summarized by Shulman2
as follows:

The controversy seems to center essentially about
the question of how much and what kind of guidance
ought to be provided to the students in the learn-
ing situation. Those favoring learning by discovery
advocate the teaching of broad principles and prob-
lem-solving through minimal teacher guidance and

 

1Hervey, Margaret A., "Childrens Responses to Two
Types Of Multiplication Problems," The Arithmetic Teacher,
VOl. 13 (April, 1960): PP. 288-292.

2Shulman, Lee S., "Psychological Controversies in the

Teaching Of Science and Mathematics," The Science Teacher,
‘VOl.35 (September, 1968), pp. 34-37, 89-90.

 

22

maximal Opportunity for exploration and trial-and-
error on the part of the student. Those preferring
guided learning emphasize the importance of care-
fully sequencing instructional experiences through
maximum guidance and stress the importance of basic
associations of facts in the service of the eventual
mastering Of principles and problem solving.

The learning objectives of the theories differ, and as such
defy comparison. Bruner,l a strong proponent of discovery,
describes the objectives of discovery as follows:
. . . a theory of instruction seeks to take account
of the fact that a curriculum reflects not only the
nature of knowledge itself--the specific capabili-
ties--but also the nature of the knower and Of the
knowledge getting process . . . To instruct some-
one in these disciplines is not a matter of getting
him to commit the results to mind, rather it is to
teach him to participate in the process that makes
possible the establishment Of knowledge.
Gagné,2 who adamantly favors the guided learning approach,
argues that to effectively solve problems the learner must
have accumulated knowledge and that this is done best by
leading students through guided learning experiences.
Gaining knowledge is one objective of guided learning.
The reasons for choosing one set Of Objectives over
another are epistemological. Bruner3 declares:
But I think we would all agree that, at the very
least, an educated man should have a sense of what
knowledge is like in some field of inquiry, to know

it in its connectedness and with a feeling for how
the knowledge is gained.

 

lBruner, Jerome S., Toward A Theory . . ., p. 72.
2 .
Gagné, op. Cit., p. 170.

3Bruner, Jerome S., "On Learning Mathematics," The
Mathematics Teacher, Vol. 53 (December, 1960), pp. 610-619.

23
Ausubel1 replies:

This miracle of culture is made possible only be-

cause it is so much less time-consuming to communi-

cate and explain an idea meaningfully to others

than to require them to re-discover it by them-

selves.

In general, research studies in curricular develop.
ment use diadactic teaching methods. Studies involving
guided instruction, by the nature of the instruction,
are easier to design, control, and the objectives can be
described in terms of observable behavior. The researcher
working with discovery methods is faced with some impos-
ing problems. Wittock2 characterizes these problems as:

(1) Conceptual Problems. Is discovery a way to learn
subject matter or is it an end in its own right? Is it
learning by discovery or learning to discover?

(2) Methological Problems. How does one control the
rate and sequencing of stimuli in treatments? What are
the dependent variables?

(3) Semangic(Inconsistencies. How can operational
definitions be developed? How can one avoid the naming

of treatments in terms of responses, i.e., rote learning

and discovery are responses, not stimuli.

 

;Ausubel, David P., "Some Psychological and Educa-

tional Limitations of Learning by Discovery," The Mathee
matics Teacher, Vol. 57 (May, 1964), pp. 290-302.

2Wittock, M.C., "The Learning by Discovery Hypo-
thesis," in Shulman, Lee (Editor), Learning py Discovery:
A Critical Appraisal (Chicago: Rand, McNally and Company.
1966), pp. 42-48. '

24

A closer look at one of these problems may bring the
difficulty into sharper focus. Consider the conceptual
problem of what does one mean by discovery teaching? For
some, it means literally placing the child in a sea of
stimuli and letting him sink or swim. For others, dis-
covery teaching implies a highly structural system of dis—
pensing stimuli leading the child in discoveries. Glaserl
takes the first approach when he writes:

. . . a learning by discovery sequence involves

induction. This is the procedure of giving exam-

ples of a more general case which permits the

student to induce the general propositions in-

volved.
Johnson2 takes the second point of view. He writes:

What we really do is provide a setting where educa-

tional experiences are intelligible and understand-

able and we guide the mind of the child, as it were,

along paths which cause him to see, not only the

correctness of the manipulation, but also the

rationale of the process.
Clearly, it is wise to heed Shulman's warning3 that one
man's discovery can easily be confused with another's
guided learning.

The research dealing with discovery teaching centers

largely around the relative effectiveness of discovery and

 

lGlaser, Robert, "Variables in Discovery Learning."
in Shulman, Lee (Editor), LearningAby Discovery: A
Critical Appraisal (Chicago: Rand, McNally and Company,
1966?: p. 15

zJohnson, Harry C., "What Do We Mean by Discovery?,"
The Arithmetic Teacher, Vol. 11 (December, 1964): Pp. 538-
539.

3Shulman, pp. git., p. 34.

25
non-discovery teaching on the accumulation of knowledge,
retention, and transfer as dependent variables. Studies
by Bassler,1 Fleckman,2 Scandura,3 Ter Keurst,4 and WOrthen5
support the claims of the advocate of discovery in that div
dactic methods lead to better results in initial testing
but that discovery methods result in better performance
on retention tests. The results further indicate that the
discovery groups transfer concepts more easily. A study by

Wilson6 shows that groups taught by discovery methods trans-

fer discovery problem solving approaches to new situations.

 

lBassler, Otto C., "Intermediate Versus Maximal Guid-
ance--A Pilot Study," The Arithmetic Teacher, Vol. 15

2Fleckman, Bessie, "Improvement of Learning Division
Through Use of the Discovery Method," Dissertation Abstracts,
Vol. 27A (April. 1967). pp. 3366-3367.

3Scandura, Joseph J., "An Analysis of Exposition and
Discovery Modes of Problem Solving Instruction," gournal
of E§perimenta1 Education, Vol. 33 (December, 196 a PP.

-5.

4Ter Keurst, Arthur J., "Rote Versus Discovery Learn—
ing," School and Communit , Vol. 55 (November, 1968), pp.
42-44.

5W'orthen, Blaine R., "A Study of Discovery and Ex-
pository Presentation: Implications for Teaching,"
Journal of Teacher Education, Vol. 19 (Summer, 1968),
pp. 223-242. 5

6Wilson, John H., "Differences Between the Inquiry
Discovery and Traditional Approaches to Teaching Science
in Elementary School," Research In Education, Vol. 4
(1969), p. 752. fi

26
Armstrong1 reports that the inductive (discovery) approach
fosters the learning of Operations, while deductive (direct-
ed) methods result in greater learning of mathematical
properties.

Kersh,2'3 a critic of the discovery method, argues
that research supports the claim that through discovery
students (a) develop an interest in the task, and (b) under-
stand what they learn and are better able to remember and
to transfer what is learned. He denies that there is any
evidence to support the conjecture that students learn
strategies for discovering new generalizations. At this
later date the criticism, in view of the studies cited,
still has some validity.

Regardless which instructional strategy one may favor
or what teaching procedures research may support, the prob-
lem of considering the effects of teaching procedures on
curriculum development is with us. Any study that investi-
gates the introduction of new curricular material should
include the results obtained by differing modes of instruc-

tion.

 

lArmstrong, Jenny Rose, "The Relative Effects of Two

Forms of Spiral Curriculum Organization and Two Modes of
presentation on Mathematical Learning," Dissertation Ab-
stracts, Vol. 29 (July, 1968), p. 141.

2Kersh, Bert Y., "Learning by Discovery: What is
Learned?," The Arithmetic Teacher, Vol. 11 (April, 1964),
p. 226.

3 "Learning by Discovery: Instruction-
al Strategies," The Arithmetic Teacher, Vol. 12 (October,
1965). pp. 414-417.

 

 

27

Summagy

A survey of the literature indicates that mathematics
educators recognize a need for introducing the algebra of
signed numbers at the elementary school level. To some
extent, this is being done at the sixth grade level in some
programs. In these cases, the crucial problem of the multi-
plication of integers is ignored.

If the algebra of signed numbers is to be a part of
the elementary school curriculum, the topic should be devel-
oped through an understanding of the structure of the mathe-
matical system rather than through the assumption of seem-
ingly arbitrary rules of operation. Research indicates
that children who learn the "reasoning" behind mathematical
concepts learn those concepts faster and retain them longer.
Further, the "reasoning" is best learned through an under-
standing of the laws which give structure to the mathemat-
ics system. Studies show that the structural laws must be
taught and that some of them, as the distributive law, are
difficult.for children to learn.

Finally, studies in learning theory have lead to the
formation of theories of instruction in mathematics.
Essentially, these theories follow one of two tracks:
guided learning or discovery learning. The proponents of
guided learning argue that their procedures provide for
more efficient learning. Those who favor discovery learn-
ing maintain that one who learns through discovery will

retain what he has learned for a longer period and will

28
more easily transfer this knowledge. Research supports
the claims of both groups. The time of investigating dif-
fering teaching strategies is here, and a study investigat-
ing the introduction of signed numbers in the elementary
school should consider the effects of different instruc-

tional procedures.

CHAPTER 3: DESIGN OF THE STUDY

The Curriculum And Its Presentation

 

The purpose of the study is to investigate the abil-
ity of elementary school children to accomplish tasks re-
lated to operations on signed numbers. In the curricular
material developed for the study signed numbers were repre-
sented on a number line as bi-directional vectors in the

following way.

 

 

k 3 I l 5 I
' v ' I 1' J 114 1):
Signed numbers were called direction numbers in their pre-
sentation to the subjects. The number line was coordinat-

ed by indicating the direction and distance a point was

located from zero.

 

6- fl 6' 6- f- e 4 4 a 4
5 4 3 2 l O l 2 3 4 5
- L L L y l l l L, 1 - Li pl 1

The operation of addition was developed by placing the tail
of the first addend vector at zero, placing the tail of the
second addend at the head of the first addend, and naming
the sum to be the vector extending from zero to the head

of the second addend vector. The following example illus-
trates this Operation.

29

3O

 

 

 

4 e e
Examp 1e.- 2 -+ 5 = 3
(u
5 _- .. second addend
I g 1 1%.. I: .
sum :_ 3 L 2 Q first addend
f' I d
E p i J . i i 5 '? J5 i
4 3 2 l O l 2 3 4 5

The subtraction of direction numbers was motivated by pre-
senting that operation as the inverse operation of addition.

-o 4.

a 4 A
Example: 3 — 2 = 5 since 5 + 2

4
=3.
4 '9

The multiplication of direction numbers of the form a x b
4 a -—-—?

was defined as a x b = a x b. The rules for multiplica—

+ 6 ‘F—--
tion of direction numbers of the forms a x b = a x b and
<-<~——-a
a x b = a x b were develOped as consequences of the dis-
tributive and additive inverse properties. The division of
signed numbers was not considered in the study.

It was assumed that the directional number approach
would provide a better visual image that children need at
this age level than would the use of "plus” and "minus”
signs. The operations of subtraction and multiplication
were developed using the structural approaches epistemolog-
ically proposed in the first chapter and somewhat empirically
supported in the second chapter.

The material was organized into seventeen achievement

tasks that could be objectively scored. The tasks were:

31

q . .
(1) Name' the points on the coordinated number line.

.d- a. -0.

2) ,CSHSTTUC“ and name a direction number given its

initial and terminating point.

3) _Name the terminating point of a direction number

I‘s

given the direction number and its initial point.
(4) .Name the initial point of a direction number
given the direction number and its terminating point.

(5) Constlpct and.pamg the sum of direction numbers

 

with the same direction.

(6) Copstpuct and name the sum of direction numbers

 

with different direction.

(7) Conspguct and name the additive inverse of a

 

given direction number.

(8) Construct and name an unknown addend given the

 

sum and the other addend.

(9) Demgpstfate the ability to restate number sen—

 

tences involving the operation of subtraction into sentences
involving the operation of addition.

(10) Construct and name the solutions of subtraction

 

problems.

(ll) Name the product of direction numbers of the

4 4
form a x b .
.4

(12) game the product a x O .

“‘-

 

lThe underlined verbs in this list are operationally
defined in AAAS Commission on Science Education, Science—-
A Process Apngache An Evaluation Model And Its Applica-
tion, Second Repggp (The ASSOCiation, 1968), pp. 7—9.

 

 

32

(13) 'Eggg the missing terms in equations illustrating
the distributive law.

(14) ggg§_the product of direction numbers of the
form 2 x‘; using the distributive law.

(15) ‘Egmg the product of direction numbers of the
form 2 x‘g using the rule.

(16) Egmg the product of direction numbers of the
form a x‘; using the distributive law.

(17) .gggg the product of direction numbers of the
form : xi; using the rule.
The seventeen tasks were organized into six lessons. Each
lesson consisted of two sets of exercises. The first set,
called the problem set for group work, was used by the
teacher for instructional purposes. The second set, called
the problem set for individual work, was used to test the
subjects ability to solve direction number problems. The

problem sets are found in Appendix.A.

Sample

The 578 subjects in the study were children enrolled
in twenty-one fourth, fifth, and sixth grade classes from
various elementary schools in southwestern Michigan. The
classes were from thirteen different schools in eleven diff-
erent cities. The cities ranged in pOpulation from 5,000
through 500,000. The teachers that participated in the
study were selected from volunteers enrolled in continuing

education mathematics courses for elementary teachers offered

33
by Western Michigan University at centers in Fremont,
Grand Rapids, and Marshall, Michigan. The teachers, all
of whom were certified and experienced, worked with their

own classes in their own schools.

Measures

 

Eight measuring devices were used in the study. The
problem sets for individual work, as previously mentioned,
constituted six of the measures. A post test covering the
curricular material in the problem sets (see Appendix A)
and the Otis Quick-Scoring Mental Ability testl made up the
remaining two. Four of the twenty-one classes in the study
were selected at random and their test scores were used to
compute reliability estimates. The equation employed for

computing reliability was

where Vc was the error variance and Vt was the individ-
ual variance of an analysis of variance upon the two classi-
fications of subject and test item. The theory and compu-

tational procedures used to find the measures of reliability

have been clearly explained by Kerlinger.2 The reliability

 

lOtis, Arthur S., Otis Qpick-Scorinngental Tests: New
Edition, Beta Test Form Em (New York: Harcourt, Brace and
WOrld, Inc., 1954).

 

 

2Kerlinger, Fred N., Foundations of Behavioral Research
(New York: Holt, Rinehart and Winston, Inc., 1964), pp.

 

34

measures are summarized in Table 3.1. Further information

on the Otis Test has been compiled by Buros.l

Table 3.1

RELIABILITY MEASURES

 

Tests

 

l 2 3 4 5 6 Post Otis

 

Measure .475 .726 .824 .536 .956 .897 .547 .953

 

 

 

 

 

 

 

 

 

The Design of the study

 

The classes were divided as classes into two treatment
groups (the pupil-discovery group and the teacher-instruc-
tion group) and one control group at each grade level.

Each class was also partitioned into four disjoint sub—
classes by sex and high and low 1.0. The median raw score
for the Otis Mental Abilities test was found for each class.
Those subjects within the class with raw scores above this
median were classified as high 1.0., and those with raw
scores below this median were classified as low 1.0.

The seventeen tasks listed in the first section of this
chapter were organized into six problem sets. The day af-

ter the subjects had a learning eXperience with a particular

 

lBuros, Oscar K., The Sixth Mental Measurements Year-
book (Highland Park, New Jersey: The Gryphon Press, 1963),
p. 381.

35
problem set, by either the discovery or instructional treat-
ment, they were given an examination on the tasks in the
problem set. The control classes were given the examination
without the learning treatments. All classes received a
skill retention examination (post—test) one month after the
sixth problem set examination.

The design permitted comparisons of a discovery type
learning experience with a didactic type learning experience,
boys with girls, high I.Q. with low I.Q.. and one grade
level with another on specific learning tasks involving
signed numbers. The post test gave the same comparisons
on retention. It was assumed that the learning due to
maturation and test experience was uniform throughout all
classes. The control classes were used to give some indi-

cation of the extent of this learning.

Treatment Procedures

A review of skills involving natural numbers on the
number line was conducted by the classroom teacher for the
purpose of defining the problems in the problem set under
consideration. Classes in all three groups (instructional,
discovery, and control) received this review. The pupil-
discovery classes then organized themselves into pupil
committees of about six mumbers each to cooperatively work
for a period of 30 minutes toward the solutions of the
task problems. The teachers in the discovery classes were

permitted to answer questions concerning the correctness

36
or incorrectness of the committee solutions to the problems.
She could offer encouragement. She did not explain why
a solation was incorrect nor suggest correct procedures.
The classes in the teacherminstruction group were conduct—
ed by the teacher. She involved the students as much as
possible in teaching the students to solve the task prob-
lems during a 30 minute period. Each teacher used her
own instructional style. The classes in the control group
received only the review of the skills in natural numbers,
and then they worked individually on the task problems with-

out any help whatsoever from the teacher.

The Hypotheses

 

The hypotheses of the study were grouped into three
classifications: those dealing with learning, those deal-
ing with retention, and one dealing with satisfactory achieve-
ment. The hypotheses related to learning were as follows.

(1) Hypothesis LA: There will be no difference in
the mean scores on tasks between boys and girls.

(2) Hypothesis 2A: The mean score on tasks at any
grade will be higher than the mean score on the same tasks
at a lower grade level.

(3) Hypothesis 3A: The mean score on tasks by child-
ren with higher intellectual ability as measured by the
Otis Mental Abilities test will be higher than the mean
score on tasks by children with lower intellectual ability.

(4) Hypothesis 3A: There will be no difference in

37
the mean scores on the tasks between groups receiving differ-
ent instructional methods.
The hypotheses related to retention were as follows.

(1) Hypothesis 1B: There will be no difference in
the mean scores on the retention of task skills between
boys and girls.

(2) Hypothesis 2B: The mean scores on retention of
task skills at any grade level will be greater than the
mean scores on retention of task skills at a lower grade
level.

(3) Hypothesis 3B: The mean scores on retention of
task skills by children with higher intellectual ability
as measured by the Otis Mental Abilities test will be great-
er than the mean scores on retention of task skills by
children with lower intellectual ability.

(4) Hypothesis 4B: There will be no difference in
the mean scores on the retention of task skills between
groups receiving different instructional methods.

The following hypothesis was related to satisfactory class
achievement.

(1) Hypothesis 5: The classes at all three grade
levels will attain satisfactory levels of achievement on

the learning of task skills.

Analysis

Just as the hypotheses were grouped into three differ-

ent classifications, the analysis of these hypotheses require

38
three different analytic procedures. An a = .05 level
of significance was used in each case to accept or reject
a hypothesis.

At first glance an analysis of variance seemed to be
an ideal vehicle for testing the hypotheses related to
learning. However, this procedure must be rejected for
good reason. The number of subjects in each cell would
vary as a result of differing class size and mix. This
leads to an unbalanced design and the assumptions of inde-
pendence (or orthogonality) would not be valid. A five
way unbalanced analysis of variance does not exist. An
analysis of variance based upon a reduction of the number
of factors, as the pooling of sex and class data, was
possible. But this procedure would not have yielded full
information on the interactions among the factors. Instead,
the analysis used in testing the hypothesis on learning was
the technique of planned comparisons as described by Hays.l
This analysis can be used when a number of particular quest-
ions, formulated prior to data collection, are to be answer-
ed separately. In this procedure the means ”1 under com-
parison are expressed as a linear combination with weights

cj , not all equal to zero, in the form

V = 21c. . .
Hi 3”:
3'

 

1Hays, William L., Statistics for Psycholo ists 9(New
York: Holt, Rinehart and Winston.—l965 , pp. 59-

39

The requirement is made that

If the cj's are selected prOperly, the YH 's will be
i

Orthogonal. In these cases the hypothesis generally tested

is

by the statistic

t:
yest. var (Y)

 

distributed as t with the degrees of freedom of the mean
square error. Since the computational procedures were
written specifically for the study they are shown in detail
in Appendix B.

The testing of the hypotheses dealing with retention
also presented their own peculiar problems. The differences
in post-test scores could have easily been tested, but the
question whether these differences were due to better re-
tention or to better initial learning would remain. To
avoid this difficulty, a multifactor analysis having repeat-
ed measures and unequal group size was used. The repeated
measures used were the post-test scores and the sum of the
task scores from the problem sets of those tasks that were
identical to the ones found in the post—test. The analysis

corrects for differences due to initial learning in the

4O

variance of the mean of the post-test scores by having each
subject used as his own control. The computational proce-
dures followed were found in Winer.l

Finally, to test the hypothesis concerned with satis-
factory achievement, it was necessary to define satisfac-
tory achievement. A class was said to have done satisfac-
tory work on a problem set if 50% of the class correctly
solved 50% or more of the problems on that set. This was
clearly an arbitrary level. However, considering the ex-
tent of the material covered in the six lessons of only 30
minutes each, and considering that the curricular material
was new to many of the teachers, the level of achievement
was believed to be reasonable. The hypothesis was to be
accepted if it could be expected that this level of accomp-
lishment would be reached 90% of the time. The statistic

used was

«2-3:».

where fi was the observed frequency of class having suc-
cess or failure and Fi was the theoretical frequency of
classes having success or failure. The two classifications
of Success or failure were represented, respectively, by

i = 1,2 . This statistic was assumed to have a X2

 

lWiner, B.J., Statistical Principles in Experimental
Desi n (New‘York: McGraw-Hill Book Company, 19627, pp. 374-

 

41
distribution with one degree of freedom. The computational

procedures recommended by Dixon1 were followed.

Summary

In this study, seventeen tasks were selected as meas-
ures of the ability of elementary school children to perform
and understand the operations of addition, subtraction and
multiplication of signed numbers. These seventeen tasks
were organized into six lessons. The subjects were members
of twenty-one fourth, fifth, and sixth grade classes select-
ed from school districts of southwestern Michigan. At each
grade level the classes were assigned to one of two treat-
ment groups (discovery or instructional) or to a control
group. Each class was partitioned into four disjoint sub—
classes by sex and I.Q. A unit on signed numbers using the
six lessons was taught to each class by the classroom teach-
er. The type of instruction or learning treatment they re-
ceived was determined by the treatment group to which the
class was assigned. An examination was given after each
lesson, and a post-test was given one month after the sixth
lesson. The design permitted comparisons of a discovery
type learning experience with a didactic type learning ex-
perience, boys with girls, high I.Q. with low I.Q., and one

grade level with another. The post—test gave the same

 

lDixon, Wilfred J., and Massey Jr., Frank J., Intro-
‘duction to Statistical Analysis (New York: McGraw-Hill
Book Company, Inc., 1957}, pp. 221—224.

comparisons on retention.

42

CHAPTER 4: ANALYSIS OF DATA

The purpose of this study was to investigate the ability
of elementary school children to accomplish tasks related to
operations on signed numbers. Twenty-one fourth, fifth, and
sixth grade classrooms were divided into two treatment gr0ups
(a discovery group and an instructional group) and one con-
trol group. The classes were given six examinations cavern
ing seventeen tasks on the operations of signed numbers-during
a training period and a post-test one month after the train,
ing period. A summary of the mean scores on these tasks is
found in Table 4.1.

A number of hypotheses on learning, retention, and a-
chievement as measured by the test scores were tested. A
planned comparisons test was developed for each of the seven-
teen tasks. For this test the classes within each treatment
group were pooled even though the 102 analysis of variances com—
paring means between classes for each task indicated differ-
ences in 38 cases, no significant differences in 48 cases, and
no analysis in 16 cases at a = .05. This pooling was rational-
ized on the basis that the sample selections were classes and
not the individual students within the classes. The analysis
of variance tables using the planned comparisons computational
procedures are found in Appendix C. The overall analysis of

variance for each of the tasks indicated differences at

43

44

 

 

 

 

 

 

 

 

 

mmm.ma mom.m aeh.m mnmom omo.m hMA¢m ad Home uuom

bvm.m Hum.m mmw.~ ~mo.~ mom.~ mom.m e ha 0

mmm. omm. moo. Hue. mum. mom. a 0d

mom.m ova.m Nao.a mum.a mmm.~ omm.a e ma m

moo. mmm. owe. van. How. cam. H ea

mmm.v mnm.a ado.m mom.~ vmh.~ emm.m m ma

ooo.m ohm.a hmm.a mom.a ohm.a mom.a N NH c

5mm.m mam.~ mm>.~ hem.m mum.~ omm.m m AH

mmm.~ HHN.H mmm.a mam. oom.a mmm.a e 0H m

0mm.m Nah. Hmm.m mma.~ mho.~ 5mm.m e m

~¢5.H mom. oa¢.a HoH.H omo.a omo.d N w

mm>.~ hmm.a eom.~ o~o.~ oo~.~ ma¢.~ m h m

moo.a mma.a m¢m.a Hom.a oma.a oom.a N o

mmm.a Ham.a vmh.a mm~.H Hom.a mmh.H m m

d>¢.H mme. H¢M.H mmo.a moo.a Hmw- m e

mam.a mom. o¢¢.H mum. «om. oeo.a m m H

mmn.m oh¢.H ona.~ mah.~ mmo.~ oom.a m N

ooova wmo. mmm. «pm. wow. mmm. a H

.HumsH .oman .HumsH .omen .HumsH .oman _ muoom use ,Hwnasz
moose nume moose numwm moose seesaw _ swam x Home

 

Emma Bmom DZ¢_m&m¢H “Oh mNMOUm 2¢N2_h0 NMfiZZDm

H.v manna

45
a = .05 for all tasks except eleven and twelve. The hy~
potheses on learning as measured by the test scores and tests
of these hypotheses were as follows.
(1) Hypothesis 1A: There will be no difference in the

mean scores on tasks between boys and girls.

Symbolically: H : u - u = O
0 Bi Gi
Legend: “B a mean score of boys on task i, i = l,...,l7.
i ,
“G. a mean score of girls on task i, i = l,...,l7.
1

Since the degrees of freedom exceeded400 in each case,
the distribution of t was considered normal. The null hy—
pothesis was rejected if t was not in the interval
-l.960 < t < 1.960 (a w .05). The tests of the hypothesis
for each of the tasks are listed in Table 4.2.

Table 4.2

THE TESTS OF HYPOTHESIS 1A ON THE COMPARISON OF LEARNING
BETWEEN THE SEXES FOR 17 LEARNING TASKS

 

—~—r #-

 

Task IMean Difference t-value ‘ Ho
1 -.032 ‘ -l.431 ns*
2 -.34s -l.718 _ he '
3 -.023 -.213* ns
4 —.156 -1.652 ns
5 .055 1.046 ns
6 .020 .355 ns
7 —.065 -.755 ns
8 -.109 -1.855 ns

;
V— "Y rw—vv wy ‘T fir. v—v 7‘

*
not significant

46

 

 

Table 4.2

Continued
Task Mean Difference tpvalue HO
9 -.158 -1.107 ns
10 -.164 -.970 ns
11 .010 .394 ns
12 .005 -.331 ns
13 -.181 w.759 ns
14 —.052 1.433 ns
15 -.105 .585 ns
16 -.007 .212 ns
17 -.l3l n.528 ns

‘v—Y ’—f fivv '_V .Wfr .Y ‘1 'v—v—rv—Y fiv—j aa—

(2) Hypothesis 2A1: The mean scores on tasks at the
sixth grade level will be greater than the mean scores on tasks
at the fifth grade level. 7

Symbolically: H a 0

3 HI ll
0 61 Si

8 p. -p, >0
”A 61 51

Legend: “6 - mean score of sixth graders on tasks i,
i
i. lpsssol7s

“5 - mean score of fifth graders on tasks i,
i

i .7100003170
Since the degrees of freedom exceeded 150 in each case, the

distribution of t was considered normal. The null hypoth-

ssis was rejected it t > 1.645 (a - .05). The tests of the

47

hypothesis for each of the tasks are listed in Table 4.3.

Table 4.3

THE TESTS or HYPOTHESIS 2A. ON THE COMBARISON 0F LEARNING
sETwEEN FIFTH AND SIXTH GRADERS son 17 LEARNING TASKS

 

fi—W

v—ri V V'Y
‘

H

 

Task n Mean Difference t-value O
1 .037 1.360 ns
2 .053 .213 ns
3 ~.l95 ' -1.473 ns
4 .257 2.211 rejected
5 .113 1.727 rejected
6 .036 .522 ns
7 .004 .040 ns
8 .060 .838 , ns
9 -.581 -3.306 ns
10 .594 2.855 rejected
11 .100 2.237 rejected
12 .007 .397 ns
13 .095 .326 ns
14 .212 4.768 rejected
15 1,143 5.203 rejected
16 .286 6.974 rejested
17 1.049 3.450 rejected

 

1
T

ww— w— r. ‘ V,—r —y 71 vv f —‘r—

(3) Hypothesis 2A2: The mean scores on tasks at the
fifth grade level will be greater than the mean scores on the

same tasks at the fourth grade level.

Symbolically: HO 3 “51 - u4. - 0
* 1

3A ' “5i ' “4i ’ °

48

Legend: "5 a mean scores of fifth graders on task i.
i

i = 1'000'170
“4 a mean scores of fourth graders on task i,
i

i = 1.....17.

Since the degrees of freedom exceeded 150 in each case, the
distribution of t was assumed normal. The null hypothesis
was rejected if t > 1.645 (a = .05). The tests of the

hypothesis for each of the tasks are listed in Table 4.4.

Table 4.4

THE TESTS OF HYPOTHESIS 2A2 ON THE COMPARISON OF LEARNING
BETWEEN FOURTH.AND FIFTH GRADERS FOR 17 LEARNING TASKS

 

 

T;;k Mean Difference fit—valhe Ho
11 ._ .1 r 1.7 Y r T 7
1 -.207 -7.462 ns
2 .169 .679 ns
3 .013 .103 ns
4 .307 2.655 rejected
5 -.238 ' —3.639 ns
6 .069 .992 ns
7 —.014 -.141 ns
8 .210 2.930 rejected
9 -.155 -.884 ns
10 -.093 —.453 ns
ll -.057 «1.274 as
12 .006 .309 ns
13 .051 .174 ns
14 —.250 —5.603 as
15 .479 1.888 rejected

 

l

1 7* ' ’ '

 

 

.
‘ "
. ,
s
s‘ .
o
u
.
» v
s
a
.
.
O
.
.
I
v . .
s
, .
,
.

49

 

 

Table 4.4
Continued
Task If Mean Difference T ‘ t—value THO I?
16 ‘ i ‘75 j-10241 Y If Y -.605 IT ha I
17 .143 .472 ns

 

(4) Hypothesis 3A: The mean score on tasks by children
with greater intellectual ability as measured by the Otis
Mental Abilities test will be greater than the mean score on

tasks by children with lower intellectual ability.

Symbolically: Ho F “Hi - ”Li = 0

= u - u > 0
HA Hi Li

Legend: 8 mean score of children with greater mental

u
Hi

ability on task i, i = l,...,l7,

“L a mean score of children with lower mental abili-
i

ty on task i, i a l....,l7.

Since the degrees of freedom exceeded 400 in each case, the
distribution of t was considered normal. The null hypoth-
esis was rejected if t > 1.645 (a - .05). The tests of the

hypothesis for each of the tasks are listed in Table 4.5.

50

Table 4.5

THE TESTS OF HYPOTHESIS 3A ON THE COMEARISON OF LEARNING
BETWEEN HIGH.AND LOW I.Q. GROUPS FOR 17 LEARNING TASKS

 

Task

‘ '

Mean Difference

—r

T

Y—v—

t-value

 

0
or T -1 r f 11.1.

1 .015 .671 ns
2 1.140 5.619 rejected
3 .259 2.399 rejected
4 .641 6.789 rejected
5 .252 4.723 rejected
6 .411 7.198 rejected
7 .535 6.183 rejected
8 .540 9.225 rejected
9 .646 4.517 rejected
10 .656 3.874 rejected
11 .078 2.128 rejected

12 .012 .795 ns
13 .868 3.647 rejected
14 .180 4.973 rejected
15 .511 2.848 rejected
16 .120 3.598 rejected
17 .567 2.287 rejected

 

ji—

v~ v—Y w,

(5) There will be no difference in the

mean scores on tasks between the discovery treatment group and

Hypothesis 4A:

the instructional treatment group.

Symbolically: H0 3 “Di w “Ii - 0

Legend: - mean score of discovery treatment group on task

“Di

it i 8 1:000:17.

51

“I = mean score of instructional treatment group
i

on task i, i w 1.....17.

Since the degrees of freedom exceeded 400 in each case the

distribution of t was considered normal. The null hypoth-
- esis was rejected if t was not in the interval -1.960 <
t < 1.960 (a a .05). The tests of the hypothesis for each

of the tasks are listed in Table 4.6.

Table 4.6

THE TEST OF HYPOTHESIS 4A ON THE COMBARISON OF LEARNING
BETWEEN DISCOVERY-INSTRUCTION GROUPS FOR 17 LEARNING TASKS

 

“r.— w a. , w rvawvv-fi qw—

ywfi w. ~— fi Y7

Task Mean Difference t-value H

 

0

l .003 .150 na
2 -.843 -4.159 rejected
3 -.685 ~6.353 rejected
4 -.563 -5.963 rejected
5 -.293 —5.487 rejected
6 -.151 -2.657 rejected
7 -.382 -4.417 rejected
8 -.382 -6.524 rejected
9 —l.l47 —8.017 rejected
10 -.483 -2.855 rejected
11 .000 —.015 ns '

12 -.005 -.331 ns
13 -.777 -3.263 rejected
14 -.124 -3.425 rejected
15 -.584 -3.257 rejected

16 -.023 -.701 ns

17 .090 -.364 ns

 

i

52

An analysis of variance for repeated measures using the
unweighted means solution was used to test the hypotheses on
retention. The repeated measures used were the post—test and
the sum of the scores on the tasks that were the same as the
tasks on the post-test. If a child had missed taking any one
of the task tests or the post-test used in the repeated mea-
sure, he had to be removed from the analysis. This require-
ment resulted in the loss of about one-fourth of the subjects.
Because of this loss, the data were pooled to form one factor
designs. The analysis was computed for each of the factors
of sex, I.Q., and instructional method for each grade level.
One further analysis was computed for grade level. The re-
sults are summarized in Tables c.18 through C.27 included as
part of Appendix C.

Each of the following hypotheses on retention was tested

by the statistic

 

[u-ulz
F g ‘_l 2

MSsubjects w groups 1 +-$—

n n
l 2

where pl and “2 were the post test means under comparison,
nl and n2 were the number of subjects in the mean groups,
and MS was the subjects within groups mean

subjects w groups
square from the analysis of variance. Since the degrees of

freedom of the denominator exceed 30 in each case, the function

53

 

 

 

V ”V
Loglo F 05 (vlvz) ~ 1'4287 v e .- (.681) 337$
' 2v v 1 2 1
«l 1 2 95
V1+V2

in Dixon1 was used as an approximation for the F distribuw

tion percentiles where v1 and v2 were the degrees of

freedom of the numerator and denominator respectively.

(6) Hypothesis 13: There will be no difference in the
mean scores on the retention of task skills as measured by
the post test between boys and girls at the same grade level.

Symbolically: H0 2 “Bi = ”Gi

Legend: a mean score on post test by boys in grade' 1.

[Jr
Bi

1 ‘ 4'5360

“G a mean score on post test by girls in grade i.'
i
i = 4,5,6.

The tests of the hypothesis for each of the grade levels are
listed in Table 4.7.

Table 4.7

THE TESTS OF HYPOTHESIS 18 ON THE CONRARISON OF RETENTION
BETWEEN THE SEXES IN GRADES FOUR THROUGH SIX

v—v' r v T *1 v—

Grade F F at a = .05 f Ho

I'—— v a v fi—f 3 fiv—Yy v—vwv

4 .288 .738 ns

Dixon, 92. ci .. p. 402.

 

54

 

 

Table 4.7
Continued
Grade F F at e s .05 Ho f
5 .86l .731 rejected
6 .079 .731 ns

(7) Hypothesis 231: The mean score on the retention of
task skills as measured by the post test at the sixth grade
levels will be greater than the same mean score at the fifth
grade level.

Symbolically: Ho : “6 - “5 = 0

a; 3 H6 - 95 > 0

Legend: “6 = the mean score on the post test by sixth graders.

“5 - the mean score on the post test by fifth graders.

The region for rejection was F a .720 (a - .05). F was com—

puted to be 8.337. The null hypothesis was rejected.
(8) Hypothesis 282: The mean scores on the retention

of task skills as measured by the post test at the fifth grade
level will be greater than the same mean score at the fourth
grade level.

Symbolically: Ho 2 ”5 - u4 - o

“A‘ “s'M’o

Legend: 95 - the mean score on the post test by fifth graders.

55

“5 = the mean score on the post test by fourth grad-

81:8.

The region of rejection was F 2 .720 (a a .05). F was some
puted to be 1.185. The null hypothesis wasrejecteg.

(9) Hypothesis BB: The mean score on retention of task
skills as measured by the post test by children with higher
mental ability as measured by the Otis Mental Abilities test
will be greater than the same mean score by children with
lower mental ability.

Symbolically: H0 :

Legend: “H- - mean score on post test by children with high-
1

er mental ability in grade i. i s 4.5.6.

“L a mean score on post test by children with lower
i

mental ability in grade 1. i a 4.5.6.

The tests of the hypothesis for each of the grade levels are
listed in Table 4.8.

Table 4.8

THE TESTS OF THE HYPOTHESIS 38 ON THE COMBARISON OP RETENTION
BETWEEN HIGH‘AND DOW I.Q. GROUPS IN GRADES FOUR THROUGH 81X

T ‘7'

Grade F F at a - .05 , Ho

v—fi Y W‘Y r— v— 1’"

fl , v 7 TT

4 28.686 .737 rejected
5 1.491 .731 rejected

— fi—fi *r‘r— V, 17—

56

 

 

Table 4.8
Continued
Grade F F at a = .05 H0
6 2.770 .731 rejected

 

Y "r ‘T v

(10) Hypothesis 4B: There will be no differences in
the mean scores on the retention of task skills as measured
by the post test between groups at the same grade level re-
ceiving different instructional treatments.

Symbolicallyz Ho : “Di a “Ii

Legend: “D a mean score on post test by the discovery group
i ' ‘ s

in grade 1' i E 40596.

pl - mean score on post test by the instruction
i

group in grade i. i a 4.5.6.

The tests of the hypothesis for each grade level are listed
in Table 4.9.

Table 4.9

THE TESTS OF HYPOTHESIS 48 ON THE COMPARISON OF RETENTION
BETWEEN DISCOVERY-INSTRUCTION GROUPS IN GRADES
FOUR THROUGH SIX

 

Grade E F at a E .05 Ho

 

4 1.724 .731 rejected

.— r f w v—r

 

.‘J__

57

 

 

Table 4.9
Continued
Grade F F at a = .05 HO
5 .046 .731 ns
15.366 .731 rejected

 

A class was said to have achieved a satisfactory level
of learning for a particular lesson if 50% of the class scored
50% or better on the test covering that lesson. The achieve-
ment levels of the classes in the study are summarized in
Tables C.28 through C.30. found in Appendix C. Further, the
classes were expected to achieve this level 90% of the time.

(5) Hypothesis 5: At each of the three grade levels.
50% of the students in a class will score 50% or higher on
the six individual tests 90% of the time.

Using the computational procedures described in the
analysis section of chapter three, page 42 the hypothesis
was rejected if X2 > 3.84 (a = .05 with one degree of
freedom).

At the fourth grade level. x2 = 16.89, the hypothesis

was rejected.

 

At the fifth grade level. x2

21.777. the hypothesis
was rejected.
At the sixth grade level. X2 = .604, the hypothesis was

not rejected.

Summarv

58

In this chapter a number of hypotheses were tested on

the ability of fourth, fifth, and sixth grade children to

learn and retain skills involving the operations on signed

numbers. A final hypothesis was tested on the level of

achievement of the classes in lessons involving operations

on signed numbers.

In tests investigating the ability of children to learn

the skills. the mean scores of seventeen tasks were compared

within classifications of sex, grade, I.Q. and method of in-

struction. Table 4.10 summarizes the findings of these com-

parisons.

Table 4.10

NUMBERS OF TESTS THAT SUPPORT THE HYPOTHESES COMPARING MEANS
(u) WITHIN THE CLASSIFICATIONS OF SEX, GRADE. I.Q. AND
METHOD OF TEACHING

 

Comparison Hypotheses
Sex uBoy h uGirl
Grade uG6 ”G5
LLGs uG4
I°Q° “Hi IQ ”Lo IQ
Method of
Instruction ”Disc = “Instr

No. of tests

supporting
Hypotheses

17

15

Total No.
of tests

l7

l7

l7

l7

l7

59
The mean scores on a post-test were used to compare the
retention of subjects classified according to sex, I.Q., and
method of instruction at each grade level and then between
the grades themselves. Table 4.11 summarizes the findings

of the comparison.

Table 4.11

SUMMARY OF TESTS OF HYPOTHESES COMPARING POST TEST SCORE MEANS
(p) WITHIN THE CLASSIFICATIONS OF SEX. I.Q.. TEACHING METHOD.

 

 

 

 

 

 

 

 

AND GRADE
. I I? . Result of Tests of Hypgthegis
Comparisons Hypothesis Grade 4 Gradevs Grade 6
Sex “Boy = “Girl supported rejected supported
I.Q. ”Hi IQ > “Lo IQ supported supported supported
Teaching
Method ”Disc = “Instr rejected supported rejected
Grade p66 > “5G : supported, “5G > “4G : supported

 

—-—r v W

Finally. the classes were said to have reached satis-
factory achievement if 50%,of students in the class scored
50% or higher on the tests given after each lesson. It was
hypothesized that this level of achievement would be reached
90% of the time. The hypothesis was rejected at the fourth

and fifth grade levels and supported at the sixth grade level.

CHAEILR 5: SUMMARY AND CCNCLUSIONS

Summarv

Elementary school mathematics has undergone dramatic
changes in both content and procedures within the past ten
years. The “modern“ mathematics revolution is continuing
with recommendations for the inclusion of still newer ideas
in the curriculum. One such topic is the study of the rules
of Operations on signed numbers. It was the purpose of this
study to investigate the ability of children to learn and
retain skills used in operations on signed numbers.

The numbers, in fact integers, were represented on the
number line as bindirectional vectors. The number line was
coordinated by indicating the direction and distance a point
was located from zero. The operation of addition was defined
as vector addition. The operation of subtraction was motiva-
ted by presenting that operation as the inverse operation
of addition. The rules for multiplication were developed
as consequences of the distributive and additive inverse
properties. The skills needed to effectively work with these
operations were organized into seventeen objectively scored
tasks. These tasks were further grouped into six lessons.

The subjects were members of twenty-one fourth, fifth.
and sixth grade classes selected from school districts of
southwestern Michigan. At each grade level the classes were

60

61
assigned to one of two treatment groups (discovery or in-
structional) or L3 a control group. Each class was parti-

tioned into four diSJSint subclasses by sex and I.Q. A unit

(1
T

on signed numbers was taught to ea 4 class by the classroom
teacher. The type of instruction or learning treatment they
received was determined by the treatment group to which the
class was assigned. An examination was given after each
lesson, and a post test was given one month after the sixth
lesson. The design permitted comparisons of a discovery
type learning experience with a didactic type learning ex-
perience, boys with girls, high I.Q. with low I.Q.. and one
grade level with another. The post test gave the same com-
parisons on retention.

The hypotheses were statistically tested at a = .05.
This analysis of the data indicated that no significant
difference existed between boys and girls in learning the
17 tasks. There was no difference in retention between the
sexes at either the fourth or sixth grade level. However,
at the fifth grade level.girls did slightly better than boys.
On 15 of the 17 tasks children with high I.Q. scored better
than children with low I.Q.. and on retention the high I.Q.
subjects scored higher than the low I.Q. subjects at all
grade levels. The subjects in the instruction group had
higher scores in general than the subjects in the discovery
group in 13 of the 17 tasks. Also, the subjects in the in-
Struction classes retained more of what they learned than

did.the subjects in the discovery classes in the fourth and

62
sixth grades. p.13 in the fifth grade was there no signifi-
cant differ nee. The differences between the grades were
more mixed. In 8 of the 17 tasks sixth graders scored higher
than the fifth graders. and on only 3 of the 17 tasks did
fifth graders score higher than iDUlth graders. However,
on retention, the sixth graders retained more of what they
learned than the fifth graders who in turn retained more of
what they learned than fourth graders. Overall, a class was
said to have reached a satisfactory level of achievement for
a particular lesson if 50% or more of the class scored 50%
or more on the test following that lesson. It was found that
sixth graders could be expected to achieve at this level 90%

of the time, whereas fourth and fifth graders could do so

only 80% of the time.

Conclusions

The following conclusions are stated as a result of the
tests of the hypothesis.

(1) No difference existed between boys and girls in
the fourth, fifth, and sixth grades in their ability to learn
tasks involving the operations of signed numbers.

(2) No difference existed between boys and girls in
the fourth, fifth, and sixth grade in their ability to re-
tain skills learned involving the operations on signed num-
bars.

(3) Higher I.Q. children in the fourth, fifth, and sixth

grades scored higher on tasks involving operations of signed

63
numbers than did later I.Q. children.
(4) Higher I.Q. children in the fourth, fifth and

sixth grades retained skills learned in the operations of

n lower I.Q. children.

M

signed numbers better th

(5) Children in the sixth grade scored higher on tasks
involving operations of signed numbers than did children in
the fifth grade.

(6) Children in the sixth grade retained skills learned
in the operations of signed numbers better than fifth grade
children.

(7) No difference existed between fifth and fourth grade
children in their ability to learn tasks involving the opera-
tions of signed numbers.

(8) Children in the fifth grade retained skills learned
in the operation of signed numbers better than did fourth
grade children.

(9) Children in the fourth, fifth, and sixth grades
who had an instructional type learning experience scored
higher on tasks on the operations of signed numbers than did
fourth, fifth, and sixth grade children who had a discovery
type learning eXperience.

(10) Children in the fourth, fifth, and sixth grades
who had an instructional type learning experience retained
skills learned on the operations of signed numbers better than
did fourth, fifth, and sixth grade children who had a dis-
covery type learning experience.

(11) Sixth grade classes attained satisfactory levels

U}

igned numbers.

, I
a I" k " (.13: mt“. V‘ f' ”1 v‘ , .. J r .: »;“ i- ‘H ,w .. v '..‘ " (‘1’. ‘ «F
4 ~“._ 5 ,, -s,; a... ‘. 1.x _,_ ..' .) ~ "Ursa -a... ‘vfi mva. .4..:n.~«.¢‘¢—.D \sJ‘.
.L

A number oi the conclusions are easily justified by a

cursory rexiew cf the analysis. (39:12.1 sion (1), that no

difference existed between s«3x;:s in their ability to learn,
was well s.pgcrted in that all 17 tests were not significant

(Table 4.2). Conclusion £3}, that high I.Q. children were

better able to learn than low I.Q. children was supported by
15 of the 17 tasks and the remainin, two, while not signifi-
cant, were positive {Cable 4.5). Similarly, conclusion (4),
that hi hIOQ. children were be ter able to retain what they
had learned was well suptorted (:able 4. 8). Conclusions (6)
and (8) on the ability of higher gr de children to better

retain what was learned are SJ$10rte dby the statistical tests.

kit.”

$1)

Conclusions i9 nd (10}, that children receiving in-
struct .ional lear nin .g treatments learned better and retained
more of wr at they learned is contrary to what was conjec-
tured in the hypotheses, namely, that no difference existed.
The analysis, however, indicated that differences did exist
in 12 of the 1? tasks in favor of instruction. The differ-
ences in four of the five nonmsivnificant cases were in the
direction of the instructional procedure (Ta.le 4.6). In
defense of t.e discoJe ery method it she J be reported that

this procedure was a new experience for both the subjects

3'1

and their teaohe 8. Even though the discovery groups did

not learn as well, they di.d lea: r as illustrated by the fact

65
that the discovery classes reached satisfactory levels of
achievement in 34 of 54 lessons. Also, it is interesting
to note how these differences due to teaching method came
about. A review of the mean scores (Table 4.1) revealed that

in general, little difference existed between the methods

at the fourth grade level, and that the differences at the

sixth grade were greater. This occurred because the means of

the instruction group increased with the grade level, and

time means of the discovery group decreased with the grade

J£NJel. The superior performance by the instructional groups

‘was uniform over both the high and low I.Q° classification
arui the boy—girl classification.

Conclusion (7), that there was no difference in learn-
irug ability between fifth and fourth grade children, was
also contrary to conjecture. Only 2 of the 17 tasks showed
significant differences and the non—significant t‘s were
positive in five cases and negative in six cases (Table 4.4).
(knuzlusion (5), that children in the sixth grade learned
better than children in the fifth grade, was not overwhelm-
inglfir supported. In only 8 of the 17 tasks was the null
hYEKIthesis rejected. However, the fact that the overall
analysis of variance indicated that the data on two tasks
contained no real differences and that 7 of the 9 non-signifi-
cant tests favored the conclusion were considered indicative

(Table 4.3).
Conclusion (2), that no difference existed between boys

and girls in their ability to retain what they learned, was

‘ ‘ ‘ - ." ._ . , .. g‘. '1. .. If
sugggrges f? 3r? age. s,s .. -ie :-ulan and sixth grade level,

bat not at 1”. .lf”“ izaia Local. When a large number of
statistical teaLo arr WJle a: r z .05, as was the case in

one errors will

'd
U:
r-‘
4
\
J.
r:
f)
l"
"<
f
'1
rd
:1
1'
a"
.1
f?
r3
(D

¢¢e significant difference

0
O
D
5:.
lv
H
“T
1)
'1
if.
n
r‘:
m
(§
1
(h
sh
w
r
,J
U:
0
“:5
(n
5"
,

.vfii was judged to be such an error.
First, the T~ratio was ;,t highly significant (Table 4.7).
Second, all other comparisons based upon sex in the study
were not significant. Clird, and most importantly, a review
of the raw data indicated that fifth grade girls had an un-

usually large share of high I.Q. subjects (54.9% compared

ade and 48.3% in the fifth grade).

*1

with 48.8% in tne fourth g
The results if the study, in some cases, supported the
findings of other researchers and, in other cases, question-
ed their findings. The problems in task 7, testing the under-
standing of the inverse property, were answered correctly
77.9% of the time, whereas the problems in task 15, testing

the understanding of distributive property, were answered

'1)

correctly only 52.1% of the t me. this clearly supported
Crawfordl who reported that the inverse property was more
easily learned than the distributive property.

No difference was found in the abilities of boys and

J

,, . a u . c - o . I 2
girls in learning or retaining :ne material. JarVis and

 

Crawford, 92, cit., p. 5728-5729.

93:31?! lfij,CLI~j ifi, L,;s ac zevcx Setter than girls on
tasks desired «-2; $97.51;?“Q3; canccpts. This was not ob-

‘ L '\ PI " 1» !‘\ ’ , ' ; - - .
served .n t‘is s 'd} 1 the. L : le~rnlng of mathemati-
cal concerts Was 11:.” 3-:t .f t‘— rea-ctng.

1 1 2 3
, N .. ~ ~ 1.3 ‘ a ‘
The c-;-m :1 sass-:r, :rec-ac Ad others that di-

dactic teaching methods lead to Jetzer ISSJltS in initial

testing was ~crted, to: the claim that discovery methods

cl

0»
t
’55
m

r perfor,ance on ret ntion tests was not

H)

result in beet

verified. Als onjectare that a discovery approach

I
|
Q

aids in the learning of operations, while directed methods
result in greater learning of mathematical properties as
reported by Ar stron g was not substa..t ated.

A study of the data revealed some unexpected observa—
tions that were not directly related to the theories discuss-
ed in the studj. Recall that in task 8 the subjects were

required to fir d a missang alisdd given a sum, in task 9 the

*1

subjects were required to restate a subtraction problem as

an addition problem, and in task 10 the subjects were requir-

,‘S

d 9 t- a solution to a subtrac-

)
* *5
5.4..

S
0.

ed to combine ta s‘es 8 a

C

child who had mastered

95

tion problem. One would think that

asks 8 and 9 woul d find task l0 easy to solve. However,

 

ParLl ey,‘gp. capo, pp. 268 2.f.
BaSSler, K209 1":to Ski-2o 357’3620
Fleckman, 3p. cit., po. 3366-3367.

u-x‘r n

4 .. ~ -
Armstrong, g3, 392., p. i4l.

. r311;-

 

I)!

p:

121.1

I?

‘2!

m7.

this was not the case, 33erall, the scores averaged 60.0%
correct on task 8, 56,3% on task 9, and 36.6% on task 10.

The subjects had difficulty in combining the two previously

In another case the learning of tasks l4 and 16 was pre-
requisite to the learning of tasks 15 and 17, respectively.
In tasks l4 and 16 the distributive prOperty and the additive
inverse property were used to justify the rules for finding
the products of numbers with negative values. In tasks 15
and 17, the subjects were to appl" tie rules they learned in
the previous task. Thus if a child had failed in task 14,
one would expect him to fail in task 15. However, this was
not always the case° Many subjects, after missing tasks l4

and 16, went on to correctly solve the problems in task 15

and 17.

Recommendations

The study of the operations of signed numbers is a topic
that could well be taught within the sixth grade mathematics
curriculum. This would include the operation of multiplica-
tion as well as the operations of addition and subtraction
already included in some programs. The order of difficulty
of the operations in the study were addition (easiest),
multiplication, and subtraction (most difficult). Since the
multiplication of signed numbers is apparently easier to

learn than subtraction, there is no reason to exclude it.

 

m:

5L

VLU‘:

a,“ H

n\&

n u

69
The fears expressed in the Cambridge Report1 about the pedam
gogical problems of teaching the multiplication of signed
numbers seem to be exaggerated. The topics could be taught
also in the fourth and fifth grades, but not with the degree
of rigor used in the sixth grade. This is evident by the
fact that while only the sixth graders achieved satisfactory
levels of achievement 90% of the time, the fourth and fifth
graders did so 80% of the time. The closeness of these per-
centages lends support to those who advocate nonwgraded
schools.

Replications or further similar studies are needed be-
fore one can judge which conclusions can be generally accept-
ed. The study contains a number of weaknesses that restrict
such generalization. First, the classes used in the study
were not selected at random, and little demographic informa-
tion is available concerning the subjects. This makes it
difficult to conjecture how other elementary school children
would do in similar studies. Also, the relatively low relia-
bility scores on some of the examinations were disappointing.
In a replication of the study, where better instruments were
developed, better results may be expected. Finally, further
study using more instructional time might result in higher
achievement. This is suggested in the mean scores (Table 4.1)
for lessons five and six. The lessons were very similar in

that they covered the multiplication of negative numbers.

1The Cambridge Conference on School Mathematics, op,

cit., p. 37.

7C
The mean scores on lesson six were higher than the mean scores
on lesson five in every case. Since the lessons were similar
the higher scores on lesson six may be attributed to the
total experience, i.e. extra time.

Further investigations in the study of the operations
on signed numbers might compare the "direction” number ap-
proach used in this study with the traditional plus and
minus representation. The direction number approach was
used because it was hoped that this would provide a better
visual image that children need at this age. If it can be
shown that the plus and minus symbols serve just as well,
then they should be used since they are universally accepted
and the students must adopt them sooner or later.

In further studies, the number system used should be
extended to include rational numbers as well as integers.
The number combinations used in this study were restricted
purposefully to the easier combinations. The desire was to
measure the ability of children to learn the concepts rather
than to measure their arithmetic ability. The extension to
include the rational numbers would permit an investigation
of the operation of division, which was excluded from this
study.

Finally, the positive results of this study should en-
courage similar investigations of other topics recommended

in the Cambridge report1 for the elementary school.

 

1‘The Cambridge Conference on School Mathematics,

_Qp. cit.

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Jarvis, 0.T., "Boy-Girl Ability Differences in Elementary
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Vol. 64 (November, 1964), pp. 657—659.

Johnson, Harry C., "What Do We Mean by Discovery?," The
Arithmetic Teacher, Vol. 11 (December, 1964, pp.
538-539°

Keedy, Mervin J., et. al., Exploring Elementary Mathematics,
.Q (New York: Holt, Rinehart and Winston, 1970).

Kemeny, John, "The Impact of the Computer on Teaching," An
address given at the Cleveland Meeting of the ‘
National Council of Teachers of Mathematics, Cleve-
land, Ohio, November, 13, 1969.

Kerlinger, Fred N., Foundations of Behavioral Research (New
York: Holt, Rinehart and Winston, Inc., 1964),
pp. 429—443.

Kersh, Bert Y., "Learning by Discovery: Instructional
Strategies," The Arithmetic Teacher, Vol. 11
(April, 1964), p. 226.

Kingston, J. Maurice, Mathematics for Teachers of the Middle
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Krich, Percy, "Grade Placement and Meaningful Learning,"
School Science and Mathematics, Vol. 64 (February,
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Miller, G.A., "Crusade Against the Use of Negative Numbers,"
School Science and Mathematics, Vol. 33 (December,

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49.

Hirh School Alaehra (New York: American

5‘... -u... -5.“ ‘4 _

Milne, Wi, .' ..‘

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Parsley, Kenneth M., “Further Investigation of Sex Differences
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76

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APPENDICES

APPENDIX A

PROBLEM SETS AND POST-TEST

(l-l)

(1-2)

(2-1)

(2-2)

(2-3)

(3-1)

(3-2)

<4-1)

PROBLEM SET 1 FOR GROUP WORK

Write the direction number at each point to tell the
distance and direction the point is from zero.

 

e 4 4
l O l 2
l 5 _l l A l L J. l

 

Choose any point you wish on the number line and name
it 0.

O
l 1 n n J_ J_ i 4 1

Complete each number line below. Use the number line
to answer the question following it.

0

g n J 4‘ I it c; .L liL
4 4
What direction number goes from 1 to 4? Ans.

O

1 J_ J_ J. 4__ 5 J, n L
Q 4
What direction number goes from 2 to 3? Ans.

 

 

 

O
4_l l 4 g, L_ 4L lLi 1 pl
4 4
What direction number goes from 4 to 2? Ans.
0
1 1 i _i L l Ii I op
G 4

The direction number 4 goes from 1 to .

 

 

0
g 5 1 1 L _Ll 5 l L _+
4 4
The direction number 3 goes from 2 to .
0
4_ 1 4_ 1 i L .L 1 L
4 ' 4

 

The direction number 5 goes from to 2.

77

 

(1-1)

(2-1)

(2-2)

(2-3)

(2-4)

(2-5)

(3-1)

(3-2)

78

PRQEiiJLIVI SET .1 FOR INDIVIDUAL WORK

Write the direction number at each point to tell the
distance and direction the point is placed from zero.

(« 4 4
l O l 2
‘L L _L _._L l L l l J . l

 

Complete each number line below. Use the number line
to find the answer to the question following it.

1 _L .1 1 1 1 1 __L k 1

What direction number

_l_ L 1 1

What direction number

.1
goes from 1

L l

.1
goes from 5

.1
to 3? Ans.

J_ 1 1
4

to l? Ans.

f Y

 

 

 

1 L 1f 1 J_ _17 l 1_ A

4 e-
What direction number goes from 2 to 1? Ans,
1 1 1 1 ‘L L 1 1 1

e. 4

What direction number

goes from 4

to 2? Ans.

 

L 1 1 L 1 _L 1'_ 1 1

e- ‘4
What direction number goes from 3 to 5? Ans.
_1 L L 1 1 l __L l _l_

 

.9

The direction number 3 goes from 2 to

s 1 ,_L L

(-

i 4

1 L a

.9

The direction number 1 goes from 4 to

:Ia. 1.1 I-flum
m. . ‘ .-
‘ ..

‘ 1.
.- 4
I'
I r
1
4

(3—3)

(4-1)

(4-2)

(4-3)

 

 

 

 

 

___..._..1__.-.._r-.-- r -V.--i,....i-__v._ ._ h L l L
e

he direction nu -‘ 2 9358 from to .
1 _1_ BL 11 1 L 4.

4 4

The direction number 5 goes from to 3.
n 1. “1f 1 1 1 4

£4 6

The direction number 2 goes from to 1.
1 1 l l L l l

4 4-

The direction number 2 goes from to 2.

(I)
(f‘

EROHLEM SET 2 FOR GROUP WORK

Use the number line to find the missing number in each number
sentence.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q' é 4» (~ é- 4 4 4 4 4 4 4
5 4 3 2 1 O 1 2 3 4 5 6 7
(5"1) 1 1 1 1 1 1_ L LL 1 1 1 L V 1—
4 4
2 + :3 =
(- (1- é- (a (L (— 4 4 4 4 4 4
6 5 4 3 2 l O l 2 3 4 5 6
(5'2) I L 1 1 _L 1 p; 1 _1f 1 _L l 1
é- e
3 + 3 =
6- 6 e e e 6 4 4 4 4 4 4
6 5 4 3 2 1 O 1 2 3 4 5 6
(6-1) L l L I l L 1 L p L L 4_ g
4 e
4 + 1 =
€- (- é (- 4- 4- 4 4 4 4 4 4
6 5 4 3 2 1 O 1 2 3 4 5 6
(6'2) 1 1 L 1_ 3 L 1 1 1 L 1_ L 1
(u 4
3 + 2 =
(- (- (— Q 6 (- 4 4 4 4 4 4
6 5 4 3 2 l O 1 2 3 4 5 6
(7‘1) 1 J 1 1 1 1 1 1 L 1 La !, 1
4 <-
4 + 4 =
e é { é 6 e 4 4 4 4 4 4
6 5 4 3 2 1 O 1 2 3 4 5 6
(7'2) 1 4 1 L 1 1 1 1 1 _l_ L 1 L
a)
2 4 == 0

 

 

 

(7-3)

(8-1)

(8-2)

8].

 

 

 

 

 

 

 

 

 

 

(- 6 4 .1 (- gt- 4 4 4 4 4 4
6 5 4 3 2 .1. O l 2 3 4 5 6
i. i 1 l L —L a La 4 L i e ‘
Q
+ 3 ==
(- (- 14 (- 4 Q 4 4 4 4 4 4
6 5 4 3 2 l O l 2 3 4 5 6
1 1 L L _L L L : 1 1 1 L _L_T
1 + = 4
(- (- 6 Q» 4 q! 4 4 4 4 4 4
6 5 4 3 2 l O 1 2 3 4 5 6
1 1 L 1 L 1 1 1 1 1 JL 1 L1
Q- 4
+ 2 =

 

 

 

LI

82

PROBLEM SET 2 FOR INDIVIDUAL WORK

Use the number line to find the missing number in each numb-
er sentence.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.— 4— 11» 4- + 4 4 4 4 4 4 4
5 4 3 2 l O l 2 3 4 6 7
(5-1) 1 1 1 _L 4 L 1 L 1 § 3 1
'4 4
3 + 4 =
1‘
6- 4— +— 6: v- o- 4 4 4 4 4
(5-2) ,6 ? f .3 f .1 9 ,1 .2 i ‘3 .5 6.
(— d-
2 + 4 =
‘- 4- c: 4- +- 4 4 4 4 4 4
(6-1) j i f 2” § ,1 ‘3 ,1 i i f’ j’ 5’
4 +, '
3 + 1 =1
. + ‘5 ‘- + O- '9 4 '9 '4 '9
6 5 4 3 2 l O l 2 3 4 5 6
(6-2) 3 l 4% 41 _L 1 1 _l_ L : : _£_
9 4 “"'
4 + 2 = I
‘- ‘I' ‘- ‘- +~ .- 4 '0 '4 '9 '0
6 5 4 3 2 l O l 2 3 4 5 6
(7-1) 1 1 1 1 W‘L 1 1 _1_ 1 _L L 1 1
4 4— f
3 + 3 = 2- -*
4- 4- 4— 4— 4- 4- 4 4 4 4 4 4
(7_2) 6 5 4 3 2 1 o 1 2 3 4 5 6

 

 

83

(743)

 

0

+2

3

 

4.41

+21

$4.1

66.1

(8-1)

.J
T C

‘31.

 

‘3

 

42

.1,

#21

+3..

‘5

L1

1

‘6

(8-2)

fi—

+1

4.3

 

 

 

 

 

PROBLEM SET" 3 FOR GROUP WORK

Rewrite each of the following subtraction number sentences
as an addition number sentence.

 

 

 

 

 

 

 

 

 

4 4 4
(9—1) 5 - 3 r: 2
4 42 At ’T
(9-2) 2 — 2 = “ "
+1 4
(9-3) 2 - = 3
+- 4-
(9—4) - 1 = 4

 

 

Rewrite each of the following subtraction number sentences
as an addition number sentence. Then use the number line to
find the missing number.

 

 

 

 

 

 

 

 

 

 

 

‘— e- 4— d- 4- 4— 4 4 4 4 4 '9
6 4 3 2 1 O l 2 3 4 6
(lo-'1) 1 Li 1 1 1 it 1 1 1 1 L 15 L
4- Q-
1 - 4 =
o— O- 4— 4-' 4- a- 4 4 4 4 4 4
6 5 4 2 l O 1 2 3 4 5
(10‘2) L3 1 L 1 1 4 1 L L FL
4- 4
1 - 3 =
4- 4- 4- .a- t— +- 4» 4 4 4 4 4
2 l 2 3 4
(lo-3) ? .5 ‘3 i . .1 9 1 . L r j ?
€- 4‘ V
5 - _fl_ = 3

 

 

 

85

86

PROBLEM SET 3 FOR INDIVIDUAL WORK

Rewrite each of the following subtraction number sentences
as an addition number sentence.

 

 

 

 

 

 

 

 

 

+~ 6a 4»
(9‘1) 4 s" 3 r: l
4 4»
(9-2) 4 - 2 ==
4- ‘4
(9—3) 1 — = 3
4 o-
(9—4) .. 4 = l

 

 

 

Rewrite each of the following subtraction number sentences
as an addition number sentence. Then use the number line to
find the missing number. ‘

 

 

 

 

 

 

 

 

 

 

4- +- +- 4», ". 4— 4 4 4 4 4 4
(10-1) 3 ? ‘3 3. i" i ‘2 i f f i f’ ‘?
4 db -
3 - l =
4- 4- 4- 4- 4- 4- 4 4 4 4 4 4
(lo-2) f” 5’ ‘3 3 i } 9 i i 1” f}. 5’ ‘3
4 1+
2 - 2 =
. 4- 4- on t— 4- l— 4 4 4 4 4 4
'(10-3) 3 ? ‘1' 3? I" } ‘3 i .2 i f} .5 ?

 

 

N?

 

 

 

 

87

44

*3

 

 

(]O~4)

 

 

 

88

PROBLEM SET 4 FOR GROUP WORK

Place a direction number in each of the following spaces to
complete the number sentence.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 4
(ll—l) 2 X 4 =
4 4
(ll—2) 3 x = 6
4 4
(ll—3) x 2 = 8
.9
(12-1) 4 x O =
4
(12-2) 5 x = O
4 4 4 --’ 4. 4
(13-1) 3 x (2 + 4) = x 2 + x 4
4 4 4 .. - 4———. _,
(13—2) 4 x (5 + l) = '4 x + 4 x
4 4 4 4 4 4
(13-3) 2 x ( x 3) = 2 x 4 + 2 x 3
4 4 4 4 4 4
(13-4) x (4 + 2) = 3 x 4 + 3 x 2
4 4 4

 

 

 

 

 

 

 

 

(13-5) x (3 + ) = 2 x + x 5

 

 

 

 

89

PROBLEM SET 4 FOR INDIVIDUAL WORK

Place a direction number in each of the following spaces to
complete the number sentences.

 

4 4
(ll—l) 2 X 3 =

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 4
(ll-2) 2 X = 4
4 4

(ll-3) X 3 = 9

4
(12-1) 5 X 0 =

.p
(12-2) 3 x = o

4 4 4 4 4
(13-1) 2 X (3 + 4) = ( X 3) + (_ X 4)

 

 

 

 

 

 

 

 

 

 

4 4 4 4 4 ‘
(13-2) 3 X (2 + 4) = (3 X }) + (3 XE)

 

 

 

 

 

4 4 4 4 4 4

(1.3-3) 4 X ( ___+ 3) == (4 X 5) + (4 X 3)
4 4 4 4 4 4

(1344) ‘_ X (3 + 5) == (2 x 3) + (2 X S)

 

 

 

 

 

 

 

 

 

 

 

r“ * '— »
(1345) x (4 *4...” .. (2 x__j) + (. x 3)

 

PROBLEM SET

90

5 FOR GROUP WORK

Place a direction number in each of the following spaces to
complete the number sentences.

(14-1)

(14-2)

uh;

q—x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What is

4 4
= 2 I 3 +
= +

r
= 7 +
r___.,

= +
What is

4 4
= 4 X 2 +
= -_7 +
= ‘F—fll +
= +

 

 

 

k)$ «at

N4

J44 b} 434

4:4;

 

fix

1n, 61? 63’

L41

 

 

 

 

:44 :44

N!

«A

Work each of the following problems using the rule you have
discovered.

(15-1)

.9
l

X

‘-
4

 

 

 

 

(15—2)

(15-3)

“3*

n)?

 

 

 

 

 

91

92

PROBLEM SET 5 FOR INDIVIDUAL WORK

Place a direction number in each of the following spaces to
complete the number sentence.

 

 

4 4 (r 4 4 4 .-
(14—1) 2 X (l 11- l.) = 2 X l + 2 X l
4 l 4 v-
2 = + 2 X l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
.__J =
F_1_

 

 

 

 

 

 

 

 

 

Work each of the following problems using the rule you have
discovered.

 

 

 

 

 

 

 

 

 

4 4-
(15-1) 3 X 3 =

4 4. ———‘
(15—2) 3 X l =

4 <-
(15-3) 3 X 4 =

4 t- -_—
(15-4) 2 X 5 =

 

 

 

‘93

PROBLEM SET 6 FOR GROUP WORK

Place a direction number in each of the following spaces to

complete the number sentence.

 

 

 

 

 

 

 

 

(16-1)
1- 4 *—
3 X (2 + 2)
Q- LI
3 X
4- 4 «e
(16-2) 2 X (2 + 2)

 

 

 

5  
_L

 

 

 

of

N9

 

 

 

 

 

 

 

 

 

 

 

1
33 +
_L.

What is
.9
x 2 +
+
+
+
4
2 +

 

 

 

 

 

 

 

 

«.34 of

of

of

to" to?

to?

 

X

N?

‘—
X 2
‘.
X 2
.-
X 2
.-
X 2
g.
X 2
X
+
2

 

 

WOrk each of the following problems using the rule you have

 

 

 

 

 

discovered.
«- 4-
(17-1) 2 X 4 =
1r (-
(14-2) 3 X l = ' l

 

 

94

.
1
. ~
its-Gilli. 2:.

(17-3)

95

PROBLEM SET 6 FOR INDIVIDUAL WORK

Place a direction number in each of the following spaces to
complete the number sentences.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

é- '6
(l6-l) What is 2 X 4
¢- 4 44 4— 4 o- 4+
2 X (4 1 4) = 2 X 4 + 2 X 4
4— i '_—l—’ 4- 4b
2 X = J + X 4
l .

o- 4-
= + 2 X 4

= +

 

 

 

 

 

 

 

 

 

Work each of the following problems using the rule you have
discovered.

 

 

 

 

 

 

 

 

 

 

 

 

 

4— 4-
(l7-l) 3 X 3 =
4. 4-
(l7—2) 2 X 3 =
4— 4-
(l7—3) 4 X 2 =
fl- ‘-
(l7-4) l X 4 =

 

 

 

96

Use the number line to find the missing direction number in
each number sentence.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4- t— t- +- e— e- 4 4 4 4 4 4
4 3 2 2
(5-1) ? ? . . i 9 ,i . ti ? ? §
4 4
3 4 4 =
+- +- G— t- t— Q" 4 4 4 4 4 4
6 2
(6-1) . 343.19}: 315;?
4 +- "—W’
3 + l =
+- +- +- +- Q- 4- 4 4 4 4 4 4
6 5 4 3 2 l 2 4
(7-1)11111_10L]1-1?1??
4 (-
3 + 3 =
Q- 4— 4— 4- e- 4- 4 4 4 4 4 4
6 5 4 3 2 l O l 2 3 4 5 6
(8-1)l£LIIJI!III¥I
4 -
2 + = 3

 

 

 

Rewrite each of the following subtraction number sentences as
an addition sentence. Then use the number line to find the
missing number.

 

 

 

 

 

4F 4- 44 ¢- 4— ‘fi- 4 4 4 4 4 4
(10-1) ? ? f ? 3 i 9 i ? 3 ? E i
4 4-
3 - 2 =
0— 4- 4— 4- +- ‘+- 4 4 4 4 4 4
(10-4.6243; .1 9 £3 3 :1 2:

 

Z _ "_—T =

u)?

 

 

 

97

Write the missing direction number in each of the following
spaces to complete the number sentence.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 4
(ll—l) 2 >< 3 =
4 4
(ll-2) 2 X = 4
4
(12-1) 5 X 0 =
4— 4
(15—3) 3 )C 2 =
4— 4-
(17-1) 2 X 4 =

 

 

 

APPENDIX B
PLANNED COMPARISONS COMPUTATIONAL PROCEDURES

The data,

tions by grade,

comparison analysis.

space code.

The first space identified the grade (4: fourth.

teaching method,

sex,

and I.Q.

for each task, were pooled into 24 classifica-
for the planned

Each group was identified by a four

5: fifth, and C: sixth), the second space identified the

teaching method (D: discovery and I: instructional), the

third space identified the sex (M: male and F: female), and

the fourth space identified I.Q. level (H: high I.Q. and

L: low I.Q.).
jects who were fifth graders, taught by the discovery method,
boys, and with low I.Q. measures.

cedures described in this section, the groups are identified

by subscripts,

i . 10.00’24.

In the computational pro-

Thus the code SDML would identify those sub?

The identification codes

and their corresponding subscripts are found in Table 3.1.

Table 3.1

SUBSCRIPT CODING FOR IDENTIFICATION OF CLASSIFICATION GROUPS

USED IN PLANNED COMPARISON COMPUTATIONAL PROCEDURES

 

r.

 

1 Code i Code 1 Code
1 4DMH 9 SDMH 17 6DMH
2 4DML 10 SDML 18 6DML
3 4DFH ll SDFH 19 6DFH
4 4DFL 12 SDFL 20 6DFL
5 41MB 13 51MB 21 GIMH
6 4IML 14 SIML 22 GIML
7 4IFH 15 SIFH 23 GIFH
8 4IFL 16 SIFL 24 6IFL

 

98

99
For each group i, the mean (mi), variance (Si) and
group size (Ni) was calculated. These statistics were
then used to compute the following values used in the analy4

sis of variance tables.

Thg_degrees of freedom for within mean squares
24
(1) DF=[Z Ni]- 24
i=1
The overall mean
24
(2) M= 2 Ni mi
i=1

The between sum of squares

 

 

24
2
(3) SB 8 Z ”1““1 - M)
i=1
The between mean sguare
SB
(4) MB '55-
The within sum of sguares
24
2
(5) SW - 2 (Ni - 1) s1

i-l

The within mean_§guare

(6) MW

The Feratio of mean square

(7) F

Sample size and overall mean

(8) N1

(9) M1

Sample size and overall mean

100

SW
DF

élé

for grade four

 

 

16
(10) N2 = 2 Ni
i=9
16 “1‘“1
(11) M2 . 2 N2
i=9

§ample size and overall mean

(12) N3 3

for_grade siX

24

‘5.

101

N

 

24
imi
(13) M3 = 2 N3
1:17

Sample size and overall mean for discovery group

fi.

(14) N4 = 2 Ni
i

N.m.

(15) M4 = 73—44
i

i = 1,2,3,4,9,10,11,12,17,18,19,20

Sample size and overall mean for instruction group

(16) N5 = 2 Ni
i
N.m
an ms :—
i

i = 5,6,7,8,13,14,15,16,21,22,23,24

§ample size and overall mean for boys

(18) N6 = 2N1
i
N m.
i

i = l,2,5,6,9,10,l3,14,17,18,21,22

102

Sample size and overall mean for girls

(20) N7 = EN1
1

 

Nimi
(21) M7 =2 N7
1

i = 3,4,7,8,ll,12,15,16,19,20,23,24

Sample size and overall mean for high I.Q. subjects

12
(22) N8 = Z N2141
i=1
12
N . m
21-1 21-1
(23’ M8 = Z ““1??—
i=1

Sample size and overall mean for low I.QL

12
(24) N9 =- 2 N21
i=1

12

N .m .

21 21

(25’ M9 = Z “375—
i=1

Difference of means and t-value forgrade 4 - 5 comparison

(26) M45 = M1 - M2

103

M45

1 1
x/MW (N1 + N2)

 

(27) T45 =

 

Difference of means and t-value for grade 5 — 6 compapison

(28) M56 = M2 - M3

M56

1 l
JMW (r: + N3"

(29) T56 =

 

Difference of means and t-value for discovery-instructional

comparisons

(30) MDI = N4 - N5

MDI

JMW (311-: +413»

(31) TDI =

 

Difference of means and t-value for boy-girl comparisons

(32) MMF = M6 - M7

(33) TMF

 

..____J!!E______
l I
J“ (raw?)
Difference of meanpwand t-value for high-low 1,9, comparisons

(34) MHL I M8 - M9

MHL

1 1
(MW (N5 + as"

 

(35) THL I

 

 

104

Difference of means and t—value for instructional method
and boy-girl interactions

(36) M68 = MDI - MMF

(37) T68 =

 

Difference of means and t—value forvI.‘Q3 and bgy—girl inter-

actipns
(38) MSIQ = MMF - MHL
MSIQ- .

1 i 1 1
4““ (N3+N'7'+N§'+'N"9')

 

(39) TSIQ =

Difference of means and t-value for instructional method ang
I.Q. interactions

(40) MGIQ = MDI - MHL

(41) TGIQ = ”939 .

1 ' 12 1 1'
JMW<N4+16§+§§+§5>
Difference of means and t-value for_grade 4 - 5 and instruc-
pional method interactions a”

(42) M456 = M45 - MDF

M45G

1 1

(43) T456 = L
‘1 1
(MW (fi+N2+NZ+'N_5')

 

m??? WWW.M~ as.”

'l

I.

105

Difference of means and t—value for grade 5 — 6 and instruc-
tional method interactions

(44) M56G = M56 - MDI

(45) T56G = M56G

l l l 1
JW (N2+N3+N4+N—5')

Difference of means and t-value forggrade 4 - 5 and boy—girl
interactions

 

 

 

(46) M458 = M45 - MMF
(47) T458 = M455
1 1 71 1
(MW (N'I+N'2'+N'6+N7)

pifference of means and t-value for grade 5 - 6 and boy-girl
interactions

 

 

 

(48) M568 = M56 - MMF
(49) T56s = “565
‘1 1 1 1
«W (N2+'N§' raw?

Difference of means and t-value for grade 4 - 5 and 1,9,
'1— j‘
interactions

 

(50) M4510 = M45 - MHL

(51) T4510 - “4539——

1 1 1 l
(“w (NI+N'2'+N§+N§)

Difference of means and t-value for grade 5f- 6 and I.Q.

interactions

(52)

(53)

T5610 =

106

M56IQ = M56 - MHL

M56Ig

 

1 1 1
J?“ (N2+fi+'N§

1
+115)

pa

APPENDIX C

TABLES

Table C.l

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK l

 

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 .207 7.462 *
Grade 5—6 -.037 -l.360
Treatment .003 .150
Sex -.O32 -l.431
I.Q. .015 .671

Interactions
Group-Sex .035 1.119
I.Q.-Sex -.O47 -1.486
GrOup-I.Q. -0011 -0367
Grade(4,5)-Treatment .203 5.698 *
Grade(5,6)-Treatment -.041 -1.149
Grade(4,5)-SeX .239 6.695 *
Grade(4,5)-I.Qo .192 5.370 *
Grade(5,6)-I.Q. -.052 -l.478
Table C.2

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 2

 

Mean Difference t-value Significant

 

Comparison
Grade 4-5 —.169 - .679
Grade 5-6 -.053 - .213
Treatment -.843 -4.159 *
Sex —.348 -l.718
I.Q. 1.140 5.619 *
Interactions
Group—Sex -.495 -1.725
I.Q.-Sex -l.488 -5.188 *
Group-I.Q. -1.984 -6.914 *
Grade(4,5)-Treatment .674 2.094 *
Grade(5,6)-Treatment .790 2.461 *
Grade(4,5)-Sex .178 .555
Grade(5,6)-SeX .295 .919
Grade(4,5)-I.Q. —1.309 -4.069 *
Grade(5,6)-I.Q. —1.l93 -3.714 *

 

107

108

Table C.3

PLANNED COMPARISONS ANALYSIS CF VARIANCE FOR TASK 3

 

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 —.013 -.103
Grade 5-6 .195 1.473
Treatment -.685 -6.353 *
Sex -.023 -.213
I.Q. .259 2.399 *

Interactions
Group-Sex -.662 -4.341 *
I.Q.-Sex -.282 -1.847
Group-I.Q. -.944 -6.189 *
Grade(4,5)-Treatment .672 3.927 *
Grade(5,6)-Treatment .881 5.151 *
Grade(5,6)-Sex .218 1.278
Grade(4,5)-I.Q. -.272 -l.593
Grad8(5,6)—I.Q. -0063 -0369

Table C.4

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 4

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 -.307 —2.655 *
Grade 5-6 .257 2.211 *
Treatment -.563 -5.962 *
Sex -.156 -1.652
I.Q. .641 6.789 *

Interactions

Group-Sex -.407 -3.047 *
I.Q.-Sex -.797 -5.969 *
Group-I.Q. -1.204 -9.016 *
Grade(4,5)—Treatment .255 1.713

Grade(5,6)-Treatment .820 5.475 *
Grade(4,5)-Sex -.151 -1.012

Grade(5,6)-Sex .413 2.758 *
Grade(4,5)-I.Q. -.948 -6.351 *
Grade(5,6)-I.Q. -.384 -2.565 *

 

Table C.5

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 5

 

 

 

Comparison Mean Difference t~va1ue Significant
Grade 4-5 .238 3.539 *
Grade 5—6 -.113 -1.727
Treatment -.293 -5.487 *
Sex .055 1.046
I.Q. .252 4.723 *

Interactions

Group-Sex —.349 —4.619 *
I.Q.-Sex -.196 -2.598 *
Group-I.Q. -.545 -7.220 *
Grade(4,5)-Treatment .531 6.289 *
Grade(5,6)-Treatment .180 2.128 *
Grade(4,5)—Sex .182 2.155 *
Grade(5,6)-Sex -.169 -2.000 *
Grade(4,5)-I.Q. -.014 -.168

Grade(5,6)-I.Q. -.365 -4.324 *

Table C.6

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 6

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 -.069 -.992
Grade 5-6 -.036 -.522
Treatment -.151 -2.657 *
Sex .020 .355
I.Q. .411 7.198
Interactions
Group-Sex -.172 -2.129 *
I.Q.-Sex -.39O ~4.837 *
Group-I.Q. -.563 -6.969 *
Grade(4,5)—Treatment .082 .912
Grade(5,6)-Treatment .115 1.273
Grade(4,5)-Sex -.089 —.993
Grade(5,6)-Sex -.056 -.629
Grade(4,5)-I.Q. -.480 -5.321 *
Grade(5,6)-I.Q. -.447 -4.953 *

 

Table C.7

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 7

 

 

 

Comparison Mean Difference twvalue Significant
Grade 4-5 .014 .141
Grade 5-6 -.004 -.O40
Treatment -.382 ~4.417 *
Sex —.C65 -.755
I.Q. .535 6.183 *
Interactions
Grouanex -.317 -2.588 *
I.Q.-Sex -.600 -4.905 *
Grade(4,5)-Treatment .397 2.903 *
Grade(5,6)-Treatment .378 2.759 *
Grade (4'5) -56X .080 0587
Grade(5,6)~Sex .061 .445
Grade(4,5)-I.Q. —.520 —3.801 *
Grade(5,6)-I.Q. —.539 -3.938 *
Table C.8

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 8

 

v+

 

Comparison Mean Difference t-value Significant
Grade 4-5 -.210 -2.930 *
Grade 5-6 .060 .838
Treatment -.382 -6.524 *
Sex -.109 -1.862
I.Q. .540 9.225 *

Interactions
Group-Sex -.273 -3.295 *
I.Q.-Sex n.649 —7.839 *
Group-I.Q. -.922 -1l.136 *
Grade(4,5)-Treatment .172 1.857
Grade(5,6)-Treatment .442 4.772 *
Grade(4,5)-Sex «.101 -1.091
Grade(5,6)-Sex .169 1.826
Grade(4,5)-I.Q. -.750 ~8.104 *
Grade(5,6)-I.Q. —.480 -5.l79 *

 

PLANNED COMPARISONS ANALYSIS OF

111

VARIANCE FOR TASK 9

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 .155 .884
Grade 5~6 .581 3.306 *
reatment -1.l47 —8.017 *
Sex ~.158 -l.107
I.Q. .646 4.517 *
Interactions
Group—Sex -.989 -4.882 *
I.Q.-Sex —2805 -3.976 *
Group-I.Q. -1.794 -8.863 *
Grade(4,5)—Treatment 1.302 5.757 *
Grade(5,6)-Treatment 1.729 7.626 *
Grade(4,5)-Sex .313 1.386
Grade(5,6)-Sex .740 3.262 *
Grade(4,5)-I.Q. -.49l —2.l72 *
Grade(5,6)-I.Q. -.065 -.288

 

Table C.1O

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 10

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 .093 .453
Grade 5-6 -.594 -2.855 *
Treatment -.483 -2.855 *
Sex -.164 -.970
I.Q. .656 3.874 *
Interactions
Group-Sex -.318 -1.331
I.Q.-Sex -.820 -3.424 *
Group-I.Q. -l.139 -4.758
Grade(4,5)-Treatment .577 2.159 *
Grade(5,6)—Treatment -.111 -.414
Grade(4,5)-Sex .258 .965
Grade(5,6)-Sex -.430 —1.601
Grade(4,5)-I.Q. -.562 -2.103 *
Grade(536) .100. -10250 ”4.660 *

 

112

Table

C.11

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK ll

 

 

Comparison Mean Difference t~va1ue Significant
Grade 4~5 .057 1.274
Grade 5H6 -.100 -2.237 *
Treatment 0.000 -.015
Sex .010 .294
I.Q. .078 2.128 *
Interactions
Group-Sex —.011 -.219
I.Q.-Sex —.067 -l.296
Group-I.Q. —.078 -1.515
Grade(4,5)—Treatment .058 .999
Grade(5,6)-Treatment -.100 -1.722
Grade(4,5)-Sex .046 .804
Grade(5,6)-Sex «.111 -1.918
Grade(4,5)—I.Q. -.020 —.349
Grade(5,6)-I.Q. -.178 -3.078

 

Table C.12

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 12

 

 

Comparison Mean Difference t-value Significant
Grade 4—5 -.006 -.309
Grade 5-6 -.007 —.397
Treatment -.015 -.934
Sex -.005 —.331
I.Q. .012 .795

Interactions
Group-Sex —.009 -.426
I.Q.-Sex -.018 -.796
Group-I.Q. -.027 -1.222
Grade(4,5)—Treatment .008 .348
Grade(5,6)-Treatment .007 .282
Grade(4,5)-Sex 0.000 —.031
Grade(5,6)—Sex -.002 -.098
Grade(4,5)-I.Q. -.019 -.740
Grade(5,6)-I.Q. -.020 -.810

 

113

Tabie C.l3

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK l3

 

 

 

Comparison Mean Diffe-ence t-value Significant
Grade 4~5 -.051 -.174
Grade 5—6 -.095 -.326
Treatment «.777 —3.263 *
Sex ~.181 «.759
I.Q. .868 3.647 *
Interactions
Group-Sex -.596 —l.770
IoQo'SeX '10049 -30116 *
Group-I.Q. -l.646 -4.886 *
Grade(4,5)—Treatment .726 1.919
Grade(5,6)-Treatment .682 1.812
Grade(5,6)-Sex .085 .228
Grade (495) -IoQo -0919 -2043]. *
Grade(5,6)-I.Q. -.963 -2.559 *
Table C.l4

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 14

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 .250 5.603 *
Grade 5-6 -.212 -4.768 *
Treatment -.124 -3.425 *
Sex -.052 -l.433
I.Q. .180 4.973 *

Interactions
Group-Sex —.C72 —1.407
I.Q.-Sex ‘0232 “4.530 *
Group-I.Q. —.304 -5.939 *
Grade(4,5)-Treatment .374 6.509 *
Grade(5,6)-Treatment -.087 -1.527
Grade(4,5)-Sex .302 5.252 *
Grade(5,6)-Sex -.159 «2.786 *
Grade(4.S)-I.Q. .069 1.212
Grade(5,6)-I.Q. -.392 -6.839 *

 

114

Table C.15

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 15

 

 

Comparison Mean Difference t—value Significant
Grade 4-5 .659 2.984 *
Grade 5-6 -1.l43 -5.203 *
Treatment —.584 -3.257 *
Sex —.105 -.585
I.Q. .511 2.848 *

Interactions

Group—Sex -.479 “1.888

I.Q.-Sex -.616 -2.427 *
Group-I.Q. —1.095 -4.317 *
Grade(4,5)-Treatment 1.243 4.370 *
Grade(5,6)-Treatment —.559 -l.970 *
Grade(4,5)-Sex .764 2.685 *
Grade(5,6)-Sex -1.038 -3.659 *
Grade(4,5)-I.Q. .148 .520

Grade(5,6)-I.Q. -1.655 -5.831 *

 

Table C.16

PLANNED COMPARISONS ANALYSIS OF VARIANCE FOR TASK 16

 

 

Comparison Mean Difference t-value Significant
Grade 4-5 .024 .605
Grade 5-6 -.286 -6.974 *
Treatment -.023 -.701
Sex —.007 ~.212
I.Q. .120 3.598 *
Interactions
Group-Sex —.016 -.345
I.Q.-Sex -.127 ~2.694 *
Group-I.Q. -.144 ~3.040 *
Grade(4,5)-Treatment .048 .912
Grade(5,6)-Treatment -.263 -4.962 *
Grade(4,5)-Sex .031 .603
Grade(5,6)-Sex —.279 ~5.270 *
Grade(4.5)—I.Q. -.095 ~1.807
Grade(5,6)-I.Q. -.407 ~7.680 *

 

Tibia Co 1-7

PLANNED COMPARISCNS ANALYSIS OF VARIANCE FOR TASK l7

 

 

Comparison Mean Difference tmvalue Significant
Grade 415 .143 .472
Grade 5~6 ~1.049 -3.450 *
Treatment «.090 -.364
Sex -.131 -.528
I.Q. .567 2.287 *

Interactions

 

Group~Sex .040 °116
I.Q.~Sex -.698 -l.99l *
Group~I.Q. -.657 ~1.875 *
Grade(4,5)«Treatment .233 .596
Grade(5,6)-Treatment -.959 -2.444 *
Grade(4,5)-Sex .274 .700
Grade(5,6)-Sex n.918 —2.339 *
Grade(4,5)—I.Q. -.423 ~l.081
Grade(5,6)~I.Q. —1.616 —4.120 *
Table C.18

ANALYSIS OF VARIANCE FOR REPEATED MEASURES ON THE FACTOR
OF SEX IN THE FOURTH GRADE

 

Source SS df MS F

 

 

 

Between Subjects 117
Sex 1.737 1 1.737
Subjects w groups 4.735.512 116 40.823
Within Subjects 118
Tests 927.334 1 927.334 85.816*
Sex x Tests 11.698 1 11.698 1.082
Tests x Subjects
w groups 1.253.598 116 10.806

 

*Significant

116

Table C.19

ANALYSIS OF VARIANCE FOR REPEATED MEASURES ON THE FACTOR
OF SEX IN THE FIFTH GRADE

 

 

 

 

 

Source SS df MS F
Between Subjects 113
Subjects w groups 2.719.864 112 24.284
Within Subjects 114
Tests 1.294.603 1 1.294.603 63.585*
Sex x Tests 15.639 1 15.639 .768
Tests x Subjects
w groups 2.280.332 112 20.360
*Significant
Table C.20

ANALYSIS OF VARIANCE FOR REPEATED MEASURES ON THE FACTOR
OF SEX IN THE SIXTH GRADE

 

 

 

 

Source SS df MS F

Between Subjects 108

Sex 8.918 1 8.918

Subjects w groups 3.892.789 107 36.381
'Within Subjects 1 9

Tests 282.319 1 282.319 28.046*

Sex x Tests .429 1 .429 .042

Tests x Subjects

w groups 1.077.122 107 10.066

 

*Significant

117

Table C.2l

ANALYSIS OF VARIANCE FUR REPEATED MEASURES ON THE FACTOR
OF I.Q. IN THE FOURTH GRADE

 

 

 

 

 

Source SS df MS F

Between Subjects 117

I.Q. 3.005 1 3.005

Subjects w groups 3.125.745 116 26.946
Within Subjects _118

Tests 12.933.281 1 12.933.281 1.187.520*

I.Q. x Tests 1.608.911 1 1.608.911 147.728*

Tests x Subjects

w groups 1.263.445 116 10.891
*Significant
Table C.22

ANALYSIS OF VARIANCE FOR REPEATED MEASURES OF THE FACTOR
OF I.Q. IN THE FIFTH GRADE

 

 

 

 

Source SS df MS F

Between Subjects 111

I.Q. 238.588 1 238.588

Subjects w groups 2.428.495 110 2.428.495
Within Subjects 112

Tests 1.466.264 1 1.466.264 77.645*

I.Q. x Tests 55.424 1 55.424 2.934

Tests x Subjects

w groups 2.077.276 110 18.884

 

*Significant

118

Table 0.23

ANALYSIS CF VARIANCE FOR REPEATED MEASURES ON THE FACTOR
OF I.Q. IN THE SIXTH GRADE

 

Source SS df MS F

 

 

 

 

Between Subjects 106

I.Q. 50.730 1 50.730

Subjects w groups 319.130 105 3.039
Within Subjects 3221

Tests 305.079 1 305.079 30.615*

I.Q. x Tests 10.691 1 10.691 1.072

Tests x Subjects

w groups 1.041.370 105 9.965
*Significant
Table C.24

ANALYSIS OF VARIANCE FOR REPEATED MEASURES ON THE FACTOR
OF TEACHING METHOD IN THE FOURTH GRADE

 

 

 

 

 

Source SS df MS F

Between Subjects 11?

Method .579 1 .579

Subjects w groups 549.229 116 4.734
Within Subjects 118

Tests 924.670 1 924.670 81.146*

Method x Tests 10.019 1 10.019 .879

Tests x Subjects

w groups 1.322.271 116 11.395

 

*Significant

119

Table C.25

ANALYSIS OF VARIANCE FOR REEEATED MEASURES ON THE FACTOR
CF 'I‘L'ACHIL'G- MEEEH’I‘D IN THE FIFTH GRADE

 

.'-~-‘ - ‘ ‘ ‘ ‘- u~-‘.4—-....a‘—~> C-vai a“- .. “M n U “I A.”_.-.uou_-4 _ A -J- sans-Aan-mu

 

 

 

 

 

Source 53 df MS F

Between Subjects 13

Method 142.056 1 142.056

Subjects w groups 2.952.893 112 26.365
Within Subjects 114

Tests 1.274.801 1 1.274.801 8.108*

Method x Tests 107.468 1 107.468 .686

Tests x Subjects

w groups 17.453.118 112 155.831

*Significant

Table C.26

ANALYSIS OF VARIANCE FOR REPEATED MEASURES ON THE FACTOR
OF TEACHING METHOD IN THE SIXTH GRADE

 

 

 

Source SS df MS F
Between Subjects 108
Method 1.411.583 1 1.411.583
Subjects w groups 2.170.239 107 20.282

Within Subjects

 

Tests 331.275 1 331.275 49.748*
Method x Tests 161.141 1 161.141 24.198*
Tests x Subjects

w groups 712.516 107 6.659

 

*Significant

‘uummwmmwvwnm-V

 

120

Table C.27

ANALYSIS OF VARIANCE FOR.REPEATED MEASURE ON THE FACTOR
OF GRADE

 

—-u-. .3

 

 

 

 

 

 

 

 

Source SS di MS F

Between Subjects 340

Grade 428.407 2 214.203

Subjects w groups 1.178.281 338 33.071
Within Subjects .341

Tests 2.351.697 1 2.351.697 18.875*

Tests x Grade 155.226 2 77.613 .622

Tests x Subjects

w groups 4.221.340 338 124.595

 

*Significant

121

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$0.00 $0.00 $0.00H $0.00 $H.m0 $N.0m $€.H0 $mm

$0.00 $0.00 * $0.00H« $K400 « $3.00 $N.00 $&.H0 $00 0
$0.00 $0.00 $0.00H $0.00 $%.H0 $0.00 $&.H0 $00

$N.0H $N.00 $0.00 $0.00 $N.Hm $0.00 $H.00 $00

$0.HH $0.m0 « $M.H0 « $0.Nm e $0.0H $0.00 $0.00 $00 0
$0.00 $0.00 $0.00 $0.00 $H.m0 $0.00 $0.00 $00

$0.00 $0.00H $0.00H $0.00H $0.00H $0.00H $H.00 $00

$0.00 0 $0.00H« $0.00H0 $0.00H« .$¢.00 a $0.00He $M.N0 $00 0
$0. 00 $0.0 $040 $0.00 $0.00 $0.00 $0.00 $00

$0.00 $0.00H _$0.H0 $0.H0 $0.00 $0.00 $0.00 $00

$0.00 $0.00 a $0.H0 $0.m0 r $0.0H $m.N0 $0.00 $00 m
$0.00 $0.00 $0.H0 $0.00 $0.00 $H.mm $0.00 $00

00.00 $0.00H $0.00H $0.00H $0.00 $0.00 $M.H0 $mm

$0.0m $0.OQA0 $m.a0 « $0.Nm « $0.00 $0.00 $0.00 $00 N
$0.00 $0.00 $m.a0 $0.00 $8.0m $0.00 _$0.vm $00

$0.0a $0.00H $0.00H $0.00 $0.00 $0.00 $0.00 $00

.fim.mo $0.a0 0 $0.00 a $0.00 « $H.¢m .$¢.0¢ $0.00 $00 A
$0.00 $0.m0 $0.00 $0.00 $0.00 $0.00 $0.00 .$00

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