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thesis entitled

ACHIEVEMENT OF FIFTH, SIXTH, NINTH AND
TENTH GRADERS IN COORDINATE GEOMETRY

presented bg
Mary Catherine Gallick

has been accepted towards fulfillment
of the requirements for I

Ph , D . degree in Elementary
Education

   

Major professor

Date May 8, 1970

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ABSTRACT

ACHIEVEMENT OF FIFTH, SIXTH, NINTH AND
TENTH GRADERS IN COORDINATE GEOMETRY

By
Mary Catherine Gallick

This study was designed to analyze the achievement
of upper elementary and of secondary school students on a
comparable unit in coordinate geometry. One hundred sixty
pupils, seventy-seven fifth graders and eighty-three sixth
graders, comprised the elementary school sample. They were
pupils whose teachers had enrolled in a coordinate geometry
workshop and consented to teach a unit on this topic to
their classes. The two hundred thirty-eight secondary
school students represented a stratified random sample
with one first year algebra class drawn from each of the
five Junior high schools and five classes from the three
senior high schools of the Lansing metropolitan area.

A unit pertinent to linear equations and their graphs
was written in the form of a set of lesson plans. The con-
tent of this unit, suggested materials, and pedagogical
techniques for its development formed the basis for the
workshop for elementary school teachers. Secondary school
students studied coordinate geometry from their regular
algebra textbooks in which the content of one chapter was

comparable to this unit.

Mary Catherine Gallick

To appraise achievement, the Test on Coordinate
Geometry (TOCG) was developed from preliminary drafts
employed in two pilot classes. The following seven sub-
tests were embedded in the instrument to study achievement
on the specific component concepts of the unit:

1. Plotting and Recognizing Points in the
Coordinate Plane

Recognizing Members of a Truth Set

Intercept Relation to Open Sentence or Graph
Slope-Graph Relation

Operations with Signed Numbers

Graph-Open Sentence Relation

Extension of Concepts

flaunt-cum

The appropriateness of the coordinate geometry unit for
elementary school pupils was examined with respect to
achievement level. The Arithmetic Concepts subtest of the
Stanford Achievement Test, Intermediate II Battery, Form X
was administered to sixth graders. On the basis of the
median score, these pupils were dichotomized into two groups,
high and low, and their achievement compared with that of
ninth and tenth graders.

Data from the TOCG were analyzed by using a three-way
analysis of variance repeated measures design with propor—
tional subclass frequencies. The independent variables in
the design were (1) Grade, (2) Class, and (3) Subtest. The
design provided for an overall grades main effect and an
interaction between Grades and Subtest, the repeated meas-
ures dimension of the design. Where main effects or inter-
actions were significant, Scheffé's post hoc comparison was

computed to test specific hypotheses.

Mary Catherine Gallick

The following conclusions were drawn from the

analysis of the data. At the .05 level of confidence,

there is no difference in achievement on a unit in coordin-

ate geometry between

(1)

(2)

(3)

(14)

(5)

fifth graders and tenth graders.
sixth graders and tenth graders.

the upper half of the six graders as measured
by a general mathematics achievement test and
ninth graders.

the lower half of the sixth graders as measured
by a general mathematics achievement test and
tenth graders.

the upper half of the sixth graders as measured
by a general mathematics achievement test and
ninth and tenth graders combined.

At the .05 level of confidence, statistical tests

rejected the hypotheses that there were no differences in

achievement on a unit in coordinate geometry between

(1)
(2)
(3)

fifth graders and ninth graders.
sixth graders and ninth graders.

fifth and sixth graders combined and ninth and
tenth graders combined.

In the three preceding comparisons, the achievement of

secondary school students was higher than that of elemen-

tary school pupils.

When achievement in coordinate geometry was analyzed

by concepts, all the component tasks were understood as

readily by elementary school pupils as by secondary school

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Mary Catherine Gallick

Though not of primary concern to this study, two
additional hypotheses were analyzed. No significant dif-
ferences existed in achievement on a unit in coordinate
geometry between fifth graders and sixth graders.

Significant differences in achievement on the
coordinate geometry unit did exist between ninth graders
and tenth graders in favor of the ninth graders. Achieve—
ment of ninth graders was higher than all other classes
except the upper half of the sixth graders. This may be
accounted for by assignment procedures which placed
students with higher mathematics aptitude in Algebra I
while those with lower aptitude first completed a general
mathematics course. Thus, ninth graders represented a
select group.

A reactionaire assessed attitude toward coordinate
geometry. Approximately two-thirds of the elementary
school pupils preferred the unit in coordinate geometry
to their regular mathematics program. Only one-third of
the secondary school students liked the chapter in coordin-
ate geometry more than the algebraic topics in their
textbook. Similar proportions of each group rated coordin-

ate geometry more interesting.

ACHIEVEMENT OF FIFTH, SIXTH, NINTH AND

TENTH GRADERS IN COORDINATE GEOMETRY

By

Mary Catherine Gallick

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Elementary Education

1970

\ Copyright by
MARY CATHERINE GALLICK

1971

 

 

 

 

 

 

 

 

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ACKNOWLEDGMENTS

Many people contributed to the successful completion
of this study. I am most deeply indebted to my major
professor, Dr. W. Robert Houston, whose counsel has been
invaluable not only in encouraging and directing every
phase of the research for this study, but throughout my
entire doctoral program. His leadership and the high
standards to which he subscribes add more meaning to the
accomplishment of my goal.

Many others also contributed to the success of this
research project. I owe special thanks to Dr. Mary Alice
Burmester for her able assistance in developing the Test
on Coordinate Geometry, to Dr. Andrew Porter for his
assistance in planning the research design, and to Dr.
William Fitzgerald for his interest and suggestions. I
extend thanks to my committee members, Dr. John Wagner,
Dr. Calhoun Collier, and Dr. George Myers. Their diverse,
complementary talents were appreciated. I am also grateful
to the participating elementary and secondary school
teachers.

Finally, disregarding his protest, I must recognize
with deep gratitude my matchless husband, Harold Gallick,

whose helping hand was ubiquitous. Without his unselfish

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TABLE OF CONTENTS

ACKNOWLEDGMENTS. . . . .

LIST OF TABLES . . . . . . . . . .

LIST OF FIGURES. . . . . . . . .

LIST OF APPENDICES. . . . . . . .

Chapter

I. INTRODUCTION. . . . . . .

Background. . . . . . . .
The Problem . . . . .

II.

III.

IV.

Importance of the Study . . . .
Theoretical Foundation of the Study
Overview of Procedures. . . . .
Design of the Study. . . . .
Organization of the Study. . .

REVIEW OF THE LITERATURE. . . . .

Readiness . . . . . . .
Grade Placement . . . . .
Related Studies . . . . .
PROCEDURES . . . . . . .

The Study Sample. . .
Treatment . . . . . .
Pilot Classes. . .
Elementary Teacher In— Service
Education . . . . . .
The Test Instruments . .
Procedures of Testing . . .
Analysis of Data. . . . .

THE ANALYSIS OF THE DATA. . .

Testing of the Hypotheses.

Comparable Difficulty by Subtest and
Grade. . . . .

Student Reaction to Coordinate
Geometry. . . . . .

iv

Page
ii
vi

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V. CONCLUSIONS AND EDUCATIONAL IMPLICATIONS 112
Conclusions . . . . . . . . . ll3
Implications of the Study. . . . . ll8

BIBLIOGRAPHY. . . . . . . . . . . . . 126
APPENDICES . . . . . . . . . . . . . 132

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Table

8.

10.

LIST OF TABLES

Distribution of Pupil Achievement by Class
and Stanine on the Stanford Achievement
Arithmetic Computation Subtest , .

Distribution of Pupil Achievement by Class
and Stanine on the Stanford Achievement
Arithmetic Concepts Subtest . . . .

Distribution of Pupil Achievement by Class
and Stanine on the Stanford Achievement
Arithmetic Applications Subtest . .

Mean Stanine by Grade in Arithmetic Sub-
tests on the Stanford Achievement Test.

Number of Years Teaching Experience and
Mathematics Preparation of Elementary
School Teachers . . . . .

Number Years Teaching Experience and
Mathematics Preparation of Secondary
School Teachers . . . . . . . .

Correspondence of Elementary and Secondary
Units in Coordinate Geometry .

Achievement of First Pilot Class by
Quartiles and Reading and Arithmetic
Subtests as Measured by the Stanford
Achievement Test, Intermediate II,
Battery X . . . . . . . .

Achievement of Second Pilot Class by
Quartiles and Reading and Arithmetic
Subtests as Measured by the Stanford
Achievement Test, Intermediate II,
Battery X . . . . . . . .

Index of Difficulty and Index of Discrim—
ination as Computed from Second Pilot
Class for the Forty Items Included in
the Test on Coordinate Geometry

Page

37

38

38

39

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54

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Table Page

11. Means and Standard Deviations on Both
Forms of the Test on Coordinate
Geometry Administered to Non-
Experimental Tenth Graders Two Days

Apart. . . . . . . . . . . . 71
12. Mean Scores by Subtest on the Test on

Coordinate Geometry for Students in

Grades 5, 6, 9 and 10 . . . . . . 77
13. Analysis of Variance Summary . . . . . 78
IA. Mean Score and Standard Deviation by

Grades for Subtest l: Plotting and
Recognizing Points in the Coordinate
Plane. 0 o o o o o o o o o o 93

15. Index of Difficulty by Item for Subtest l:
Plotting and Recognizing Points in the
Coordinate Plane . . . . . . . . 93

16. Mean Score and Standard Deviation by
Grades for Subtest 2: Recognizing

Members of a Truth Set . . . . . . 95
17. Index of Difficulty by Item for Subtest 2:

Recognizing Members of a Truth Set . . 95
I8. Mean Score and Standard Deviation by

Grades for Subtest 3: Intercept

Relation to Open Sentence or Graph . . 96
19. Index of Difficulty by Item for Subtest 3:

Intercept Relation to Open Sentence or

Graph. . . . . . . . . . . . 97
20. Mean Score and Standard Deviation by

Grades for Subtest A: Slope-Graph

Relation. . . . . . . . . 98
21. Index of Difficulty by Item for Subtest A:

Slope—Graph Relation. . . . . . . 98
22. Mean Score and Standard Deviation by

Grades for Subtest 5: Operations with

Signed Numbers. . . . . . . . . 100
23. Index of Difficulty by Item for Subtest 5:

Operations with Signed Numbers . . . lOO

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25.

26.

27.

28.

29.
30.

Mean Score and Standard Deviation by
Grades for Subtest 6: Graph—Open
Sentence Relation. . . . . . .

Index of Difficulty by Item for Subtest
Graph-Open Sentence Relation . .

Mean Score and Standard Deviation by
Grades for Subtest 7: Extension of
Concepts. . . . . . . . . .

Index of Difficulty by Item for Subtest
Extension of Concepts . . . . .

Mean Score and Standard Deviation by
Grades for Total Test on Coordinate
Geometry. . . . .

Student Reaction to Coordinate Geometry

Student Interest in Coordinate Geometry

viii

Page

102

102

10“

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107

109

110

LIST OF FIGURES

Figure Page
I. A Three-way Analysis of Variance
Repeated Measures Design with

Proportional Subclass Frequencies . . l6

2. Pegboard Divided Into Quadrants. . . . A7

ix

Appendices

A.

B.

LIST OF APPENDICES

Teacher Lesson Plans for Elementary
School Unit on Coordinate Geometry.

Pupil Exercises for Elementary School
Unit on Coordinate Geometry . .

Synopsis of Pilot Classes for
Elementary School Unit on
Coordinate Geometry. . . . .

Test on Coordinate Geometry (TOGO)

Page

133

160

182

202

CHAPTER I

INTRODUCTION

Background

 

Today's mathematics curriculum reflects current
understanding of learning theory, instruction, and
mathematics. Major changes in the elementary school
mathematics program occurred during the 1960's as a
result of extensive efforts by mathematicians, educa—
tors, and psychologists, and followed similar endeavors
directed toward improving the secondary school program.

Examination of the new programs revealed at least
two trends: (1) introduction of mathematical tOpics
previously not taught in the traditional curriculum,
and (2) introduction of topics at earlier levels in the
mathematics program. The mathematics program continues
to change, influenced by the work of Edith Biggs, Bert
Kaufman, Robert Davis, Jean Piaget and others, and
by projects such as The Madison Project, the Nuffield
Project, and the School Mathematics Study Group. In a
dynamic, changing society, the mathematics content and
instructional procedures must evolve continually.

One vision of a rigorous program for the future
was provided by twenty-nine mathematicians, scientists,

l

psychologists, and mathematics educators who met for the
Cambridge Conference on School Mathematics (CCSM).l In
this recommended program, the high school student would
study the equivalent of three years college mathematics
in today's curriculum. Junior high school students would
study algebra and geometry while elementary pupils would
encounter such topics as groups, matrices, logic, and
geometry,2 all of which represent radical changes from
present programs.

Prior to recent endeavors to revise the mathematics
program, little geometry was included in the elementary
school curriculum. In fifth and sixth grades, pupils
formed a nodding acquaintance with perimeters and areas
of familiar polygons, chiefly squares and rectangles.

An analysis of contemporary material, however,
revealed an increased emphasis on geometry at the
elementary school level. In the primary grades, many
informal activities develop such concepts as congruence,
symmetry, and topics from non—metric geometry. One
fourth grade textbook introduces measurement concepts of

length, perimeter, volume, and surface area.3 Other

 

1Educational Services Incorporated, Goals for School

 

Mathematics, The Report of the Cambridge Conference on
School Mathematics (Boston: Houghton Mifflin Co., 1963),
p. 6.

 

2Ibid., pp. 3l-67.

3Robert E. Eicholz, et al., Elementary School
Mathematics, Book A (Reading, Mass.: Addison-Wesley
Publishing Co., 196A), pp. 1—25.

 

 

geometric topics in the upper elementary school cur—
riculum include parallelism, parallelograms, polygons
and diagonals, right triangles, triangular pyramids,
circles, and central and inscribed angles. These topics,
many of which did not appear in traditional textbooks
until the tenth grade, are evidence that many branches
of geometry have become an established part of the
elementary school mathematics program. They represent
simultaneously new topics for the elementary school
level and an earlier introduction of topics once
delegated exclusively to higher levels.

Accompanying the innovations by curriculum
revisers were the pleas of educators and researchers for
criticism and evaluation of new content and its grade
placement.“ Research by Piaget and others substantiated
that children develop through growth stages. To what
extent do these stages influence the age placement of
topics in the elementary school? On the other hand,

Bruner, in an often quoted thesis, hypothesized that any

 

“Carl B. Allendoerfer, "The Dilemma in Geometry,"
The Mathematics Teacher, LII (March, 1969), 165; Kenneth
E. Brown and John J. Kinsella, Analysis of Research in
the Teaching of Mathematics: 1957-1958, Bulletin 1960,
#8 (Washington, D. C.: United States Government Printing
Office, 1960), pp. 23-26; J. Fred Weaver, "Non—metric
Geometry and Mathematics Preparation of Elementary School
Teachers," American Mathematics Monthly, LXXIII
(December, l966), 1115-1121; and Otto C. Bassler,
Research Workers in Mathematics Education," American
Mathematics Monthly, LXXIV (September, 1967), 859.

 

 

 

 

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subject can be taught effectively in some intellectually
honest form to any child at any stage of deveIOpment.5
With the introduction of new ideas in elementary school
curriculum, research which studied pupil learning at
various stages of development was important. One of the
topics introduced into the elementary school curriculum
was coordinate geometry.

The present investigation studied the achievement
of fifth and sixth grade pupils on a unit in coordinate
geometry, a tOpic heretofore generally not taught at
this level. How did their achievement on this unit

compare with that of ninth and tenth graders to whom

the topic had been taught traditionally?

The Problem

 

The purpose of this study was to examine and
evaluate the achievement ofeflementary and secondary
students on a comparable unit in coordinate geometry.
Which concepts, if any, could elementary school pupils
understand as well as secondary students? Could all
elementary school pupils learn the concepts or should
the topic be treated as enrichment material for more
able pupils?

Specifically, the study investigated the following

hypotheses:

 

5Jerome S. Bruner, The Process of Education
(Cambridge: Harvard University Press, 1962), p. 33.

 

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

There is no difference in achievement
on a unit in coordinate geometry between
fifth graders and ninth graders.

There is no difference in achievement on a
unit in coordinate geometry between fifth
graders and tenth graders.

There is no difference in achievement on a
unit in coordinate geometry between sixth
graders and ninth graders.

There is no difference in achievement on a
unit in coordinate geometry between sixth
graders and tenth graders.

There is no difference in achievement on a
unit in coordinate geometry between fifth
and sixth graders combined and ninth and
tenth graders combined.

There is no difference in achievement on a
unit in coordinate geometry between the
upper half of the sixth grade students as
measured by a general mathematics achieve-
ment test and ninth graders.

There is no difference in achievement on a
unit in coordinate geometry between the
lower half of the sixth grade class as
measured by a general mathematics achieve—
ment test and tenth graders.

There is no difference in achievement on a
unit in coordinate geometry between the
upper half of the sixth grade as measured
by a general mathematics achievement test
and ninth and tenth graders combined.

To answer the pertinent questions related to

distinct concepts in coordinate geometry, achievement was

further analyzed by the seven subtests listed below.

For each subtest, Hypotheses A through H were tested for

significant differences.

Subtest

Subtest

Subtest

Subtest

Subtest

Subtest

Subtest

NQU'I

Plotting and Recognizing Points
in the Coordinate Plane

Recognizing Members of a Truth Set

Intercept Relation to Open Sentence
or Graph

Slope—Graph Relation
Operations with Signed Numbers
Graph-Open Sentence Relation

Extension of Concepts

Importance of the Study

 

Dictated by tradition, concepts of coordinate

geometry related to graphing linear equations have been

reserved for first year algebra courses. However,

experimental programs such as the Madison Project,6 for

example, have introduced certain concepts of coordinate

geometry to fifth and sixth grade pupils. Since such

experimental projects are frequently the harbingers of

the content in future commercial textbooks, an analysis

of the achievement of fifth and sixth grade pupils on a

unit in coordinate geometry should provide information

helpful in determining if the topic, or a portion of it,

may be appropriate for these levels.

In this research project, a unit in coordinate

SEOmetry was taught to fifth and sixth graders and their

»

6

Robert B. Davis, Discovery in Mathematics
(Reading, Mass.:

 

Addison-Wesley Publishing Co., 196A),

Chapters 10, ll, 17 and 18.

achievement was compared with that of first year algebra
students. Such studies are important prerequisites to
more general dissemination of new content in the mathe-
matics curriculum.

Another aspect of this study relates to the quali-
fications of teachers. Coordinate geometry, as a part
of the first year algebra course, is taught by teachers
who have majors or minors in mathematics. Theoretically,
they possess the capability to handle these and more
advanced concepts. On the other hand, fifth and sixth
grade teachers are elementary generalists who may
possess the most meager of mathematical backgrounds.
Thus, an obvious, essential question must be asked:

Are elementary teachers prepared to teach meaningfully
the increasing amount of geometry appearing in the
curriculum?

The fifth and sixth grade teachers in this study
participated in a six-week workshop in coordinate
geometry designed to teach content and to suggest
methodology for an activity—oriented, discovery-learning
program. Is such brief in—service training adequate to
meet the needs of the elementary teacher so that the new

topic can be taught effectively?

Theoretical Foundation of the Study
Hypotheses generated for this study stem from the

theoretical framework of curriculum building vis a vis

grade placement and the sequence of topics, and pupil

readiness. Both content and its sequence, perennial

problems in all fields of education, have been influenced

by pedagogical theories in vogue at a specific time.7

A brief review of these theories should provide insight

into the motivating forces behind current mathematics

curricular endeavors .

When the Faculty Theory was extant, content which

was difficult and purported to strengthen the faculties
of memory and reason comprised mathematics courses.
Little thought was given to selecting topics relevant to

the child's needs, of intrinsic interest, or which might

whet the child's mathematical appetite. A hierarchy of

difficulty determined the sequence of content instead of

child maturation.

The Child Psychology Theory, in contrast to the

harshness of the faculty school, greatly affected both

the content and sequence of arithmetic. The eXpressionist

wing of this movement favored inclusion of only those

;()pics which related to the child's interest and needs.

be essentialist wing favored the teaching of only those

>pics that were essential to the social utility of

alts . Since little need for arithmetic was seen for

early elementary child and his readiness to deal with

3
p

I:

7M. Vere Devault and Thomas Kriewall, Perspectives

(Elementary School Mathematics (Columbus, Ohio:
Lrles E. Merrill Publishing Co., 1969), pp. A1-7l.

UnemaUmmatics was questioned, the teaching of arith-

methzrmarly disappeared in many schools at the primary

leveltmtween 1930 and 1960.8 To the child psychologist,

thermnmal hygiene of the learner was the most important
fecun‘in choosing content, with maturation providing

tmm basis for sequence and social need the basis for

grade placement.
Subsequent theories had little impact on content

and grade placement and neither underwent much change

until the late 1950's. Thorndike's Stimulus-Response

Theory offered the notion of hierarchies as the basis

of sequence. Gestaltism exercised its greatest

influence on method, increasing emphasis on discovery-
criented teaching and sustaining the meaning theory of

instruction. While the major impact of the Neo—

behaviorists has been at the experimental and research

level, their emphasis on the use of programmed learning

materials and the statement of classroom goals as

behaviorial objectives influenced teachers and instructors

of teachers throughout the nation.

Massive curriculum reform projects of the 1960's

:tinunhated a renaissance of American interest in the work

>f J€%fl1 Piaget. Piaget, a Swiss zoologist and psycholo—

;ist, (nonducted experiments with children for more than
zalf aitcentury. He investigated verbal and conceptual

 

81bid.,§L 53.

 

10

:uxmctscfi‘a.child's thought, the organization of the
senmnwamotor schemata for assimilating intelligence,

and Unadevelopment of operations which give rise to

rummer and continuous quantity.9 Piaget theorized that

nannmaand nurture interact in a dual way. Environment
serves as nourishment for mental growth whose pattern

of development follows a course laid down by genes.

With respect to growth of abilities, Piaget believed
that nature provides the pattern and the time schedule
of its_unfolding, while nurture provides the nourishment
for the realization of this pattern. With respect to
the content of knowledge, the reverse is true; nurture
determines what is learned while nature provides the
requisite capacities.

According to Piaget, the development of intel-

lectual capacity proceeds through stages whose order is

constant but whose time intervals vary by individual and
society. These maturational stages in the thought

process determine readiness for cognitive learning.

ieadiness factors underlie any and all efforts to

 

9Jean Piaget, The Child's Conception of Number
'New York: Norton.and Co., 19627; and Jean Piaget,
Sarinfil Inhelder, and Alina Szeminska, The Child's Con-
;eptitni of Geometry (New York: Basic Books, 1960).

10David Elkind, "Piaget and Montessori," Harvard
gygcatitni Review, XXXVII (Fall, 1967), 538-5AA.

11

hmnpvetme school program. The notion that a child

nmvestfluough levels of development with a timing that
is difonflt to accelerate has focused attention on the
probhm10f how, when, and what to teach in mathematics.
fdagetksfindings raised questions about recent cur-

riculum reform, for various innovative groups (such as

SMSGznuiihe Madison Project) have accelerated intro-

duction of topics in the elementary school with apparent

disregard for children's cognitive development. The

successful teaching of these topics to younger children
forces the educator to consider the investigations of

J. S. Bruner and his efforts to construct an alternate

theory of cognitive growth.
Bruner, a professor of psychology at Harvard
University, also believed that intellectual development

moved through stages. Even though his theory reflects

the strong influence of Piaget, Bruner nevertheless
cautioned against rigid acceptance of the concept of

stages. Iiis previously quoted statement that any subject

:arttxa taught effectively to any child at any age in
mums intellectually honest form appears to contradict his
.cceptance of'Piaget's consideration for children's

eadirmnss for particular learnings on the basis of their

:revaJLLing stages of development. Some educators who

 

LllHaITW’S. Broudy, Othaniel B. Smith, and J. R.
irwuettea, "New Look at Readiness," Theory Into Practice,

[I (December 28, 1963), A2A-A29.

12

interpreted Bruner's statement literally, disagreed.l2

Ausubel termed it a "generalization that has wrought incal-

culable mischief in an entire generation of over-eager

O 1
curriculum reform workers." 3 Shulman viewed Bruner's

remark as a suggestion that the conception of readiness

should be modified to include not only the child but also

the subject matter.lu Research exists supporting both

Piaget's and Bruner's views;15 more will be required to

resolve the conflict.

In his research with elementary school children,

Robert Davis, Director of the Madison Project, practiced

tenets suscribed to by both Piaget and Bruner. Davis

believed, as did Piaget, that good pedagogy must involve
presenting the child with situations in which he eXperi—

Tents, manipulates objects and symbols, poses questions,

weeks answers and compares findings. Like Bruner, Davis

as a proponent of the discovery method. He was firmly

 

;

12David P. Ausubel, "Can Children Learn Anything That

iults Can—-and More efficiently?" ElementarySchool
Durnal, LXII (February, 1962), 270-271; and Alice Keliher,

£ditorial," Childhood Education, XLII (May, 1966), 527.

13David P. Ausubel, "Review of Toward a Theory of
struction, by Jerome S. Bruner," Harvard Education

”view, XXXVI (Fall, 1966), 338.

 

1“Lee S. Shulman, "Perspectives on the Psychology of
arning and the Teaching of Mathematics," in Improving
ghematics Education, ed. by W. Robert Houston (East
using: Michigan State University, 1967), pp. lA-l7.

 

15Ausubel, "Can Children Learn Anything That Adults
:——and More Efficiently?" o . cit.; and Arthur F. Coxford,
e Effects of Instruction on the Stage Placement of
ldren in Piaget's Seriation Experiments," The Arithmetic

Cher, XI (January, 196A), A-9.

 

13

cmndhcmithat the intellectual springs of children,
wheUnn=they were culturally disadvantaged, normal, or
gHTed,lmye not been tapped by our traditional arith—
rmmic mnmiculum.l6 Of similar conviction, the researcher
rmflelhxithe coordinate geometry unit utilized in the
curman;study on Davis' theory of instruction as he

statmiit in The Madison Project's Approach to a Theory

of Instruction:

The Madison Project attempted to bring the
elementary student in contact with mathematics
. and has chosen topics with the following
aims:
1. Students must be ready for ideas and take
an active role in developing them.
2. Concepts must rise naturally from some
problem solving situation.
3. The concepts must be related to some
fundamental mathematical ideas.
A. The concepts must lead to some significant
patterns of generality.
5. The topics must be appropriate to the age
of the child and must appear, in toto, to
observer to amount to a significant

experience.17

In teaching the pilot classes and conducting the
in—service workshop, the researcher used a "low-keyed

discovery approach"18 emulating Davis' technique.

 

16Robert B. Davis, "Mathematics for Younger
Hiildrwnr-—The Present Status of the Madison Project,"
Iew Ytudc State Mathematics Teachers Journal, X (April,

17 .
fhabert B. Davis, "The Madison Project's Approach
0 a Tfluasry of Instruction," Journal of Research in Science

gachirug, II (March, 196A), 1A8.

 

18DeVault, o . cit., p. 63.

1A

Overview of Procedures
A synopsis of the procedures used to test the

hypotheses of the study is presented below and described

in detail in Chapter III.

Unit for Pupils and Teachers

A unit in coordinate geometry, focusing on linear
equations and their graphs and traditionally a part of
first year algebra, was designed for the upper elementary
school level. Elementary school teachers taught the
unit. Because they had not studied the concepts involved

in coordinate geometry, a six—week workshop was organized.

In the workshop both content and methodological approach

were emphasized.

Sample
Pupils of the workshop participants comprised the

elementary sample. Their counterparts, students of

randomly selected algebra teachers, made up the

secondary sample.

Pilot Class and Instrument

The researcher piloted the coordinate geometry
unit with two elementary classes, a sixth grade and a
combirmni fourth—fifth grade. These trials indicated

that the unit would require about four weeks of classroom

instruction.

~. l n \k. VA
.. l4 . x “a 1» .l‘x rt ut~ .. ”rm. W«w ”A L. [K
. . l. t t v at. l, a p. . . s \f H
VP 1 Y. .I. v. FL. L C a a H , L
l we. a i y . . .L a J P. .. no .. ... , C
. I it T. .u \ ptr h.~.... V.

 

 

15

Pranminary forms of the instrument used to measure

pupfl.adflevement in coordinate gebmetry were administered

M)these;fllot classes. An item analysis of these pre—

limhunw forms provided the basis for selecting those

questhnm which were included in the final form of the

Test<n1Coordinate Geometry (TOCG). To study achieve—

ment<n15pecific concepts by grade level, the TOCG used
lilthiS study was subdivided into seven subtests, pre-

viously identified in the statement of the problem of

the study.
The TOCG was administered to elementary classes

upon completion of the unit and to first year algebra
students when they finished the chapter, "Graphs of

Linear Equations and Inequalities," in the regular

algebra text.l9

Design of the Study

A three-way analysis of variance repeated measures

design with proportional subclass frequencies was

employexi‘to analyze the data. The scheme of this design

is prwnyented.in.Figure 1. Independent variables included

knade, (llass, and Subtests, while the dependent variables

'ere EMHItest and Total Achievement on the TOCG.

 

'lgrknnovan A. Johnson, John J. Kinsella, and Herman
Algebra: Its Structure and Application (New

asenberg,
)rkw TTue Macmillan Co., 1967), Chapter 9.

I5

Pranminary forms of the instrument used to measure

pmfll adfievement in coordinate geometry were administered

M)these1filot classes. An item analysis of these pre-

limmmnw forms provided the basis for selecting those

questhnw which were included in the final form of the

TestcnuCoordinate Geometry (TOCG). To study achieve-

mentcnlspecific concepts by grade level, the TOCG used
irtthis study was subdivided into seven subtests, pre-

viously identified in the statement of the problem of

the study.
The TOCG was administered to elementary classes

Upon completion of the unit and to first year algebra
students when they finished the chapter, "Graphs of

Linear Equations and Inequalities," in the regular

algebra text.19

Design of the Study

A three-way analysis of variance repeated measures

design with preportional subclass frequencies was

employemi‘to analyze the data. The scheme of this design

is prennented.in Figure 1. Independent variables included

lrade, (Lhass, and Subtests, while the dependent variables

rere Ehflitest and Total Achievement on the TOCG.

 

ls9Donovan A. Johnson, John J. Kinsella, and Herman
osenberg, Algebra: Its Structure and Application (New

ordc: TTue Macmillan Co., 19677, Chapter 9.

l6

 

_ Subtests
Grade Class .

 

S S S

1 2 3 SA S S6 1

 

l
l
T
f

 

?

X l

 

 

I High

 

2 High

 

 

3 High

 

1 Low l

 

 

2 Low

 

3 Low

 

 

 

 

 

 

qr»..~...- .1 .. any... .

 

 

 

 

IO 3

 

N
1... -.- A.-. 1.1.-

 

-__..1,..

 

 

 

‘JI

 

 

 

 

 

J n

 

Iiig. .l.--A three-way analysis of variance repeated
.SLU?€S caesxign with proportional subclass frequencies.

l7

'Hidetermine the appropriateness of the content of

the mxndinate geometry unit for achievement levels of

elmmnmary school pupils, the Arithmetic Concepts subtest

oftmm Stanford Achievement Test, Intermediate II

lumtery, Form X, was administered to the pupils in the

sixth grade sample. They were dichotomized into groups,

high and low, on the basis of the median score of the

Concepts subtest.
The class mean was used as the experimental unit

rather than individual pupil scores. One of the assump-

tions of the analysis of variance is the independence of

experimental units. Were individual pupil scores

utilized in the study, this assumption could not be met
and the resulting statistics potentially invalid. The
assumption that pupils within a class have learned

geometry independently of each other cannot be made;

whereas, between—classes independence can be assumed.

Therefore, class means were computed and used as the

lependent variable in the study. While this procedure

wmduces the power which the increased number of indi-

dchurl scores would generate, the resulting statistics

effljurt greater precision of measurement.

1TH? analysis design outlined in Figure I provided
ests time an overall Grades main effect and an interaction

2tweeni<3rades and Subtest, the repeated measures dimen—

'«n1 of‘tfldis design. Where main effects or interactions

18

weresfigmificant, Scheffé's post hoc comparison was com-

putedTX)test the specific hypotheses of the study. The

.OSlevel<u‘confidence was selected as being suffi—

cienthyrdgorous for the purposes of this investigation.

Organization of the Study

Research studies related to the present investiga-

tion and upon which the generated hypotheses were based

are reviewed in Chapter II. Chapter III describes the

procedures in detail while the results of statistical

analyses are included in Chapter IV. Conclusions and

educational implications are the focus of Chapter V.

CHAPTER II

REVIEW OF THE LITERATURE

Readiness
Increasing demands of the knowledge explosion
stimulated curriculum reform in all the disciplines at

all levels of the school's program during the late

fifties and throughout the decade of the sixties. Since

the curriculum depends on the child's "readiness," this

concept assumed increased importance.

Cronbach defined a pupil's readiness for any situa—

tion as the sum of all the characteristics which make
him more likely to respond in one way than another.20

Whereas some earlier psychologists regarded readiness

as solely a function of physical maturation, l contemporary

psychologists and learning theorists believe that in addi—
tion to maturation, readiness is a function of the child's

needs, goals, learned ideas, and skills. Moreover,

readiness changes day by day as a result of experience,

 

20Lee J. Cronbach, Educational Psychology (2d ed.;
New Ytuflt: Harcourt, Brace and Co., 1963): p. 89.

21.Arnold L. Gesell, "The Ontogenesis of Infant
Behavicnc," in Manual of Child Psychology, ed. by Leonard
Carmichael (New York: John Wiley and Sons, 195A), pp.
355—356; and George E. Coghill, Anatomy and the Problem
9: Behavior (New York: Macmillan and Co., 1929), p. 113.

19

biological development, and development of the nervous

system. Changes in one facet can alter the whole system

of responses.

The basic idea involved in readiness is that an
optimum time exists for the initiation of new concepts.22
Different views of readiness evolved from the theories
of learning referred to in the previous chapter and
from research in child development.

In support of a curriculum designed to discipline
the mind, the faculty theorists totally disregarded child
readiness. Child psychologists used the lack of readi-
ness, which they viewed as a product of maturation, to
support the doctrine of educational postponement. In a
recent yearbook on Theories of Learning, McDonald
reviewed two general theories of learning: stimulus

(S-R) theories and cognitive insight theories.23

response

In general, S-R psychologists seldom considered
readiness for several reasons. Among them, Tyler cited
the following: (1) S-R theorists who investigated

learning sought materials devoid of association value; in

such a case questions about readiness were reduced to

 

'3
2“Wilbur H. Dutton, Evaluating Pupils' Understanding
of Arithmetic (Englewood Cliffs: Prentice-Hall, Inc.,

196“), p. 23.

23Frederick J. McDonald, "The Influence of Learning
Theories on Education," (190041950), Theories of Learning
and Instruction, Sixty-third Yearbook of the National

 

Society for the Study of Education, Part I (Chicago:
University of Chicago Press, 196“), pp. 1—26.

 

 

21

questhnm about learning sets, topics which they had not

.hwestnymed. (2) The concept of "mediation" used in

descruflions of rote verbal learning also implied some

nothn1of readiness because the learner's experiential

background affected the mediators he employed”

Recently, some neobehaviorists have become

interested in processes of cognitive learning. Gagné

and his associates conducted extensive research on the

acquisition of knowledge, in particular of mathematical

knowledge. Gagné believed that the attainment of any

entity of knowledge depended upon prior attainment of
relevant subordinate knowledge.25 To teach a particular
concept, Gagné first determined the desired terminal

behavior. He analyzed the knowledge necessary for its

accomplishment and then constructed a pyramid of pre-

requisites to prerequisites to the objective or terminal

behavior.
Gagné viewed readiness as a function of pre-

requisite knowledge. The child who could not attain the

stated goal simply did not possess the necessary prior

experience. Given those experiences, Gagné believed the

 

2“Frederick Tyler, "Readiness," Encyclopedia of
Educatitnnal Research (4th ed.; New York: The Macmillan

30., 1969), p. 1063.

25Robert M. Gagné, "The Acquisition of Knowledge,"
:syckKXhogical Review, LXIX (July, 1962), 355-365.

22

chfhicould accomplish the goal. He was unconcerned with

theckwelopment considerations of the cognitive psycholo—

26

gists.
Foremost among the cognitive approaches to develop—

nmnt are the theories of Jean Piaget. Piaget held that

mental development was an extension of biological growth.

DeveIOpment progressed through four concept stages.
The first is the sensory—motor stage lasting approximately

the first two years of life. The concept of permanence

of an object develops during this stage. During the

preoperational or second stage, from two to seven years

of age, language, symbolic function, and thought develop.

In the third or concrete operational stage (from seven to
eleven years of age), the child develops number concepts

in terms of a collection--substance, weight, and volume

which are always discovered in that order. In the

fourth, the formal operational stage (from twelve to
fifteen years of age), the child performs operations on

abstract ideas. Mathematics can be understood and

the child is capable of constructing chains of deduc—

tive reasoning. Four factors contribute to the develop-

ment of these stages: (1) maturation of the nervous

géshulman, op. cit., p. 27.

23

system, (2) encounters with experience, (3) social
transmission, and (A) equilibration or auto regulation.27

Piaget recognized that the time of appearance of
these stages varied from culture to culture, but he
accumulated a vast amount of data substantiating that
the sequence is unaltered. These stages aid in assessing
the readiness of a child to learn particular concepts.
Upon identification of the stage of intellectual develop-
ment attained by pupils, both content and method of pro-
viding related experience can be selected. Piaget's
theory redirected attention toward the use of concrete
objects to develop basic mathematical ideas. The use
of materials such as the Stern blocks, Cuisenaire rods,
and Dienes' multiphasic blocks in the classroom has
provided multisensory experiences for the child. Rather
than teaching the structure of the subject area, Piaget
favored providing the child with situations where he was
active and created the structure.

Piaget doubted that these stages could be accelerated.
in his address at a conference at Rochester, New York in

196A, he said that he asks investigators who have

succeeded in accelerating the Operations structure the

following three questions:

 

27Jean Piaget, "Development and Learning," Journal
of Research in Science Teaching, II (March, 1964), 176—

178.

2A

1) Is this learning lasting?

How much generalization is possible?

In the case of each learning experience, what
was the operational level of the subject before
the experience and what more complex structures
has this learning succeeded in achieving?2

Nevertheless, a growing body of research supports

the claim that young children can learn more difficult

concepts than presupposed.29 An outspoken proponent

of this tenet was Jerome Bruner who presented another

view of cognitive development. He theorized that all

cognitive activity was dependent upon the process of

categorizing.30 Although he subscribed to levels of
development (enactive, iconic, and symbolic, which
paralleled Piaget's pre-operational, concrete and formal

stages), he felt Piaget did not recognize the role of
environmental forces.

An articulate advocate of the discovery method,
Bruner believed that new concepts should be introduced

by means of concrete manipulative activities with ample

provision for self exploration. He emphasized the

 

28
Ibid., p. 183.
29Betty Estes, "Some Mathematical and Logical
Concepts in Children," Journal of Genetic Psychology,
LXXXVIII (June, 1956), 219-222; and Marilynn J. Adler,
"Some Educational Implications of the Theories of J.
Piaget and J. S. Bruner," Canada Education and Research

Digest, IV (December, 196”), 299.

 

3OJerome S. Bruner, Jacqueline Goodnow, and G. A.
Austen], A Study in Thinking (New York: John Wiley and

Sons, 1956),IL 236.

25

Hmnntance of a spiral curriculum in which the basic
:ummctural concepts of each subject are presented early

:nkireturned to again and again at higher cognitive

levels as the child develops.31

Readiness, for Bruner, could be modified to include

not only the child but also the subject matter. As the

child grows, the same subject matter must be presented

at a manipulative or enactive level, an iconic or
representational level, and finally at a symbolic level.32

Bruner argued that readiness is a function not so much
of maturation but rather of the intentions and skill of

the teacher to translate ideas into the language of

those being taught. He warned that these intentions

must be plain before teachers decide what can be pre—

sented to children of certain ages. "When we are clear

about what we want to do . . . , I feel reasonably sure
that we will be able to make rapid strides ahead in
"33

dealing with the pseudo—problem of readiness.

Davis' perception of readiness incorporated ideas

from both the S-R and cognitive insight schools of

thought. Like Gagné, he believed that meticulous

attention should be paid to prerequisites and that new

 

31Jerome S. Bruner, "On Learning Mathematics,"
EEMBIWathenmtics Teacher, LIII (December, 1960) 617.

 

32Shulman, op. cit., p. 31.
33Bruner, op. cit., p. 619.

‘26

(unwepts should be subdivided into their "atomic con—
stituents." But like Piaget and Bruner, he stressed
that the child should be involved in a variety of con-
crete eXperiences relevant to new learning.314
Portions of all the preceding views of readiness
are reflected in this study. The final task and all of
the prerequisite skills were stated as behavioral
objectives. Kinesthetic experiences were provided for
pupils because they were in the concrete or enactive
stage of development. By translating the ideas into
iconic terms, the investigator sought to determine
whether younger pupils could learn concepts tradition—
ally taught to older students, thus raising the problem

of grade or age placement, a problem which is inextricable

from readiness.

Grade Placement
The most comprehensive research on grade placement
was done by the Committee of Seven from 1924 to 1931.
Using data obtained from controlled eXperiments in 255
cities, the Committee attempted to determine the Optimal
stage in child development at which various processes
should be presented. When the Committee had ascertained

the approximate grade placement from a comparative

 

3”Robert B. Davis, "The Madison Project's Approach
to a Theory of Instruction," op. cit., p. 148. '

27

analysis of courses of study, it was taught at that

Evade, one grade lower, and one grade higher. Pre—

and posttests were administered. After an analysis of

test scores, the tOpic was placed at the grade level at
which three-fourths of the children achieved 80 per cent
accuracy.35 Their recommended placement of arithmetic
topics affected courses of study and textbook con—
struction throughout United States and Canada until
the current reappraisal in mathematics education. The

work of the Committee of Seven was followed by many

other studies dealing with the grade placement of
36

arithmetic topics.

Gibney and Houle pointed out that with the
increased attention to geometry in contemporary mathe-
matics texts, the topic_of geometric readiness has been

slighted.37 D'Augustine attested to the need for

 

35Carleton W. Washburne, "The Grade Placement of
Arithmetic Topics," Twenty-ninth Yearbook of The National
Society for the Study of Education, Part II (Chicago:
University of Chicago Press, 1930), pp. 641-670.

36Foster E. Grossnickle, "Experiment with One
Figure Divisor in Short and Long Division," Elementary
School Journal, XXXIV (March, 193A), “96-506; William A.
Brownell and Harold Moser, "Meaningful vs. Mechanical

Learning; A Study in Grade III Subtraction," Research
Monoggmufla No. 8 (Durham: Duke University Press, 1939);

euui WilliafllAu Brownell, "Arithmetic Readiness as a
fawuitical Classroom Concept," Elementary School Journal,

Jill (September, 1951), 15-22.

37Thomas C. Gibney and William W. Houle, "Geometry
Fkxuiiness in the Grades," The Arithmetic Teacher, XIV
(October, 1967), U71. ’

 

28

(Refinitive research to answer the questions: (1) At
what grade levels are certain topics learned with a high
degree of efficiency in terms of time and effort spent?
(2) Which geometric and topological topics are

appropriate at various levels to clarify and simplify

other mathematical concepts? (3) What factors relate to

student achievement with geometrical and topological
38

topics?

In the past, the only geometry included in the cur—

riculum was metric in nature. Mensuration, for example,
was the only geometric concept sequenced into the
mathematics program by the Committee of Seven. But
Piaget's work indicated that children learn topological
concepts first, projective concepts next, and demon-
strative concepts last——exactly the opposite order in
which geometric concepts are taught in American schools.

Piaget described the following three stages in a
child's conceptual construction of Euclidean space:

1. The child understands qualitative operations
of distance, length, area, volume, and congruence. Stage

one occurs when the child can distinguish between space

as a container with fixed sides and space occupied by

moving objects.

 

38Charles H. D'Augustine, "Factors Relating to
Achievement with Selected Topics in Geometry and Topology,"
ThmzArithmetic Teacher, XIII (March, 1966), 192-197.

20

a

2. The child understands simple material operations
of one, two and three dimensions, conStruction of
coordinate systems, measures of angles, and areas.

This stage depends on the ability of the child to apply
Operations of subdivision and change of position with
recognition that wholes are conserved by these Operations.

3. The child understands how to calculate areas and
volumes. This stage is attained when the child compre-

hends that an area or volume is unaffected by surrounding

39

space and conceives of space as a continuum.
Piaget's findings in relation to the sequence in

which children learn geometric concepts, along with the

need for research on the placement of geometric topics

cited in current literature, support an investigation

such as the present one. Several studies which have

examined the grade placement of geometric concepts will

be described in the following section.

Related Studies
In investigating questions related to the hypotheses
of this study, several researchers examined the teaching
of a geometric unit at different grade levels. One of
these studies was conducted by Corley who taught a one-
week unit on reasoning followed by a four—week unit on

demonstrative geometry to grades 6, 7, 8, 9 and 10.

 

39Piaget, Inhelder and Szeminska, pp. 389-“08,
passim.

3O

Him evaluative instrument was divided into three parts:

knowhxmfi of concepts, understanding of geometric proofs,
and ability to use the syllogism for reasoning. Analysis
of variance using the treatments by levels design“

tested the hypothesis that there was no difference in
the mean achievement of the grades on each part of the

geometry test as well as the total test. Corley's con—

clusions pertained to the comparative ability of pupils

grade-by-grade to perform on this test. He concluded

that sixth graders could learn simple geometric terms

and concepts. A marked improvement in ability to learn

methods of reaching conclusions and to understand
logical systems occurred in seventh grade. He did not

offer conclusions as to whether differences in the mean

achievement of pupils by grade did or did not exist.”0

Fitzgerald taught three units, one on numeration,
one on negative numbers and one on non—metric geometry
to fifth, seventh, and ninth graders. Only the results
of the latter unit are of direct relevance to this study.
Fitzgerald's primary interest was to study the degree to
which classes of different grade levels were alike or

different with respect to achievement on the experimental

 

qulyn J. Corley, "An Experiment in Teaching Logical
'Phinking and Demonstrative Geometry in Grades 6 through
10" (unpublished Ph.D. dissertation, George Peabody
Universityg 1959), Dissertation Abstracts, XX, p. 1375.

31

1ndts. He also investigated the relative effect of
various factors such as height, chronological age,
mental age, educational age and total experimental
success on the learning of mathematics. His treatment
of data was purely descriptive. Test scores for the
three grades were placed into the same distribution
which was separated into six intervals by one-half
standard deviations. The percentage of students from
each class in each interval was computed. The mean

achievement of the fifth graders on the geometry unit

slightly, but not significantly, higher than that of

VA
.v.

wa
the seventh or ninth graders.

Fitzgerald found that the highest 10 per cent of

the fifth grade learned an amount superior to that learned

by the bottom ten per cent of the ninth grade, the bottom

thirty per cent of the seventh grade was inferior to the

top thirty per cent of the fifth grade; and that the top

thirty per cent of the seventh grade was superior to the

bottom thirty per cent of the ninth grade. He concluded

that nonmetric geometric concepts were sufficiently easy

for most upper elementary students to understand and that

many elementary school age children could learn more

some of the basic ideas related to geometry than

they were generally provided the opportunity to learn.”1

about

 

ulWilliam Fitzgerald, "A Study of Some of the
Factors Related to the Learning of Mathematics by Children
in Grades 5, 7, and 9" (unpublished Ph.D. dissertation,

University of Michigan, 1962).

32

Other studies support Fitzgerald's conclusion about
theznnlity of elementary students to learn geometry.
Iknmerk and Kalin concluded that fifth grade pupils learn

a satisfactory amount of skills and concepts from the

Hawley and Suppes' workbooks dealing with constructions.

They found that the content of Book I was easy for the
pupils as indicated by a mean score which represented
78.6 per cent of the total possible on the criterion
test. In their opinion the class mean of 50 which
represented 56.8 per cent of the total possible on the
criterion test was not satisfactory in relation to the
amount of instruction time used.

On the basis of their study, Denmark and Kalin con-

with the opinion of national curriculum groups such

curred
as the Greater Cleveland Program and the School Mathe-
matics Study Group that more geometry could be taught

in elementary school than was then taught. The researchers

further speculated that most of Book I and some of Book II

could be learned by either younger or less intellectually

capable pupils than the children in the experimental class

was 126. Trial of these materials in an

’)
(-

whose mean I . Q.

average class would seem desirable even for speculation.

 

ugThomas Denmark and Robert Kalin, "Suitability of
lknuflding Geometry in Upper Elementary Grades: A Pilot
Study," The Arithmetic Teacher, XI (February, 196A),

73-80.

33

[flAugustine investigated the factors which relate

tozizmudent's achievement with geometrical and topological

topics. Fifth, sixth, and seventh grade pupils studied

a pmxyemmed text which he developed. The text presented

:nufiitopics as paths and their properties, simple closed

paths,;xflygons, and classification of polygons based on

nonmetric properties. An analysis of variance—covariance

performed on test scores and scores from standardized

achievement tests isolated reading and arithmetic as

significant factors in achievement with topics of topology

and geometry when taught with a programmed text. Mental

age and chronological age did not significantly affect

achievement. D'Augustine found no significant differences

in the mean levels of achievement on the criterion post-
test attributable to grade seven after the mean criterion

posttests had been adjusted for mental age, chronological

age, and reading and arithmetic achievement. D'Augustine,
who pooled test results into one score, pointed to the
need to investigate each independent subtopic.”3 The

current study investigated the subtopics involved in the

cuxnwiinate geometry unit, and compared the achievement of

{diufli.and sixth grade elementary pupils to that of ninth

and tefufli grade first year algebra students.

,—

LERD'Augustine, op. cit., pp. 192-197.

3A

lrlthe study most closely related to the present one,
:kni'Ali Shah conducted research to determine to what
exhnu;certain content in geometry was satisfactory for
cfiflldren within the age range seven to eleven years.
{Nuénmthematical content included the geometric concepts
of:

l. Plane figures with three to twelve sides.

2. Nets of figures (obtained by opening models

of cubes, rectangular blocks, etc.).
3. Symmetry of figures about a line.
A. Reflection of figures in a plane mirror.

Rotation of figures about a point.

L“

Translation.

6

7. Bending-stretching of figures with no cutting.

8 Networks.

Shah reported satisfaction with the results of the
tests on the various concepts listed above and felt that
the results suggested the kind of geometry which children
could study successfully. While the performance of the
seven-to-eight age group was sometimes low, the performance
by tine eight-to-eleven age group was on the whole satis-
Iflrctory and in some areas was high. He observed that
performance became better as the age of students increased.

However, his main concern was the reactions of the
CWTIldITNl toward the content discussed. "Thus, to estimate

the rmmnztions, we used (1) distributions of percentage of

35

candrkmes obtaining a given score interval, and (2) the
AA

meanscfl‘scores obtained by different age groups."
ihesechfia, which reflected achievement rather than

reacthnn were presented for each of the nine tests on

topohnfical concepts. The results described offer in

detailthe performances of each age group on each concept.

Efimme the data were not subjected to statistical analysis,

two questions remain unanswered: (1) Does the performance

of one age group differ from another on any or all of the

concepts? (2) Is the difference, if any, significant?

The present study examined the performance of
students of four different grade levels on various con-
cepts of coordinate geometry-—those pertinent to graphing

linear equations. The results of the criterion instru-

ment were subjected to analysis of variance to test for

possible differences between grade levels and to

Scheffé's post hoc test to determine if any existing

differences were significant, thus extending Shah's

;m4niy in the areas he considered as the most logical and

necessary'fEH'Iurther understanding of the curriculum and

childrwni's learning.

 

uuSair'Ali Shah, "Selected Geometric Concepts
kwight txa Children Ages Seven to Eleven," The Arithmetic

XVI (February, 1969), 122.

 

‘eacher,

CHAPTER III

PROCEDURES

The purposes of this study was to evaluate the
relative achievement of elementary and secondary students

on a comparable unit in coordinate geometry. This topic

was being incorporated into eXperimental courses in

elementary schools at the time of this study. The feasi-

bility of introducing this topic four to five years

earlier than students traditionally had studied it in

first year algebra was explored. Eight hypotheses were

posed and tested in the investigation.

The Study Sample

Elementary School Pupils

One hundred sixty of the three hundred ninety-eight

students who comprised the sample population were elementary

school pupils. Seventy-seven were fifth graders and

eighty-three were sixth graders. They were pupils whose

teachers enrolled in a coordinate geometry workshop and
(unmented to teach a unit on this topic to their classes.

Thmxzclasses, one fifth and two sixth grades, were from

schOthlocated in a low to lower-middle socioeconomic

36

'esicierniial area. Three classes, two fifth and one
:ixtni grwrde, were from schools which represented middle
LC)llpp€H‘ socioeconomic residential areas.

'Tables 1, 2 and 3 describe the mathematics
achiennmnent of the pupils in these classes as measured
by tfiu: Stanford Achievement Test, Intermediate II

Batttnqu Form X. Sixth grade data were from the test

 

battery adndnistered in October, 1968. Fifth grade data
were fkmnn the test battery administered in October, 1967,

when pupils were in fourth grade.

TABLE 1

DISTRIBUTION OF PUPIL ACHIEVEMENT BY CLASS AND STANINE
ON THE STANFORD ACHIEVEMENT ARITHMETIC
COMPUTATION SUBTEST

 

 

Number of Pupils by Stanine Class

Class Mean
1 2 3 A 5 6 7 8 9 Stanine

6—1 3 7 5 3 5 3 0 O 0 3.3L!
6-2 3 '2 8 9 2 l 1 O O 3.33
6—3 1 2 3 A 5 8 2 2 O A.92
5—1 9 2 o 3 6 3 5 2 0 5.148
5-2 1 l4 5 O '7 3 l. O O A.OO
5-3 2 8 2C 1 5 O O O O A.ll

 

38

TABLE 2

:XISTHRIPMJTION OF PUPIL ACHIEVEMENT BY CLASS AND STANINE
(3N THE STANFORD ACHIEVEMENT ARITHMETIC
CONCEPTS SUBTEST

-—a.- -

 

 

 

 

11333 Number of Pupils by Stanine giggs

1’ 2 3 A 5 6 7 8 9 Stanine
5—1 O 3 2 A 10 2 3 2 0 A.73
3—2 1 3 3 10 3 3 A 0 0 A.A8
6—3 0 l 7 9 5 l 2 2 O A.AA
5-1 1 1 o A 6 3 3 2 l 5.38
5-2 1 O 2 7 l 2 A 3 1 5.00
5-3 1 5 5 5 8 1 1 o o 3.81

TABLE 3

DISTRIBUTION OF PUPIL ACHIEVEMENT BY CLASS AND STANINE
ON THE STANFORD ACHIEVEMENT ARITHMETIC
APPLICATIONS SUBTEST

Number of Pupils by Stanine Class
Class Mean
1 2 3 A 5 6 7 8 9 Stanine

 

 

6—1 3 2 1 3 5 8 l 3 O A.8A
6-2 1 2 A 7 5 3 5 O O 1A.S6
6-3 0 A 2 6 7 A 3 l 0 14.67
5—1 0 O 3 O '7 7 l 1 2 5.66
5-2 3 7 2 9 2 2 3. O O 3.38
5-3 0 2 A A 2 2 2 2 2 14.85

 

39

The achievement scores of pupils in five of the six
classes were skewed to low average in all three subtests;
computations, concepts, and applications. In only one
class, 5-1, did more pupils score above than below stanine
five. More pupils scored above than below stanine five in
class 6-1 and in class 6-3 on the applications and computa-
tions subtests respectively. The class mean stanine fell
in the average range (fourth, fifth, and sixth stanines)
on the applications and concepts subtests, while one class
was below average in each of the subtests. Class 5—2
scored below average in the concepts subtests. The mean
stanine of two classes, 6-1 and 6-2, were below average
in the computation subtest, while the mean stanine of the
four remaining classes was in the average range.

The mean stanines by grade are found in Table A.
Both the fifth and sixth grades placed in the low average

range on all three subtests.

TABLE A

MEAN STANINE BY GRADE IN ARITHMETIC SUBTESTS
ON THE STANFORD ACHIEVEMENT TEST

 

 

Arithmetic Subtest

 

Grade
Computation Concepts Applications

A.53 A.76 A.63

U"!

6 3.86 A.55 A.69

 

 

A0

The mathematical background and teaching experience
of the teachers of these elementary classes are found in
Table 5. A

One of the elementary teachers who enrolled in the
workshop was a beginning teacher. The remaining partici-
pants were experienced teachers who had taught from 6 to
18 years. None had taught at their currently assigned
grade levels for more than 6 years.

Four of the teachers had studied mathematics for
two years in high school and one for two and one-half
years, while one teacher had not studied any mathematics
in high school. Four teachers had earned 3 term hours
of credit in college mathematics, one had earned 7.5
term hours and one had earned credit for 15 term hours
of college mathematics. None of the teachers had

studied geometry in college.

Secondary Students

 

To select a representative secondary student
sample, a stratified random sample was drawn from
each secondary school in the school district. Within
each school, algebra classes were assigned identifying
numbers. For the study one class was selected from each
of the five junior high schools and five from the three
A5

senior high schools using a table of random numbers.

 

"5M. N. Downie and R. w. Heath, Basic Statistical
Methods (New York: Harper and Row Publishers, 1965),
p. 316.

 

Al

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A2

Of the 238 secondary students included in the study,
135 were ninth graders and 103 were tenth graders.

The overwhelming majority of students in the
Lansing school system studied algebra in either the
ninth or the tenth grade. Mathematics was required in
ninth grade where three courses (algebra, general
mathematics, and basic mathematics) were available.

Eighth grade students were scheduled for one of
these courses according to the recommendation of their
eighth grade mathematics teacher. In placing a student
in algebra in ninth grade, the teacher's recommendation
was guided by the following factors:

1. The student's overall Stanford Achievement

score (6th stanine or above)

2. The student's grades in seventh and eighth
grade mathematics (A or B)

3. The student's average of concepts, computation
and application scores on the mathematics
subtests of the Stanford Achievement Battery
(6th stanine or above)

A. Other factors (e.g., parental desires)

Ninth grade students were scheduled to study algebra
in tenth grade on the basis of their performance in
general mathematics. Students who studied basic mathe-
matics seldom selected algebra in tenth grade. Conse-

quently, ninth grade algebra students generally represented

“3

the upper third of the class, while tenth grade algebra
students represented the middle third of the class in
mathematical aptitude.

The mathematical background and teaching experience
of the secondary teachers whose classes were included in
the study are found in Table 6.

The number of years of teaching experience of the
secondary teachers ranged from U to 17. Five of the
teachers had studied mathematics four years in high
school, and the remaining five from one to three and
one-half years. College mathematics preparation ranged
from 30 to 120 term hours. All ten secondary teachers
had studied at least one college geometry course with a
range of 3 to 11 hours. When the preparation of
secondary school teachers is compared with that of the
elementary school teachers, the far superior training

in mathematics is evident.

Treatment

 

Elementary pupils, guided by their teachers, studied
a unit in coordinate geometry which was designed by the
researcher using the Madison Project's treatment of the
topic as a model. The content of this unit, suggested
materials, and pedagogical techniques for its development
were presented in a workshop for elementary school
teachers. Workshop participants volunteered to teach

the content to their pupils. Four school weeks of

 

 

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“5

forty—five minute periods per day were allowed for
classroom instruction.

Secondary students who elected algebra studied
coordinate geometry as part of the regular course in
Chapter 9, "Graphs of Linear Equations and Inequalities,"
in their textbook.“6 An average of eighteen fifty-

minute periods was devoted to studying the chapter.

The Elementary Unit

 

A unit was written with the objectives of teaching
fifth and sixth grade pupils to graph linear equations
and to discover the slope and intercept relations
betweeen graphs and open sentences. This unit was written
in the form of lesson plans for teachers and employed the

“7 Each lesson plan included student

discovery approach.
objectives in behavioral form, an overview for teachers
(eXplaining the mathematics necessary for each lesson
and indicating its present and future importance),
suggested procedures, activities for pupils, and a set of
exercises for each pupil. The objectives of the ten
lessons are listed below:

Lesson 1. Pupils will learn to plot ordered pairs

representing points in all four quadrants
of the plane.

 

H6

Johnson, loc. cit.

“7The complete set of lesson plans for teachers is
found in Appendix A.

Lesson

Lesson

Lesson

Lesson

Lesson

Lesson

Lesson

Lesson

10.

M6

Pupils will list some elements of the
truth set of an open sentence with two
variables by two methods; (a) by
listing ordered pairs as elements of

a truth set, and (b) by constructing

a table.

(a) Pupils will plot the graph of the
truth set of an open sentence with two
placeholders. (b) Pupils will recognize
that the graph of an open sentence of
the form A = [1+ K is a straight line.

Pupils will discover the pattern of the
graph of an open sentence.

(a) Pupils will discover the relation
of the geometrical pattern of the graph
and the open sentence. (b) Pupils will
identify the intersection of the graph
and the A axis from the open sentence.

Pupils will discover the Open sentence
when the graph of the truth set is
displayed.

Pupils will compute sums of signed
numbers.

Pupils will compute products of signed
numbers.

Pupils will discover the pattern of the
graph of an equation in which the A
coordinate decreases when the C]
coordinate increases.

Pupils will plot the graph of an equation
as a continuous line instead of a set
of discrete points.

Pedagogical approach.--Content of the coordinate

 

geometry unit for elementary school pupils was developed
by posing questions and creating challenging situations
designed to guide pupil discovery by following the

Madison Project's strategy:

”7

1. Begin, if necessary, by recalling key words,
experiences, notations, etc., from previous
lessons that will be crucial in today's lesson.

2. Do something. Have the child actually carry
out some sort of activity or happening.

3. As the occasion arises and as it becomes
appropriate, discuss what the class has done.“8

Pupils were involved in as many multisensory

experiences as possible. The most valuable instructional
aid was a 2 x 4 foot pegboard used to represent the
coordinate plane. Yarn looped around golf tees inserted
in the pegboard served as axes which could be moved to
accommodate the teaching of particular mathematical

concepts. Figure 2 illustrates the pegboard as it was

used to represent the quadrants of the coordinate plane.

 

 

 

 

 

Figure 2.--Pegboard Divided Into Quadrants.

 

u8Davis, Discovery in Mathematics, op. cit.,

pp. 19-220

 

NB

The pegboard afforded a means of involving pupils
in many learning activities. They "plotted points" by
inserting golf tees into specified holes. They discovered
that the graph of the truth set of an Open sentence with
two variables was a straight line. They described the
slopes of these lines by observing the pattern alignment
of the golf tees. They saw the difference between a line
composed of discrete points (whose domain was the set
of integers) and a continuous line (whose domain was the
set of real numbers) which was represented by looping
yarn around the golf tees. When pupils participated in
games, different colored tees conveniently identified
teams. Other visual aids included grids for an overhead

projector and a latticed chalk board.

The Secondary Unit

The content of one chapter (Graphs of Linear
Equations and Inequalities) in the textbook studied by
first year algebra students was comparable to that
included in the elementary school unit. Table 7 compares
the content of this chapter (by section) with that of the

ten lesson plans constructed for the elementary classes.

Pilot Classes

 

The coordinate geometry unit was piloted in two
elementary classes prior to use with the experimental

Population. Through such trials, the scope and sequence

A9

TABLE 7

CORRESPONDENCE OF ELEMENTARY AND SECONDARY UNITS
IN COORDINATE GEOMETRY

 

Coordinate
Geometry,

Chapter 9,
Algebra Text,

 

lopic Secondary School Elementary School
Unit* Unit

Graphs of Integers

on Number Line Section 1 Lesson 1

Equations with

Two Variables Section 2 Lesson 2

Graphs of Pairs of

Integers; The Lattice Section 3 Lesson 1

Coordinates for

Lattice Points Section A Lesson 1

Lattice Graphs

of Equations Section 5 Lesson 3

Rational Numbers

and Graphs Section 6 Lesson 9

Drawing Graphs in

the Rational Plane Section 7 Lesson 3 and 9

Linear Equations Section 8 Lesson 3, A, 5

The Intercepts

of Lines Section 9 Lesson 5

Some Unusual

Equations Section 10 Lesson 9

The Slopes of

Lines Section 11 Lesson 5, 6, 9, 10

 

* ,
Johnson, 100. cit.

50

could be tested with elementary pupils to assess the
pacing of the unit. In addition, children's reaction to
and interest in exercises and other materials could be
ascertained. Preliminary editions of test items could
be evaluated. Also, notes and ideas for elementary
teachers could be gleaned from the sessions in pilot

classes.

First Pilot Class

 

The first pilot group was a sixth grade class
composed of thirty-one pupils, twenty-one of whom were
boys and ten were girls. The school they attended was
located in a low-income residential area. The general
achievement of the pupils is described in Table 8.

The median percentile achievement in word meaning
was 18.5, with a class range of from the fourth to the
ninety-ninth percentile. The median percentile achieve-
ment for paragraph meaning was the seventeenth percentile
with a class range of from the first to eighty-sixth
percentile. The median percentile achievement on the
arithmetic subtests measuring arithmetic computation,
concepts and applications was 17.5, 25.5, and 22.5,
respectively. On the basis of these low general achieve-
ment data, this class could adequately test the impact of

a unit in coordinate geometry for low achievers.

 

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The class studied the unit on coordinate geometry
for A5 minutes each day for 13 consecutive school days.
During this period, pupils explored the first five
lessons in the unit. On the fourteenth day, a posttest
was administered to measure their understanding of unit
concepts. A summary of daily lessons is presented in
Appendix C.

Revision of material after the first pilot class.--

 

Because this pilot class was comprised of low achievers
and the entire unit which had been prepared was not com-
pleted, a second pilot class was utilized during the
trial period.

Modifications in lessons were made as a result of
the experience gained from the first pilot class.
Lesson 1, Judged too long for an average class to com-
prehend in one day, was divided into two parts. Part
one included leading students to see the necessity for a
vertical axis and of two coordinates to locate a point.
The importance of an agreement about the order of two
points was also included. Part two developed the need
for extending the axes and included practice in plotting
points in all four quadrants.

Consideration was given to interchanging Lessons A
.and 5 because only one student discovered the relationship
between an open sentence and the slope pattern and inter-

cept of the graph. However, the order was maintained to

53

see if the lack of discovery was due to the low mathe-
matics achievement of the students.

Revision of the unit included several new features.
A 2 x A foot pegboard and golf tees to represent points
in the coordinate plane were employed. "For Fun"
exercises were added. In these exercises, ordered pairs
were listed which when plotted and Joined consecutively

produced familiar objects or optical illusions.

Second Pilot Class

To provide additional data on the unit selected for
testing in this research study, a fourth-fifth grade com—
bination class was selected as the second pilot group.
In this class of twenty-two pupils, sixteen were fourth
graders and six were fifth graders. The school that
these pupils attended was located in a middle-class
neighborhood. Table 9 describes the range and percentiles
for the second pilot class by quartiles.

These statistics were computed from scores on
the Stanford Achievement Test administered to the fourth
.graders of this class in October, 1968 and to the fifth
graders in October of the previous year. The median
percentile for word meaning was 30.5 and for paragraph
meaning,28.5. The median percentiles on the arithmetic
subtests were 18.5, 50.7, and A2 for arithmetic compu-

tation, concepts, and applications, respectively.

5A

 

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55

Comparison of the scores of the two classes indicated
that the second pilot class achieved at a higher level
than the first.

The class studied the coordinate geometry unit for
A0 minutes each day for twenty days while an additional
three days were reserved for testing. During this period,
the class completed all ten lessons and the accompanying
exercises (Appendix B) in the unit.

An observer who attended the second pilot class
tape-recorded the sessions and offered valuable criticism.
The taped lessons, and his constructive suggestions
were beneficial in teaching that particular class,
revising the lesson plans, and preparing the workshop for

teachers.
Elementary Teacher In—Service Education

Recruiting and Selecting Teachers

Announcements of the six-week workshop in coordinate
geometry were sent to each elementary school where notices
were posted on the teachers' bulletin board. The notice
also stipulated that teachers who completed the workshop
could participate in a research project that would culmi-
nate in a dissertation. Principals were asked to encourage
fifth and sixth grade teachers to enroll. Of the nine
teachers who registered, seven completed the workshop.

One of these teachers was involved in a team teaching

situation where mathematics was taught by the other
member of the team, thus only six participants taught
the material. The standard cost for an in-service
workshop, $2.00, was paid by the teacher. One local
school district professional growth unit was accredited

to each teacher who completed the in—service training.

The Workshop Plan

 

Six two-hour sessions were planned which encompassed
the content of ten pupil lessons, outlined in the previous
section and detailed in Appendix A. The mode of
instruction paralleled that utilized with the pilot
classes. This method was chosen by the teachers who
wished to eXperience the unit from their pupils' view-
points. The choice met with the approval of the
researcher, the instructor, who was afforded the oppor-
tunity to lead "teacher discovery" and to suggest
pedagogical techniques.

Lesson plans were prepared, but were not distributed
until teachers had experienced the lesson in the workshop.
The first three sessions consisted solely of content and
approaches to its development. After the third session,
plans for lessons 1 and 2 were distributed to the teachers
who began teaching the unit the following week. The
remaining lesson plans were distributed at subsequent

sessions.

57

Part of each of the last three sessions was devoted
to a discussion of the teachers' experiences with their
classes during the week. Teachers exchanged information
about their problems, progress, successes and pupil
reaction to the materials. The other part of these
sessions continued with the explanation of content. For
broadening teachers' backgrounds, concepts were extended
beyond the content required of pupils. For example, the
relationship of slopes of parallel and perpendicular
lines, use of different visual aids, and other uses for
the pegboard and golf tees, such as for finding perimeters
and areas of polygons were discussed.

Description of Content
and Presentation Mode

 

 

Session l.--Nine teachers attended. At the begin-

 

ning of the session teachers were told that the workshop
had two purposes: (1) to introduce the content of a unit
in coordinate geometry, new to the elementary program,
using concrete materials and to suggest techniques for
instruction in an activity-centered, classroom setting,
and (2) to determine the feasibility of teaching selected
topics in coordinate geometry to fifth and sixth graders.
All of the teachers agreed to teach the material to

their pupils and granted the instructor permission to test
their classes and use the data for a research project.

During the week, one teacher dropped the course because

58

he did not have adequate time and another teacher dropped
because she had already taught similar material in her
class. The content of student Lessons 1 and 2 was pre-
sented. Points were located on the horizontal number
line, already familiar to teachers, and the need for a
vertical number line was developed by placing a point
above the number line. A series of leading questions for
teachers to use in stimulating children to suggest the
introduction of a vertical number line was offered.

After the need for intersecting perpendicular lines was
established, three golf tees were inserted into holes

at the left top and both bottom corners of a 2 x A foot
pegboard and yarn was wrapped around these golf tees to
form a pair of axes perpendicular to each other. These
axes, commonly called the x and y axis, were named the
box (CD and triangle (A) axis respectively, using the
Madison Project notation. Having demonstrated that the
location of a point required two coordinates to designate
its distance from the vertical (y) axis and its distance
from the horizontal (x) axis, it was then illustrated that
the location of a specific point depends upon the order
in which these distances were stated. Teachers plotted
points by inserting golf tees into the holes of the peg-
board. Only points in the first quadrant were used until
all teachers had grasped the concept of naming the

coordinates in the proper order. Since points lie to the

59

left and below these axes, the tees were placed so that
the pegboard was divided into four quadrants. Yarn
wrapped around the tees again formed the axes. Since
teachers were familiar with negative numbers, locating
points in quadrants II, III and IV followed readily.

The session closed with a discussion Of mathematical
sentences. Teachers gave examples of true, false, and
Open sentences.

Session 2.——The first session was reviewed by

 

having teachers complete the exercises included in student
lessons 1 and 2. Utilizing the pegboard, teachers plotted,
with golf tees, sets of points listed in one of the pupil
"for fun" exercises. The union of these consecutive
"points," when joined with yarn, formed the image of a
kangaroo.

Mathematical sentences were discussed. True, false
and Open mathematical sentences were illustrated. Open
sentences were written in the form, A = Cl+ K, where K was
limited to 0 or a positive integer. The necessity for
consistency in naming the order of coordinates was stressed
with the agreement to first name EL the horizontal
distance, followed by A, the vertical distance.

Since mathematicians are most interested in values
that make an Open sentence true, concise methods for
displaying the truth sets were desirable. Teachers were

asked to suggest convenient methods for organizing pairs

60

of coordinates that make a specific sentence true. The
aim was to elicit the following suggestions: (1) the

use of set notation, (2) the use Of a table, and (3)

the construction of a graph. Various teachers were asked
to choose an Open sentence and use the overhead pro-
jector to display its truth set by the three methods
listed above. They noticed that the graph of the points
was a straight line.

Teachers were eager to begin the unit with their
pupils. One teacher had already bought a pegboard. All
planned to finish the topics which their classes were
studying so that they could begin the coordinate geometry
unit after the next session.

Session 3.--Student lesson plans I and 2 and a set

 

of exercises for each student were distributed to teachers.
Content introduced in the two previous workshops was
reviewed by discussing these lesson plans and the pro-
cedures suggested for their implementation. Concepts to
be emphasized and some possible pitfalls, based on the
instructor's eXperiences in pilot classes, were enumerated
and demonstrated by playing portions Of the tape of lessons
recorded in the second pilot class. Lesson 1 was divided
into two parts with the plotting of points in the first
quadrant concluding part 1.

Teachers were taught to play a modified version Of

the Madison Project's game, tic-tac-toe, which when played

61

by pupils served to strengthen, in an entertaining
fashion, their ability to plot points.

An equation from pupil exercise 3 was plotted on the
pegboard. Teachers were asked to add points to this line
without doing any arithmetic. When a point was added and
accepted by the class, its coordinates were named and
substituted into the Open sentence to verify that the
ordered pair belonged to the truth set. As they graphed
other equations on the pegboard, they quickly perceived
the pattern of each. Informed that the pattern of a graph
could be determined without plotting a single point,
teachers were asked to do student exercise A to see if
they could detect the clue. Five of them comprehended that
the multiplier Ofl] revealed the slope pattern. Pupil
exercise 5 was distributed with the encouragement that it
contained stronger hints. These hints proved to be
sufficiently plain so that upon completion of the exercise
all teachers had discovered the relation between the
number sentence and slope as well as the relation between
the sentence and the A intercept.

All of the teachers planned to begin teaching the
unit the following week.

Session A.--This session began with teachers volun-

 

tarily discussing the experiences of their classes. All
reported that their pupils were highly receptive of the

material and that pupils especially enjoyed the "For Fun"

62

exercises. In particular, they liked the picture of
Snoopy. All teachers had completed Lesson 1 in their
classes. One teacher's pupils had spent some time
playing tic-tac-toe.

The topic which had terminated the previous in-
service education session, the relation between the
equation and the slope and A intercept Of its graph, was
continued. Teachers saw that the intercept and lepe
provided sufficient information to graph the sentence.
The process was then reversed. Golf tees were inserted
in the pegboard and teachers named the equation so
represented.

Lesson plans A, 5, and 6 with pupil exercises were
distributed.

Session 5.--The discussion of lesson plans A, 5, and

 

6 afforded the Opportunity to review their content. Some
teachers felt that pupils would need more exercises,
especially for lesson 6. Others felt that the set of
exercises was sufficient if, in addition to this set,
they used the pegboard or overhead projector to represent
graphs and let the pupils write or state their equations.
Most classes were finishing Lesson 3. Teachers
reported that pupil interest continued high. They claimed
that pupils wanted to play tic—tac-toe longer than their
plans allowed. The most frequent problem reported was the
tendency of students to name a point on one axis with

only one coordinate. For example, they said some children

63

were inclined to call (0,1) just 1, or (2,0) just 2,
forgetting to name the 0 coordinate. In answer to one
teacher's question about the slopes of the axes, the
slopes of vertical and horizontal lines in general,
which included the axes, were discussed. Teachers were
asked to plot pairs of parallel lines and investigate
the relation of the slopes of these lines. They also
plotted perpendicular lines and "discovered" the relation-
ship of their slopes.

In preparation for plotting lines with negative
slopes, the Madison Project Postman Stories were intro-
duced. In these stories the postman brings envelopes
containing checks or bills. Considering his daily mail
delivery as the only monetary transactions, pupils
determine the family's immediate financial state which
requires addition or subtraction of signed numbers.

Session 6.--In the discussion of the week's classes,

 

three teachers felt that the content required "too much
discovery" by children. Two teachers interchanged
Lessons A and 5 for this reason. Students could see the
pattern of a graph, they reported, but very few were able
to relate the pattern and the open sentence. Nonetheless,
student interest was still high.

After reviewing the "rules of the game" for Postman
Stories that required adding signed numbers, Postman

Stories that required multiplying signed numbers were

6A

created. Portions of the tape of Lessons 7 and 8 of the
second pilot class were played so that teachers could
hear typical questions and responses. Teachers examined
the effect of negative multipliers oflfl on the graphs of
open sentences.

Rational numbers were then used as replacements for
C]. Ordered pairs were plotted on a grid projected on a
screen to demonstrate that the graph of a linear equation
is a continuous line if values selected for box were
closer and closer together. To show that a line was
continuous when discrete points were plotted on the peg—
board, yarn was looped around the golf tees. This topic
completed the content of the ten lessons to be presented
to pupils.

Session 7.—-Teachers reported losing student interest

 

with the computation of the postman stories. Others
related humorous stories told by their children. Arrange-
ments were made with each teacher to administer the test

on the unit of coordinate geometry within two weeks.

The Test Instruments

 

Two instruments were employed in the study: The

Stanford Achievement Test and a Test on Coordinate

 

 

Geometry (TOCG). To test Hypotheses F, G, and H, the
sixth grade sample was divided into two groups, high and
low, on the basis of their scores on the arithmetic con-

cepts subtest of the Stanford Achievement Test. The

 

65

Test on Coordinate Geometry provided data for the dependent

 

variables in the study (total score and seven subtests).

The Stanford Achievement Test

 

The Stanford Achievement Test is a series of compre-

 

hensive tests developed to measure the knowledges, skills,
and understandings commonly accepted as goals of the major
branches of the elementary school curriculum. First
published in 1923, the test was revised in 1929, 19AO,
1953, and 196A. TO provide normative data descriptive

of the current achievement of the nation's schools, more
than 850,000 students from 26A school systems drawn from
the 50 states were tested.

The Stanford Achievement Test measures reading,

 

language, spelling, social studies, science and mathe—
matics. The mathematics section consists of three sub-
tests: arithmetic computation, arithmetic concepts, and
arithmetic applications. The concepts subtest was chosen
for dichotomizing the sixth grade into two groups because
its measure of understanding was more relevant to the
unit in coordinate geometry which did not require exten—
sive computation or the application of arithmetic to
practical problems. The reliability of the arithmetic
concepts subtest obtained by the odd-even split-halves
method for grade six was .85. Content validity was
established by comparison of the test's content with the

curriculum of the school.

6.6

The Test on Coordinate Geometry

 

To test the major hypotheses Of the study, a
reliable instrument with parallel forms valid for both
the elementary and secondary content was required. A
search for such an instrument was unsuccessful. Nor could
a portion of any existing instrument be located which met
the specifications of the research design; therefore, a
suitable test was deveIOped. In its final form, the
instrument consisted of forty items selected from four
preliminary forms. The pilot groups of elementary classes
were used to test the preliminary drafts of the instrument.

Test administered to the first pilot class.--A

 

preliminary instrument covering the content Of Lessons 1
through 5 was administered to the first pilot class.
Different types of objective questions were included to
help determine the most desirable format for the instru-
ment. These types included: (a) the plotting of points,
(b) the completing of a table, (c) the graphing of an
equation, and (d) responding to multiple-choice questions.
This test was scored by the researcher. Several
types of items were eliminated from future revisions of
the instrument because of subjective decisions involved
in scoring. For example, in several instances, pupils
located points in such a way that the response could be
deemed either correct or incorrect. Legibility of numerals

was another problem. After review of the administration

67

and scoring problems, multiple-choice items seemed the
most appropriate for future editions.

Tests administered to the second pilot class.—-

 

Three instruments were administered to the second pilot
class: Test 1, consisting of 25 items, was administered

at the end Of Lesson 2; Test II, 20 items, was administered
at the end of Lesson 6; and Test III, 25 items, at the end
of Lesson 10.

Pupils marked their answers to all instruments on
machine-scored 5-choice response sheets. Because fourth
and fifth graders had limited experience with this means
of response detailed instructions for marking the answer
sheet were given. That the answers were numbered hori-
zontally was emphasized. The rows of blanks for the
answers were so close together that each pupil was given
a sheet of paper to lay along a row in order to help him
keep his place on the response sheet. Tests were scored by
the Michigan State University Office of Evaluation Services.

The Data Processing Center Of the University then
computed an item analysis Of each of these preliminary
instrument drafts. The basic item statistics derived from
this analysis included the index Of difficulty and the
index Of discrimination. The index Of difficulty is the
proportion of the total group who answered each item
incorrectly. The index of discrimination is the difference
between the proportion of the upper and lower groups who

answered each item correctly. Optimal discrimination was

68

obtained by including 27 per cent of the total group in
A9

each of the upper and lower sub—groups.

Choice of items for the instrument.-—Forty items

 

were selected from the three tests that were administered
to the second pilot class. Several factors guided the
selection of each item. Foremost, the question had to
test at least one of the objectives of instruction. The
indices Of difficulty and discrimination were considered
with the realization that these statistics may be unstable
because they were computed from a small group. The number
of items selected which focused on one concept was a
function of the teaching time devoted to that concept.
Table 10 lists the index of difficulty and the index of
discrimination of the forty items selected:

Two items with negative discriminations were
retained in the instrument. A correct response to these
questions required pupils to extend and apply concepts
they had learned to situations which had not been dis—
cussed in class. These items were retained because it
was anticipated that correct solutions would be given by
secondary students, but not by elementary pupils. Each
of the questions presented a different challenge. Because
spacial concepts were involved in Item 11 (shown on the

following page), it was replaced by another item.

 

Aq
’Office of Evaluative Services, Item Analysis

(East Lansing: Michigan State University, 1965), p. A.

 

 

69

ON ON 0: ON no om

 

OO Om Om OO Om OH
OO mO Om Om OH OH
O Om Am 3mg 20: NH
Om Om Om OO O; OH
Om OO mm OO m: mH
OO OO Om Om: Ow OH
O mm mm OO O: mH
OO mO mm OOH OO NH
O: mm Hm Om: OO HH
OO Om om Om O OH
OO Om mm Om mm O
Om: we Om OO mH O
O: as am OO Om s
OO mm mm OO mm O
OO me mm OOH OO O,
O: OH em OO OO O
OO mm mm OO mm m
OOH O: mm OO Os N
O HO Hm Om Om H
OOHOmcHsHsomHO spHOOHOOHO smpH OOHOmcHsHsowHO sOHOOHOOHO EOOH
mo meCH mo xmch pcmESmecH mo meCH . mo meCH unmezppmcH

 

Mmemzomo mB<zHQmOOo zo BmMB mmB zH
QMDDQQZH mszH MBmom mmB mom mmddo Bqum Ozoomm 20mm
QmBDmEOD m¢ ZOHB¢ZHSHmomHQ mo XMQZH 02¢ MBqDonmHQ mo meZH

OH mqm<e

70

11. The location of a point in space requires

A. 2 coordinates

B. 3 coordinates

C. l coordinate

D. None of these
Item 1A, listed below, was included to determine if the
student could extrapolate from the specific instances
to which he had been exposed in class to the more general

1A. In the figure at the right, the coordinates

 

 

 

 

of P are
A. (a,d) (AP S(c,d)
B. (a,c)
C. (c,a)
D. (d,a) n
E. None of these Q‘(a b) R
.J , x
W

 

Item 28 required close scrutiny by the student of each

point represented on the graph.

28. Does the graph at the right [below]
represent the Open sentence

A = 2 x Cl- 2 (elementary form)
y = 2 x - 2 (secondary form)
A. Yes
B. NO

 

The instrument used in this study is found in Appendix D.

.71

Notation of variables employed in the instrument.--
Although elementary and secondary classes studied the
same content, they used different notations for the axes
and variables. Elementary school pupils used the Cl- A
notation of the Madison Project, while secondary school
students used the conventional Cartesian x-y notation.

To determine if the different notations influenced
the instrument, both forms were administered to two non-
experimental tenth grade classes two days apart following
their study Of the chapter on linear equations and their
graphs. A cover sheet on the test explained the use and
definitions of the [land A symbols. The means and vari—

ances of the results are reported in Table 11.

TABLE 11

MEANS AND STANDARD DEVIATIONS ON BOTH FORMS
OF THE TEST ON COORDINATE GEOMETRY
ADMINISTERED TO NON-EXPERIMENTAL
TENTH GRADERS TWO DAYS APART

 

 

 

 

Form f O
[3 — A Test 2A.55 6.56
x — y Test 25.71 7.59

 

 

A t-test between these two means did not show any
significant difference. The correlation between the two

tests was .79. This correlation was similar to that

72

expected in test—retest situations, and within expected

tolerances for parallel tests measuring the same concept.
Thus, the conclusion was drawn that the use of x - y and
A -[3 notation would not adversely affect study results.

Reliability Of the instrument.-—The reliability for

 

both forms of the test taken by the two non-experimental
classes was computed by the Kuder-Richardson Reliability
#20. The reliability of the A -[3 Test was .80. The
reliability of the x - y Test was .81.

Validity of the instrument.--A professor of mathe-

 

matics at Michigan State University examined the test for

face validity vis a vis the unit in coordinate geometry

which had been prepared for elementary teachers to use

in their classes. A junior and a senior high school

teacher examined the content of the test relevant to the

content in the first year algebra textbook and judged

that it tested the concepts included in Chapter 9,

Graphs of Linear Equations and Inequalities.50
Subtests.--Seven subtests, determined by content

groupings were embedded in the instrument. These cate-

gories and the relevant test items are listed below.

 

50

Johnson, loc. cit.

73
Subtest l. Plotting and Recognizing Points in the
Coordinate Plane
Items 1, 2, 3, A, 6, 7, 8, 9, 10, 12, 26
Subtest 2. Recognizing Members of a Truth Set
Items 5, l6, 17, 20, 3A
Subtest 3. Intercept Relation to Open Sentence or Graph
Items 15, 19, 23, 25, 30, AO
Subtest A. Slope-Graph Relation
Items 18, 2A, 37, 39
Subtest 5. Operations with Signed Numbers
Items 11, 29, 31, 36
Subtest 6. Graph-Open Sentence Relation
Items 21, 22, 28, 32, 33, 35
Subtest 7. Extension Of Concepts

Items 13, 1A, 27, 38

Procedures of Testing

 

The test on the unit in coordinate geometry was
administered to four junior high school classes (ninth
grade) and two senior high school classes (tenth grade)
on the same day. For uniformity of administration, the
same individual administered all the tests. The following
day, the test was administered to the remaining secondary
classes. These tests were scheduled in the beginning Of
March, 1969, after the completion of the chapter on

coordinate geometry.

7A

The test on the coordinate geometry unit was
administered to the elementary classes within a two-day
period during the third week in March, 1969, following
the experimental treatment period in which their teachers
guided their learning the concepts of coordinate geometry.
All tests were administered by the same individual to
eliminate differences due to testing procedures. Since
the instrument was designed as a power rather than a
timed test, no time limits were set for completion;
however, the typical time for completion was approximately

A5 minutes, with a range from 35 to 55 minutes.

Analysis of Data

 

51

A three-way analysis of variance was used to test
the hypotheses of the study. The .05 level of significance
was accepted as being sufficiently rigorous for the
purposes of this study. Thus, the probability that any
differences which were found were real rather than

spurious was 95 out of 100 cases. Where significant

c

differences occurred, Scheffé's post hoc comparison“ was

used to test the specific hypotheses involved.

 

leilliam Hays, Statistics for Psychologists (New
York: Holt, Rhinehart, and Winston Publishing Co., 1963),
p. A8A.

52S. W. Greenhouse, "0n Methods in the Analysis Of
Profile Data," Psychometrika, XXIV (June, 1959), 9A-112,

 

 

 

CHAPTER IV

THE ANALYSIS OF THE DATA

Testing of the Hypotheses

 

The purpose of this study was to examine and
evaluate the achievement Of upper elementary and secondary
school students on a comparable unit in coordinate
geometry. The tOpic was traditionally studied in first
year algebra, but had recently been introduced to elementary
school pupils. To examine the feasibility Of introducing
this topic three to four years earlier than was customary,
eight hypotheses were examined to determine differences
in performance between elementary and secondary students.
Achievement in coordinate geometry was further analyzed
by seven subtests to determine if elementary school pupils
could understand all or a part of the specific concepts
-embedded in the coordinate geometry unit as well as
secondary school students.

Another facet, the degree of difficulty for individual
items by subtest, was examined to determine if topics were
appropriately placed. Finally, the results of a student
questionnaire reflecting their interest and preferences
for this material as opposed to the usual topics was
analyzed.

75

76

The following eight hypotheses were tested:

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

Hypothesis

A.

H.

There is no difference in achievement on
a unit in coordinate geometry between
fifth graders and ninth graders.

There is no difference in achievement on
a unit in coordinate geometry between
fifth graders and tenth graders.

There is no difference in achievement on
a unit in coordinate geometry between
sixth graders and ninth graders.

There is no difference in achievement on
a unit in coordinate geometry between
sixth graders and tenth graders.

There is no difference in achievement on
a unit in coordinate geometry between
fifth and sixth graders combined and
ninth and tenth graders combined.

There is no difference in achievement on

a unit in coordinate geometry between the
upper half Of the sixth graders as measured
by a general mathematics achievement test
and ninth graders.

There is no difference in achievement on

a unit in coordinate geometry between the
lower half of the sixth graders as measured
by a general mathematics achievement test
and tenth graders.

There is no difference in achievement on

a unit in coordinate geometry between the
upper half Of the sixth graders as measured
by a general mathematics achievement test
and ninth and tenth graders combined.

These hypotheses and achievements by subtest were

tested using analysis of variance and Scheffé's post hoc

test. Table 12 summarizes the mean scores for the

coordinate geometry test by subtests for grades 5, 6 High,

6 Low, 9 and 10. Table 13 reports the results of the

analysis of variance of the data presented in Table 12.

77

 

 

 

 

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mm.a Hm.a mo.m Om.a OH.H mm.a mH.H mpdoocoo mo OOHmcouxm .s

mm.m Hm.m om.m :o.m mm.m om.m mm.m coapmaom
oocopsom conclzdmso .m

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oocwflm OOH; mdoapmpoqo .m
mw.H Hm.H mo.m mm.a mm.H :o.m m©.H coflpmaom compouoaoam .:

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Como Op coapmaom OQOOLOOQH .m

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m mo washes: mcflwflcwooom .m

m:.s mm.s mm.m HH.N mm.© mm.s mm.w ocmam

oomcflosooo one CH mpcflom
mcflmficwooom new mcflppoam .H

Hmpoe 30H OmHm
Hmooe OH m m m omoonsm
compo

 

OH Ucm m no .m mmo<mo 2H mBzmmme mom
wmemzomo medzHQmooo zo BmMB mmB ZO BmmBmDm wm mmmoom z<mz

ma mqm<H

78

TABLE 13

ANALYSIS OF VARIANCE SUMMARY

 

Sum of Mean

 

Source df Squares Square F
Grade A 21.7A37 5.A359 5.092*
Class/Grade 1A 1A.9AA5 1.0675
Subtest 6 A61.687A 76.9A79
Subtest/Grade 2A 8.7659 .3652 l.9lA
Subtest/Classes/Grade 8A 16.02A2 .1908

TOTAL 132 523.1657 39.6333

 

*Significant at the .05 level of confidence.
Table 12 indicates that the mean scores of the
ninth graders exceeded the mean scores of fifth, low
sixth, and tenth graders on all seven subtests. The
mean scores of the ninth graders exceeded the high sixth
on Subtests I, V, VI, and VII, and were inferior to the

high sixth graders on Subtests II, III, and IV.
Results were significant at the .05 level Of con-
fidence in one of the main effects, Grade. Differences
between subtests and differences among classes, the other
two main effects,were not a primary concern in this study.

Subtest—grade interaction was not significant.

79

To determine the source of significant differences

among grade levels, each null hypothesis (HO) was tested

53

by the Scheffé confidence interval:

 

// -—i

A C o

'0 / l A
C {v - (J-1)(F)(MSE>(2nJ ) 3 w 1 w

 

2
c

 

+ /(;-1><F)<MSE>< >}: 1 - a

n

J, the number of groups, equals 5
F, the critical value with A and 1A degrees

of freedom, equals 3.11

MSE’ the mean square error, equals 1.0675, and
i is the weight given the ith group (grade)
nj is the number of classes in the jth group

(grade)

a, the level of confidence, equals .05

If the above interval spans zero, the test fails to reject
HO. If the above interval does not span zero, the test

rejects H
o

 

53Hays,'loc. cit.

80

Hypothesis A. There is no difference in achievement on
a unit in coordinate geometry between
fifth graders and ninth graders.

Symbolically, HO: w = “G5 - “G9 = 0
Estimated by w = KGB - YG9
For this case: O = 18.A9 - 2A.73

—.62A

'6)
ll

When tested by the Scheffé confidence interval, where

c 2 2 2

X (—l—)
n
J

loo WIH
+
Ulli—J

H
U7

 

c {—6.214 — ./ I473.11)(1.0675)T8/15) _<_ w 5 - 6.214

 

+ /'A (3.ll)(l.O675)(8/15)} 1 l - .05

C{- 6.2A - 2.66 i w i - 6.2A + 2.66}: .95

C{- 8.9 3 w i — 3.58} 3 .95
reject the null hypothesis

Since the interval does not span zero, the null
hypothesis was rejected. A significant difference in
achievement on the unit in coordinate geometry did exist

between fifth graders and ninth graders, favoring ninth

81

graders. This result was consistent since the fifth
grade sample included all fifth graders, whereas the
ninth grade sample represented approximately the upper
third of all ninth graders. They had been screened to
study algebra on the basis of the overall Stanford
Achievement Score (6th stanine or above), and the average
of their seventh and eighth grades mathematics marks

(A or B). Consequently, fifth graders representing the
entire range of achievement could not be expected to
equal ninth graders who not only represented the upper
third of their class in mathematics achievement but had
also completed four additional years of mathematics.
Hypothesis R. There is no difference in achievement

on a unit in coordinate geometry between
fifth graders and tenth graders.

Symbolically, HO: w = “G5 — “G10 = 0
Estimated by w = XG5 - XGlO
For this case: w = 18.A9 — 20.16

- 1.67

G)
II

When tested by the Scheffé confidence interval, where

2
Ci

2 (IT-f )
J

l
w|+—'
+
U‘l|l-’

IO

82

 

c {—1.67 — /'u <3.11)(1.0675)(8/15) : w i — 1.67

 

+ /53 (3.ll)(l.0675)(8/15)} 3 l - .05
C{— 1.67 - 2.66 i w i - 1.67 + 2.66}: .95

c{— 14.33 _

A
‘6-

A
\O
\O
H4

V
\0
U1

fail to reject the null hypothesis

This interval spans zero. The test therefore failed
to reject the null hypothesis. Even though the elementary
sample included all fifth graders, the secondary sample
included only the second third (between the thirty-third
and sixty-sixth percentiles) of the tenth grade population
(the more able tenth grade students had studied algebra a
year earlier). In contrast, fifth graders were challenged
and motivated by the unit in coordinate geometry which
was different from the content of their regular arithmetic
book.

Hypothesis C: There is no difference in achievement on

a unit in coordinate geometry between
sixth graders and ninth graders.

Symbolically, HO: w = uG6 - pGg — 0

Estimated by w = XG6 — XG9

For this case: 8 = 2l.A6 - 2A.73
IP=-3.27

When tested by the Scheffé confidence interval, where

C,2 l2 l2
Z<T)—§' +5-
J
_ _§
-15

 

c {—3.27 — /’u (3.11)<1.0675><8/15) : w 5 - 3.27

 

+ /“u (3.11(1.0675)(8/15)} 3 1 - .05

C{- 3.27 — 2.26 i w i — 3.27 + 2.26}: .95

c{- 6.53 5 w i — 1.01} 3 .95

reject the null hypothesis

This interval did not span zero and the hypothesis
that there was no difference in achievement on the unit in
coordinate geometry between sixth graders and ninth
graders was rejected. Ninth grade students achieved at
a higher level than sixth grade students in the unit on
coordinate geometry. The rationale pertinent to Hypothesis

A applies to this hypothesis.

8U

Hypothesis D. There is no difference in achievement on
a unit in coordinate geometry between
sixth graders and tenth graders.

Symbolically, HO: w = “Go — UGlO = 0

Estimated by 0 = Xéé — X010

For this case: @ = 2l.u6 — 20.16
0 = 1.30

When tested by the Scheffé confidence interval, where

 

Z ( i ) — % + %
J
_ _§
_ 15

 

0 {—1.30 — /'u (3.11)(1.0675)(8/15) 3 w i — 1.30

 

+ /Tu (3.11)(1.0675)(8/15)} 3 1 — .05
c -{1.30 — 2.66 g 0 g - 1.30 + 2.66}: .95

c{- 3.96 3 w

I A

1.36} 1 .95
fail to reject the null hypothesis

The Scheffé confidence interval failed to reject
the null hypothesis. Sixth graders achieved as well as
tenth grade algebra students on the coordinate geometry

unit.

Hypothesis E. There is no difference in achievement
on a unit in coordinate geometry between
fifth and sixth graders combined and
ninth and tenth graders combined.

Symbolically, HO: w =

'6)
ll

?stimated by

'6)
II

For this case:

The difference between

Scheffé confidence interval,

ffi)=i2+£2+£2
‘nj 3 3 5
-16.
-15

“05 + “06 ‘ “09 ' “010 =

X05 + X06 ' X09 ’ XG10

18.U9 + 2l.u6 - 2u.73 - 20.16

— 3.9M

these means was tested by

where

U‘tll—J

 

c {—3.9u —/'u (3.11)(1.

O675)(l6/15) 3 w i - 3.9a

 

+/Th (3.11)(1.
c{- 3.9M — 3.76

C{- 7.70

2675)(l6/15)} 3 l - .05
i w i — 3.9M + 3.76} 3 .95

i w i — 0.18} 3 .95

reject the null hypothesis

The null hypothesis was rejected at the .05 level

of confidence. Secondary students achieved at a higher

level than elementary pupils.

86

Hypothesis F. There is no difference in achievement on
a unit in coordinate geometry between the
upper half of the sixth graders as
measured by a general mathematics achieve—
ment test and ninth graders.

symbolically, HO: w = UG6H - “39 = 0
Estimated by w = XG6H — XG9
For this case: $ = 2u.72 — 2u.73

— .01

G)
H

Tested by the Scheffé confidence interval, where

2
c 2 2
i l l
E ('6‘?) "3“ +6
J
_ 8
”.15

 

C {- .Ol — / H (3.ll)(l.0675)(8/15) : w i - .Ol

 

+ / u (3.11)(l.0675)(8/15)} : 1 - .05
c{- .01 — 2.66 i w i — .01 + 2.66}: .95

C{- 2.67

| A

w i 2.65} 3 .95

fail to reject the null hypothesis

87

The test failed to reject Ho because the interval
spanned zero. Any difference in achievement in coordi—
nate geometry between the upper half of the sixth
graders and ninth graders can be assumed to be chance
occurrences. With respect to achievement, data collected
in this study indicated that the upper half of the sixth
grade class was as successful as ninth graders in the
study of a comparable unit on coordinate geometry.
Hypothesis G. There is no difference in achievement on

a unit in coordinate geometry between the
lower half of the sixth graders as measured

by a general mathematics achievement test
and tenth graders.

Symbolically, HO: w = “G6L - uGlO = 0

Estimated by: 8 = xG6L - xGlO

For this case: 8 = 18.193 — 20.162
113= - 1.969

When tested by the Scheffé confidence interval, where

M
P\
It
.4
u
ch
+
uuw

H
moo

88

 

c {-1.97 — /fiu <3.11)(1.0675)(8/15) : w 3 - 1.97

 

+ /Ffi (3.11)(1.0675)(8/15)} 3 1 — .05

C{- 1.97 - 2.66

[A

w i - 1.97 + 2.66}: .95

C{— 8.63

| A

w 3 .69} 3 .95
fail to reject the null hypothesis

The test failed to reject the null hypothesis;

the interval spanned zero. The lower half of the sixth

graders and the tenth graders did not differ signifi-

cantly in achievement on a unit in coordinate geometry.

Hypothesis H. There is no difference in achievement
. on a unit in coordinate geometry between
the upper half of the sixth graders as
measured by a general mathematics
achievement test and ninth and tenth
graders combined.

' ° = -12 -1 =
Symbolically, HO. w UG6H 2UG9 ’uGlO O
.n . =— _1— _1—
Estimated by w XG6H 1X69 1XG10
For this case: W = 2U.73 - %(2U.73) - %(20.l6)

2.28

'6)
II

The difference between the means for the upper half
of the sixth grade and the means of the ninth and tenth
grades combined was tested by the Scheffé confidence

interval, where

89

M
A
5 it—
C.:
v
I
WIH
4.
WM"
+
umr
ll
WIT-J
o co

 

c { 2.28 — /ru (3.11)(1.0675)(13/30) : w i 2.28

 

+ /rfi (3.11)(1.0675)(13/3O)} 1 l - -05

c{2.28 — 2.uo 2.28 +2.2u}: .95

[A
‘6
[A

C{- .12

IA
6
A

_ 4.52} 1 .95
fail to reject the null hypothesis

Because the interval spanned zero, the test failed
to reject the null hypothesis. No significant differences
in achievement on a coordinate geometry unit existed
between the upper half of the sixth grade class and the
ninth and tenth grades combined. The upper half of the
sixth grade pupils comprehended the concepts involved in
the coordinate geometry unit as well as first year algebra
students.

Two additional hypotheses were posed and tested to
complete the analysis of possible differences between
classes. The first:

There is no difference in achievement on a unit in

coordinate geometry between fifth graders and sixth
graders.

9O

Symbolically, HO: w = “G5 - UG6 = 0

Estimated by w = XG5 — XG6
For this case: W = l8.U9 - 21.46
W = - 2.97

The difference between the fifth and sixth grade

means was tested by the Scheffé confidence interval, where

0.2 2 2
““i’)‘% +%-
J
= .3
3

 

c { -2.97 - /fu (3.11)<1.o675>(2/3) 3 w 3 - 2.97

 

+ /’u (3.11)(1.0675)(2/u)} 3 1 - .05
c{- 2.97 - 2.98 3 w i - 2.97 + 2.98}: .95

C{- .95 i W j .01} 1 .95
fail to reject the null hypothesis

The interval spanned zero and the test failed to
reject the null hypothesis.

The final hypothesis tested was:
There is no difference in achievement on a unit in

coordinate geometry between ninth graders and tenth
graders.

91

Symbolically, HO: 8 = UG9 - “G10 = 0

Estimated by 8 = xcg — x610

For this case: 8 = 2u.728 — 20.162
8 = 4.566

When tested by the Scheffé confidence interval, where

4.
Ulli—J

M
(N
:,i
C.:.
V
I
\fil H

 

c { n.566 - /fu (3.ll)(l.0675)(2/5) : 8 g 4.566

 

+ / u (3.11)(1.0675)(2/5)} 3 1 — .05
C{4.566 - 2.30“ i 8 i 4.566 + 2.304}: .95

c{2.262 i 8

[A

6.870} 1 .95
.3 reject the null hypothesis

This interval did not span zero and the null
hypothesis was rejected. Significant differences existed
between ninth and tenth graders in achievement in

coordinate geometry, favoring ninth graders.

92

Comparable Difficulty by
Subtest and Grade

 

 

The analysis of variance indicated no significant
interaction between Subtests and Grade. Therefore, it
was not necessary to test the difference in means between
grades on each subtest for significance. Achievement
scores by students at different grade levels verified no
variations in mean subtest scores; thus no concepts
were more difficult for students at one level than
another. The remainder of this section examines the
performance of each grade on specific concepts by subtest.
Two tables are included for each of the seven TOCG sub-
tests. The first lists the mean score, mean per cent and
standard deviation by grade. The index of difficulty by
item and grade are presented in the second table. Sixth
grade data are presented by the groups 6 High and
6 Low, in order to compare the performance of the upper
half of the sixth graders with that of the ninth graders
and the performance of the lower half of the sixth graders
with that of tenth graders. This comparison is of interest
because the upper half of the sixth grade pupils will
probably study algebra in ninth grade and the lower half
in tenth grade.

The sixth high elementary pupils and the ninth
grade algebra students scored the highest on Subtest l,
as indicated in Table 14. The mean score of the sixth-low

elementary pupils was the lowest.

93

TABLE 14

MEAN SCORE AND STANDARD DEVIATION BY GRADES
FOR SUBTEST l: PLOTTING AND RECOGNIZING
POINTS IN THE COORDINATE PLANE

 

 

 

 

 

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 6.65 7.99 6.23 8.27 7.53
Mean Per Cent 60.5 72.7 56.6 73.2 68.3
Standard Deviation .9“ .07 .27 .Al 1.16
TABLE 15
INDEX OF DIFFICULTY BY ITEM FOR SUBTEST 1:
PLOTTING AND RECOGNIZING POINTS IN
THE COORDINATE PLANE
Item Grade
Number 5 6 High 6 Low 9 10
1 53 38 96 29 H2
2 39 21 19 21 17
3 56 41 51 26 21
U 53 29 86 35 85
6 23 19 38 11 16
7 19 I“ 29 13 23
8 21 8 26 l2 l9
9 18 1A 29 6 2A
10 U9 UN 61 2A 42
12 5A Al 62 U2 A6
26 71 58 71 N9 62

 

9A

in the eleven items of Subtest 1, five items, 6
through 10, asked students to choose the coordinates of
representations of points plotted on a graph. Three
items, 1, 2, and 12, required the location of points in
one of the four quadrants. The remaining items, 3, A,
and 26, related to the order of a pair of coordinates.
Table 15 designates the index of difficulty for each
item in Subtest l for each grade level. The concept,
naming the coordinates of points was understood best
and the concept related to the order designated by a
pair of coordinates, least by students of all grades.

With a mean 70 per cent correct, the sixth—high
elementary students scored the highest on Subtest 2.

The performance of the fifth, sixth—low, and tenth
graders was about the same, but the standard deviation
of the sixth-low elementary pupils was the greatest.

All five items of Subtest 2, Recognizing Members
of a Truth Set, involved the identification of ordered
pairs that made an open sentence true. Data in Table 17
show that this subtest was easiest for the high-sixth
graders and most difficult for the fifth graders. The
greater mathematical eXperience of the secondary students
did not aid them in choosing the sentence that contained
the origin (item 20). Nor did their additional algebraic
eXperience give them as much advantage as anticipated in
choosing the correct equation for the table given in

item 17, or for the point plotted in item 34.

 

 

 

 

 

95

TABLE 16

MEAN SCORE AND STANDARD DEVIATION BY GRADES
FOR SUBTEST 2: RECOGNIZING MEMBERS
OF A TRUTH SET

 

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 2.31 3.53 2.59 3.03 2.52
Mean Per Cent A6.00 70.6 50.8 60.6 50.A
Standard Deviation .A5 .3“ .76 .A0 .59

 

TABLE 17

INDEX OF DIFFICULTY BY ITEM FOR SUBTEST 2:
RECOGNIZING MEMBERS OF A TRUTH SET

 

 

 

 

Grade
Item
Number 5 6 High 6 Low 9 10
5 “9 15 33 33 “2
16 38 13 32 2O 3“
17 55 39 AU 38 57
20 66 56 67 55 68

3A 6A 3A 57 AD 58

 

96

Both the mean score and the standard deviation for
the sixth-low graders on Subtest 3 were the lowest.
The largest standard deviation occurred in the tenth
grade. The upper half of the sixth graders scored
higher than all other grades with a mean of 3.81 or

63.6 per cent.

TABLE 18

MEAN SCORE AND STANDARD DEVIATION BY GRADES
FOR SUBTEST 3: INTERCEPT RELATION
TO OPEN SENTENCE OR GRAPH

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 2.72 3.81 2.21 3.39 2.63
Mean Per Cent 45.4 63.6 36.8 56.4 43.9
Standard Deviation .45 .45 .30 .39 .59

 

Two items, 19 and 25, related to the y—intercept
and the graph of the open sentence, and item 40 related
to the x—intercept and the graph. Previous to this unit
or to the comparable chapter in the algebra text, no
class had had prior experience with this concept. Tables
18 and 19 show that the upper half of the sixth graders
performed best. Both tables indicate the concept was
equally difficult for the fifth and tenth graders and

hardest for the lower half of the sixth graders.

97

TABLE 19

INDEX OF DIFFICULTY BY ITEM FOR SUBTEST 3:
INTERCEPT RELATION TO OPEN SENTENCE OR

 

 

 

GRAPH
Grade

Item

Number 5 6 High 6 Low 9 10
15 59 40 62 6O 71
19 36 18 42 26 42
23 54 36 64 58 72
25 61 44 69 42 53
30 58 “3 72 43 57
4O 64 47 73 42 51

 

The mean score of the ninth grade, 2.42 (61 per
cent of the total), was the highest on Subtest 4. The mean
score of the upper half of the sixth grade, 51 per cent,
was second highest for this subtest. The scores of the
lower half of the sixth graders were the least dispersed
from the mean, and those of the ninth grade were the
most dispersed.

Subtest 4 required the student to determine the
slope of a line from its graph. Items 18 and 24 repre-
sented lines with positive slopes and items 37 and 39
represented lines with negative slopes.

Comparison of the index of difficulty for items 18
and 24 is interesting in that both graphs displayed sen-

tences with the same slope but different domains. For

Q8

/

TABLE 20

MEAN SCORE AND STANDARD DEVIATION BY GRADES
FOR SUBTEST 4: SLOPE-GRAPH RELATION

 

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 1.62 2.04 1.86 2.42 1.21
Mean Per Cent 40.6 50.8 46.6 60.6 30.2
Standard Deviation .43 .25 .12 .65 .37

 

TABLE 21

INDEX OF DIFFICULTY BY ITEM FOR SUBTEST 4:
SLOPE—GRAPH RELATION

 

 

 

Grade
Item
Number 5 6 High 6 Low 9 10
2a 65 51 57 51 69
37 51 Q9 63 52 63
39 81 71 83 59 85

 

item 18, the domain was the set of integers, thus its
graph was a series of discrete points. For item 24,

the graph was a continuous line; its domain was the set
of real numbers. Slopes of continuous lines were harder
for students to find than $1Opes of lines of discrete

points. Items 37 and 39 represented graphs of lines

99

with negative slopes. These two items were relatively
harder for all grades than the items (18 and 24) which
represented graphs of lines with positive slopes. The
slopes of the graphs for items 18, 24, and 37 were
rational numbers whose denominators were one (i.e.,

the change in x was one). The slope of the graph in
item 37 was 5/2 (i.e., the change in x was two). This
item was the most difficult of the subtest for students
of all grades.

On Subtest 5, the performance of the upper half of
the sixth grade pupils compared well with that of
secondary students who were much more experienced with
the concept (operations with signed numbers) tested.
Both grades of secondary students had studied a chapter
in the regular algebra textbook treating operations with
signed numbers, as well as having had extensive experi-
ence with signed numbers acquired from the solution of
equations. In addition, these students were introduced
to signed numbers in the eighth grade textbook and tenth
graders had studied the topic again in general mathe-
matics. The means of the elementary students, except
the sixth—high graders, were less than 50 per cent; that
of secondary students and the sixth-high graders was
above 50 per cent.

The greater mathematical experience of secondary

students with signed numbers was evident in Subtest 5

100

TABLE 22

MEAN SCORE AND STANDARD DEVIATION BY GRADES
FOR SUBTEST 5: OPERATIONS WITH
SIGNED NUMBERS

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 1.38 2.11 1.43 2.63 2.15
Mean Per Cent 34.5 52.8 35.8 65.8 53.8
Standard Deviation .34 .27 .37 .28 .42

 

TABLE 23

INDEX OF DIFFICULTY BY ITEM FOR SUBTEST 5:
OPERATIONS WITH SIGNED NUMBERS

 

 

 

Grade
Item
Number
5 6 High 6 Low 9 10
ll 61 38 50 15 37
29 63 41 80 25 38
31 63 51 60 57 62
36 80 74 80 36 56

 

in which item 11 required addition of signed numbers,
item 29 required multiplication of signed numbers, and
items 31 and 36 used both operations. The limited
experience of elementary pupils should be considered when

appraising their performance on this subtest. From this

101

standpoint, the upper half of the sixth grade pupils
performed creditably.

The graph of an open sentence was a new concept for
all students. Achievement on Subtest 6 of the sixth-high
and the ninth graders was comparable. The performance of
the fifth, sixth-low, and tenth graders was comparable.
The standard deviation for the upper half of the sixth
graders was much greater than for any of the other
grades, indicating greater dispersion of scores.

In Subtest 6, items 21 and 33 displayed graphs for
which students were required to select the correct open
sentence. The graph of the open sentence for item 21,
discrete points, was somewhat easier than the graph for
item 33, a continuous line. Items 22, 32 and 35 gave
an open sentence for which students selected the correct
graph. The graph for the open sentence of item 35 was
a continuous line and the other two graphs represented
discrete points. Item 28 asked students to decide if
the graph correctly represented the given open sentence.
Table 23 lists the mean scores for this subtest for each
group in the study.

Selecting the correct graph for an open sentence
(items 22, 32, and 35) was easier for all students than
selecting the correct open sentence for a graph (items 21
and 33). In the latter case, except for the lower half

of the sixth grade, students found the concept slightly

102

TABLE 24

MEAN SCORE AND STANDARD DEVIATION BY GRADES

FOR SUBTEST 6: GRAPH—OPEN
SENTENCE RELATION

.m—...- w-“ .—
_.- -—

 

 

 

 

 

 

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 2.62 3.26 2.82 3.30 2.61
Mean Per Cent _ 43.7 54.3 47.0 55.1 43.4
Standard Deviation .58 1.15 .48 .59 .71
TABLE 25
INDEX OF DIFFICULTY BY ITEM FOR SUBTEST 6:
GRAPH-OPEN SENTENCE RELATION
Grade
Item
Number 5 6 High 6 Low 9 10
2- 64 45 72 49 63
22 57 27 36 41 59
28 40 2O 27 33 39
:2 53 33 64 33 59
33 73 61 69 55 61
35 53 54 57 56 72
easier for a graph of discrete points than for the
graph of a continuous line (items 21 and 35). When the

Open sentence was given and students asked to select

the correct graph, the continuous line was somewhat

103

harder for all students except for the fifth graders for
whom both concepts were equally difficult.

The mean of the ninth graders, who were the best
performers on Subtest 7,was 2.09, and only 52 per cent
of the total possible score. The mean of the upper
half of the sixth graders (1.98) was not quite half of
the total. The fifth graders had the lowest mean, 1.19
or 29.8 per cent. Table 27, likewise, reflects the low
achievement on this subtest. The scores of the upper
half of the sixth graders were the most dispersed and
those of the lower half of the sixth graders the most
homogeneous.

Each item of Subtest 7 extended one of the basic
concepts of the unit. Items 13 and 14 related to naming
the coordinates of a point, item 27 related to finding
the truth set of an open sentence and item 38 explored
the relation of the y-intercept and the open sentence.

Item 13 was intended to be the easiest of this
subtest. A right triangle, for which the coordinates
of the vertices of the acute angles were given, was
pictured in quadrant III. Students were asked to choose
the coordinates of the vertex of the right angle. The
knowledge necessary to making the correct choice was
tantamount to naming the coordinates of a point, yet more
than 50 per cent of the fifth, sixth-low, and tenth

graders missed the question.

104

TABLE 26

MEAN SCORE AND STANDARD DEVIATION BY GRADES
FOR SUBTEST 7: EXTENSION OF CONCEPTS

 

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 1.19 1.97 1.10 2.09 1.52
Mean Per Cent 29.8 49.4 27.5 52.2 37.9
Standard Deviation .42 .63 .18 .23 .34

 

TABLE 27

INDEX OF DIFFICULTY BY ITEM FOR
SUBTEST 7: EXTENSION OF

 

 

 

CONCEPTS
Grade

Item

Number 5 6 High 6 Low 9 10
13 59 36 62 21 53
14 83 56 85 70 91
27 65 64 72 41 49
38 77 59 69 57 61

Item 14 pictured a rectangle in quadrant I, the
coordinates of whose opposite vertices were listed as
Pairs of letters. Students were asked to select the

coordinates of one of the other vertices. The choice

105

required complete understanding of the meaning of ordered
pairs of numbers. The anticipated outcome was that
secondary students who had a semester's experience
operating with variables, would be capable of the
abstract thinking required to make the correct choice,
but that elementary pupils would not. Instead, the item
was easiest for the upper half of the sixth graders and
most difficult for tenth graders for whom the number of
correct responses may be attributed to chance alone.
This item was the most difficult one of the subtest for
every class except the upper half of the sixth graders.

Item 27 listed an open sentence with a positive
slope. Students were asked to decide if y increased
when x increased, decreased when x increased, etc. They
could either substitute values for x and determine y,
using their knowledge about open sentences, or visualize
the graph of the open sentence. This item was easier for
secondary students than for elementary pupils. It was
the most difficult item of the subtest for the upper half
of the sixth grade.

Item 38 listed an open sentence with letters for
the coefficient of x and for the constant term instead of
numbers. Students were asked to choose the coordinates
of the y—interCept. Again, the anticipated outcome was
that secondary students having had greater experience with

variables and generalization would significantly outscore

106

elementary pupils whose experience with letters as
variables was meager. However, more than half of the
students of all the grades missed the question. The
upper half of the sixth graders performed as well as
the ninth and tenth graders. Generalizing concepts
appears to be as difficult for secondary students as
for elementary pupils according to the results of
Subtest 7 .

When considering the mean achievement by grades
for the complete test, the most outstanding result was
that, on the same test, the upper half of the sixth
graders emulated the upper third of the ninth graders
who represented a more select subset of their class
with three more years of mathematical experience.
Moreover, these three years included one—half year of
algebra.

Comparison of the means of the fifth, tenth, and
lower half of the sixth grade may not be so obvious as
the comparison of means of the upper half of the sixth
grade with the ninth. However, the analysis of variance
established that there was no significant difference in
achievement between the fifth and tenth grades or between
the lower half of the sixth grade and the tenth grade on

this test on coordinate geometry.

107

TABLE 28

MEAN SCORE AND STANDARD DEVIATION BY GRADES
FOR TOTAL TEST ON COORDINATE GEOMETRY

 

"—=

 

 

 

Grade
5 6 High 6 Low 9 10
Mean Score 18.A9 2U.73 18.19 24.73 20.16
Mean Per Cent “6.2 61.8 45.5 61.8 50.4
Standard Deviation 8.76 3.32 3.57 2 U6 1 98

 

Student Reaction to

 

Coordinate Geometry

 

In order to appraise the reaction of the elementary

pupils to the unit in coordinate geometry,

two questions

were reproduced and distributed with the final test. The

questions were:

1. How well did you like the unit in coordinate

geometry?

a. More than mathematics from the regular

textbook.

b. As well as mathematics from the regular
textbook.

c. Less than mathematics from the regular
textbook.

2. How interesting did you find coordinate

geometry?

a. More interesting than the mathematics in

the regular textbook.

108
b. As interesting as the mathematics in the
regular textbook.

0. Less interesting than the mathematics in
the regular textbook.

Students were asked to respond honestly to the questions
in order to evaluate the suitability of the materials for
elementary pupils. For comparative purposes, secondary
students were likewise asked to respond to similar ques-
tions with appropriately modified responses:

1. How well did you like the chapter on coordinate
geometry?

a. More than the previous chapters in the
algebra textbook.

b. As much as the previous chapters in the
algebra textbook.

c. Less than the previous chapters in the
algebra textbook.

2. How interesting did you find the chapter on
coordinate geometry?

a. More interesting than previous chapters in
the textbook.

b. As interesting as previous chapters in the
textbook.

c. Less interesting than previous chapters in
the textbook.

Table 29 presents the results of student response
to the question "How well did you like the unit in coordi-
nate geometry?"

The response of elementary pupils to the unit in
coordinate geometry was much more favorable than that of

secondary students to the comparable chapter in their

109

TABLE 29

STUDENT REACTION TO COORDINATE GEOMETRY

 

 

 

Grade
Reaction
5 6 High 6 Low 9 10
Liked more than regular
mathematics 63.6 62.5 64.1 27.4 29.7

Liked as well as
regular mathematics 23.4 25.0 20.5 34.1 34.7

Liked less than regular
mathematics 13.0 12.5 15.4 38.5 35.6

 

textbook. While consideration must be given to the sub—
jectivity of this appraisal and to factors which could
have influenced opinions, such as teacher approval and the
Hawthorne effect, still the results of the reactionaire
reflected a highly positive attitude on the part of ele-
mentary pupils. About 60 per cent of the elementary
pupils indicated they liked the coordinate geometry unit
more than the mathematics in their regular textbooks. The
fact that the unit differed from their regular mathematics
program undoubtedly accounted for much of its appeal to
the younger pupils. Less than 30 per cent of the secondary
students liked the coordinate geometry chapter more than
other chapters in the algebra textbook.

The favorable reaction of the elementary pupils

agrees with teachers' Opinions about the reception of the

110

unit. They reported at workshOp sessions that children
were very enthusiastic about the unit and enjoyed it im-
mensely. Moreover, even the sixth-low elementary school
pupils participated with interest and learned the con—
cepts. Sixty—four per cent of this group liked coordi-
nate geometry more than their regular mathematics program.
Table 30 presents the results of student response
to the question "How interesting did you find coordinate

geometry?"

TABLE 30

STUDENT INTEREST IN COORDINATE GEOMETRY

 

Percentage by Grade

 

Interest Rating
5 6 High 6 Low 9 10

 

More interesting than
regular mathematics 68.8 62.5 59.0 28.9 37.6

As interesting as
regular mathematics 22.1 20.0 30.7 33.3 35.6

Less interesting than
regular mathematics 9.1 17.5 10.3 37.8 26.7

 

About 65 per cent of the elementary pupils indicated
that the coordinate geometry unit was more interesting than

their regular mathematics. Tables 29 and 30 indicated that

111

elementary pupils reacted more positively to the unit in
coordinate geometry than did secondary students to the
coordinate geometry chapter in their algebra textbook.
The final chapter of this dissertation will sum-
marize the findings of this study, draw conclusions, and

consider the educational implications.

CHAPTER V

CONCLUSIONS AND EDUCATIONAL IMPLICATIONS

The present study compared the achievement of ele-
mentary and secondary students on a comparable unit in
coordinate geometry, a topic new at the elementary school
level. A unit was written utilizing the discovery ap—
proach and piloted in two elementary school classes. Ele-
mentary school teachers who volunteered for a workshop
taught by the researcher, introduced the material to their
pupils. A test was developed and administered to these
pupils upon completion of the unit. The same test was
administered to the students of randomly selected first
year algebra classes upon completion of the chapter in
the first year algebra textbook dealing with the same
concepts of coordinate geometry that were encompassed in
the elementary school unit. The results were subjected to

analysis of variance and Scheffé's post hoc test.

112

113

Conclusions

 

Achievement in Coordinate

 

Geometry

Eight hypotheses were tested. At the .05 level of

confidence, statistical tests failed to reject the follow-

ing five hypotheses:

Hypothesis B.

Hypothesis D.

Hypothesis F.

Hypothesis G.

Hypothesis H.

At the

There is no difference in achievement on a
unit in coordinate geometry between fifth
graders and tenth graders.

There is no difference in achievement on a
unit in coordinate geometry between sixth
graders and tenth graders.

There is no difference in achievement on a
unit in coordinate geometry between the
upper half of the sixth grade students as
measured by a general mathematics achieve-
ment test and ninth graders.

There is no difference in achievement on a
unit in coordinate geometry between the
lower half of the sixth grade class as
measured by a general mathematics achieve-
ment test and tenth graders.

There is no difference in achievement on a
unit in coordinate geometry between the
upper half of the sixth grade as measured
by a general mathematics achievement test
and ninth and tenth graders combined.

.05 level of confidence, statistical tests

rejected the following three hypotheses:

Hypothesis A.

Hypothesis C.

There is no difference in achievement on a
unit in coordinate geometry between fifth
graders and ninth graders.‘

There is no difference in achievement on a
unit in coordinate geometry between sixth
graders and ninth graders.

114

Hypothesis E. There is no difference in achievement on
a unit in coordinate geometry between
fifth and sixth graders combined and
ninth and tenth graders combined.

In these three cases, secondary students achieved at a

higher level than elementary school pupils.

Though not a prime concern of the study, the achieve—
ment of the fifth graders was compared to the sixth
graders and the achievement of the ninth graders was com-
pared with the tenth graders. Statistical tests failed to
reject the hypothesis that there is no difference in
achievement on a unit in coordinate geometry between fifth
and sixth graders. On the other hand, statistical tests
rejected the hypothesis that there is no difference in
achievement between ninth and tenth graders on a compar-
able unit in coordinate geometry. Ninth graders achieved
at a higher level than did tenth graders, due perhaps to
the selection and placement procedures which assigned
students with higher achievement to algebra in the ninth
grade, while other students completed general mathematics
in the ninth grade and algebra in the tenth.

Evidence compiled in this research project appeared
to warrant the introduction of coordinate geometry into
the upper elementary school mathematics curriculum. The
upper half of the sixth graders, who in all probability

would study algebra in the ninth grade where coordinate

geometry has been taught traditionally, did just as well

115

as ninth graders on the TOCG. In fact, the median for the
upper half of the sixth grade was only .01 of a point
below the median of the ninth grade. Not only were the
upper half of the sixth graders as successful as the

ninth graders, but the lower half of the sixth graders,
those students who would probably study algebra in the
tenth grade, did as well as tenth graders. Even the fifth
graders, with still less mathematical experience, did not
differ significantly in achievement from the tenth graders.
These data indicate that upper elementary school pupils
learn that part of coordinate geometry dealing with the
graphs of linear equations as well as first year algebra

students.

Achievement by Subtest

 

Achievement in coordinate geometry was further
analyzed by concepts. Seven concepts were identified as
the component tasks of the coordinate geometry unit. The
instrument to measure achievement was constructed so that
it could be divided into seven subtests related to the
following concepts:

Subtest l: Plotting and Recognizing Points in the
Coordinate Plane

Subtest 2: Recognizing Members of a Truth Set

Subtest 3: Intercept Relation to Open Sentence
or Graph

Subtest 4: Slope-Graph Relation

116

Subtest 5: Operations with Signed Numbers

Subtest 6: Graph-Open Sentence Relation

Subtest 7: Extension of Concepts

In writing the unit and constructing the test, the
anticipated outcome was that elementary school pupils
would find plotting points in the coordinate plane and
recognizing members of a truth set (Subtests l and 2)
easy, but that they would find such concepts as slope of
a graph or the slope-intercept form of an open sentence
(Subtests 4 and 6) too difficult. This hypothesis was
accompanied by the speculation that all seVen concepts
would be comprehended more fully by secondary students.
However, an analysis of variance established that there
was no significant interaction between Subtests and
Grade. In other words, elementary school pupils scored
as well as secondary students on the Subtests.

Examination of the index of difficulty of indi-
vidual items by subtest revealed some interesting simi-
larities between elementary and secondary school students.
For example, identifying the slope of the graph of the
same open sentence was more difficult for both sets of
students when the graph was represented by discrete points.
Selecting the graph of an open sentence was more diffi-
cult for all students than selecting the correct open
sentence for a graph. Likewise, graphs with negative

slopes and graphs in which the change in x was not one

117

was as hard for secondary school students as for ele-
mentary school pupils. However, operations with signed
numbers were easier for secondary students who had much
more experience with this concept than for the elementary
school pupils who had no previous experience operating
with signed numbers.

Contrary to expectation, all of the component con-
cepts of the unit in coordinate geometry were understood
as readily by elementary school pupils as by secondary
school students. Caution should be exercised, however,
in generalizing these results because of the number of

items in some of the subtests.

Attitude Toward Coordinate
Geometry

In responding to a reactionaire, approximately two-

 

thirds of the elementary pupils preferred the unit in
coordinate geometry to their regular mathematics program.
Only one-third of the algebra students liked coordinate
geometry more than algebraic tOpics. Likewise, a similar
prOportion of each group rated coordinate geometry more
interesting.

Some reservations must be considered when evaluating
these responses. Elementary school pupils may have re-
sponded as they perceived their teachers might hope that

they would. The appeal of studying something different

 

lk .‘ - .
'

3 Wk.

.bn»

o. . shri—

 

118

from their textbooks and the operation of the Hawthorne

effect may have influenced their opinions.

Implications of the Study

 

Implications for the Mathe-
matics Curriculum

 

 

Data from this study indicated that the vision of
the Cambridge Conference on School Mathematics (CCSM)
may not have been unrealistic. In the recent past, geo-
metry had not been viewed as a salient area of study for
elementary school pupils. However, in this study, the
upper half of the sixth graders performed as well as ninth
graders and the lower half of the sixth graders performed
as well as the tenth graders on comparable coordinate geo—
metry concepts. Previously cited studies concluded that
elementary school pupils were capable of learning other

54

geometric concepts. These studies supported the recom-

mendations of many mathematics educators who favored the

inclusion of more geometry in the elementary school cur-

55

riculum. With the recognition that young children can

 

5"Fitzgerald, loc. cit.; Sair Ali Shah, loc. cit.
55Irvin Brune, "Geometry in the Grades," Enrichment
Mathematics for the Grades, Twenty-seventh Yearbook of the
National Council of Teachers of Mathematics (Washington,
D.C.: National Council of Teachers of Mathematics, 1963),
pp. 134-147; Lenore John, "Geometry for Elementary School
Teachers," American Mathematics Monthly, LXVII (April,
1967), 374-376; Anne Peters, "Articulating Geometry Be—
tween Elementary and Secondary School," National Associa-
tion of Secondary School Principals Bulletin, XLIII (May,
1957), 131-133.

 

 

 

119

learn more than had been supposed, such non—metric tOpics
as matrices, probability, and modular arithmetic, which
the CCSM proposed for the elementary curriculum and which
teachers had questioned as appropriate, may be within the
comprehension of elementary school pupils.

Introducing coordinate geometry at the elementary
school level may be justified since it relates counting
and number, bridges the gap between adding and subtracting
and the properties of number, offers first-hand experience
with positive and negative numbers and provides an informal
introduction to important mathematical concepts as variable,
slope, and function. Coordinate geometry also affords an
opportunity for students to learn by discovery and to dis-
cover patterns. Both of these activities encompass the
very essence of "modern mathematics" as this term applied
to the K-12 curriculum.

Examination of the achievement of secondary students
on the coordinate geometry chapter in the first year
algebra textbook as measured by the TOCG showed that the
mean of ninth graders, who represented the upper third of
the class, was 24.73, or 60 per cent of the possible total.
The mean of the tenth graders, who represented the middle
third of the tenth grade class, was 20.16, or 50 per cent.
These values may be low if the current emphasis on "quality
education" also incorporates "quality performance." On

the same TOCG, the mean of the upper half of the sixth

120

graders, as determined by a general achievement test, was
as high as that of the ninth graders and the mean of both
the lower half of the sixth graders and of the fifth
graders was not significantly different from the tenth
graders. Thus previous exposure to coordinate and other
geometric topics, beginning at the elementary school level
and continuing through junior high school, could provide
a foundation of knowledge which could be expanded and
developed in greater depth and perhaps thereby increase
achievement of secondary school students. Spiralling
geometry throughout elementary school appears to be a
practical and efficient procedure to increase understand—
ing and competence of students.

At the time of this study, secondary school students
studied coordinate geometry in the first year algebra
course and Euclidean geometry a year later. Both of these
courses were presented by the lecture method even though
secondary students had had little previous experience
with geometric concepts. Thus, secondary students met
these concepts for the first time at the formal Operations
stage of learning according to Piaget's classification.
Developments in learning theory indicate the desirability
of modifying pedagogical techniques at the secondary
school level so that students are encountering new con-

cepts at the concrete and earlier stages of learning.

121

Implications for In-Service
Education for Teachers

 

57

Research conducted by Hammond,56 Houston, and

Ruddell and Brown58

documented significant improvement in
the achievement of pupils whose teachers were participat-
ing in in-service workshOps. Weaver cautioned mathematics
educators to be realistic in designing teacher preparatory
programs. Materials must be developed that are related to
the textbooks used by elementary personnel. These mat-
erials must begin at the prevailing level of understanding
and be paced accordingly.59
Recent research implied that in-service education

for elementary school teachers warranted treatment dif—

ferent from that of mathematics education courses designed

 

56Harry Hammond, "Deve10ping Teacher Understanding
of Arithmetic Concepts Through In-Service Education"
(unpublished Ph.D. dissertation, University of California
at $05 Angeles, 1964), Dissertation Abstracts, XXV,
513 . .

57W. Robert Houston, "Mathematics In-Service Educa-
tion: Teacher Growth Increases Pupil Growth," Arithmetic
Teacher, X (May, 1963), 243-247.

58Arden Ruddell and Gerald Brown, "In-Service Edu-
cation in Arithmetic; Three Approaches," Elementary School
Journal, LXVII (April, 1964), 417.

59Fred Weaver, "Non-Metric Geometry and the Mathe-
matics Preparation of Elementary School Teachers," The
American Mathematics Monthly, LXXXIII (December, 19667,
1115-1121.

 

122

for preservlce teachers}0 The present study indicated
that when in-service training was planned to deal with
content, methods and materials that teachers could use
directly in their classroom programs, their students
learned the intended concepts. The elementary school
teachers who participated in the workshop conducted for
this study did not have particularly strong mathematics
backgrounds. Yet these teachers, with the help of a
mathematics specialist, broadened their mathematical knowl-
edge on a specific topic and developed and used materials
so effectively that the achievement of their pupils was
almost as great as the achievement of secondary students
(whose teachers held mathematics majors or minors) on

the same topic.

A potential implication of the findings of this study
is that subject matter specialists might provide a method
for improving classroom effectiveness until teacher training
programs, such as those supported by CUPM and CCTT, can be
effected or until some modification in the structure of
elementary schools, such as team teaching, enables teachers
to become specialists in certain fields and pool their
strengths.

There is no reason to believe that the achievement

of the pupils whoSe teachers participated in this workshop

 

60Thomas c. Gibney, John L. Ginther, and Fred L.

Pigge, "The Mathematical Understandings of Preservice and
In-Service Teachers," The Arithmetic Teacher, XVII (Feb-
ruary, 1970), 155-162.

 

123

was peculiar to this special unit. The cooperative en-
deavors of elementary school teachers and subject matter
specialists could improve the mathematics program by
broadening the teacher's knowledge and keeping him abreast
of current materials and methods just as the recent coop-
eration of mathematicians, educators, and classroom
teachers improved the content, itself, of the mathematics
curriculum.

Implications for Further
Research

 

The fact that a specific topic in coordinate geometry
can be learned as efficiently by fifth and sixth graders
as by ninth and tenth grade algebra students does not imply
that it should automatically be assigned to these elemen-
tary grades. This philosophical issue is so important and
encompassing that it must be resolved before delegating
this concept to the fifth or sixth grade. The achievement
of pupils in other elementary grades on the graphing of
linear equations and other geometric concepts should be
researched. Nor should the introduction of new material
at the elementary level be limited to geometry. The con-
cepts of matrices, logarithms, trigonometric functions,
and conic sections, to name but a few, are also worthy
of investigation. Component or subordinate tasks of these
concepts should be identified and assigned to the level

appropriate to the child's comprehension.

124

Secondary school students studied a unit in coor-
dinate geometry taken directly from their textbooks while
elementary school pupils studied from specially produced
materials. This procedure, recognized as a potentially
limiting factor, may have influenced results through
operation of what is commonly referred to as the "Hawthorne
Effect." A replication of this study using special mater-
ials for the secondary sample should be undertaken to test
the interaction between student and specially produced
materials.

In this study, the alarmingly low achievement (50
to 60 per cent) of the secondary students on coordinate
geometry may reflect the inadequacy of a one-chapter
encounter with Cartesian geometry. The presentation of
geometric topics at the elementary level, even if limited
to the most rudimentary concepts, may favorably influence
student achievement in subsequent grades. Therefore, a
longitudinal research project is recommended to investi-
gate the effect of earlier introduction of coordinate
geometry on pupil achievement when they again encounter
this topic in high school algebra.

The instrument utilized in the study was developed
for one geographical area, and primarily with students
whose general mathematics achievement was slightly below
the national test norms. Data in this present study indi-

cated that the instrument might be more widely applicable.

 

 

 

 

 

However, full range of its use and usefulness has not been
tested, thus suggesting further study and refinement of
instrument sub-categories and items.

The study of curriculum and the placement of topics
are of utmost importance. Some learning theorists have
hypothesized that topic placement is a function of prior
learning in a particular field. This is especially true
for mathematics, where a relatively linerar curriculum
pattern pervades most textbook series, and continuous
progress plans are being strongly recommended.

In contrast, other learning theorists profess that
the method of learning is most important, and that topic
placement is not the vital concern of the curriculum
specialist. While the present study neither supports
nor denies either position, it does focus attention upon
the undergirding problem. Research must be undertaken
to test these theories. Only through careful study of
the many facets of curriculum and content placement can

a more viable mathematics program evolve.

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126

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APPENDICES

132

APPENDIX A

TEACHER LESSON PLANS FOR ELEMENTARY

SCHOOL UNIT ON COORDINATE GEOMETRY

133

LESSON PLANS IN COORDINATE GEOMETRY
FOR ELEMENTARY TEACHERS

ILESSON 1. PLOTTING POINTS

Pupil Objective
Pupils will learn to plot ordered pairs representing

points in all four quadrants of the plane.

Overview for Teachers

Introducing the student to Cartesian coordinates (named
after Descartes, who developed analytic geometry) may involve
simultaneously, his first mathematical experience with signed
numbers. Thus, two important concepts must be developed:

a. The concept that a sign is a vital part of a number, and
b. The concept of an ordered pair.

The union of a horizontal number line and a vertical
number line forms a pair of axes which divide the plane into
four quadrants. These quadrants are numbered counterclock-
wise, I through IV, beginning with the upper right
quadrant. Numbers with positive signs lie to the right of 0
on the horizontal number line and above 0 on the vertical
number line. Negative numbers lie to the left of 0 on the
horizontal number line and below 0 on the vertical number

line.

 

134

135

In advanced courses the horizontal number line will be called
the abscissa and the vertical number line the ordinate.. The
intersection of the axes, (0,0), is called the origin.
Relevant to order, mathematical convention dictates that

horizontal distance be cited first and vertical distance
second. Thus, an ordered pair, called the coordinates of a
point, is conventionally enclosed in parentheses and the
numbers separated by a comma. For example, the ordered pair
2,3) designates the point which lies two units to the right
of the ordinate and three units above the abscissa and is

plotted:
0 (2,3)

This is not the same point as represented by the ordered pair
(3,2) which lies three units to the right of the ordinate

and two units above the abscissa:

0 (3:2)

Suggested Procedure

Let students review the location of points on a number
line. Draw a number line on the board and have students
represent points, such as 3, 5. 1, 7 and 4, on the number
line. Call attention to the fact that for each number there
corresponds one point and that for each point there

corresponds only one number. Represent several points on the

136
number line and let students name the coordinate. Now place
a point P one unit above the number line as follows and ask
students to name its coordinate}1
o

- P
4 A j A n +-

 

If students respond 3, put a dot 93 the number line at 3.
Ask if they think 3 adequately locates both points. If they
answer yes, ask for the coordinate of Q, which is one unit
above P and two units above the axis. Try to stimulate
students to suggest a vertical number line if the suggestion
has not arisen by now. You may want to remind them of the
one-to-one correspondence between points and numbers. When
the construction of a vertical number line has been suggested,
name several points such as P with coordinates (3,1) and

Q with coordinates (3,2)-
Q 0 (3,2)
P o(3,|)

l A l \
r

' V "

Ask a student to plot (1,2). If he plots
see if the class agrees. Possibly someone will plot this

point:

A I

Here are two differentIdeas for the position of one set of
coordinates. When the coordinates of a point are named, we
must be sure that everyone plots the same point. Therefore,
mathematicians have agreed to name the horizontal distance
first and the vertical distance second. To emphasize that the

order of naming the coordinates determines different positions

137

for the point represented, have pupils plot several more
points, interchanging their coordinates, as (4,3), (3,4);
(1,5),‘ (5,1), until they recognize the importance of
order.

When students can plot points in the first quadrant,
ask if they think that the horizontal number line ends at
zero. (They probably already know that it continues to
the left of zero). To motivate continuing the vertical
number line below 0, try to guide the discussion to the
consideration of temperature. When pupils suggest "2
below zero," ask if they know the mathematical names for
these numbers. 'If they do not, name them "negative num-
bers" and discuss possible applications as below zero
temperatures, below sea level, debit, and other ideas
they should suggest. Treat positive and negative num-
bers as opposites--if positive numbers indicate "up," then
negative numbers indicate "down." If positive numbers in-
dicate "gain," than negative numbers indicate "loss."

Continue to emphasize order. Stress the location of
positive numbers to the gighg of the ordinate and gbgyg the
abscissa and the location of negative numbers to the lggt,
of the ordinate and bglgw the abscissa. Thus, ("1.3) is
located 1 unit to the left of the ordinate and 3 units above
the abscissa; ('1,’2) is one unit to the left of the

ordinate and 3 units below the abscissa.

Activities for Pupils

1. Send pupils to the blackboard or overhead projector to

138

plot points.

9. Distribute geoboards, pegboards, or checkerboards so
that a small group of pupils (3 or 4) has access to one of
these devices. Cut plastic straws into little_cylinders
for use with geoboards. Place rubber bands around the
middle row and center column of pegs to represent the axes.
Use straight pins for the checkerboards and golf tees for
the pegboard. Call out ordered pairs, as (3.4). Pupils
with geoboards can place a plastic cylinder over the nail

3 units to the right of the vertical axis and 4 units above
the horizontal axis. Pupils with pegboards can insert golf
tees into the proper hole while pupils with checkerboards
can stick a straight pin into the intersection of the
correct horizontal and vertical line.

3. Distribute Exercise 1. Have pupils complete this

exercise during the class period.

139

LESSON 2. LISTING TRUTH SETS

Pupil Objective

 

Pupils will list some elements of the truth set of
an Open sentence with two variables (1) by listing ordered
pairs as elements of a truth set, and (2) by constructing

a table.

Overview for Teachers

 

This lesson prepares the pupil for graphing linear
functions. The truth set of an open sentence with two
place holders (‘Zi and [3 represent the two variables)
consists of an ordered pair. The ordered pairs which make

a particular equation true can be listed as the elements

of a set. For example, {(-1,-3),(0,0), (l,3),(2,6), (3,9)...}

are a few elements of the open sentence Z§.= 3 x [3.
A table is another convenient device for displaying the

truth set in an open sentence.

 

D A
3
6
9

 

l
2
3

Suggested Procedure

 

If necessary (depending upon the class' background)
discuss open sentences and their classification as true
or false depending upon the numbers substituted for the
placeholders. Ask pupils to consider the open sentence

[3 = [j + l where [X and E] both represent numbers.

lUO

Replacing the box and triangle requires a pair of
numbers, one of which replaces box; the other replaces
triangle. We agree to name the number that replaces

box first and the number that replaces triangle second.
Ask a pupil to name a pair of numbers. If he names
(2,0), then 2 replaces box and U replaces triangle. The
sentence becomes: u = 2 + l. A convenient method of
demonstrating the replacement of Z: and E] with numbers
is to write the numbers within the symbols, as

Z§5= [:]+ 1. This sentence is false. Let other pupils
name pairs of numbers. Substitute the first number
named into the box, the second into the triangle and
classify the sentence as true or false accordingly.
Tell pupils to look for values that make the sentence
true. Continue to emphasize the importance of order as
pupils suggest pairs of numbers. The pair, (“,5),
produces a true statement for the above open sentence:
ZQ§= [:]+ l, or 5 = 5. But the ordered pair (5,”) makes
the open sentence (£3: [:]+ l, or u # 6, false. Call
the pairs of numbers which make the sentence true the

truth set of the open sentence.

 

A table provides another convenient method for
listing the members of a truth set. Consider another
open sentence such as [3 = E] + 3. Let the pupils make
a table of ordered pairs of numbers which make this

sentence true .

lul

 

MHOD
U'IJrLAJD

 

Activities for Pupils

1. Relay game. Divide the class into teams. If
pupils are arranged in rectangular arrays, let each column
be a team. (Shift pupils so that each column contains the
same number.) Write the same open sentence on a piece
of paper—~one for each column. Include a blank table.
Let each pupil fill in a pair of numbers on the table and
pass it to the pupil sitting behind him. The column that
completes the table first with every pair a correct
element of the truth set wins the relay.

2. Distribute Exercise 2. Let pupils complete

exercise during the class period.

1A2
LESSON 3. GRAPHING THE TRUTH SET OF AN OPEN SENTENCE

Pupil Objective

(1) Pupils will plot the graph of the truth set of
an Open sentence with two placeholders. (2) Pupils will
recognize that the graph of an open sentence of the form

z: = x []‘+ __ is a straight line.

Overview for Teachers

 

Pupils have already represented the truth set of an
open sentence with two variables by listing the ordered
pairs as elements of a set and by making a table. A third
way to represent these truth sets is by graphing. Let [3
denote the abscissa (horizontal axis), and [5 the ordinate
(vertical axis). After plotting several graphs, pupils
should realize that the graphs of the truth sets suggest

straight lines.

Suggested Procedure

 

1. Review the methods for writing the truth set of
an equation, such as A = C] + 2 by
a. Listing the truth set of ordered pairs as
elements of a set:
{(0,2), (1,3), (2.“).(3,5)...}
b. Making a table of some of the elements of the

truth set.

1113

 

semi—JO D
mzwm D

 

2. Ask pupils if they can think of another way to
represent the truth set of the open sentence, A= D + 2.
If dead silence ensues, suggest that there is a method
which relates to the new information they learned in the
last few days. If further hints are needed, remind them
of plotting points on the number line and the agreement to
plot the horizontal distance first and the vertical
distance second. Ask if any other agreements were made
about open sentences. Hopefully, some pupil will recall
that in finding the truth set of an open sentence, E]
was named first and Z3 was named second. Ask if they
can relate the two agreements. Try to stimulate pupils
to suggest naming the horizontal axis, C], and the
vertical axis, [3.

3. Consider the open sentence Z§w= 2.x E] . Send
various pupils to the board to plot points representing
elements of the truth set. Plot a few additional graphs,
and leave all of the graphs on the board. Ask pupils what
the picture of an open sentence of the form

[3 = __ X E] + __ would be.

14“

Activities for Pupils

 

1.

Graph several open sentences by sending pupils

to the blackboard to plot elements of the truth set. (Or

mark points on an overlay.)

2.

a.

3,.

Play Tic-Tac-Toe.

On the blackboard.

With teams of boys against girls. Arrange

desks or chairs in rectangular array. Number
rows and columns. Vacate desks. Team captains
will name coordinates of point. Team members, in
turn, will sit at designated desks. Team placing
four boys or four girls in row, column or
diagonal wins.

Distribute Exercise 3. Have students complete

the problems during the class period.

1&5

LESSON U. DISCOVERING SLOPE PATTERN

Pupil Objective

 

To discover the pattern of the graph of an open

sentence.

Overview for Teachers

 

Pupils have discovered that the truth set of an
open sentence (equation) which has two variables, E]
and [3 (x and y), can be displayed by a graph which will
be a straight line. This graph has a "pattern." The
pattern, which is called the slope of the graph, is not
always the same. The slope depends upon the open sentence.
In analytic geometry the slope pattern is defined as the
ratio of the vertical change compared to the horizontal
change. Since it is easier to see the slope pattern if
only whole numbers are used, use whole numbers only when
first introducing the concept. In the graph of
A: =:[] + 2, the slope pattern is: over one square to

the right and up one square.

A

t

 

 

 

E] A

0 2

l 3

2 u " ¥

3 5 Th .__’_
_J

 

1M6

Suggested Procedure

 

List an open sentence, suchvas Z§== E] + 2, with
a table containing about three elements of the truth set.
Have pupils verify that these elements do belong to the
truth set. Plot the points on a pegboard with golf tees
after verification. Then ask a pupil to "place" another
point without doing any arithmetic. After a point has
been plotted, let pupils determine its coordinates and
verify that the point is an element of the truth set by
substituting the coordinates into the equation.

For [3 = E] + 2, the pattern is "over one square to
the right and up one square." Do not tell your pupils
this pattern. Lead them to discover it by asking some
pupil to place another golf tee on the graph simply by
observing the pattern without doing any arithmetic.

After a point is placed, have another pupil name its
coordinates and substitute them into the equation to see
whether the resulting statement is true or false. Test

all points. This process also enables the pupil to see

the relation between the points on the graph and the numbers
substituted for D and A .

Graph the truth set of the open sentence,

[3 = 2 x E] - 1. Have pupils determine if the graph is
correct by verifying each point. Have a pupil plot a point
on the pegboard by observing the pattern and not doing any

arithmetic. (Pattern is one square right, one square up.)

1H7

>4>

 

 

 

 

 

 

 

 

 

 

 

 

 

>-C]

Verify that the coordinates of this point makes the sentence
true; e.g., the coordinates of Q when substituted into the
sentence results in u = 2 x 3 — l, or U = 5, which is
false. The coordinates of P, (3,5), when substituted into
the equation, gives: 5 = 2 x 3 - l, or 5 = 5, a true
sentence.

Do not tell the pattern to pupils. Moreover, as each
child discovers the pattern, encourage him to keep it a
secret so that the other pupils can discover it for them-

selves.

Activities for Pupils

 

1. Prepare several graphs for overhead projection.
Have pupils place additional points on these graphs by
determining the "pattern."

2. Plot the graph of an open sentence on a pegboard.
Have pupils place golf tees to represent additional points
without doing any arithmetic and then substitute the
coordinates into the open sentence to verify that the point
belongs to the truth set.

3. If floor is tiled, agree on number of tiles per

unit. List open sentence and a table of a few elements of

1H8

truth set on the blackboard. Let pupils be "points"
and stand on intersections of tiles corresponding to the
coordinates of the points listed. Then ask another
pupil to stand on intersection of "tile lines" so that
he will be a member of the truth set (by observing the
geometry of the slope pattern and not doing any arith-
metic).

H. Distribute Exercise u. Have pupils complete the

problems during the class period.

1&9

LESSON 5. DISCOVERING RELATION OF SLOPE PATTERN AND A"

INTERCEPT TO THE OPEN SENTENCE

Pupil Objective

 

(l) Pupils will discover the relation of the
geometrical pattern of the graph and the open sentence.
(2) Pupils will identify the intersection of the graph

and the [X axis from the open sentence.

Overview for Teachers

 

‘The pattern of the graph of the truth set of an
open sentence can be determined without plotting a single
point. One can also determine the point of intersection
of the graph and the [5 axis without plotting points.

The following exercises are constructed to direct the
pupil's attention to the relation between these two

concepts about the graph and the open sentence.

Suggested Procedure

 

Distribute Exercise 5. Each pupil will construct
tables and plot the graphs on the exercise. Encourage
pupils to work individually. Pupils will study each Open
sentence and its graph to see if they can see a relation

between the two.

Activities for Pupils

Distribute Exercise 5.

150

LESSON 6. DETERMINING THE OPEN SENTENCE FROM A GRAPH

Pupil Objective

 

Pupils will discover the open sentence when the

graph of the truth set is displayed.

Overview for Teachers

 

Pupils who have discovered the relation between the
open sentence and the slope of its graph, and the relation
between the open sentence and the ZX-intercept, should be
able to look at a graph and write its equation. (For
those who have not made this discovery, here is another
chance to make it.) The equation will be in the form
[5 ==___x [J + __, or y = mx + b, known in analytic
geometry as the slope—intercept form. Identification of
the pattern of the graph gives the slope coefficient-—
the number preceding box. The multiplier of box deter-
mines the pattern (slope) of the graph. In the equation
[5 = 2 x E] + l, the slope is found on the graph by
counting over 1 to the right and up 2. Two, the multiplier
of E] is the s10pe. The constant in the open sentence
indicates where the truth set intersects the Z: axis. In

this case, it is 1.

Suggested Procedure

 

1. Use overlays on overhead projector. Project a

few graphs. See if pupils can find their equations.

151

2. Plot graphs of lines on the pegboard with

golf tees. Have pupils find the equations.

Activities for Pupils

 

Distribute Exercise 6. Have pupils complete the

exercise in class.

152

LESSON 7. ADDING SIGNED NUMBERS

Pupil Objective

Pupils will find sums of signed numbers.

Overview for Teachers

 

Graphs whose 13 (y)—intercepts are less than 0
require some understanding of sums with negative addends.
Addition of signed numbers can be introduced in several
ways. In one method, a cricket hops on the number line.
Another method measures directed distances on the number
line. We will use still another method, the Madison
Project's "Postman Stories." In this scheme, (1) a check
is represented by a positive number, a bill by a negative
number. (2) Something brought to you is represented with
an addition sign, something taken away is represented by
the subtraction sign. (3) In a sum, something happens and

then something else also happens.

Suggested Procedure

 

l. Invent an arithmetic for numbers with signs, i.e.
the numbers left of zero on the horizontal axis, below
zero on the vertical axis, and positive numbers that the
pupils have been using. To aid this invention, make up
postman stories. For example, suppose the postman brings
a check for $4. We can represent this outcome as +H. If

he brings a bill for $2, we can represent this as -2. A

153

story based on this information might be: the postman

brings a check for $u and a bill for $2. Are you richer

or poorer? By how much? Or, if the postman brings a

bill for $2 and a bill for $5, are you richer or poorer?
2. Let students make up postman stories about

several problems.

Activities for Pupils
Distribute Exercise 7. Have pupils create stories

for each problem and tell the stories to the class. Have

pupils write the sum.

ISM

LESSON 8. MULTIPLYING SIGNED NUMBERS

Pupil Objective

 

Pupils will find products of signed numbers.

Overview for Teachers

 

Thus far, we have substituted only positive numbers
for box and have considered only equations with positive
slopes, i.e., as E] increased [3 also increased. In
order to graph any equation with two variables, pupils
should be able to substitute negative integers for box
and consider equations with negative slopes. Both
situations force the consideration of products with at
least one negative factor. The substitution of a negative
number into an open sentence with a negative slope makes
finding a product with two negative factors necessary.

Again, the Postman Stories of the Madison Project will be

a. A check is represented by a positive number

b. A bill is represented by a negative numberi‘

c. Something brought to you is represented by an
addition sign

d. Something taken away from you is represented
by a negative sign

e. In a product the second factor is the bill or
check, the first factor is how many bills or

how many checks.

155

Suggested Procedure

 

Make up a Postman Story to suggest the use of signs.
Example: In today's mail, suppose mother found that the
postman had brought 3 checks for $5 each. We could make
up this numerical problem about the checks: +3 x +5. Is
mother richer or poorer? (Richer). By how much?
+3 x +5 or +15. How did you get the answer? Suppose the
postman brings 2 bills for $6 each. Are we richer or
poorer? (Poorer). By how much? $12. How did you get

the answer? (+2 x -6).

Activities for ngils

 

l. Dictate some postman problems for students to
multiply and answer orally.

2. Distribute Exercise 8. Have students complete
the exercise in class.

3. Have students create a Postman Story for each

problem.

156
LESSON 9. GRAPHING SENTENCES WITH NEGATIVE SLOPES

Pupil Objective

 

Pupils will discover the pattern of the graph of an
open sentence in which the [X coordinate decreases when

the E] coordinate increases.

Overview for Teachers

 

Here we take a second look at the pattern of a graph.
This time, after the postman stories, we investigate
negative slope patterns in which [5 decreases as E]

increases.

Suggested Procedure

 

Plot several graphs on the pegboard having negative
slopes. Ask the children to find the slope patterns of
these graphs. In the following example, the SIOpe is

over one to right and down one.

 

 

 

 

 

 

All the graphs plotted so far have had positive slope
coefficients. That is, as [3 increased, [5 increased. Plot
the graph of £3 = 2 x C] + 3. Ask pupils what happens to

values of [S as values for E] increase. Suggest using the

157

:1r11,hnu3ti<: ol' ttu? lknstnuin thoxfiiezs ta) plx)t 1.h6? gIQipi) of‘
[X = - 2 x E] + 3. Now investigate what happens to

values of [X when values for [3 increase.

Activities for Pupils

 

1. Send students to the pegboard to plot graphs
with negative slopes.

2. Distribute Exercise 9.

158

LESSON IO. PLOTTING GRAPHS AS CONTINUOUS LINES

Pupil Objective

 

Pupils will plot the graph of an equation as a

line instead of discrete points.

Overview for Teachers

 

By substituting fractions as well as whole numbers
for box, the points of the graph get closer and closer
together. If we allow all kinds of numbers—~whole numbers,
fractions and irrational numbers--the graph would become
a continuous line. If any pupil suggests plotting

intercept and then the pattern, hurrah!

Suggested Procedure

 

Use overlays or the balckboard to graph the equation

[3 = 2 x E] + 3.
1. First with whole numbers
2. Then use halves

3. Then fourths

A. Then eights.

The slope pattern of a continuous line can be
determined more easily from the graph when only whole
numbers are substituted for box. To illustrate a con-
tinuous line, stick pins with large heads into the
pegboard at appropriate distances between the golf tees.

Perhaps pupils may see that they need only two

points to draw the graph of a continuous 1ine——two very

159

easy points are tkma [Ev—intercept and the slope
coefficient. If they do not make this observation, don't

tell them.

Activities for Pupils

 

Distribute Exercise 10.

APPENDIX B

PUPIL EXERCISES FOR ELEMENTARY
SCHOOL UNIT ON COORDINATE

GEOMETRY

160

161

EXERCISE I
I. Plot the following points.
a. i3,|) e. (2.5)
b. (4,3) f {-5,-2)
c. (-l,-3) g. (-l,-2)
d. (|,-3) h (2, -3)

 

2. The letters for the sets of
coordinates listed below spell a
special message for you. See if
you can find the message.

(l,|), {-5,-3l, (3,4), (-5,4)

(2,-ll, (4,0)

(O,-3), (5,3), (-2,|)

 

162

3. Write the coordinates of the
ordered pair representing:‘
a. Apple

 

Banana

 

Ring

 

Triangle

 

09.00-

Box

 

 

163

FOR FUN EXERCISES

That is, join a to b,

Plot the points and join them with lines in

n
O
O
.S
d
6d
tn
ta
0
1’
pd
80
Ct
n
EC
U
q,
SC
5
0
Gt
.0
Lb

Directions:

 

Exercise 2

Exercise 1

 

 

a
W)
3.1
w...
)\./\/ do \/ \.l )\./\..l
232 t \./ 2)))2 )))\/\I/ 212) \.I
111) )\./t 2 )13211 23219.. 11.1.17 2
. . .u)\/2hfi 010)]. 2 .111. )\)lllll ....\./l
“.2 n 2. ... 75. . . .. 7.
’33, ’3 m C., 9 3”!
5“».33 3,1300 3, ll 39.31 !!!!!!! 3.“..55 3,
....ll..drll _.\./.|.O. 355143145 ....33
(((((((/\(f(( (((((( ((((/\/I\/.\ (((l.\/\/\
Die P P
oooooooo on O o o o o 0 CO 0 o o o o 0 GO 0 o o o o o
abcdefeEEIItJTklmn0n$23rPtuvaxyzabc
81 S 8 abc
\.l \/
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lidqliz \sz) léliiQ?O\/)22Mw2\)\/4qcafib
,oxilFOifi4\/M._. _ .Onu_ . .0nu
’3’ 95 2- ’3’,
012 , 3” 33” ,,,,, ’32u5u

1118 3 3526 3 all/06213325571111
....1912150_.._.._._......

((((((((((((((((((((((((((

,Q'U O D S F‘KTa-P°SWN “30 9.0 C‘Q
o o o o o o o o o o o o o o o o 0

STOP:

X 2 < C d-m H

F’xTA-H‘Swm Hlm Q.O U‘m
oooooooooooo

STOP:

16“

Exercise 3.

 

(1, 11 1/3)

(3 1/2, 11)
(u, 10)

(3 2/3, 9 2/3)
(2 1/2, 10)
-1/2, 10)
'B/A: 12)

-1, 12 1/2)
-1 1/2, it 1/2)
-2 1/3, 15)
—2, 15 1/2)
-3, 15 1/3)
-2 2/3, 15)

between q and r.

(8, 10)

(9 1/2, 10)
(11 1/2, -1/2)
(8 1/2, -1)

(8 1/2: ‘7)
(-6 1/2, -7)
(-6 1/2, -1)

Do not draw line

aa.
bb.
cc.

STOP:

dd.
ee.
ff.
ge-
hh.
ii.
33-
kk.
11.
mm.
nn.
00.

PP-

QQo
IT.

53.
tt.

uu
VV.
WW

Exercise h.

 

('6: 2 1/2)

Do not draw line
between 1 and m.

m.
n.

Do not
betwee

(h, 10)
(8, 10)
(7 3/b,
(6 3/u,
(6, 12 3
(6 3/h,
(5 1/3,
(t 2/3,
(3 2/3:
(2 1/3,
(1 1/2,

-1/2)
10)

1o)

11)

12 1/2)

draw line
n cc and dd.

12 3/h)
13 1/2)
/b)
11)
11)
11 3/h)
13)
13 2/3)
13 1/2)

(0. 12)
-3/h, 12)
-1/2, 10)

-3 1/u, 10)
1/3, 7 1/2)
1/2, 7)

' g 7 1/2)
10)

STOP: Do not draw line

0.
p.

between n and o.

(6, 0)
(e, -2 1/2)

STOP: Do not draw line

d'm H.Q

between p and q.

165

 

s.0*o o S S lewn.H-:wm Ham o.o C‘m
O 0 O O O O O O I O O O O O O I I I

 

Exercise 54 Exercise 6.

(-22, 9) a. (- 3 1/2, -1o)
(~21, 8) b. ( -3 1/2, -lS)
(-17, -6) c. ( 5 2/3, -17)
(15’ ’6) do ( 5 2/3, 12)
(2h, 6) e. ( -3 1/2, -10)
(ll, 6) f. ( -3 1/2, -6)
(9, 10) g. (- 2 3/h, -S)
(S, 10) . h. (- 2 1/2, 2 1/2)
(b, 1h) i. ( 2, 3 1/2)
(-1, 1h) j. ( -1 1/2, 12 1/2)
(0, 10) k. (0,1b 1/3)
(-3, 10) 1. (1 1/2, 12 1/2)
("by lb) m0 (2: 3 1/2)
(~9, 1b) n. (2 1/2, 2 1/2)
(‘8, 10) 00 (2 B/h, ’5)
('12: 10) p. (3 1/2, -6)
(~1h, 8) , q. (3 1/2, ~15)
(-21, 8) r. (5 2/3, ~17)

s. (5 2/3, -12)

t. (3 1/2, -10)

u. (3 1/2, -1h)

v. (-3 1/2, -1u)

166

EXECISE 2

Find several pairs of numbers that
will make the following open
sentences true. List your answers
as truth sets.

I. A= [3+ 2 Answer
A= 3 x El Answer
A=2 xEl +| Answer
If E] is represented by each of
the following numbers, what is A:
using the open sentence ZX=EJ+2?

E] 23

 

 

 

#(JON

 

O
I
2
3

4

 

Make tables of the truth sets for the
following open sentences:

5. £1= 2 x El+|
E] [X

 

 

6 .

A=3XD+|

167

 

 

ma
EXERCISE 3

Make a table and graph of the truth
sets of the following Open sentences.

I. £1=EJ

A:

E] [k

 

 

 

169

2. A=2xEl+3

 

 

 

3.A=2xl:l+l

 

 

 

 

170

3xEl-2

A

4.

 

 

 

171

EXERCISE 4

At the right is the
graph for the truth
seto1c A=2xEl.
Is this correct?

[X

Can you mark three
more points on the
graph without doing
the arithmetic?

(By looking at [3
the geometric
pattern.)

 

Check your points by substituting.
Are they right?

At the right is
the graph for
the truth set of

Z§= [1+ 5.
Is the graph
right?

 

IO.

I2.

N2
At the right is the
start of a graph for
the truth set of

ZX= 3 x [1+ 2.

Does the point work?
Can you mark three
more points on the
graph without doing
the arithmetic?
Check your points [3
by substituting

into the equation.
Were your points right?

At the right is the
start of a graph for
the equation

ZX= 5 x E]- 2.

Does the point work?
Can you mark two
more points on the
graph without doing
the arithmetic?
Substitute into the
Open sentence. [3
Do you get a true
statement?

Were your points right?

.A

 

A:

 

l3.

l4.

l5.
l6.

N3
Below is the start of a graph for

A= 2 x [j + 3. Do you agree?
Can you mark three more points
on the graph without doing the
arithmetic?
Substitute into the equation.
Were your points right?

23

 

1714

EXERCISE 5

Complete the table and plot the graph
for each of the following equations.
Can you find the pattern for each
graph?

Can you see the relationship between
the pattern of each graph and its
equation?

 

 

l.A=2xI:l A
DA
0
I.
2
3
2.11= 3 XI]
DA

 

(JON—0'

 

 

175

 

C]
I
2
3

 

 

Complete the table and plot the graph
for each of the following equations.
Where does the graph cross the Z:
axis?

Do you see a relationship between the
intersection and the equation? V

4. A.=I] + I A;

 

E]
O
I
2
3

 

 

176

EXERCISE 6

Complete the open sentences for each
of the following graphs.

I. £1= __,x [1+ __ 2.1§= XI] +

A: Z:

 

177

=El+2

A

5.

 

 

 

0'23

=I:I+3

A

6.

 

 

 

0'23

178

EXERCISE 7

Make up stories about the following
number sentences and compute the
outcome (result) of your story.

You may tell your stories orally and
then write the answers.

I. *5 + -2=
2. -2 + -2=
3. +5+ -6=
4. -7 + +9=
5. -5+ "'l =
6. -3 + +o=
7. -2 + -5=
8. +6 + *3 =
9. +8 + -I =
IO. -3 + '6 --
II +I+ +I2=
I2 +7 + -9=
I3 +2 + +I7 -
l4 *5 + -Io=

In"
4.
23
+
I
01
II

179

EXERCISE 8

Make up stories about the following
number sentences and compute the
outcome (result) of your story.

You may tell your stories orally and
then write the answers.

I. +2 x ‘l =

2. '3 x +2 =

3. '3 x +I =

4. “I x +6 =

5. +I x ’3 =

6. +2 x +l2 =
7. '2 x ‘4 =

8. 'I x '6 =

9. +3 x '8 =

ID. ‘2 x +7 =

II. +3 x ‘5 =

I2. +4 x '7 =
l3. +3 x +ll =
l4. ’4 x +5 =

I5. +6 x ’7 =

180'

EXERCISE 9

Plot the graphs of the following
equations.

I. A=42xD+-I 2. A="2xl:l+'l
[3 AS

 

3.A=+IxEl++lk4.A='IxE]++l

 

181

EXERCISE IO

Draw continuous graphs of the following
equations.

I. = ‘2 x EI+ +3- 2..A= +3 X [1+ '2
A A

 

3.A= "BXD 4.A= +|xl:l+'2

 

APPENDIX C

SYNOPSIS OF PILOT CLASSES FOR
ELEMENTARY SCHOOL UNIT ON

COORDINATE GEOMETRY

182

PART I

FIRST PILOT CLASS

183

APPENDIX C

SYNOPSIS OF DAILY SESSIONS

Part I: First Pilot Class

 

Session 1

 

Pupils were familiar with the horizontal number line.
One student drew a number line on the board. Individual
pupils went to the board and made dots with colored chalk
to represent points designated by the instructor. The
instructor marked points on the number line and asked
pupils to name these points. Attention was directed to
the fact that each point corresponded to one number and
that to each number there corresponded one point.

When pupils could place points and name them un-
erringly, a point was placed approximately one unit above
(3,0). When asked to name this point, pupils answered
three. Reminded of the one-to—one correspondence between
points on the number line and numbers, pupils decided to
draw a line through the new point parallel to the original
number line. This line was drawn and a point placed
about one unit above the 3 on the newly constructed number
line. If the point which represented 3 was to be dis-

cussed, the class was asked, "which point would you

18“

185

choose?" One student suggested that the original point be
called "3 on number line 1," the second, "3 on number line
2," and the third, "3 on number line 3." By questioning
the labelling of a point midway between the first two
number lines, the first number line was renamed from 1 to
O. From this suggestion, the idea of a vertical number
line emerged. Through these and other leading questions,
pupils grasped the requirement of two numbers to locate a
specific point on a plane. Stimulating pupils to build a

vertical number line required the entire period.

Session 2

 

Using an overhead projector, a 7 x 7 grid of the
first quadrant was projected on a screen. A student was
asked to place a dot representing the point (1,2). He
placed a dot (Pl) on the grid as shown in Figure 1. Many
pupils agreed that the point was located correctly. One
of the few who did not agree, marked a second point (P2)
on the grid (Figure 2). The importanCe that a specific

point, such as (1,2) has a single location was evident.

A

‘I
f * a

Figure 1 Figure 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

186

The class was told that consent as to which point should
designate (1,2) had already been reached by mathematicians
who agreed to name the horizontal distance first and the
vertical distance second. The word "coordinates" of a
point was defined, the notation described, and the signifi-
cance of each number in the pair stressed.

Placing a point underneath the horizontal number
line elicited the idea of extending the vertical number
line below 0. Pupils called the numbers below 0, minus 1,
minus 2, etc. Because "minus" implies subtraction and may
cause confusion, students were informed of the accepted
negative 1, negative 2, etc. nomenclature. The lesson
ended with a discussion of positive and negative numbers

connotating opposites.

Session 3

 

Content of the two previous lessons was reviewed.
Pupils were asked to plot specific points in quadrants I
and IV and to explain the meaning of each number in the
pair of coordinates. Pupils saw that the location of
points to the left of the vertical axis required the
extension of the horizontal number line to the left of
O. The division of the plane into four sections by the
number lines was pointed out. These sections were named
quadrants I through IV in counterclockwise direction and

each number line (horizontal and vertical) was referred to

187

as an axis. Identifying points in all four quadrants was
accomplished by having students plot points on a 16 x 16
grid on the overhead projector andcalling on other stu-
dents to name them.

Pupils in this class were seated with their desks
pushed together in groups of four. Each group of pupils
was supplied with four geoboards and small cylinders made
by cutting straws into pieces. The geoboards were laid
adjacent to each other so that each represented one
quadrant. Pupils took turns placing the straw cylinders
over the nails representing the points whose coordinates
were called..

,A distinct disadvantage accompanied the use of four
geoboards to simulate the coordinate plane. The suggested
line segments where the boards joined were considered to
be the axes. As there were no nails on these "lines,"
points lying on the axes could not be plotted. If only
one geoboard were used, a different difficulty resulted.
Each geoboard was only 5 x 5. If rubber bands were
stretched around the middle row and column to represent
axes, little latitude remained for the choice of points.
Because both situations were undesirable, geoboards were

not used again even though pupils enjoyed this activity.

188

Session A

 

Exercise 1 was distributed. The low reading level‘
of the pupils was overwhelmingly evident. Without reading
directions, students, with hands raised at every table,
said they did not know what to do. Both the regular
teacher and the researcher moved from pupil to pupil for
the entire time allotted. The majority of the class
could not work independently. Only a few worked without
assistance. They were allowed enough time to complete
Exercise 1 which was collected for diagnostic purposes.

Sentences, true, false, and those which were impos-
sible to judge as either, were discussed. The latter were
classified as open sentences. Pupils readily volunteered
examples of all three kinds. The discussion was guided
toward mathematical sentneces. Examples, as 3 x 5 = 15
and 2 + l = A, were written on the board and classified
as true or false. The open sentence, [S = C] + l, was
introduced and students suggested pairs of numbers for
[k and E]. The discussion was guided toward choosing
pairs of numbers whose coordinates differed by l. Pupils
could see that for a specific pair such as (2,3), if 2
replaced [3 and 3 replaced [3 , a true sentence was pro—
duced, whereas if 3 replaced [3 and 2 replaced A , a
false sentence wasproduced. Therefore, an agreement was
made whereby pupils named the number replacing (:1 first

and that replacing A , second. Those pairs of numbers

189

that made a true statement were identified as belonging to
the truth set of the open sentence. Students were en—
couraged to find as many pairs as they could that would
make the sentence true. Consequently, the truth set was
shown to have so many members that all of them could not

be enumerated. Since pupils were unfamiliar with set nota-
tion, the use of braces to enclose the truth set, the
separation of these elements by commas, and the use of
ellipsis (...) to indicate that the set did not end was

explained.

Session 5

 

Exercise 1 was returned and discussed. Pupils had
especially enjoyed decoding the "secret message." Five
pupils missed six or more of the 22 parts. An open sen-
tence was considered and some members of its truth set
listed in set notation. The possibility of other methods
for displaying truth sets was posed. One pupil suggested
making two columns, one for the pairs of elements that
made the sentence true, the other for elements that made
the sentence false. This suggestion was followed. Then
it was explained that the true and not the false sentences
were the main interest. Forgetting the column of elements
that produced false sentences, the other column offered a
second method of displaying the truth set of a sentence,
namely, a table. Tables for several open sentences were

constructed on the blackboard.

190

Exercise 2 was given to students to complete in
class. Again, many pupils depended on the teacher for
additional directions and encouragement. The greatest
difficulty lay in finding a number for E] . (Some were
concerned because their work differed from that of their
neighbors, yet both were correct.) They hesitated tO per-
form the process--select any number, "plug" it into [3 ,

and find [A .

Session 6

 

Pupils were allowed time to finish Exercise 2 which
was collected. Various tables, randomly selected from
this set Of papers, were copied On the board and verified
for accuracy.

The class could not visualize a method other than
listing the elements in set notation and the use Of tables
for representing truth sets of Open sentences until asked
if they thought Open sentences and plotting points could
be linked. The same student who Offered the idea that led
to constructing a table immediately suggested plotting the
number pairs listed in the tables on a grid. The set Of
elements for each table was plotted. Such a display Of
points, students were told, was called a graph. They now
knew three methods for representing the truth set Of an

Open sentence.

Session 7

 

Exercise 2 was returned. The three pupils who did
poorly were given individual assistance later in the
period. Constructing a graph for an Open sentence was
reviewed. Exercise 3 was given to the class to work for

the remainder Of the period.

Session 8

 

The class was divided into two teams for the purpose
Of playing a modified version Of the Madison Project's
tic—tac—toe. The game, which took the entire period, was
enthusiastically accepted by the pupils whose display Of

competitive excitement was gratifying.

Session 9

 

Exercise 3, collected at the end Of Session 7, was
returned. One student copied his table for A = 2 x
[J + 3 (NO. 2 from Exercise 3) on the board and proved
that each ordered pair was a member Of the truth set Of
this equation. Another student plotted the points on a
. grid which was projected. The class was asked if anyone
could place another point on the graph without doing any
arithmetic. A point was plotted by one pupil, its coordi-
nates were noted and used as replacements for [5 and E]
for verifying the sentence. A second pupil added another
point and the process was repeated. These pupils were

asked to tell the class the pattern that determined where

192

they placed a new point. ~Pupils placed points confidently
because, as they remarked, the graphs for exercise were
straight lines.

Exercise A was distributed. The questions contained
therein were designed for challenging pupils to find the
"secret" Of correctly placing additional points on graphs
without performing arithmetic. Verbal assistance was not
given. Pupils able to discover this secret were urged

not tO tell their classmates.

Session 10

 

Pupils continued to work Exercise A. Most Of the
pupils could add points to a graph that contained more
than one point, but they were unable to complete the
three graphs in the exercise that contained only one
point. Some pupils disregarded directions and completed

the graphs by substituting numbers for [3

Session 11

 

Pupils were given Exercise 5 which directed their
attention from a graph tO its sentence. The class per-
formed the activities suggested in the exercise and
answered the questions, but only one pupil perceived that
the multiplier Of 1:] determined the pattern of the graph

and that the constant denoted the Z§-intercept.

193

floss Ion 12‘

 

The pupil who saw the relationship between the graph
and its Open sentence told the class this "secret."
Several pupils added points tO graphs taken from Exercises
A and 5. They then completed the exercises. "Telling"
the class this information violated the Madison Project
technique Of proceeding to a different activity and re—
turning tO the same topic later. Revealing "discovery"
information was justified as the limited time allotted for
presenting materials did not permit the redevelopment Of

ideas.

Session 13

 

Using the overhead projector and chalk board, stu-
dents graphed linear equations by plotting the A-inter-
cept and counting out the slope which they determined

from the sentence. The entire unit was reviewed.

Session 1“

 

A test on the material covered in the first five
lessons was administered, completing the first pilot

class.

PART II

SECOND PILOT CLASS

191)

195

Part II: Second Pilot Class

 

An Observer attended the second pilot class sessions,
recorded them, and Offered valuable criticism. The taped
lessons, and his constructive suggestions, were beneficial
in teaching that particular class, revising the lesson
plans and preparing for the workshop for teachers. Por—
tions Of the tape were played in some sessions Of the
workshop to demonstrate typical situations that teachers
might encounter.

In describing the second pilot class only differences
between it and the first pilot class will be elaborated,
including those revisions which contributed to improving
the presentation Of material. Lessons 6 through 10, which
were not covered in the first pilot class, will be dis-

cussed in detail.

Lessons 1—3

 

A number line was drawn on the chalk board as in the
first pilot class. However, when pupils suggested that a
vertical line be constructed, three golf tees were in-
serted into the pegboard and positioned so that when yarn
was wrapped around them, perpendicular line segments were
represented. The tees were placed such that a right angle
was formed whose interior was the first quadrant. When
the thought Of plotting points in the exterior Of this
angle occurred to the students, the golf tees were removed

and inserted midway on each of the four edges and in the

196

center of the pegboard. Yarn was wrapped around the upper
and lower tees and around the two side tees to form axes.
The interiors Of the four right angles represented
quadrants I, II, III, and IV. The pegboard stimulated
interest and proved to be a much more effective aid than
either a latticed chalk board or a grid for the overhead
projector. Moreover, it created an activity that children
thoroughly enjoyed.

Red, yellow, white, and blue golf tees made possible
the representation Of points in each quadrant by a dif-
ferent color. When playing games, such as tic-tac—toe,
the use Of different colors made the identification of
teams easy.

An unanticipated problem arose from the use Of the
pegboard. When plotting points on graph paper, a few
pupils did not perceive their loci accurately. Some
pupils plotted points in the middle Of spaces when the

coordinates were integers, as e.g., (1,2) in Figure l.

A

 

 

 

 

 

 

 

 

4)-
Figure 1

Others represented the points in the upper right corner

Of the space, as in Figure 2.

197

A

+_1_

I

 

 

 

 

 

3.

Figure 2

Two corrective measures eliminated the confusion: (1)
horizontal and vertical lines were painted on the peg-
board such that the holes fell on their intersections,
and (2) pupils were shown that the loci Of points with
integral coordinates were the intersections Of lines
parallel to both axes at the distances indicated by the
ordered pairs.

The nine days spent on Lessons 1, 2, and 3 included
a day for the administration Of a 25 item multiple—choice

test upon the completion Of Lesson 2.

Lessons A and 5

 

Lessons A and 5 were more successful in the second
than the first pilot class. Exercise 3 was returned and
discussed individually while pupils worked on Exercise A.
One pupil quickly discovered the relationship between the
slope pattern and the equation. Another pupil perceived
the clue the second day he worked on Exercise A (11th
session). With the help of Exercise 5, two additional

pupils found the secret.

198

If a graph contained at least two points, most pupils
could find the pattern and place additional points cor-
rectly. In fact, most pupils completed both exercises.

In answer to the question in Exercise 5, "Can you see the
relationship between the pattern Of each graph and its
equation?", they answered, "yes." When asked, however,
they could not describe the relationship. Therefore
another question, "What is this relationship?", was added
to Exercise 5 for future classes.

After working two and one-half class periods on
Exercises A and 5, the first pupil who discovered that the
multiplier Of box indicated the graph's slope was permitted
to relate the "secret" to the class. The second dis-
coverer explained the relation between the constant in
the equation and the graph. Again, "telling" the class
was accepted for the reason stated in the first pilot

Class.

Lesson 6

 

Lesson 6 was completed in one session. Graphs were
plotted on the pegboard. Pupils stated the equations Of
these graphs using the information they had discovered
(or were told) in Lessons A and 5. Exercise 6 involved
the same activity in written form.

Some graphs whose slopes were not integers were

plotted. Students could determine the slope pattern, but

199

Vlack Of experience with rational numbers was very evident.
TO describe the slope Of the graph in Figure 3, students

said, "Write 2. Draw a line. Put a three underneath it."

.I
(I—

 

 

 

firs

 

 

 

 

 

 

 

 

 

Figure 3

A 20 item multiple choice test was administered on Lessons

A, 5, and 6 during the lAth class session.

Lesson 7

Addition Of signed numbers was presented through the
use Of the Postman Stories suggested in the Madison Pro—
ject. Pupils participated in two different activities:
(1) they figured the amount Of money resulting from the
postman's delivery Of checks and bills, and (2) they made
up stories that involved positive and negative numbers.
Their first stories were similar to the stories Of the
instructor. But as they grasped the ideas, their stories
displayed more imagination. One boy described a "weird"
baseball game in which a team scored -A runs. When his
classmates protested, he retorted, "Well, I said it was

weird, didn't I?"

200

The Postman Stories led to the plotting Of equations
with a negative constant ( Z§-intercept). Two class ses-

sions were devoted to Lesson 7.

Lesson 8

 

Replacing box with negative numbers required the
ability to multiply signed numbers. Again the Postman
Stories were used to convey the concepts. Pupils par—
ticipated in the same activities for multiplying signed
numbers as they did for adding signed numbers. The two
activities are listed in Lesson 7. In pupil Exercises 7
and 8, stories were related orally and numerical results
were written.

Operations with signed numbers were unquestionably
difficult for some pupils. Insufficient command Of
arithmetic combinations was a stumbling block.

Two class sessions were spent on Lesson 8 and one
additional class session reviewed the combination of

Lessons 7 and 8.

Lesson 9

 

Pupils practiced graphing equations with negative
slopes and negative intercepts. Equations which were
dictated were graphed by one pupil on the pegboard and
simultaneously by another pupil on the chalkboard. They
enjoyed working with the pegboard again. It had not lost

its appeal.

201

Lesson 10

 

The elements, (0,0), (1,8), and (2,16), Of the
number sentence [5 = 8 x C] were plotted on the peg-
board. Students were asked to compute the value for [A
if D were 1/2. The locus, (1/2, A), was estimated and
a pin with a large red head was stuck into the pegboard.
Similarly, the points (l/u, 2) and (1/8, 1) were plotted.
The class approximated A when values for D were chosen
between two known points. They could see that if many
points were chosen, the pins would soon touch each other.
The conclusion Of the class that the graph would be a
continuous line was accepted because they had no knowledge
of irrational numbers. When students plotted other equa-
tions, yarn was looped around the peg that represented the
[:1— intercept and the last inserted peg (after the slope
had been counted out several times) to show that the graph
was a continuous line if box were not restricted to inte-
gers.

The class spent the rest Of the session on Exercise
10. This exerciSe was discussed and Lessons 7 through 10
were reviewed during the twenty—second session. A test
on these lessons was administered in the following (last)

session.

APPENDIX D

TEST ON COORDINATE GEOMETRY

(TOGC)

(‘0
O
I‘d

PART I

ELEMENTARY SCHOOL TOCG

203

COORDINATE GEOMETRY TEST
FOR ELEMENTARY STUDENTS

DIRECTIONS: Choose the response that
Correctly completes the sentence.
Mark the letter corresponding to this
response on the answer sheet.

I. Point P whose coordinates are I2.-I)

lies in
A. Quadrant I , B. Quadrant II
C. Quadrant Ill D. Quadrant IV

E. None of these

2. What is the Elcoordinate of every
I point on the Ataxis?

A. l B. -|

C. 0 D. None of these

3. In plotting a point, we agree to
name first the
A. A intercept B. A coordinate
C. sl0pe D. Elcoordinate

4. Point P whose coordinates are I-2,|)

lies in
A. Quadrant l B. Quadrant II
C. Quadrant lIl D. Quadrant IV

E. None of these
201)

205

Some members Of the truth set of the
open sentence [i= 2 x [1+ 4 are

A. (5.l4) B. (0.4)
C. (6,l6) D. A and C
E. All Of these

Answer questions 6 through ID from the
graph at the right.

6.

IO.

The coordinates Of
point R are

A. (2.3) B. I-I,Il
C. (3.2) D. (I,-|)
E. None of these

 

The coordinates of
point 8 are

A. (-3,3) B. I-I,l) C. (I.-I)
D. (|,I) E. None of these
The coordinates Of point T are
A. (392) Bo (“'ol) 0. (232)
D. (2,3) E. None of these

The coordinates Of point U are
A. (2,-3) B. (3,2) C. (3,-3)
D. I-3,-3) E. None of these

The coordinates of point Q are
AI. (092) _ Bo (290) ‘ Co (09-2)
D. None Of these

I2.

I3.

I4.

206

The coordinates Of point A are
(0.8). The coordinates of point
B are (0,-5). The distance from
A to B is

A. 3 B. I3 C. 5

0. None Of these.

A point whose Elcoordinate is
negative lies in

A. Quadrant I or II

B. Quadrant II or III

C. Quadrant III or IV

D. Quadrant I or IV

The coordinates of B

 

 

 

in the figure at the _6 119—2
right are ‘* " " _

A. (-5,-2) ”“

B (-I,-5) . ,1

c. {-2.-5) 5H”) B

o (-5,-I) _» I

E None of these

In the figure at the
'right,.the coordinates P sea)
of P are ’

A. Ia,d) B. (a,c) ‘ Q(a.lo)‘ Rg
C. (c,a) D. (d,a) " ’ '
E. None of theSe

 

207

If the Aiintercept is 3, the Open
sentence is

A. Ai= 3 x[] B. Ai= 3
C. )3: D + 3 D. Ai= 3
E. None Of these

x EI-3
x EI+I

Which Of the following ordered
pairs is not an element of the
truth set Of A= 2 x I] + I?

A. (0,I) B. (2,3)

C. (3,7), D. (I,3)

E. None Of these

Study the following table. The
Open sentence for the table is

D .A
2 3
3 5
4 7
A. £i= EI+|
B. [i— [1+2
C. Ai= 2 x E]- I
D. Zi= 3 x E]- 3
E. None of these

208

I8. In the graph at the right, the
slope pattern is

A. Over one to the
right, Up one

B. Over one to the
right, up two

C. Over one to the
left, up two
D. None Of these

 

I9. In the graph for number I8, the
A intercept is ‘
A. (0,|) B. (-I,0)
C. (0.2) D. None of these

20. Which of the following represents
a graph Of an Open sentence that
does not contain the origin?

A.A=2x'[l-I B.A=2x[]
C.A=3x[]-'2 D. AandC
E. A, B and C

2|. The equation of the graph
at the right is
A. [i= 2 x [I- I

 

B. Ai= I x [1+ 2
C. Ai= l x [I- 2
D. Zi= 2 x [1+ 2
E. None Of these

22.

23.

24.

 

209

Which Of the following is the

graph Of the Open sentence

[1= 3 x D +I?
B. C.

   

Which of the graphs of the follow-

ing open sentences has a
intercept of 3?

A. Ai= D
B. Zi=
C. 3
D. 2 x XAS=
E. None of

Dim
Il><

fl mljljlu
+

m [JnJOJ
+

x
has
The slope Of the graph at the
right is
A. Over one to the
right, up one
B. Over one to the
right, up two
0. Over two to the
right, up one
0. Over two to the
right, up two
E. None of these

 

25.

26.

27.

28.

210

The [lintercept for the graph in
number 24 is

A. (0.2) B. (I,O)

C. (2.0) D. (-2,0)

E. None of these

In the graph at the

 

 

 

 

 

 

 

 

 

right, the z: coord- I

. . II I
Inate Is always

A-O. « III .
B. Any number I11

C. 3

D. None Of these

Inthe open sentence Zi= 2 x [I- I
A..A increases when [lincreases
B. Aiincreases when [Idecreases
C. Aidecreases when [lincreases
0. None of these

Does the graph at the
right represent the
Open sentence
Zi= 2 x [1- 2?

A. Yes

B. No

 

29.

30.

3|.

32.

C.
D.

211

product of ‘2 x '5 is
’7 B. +7
‘IO D. +IO
None of these

Aiintercept of the equation
3 x [1- 2 is

The origin B. (3,0)

(0,2) D. (-2,0)
None of these

the graph of Zi= ‘3 x [1+ 2,

A.increases as [lincreases
Aidecreases as [Jdecreases
Aidecreases as [lincreases
None of these

Which Of the following is the
graph of the equation
Ai= 3 x [I- 3

 

B. C. D.

   

33.

34.

 

212

The equation Of the
graph at the right is
A. li= 2 x [1+ 2

B. £l= '2 x [1+ l

C. At: ‘2 x [1- 2

D. Ai= 2 x [1+ I

E. None Of these

 

The point represented on the graph

at the right
truth set Of

A. Ai= 2 x [1+ I
B. Zi= [1+ I

C. A(=[]

D. [i= ‘2 x [1+ I

E. None of these

Which of the following
graph of the equation

xl] - 2?

3
A‘é

 

 

 

 

 

 

 

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

is a member of the

 

is the

 

36.

37.

38.

39.

40.

213

In the open sentence A= ‘4x I] -2,
if []= 4, £i=

A. -I8 B. )4

C. l8 D. -I4

E None of these

In the graph at the L
right, the slope - 4‘
I
I
I

 

 

 

 

pattern is g

A. -4 B. 4
I t

C. - 4 D. None of these

 

 

 

 

 

 

 

_t—d_

 

 

f

 

 

In the open sentence £5=T XEJ+ P,
the AIintercept is

A. (O,T) B. IT,O)

C. IT,P) D. None of these

In the graph at the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

right, the slope is ‘
-5 5 E

A. é B. E i ”as :-
2 -2 .,

c. 5 o. 5

The [Iintercept for the graph in
number 39 is

A. (2,0) B. (5,0) _

C. (0,5) D. None of these

PART II

SECONDARY SCHOOL TOCG

211)

215

COORDINATE GEOMETRY TEST
FOR SECONDARY STUDENTS

DIRECTIONS: Choose the response that correctly completes the
sentence. Mark the letter corresponding to this response on
the answer sheet.

1. Point P whose coordinates are (2,-1) lies in

A." Quadrant I . B. Quadrant II
C. Quadrant III D. Quadrant IV
E. None of these

2. What is the x coordinate of every point on the y axis?
A. 1 B. -1 C. 0 D. None of these
3. In plotting a point, we agree to name first the

A. y intercept B. y coordinate
0. slope D. x coordinate

4. Point P whose coordinates are (-2,1) lies in

A. Quadrant I B. Quadrant II
C. Quadrant III D. Quadrant IV
E. None of these

5. Some members of the truth set of the open sentence
y = 2x + 4 are

A. (5.14) B. (0,4) 0. (6,16)
D. A and C B. All of these

Answer questions 6 through 10 from the
graph at the right.

6. The coordinates of point R are

A. (2,3 B. -I,I)
C. (332 Do 19")
E. None of these

7. The coordinates of point S are

A. -333; B. (
Bo 1,-1 (
E. None of these

 

‘0
O

10.

II.

12.

13.

14.

216

The coordinates of point T are

A. (332) B0 (-131) C. (2,2)
D. (2,3) E. None of these

The coordinates of point U are

A. (2.-3) B. _(3,2) 0. I3.-3)
D. (-3,-3) E. NOne of these

The coordinates of point Q are

D. None of these
The coordinates of point A are (0,8). The coordinates
of point B are {0,-5). The distance from A to B is

A. 3 B. 13 0. 5
D. None of these

A point whose x coordinate is negative lies in

 

A. Quadrant I or II B. Quadrant II or III
C. Quadrant III or IV D. Quadrant I or IV
The coordinates of B in the figure
at the right are -6 (O O)
4 l 1 LI J L
A- (‘59'2’ B. (~1,-5 7E ' I II I
0- (~1,-5) D- (‘59'1 (-I;2)Ri

E. None of these

 

 

 

 

 

In the figure at the right, the AI
coordinates of P are P Shad)
A. (a,d) ’ B. (a,c)
E. 11:13:)“ these D. (d,a) 4 Q(&,b) R >
V

 

If the y intercept is 3, the Open sentence is

A.y=3x B. y=3x-3
C. y = x + 3 D. y = 3x + 1
E. None of these

I6.

17.

18.

19.

20.

21.

217
Which of the following ordered pairs is not an element of
the truth set of y = 2x + 1?

A. (0,1) B. (2,3) 0. (3.7)
D. (1,3)- E. None of these

Study the following table. The open sentence for the
table is .

 

 

X Y

2 3

3 5

4 7
A. y = x + 1 B. y = x + 2 O. y = 2X - 1
D. y = 3x - 3 E. None of these

In the graph at the right, the
slope pattern is

A. Over one to the right, up one
B. Over one to the right, up two
O. Over one to the left, up two
D. None of these

 

In the graph for number 18, the y intercept is

A. (0’1) Be (-130 C. (0,2)
D. None of these

Which of the following represents a graph of an open
sentence that does not contain the origin?

A. y = 2x - 1 B. y = 2x 0. y = 3x - 2
D. A and C D. A, B and O

The equation of the graph at the
right is

x + 2
2x + 2

A. y 2X “'1 B. y
C. y x - 2 D. y
D. None of these

II II

 

22.

218

Which of the following is the graph of the open sentence
y = 3x + 1?

 

24.

Which of the graphs of the following open sentences has
an intercept of 3?

A. y : x - 2 'B. y = 2x + 3 0. 3y : x - 2
D. 2y : 3x + 1 E. None of these

The slope of the graph at the right
is

A. Over one to the right, up one
B. Over one to the right, up two
0. Over two to the right, up one
D. Over two to the right, up two
E. None of these

 

25. The y intercept for the graph in number 24 is

26.

A. (0,2) B. (1,0) 0. (2,0)
D. (~2,0) E. None of these

In the graph at the right, the
y coordinate is always

A. O B. Any number
0. 3 D. None of these

 

27.

28.

29.

30.

31.

32.

In the graph of y

In the Open sentence

219

y:2x-19

A. y increases when x increases
B. y increases when x decreases
O. y decreases when x increases

D. None of these

Does the graph at the right
represent the open sentence

y:2x-2? “
A. Yes B. No

 

The product of -2 x -5 is

A. -7 B. +7

C. ~10 D. +10

The y intercept of the equation y = 3x - 2 is

A. The origin
Do (“2,0)

A. y increases as
B. y decreases as
C. y decreases as
D. None of these

Which of the following
y = 3x - 3?

A. B.

 

II
I
U)
N

Bo (3,0) 00 (092)
E. None of these

+ 2,
x increases

x decreases
x increases

is the graph of the equation

C. D.

   

220

33. The equation of the graph at the

right is
B. y = -2x + 1

D. y _ 2x + 1
E. None of these

 

34. The point represented on the graph
at the right is a member of the
truth set of

A. y = 2x + 1
Co y=X

D. y -2x + 1
E. None of these

 

35. Which of the following is the graph of the equation

3
yz-X-2?
2

A. B.

     

36. In the Open sentence y = -4x - 2. if X = 4. Y =

A. -18 B. 14 c. 18 D. -14
E. None of these

37.

38.

39.

40.

221

In the graph at the right,
the slope pattern is

A. -4

B. 4
1

Co "Z

D. None of these

In the open sentence y = Tx + P,

A. (O,T) B. (T,0)

D. None of these

In the graph at the right,
slope pattern,is

5 5
A. ""2- Bo '2'

2 2
CO 5" D. -g

 

 

y intercept is
C. (T’P)

The x intercept for the graph in number 39 is

A. (2,0) ‘ B. (5,0)

D. None of these

0. (0.5)

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