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dissertation entitled
Achievement in Similarity Tasks: ' Effect of

Instruction, and Relationship with Achievement in

Spatial Visualization at the Middle Grades Level

presented by

Alex Friedlander

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Teacher Education

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Major professo/d \

Dr. William M. Fitzgerald
Date June 27, 1984

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ACHIEVEMENT IN SIMILARITY TASKS:
EFFECT OF INSTRUCTION, AND RELATIONSHIP WITH
ACHIEVEMENT IN SPATIAL VISUALIZATION AT THE MIDDLE
GRADES LEVEL

By

Alex Friedlander

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Teacher Education

1984

")Icl'r/V

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37"".

ABSTRACT

ACHIEVEMENT IN SIMILARITY TASKS:
EFFECT OF INSTRUCTION AND RELATIONSHIP WITH
ACHIEVEMENT IN SPATIAL VISUALIZATION AT THE

MIDDLE GRADES LEVEL

BY

ALEX FRIEDLANDER

Purpose

This study had three related purposes: (1) to determine any existing
differences in similarity achievement by grade level and by sex, prior
to, and after instruction pertaining to similarity, (2) to ascertain
any existing differences between a verbal and a ‘visual presentation
of similarity tasks, and (3) to study the relationship between performance

on similarity tasks and performance on spatial visualization tasks.

The sample (N==67S) was drawn from five schools with a suburban,
middle-class, midwestern, predominantly white population and with
teachers that volunteered to teach the Similarity Unit--an instructional
unit developed by the Middle Grades Mathematics Project for grades six,

seven and eight.

Analysis of Variance, and Analysis of Repeated Measures served as

the main statistical tools to test the research hypotheses.

Main.Finding§

 

Pre, and post instructional performance on four similarity-related
topics were analyzed: (1) basic properties of similar shapes, (2)
proportional reasoning, (3) area relationships of similar shapes, and
(4) applications. ,At a significance level of .05: no sex differences
in pre-instructional achievement or gains were observed; achievement
increased as a function. of grade level; and seventh. graders gained
significantly more when compared to the sixth and eighth graders. Pre-
instructional performance on area growth tasks was uniformly poor (18-22
percent) and the gains of the sixth graders as a result of instruction

were particularly low (5 percent).

Performance on four similarity tasks presented verbally was compared
with performance on four equivalent tasks accompanied by figures.
Although no overall difference between the two presentation modes could
be detected, different presentation modes seemed to be favored for

different similarity-related topics.

Student performance on a spatial visualization test before and after

instruction on similarity indicated significant gains.

For a restricted sample (N=161) performance in similarity tasks
of students that underwent instruction in spatial visualization one year
before this study was compared with performance of students that did
not undergo this kind of instruction. In this case, instruction in
spatial visualization did not have a significant effect on achievement

in similarity tasks.

Sara, Amit, Ronen -- my family

Professors William Fitzgerald (Chair), Richard Houang, Glenda Lappan,

Perry Lanier, and Bruce Mitchell -- my doctoral committee.

David Ben-Haim and many other Israeli and American friends.

Sharon Tice -- my typist.

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . .
LIST OF FIGURES . . . .'. . . . .
Chapter

I. INTRODUCTION .

II.

III.

IV.

Background .
The Study .

REVIEW OF LITERATURE .

Introduction . . . . . . .

The Development of the Similarity Concept

The Development of Proportional Reasoning
The Concepts of Area and Area Growth .
Geometry and Spatial Visualization .

Sex Differences . . . . . .

The Effect of Instruction

DESIGN OF THE STUDY . . . .

Purposes of the Study

Population and Sample
Instrumentation . . .
Instructional Material .

Procedure and Data Collection . .
The Statistical Design of the Study

ANALYSIS OF DATA AND RESULTS
Introduction . . . . . . . . . .
Ere-Instructional Performance in Four

Similarity-Related Areas .

Effect of Instruction in Four Similarity-
Related Areas

iii

Page“

vii

H

13

13
13
19
31
36
41

45

57
57
58
61
64
67
68
7O
7O

72

76

Chapter

Pre-Instructional Performance on Tasks
Presented in a Verbal, and a Visual Mode . . . . .
Effect of Instruction on Performance

in Verbal.and in Visual.Task . . . . . . . . . . .
Effect of Instruction in Similarity on Spatial
Visualization . . . . . . . . . . . . . . . . . .

Effect of Instruction in Spatial Visualization
on Performance in Similarity Tasks . . . . . . . .

V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS-.

Significance of the Study . . . . . . . . . . .

Design of the Study . . . . . . . . . . . . . .

Main Findings and Conclusions . . . . . . . . . . .

Pedagogical Implications . . . . . . . . . . . . .

Recommendations for Further Research . . . . . . . .
APPENDICES

Appendix

A. MGMP SIMILARITY TEST: ITEMS INCLUDED IN THE FOUR
SUBTESTS: ITEMS INCLUDED IN THE TWO PRESENTATION
MODES . . . . . . . . . . . . . . . . . . . . . . . .

B. MGMP SPATIAL VISUALIZATION SUBTEST: SAMPLE ITEMS

C. THE MGMP SIMILARITY TEST AND THE MGMP SPATIAL VISUALI-
ZATION TEST: RELIABILITY COEFFICIENTS, AND PRETEST-
POSTTEST CORRELATION COEFFICIENTS . . . . . . .

D. THE MGMP SIMILARITY UNIT: OVERVIEW .

E. MEAN SCORES, STANDARD DEVIATIONS AND CORRELATIONS
RELATED TO THE RESEARCH QUESTIONS . . .

REFERENCES . . . . . . . . . .

iv

Page

82

86

92
98
104
104
105
107

119
121

124

133

135

137

139

146

’ Table

2.1

3.1

3.2

3.3

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

LIST OF TABLES

The Van Hiele Levels of Development in Geometry .

Distribution of Participating Classes and Teachers
by Grade Level and by School . . . . . . .

Sample Ae-Distribution of Students by Grade Level
and by S ex 0 O O O O O O O O O O O O O O O 0

Sample B--Distribution of Students by Grade Level
and by Experience in Spatial Visualization (SV)
Instruction . . . . . . . . . . . . . . . . .

Design of the 3 x 2 Multivariate Analysis of Variance
for Hypothesis 1.1 . . . . . . . . . . . . . . .

Summary of the 3 x 2 Multivariate and Univariate
Analysis of Variance for Hypothesis 1.1

3 x 2 Multivariate Repeated Measures Design
for Hypothesis 1.2 . . . . . .

Summary of Multivariate and Univariate Analysis
of Repeated Measures for Hypothesis 1.2 . . . .

Design of the 3 x 2 Bivariate Analysis of Variance
for Hypothesis 2.1 . . . . . . . . . . . . .

Summary of the 3 x 2 Bivariate Analysis of Variance
for Hypothesis 2.1

3 x 2 Multivariate Repeated Measures Design for
Hypothesis 2.2

Summary of Analysis of Repeated Measures for
Hypothesis 2.2 . .

3 x 2 Multivariate Repeated Measures Design for
Hypothesis 3.1 . .

Summary of Analysis of Repeated Measures for
Hypothesis 3.1

Page

47

S9

60

60

73

74

77

79

83

84

87

89

93

96

Table

4.11

4.12

C.1

0.2

E.1

E.2

E.3

E.4

E.5

E.6

2 x 2 x 2 Univariate Repeated Measures Design for
Hypothesis 3.2 . . . . . . . . . . . . .

Summary of Analyses of Repeated Measures for
Hypothesis 3.2 . . . . . . . . . . . . . . . .

Reliability Coefficients--Cronbach a for the MGMP
Similarity Test and for the MGMP Spatial Visualia
zatiOn Subtest by Time, by Grade, and by-Sex

Pearson Correlation Coefficients between Pretest
and Posttest Scores on the MGM? Similarity Test
and on the MGMP Spatial Visualization Subtest
by Grade and by Sex . . . . . . . . . . .

Mean Scores (in percent) and Standard Deviations
on Four Similarity-Related Areas for Sample A
by Grade Level and by Sex . . . . . . . . . .

Correlations among Pre-instructional performance
on Four Similarity-Related Areas for Sample A
by Grade 0 O O O O O O 0 O O O O O O O O O 0

Mean Scores (in percent) and Standard Deviations
on Test Items Presented in a Verbal and a Visual
Mude for Sample A by Grade Level and by Sex .

Correlations among Pre, and Post-instructional
Performance on Similarity and Spatial Visuali—
zation Tasks for Sample A by Grade

Mean Scores (in percent) and Standard Deviations on
Similarity and on Spatial Visualization for Sample
A by Grade Level and by Sex . . . . . . .

Mean Scores (in percent) and Standard Deviations on
Similarity and on Spatial Visualization for Sample
B by Level of Instruction in Spatial Visualization
by Grade and by Sex

vi

Page

100

101

135

136

139

140

141

142

143

144

Figure

4.1

4.2

4.3

LIST OF FIGURES

Properties Invariant Under Transformation Group

Performance by Grade Level in Four Similarity-
Related Areas, Prior to, and After Instruction

Performance by Grade Level in Tasks Presented
in a Visual, and a Verbal Mode, Prior to, and
After Instruction . . . . . . . . . . . . .
Performance by Grade Level on the Similarity Test
and on the Spatial Visualization Subtest, Prior
to, and After Instruction . . . . . . . . .

Gains in Four Similarity—Related Topics--Sample A

vii

Page

14

78

88

95

111

CHAPTER I

INTRODUCTION

Background

 

The Concgpt of Similarity

The acquisition of the similarity concept is important to the
development of children's geometrical understanding of their environment,
and of prOportional reasoning. Phenomena that require familiarity with
enlargement, scale factor, projection, area growth, indirect measurement
and other similarity-related concepts are frequently encountered by
children in their immediate environment and in their studies of natural

and social sciences.

Since Felix Klein's Erlanger Programm--a classification of
geometrical transformations, similarity has been recognized as an

important mathematical concept.

Piaget's (1960) developmental theory of the child's understanding
of geometrical concepts added a cognitive aspect to research on the
concept of similarity. Piaget viewed the ability to make similarity
judgments as an intermediate stage in children's developmental path from
a topological to a Euclidean perception of the environment. Many
researchers (e.g., Martin, 1976b; ‘Lesh, 1976; Schultz, 1978) raised
serious questions about Piaget's hierarchical view of the child's
construction of his spatial reality, and about the relevance of his
theory to mathematics education. However, researchers seem to agree

that around the age of 9, children may make perceptual judgments of

similarity (as for example, in comparing two triangles) and that around
the age of 11 children may have the mental ability to make a gradually
increasing use of proportions (as expressed in judging similarity of

rectangles).

A poor performance on similarity tasks (Carpenter et a1., 1981)
may indicate existing deficiencies in the instruction of geometry in
general (Wirszup, 1976), and of the similarity concept in particular.

Fuson (1978) points out the need for instruction in similarity:

Similarity ideas are included in many parts of the school
curriculum. Some models for rational number concepts are
based on similarity; thus, part of student's difficulty with
rationals may stem from problems with similarity ideas.
Ratio and proportion are part of the school curriculum from
at least the seventh grade on, and they present many
difficulties to the student. Standardized tests include many
proportion word problems. Verbal analogies (a:b::c:d) form
major parts of many intelligence tests. Similar geometric
shapes would seem to provide a helpful mental image for
other types of proportion analogy situations. Training
studies of teaching experiments concerning ways to teach
geometric similarities and ways to generalize the solution
of geometric proportions to other types of proportion would

be valuable. (p. 259)

Proportional Reasoning

The concept of similarity is an instance of proportionality.
Proportional reasoning is frequently required in mathematics, natural
science and everyday life. Chiapetta and McBride (1978) for example,
found among a sample of ninth graders a positive relationship between
the ability to reason proportionally and knowledge and understanding
of simple machines, concepts on the structure of matter, and applications

of equivalent fractions.

The importance of prOportional reasoning in a child's intellectual
development can hardly be overemphasized. Inhelder and Piaget (1958)
consider it one of the six abilities that characterize the formal-

operational thinker.

Due to its wide-spread use, the concept of proportionality has
been investigated more systematically than similarity. The Karplus
studies would be an instance of a thorough analysis of proportional
reasoning as a function of age, sex, social status, and nationality.
Besides attempts to verify Piaget's cognitive stages in the deve10pment
of proportionality, more detailed studies suggest that task-related
variables are equally important when one investigates performance in
proportionality. The variables that have been recommended for
consideration are: (1) level of abstractness (Wollman & Karplus, 1974;
Portis, 1973), (2) ratio versus fraction (i.e., part-to-part versus part-
to-whole) presentation (Wachsmuth, Behr & Post, 1980), (3) level of
numerical difficulty (Abramowitz, 1975; Karplus, Pulos & Stage, 1980),

(4) task sequencing (Karplus, 1978), and (5) irrelevant information

(Collea, & Numadel 1978). Furthermore, sex differences on performance
of proportionality tasks in form of male superiority are also indicated

by some studies (Keating & Schaefer, 1975; Stage, Karplus & Pulos, 1980).

Several studies indicate a generally poor performance on
proportionality tasks at various ages and stress the need for
instructional interventions (e.g., Lovell & Pumfrey, 1966; Renner &
Paske, 1977; Pagni, 1983). Consequently, units on proportional reasoning
have been designed and evaluated from the middle grades level (Wollman
& Lawson, 1978) through high school (Kurz & Karplus, 1977), and even

at the college level (Pagni, 1983).

Research indicates that in most cases mastery of abstract
proportional reasoning may be expected only above the age of 14-15,
if at all (Lovell, 1972). Piagetian research (e.g., Piaget & Inhelder,
1967) shows that similarity tasks may be mastered two or three years
earlier than other proportionality schemes. Therefore, the concept of
geometrical similarity may be a step towards an understanding of
proportionality and research findings in either of these fields should

be relevant to the other.

The Middle Grades Mathematics Project

 

One of the four instructional units designed by the Middle Grades
Mathematics Project (MGMP) is on the concept of similarity. The purpose

of the MGMP Similarity Unit is to provide a response to the present

 

deficiencies in the instruction of geometry. By offering a rich variety
of experiences designed to fit the first two levels in the Van Hiele

model of development in geometrical understanding, the unit attempts

to build a solid base for further advance in the understanding of geometry
in general, and of similarity and prOportionality in particular (Lappan,

1983).

At the middle grades level, the importance of teaching informal
geometry is not contested. Due to the present deficiencies in the
understanding of the most basic geometrical concepts (Carpenter et al.,
1981), and to the recommendations mentioned before, a problem-solving
oriented, activity-based, informal geometry unit of instruction seems

warranted.

The Study

The'Purpose

This study is concerned with the performance of middle-grade students

in similarity tasks prior to instruction with the MGMP Similarity Unit,

 

and with the impact of instruction of the unit on their performance.
The basic goal of the study is to detect the extent to which the MGMP
instructional intervention helps to overcome the cognitive difficulties

that exist at this critical transitional age.

There are three purposes for this study. The first is to determine
any existing differences in similarity achievement in general and in
its four identified content components by grade level and by sex, prior
to, and after instruction pertaining to similarity. Second, to ascertain
any existing grade level and sex differences in performance on similarity
tasks presented in a verbal or a visual mode. The third purpose is

to study relationships between performance on similarity tasks and

performance in spatial visualization. Accordingly, the study will

examine performance in similarity tasks from three different aspects:

1.

2.

A classification by content will distinguish among tasks requiring:
(1) recognition of similar shapes and of their properties, (2)
proportional reasoning, (3) use of the area relationship between

similar shapes, and (4) applications of the similarity concept.

Two research questions will be asked on the performance of sixth,
seventh, and eighth graders in similarity tasks. The first question
is concerned with grade level and sex differences in pre-instructional
performance, whereas the second inquires about the effect of
instruction (i.e., the existence of gains) on achievement in the

four similarity-related topics mentioned above.

A classification by presentation mode will distinguish between (1)

 

visual tasks which include drawings of the involved geometric shapes,
and (2) verbal tasks which contain only a word description of the

geometric situation.

In this study, two research questions are related to pre-instructional
performance and gains on the verbal and the visual subtests. These
questions are concerned with grade level and sex differences in

achievement on the items presented in the two modes.

3. The relationship with spatial visualization will be examined by
analyzing performance on spatial visualization tasks in parallel to

performance in similarity tasks.

Two research questions are concerned with the possibility of a
relationship between achievement in similarity tasks and spatial
visualization. The first question inquires about the effect of
instruction in similarity on spatial visualization, whereas the second
deals with the effect of instruction in spatial visualization on

performance in similarity tasks.
Significance of the Study

One of the goals of this study is to determine patterns in the
performance of middle-grade students in geometrical similarity tasks.
Investigations on geometrical similarity that were conducted in school
settings are rare. Grade level differences (i.e., level of performance
increasing with age) would confirm the developmental nature of
understanding the concept of similarity. This study may contribute to
the present knowledge on the complex issue of sex differences in
mathematical performance: male superiority among students of this age
has been indicated in spatial visualization (e.g., Ben-Haim, 1983), and
in proportional reasoning (e.g., Brendzel, 1978). In regard to geometry,
the issue of sex differences is less clear: some studies tend to detect
sex differences in relation to informal geometry tasks (Shonberger, 1976;
Werdelin, 1961), whereas other studies indicate contradictory results

(Olson, 1970; Thomas, 1977; G. D. Peterson, 1973).

Another goal of this study is to determine the effect of instruction
with the MGMP SimilaritygUnit on the performance of middle-grade students

in similarity tasks .

Significant gains in performance in similarity and/or spatial
visualization would indicate the effectiveness of the instructional
intervention and strengthen the claims that deficient geometrical
understanding may be improved by using apprOpriate teaching strategies

and materials .

The use of concrete models and of a problem-solving orientation
appears to be superior in the instruction of geometry (Bring, 1972;
Buchert, 1980; Hempel, 1981). These observations support the educational
implications of cognitive research on the acquisition of geometrical
concepts. However, cognitive research also sets limitations on the effect
of instruction: instructional interventions may accelerate cognitive
development, but cannot substitute it (Montangero, 1976). Some studies
support this view by indicating a lack of instructional effect as a
result of their subjects' low cognitive level (Young, 1975; Wirszup,

1976).

P.M. and Dina Van Hiele identified five cognitive levels in the
development of geometrical understanding, and more importantly, showed
how to adapt the instruction to the limitations set by these levels.
Present deficiencies in the understanding of geometry in the United
States are attributed by Wirszup (1976) to instructional, rather than
cognitive limitations. He presents as evidence the significant results

obtained by a Soviet reform in the geometry curriculum that relied on

 

the Van Hiele model.

The teaching of geometry in an informal manner is generally accepted
at the elementary' and ‘middle grades level, and recommended. by' some
mathematics educators even at the high school level (J. C. Peterson,
1973). Transformational geometry was also considered as an alternative
for the traditional high school curriculum. However, the results of
some related evaluation studies were disappointing (Olson, 1971; Usiskin,
1972; Durapau, 1979)--a fact that can be explained in part by observing
that new ways of instruction were used without changing in any significant
way the instructional goals or the evaluation tools of the traditional

curriculum.

A third goal of this study is to determine the effect of instruction
with the MGMP Similarity Unit on the performance of middle-grade students

in spatial visualization.

If a relationship between spatial visualization and performance in
similarity tasks is indicated by this study, a certain sequence between

the MGMP Similarity Unit and Spatial Visualization Unit may be

 

 

recommended. More general conclusions on the role of spatial

visualization in informal geometry tasks may also be drawn.

There are strong cognitive and pedagogical reasons to assume a
close relationship between the ability of spatial visualization and the
ability to acquire geometrical abilities (Piaget, 1964; Hoffer, 1977).
Some studies investigated the relationship between Spatial visualization
and geometrical abilities in general. In sharp contrast to spatial

visualization, geometrical abilities do not have a clear definition,

 

10

and have not been widely investigated. Studies that analyzed achievement
in plane Euclidean geometry (e.g., Holzinger & Swineford, 1946; Werdelin,
1961; Hanson, 1972) are more likely to find connections with verbal
or reasoning abilities rather than with spatial visualization. On the
other hand, studies that consider less formal aspects of geometry were
able to show a connection between geometry and spatial visualization.
This relationship reveals itself, at least from the middle grades level
and on, through the improvement in spatial visualization that was shown
to occur as a result of instruction in informal geometry (Van Voorhis,

1941; Brinkmann, 1966; Battista, Wheatley & Talsma, 1982).

Population, Sample and Instrumentation

 

The sample used in this study consisted of sixth, seventh, and
eighth graders from five schools in the area of Lansing, Michigan. All
five schools have a middle-class, predominatly white student pOpulation

and may be considered as typical, midwestern suburban schools.

The tests used in this study have been developed as evaluation
tools for two MGMP units: The MGMP Similarity Test consists of 25
multiple choice items that assess a variety of similarity-related tasks
(for sample items, see Appendix A); the MGMP Spatial Visualization Subtest
(Appendix B) consists of 15 multiple choice items chosen from the original

test that has been designed to evaluate the MGMP Spatial Visualization

 

Unit.

Both tests were administered to all the sampled students prior
to, and after a two to three-week-long instruction with the MGMP

Similarity Unit .

11

Limitations

In order to allow the evaluation of relatively large number of
students, this study used paper-and-pencil tests. Although the items
are concerned with a variety of concepts related to similarity, the
presented situations resemble closely the ones presented in the
instructional unit. Consequently, tasks that require proportional

reasoning in other than geometrical settings have not been considered.

Research indicates that many task-related variables may influence
the performance in prOportionality tasks. Not all of them can be
controlled in a single study. The level of abstractness varies according
to the visual or verbal presentation of items; manipulatory aids were
not included in the process of testing. However, all tasks relate to
geometrical situations that are considered more concrete than some other
proportionality tasks (e.g., the Balance Task, numerical proportions,

or some verbal story problems).

The computations involved in the similarity tasks tested are of
a numerically moderate level of difficulty. Sequencing of tasks according
to level of difficulty or inclusion of irrelevant or redundant information

has not been employed in the test items.

Although socio-economical status (SES) was indicated as a variable
that influences performance in proportionality tasks (Karplus & Peterson,
1970; Karplus, Karplus & Paulsen, 1977), the sample used by this study

is limited mainly to middle-class students from suburban areas.

12

The conclusions that can be drawn from this study will be therefore
limited to the effect of a specific instructional intervention (i.e.,
the MGMP SimilarigLUnit), at the middle-grades level, on middle-class
students from a typical midwestern suburban area as expressed in two
paper-and-pencil achievement tests: (1) the MGMP Similarity Test, and

(2) a selection of items from the MGMP Spatial Visualization Test.

 

 

. CHAPTER II

REVIEW OF LITERATURE

Introduction

In this chapter, literature on different aspects of geometrical
similarity will be reviewed. The first three sections examine the
deveIOpment in children's understanding of the similarity concept itself,
of proportional reasoning, and of the area relationship of similar shapes.
Existing research on the relationship between geometrical ability and

spatial visualization will be examined next.

This study examines the influence of grade level and sex on
performance in similarity tasks. From the reviewed literature on
cognitive development in areas related to similarity, conclusions on
grade level (i.e., age) differences may be drawn. » Literature on sex
differences in performance on similarity-related fields will be reviewed

in a separate section.

The rationale behind this study's examination of the effect of
instruction on similarity will be presented in the last section of this
chapter. This section will review studies that have examined

instructional interventions in fields connected to similarity.

The Development of the Similarity Concept

 

The Mathematical Aspect

 

In 1872, Klein established a new structure for the analysis of

geometrical concepts, which is known also as the Erlanger Program.

13

14

The Program includes a classification of geometrical transformations
according to the transformation groups involved and the invariant
prOperties under that group (Fig. 2.1). The similarity transformation
determines one of the "middle geometries", becoming thus an important

stage in Klein's hierarchy of geometrical concepts.

Invariant Properties

Group:
- . r‘ p
a F . openness (closedness) of curves
8 ,3 . interior, exterior, boundary point
<5 0 linear order, cyclic order
0

. straightness of lines
convexity of figures
. parallelism of lines
8. ratios of distance
9. measure of angles
'__—__—_'10. length

1
2
3
4. connectedness
5
6
7

Similitude
Affine

 

Euclidean

 

 

Transformation

 

Figure 2.1 Properties invariant under transformation group.
(adapted from Martin, 1976b, p.95)

Piagetian Research

Piaget's research on the child's perception of space fits Klein's
model of classification quite smoothly. Both, Klein and Piaget emphasized
invariability under transformation. Piaget (1970) opposed the view that
knowledge is a passive copy of reality. According to him, to know
reality, one must assimilate reality into a system of transformations

which attempts to model isomorphically the transformations of reality.

ihi his space books, Piaget (1960, 1967) asserts that projective

and Euclidean concepts develop concurrently. However, the design of

 

15

the chapters, the research reported, and other comments suggest that
after the deveIOpment of certain basic topological notions, spatial
concepts tend to develOp from projective, to affine, to similarity,
to Euclidean. Fuson (1978) observes that most of Piaget's analyses
focus on topological concepts, or on Euclidean concepts, leaving thus
the "middle geometries" relatively neglected. Nevertheless, Piaget
(1967) describes five experiments related to the concept of similarity:
(1) drawing similar triangles, (2) sorting similar cardboard triangles,
(3) choosing and drawing similar rectangles, (4) drawing a similar
configuration of line segments, and (5) c0pying supplementary angles.
Piaget's categorization of results indicates that at the age of 9-11,
his subjects were able to perform perceptual comparisons (considering
slopes, parallelism, or angles), began to use simple proportions (1:2),
and made measurements, whereas they could use proportions at the age
of 11 or above. The failure to perform well in these tasks at earlier
ages is attributed to the child's perceptually centered nature of
preoperational thought, and later to the inability to use mental
constructs (i.e., ratios and proportions) which have no direct concrete-

empirical representation.

Mathematics educators generally recognize both the advantages and
the limitations of Piaget's research in the field of mathematics
education. In a mathematical analysis of some of Piaget's topologi'cal

tasks, Martin (1976a) concludes:

Piaget's objective is not to study the development of

mathematical concepts of the child but to study the

16

development of the child's concept of space. The models
provided by mathematics are merely the means to an end.
As a developmental psychologist, Piaget does not always use
mathematical language as precisely as the mathematician might

desire. (p.24)

The issue of topological-to-Euclidean spatial development was also
questioned. Martin (1976b) considers the evidence presented by Piaget's
research and states that "it is premature to claim that any particular
hierarchy models the sequence or the structure of the child's construction
of his spatial reality" (p.112). Martin further suggests that "on the
basis of evidence now available" it seems that topological, projective,
and Euclidean concepts develop in parallel rather than sequentially.
Relating to the concept of similarity, Martin asks whether it is
psychologically sound to expect a child to develOp the concept of a
variable constant of proportion (i.e., similarity) before the concept

of a fixed constant of proportion of one (i.e., congruence).

Variables Related to the Acquisition of Similarity Concepts

 

Schultz (1978) showed in her experiments that operational structure
is not the only determining factor in tasks that involved geometrical
transformations. Attributes of fixed states (e.g., familiarity and size
of the involved figures) and the features of the operation itself tend
to have a significant influence. ‘Martin (1976a) suggests that the child's
ability to order, organize and coordinate his actions might offer a

better framework for his spatial development.

 

17

The model built by Pascual-Leone (1970) considers the informational
content and the processing load required by a task as determinant
variables in the logical thinking of a child. He defines the number
of schemes, rules, or ideas that a child can handle simultaneously as
M-capacity. According 1x) him, this capacity increases regularly in an
all-or-none manner from age 3 through 16 at a linear rate of one chunk
every two years. Thus, the derivation of the proportionality rule that
requires simultaneous manipulation of four variables, is not possible

before the age of 9.

Leon (1982) applied the information processing theory to the child's
development of some geometrical concepts such as similarity, and area.
He reexamined Piaget's experiment on similarity judgments of rectangles
and suggests that the emphasis should be not on the cognitive structures
necessary to use prOportions, but rather on the prerequisite quantitative
concepts, i.e., the logically implicit response rule requiring

proportions.

Another limitation on Piaget's theory lies in observed differences
in ages at which children come to master operationally equivalent tasks.
Piaget calls this phenomenon decalage. According to many researchers
(e.g, Laurendeau and Pinard, 1970; Fuson, 1978), Piaget makes a too
frequent use of the decalage as an explanation of arising difficulties.
Referring to the "topological to projective to Euclidean" developmental

path, Lesh (1976) states:

The most important challenge to Piagetian theory is not the

possibility of 21 different hierarchy-but that often

18

operationally isomorphic tasks vary so much in difficulty
that it may be meaningless to classify concepts on the basis

of operational structure. (p.235)

To summarize, not much research has been done on the growth of
the similarity concept. There is disagreement on the exact nature or
the sequential order of the cognitive development of geometrical concepts
in general, and of similarity concepts in particular. However,
researchers agree that at the age of 11, a more general and conceptual
use of proportions (as expressed in judgments on similar rectangles)
occurs. Perceptual judgments of similarity (e.g., judgments on similar

triangles) is possible about two years earlier.

Performance on Similarity Tasks

 

Most of the above mentioned studies were based on intensive
interviews conducted with a small number of children. Statistical support
for many of these claims is even scarcer. Young (1975) examined the
performance of 791 children from grades K through 3 on tasks related
to ten geometrical concepts--one of them being similarity. In view
of the poor results on this specific task, Young recommends introducing

the concept of similarity "above the third grade".

Carpenter et al. (1981) analyze the results of the 1977-78 second
mathematics assessment of the National Assessment of Educational Progress
(NAEP). The answers of 70,000 students, ages 9,13, and 17 (Hi the
similarity items follow the general pattern of performance in geometry:
Students have some knowledge of certain basic concepts, but have little

knowledge of the properties associated with these concepts, and little

 

19

abililty to apply these prOperties. In the case of similarity, 94 percent
of the 13-year-olds recognized two similar triangles. However, only
about 33 percent of the 13 year-olds, and 44 percent of the 17—year-
olds knew that similarity does not require congruent sides, and less
than 33 percent of the subjects knew that the angles of similar shapes
must be congruent; a third of the 13-year-olds and half of the 17-year-
olds could use indirect (shadow) measurement to determine the height

of a tree.

This gap between the potential achievements shown by cognitive
research and the observed deficiencies in the acquisiton of geometrical
concepts implies that instruction may play a crucial role, particularly
if it is conducted at the transitional stages. Research on cognition
shows that children at the middle grades level are at such a transitional
stage in the growth of concepts related to similarity. This suggests

that more detailed research, particularly at this age, is needed.

The Development of Proportional Reasoning

 

Similarity and Proportional Reasoning

 

The ability to handle metric proportions and the concept of geometric
similarity are related to each other: As mentioned in the previous
section, at the mastery level of the similarity concept, proportions
are recognized as the inherent rule that governs the situations in which
similarity is encountered. Piaget, Inhelder and Szeminska (1960) argue

that it is easier to study the growth of the concept of proportion

 

20

in geometric, rather than in non-geometric form, since before a child
can think about proportions, he can perceive whether two different shapes
are similar or not. On the other hand, Lunzer and Pumfrey (1966) showed
in their study, that success in geometrical situations that require
proportional reasoning (i.e., building a "wall" of a given length with
Cuisinaire rods, or using a pantograph) should not be interpreted as
mastery of the underlying proportional rule. Fifty percent of the nine-
year-olds could handle the Pantograph Task with simple pr0portions,
whereas less than 50 percent of the 15-year-old subjects were successful
in the Balance Task that required the recognition and the application

of proportionality in an abstract way.

The section on the development of proportional reasoning will follow
the same path as the review of the similarity studies. Piagetian research
that analyzed mental Operational structures needed in proportional
reasoning will be followed by studies that consider other variables
that may influence performance on proportionality tasks. Studies on
children's level of performance in proportional reasoning will be
reviewed next, whereas attempts to improve proportional reasoning through

instruction will be presented in a different section.

Piagetian Research

 

Inhelder and Piaget (1958) identify the understanding of
proportionality with the stage of formal operational reasoning, which
according to them emerges at 12 to 13 years of age. They investigated
the child's acquisition of proportionality by examining children's

reactions to situations such as equilibrium on a balance and prediction

21

of shadow size. 0n the basis of this inquiry the claim was made that
pr0portional reasoning appears at the age of 11-12 years, but initially
only in a qualitative form, and later (ages 13-17) it evolves into
abstract reasoning involving the formulation of the law of proportions
and the ability to operate on this law in a quantitative form. They
identified proportional reasoning as one of the eight schemes that develop
at the stage of formal thought. Martorano (1974) tested six out of
these eight schemes and found that the proportionality scheme was more
difficult than all others, except mechanical equilibrium. Linn and Swiney
(1978) found a: strong relationship between proportional reasoning and
general ability. Lunzer (1968) argued that the equality of two ratios
always constitutes a second order relation (i.e., a relation between
relations), and even a system of such relations (i.e., a/b = c/d implies

ad 8 be and a/c = b/d).

Studies with British children (Lovell, 1961; Lunzer, 1965; Lovell
& Butterworth, 1966) and with American children (Steffe & Parr, 1968;
Gray, 1972) confirmed basically the developmental categories identified
by Piaget, but noticed a difference in age distribution. Lovell (1972)

sums up these findings as following:

Apart from very able twelve-years-olds, it is from 12 years
of age onwards, the actual age depending on the ability
of the pupil, that facility is acquired in handling metric
proportion. Many pupils may not be able to do this until

14 or 15 years of age, and some never. (p.8)

22

In order to check the linkage between proportional reasoning and
the formal operational stage, Chapman (1975) compared the performance
of first, third, and fifth graders on conservation of ratio tasks
(presented in a probabilistic manner) with that of college students.
He concluded that even 10, or ll-year-old children do not discriminate
proportions, whereas most of the items were anwered correctly by college

students.

Proportional reasoning has been widely investigated by Robert
Karplus, who devised a test to determine the level of reasoning children
use (Karplus & Peterson, 1970). In his Mr. Tall/Mr. Short task, the
subject is presented two drawings: Mr. Short, whose height is measured
in large and in small paper clips, and Mr. Tall, measured in small
paper clips only. The subject is asked to predict Mr. Tall's height
as expressed in large paper clips. Later, Karplus modified the task,
by eliminating the drawing of Mr. Tall in order to prevent children

from relying on perceptual cues.

Karplus and Peterson (1970) categorized students' explanations
(strategies) given in solving the Mr. Tall/Mr. Short proportional

reasoning problem in the following way:

N. no explanation or statement "I can't explain" given.
I. intuition. Referring to estimates, guesses, appearances

or extraneous factors, without data.
IC. Intuitive computation. Use of data haphazardly and in
an illogical way.

A. addition. Applying the difference rather than the ratio.

 

23

S. scaling. Not relating to the scale inherent in the data,
thereby failing to see the whole problem-displays a
tentative attitude toward his/her estimate.

AS. addition and scaling. Focuses on the difference rather
than ratio, but scales it up.

P. proportional reasoning. Uses proportionality. May or

may not use word ratio.

In a later follow-up study, Karplus and Karplus (1972) found it
convenient to collapse these seven categories into three categories

representing conceptual levels:

Intuitive Level I = I + IC
Concrete-operational Level II = A + S
Formal-Operational Level III = AS + P

In this latter study, the performance of 153 students was compared
to their performance on the same task, two years earlier. It was found
that during this period, 65 percent of those who were at level I moved
up, whereas more than one third of the subjects showed no change in
category. In the investigators' view the results prove the developmental
nature of the acquisition of the concept of proportionality. (No clear

order could be established between the substages of level II.)

Further Karplus studies reveal that the additive strategy (defined
above as A) is used systematically by many seventh and eighth graders
(Wollman & Karplus, 1974), and even by 20 percents of the college students
that participated in another experiment (Karplus, Adi & Lawson, 1980).

The use of this strategy is attributed to a concrete-operational way

24

of thinking combined with inadequate instruction.
Variables Related to PrOportional Reasoning

Many alternative factors in the development of proportional
reasoning have been suggested in addition to or instead of Piaget's

scheme of cognitive development.

A comprehensive study in this direction has been conducted by Pulos,
Stage and Karplus (1980). Eighty-seven sixth and eighth graders were
tested for: (1) proportional reasoning (measured by their Lemonade Task
that involved ratios of sugar to concentrated lemon juice), (2) M-capacity
(the number of schemes that a child is able to handle simultaneously),
(3) fluid intelligence (measured by a wide range series completion test),
(4) crystallized intelligence (measured by a wide range vocabulary test),
(5) cognitive restructuring (measured by the FASP Embedded Figures Test)
(6) field dependency (measured by Pascual-Leone's Water Level Task),
(7) formal reasoning (measured by Piaget's Conservation of Volume Task),
and (8) divergent thinking (measured by an. alternative uses test).
Stepwise multiple regression analyses were employed to measure the
relationship between proportional reasoning and the other seven cognitive
variables. The results of the study suggest that proportional reasoning
is significantly related to only two of the variables: (1) information
processing (M) capacity (as suggested also by de Ribaupierre and Pascual-
Leone, 1979; Case, 1979; and Furman, 1980), and (2) encoding, or cognitive
restructuring (as suggested also in Siegler, 1978). In view of the
lack of relationship between proportional reasoning and conservation

of volume, Pulos, Stage and Karplus conclude that "the results do not

 

25

seem to support the hypothesis that a formal reasoning structure, as
defined by Inhelder and Piaget, is a necessary prerequisite for

proportional reasoning" (p.148).

Other variables that seem to influence performance in proportional

reasoning tasks are:

1. Level of abstractness. In a comparison of six proportionality
tasks, Wollman and Karplus (1974) report that seventh and eighth graders
(N=450) were more successful in concrete tasks (Mr. Tall/Mr. Short,
Ruler, Shading Fractions) as opposed to more abstract ones (Candy,
Numerical). Similarly, Portis (1973) indicates a significantly better
performance of fourth, fifth, and sixth graders (N=138) in a
proportionality test using physical and pictorial aids, as compared to
an equivalent test that used only symbols. Lunzer and Pumfrey (1966)
report a much better performance on the concrete Cuisenaire Rods, and
Pantograph Tasks than in the Balance Task. They also remark that in
the first two tasks, their subjects rarely' applied. specifically' the
proportionality rule. Moreover, Wollman and Karplus (1974) found that
in abstract tasks most subjects who reason proportionally in an incomplete
manner (i.e., incorrectly or concretely) "regressed to additive
reasoning". 0n the other hand, increasing the level of abstraction may
have also :a positive effect: when Karplus's subjects were denied the
opportunity to see Mr. Tall in the Mr. Tall/Mr. Short Task, the number

of intuitive-perceptual guesses dropped significantly.

2. Ratio versus fraction. Wachsmuth, Behr and Post (1980)

 

contrasted performance of fifth grade students (N=15) on Ink-Mixture

 

 

 

26

Tasks presented in a ratio (i.e., part-to-part) format with the
performance in the same tasks presented in a fraction (i.e., part-whole)
format. They indicated a greater success (72 percent) for the ratio
format than for fractions (58 percent). Noelting and Gagne (1980) found
low correlations between the Orange Juice Experiment bearing on ratios

and the Sharing Cookies Experiment bearing on fractions.

3. Numerical context. Abramowitz (1975) raised the possibility
of a relationship between the numbers employed in a proportion and the
rate of success, since children may "have an intuitive understanding
of proportionality without concurrently having the mathematical facility
to solve proportion problems "(p.25). She checked three numerical
characteristics: (1) equal or unequal differences (i.e., presence or
absence of a repeated difference among the three given numbers), (2)
size of the unknown number (i.e., whether the unknown number is larger
or smaller than the three given numbers), and (3) types of ratio (i.e.,
whole numbers or fractions). The results obtained from 32 seventh graders
indicate that type of ratio and size of the unknown number have a
significant effect, but the difference between numbers does not.

Abramowitz (1975) also observed that

subjects, especially those transitional between concrete and
formal Operational thinking...may be quite capable of
reasoning through proportions of moderate difficulty.
However, when faced with a more demanding task, these same
subjects may revert to the use of patterned concrete

strategies. (p.26)

 

27

Karplus, Karplus, and Wollman (1974) also report that the value
of the ratio influences performance: a ratio between 1 and 2 attracts
more errors (specifically, more additive responses) as compared to ratios
smaller than 1 or greater than 2. Quintero (1983) tested the performance
of 36 fifth, sixth, and seventh grade Puerto-Rican children on verbal
problems involving the basic proportion a/b = c/x. She found that one
third of her subjects could solve problems in which both a/b and a/
c are integers, whereas 20 percent could handle problems in which only
one of these ratios is integer. The same phenomenon was reported by
Karplus, Pulos and Stage (1980) with 120 sixth and eighth graders: a
success rate of about 60 percent was observed in the Lemonade Task
when both a/b and a/c or only a/b were integers, but the rate of
success was much lower when only a/c or no ratio were integers (38

and 18 percents respectively).

The British project "Concepts in Secondary Mathematics and Science"
(CSMS) administered a Ratio and Proportion Test to over 2200 eleven
to sixteen-year-old students. In analyzing the results, Hart (1982)
identified five levels of performance: At level zero, the student is
unable to make a coherent attempt at any of the questions. At the
next four levels, the student is able to solve items of increasing
difficulty. Her categorization of the items was:

I. No rate is needed or given; the answer may be obtained
through multiplication by 2, 3 or taking half.
II. Rate is easy to find or answer can be obtained by taking

an amount and then add half again as much.

 

28

III. Rate must be found and is harder to find than above;
fraction Operation may also be needed.
IV. Ratio is needed; the questions are complex in either

numbers needed or setting.

4. Sequencing. Karplus (1978) reported a 51 percent rate of success
when a prOportional reasoning task was preceeded by a simple ratio task,
while only 12 percent answered correctly when the first task was more

difficult.

5. Irrelevant and redundant information. Collea and Numadel (1978)
found that when a problem contained both irrelevant and redundant

information, a significantly poorer performance resulted.

All these findings suggest that "consistent use of proportional
reasoning is not a developmental outcome, but depends instead on
overcoming task related obstacles" (Karplus, Pulos & Stage, 1980, p.141).
Reaching the same conclusions, Abramowitz (1975) contemplates that it
may be that "the use of proportionality occurs in.a developmental sequence
across a certain set of tasks." This argument "has implications for
when it would be best to teach various concepts requiring an understanding
of proportionality" (p.27). These instructional implications will be

discussed in a separate section.

Socio-economic status (SES) and sex are other variables that have
to be considered in performance on prOportionality tasks. Particularly
poor proportional reasoning by low SES children was shown in two Karplus
studies: Karplus and Peterson (1970) indicate that although urban and

suburban children perform equally low on the Mr. Tall/Mr. Short Task

 

 

29

at the sixth grade level, by the end of high school most of the suburban
subjects (80%) had mastered the task, whereas only 9 percent of their
urban counterparts did so. 'Thirty-six hundred seventh and eighth graders
from seven countries were tested on the same task (Karplus, Karplus
& Paulsen, 1977), and only small differences in overall achievement
among countries were detected. However, the results show that groups
of low SES American students performed at the lowest level (i.e.,

primarily intuitive).
Difficulties in Progortional Reasoning

In the section on the concept of similarity, it was shown that
generally, at the age of 11 or 12 years, children are able to handle
proportion tasks if they are accompanied by physical action. Studies
that used paper-and-pencil tests or tasks that required the use of the
prOportional rule in a generalized form, report that the ages at which
a significant portion of the population could master the tasks is higher-

-about 15 years.

Lunzer and Pumfrey (1966) report a success rate of less than 50
percent for his 15-year-Old subjects in the Balance Task (see also Lovell
& Butterworth, 1966). Similar trends are reported by the Karplus studies:
Karplus and Peterson (1970) found that 32 percent of the suburban 8-10th
graders and 80 percent of the suburban 11-12th graders could reason
proportionally on the Mr. Tall/Mr. Short Task. The corresponding numbers
for their urban peers were respectively five and nine percent. Wollman
and Karplus (1974) found that only 20 percent of the 450 seventh and

eighth graders tested on five proportionality tasks applied proportional

 

 

30

reasoning in a consistent way. In their survey of seven countries,
Karplus, Karplus and Paulsen (1977) discovered that 25 percent of their
subjects (N =- 3600, grades 7-8) used proportional reasoning. The same
rate of success among American seventh graders was reported by Abramowitz
(1975). Moreover, the rate of develOpment seems to be very slow, at
least before the age of 15 years (Karplus & Karplus, 1972). Sudden
"jumps" in the level of performance were noticed later in high school

(Karplus, Adi & Lawson, 1980; Karplus & Peterson, 1970).

Studies report a poor level of proportional reasoning even at the
college level. According to Lawson and Wollman (1976), Fuller and
Thornton (reported in Pagni, 1983), and Renner and Paske (1977), only
about 50 percent of their samples of college students were able to
reason proportionally in a formal way. The figures reported by Karplus,
Adi and Lawson (1980) for college freshmen, and by Karplus and Peterson
(1970) for suburban 12th graders are higher: 74 and 80 percent

respectively.

Conclusion

 

There are various development models for the acquistion of the
concepts of geometrical similarity and of proportional reasoning; many
of them are complementary rather than contradictory. By attempting to
superimpose the literature on similarity and on prOportionality, it may
be concluded that, the developmental stages in the acquisition of the
similarity concept and of proportional reasoning are quite similar, and
any explanation that seems valid in one field can be applied in the

other. Chronologically, however, the concept of similarity (and possibly

31

some other concrete applications of proportionality) seem to precede

a more general and abstract use of prOportionality.

The Concepts of Area and Area Growth

Definition of the Problem

The area. growth. of similar shapes, requires the recognition. of
the fact that the enlargement of a figure by a scale factor of n
will increase its area by n2. The concepts of similarity and of area
are, therefore, prerequisites to an understanding of area growth. In
this section, the literature on the concept of area will precede the

description of a few studies that can be related directly to the topic

of area growth.

Concept of Area

 

Conservation of area. According to Piaget, Inhelder, and Szeminska
(1960), area conservation (i.e., considering area as a stable attribute
independently of the shape of a figure) is attained at the early concrete-
Operational stage--around the age of 7 1/2 years. Beilin (1964), however,
objected to the fact that in Piaget's experiments on area conservation,
the transformations of the figures were performed in front of the
subjects. He defines "quasiconservation" as the ability to determine
that two noncongruent regions have equal areas when the child is presented
with the end product of a rearrangement. Beilin tested 316 children
in grades K through 4 and found a correct response level of less than

50 percent of his fourth-grade subjects. Two other larger-scale studies

32

(Renner, 1971; Hademenos, 1974) indicate that area conservation as
defined by Piaget is generally achieved at a later age of about ten
years. Similarly, in studying the conception of area measure with 75
eight, ten, and eleven-year-old children, Wagman (1975) concludes that
"about a third of the ten, and 11-year-olds in the sample either failed
to apply at least one of the neglected axioms (area, congruence, and
additivity) even in perceptually easy cases, or did not conserve area"

(p.109).

Calculation of area. The quantification skills necessary to derive

 

the area (e.g., height x width for rectangles) seem to evolve also
at a later age than Piaget's theory of cognitive develOpment indicates.
Anderson and Cuneo (1978) showed that the adding rule (i.e., height
+ width) describes five, and six-year-Olds' judgment of rectangular area.
Leon (1982) shows that seven-year-old children use the "linear extent"
rule to judge area--which in the case of rectangles would be the length
of the diagonal, and that only by the ages 8-9 is the linear extent

rule replaced by the multiplicative rule.

The difficulties in the calculation of area seem to persist even
at a later age. Carpenter et a1. (1981) analyzed the results on the
area items of the second mathematics assessment of the NAEP and Observed
that (1) few nine-year-olds have any knowledge of even basic area concepts
(28 percent could find the area of a rectangle divided in square units);
(2) among the 13-year-olds, 51 percent could find the area Of a rectangle
by the dimension of its sides, 12 percent could find the area of a

square with a given side, and even fewer could find the area Of a

33

right triangle; (3) 74 percent of the 17-year-olds in the sample
calculated correctly the area of a rectangle, 42 percent--the area of
a square, and 20 percent--the area of a parallelogram or of a right
triangle; (4) the success rate in application exercises was even lower-
-16 percent of the 17-year-olds could find the area of a region made

up of two rectangles. Carpenter et a1. (1981) conclude:

Performance Of perimeter, area, and volume exercises was
among the poorest of any content area on the assessment.
Not only was performance extremely low on exercises at the
application level, but many students at all ages appeared
to have no understanding of the most basic concepts Of

perimeter, area, and volume. (p.98)

In a survey of geometric concepts possessed by 198 sixth graders
on leaving elementary school, Schnur and Callahan (1973) report similar
results: finding areas of rectangles was "marginally easy" (i.e., a
difficulty index between .50 and .69), areas of squares and parallelograms
were "marginally difficult" (i.e., a difficulty index between .30 and
.49), and calculating the area of a triangle was "very difficult"(i.e.,

a difficulty level of less than .10).

Perimeter and area. The confusion of area and perimeter has been
Observed in all the studies that related to the acqusition of these
concepts. Piaget, Inhelder, and Szeminska (1960) observe that in their
area tasks, some subjects related to the perimeters of the two regions,
and consider the reliance on linear terms characteristic of the concrete

level (III A). Wagman (1975) reports that approximately one third of

 

34

their subjects (ages 8, 10, and 11 years) confused area and perimeter
at some time during their interview. In their analysis Of the second
NAEP results, Carpenter et a1. (1981) indicate that 23 percent of the
13-year-Olds and 12 percent of the 17-year-olds calculated the perimeter
of a rectangle instead of its area, whereas half of the 13-year-olds,
and a third of the 17-year-olds did the same in the case of a square

and a right triangle.

Area Growth of Similar Shapes

 

After three weeks of instruction that focused on growth and shrinking
of figures and objects, Fitzgerald and Shroyer (1979) report a low level
of performance among their sample of 350 sixth graders: 35 percent could
master a task (Hi the growth of a square presented in a concrete mode,
but only half of these answered correctly when the task was presented
in an abstract mode; the rate of success for tasks that required the

application of the area growth principle varied between 3 and 13 percent.

McGillicudy-DeLisi (1977) related the performance of six through
13-year-old children on tasks that involved enlarging rubberband figures
on a pegboard with the cognitive level of her subjects (N=75). By
a qualitative analysis of her subjects' strategies and types of movement,

she found that the rate of success increased with operative level.

An explanation for the difficulties encountered in area growth tasks
can be deduced from the research conducted by Bang in France (reported
in English by Montangero, 1976) and by Lunzer (1968, 1973) in England.
The tasks involved inquiries about area and perimeter under two kinds

of transformations: (1) a series Of rectangles with a fixed perimeter

 

 

35

(but a decreasing area) created from an initial square, and (2) a series
of figures with a fixed area (but an increasing perimeter) created by
cutting a triangular section from a lower corner of a square and
transferring it vertically until it is attached to the upper corner

of the original figure.

Lunzer (1968) argues that at the concrete-operational level of
reasoning, children employ "false conservation": realizing that
something was preserved, they resist the evidence of perception, and
regard the conservation of both area and perimeter as logically necessary
in both cases (1) and (2). They reason about the figures as Objects
and about area and perimeter as essential characteristics of the object
that must "go together". According to Lunzer, only at the formal level,
area and perimeter are clearly disassociated, and both are assumed to
vary according to some (but not necessarily the same) law. He argues
that since area snd perimeter are well-defined relations existing in
figures, the discrimination between them requires the ability to use
second-order relations--an ability that can be found only at the formal

level of reasoning.

This argument may provide a valid explanation to the reported
difficulties in the acquisiton Of the concepts of perimeter, area, and
area growth. The understanding of the area growth requires the
recognition of the fact that for similar shapes linear dimensions and

area vary according to different rules.

36

Geometry and Spatial Visualization

Spatial ability

 

Although descriptions and measurements of aspects of intelligence
started earlier, only in the 1940's did spatial ability start to gain
more attention. Wolfle (1940) reviewed the factorial studies up to
1940, and stated that verbal ability and space ability were the two
most frequently identified factors. Factor analyses conducted by a group
of American psychologists in the Aviation Psychology Program (Fruchter,
1954; Zimmerman, 1954; Michael, 1954) identified in spatial ability at
least two factors: (1) Spatial Relations and Orientation, and (2) Spatial
Visualization. Ekstrom et al. (1976) observe that there has been some
difficulty in explaining the difference between these two factors. They
suggest that the figure is perceived as a whole in spatial orientation,
but must be mentally restructured into components for manipulation in
spatial visualization. More specifically, they define spatial
visualization as "the ability to manipulate or transform the image of
spatial patterns into other arrangements" (p.173). In his survey of
studies on spatial visualization, Ben-Haim (1983) argues that the nature
of this ability is still a controversial issue--especially for children
younger than 11 years. He points out that "the picture in the eighties

is still unclear,‘ and quotes from Harris (1981):

Our attempts to identify the critical components of various
"spatial" tests are still part guess work, particularly where

we lack factor analyses involving both standard and

 

 

 

37

nonstandard tasks...Furthermore, consensus is still lacking
on the meaning of the two factors--orientation and
visualization...factor analysts continue to have difficulty

in differentiating and interpreting these factors. (p.23)

There is an abundance of spatial visualization tests (see for example
Smith, 1964; Ben-Haim, 1983), but only two will be mentioned here: (1)
the Spatial Relations of the Differential Aptitude Test (DAT) which
requires the subject to identify the figure of a solid that can be
Obtained when a given pattern is folded (Bennett et al., 1966), and
(2) the Middle Grades Mathematics Projects (MGMP) Spatial Visualization
Test which requires the subject to rotate cube constructions mentally
and to identify them from a different perspective (Lappan, 1983). The
DAT test was frequently used in studies that compare spatial visualization
with geometric abilities, whereas data from the MGMP test will be used
in this study to assess the relationship between spatial visualiztion

and achievement in similarity concepts.

Geometry and spatial visualization

 

One of the goals of this study is to examine the relationship
between the acquisition of similarity concepts and spatial visualization.
Since studies on this specific topic have not been reported, the issue
will be extended to include the relationship between spatial
visualization and geometrical abilities in general. Research in this
area, however, has limitations of its own. In his study on the
relationship between geometrical, and spatial ability, Werdelin (1961)

observes:

 

38

Very few studies have dealt with the structure of geometrical
ability, especially in comparison with the number of
investigations of, for example, the verbal and the visual-
perceptual field. With a few exceptions, the investigators
of the field seem to have considered it to be an unitary
part Of mathematics, or devised only one test to cover it.
Quite often, geometrical ability and spatial ability have
been confused with each other, which of course is easily
done, but which does not facilitate the solution of the

problem. (p.33)

The source Of this confusion is the fact that there is no clear
definition of geometrical ability. Most of the relevant factor analyses
and regression studies relate to achievement in plane Euclidean geometry
at the high school level. Euclidean geometry is a clearly defined field,
but on the other hand, it also requires a great deal of formal knowledge
and of logical and verbal ability. Whether conclusions can be drawn
from these studies to the relationship between spatial visualization

and a more informal geometrical ability, is a debatable issue.

There are strong cognitive and pedagogical reasons to assume a
close relationship between informal geometry and spatial ability. Some
of the visual aspects of similarity and prOportional reasoning tasks
have been already discussed. From a pedagogical point of view, Hoffer
(1977) lists seven aspects of the visual perception ability that are
essential in the geometrical development of a child at the elementary

and middle grades level: (1) visual-motor coordination (i.e.,

39

coordinating vision with movement of the body), (2) figure-ground
perception (i.e., distinguishing foreground from background), (3)
perceptual constancy (i.e., recognizing an object out of its original
context or from a different vieWpoint), (4) position in space (i.e.,
determining the relationship of an object as compared to the observer),
(5) spatial relationship (i.e., perceiving the position of two or more
objects both in relation to the observer and in relation to each other),
(6) visual discrimination (i.e., distinguishing similarities and
differences between Objects), and (7) visual memory (i.e., recalling
accurately an Object no longer in view and relating its similarities

and differences to other items in view or not in view).

The possibility of a connection between the acquisition of similarity
concepts and Spatial visualization is strengthened 'by results from a
study that revealed a relationship between prOportional reasoning and
spatial ability: Brendzel (1981) used a sample of 400 ninth and eleventh
graders, and by conducting an analysis of covariance with the 1.0. as
a covariant, she found that ability in spatial visualization accounted
for 62 percent of the variation in achievement in proportional reasoning

(as measured by the Karplus, Rund, and Piaget's tasks).

The findings of studies on the relationship between high school
geometry achievement and Spatial visualization are inconclusive.
Holzinger and Swineford (1946) report a high multiple correlation (.71)
for predicting plane geometry achievement by the best three predictors
out Of an initial battery of nine tests: one of them was connected

to the general factor (as measured by a series completion test) and

 

 

40

two others to spatial ability (as measured by the Visual Imagery, and
Punched Holes-Verbal tests). Werdelin (1961) conducted a comprehensive
factorial analysis of mathematical abilities in three highly selective
Swedish schools, and concludes that "the Space factor or factors are
of essential importance to the different aspects of geometry; primarily
to geometrical construction and abstraction and also to problem solving"
(p.122). In an earlier study, Werdelin (1958) reanalyzed the data
published by Rogers (1918), and showed that contrary to the initial
findings of this study, geometrical ability depends on spatial ability
and also, but to a smaller degree on verbal ability. Werdelin (1958)
notes that the geometrical tests in Rogers' study were "quite similar"
to space tests. This Similarity may be considered a potential source
of confusion between geometrical and spatial abilities, but on the other
hand, it may be considered a valid evaluation mean of a less formal

geometrical ability.

In an investigation of the informal aspects of geometrical abilities
at the college level, Blade and Watson (1955) indicate that achievement
in spatial visualization among engineering freshmen correlates positively
with grades in descriptive geometry (r=.54) and in engineering drawing

(r=.34).

0n the other hand, there are many studies (Murray, 1949; Weiss,
1955; Werdelin, 1958; French, 1964; Bennet et al., 1966) that seem to

indicate that geometrical ability is independent of the space factors.

The ambivalence of these findings is reflected also in a study

by Hanson (1972): he found some connection between visual perception

41

ability and geometry for his ninth-grade subjects (11 percent of the
variance in geometry achievement could be accounted for), but did not
find a Significant correlation for the tenth graders. Werdelin (1961)
observes that "the question whether mathematical (geometrical) ability
is dependent on the visual factor(s) has not been definitely answered"
(p.38). This Observation seems to be valid in view of the contradictory
findings in the case of Euclidean geometry, and of lack of evidence

in the case of informal geometry.

Sex Differences

 

Sex Differences in Geometric ability

Sex differences in mathematical abilities is a widely debated issue.
For many years it was accepted that sex differences in mathematical
abilities, favoring males emerge at adolescence (Maccoby and Jacklin,
1974). More recently, counter-evidence has been provided by Fennema
and Sherman (1978) who investigated the achievement Of high school
students in light of their mathematical backgrounds and found that males

and females with similar course enrollment had Similar achievements.

As compared with studies on sex differences in spatial visualization,
the research related to geometric abilities is Significantly scarcer.
Moreover, the two fields are sometimes hardly distinguishable. For
example, a study on the effect of instruction in Tangram puzzles (Smith
& Shroeder, 1979; Smith & Litman, 1979) is discussed as a study in

Spatial visualization, but could be related to informal geometry as

 

42

well.

The issue of sex differences related to the concept of similarity
has not been considered as a separate tOpic. An indirect attempt to
address it was made in 1925 by Cameron. By using rather rudimentary
statistical methods and a battery of nine tests, she analyzed sex
differences in mathematical aptitudes in a sample of 13 through 17-year-
oldS. One of her tasks required the completion of a construction of
a rectangle similar to a given one, when one of its Sides is also
a given. Cameron (1925) concludes that the performance of boys and
girls was slightly in the boys' favor , but "fairly equal". Due to
the lack of systematic research on the concept of similarity, studies
on sex differences in geometric abilities in general and in proportional

reasoning may be considered relevant.

Three studies that relate to informal, or to transformational
geometry have detected no sex differences in performance: Olson (1970)
at high school level, and Thomas (1977) at the grade levels 1, 3,
6, 9, and 11, reached this conclusion in studies on transformation
geometry; G.D. Peterson (1973) made this Observation after administrating
a 50-item geometry achievement test to 725 fourth, fifth, and sixth

graders.

When the connection with spatial ability is made, the results become
less clear. Shonberger (1976) indicates that practical and geometrical
problems involving Spatial components are the source for sex differences
in mathematical abilities. In his study on geometric and spatial

abilities Of Swedish boys and girls at the high school level, Werdelin

 

43

(1961) found a male superiority in less formal aspects of geometry such
as geometrical construction and abstraction, and geometrical problem
solving, whereas girls had a better ability to prove theorems. These
findings remained basically unchanged, when he compared 143 boys with
the same number of girls matched by general reasoning, numerical ability,
and age. Werdelin also refers to another Swedish study conducted in
1944 by Siegwald. This study indicated male superiority in dynamic
visualization (i.e., mental imagery manipulations), space relations,
geometrical imagination, construction of difficult, unfamiliar
geometrical figures, and in geometrical problems, but not in perception
of form, construction of simple, familiar shapes, and in static

visualization.

With regard to the effect of geometry instruction on spatial
abilities as a function of sex, Brinkmann (1966) found that eighth-
grade boys and girls performed and gained similarly on a spatial
visualization test after an intervention of ten programmed units in

informal geometry .

A four-hour-long instruction based on Tangram puzzles caused
Similarly good results for both sexes at the fourth grade level (Smith
& Schroeder, 1979). However, the same intervention erased the pre-
interventional female superiority at the sixth and seventh grade level
implying that at the Stage of early adolescence, girls profit less from

this kind of instruction (Smith & Litman, 1979).

Most of the studies on sex differences in high school Euclidean

geometry imply that as compared to boys, girls have similar or higher

44

abilities in tasks that require proving theorems (Touton, 1924; Perry,

1929; Werdelin, 1961; Senk.& Usiskin, 1982).

To conclude, the findings in geometry seem to confirm a trend that
has been Observed in other mathematical fields: when experience is
controlled (as for example, in.Euclidean geometry or in a tranformational
but formal approach to geometry) girls and boys perform equally well.
On the other side, in mathematical tasks that also rely on out-of-
School experience (e.g., some spatial tasks) sex differences tend to

Show up.
Sex Differences in Proportional Reasonipg

Studies on sex differences in proportionality tasks suggest that

boys tend to outperform girls at any age.

Chapman (1975) showed that even at the third and fifth grade levels,
boys scored significantly (p<.01) higher than girls in ratio comparison
tasks and exhibited greater frequency of verbal explanations referring

to proportions.

In his prOportional reasoning study of seven countries, Karplus
et al. (1977) Observe that eighth and ninth grade female subjects tended
to use additive responses (which are of a lower order) more Often than

males.

Brendzel (1981) refers to a study conducted in 1977 by Piburn at
Rutgers University. Piburn detected a male superiority in proportional

reasoning tasks among his subjects ranging in age from 13 tO 23.

45

Keating and Schaefer (1975) found that fifth and seventh-grade
psychometrically bright boys outperformed Sixth and eighth-grade bright

girls on three Piagetian tasks--one of them.being the Balance Task.

Some studies also indicated sex differences as a function of task-
related variables. Besides a general male superiority, Stage, Karplus,
and Pulos (1980) found, that among their Sixth, and eight-grade subjects,
younger and female students have additional difficulties with structural
changes that make tasks more difficult for all students (such as a
more difficult numerical context). At the ninth grade level, Brendzel
(1978) found evidence of sex differences favoring males in proportional
reasoning tasks, particularly when unfamiliar measuring units were
employed. Garrard (1982) concludes that at the eighth grade level, visual
adjuncts were more effective (p<.10) for girls in the solution of twenty

non-routine proportionality tasks.

The male superiority in proportional reasoning, and the inconclusive
findings on sex differences in geometrical tasks, do not allow for
predictions on the existence of, or lack of sex differences in geometrical

Similarity tasks that combine these two aptitudes.

The Effect of Instruction

 

Influencing Cognitive Development

Relatively few attempts have been made to study the influence of
instruction on the acquisition of geometrical concepts, and among these

studies few could Show positive results. Since no studies have

 

46

concentrated specifically on instruction related to the concept of
similarity, three related tOpics will be discussed: (1) the teaching
of geometrical concepts in general, (2) the teaching of proportional
reasoning, and (3) the effect of instruction in geometry on spatial

ability.

The Piagetian research on these topics is mainly descriptive, and
its instructional implications are usually stated as speculations in
separate studies. More recently, Piagetian researchers recognize the
potential of instruction as an accelerator of cognitive processes.

Montangero (1976) states:

Learning must be subordinated to the laws of development
Since operational structures do not derive from structures
that might exist outside the child, but stem from the
coordination of internalized actions...In brief, training,
according to the Genevan conception, consists in trying to

accelerate cognitive develOpment. (p. 126)

Some empirical studies showed that instruction on a certain concept
at a certain age failed to bring the desired results, and the failure
was related to the subjects' functioning at a lower mental-Operational
stage than the level requested by the presented task. Young (1975)
for example, examined the relationship between performance on tasks
related to ten geometrical concepts and instruction. By use of Chi-
Square scores in a sample of 791 kindergarten through third graders,
Young concluded that performance was independent from received

instruction. Boulanger (1974) attempted to teach prOportionality to 51

47

third graders that were at the concrete-operational level, and concluded
that the training was neither retained over time, nor did it transfer
to other Simple proportion tasks. Lawson and Wollman (1980) also
experimented with teaching proportional reasoning in a class of seventh
graders, and concluded that the extent of success depended on each
subject's operational level, and that formal operational level is a

prerequisite for teaching proportional reasoning.

An important model that takes into account both the cognitive and
the instructional aspects in children's understanding of geometry, was
designed by P.M. and Dina van Hiele (for a good description in English,
see Wirszup, 1976). The Van Hieles identified five stages of geometrical

development (Table 2.1) and propose that the levels cannot be skipped.

Unlike Piaget, they argue that the levels develop primarily under

the influence of school instruction. Therefore, instruction should be

Table 2.1 THE VAN HIELE LEVELS OF DEVELOPMENT IN GEOMETRY
(adapted from Burger et al. 1981)

 

 

 

 

 

LEVEL CONTEXT FORM OF REASONING
0 Basic shapes and figures in Visual identification and comparison
some geometrical space of shapes
I Properties of shapes in the Informal analysis of shapes in terms
space of their properties
II Relationships among proper— Logical partial ordering of pro-
ties of shapes perties
III An abstract geometrical system Formal deduction (proof) of theorems
within the system
IV Various geometries Rigorous mathematical study of the
geometries

 

 

 

 

48

geared to lead students deliberately from one level to the
next. Wirszup (1976) reports on two Soviet researchers who
adapted a program of geometry instruction to the'Van Hiele

model with striking success.

Wirszup (1976) also uses the Van Hiele model to explain the present

deficiencies in the teaching of geometry in the United States:

Only a very small number of the elementary schools offer
any organized studies in visual geometry, and where they
are done, they begin with measurements and other concepts
which correspond to Levels II and III of thought develOpment
in geometry. Since Level I is passed over, the material
that is taught even in these schools does not promote any
deeper understanding and is soon completely forgotten, Then,
in the 10th grade, 15 and 16 year old youngsters are
confronted with geometry for almost the first time in their
lives. The whole unknown and complex world of plane and
Space is given to them in a passive axiomatic or pseudo-
axiomatic treatment. The majority of our lflgh school
students are at the figgp level of develOpment in geometry,
while the course they take demands the fourth. level of
thought. It is no wonder that high school graduates have
hardly any knowledge of geometry, and that this irreparable

deficiency haunts them continually later on. (p.96)

Thus, according to this view, instructional, rather than cognitive

deficiencies are the main reason for a poor performance in geometrical

 

49

tasks.
Instructional Interventions

Geometry. Most of the studies about the effect of instruction on
achievement in geometry compare the effect of different modes of
instruction on a wide variety of concepts, rather than examine the effect
of instruction on the acquisition of a specific concept. The topics
of reported comparisons are: (1) concrete-activity orientation versus
paper-and-pencil approach, (2) problem-solving orientation versus
presentation of the end-product, and (3) formal (i.e., Euclidean) versus

informal geometry .

1. Concreteness. In a survey of Studies that compared the use
of concrete materials with reliance on symbolic representations in the

teaching of mathematics, Fennema (1972) concludes:

Learning of mathematical ideas is likely to be facilitated
by a predominance of concrete models in the early grades
and a gradually increasing proportion Of symbolic models
as children move through the elementary school...Piaget
placed children up to twelve years of age in the concrete-
operational stage of cognitive development. Children in this
stage are capable of learning with symbols, but only if
these symbols represent actions the learners have done

previously. (p. 637)

Relating specifically to geometry, Bring (1972) showed that a unit

for fifth and sixth graders on volume, congruence, symmetry and isometrics

50

gave significantly better results when presented in a concrete-activity

mode as compared to a control group that used only paper and pencil.

2. Problem solving, Mathematics educators agree on the advantages
of emphasizing the evolution of mathematical models from real-life
situations, as opposed to an expository teaching that presents a ready-
made product. Two studies that addressed this issue in relation to
the teaching of informal geometry were conducted by Scott, Fryer, and
Klausmeyer (1971), and by Hempel (1981). Both indicate that expository
teaching gives better short-term results, but the discovery approach
proved superior in retention and transfer. Buchert (1980) also compared
the two 'modes of instruction. of informal geometry ‘with. 108 seventh
graders, and found that the "mathematized" (i.e., process-oriented)
instructhmn was clearly superior (p<.0001) for students of all

arithmetical abilities.

3. Informalitl. In the last eighty years, many committees
recommended that geometry be taught informally at the K-8 grade levels.
J. C. Peterson (1973) suggests to extend this recommendation to the
high school level as well. Several studies conducted in high schools
compared the teaching of formal Euclidean geometry with a less formal

approach, but their findings are not clear.

From a pedagogical point of view, Peterson presents the advantages
of teaching informal geometry at 33.31 grade level. According to him,
"informal geometry is more than a list of topicsr-it is a method of
teaching geometry. Informal geometry at its best makes use of discovery

methods of teaching, inductive reasoning and the student's

 

51

inquisitiveness" (p.60).

However, difficulties arose ‘when.:more rigorous studies compared
achievement of students who learned an experimental informal geometry
course for the 10th, and 11th grade level with a control group that
studied the traditional high school geometry course. Cox (1980) concludes
that his results are "not clear": the control group did better on the
ETS Cooperative Test in Geometry-Part 1 and also on problems in solving
multiple concepts and properties, and more complex constructions. 0n
the other side, the experimental group showed better attitudes towards

geometry.

Three studies experimented with interventions based on
transformation geometry at the high school level: Olson (1971) reports
disappointing results-especially on a four-problem proof test; Usiskin
(1972) found that his control group outperformed (p<.01) the experimental
group in standard geometry content; Durapau (1979) indicates similar
achievements for his control and experimental groups, but a: better

performance of the latter (p<.01) on transfer questions.

A possible explanation of these results is the fact that in order
to allow for comparisons, new experimental approaches have been evaluated
with tools that measured Skills characteristic to the content and the

goals of a traditional geometry course.

Proportional reasoning. Lunzer (1973) tries to strike a balance between

 

the limitations set by' the cognitive-Operational. ability' of a: child

in terms of proportionality and the need for instruction. 'He states:

52

It has not been shown that the realization of proportionality
comes about spontaneously as a result of maturing logic...It
is more probable that these relations need to be taught.
But the evidence suggests that even given good teaching,
they will not be applied spontaneously to new problems until
the student's powers of reasoning have reached an advanced

level of development. (p.13)

After reporting SES- rather than nation-related differences of
performance in proportional reasoning, Karplus et al. (1977) conclude
their international survey by arguing that "teaching makes a difference
at this age" (i.e., 13-15 years), and recommend an instruction that
provides concrete experiences and emphasizes active participation in
learning. Similar conclusions are drawn by Wollman and Lawson (1978)
in a comparison of a verbal approach with a physical-action orientation
in teaching seventh graders to reason proportionally. By using a three-
week-long instruction unit, Kurz and Karplus (1977) showed that
proportional reasoning can be taught effectively at the high school
prealgebra level: after instruction, more than 50 percent of the
experimental students, but less than ten percent of the control group
scored highly on proportional reasoning tasks. However, in a comparison
of two instructional modes for the same unit, they could not detect
a significant difference in achievement: the rate of success in the
"manipulative" group went up as a result of instruction from 15 percent
to 70 percent, whereas the "paper-and-pencil" group advanced from 17

to 62 percent.

 

53

Spatial ability. Presently there is no evidence that instruction in
geometrical Similarity can cause improvement in spatial visualization,
the issue can be addressed indirectly by analyzing the studies that
attempted to relate instruction in geometry in general to Spatial
visualization. The studies that deal with transformational geometry are
particularly relevant, since the concept of similarity fits naturally

into a transformational approach to geometry.

Theoretically, the connection between transformational geometry and
spatial ability is quite straightforward: the study of Slides, flips,
and turns on objects or drawings should increase the ability to perform
mental manipulations of figures. Piaget (1964) argues that imagery
manipulations can be described in terms of geometrical transformation.
However, findings of some empirical studies, do not support this

argumentation.

Perham (1977) reports that after an ll-session-long instruction in
transformation geometry, his experimental group of first graders gained
significantly' more than. the control group (N==72) in transformation
geometry but not in spatial visualization (as measured by a test on
horizontial-vertical and left-right orientation, and on figure folding).
A similar failure to achieve transfer from transformation geometry to
mental imagery manipulation after a 12-session-long instruction was
reported by Williford (1972) at the second and the third grade level.
Contrary results were reported by Ekman (1968), who detected significant
improvement in spatial ability during the summer, as result of instruction

in geometrical measurement and calculations conducted towards the end

54

of the fourth grade. This study, however, has no control group that

would assure that the improvement is not a result of maturation.

Brinkmann (1966) experimented at the eighth grade level with a
10-session-long programmed course in geometry that emphasized figure
differentiation (n: discrimination, pattern folding, and object
manipulation in a problem-solving approach. Wolfe (1970) basically
replicated Brinkmann's study at the grade levels seven, eight, and nine.
Both studies Show significant gains in spatial visualization (as measured
by the DAT Space Relations Test). Wolfe, however notes that little
transfer was observed in tasks that were different from the ones

encountered during instruction.

At high school level, Ranucci (1952) did not detect any Significant
differences in space perception abilities between a group that studied
a course in solid geometry and a control group that did not take the
solid geometry course. Similarly, Brown (1955) compared performance in
spatial visualization (as measured by the DAT Space Relations Test)
of a control group that studied one year of plane geometry, with a
group that studied a combined course of plane and solid geometry. He
concluded that the addition of solid geometry did not prove itself as
a facilitator for spatial visualization. Studies by Myers (1958) and
Sedgwick (1961) indicate respectively that neither a high school course
in mechanical drawing nor a college course in descriptive geometry could
cause a significantly different performance in Spatial visualization

tasks as compared to control groups that did not study these courses.

55

However, instruction that uses a less formal approach to geometry
has a stronger influence on spatial visualization. Van Voorhis (1941)
conducted a first-year college course that included estimating linear
extent, angles and areas, three-dimensional tick-tack-toe, and various
other visualizing experiences. A study by Battista, Wheatley, and Talsma
(1982) involved a geometry course for prospective elementary teachers
that included manipulating concrete models, paper folding, tracing, using
Miras, constructing polyhedra, and transformational geometry. Both
studies report a significantly better performance in spatial
visualization: the first as compared to a control group (on the "Cards"
and "Figures" sections of the Thurstone Test for Primary Mental
Abilities), and the second as compared to pre-interventional performance

(on the Purdue Spatial Visualization Test: Rotations).

To summarize the presented evidence, at the elementary level,
instruction in geometry did not have a Significant effect on spatial
visualization. Starting from the middle grades level, this effect
manifests itself in interventions that emphasize the visual aspect of
geometry. Studies on traditional courses in geometry could not Show
similar effects. Brinkmann (1966) offers an explanation for this

phemomenon:

When one realizes that the emphasis in the teaching of
geometry is usually on develOpment of formal proofs based
on a certain type of "givens", the failure to add to the
performance on spatial visualization should not be

surprising. The behaviors demanded are simply different. (p.

180)

PLEASE NOTE:

Page 56 is lacking in number
only per author. NO text is
missing. Filmed as received.

UNIVERSITY MICROFILMS INTERNATIONAL.

CHAPTER III

DESIGN OF THE STUDY

Purposes of the Study

This study was designed to investigate three areas that are related

to the concept of similarity: (1) performance in four Similarity-

related areas, (2) performance on tasks presented in a visual or a

verbal mode, and (3) relationship between performance on similarity and

Spatial visualization.

1.

Performance in four similarity-related areas.

1.1 To investigate prior to instruction in similarity the possibility
of grade level, or sex differences in performance of sixth, seventh,
and eighth graders in tasks that require the use of (1) 'basic
properties of similar Shapes, (2) proportional reasoning related to
similarity, (3) the principle of area growth, and (4) applications

of similarity in scaled drawings and in indirect measurement.

Performance on tasks presented in a visual or a verbal mode.

1.2 To investigate the effect of instruction of the MGMP Similarity

 

Unit on the performance in the four topics mentioned above.

2.1 To investigate grade level enui sex differences in pre-
instructional performance on Similarity tasks presented in a verbal

mode, and on equivalent tasks accompanied by drawings.

2.2 To investigate the effect of instruction pertaining to similarity
57

58

on similarity tasks presented in a verbal mode, and on their visual

counterparts.

3. Relationship between performance on Similarity tasks and spatial

visualization.

3.1 To explore any possible effect of instruction in similarity

on performance in Spatial visualization tasks.

3.2 To explore any possible effect of instruction of the MGMP Spatial
Visualization (one year before this study) on performance in

similarity tasks.

Population and Sample

The participant subjects in this study were students from five suburban
middle schools :hi the Lansing, Michigan area. All five schools have
a middle-class, predominantly white student pOpulation. The selection
of the schools was made according to the teachers' willingness to

participate in the experimental instruction of the MGMP Similarity Unit:

 

schools that had more than one volunteering teacher have been selected
to participate in the study. Table 3.1 presents the distribution of

classes and teachers by school and by grade level.

Since School 5 participated a year before this study' (Winter,
1982-83) in the instruction of the MGMP Spatial Visualization Unit,

the sample was divided into two parts:

59

Sample A consisted of Sixth, seventh and eighth-grade students from
Schools 1, 2, 3, and 4. This sample was used to explore issues related
to students who did not undergo an instruction in spatial visualization.
Although most of the questions were concerned with performance in
similarity, School 5, has been omitted from this sample to prevent a
contamination of the results due to a potential influence of instruction
in spatial visualization on performance in Similarity tasks. Table 3.2
presents the distribution of the subjects in Sample A by grade and

by sex.

Sample B consisted of seventh and eigtht-grade students from School
5. This sample was used to explore issues related to the comparison

of students who learned the Sgatial Visualization Unit and those who

 

did not. Students of both kinds could be found in this sample. The
distribution of the subjects from this sample by grade level and by

experience in instruction in spatial visualization is presented in Table

3.3.

Table 3.1 DISTRIBUTION OF PARTICIPATING CLASSES AND TEACHERS

BY GRADE LEVEL AND BY SCHOOL

 

 

N N
Classes Teachers

Gr. 6 Cr. 7 Cr. 8 Cr. 6 Cr. 7 Gr. 8
School 1 6 2 - 2 2 -
School 2 3 3 - 3 3 -
School 3 - - 3 - - 2
School 4 2 2 1 1 1 1
School 5 - 3 5 - 2 2

 

Total 11 10 9 6 8 5

 

60

Table 3.2 SAMPLE A--DISTRIBUTION OF STUDENTS BY GRADE LEVEL AND BY SEX

 

 

 

 

 

 

 

N N N
Male Female Total
Grade 6 137 111 248
Grade 7 87 83 170
Grade 8 54 42 96
Total 278 236 514
Table 3.3 SAMPLE B-—DISTRIBUTION OF STUDENTS BY GRADE LEVEL AND BY
EXPERIENCE IN SPATIAL VISUALIZATION (SV) INSTRUCTION
N N N
Students with Students without Total
SV instruction SV instruction
Grade 7 29 29 58
Grade 8 69 34 103
Total 98 63 161

 

61

Instrumentation

 

The instruments used in this study were a Similarity performance

test and a spatial visualization performance test.
Similarity

The similarity performance was measured by the Middle Grades
Mathematics Project Similarity Test (MGMP SIMT). This test was developed
by the Middle Grades Mathematics Project staff, including this
investigator, in the Mathematics Department of Michigan State University.
During Fall 1982, a pilot test was administered to approximately one
hundred Sixth and seventh grade students in two schools, prior to, and

after instruction with the first draft of the MGMP Similarity Unit.

 

Test validity and reliability were considered: the test was revised
in view of student results and observations made by the project's staff

and the participating teachers.

The MGMP Similarity Test consists of 25 multiple choice items with
five Options for each item. The presented Options have been selected
according to the most popular misconceptions indicated by cognitive
research on similarity concepts and on proportional reasoning. The items
were scored by assigning a 1 for a correct response and a 0 Otherwise;
no correction was made for guessing. The test was not timed, but usually
did not exceed 25-30 minutes. The total score on the test was considered

as a general indicator of the level of performance in similarity tasks.

The Cronbach reliability coefficients calculated for the test

ranged from .53 to .82. Table C.1 in Appendix C includes the reliability

62

coefficients for the MGMP SIMT by grade and by sex.

Another indicator of the quality of the instrument was the
significant high correlations (P < .001) between the pretest and the
postest scores. The corresponding Pearson correlation coefficients

(Table C.2 in Appendix C) ranged from .46 to .79.

Twenty items out of the 25 were clustered into four subtests
(Appendix A), in order to analyze student performance on four tOpics:
(1) basic properties of similar shapes (i.e., recognition of similar
shapes and their properties), (2) prOportional reasoning (i.e., finding
the fourth number in a proportion that represents lengths of sides of
similar shapes), (3) the principle of area growth in similar shapes,
and (4) applications of similarity in scaled drawings and in indirect
measurement. Subtests (1) and (4) include six items each, and subtests
(2) and (3) include four items each. In the original test, the items
were not presented in a clustered form; the item numbers in Appendix
A represent their original sequence. The scores on these subtests were
used as indicators of student pre, and post-instructional performance

on the considered similarity-related topics.

In order to examine student performance in similarity tasks presented
in a verbal or in a visual mode, eight items were clustered into a
visual and a verbal subtest of four items each (Appendix A). Although
twenty out of the 25 items were accompanied by drawings, only four
of these were selected so that the selected visual items had strictly
equivalent counterparts stated in a purely verbal form. The four pairs

of equivalent items included one pair on the concept of similarity,

63

one pair on indirect (shadow) measurement, and two pairs on area growth.

Spatial Visualization

 

The performance in spatial visualization was measured by a 15-item
subtest of the MGMP Spatial Visualization Test. The original 32-item
test was developed by the MGMP staff during summer and fall 1981 and
was administered to about 1500 students in two evaluation studies of

the MGMP Spatial Visualization Unit (Ben-Haim, 1983 ; Lappan, 1983).

Fifteen representative items have been selected to form the MGMP
Spatial Visualization Subtest (MGMP SVST) used in this study. The SVST
consists of items requiring three-dimensional visualization and is

unrelated to topics or notations developed in the Similarity Unit.

 

Appendix B presents a representative sample of six items from the SVST.
The items of this test are also multiple choice, with five options
for each item, and with a 0/1 scoring. The total score on the test
was considered as an indicator of the level of performance in spatial

visualization.

The Cronbach OL reliability coefficients of the SVST ranged from
.59 to .80, whereas the pre-post Pearson correlation coefficients ranged

from .66 to .80 (Tables C.1 and C.2 in Appendix C).

The SIMT and the SVST were both administered during the same class
period, and the total administration time did not exceed 45-50 minutes.
In order to neutralize a possible effect due to a certain sequencing
of the two tests, about half of the classes received the SIMT first,

and the SVST second, while the other half received the two tests in

64

reverse order. The order of testing was randomly attributed to whole

classes.

Instructional‘Material

The MGMP Instructional Model

The Middle Grades Mathematics Project developed four instructional
units. Each of these units requires two to three weeks of instructional
time, and concentrates on a specific concept that is considered to be
important in the mathematical develOpment of the middle grades level
student. Two MGMP units will be briefly described here: (1) the
Similarity Unit that formed the basis for the instructional intervention
in this study, and (2) the Spptial Visualization Unit that was taught

in several experimental classes in School 5 one year before this study.

Each of the MGMP instructional units provides a carefully sequenced
set of challenging and exploratory activities designed to lead the student
to the understanding of a mathematical concept, and a carefully developed
and very detailed instructional guide for the teacher. The units utilize
an instructional model developed by Fitzgerald and Shroyer (1979). The
model consists of three phases: launch (introducing new concepts,
clarifying definitions, reviewing old concpets, and issuing a challenge),
exploration (individual work in gathering data, sharing ideas looking
for patterns, making conjectures, or developing other types of problem-
solving strategies), and summary (demonstrating ways to organize data
discussing the used strategies, and refining these strategies into

efficient pmoblem-solving techniques). The launch and the summary

65

are conducted in a whole-class mode, while at the exploration Stage,
the teacher becomes a fellow investigator who helps individual students

to follow their own exploratory path of thinking.
The MGM Similarithnit

The concrete-activity orientation of the MGM Similarity Unit

 

follows recommendations of the Piagetian research on geometrical

development, and the instructional implications of the Van Hiele studies.

The unit presents the concept of similarity at the Van Hiele levels
I (properties of shapes) and II (relationships among properties of shapes)
of geometrical development. The present neglect in the consideration
of the Van Hiele levels in the teaching of geometry in the United
States is believed to be a reason for present deficiencies in many
children's understanding of geometrical concepts (Wirszup, 1976;

Carpenter et al. , 1981).

The MGM Similarity Unit is a response to deficiencies in the

 

teaching of geometry. By offering a rich variety of experiences designed
to fit the Van Hiele levels I and II, the unit contributes to the
building of a solid base for further advance in the understanding of
geometrical concepts. Moreover, the launching-exploration-summary model
corresponds to Van Hiele's three steps required to help the child move
to a higher developmental level: information, directed orientation,

explanation.

The unit contains nine activities that require 1-1 1/2 class periods

each:

66

-figure enlargement by rubber bands

-figure enlargement by point coordinates
~comparing rectangles

-families of similar rectangles

-comparing triangles

-creating similar figures with repeating tiles
-figure enlargement by point projections
-enlarging pictures

-applications of similar triangles.

A more detailed overview of the unit is presented in Appendix D.
The nine activities use concrete manipulatives to help provide the
transition from the student's concrete to abstract thinking. The four
Similarity-related topics of this study (basic properties, proportional
reasoning, area growth, and applications) are encountered throughout

the unit in a Spiral way.

The MGM Spatial Visualization Unit

 

The Spatial. Visualization. Unit includes. ten. carefully' sequenced
activities on the representation of three-dimensional objects in two
dimensional drawings, and vice versa, on the construction of three-
dimensional objects with blocks from their two dimensional
representations. The activities deal with the flat views of buildings
as well as with the isometric drawings on dot paper (paper with dots
arranged on diagonals rather than rows). In most of the activities
the students are asked to perform some fairly demanding orientation

and visualization tasks. They are asked to mentally rotate a building

67

and draw either flat views of the other sides or isometric drawings
from other corners. Cubes are always available to help a student who

needs to see the concrete object to be successful.

Procedure and Data Collection

The experimental instruction of the Similarity Unit and the data
collection were conducted from January to May 1982. The whole sample
of 695 students was tested on the Similarity Test and on the Spatial

Visualization Subtest both before, and after instruction.

The classroom teachers were provided with all the instructional
materials, the testing material and written instructions concerning the
administration of the tests. The tests were administered during the
regular school day by the classroom math teachers. For all the teachers,

this was the first time that they taught the Similarity Unit, although

 

many of them have already taught another MGM instructional unit. Prior
to instruction, a two-hour workShOp for the teachers was conducted in
each of the five participating school, but no direct intervention by
the staff of the project has been provided during instruction. The
workshop acquainted the teachers with the nine activities in the unit,
recommended instructional strategies, and presented a more in-depth view

of the geometrical concepts involved.

 

 

 

68

The Statistical Design of the Study

To analyze the data collected during the study, analyses of means,
standard deviations, correlations, Multivariate and Univariate Analyses
of Variance and ANOVA with Repeated Measures were conducted. A detailed
description of the hypotheses tested and the statistical procedures used
for each purpose of the study will be given in the next chapter, where

they will be followed by a description of the findings.

For the significance tests, the assumptions were that the subjects
were responding independently of one another and that the error vectors
had a multivariate normal distribution with mean zero and a common
variance covariance matrix. The fact that this study was conducted
in a regular school setting made the assumption of independence and

normality feasible.

The effects of instructional intervention were examined by using
design 6 from Campbell and Stanley (1963): 01 X 02, where 01 and 02
are the scores collected before and after instruction, and is the
treatment (instructional intervention). A control (no-intervention)
group was not employed in this study, due to (1) physical difficulties
of recruiting and testing a group of equivalent size and structure,
and (2) the relatively small possibility of confounding effects as a
result of the Short duration of instruction and the wide variety of

sampled subjects.

The collapsing of students from four different school buildings

into one sample would probably increase to a certain degree the

69

heterogeneity of the pOpulation. However, the purpose of the study
was to examine the effect of instruction pertaining to similarity in
heterogeneous, middle-class suburban school conditions, rather than in

an artifically homogeneous environment.

There are two alternative units of analysis for this study: teacher
versus student. Hopkins (1982) compares these two possibilities and
concludes that taking the teacher as a unit ignores the fact that the
observational unit is the student and actually only exchanges the
assumption of independence among students with another not necessarily
more true assumption of independence among teachers. Moreover, post-
hoc examinations of the results did not reveal any outstanding
discrepancies in the results by school building or by teacher. Thus,

this study will consider the student as unit of analysis.

All the analyses were carried out on the CYBER 170, Model 750
computer at the Michigan State University Computer Center, using SPSS

(Statistical Package for the Social Sciences) programs.

CHAPTER.IV

ANALYSIS OF DATA AND RESULTS

Introduction

 

This chapter presents the findings that correspond to the six
research questions of this study. Each of the six questions that were
posed in the introductory chapter will be restated here as a hypothesis.
Each hypothesis will be followed by a description of: (1) the statistical
design and procedures used to examine the data, (2) a description of
the obtained results, and (3) a short summary of results related to

the initial question.

Basically, the Six hypotheses stated in this chapter have been
tested by employing Analysis of Variance. The means, standard deviations
and correlation coefficients computed allowed for further interpretations

of the results.

The Analyses <mf Variance employed ix: this chapter have two

limitations:

1. Since for Sample A the grade effect has three levels (i.e., two
degrees of freedom), only two orthogonal contrasts were formed: grade
six versus grade seven, and grade six and seven combined versus

grade eight.

2. Since the design is unbalanced (i.e., the cells do not contain equal
numbers of cases), the testing of hypotheses within a certain ANOVA

is ordered. The order of testing is presented in the ANOVA tables

70

71

from the bottom to the tOp: each effect controls for all the effects
that are presented in the summary table above it, but is confounded
with the effects that appear below it. Failure to reject a null
hypothesis implies that the parameter corresponding to that effect
is assumed to be zero, allowing thus to test the next hypothesis
(presented in the table above it). The rejection of a certain
hypothesis, means that the corresponding parameter may not be assumed
to be zero. In this case, the testing of the next hypothesis is
not possible because of the confounding effect of the non-zero

parameter tested before.

These difficulties may be sometimes overcome by changing the order
in which the effects are introduced, or by conducting a secondary analysis
with a different partition of the effects. However, by conducting a
multiple number of tests the probability of a Type I error would be

increased accordingly.

Further inferences can be drawn by an examination of group means.
Cohen (1969) suggests using the estimated standardized difference of
population means1 as a primary index for the effect size. He suggests
that in behavioral sciences, standardized differences of .2, .5, and
.8 standard deviation units may serve as operational definitions for
respectively "small", "medium", and "large" effects. In order to allow
for judgments of effect sizes, this study reports occasionally estimated

standardized differences of means.

 

1
The estimated standardized difference of population means is

12
(m1 - m2)/MSe / where m1, mzare the group means, and MSe is the

mean square error.

72

Pre-instructional Peformance in Four

Similarity-Related.Areas.

Question 1.1: Prior to instruction in similarity, what patterns can
be observed in the performance of sixth, seventh, and
eighth graders in tasks that require the use of: (1)
basic properties Of similar shapes, (2) proportional
reasoning related to similarity, (3) the principle of
area growth, and (4) applications of similarity in

scaled drawings and in indirect measurements?

Hypothesis 1.1: In each of the four similarity-related areas, there
will be no sex or grade level differences in the SIMT
performance, and there will be In) grade by sex

interaction.

The hypothesis has been tested by a Two-Way Multivariate Analysis
of Variance. Table 4.1 presents the design for the Analysis of Variance,
and Table 4.2 is a summary of results. Mean scores of student performance
and correlation coefficients among the four topics (Tables E.1 and E.2,

Appendix E) allow for further interpretation of the data.

The following inferences may be drawn from the Multivariate and
Univariate Analysis of Variance for Hypothesis 1.1 at a significance

level of .05:

1. There are no Significant grade by sex interactions for the pre-

instructional performance on the four similarity-related topics

73

either when tested together in.a multivariate form, or when considered

separately in a univariate form.

2. No significant sex differences in performance on the four tOpics
were detected in the multivariate, or in the univariate form.
Analysis of the means in Table E.1 reveals, however, that in a
majority of cases (eight out of twelve) the performance of the boys

slightly exceeds that of the girls.

3. There are significant grade differences in the level of performance

on the four topics (both in the multivariate and in the univariate

Table 4.1 DESIGN OF THE 3 x 2 MULTIVARIATE ANALYSIS OF
VARIANCE FOR HYPOTHESIS 1.1

 

 

Basic Proportional Area Applications
Properties Reasoning Growth
Grade 6 ' Males
x1 y1 21 U1
Females
x1 y1 21 U1
Grade 7 Males
X1 y1 21 U1
Females
x1 y1 21 L11
Grade 8 Males
x1 y1 21 U1
Females
x1 y1 21 U1

 

74

Table 4.2 SUMMARY OF THE 3 x 2 MULTIVARIATE AND UNIVARIATE ANALYSIS
OF VARIANCE FOR HYPOTHESIS 1.1

 

 

 

Source of Multivariatea Univariate
Variation D.F. F P< F P<
Constant 1 602.12 .0001 scprb 1530.59 .0001
PROPC 622.13 .0001
AREAd 455.40 .0001
APPLe 1444.51 .0001
Grade 2
Gr2 (6, 7 VS. 8) (1) 58.82 .0001 BCPT 174.13 .0001
PROP 128.89 .0001
AREA 29 6O
APPL 92.61 .0001
Grl (6 VS. 7) (1) 13.87 .0001 BCPT 29.02 .0001
PROP 42.43 .0001
AREA 4.05 05
APPL 10.86 .001
Sex 1 .98 .42 BCPT 3.54 .06
PROP .31 .58
AREA 62 44
APPL .88 .35
Grade by Sex 2 1.53 .15 BCPT 1.88 .16
PROP .20 .82
AREA 2.63 08
APPL 1.20 .31
Between Groups 6
Within Groups 508 Univariate MS error: BCPT 464-52
PROP 935.94
AREA 442.30
AppL 435.57

 

a Multivariate D.F. = 4 and 505 for main effects
8 and 1012 for interaction

Basic Concepts

c . .
Proportional Reasoning
Area Growth

e Applications

5.

75

form) between sixth, and seventh graders favoring the latter. Mean
score differences of .53, .65, .20 and .33 in standard deviation
units were Observed in the topics of basic properties, proportional

reasoning, area growth, and applications respectively.

Grade level differences between seventh, and eighth graders (in favor
of the latter in three out of four cases) may be inferred from
Table E.1 (Appendix E) by arguing that the seventh-eighth grade
performance gap is even larger than the performance gap between sixth
and seventh graders. In the same order as in 3. the standardized
mean score differences in this case were 1.17, .90, .06, and .89
standard deviation units. An additional Multivariate Analysis of
Variance that partitioned the grade effect into the "7 vs. 8", and
the "6 vs. 7, 8" contrasts also revealed significant differences
between seventh and eighth graders (Multivariate F= 27.67, p< .0001).
Both tests combined together are within the overall level of

Significance.

The univariate analysis of the performance on the topic of area
growth reveals nonsignificant differences between seventh and eighth
graders. An analysis of the mean scores (Table E.1, Appendix E)
reveals that in this topic eighth graders performed at a slightly
lower level (.06 standard deviations) than seventh graders, and
actually all three grades performed in an almost uniformly poor manner
(an average of 18-22 percent). The outstanding status of the area

growth topic is also revealed by analyzing the correlation

76

coefficients among topics (Table E.2, Appendix E): all the
coefficients except those related to area growth, are significant
(p< .01) and range from .255 to .529. On the other hand, the
correlation coefficients that involve the topic of area growth are

either nonsignificant or low (less than .25).

To summarize the findings based on the pre-instructional performance
in four similarity-related areas, at a Significance level of .05, there
are no sex by grade interactions and no Significant sex differences;
in most cases, level of performance increased significantly with level
of grade; performance on tasks related to area relationships of Similar
shapes was lower than in other similarity tasks, and did not Show the

same increasing pattern as a function of grade level.

Effect of Instruction in Four

 

Similarity-Related Areas

 

Question 1.2: What is the effect of instruction pertaining to

 

similarity on the performance of sixth, seventh, and
eighth graders in tasks that require the use of: (1)
basic prOperties of Similar shapes, (2) proportional
reasoning related to similarity, (3) the principle of
area growth, and (4) applications of similarity in

scaled drawings and in indirect measurements?

Hypothesis 1.2: There will be no differences among gain scores as a

 

function of grade level or sex, and there will be

no grade by sex interactions.

77

The hypothesis has been tested by a Two-Way Analysis of Repeated
Measures. ‘Table 4.3 presents the design that was used for this analysis,
and Table 4.4 is a summary of results. The pre, and post-instructional
mean scores presented in Table E.1 (Appendix E) and in Figure 4.1 allowed

for further interpretations of the data.

The following trends could be observed at a significance level

of .05:

1. There are no significant grade by sex interactions or sex differences

in gain scores in any of the four similarity-related topics.

Table 4.3 3 x 2 MULTIVARIATE REPEATED MEASURES
DESIGN FOR HYPOTHESIS 1.2

 

 

 

 

 

Basic Prgportional Area Applications
Properties Reasoning Growth
'Pre Post Pre Post Pre Post Pre Post
Grade 6 Males x1 x2 "1 y2 21 22 111 112
Females x1 x2 y1 y2 Z1 22 u1 u2
Grade 7 Males x1 x2 y1 y2 21 22 ul u2
Females x1 x2 y1 y2 z1 22 111 112
Grade 8 Males x1 x2 y1 y2 21 22 111 112

Females x x y1 y2 z z u u

 

 

 

 

 

 

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79

Table 4.4 SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS OF REPEATED
MEASURES FOR HYPOTHESIS 1.2

 

 

. . a .
Multivariate Univariate
Source of Variation D.F. F P< F P<
Constant 1 1128.08 .0001 Con. Avg.b 3753.32 .0001

Prop. Avg.c 1786.92 .0001
Area Avg.d 891.65 .0001
Appl. Avg.e 1895.50 .0001

 

 

Grade 2
Gr2 (6, 7 vs. 8) (1) 55.91 .0001 Con. Avg. 167.42 .0001
Prop. Avg. 109.70 .0001
Area Avg. 1.69 .20
Appl. Avg. 141.89 .0001
Grl (6 vs. 7) (1) 19.16 .0001 Con. Avg. 40.22 .0001
Prop. Avg. 58.65 .0001
Area Avg. 24.75 .0001
Appl. Avg. 38.46 .0001
Sex 1 1.18 .32 Con. Avg. 1.73 .19
Prop. Avg. 1.60 .21
Area Avg. 1.04 .31
Appl. Avg. 4.52 .04
Grade by Sex 2 .93 .49 Con. Avg. .81 .44
Prop. Avg. .58 .55
Area Avg. .20 .82
Appl. Avg. 1.09 .34
Between Groups 6
Time 1 220.53 .0001 Con. Diff 577.79 .0001
PrOp. Diff 369.53 .0001
Area Diff 53.14 .0001

Appl. Diff 162.21 .0001

 

Table continued

Table 4.4 (Cont'd)

80

 

 

 

Multivariatea Univariate
Source of Variation D.F. F P< F P<
Grade by Time
Gr2 (6,7IEL 8)by Time 2 8.68 .0001 Con. Diff 15.44 .001
Prop. Diff 20.21 .0001
Area Diff. .71 .40
Appl. Diff .65 .42
Grl (6 vs. 7) by Time (1) 5.13 .0005 Con. Diff. .01 .96
Prop. Diff. .01 .93
Area Diff. 10.80 .001
Appl. Diff 10.70 .001
Sex by Time 1 1.42 .23 Con. Diff. 1.93 .17
Prop. Diff. .83 .36
Area Diff. .03 .86
Appl. Diff. 2.07 .15
Grade by Sex by Time 2 1.27 .26 Con. Diff. 1.54 .22
Prop. Diff. .41 .67
Area Diff. 2.72 .07
Appl. Diff. .32 .73
Within SubjeCts 508
Univariate Con Avg. 676.76 Con. Diff. 278.72
MS Error: Prop Avg. 1333.86 Prop. Diff. 584.42
Area Avg. 710.80 Area Diff. 490.28
Appl. Avg. 627.46 Appl. Diff. 315.22

 

Multivariate])J?.= 4 and 505 for main effects

8 and 1012 for interactions

0- 0 0"

(D

Area Growth averaging over time (Pre + Post)-

Applications averaging over time (Pre + Post)

Basic Concepts averaging over time (Pre + Post)

Proportional Reasoning averaging over time (Pre + Post)

Basic Concepts time effect over subjects (Pre - Post)

3 Proportional Reasoning time effect over subjects (Pre — Post)

1 Applications time effect over subjects (Pre - Post)

Area Growth time effect over subjects (Pre - Post)

81

2. Seventh graders gained significantly more than the sixth graders.
These differences are due to the tOpics Of area growth and
applications(standardized gain differences of .46 in both topics);
in the other two topics no significant gain differences could be

detected.

3. A comparison of gains between seventh and eighth graders can be
made by observing the graphs in Figure 4.1, and by conducting an
additional Analysis of Repeated Measures with "7 vs. 8" as one of
the grade level contrasts. Both strategies reveal an overall
significant (Multivariate F = 6.18, P< .0008) difference in gains
favoring the seventh graders. The univariate analysis shows that
this difference is due to gain differences in the topics of basic
similarity properties and prOportional reasoning (gain differences
of .63 and .71 standard deviation units respectively). As in the
case of Hypothesis 1.1, the combined level of Significance of the

analyses conducted in 2. and 3. is Still very low.

To summarize the findings based on performance in four similarity-
related areas prior to and after instruction, at a Significance level
of .05, no sex differences in gains could be detected, and seventh
graders seem to have profited the most from the instruction: their
gains were Similar to the sixth graders' and higher than the gains
of the eighth graders on area growth and applications; on the other
side, they gained as much as the eighth graders and more than the
sixth graders on the basic properties of similar shapes and on

proportional reasoning related to Similarity tasks.

82

Pre-instructional Performance on Tasks

Presented in a Verbal and a Visual Mode

Question 2.1: Prior to instruction on similarity, what patterns can
be observed in the performance Of Sixth, seventh, and
eighth graders in tasks that are presented in a verbal

mode or in a visual mode?

Hypothesis 2.1: By analyzing the level of performance on the two modes
combined together, and the difference in performance
between the two modes: there will be no grade by
sex interaction, and there will by no sex or grade

differences.

In order to test this hypothesis, a Bivariate Analysis of Variance
has been used, by using the pretest scores on the items presented in
a verbal mode and a visual mode of each student as variables (Table
4.5). The sum of the two scores has been used as an indicator of
the overall level of performance in the two modes, and the difference
of the scores has been used as an indicator of differences in performance
between the two modes. Table 4.6 presents a summary of the Analysis
of Variance results. Mean scores (Table E.3, Appendix E) have been

examined to detect further trends in the obtained data.

 

 

Table 4.5 DESIGN OF THE 3 x 2 BIVARIATE ANALYSIS OF VARIANCE
FOR HYPOTHESIS 2.1
Verbal MOde Visual MOde
(Pretest) (Pretest)
Grade 6 Males x1 y1
Females x1 Y1
Grade 7 Males x1 y1
Females x1 y1
Grade 8 Males x1 y1
Females x1 y1

 

The following inferences may be drawn from the Analysis of Variance

for Hypothesis 2.1 at a significance level of .05:

1. There are no significant sex by grade interactions in the analysis

of the combined performance in the visual and the verbal modes,

or of the score difference between modes of presentation.

2. There are no significant sex differences in the analysis of the

combined performance in the visual and the verbal modes or the score

difference between the two modes of presentation.

84

Table 4.6 SUMMARY OF THE 3 x 2 BIVARIATE ANALYSIS OF VARIANCE FOR
HYPOTHESIS 2.1

 

 

 

 

 

SOURCE OF VARIATION D.F. F P<
Constant 1 1103.48 .0001
Grade 2
Gr2 (6, 7 vs. 8) (1) 77.81 .0001
Grl (6 vs. 7) (l) 15.63 .0009
Sex 1 1.57 .21
Grade by Sex 2 .41 .67
Between Groups 6 MS errora: 592.49
Mode 1 2.33 .13
Grade by Mode 2
Gr2 (6, 7 vs. 8) by Mode (l) .05 .83
Grl (6 vs. 7) by Mode (1) .38 .54
Sex by Mode 1 .01 .95
Grade by Sex by Mode 2 .65 .53
Within Subjects 508 MS errorb: 357.01

 

a for Verbal (pre) + Visual (pre) score

b for Verbal (pre) — Visual (pre) score

4.

85

There is a significant difference between the performance of sixth,
and seventh graders (in favor of grade seven) in their performance
on the tasks included in one of the two modes, but there is no
Significant grade effect in the analysis of the differences between
the two modes. The means in Table E.3 (Appendix E) reveal that
for all grade levels, the mean differences between pre-instructional
performance in the verbal and visual modes are uniformly low (less
than 4 percent) with a slightly higher level of performance in the

verbal mode.

A second Analysis of Variance with the grade effect partitioned in
the "7 vs. 8" and "6 vs. 7, 8" contrasts shows a significant (F
= 35.89, p< .0001) difference favoring the seventh graders in the
verbal and visual mode combined but, as shown also in Table E.3

of means, not in the score difference between the verbal and the

visual mode (F = .01, p< .93)

To summarize, at a significance level of .05, no sex by grade

interaction, sex or grade level effects could be detected in the analysis

of the difference between performances on tasks presented in a visual,

and a verbal mode. If the pre-instructional performances on the two

modes are combined, the same pattern of grade differences that was

detected in the test of Hypothesis 1.1 reappear: the level Of performance

increases significantly with grade level.

 

86

Effect of Instruction on Performance in

 

Verbal and in Visual Tasks

 

Question 2.2: What is the effect of instruction pertaining to

 

similarity on the performance of sixth, seventh, and
eighth graders in Similarity tasks that are presented

(1) in a verbal mode, and (2) in a visual mode?

Hypothesis 2.2: a) In each of the two modes, there will be no

 

differences among gain scores as a function of grade
level or sex, and there will be no sex by grade

interaction.

b) Comparing the gain scores in the two modes one
with the other, there will be no significant
differences by grade level or by sex, and there will

be no grade by sex interaction.

Table 4.7 presents the Repeated Measure Design that has been used
to test Hypothesis 2.2, and Table 4.8 is a summary of results of the
Analysis of Repeated Measures that has been conducted to test this
hypothesis. The gain scores are defined as the pretest-posttest
difference of scores for each presentation mode. For testing Hypothesis
2.2a, the gain scores are combined (i.e., Verbal gain + Visual gain)
and the analyzed effects are Grade by Time, Sex by Time, and Grade
by Sex by Time. For testing Hypothesis 2.2b, the gain scores are compared

(i.e., Verbal gain-Visual gain) under the Mode by Time interaction:

87

Grade by Mode by Time, Sex by Mode by Time, and Grade by Sex by Mode
by Time. Table E.3 (Appendix E) of means and standard deviations, and
Figure 4.2 of graphs that illustrate pre, and post-instructional
performance in the verbal and the visual modes allowed for further

interpretations of the data.

The analyses rendered the following results at a significance level

of .05:

1. There are In) significant grade by sex interactions in gain scores
of the verbal, and visual scores either when combined together or

when compared one versus the other.

Table 4.7 3 x 2 MULTIVARIATE REPEATED MEASURE DESIGN FOR
HYPOTHESIS 2.2

 

 

 

 

Verbal MOde Visual Mode
Pre Post Pre .Post
G
rade 6 Males x1 x2 y1 Y2
F l
ema es x1 x2 y1 y2
G d
ra e 7 Males x1 x2 y1 Y2
F
emales x1 x2 y1 y2
G d 8
ra e Males x1 x2 y1 Y2
Females x1 x2 y1 y2

 

88

Percent

551%

50 -

45 L

35 -

 

20 b.

10 -

 

L l l l_
PRE - POST PRE POST
VERBAL VISUAL

Figure 4.2 Performance by grade level in tasks presented in a visual and
and a verbal mode, prior to, and after instruction.

 

 

89

Table 4.8 SUMMARY OF ANALYSIS OF REPEATED MEASURES FOR HYPOTHESIS 2.2

P<

 

 

 

Source of Variation D.F. F
Constant 1 2128.53 .0001
Grade 2

Gr2 (6, 7 vs. 8) (1) 73.45 .0001

Grl (6 vs. 7) (1) 49.94 .0001
Sex 1 2.85 .10
Grade by Sex 2 .03 .98

Between Groups 6 (MS error 3: 1083.98)
Time 1 292.35 .0001
Grade by Time 2

Gr2 (6, 7 vs. 8) by Time (1) .96 .33

Grl (6 vs. 7) by Time (1) 19.34 .0001
Sex by Time 2 .32 .57
Grade by Sex by Time 2 .86 .43

(MS errorb : 444.13)

Mode 1 5.18 .03
Grade by Mode 2

Gr2 (6, 7 vs. 8) by Mode (1) 1.50 .22

Grl (6 vs. 7) by Mode (1) .81 .37
Sex by Mode 1 1.02 .32
Grade by Sex by Mode 2 .51 .61

(MS errorC: 292.35)

 

Table continued

 

 

90
Table 4.8 (Cont'd)

 

 

 

Source of Variation D.F F P<
Mode by Time 1 .16 .69
Grade by Mode by Time 2

Gr2 (6,7vs.8) by Mode by Time (1) 3.15 .08

Grl (6vs.7) by Mode by Time (1) 3.91 .05
Sex by Mode by Time 1 1.19 .28
Grade by Sex by Mode by Time 2 .42 .66

(MS errord: 319.91)

Within Subjects 508

 

0‘

for overall means (Verbal Pre3+Verbal Post-+Visual Pre-+Visual Post)

for combined gains (Verbal Pre-—Verbal Post-+Visual Pre-—Visual Post)

C for comparison of modes (Verbal Pre-+Verbal Post-—Visual Pre-—Visual Post)

0..

for comparison of gains (Verbal Pre-—Verbal Post-—Visual Pre + Visual Post)

2.

3.

6.

91

There are no significant sex differences in gain scores of the verbal
and visual presentation modes either when combined together or when

compared one versus the other.

In both modes, seventh graders gained significantly more than the

sixth graders.

In the examination of gain differences between the two modes, there
are significant differences between sixth and seventh grade level.
An analysis of the means Table E.3 (Appendix E) and the graphs
in Figure 4.2 reveals that seventh graders gained slightly more (by
about 2 percent) in the visual mode as compared to their gain in
the verbal mode, whereas the sixth graders gained 5 percent (or
.27 standard deviations) more in the verbal mode as compared to

their gain in the visual mode.

By conducting a second test that compares seventh, with eighth
graders, significant differences (F = 8.42, p< .004) were detected:
seventh grades gained more than the eight graders in both presentation
modes combined together (see also slope of the corresponding lines

in Figure 4.2).

The second test did not detect significant gain differences between
modes (F a .43, p< .52) for the seventh graders compared with the
eighth graders: at both grade levels the gains in the visual mode
were higher than in the verbal mode (see Table E-3). The combined

level of significance of the two tests is still very low.

92

To summarize, the analysis of the gain scores (i.e, the effect
of instruction) in the two presentation modes at a significance level
of .05, detected no sex by grade interaction and no sex differences;
seventh graders gained significantly more than sixth or eighth graders;
sixth graders gained more in their verbal scores than in the visual
scores, whereas seventh and eighth graders revealed the opposite pattern

of gains (i.e. higher gains in visual scores than in the verbal ones).

Effect of Instruction in Similarity

on Spatial Visualization

Question 3.1: What effect does instruction in Similarity have on

performance in spatial visualization tasks?

Hypothesis 3.1: There will be no gains in performance on the Spatial
Visualization Subtest as a result of instruction in
similarity --i.e., there will be no Significant
differences in SVST scores prior to, and after
instruction, or if there are such gains, they will

be unrelated to performance on the Similarity Test.

In order to test this hypothesis, pre, and post-instructional
performance on the SIMT and on the SVST has been analyzed for Sample
A (Table 4.9). As mentioned before, this sample consists of students

that underwent instruction with the MGM Similarity Unit, but not with

 

the MGM Spatial Visualization Unit. A Multivariate Analysis of Repeated

 

Measures has been employed to investigate the gains in similarity and

93

in spatial visualization in one test. The correlation coefficients among
Similarity and spatial visualization scores (Table E.4). range from .270
to .583 (p< .001), indicating :3 considerable relationship between
performances in the two fields. ‘Therefore, a Univariate Stepdown.Analysis
of Variance was also conducted to account for this relationship. In
other words, gains in spatial visualization have been examined after
gains in similarity (attributed directly to instruction) have been taken

into account.

Table 4.9 3 x 2 MULTIVARIATE REPEATED MEASURE DESIGN
FOR HYPOTHESIS 3.1

 

 

 

 

Similarity Test Spatial Visualization
Subtest
Pre Post Pre Post
Grade 6 Males x1 x2 y1 Y2
Females x1 x2 y1 y2
Grade 7 Males x1 x2 y1 y2
Females x1 x2 y1 y2
Grade 8 Males x1 x2 y1 y2
Females x x

 

94

Table 4.10 presents a summary of results of the Multivariate,
Univariate, and Univariate Stepdown Analyses of Variance that were
conducted to test Hypothesis 3.1. Mean scores and standard deviations
(Table 13.5) and their graphic representation (Figure 4.3) will

facilitate interpretations of the data.

The results of the analyses indicate the following trends, at a

significance level of .05:

1. There is no grade by sex interaction and there are no sex differences

in gain scores neither in similarity nor in spatial visualization.

2. Generally, seventh graders gained significantly more than sixth
graders. From the univariate ANOVA and Figure 4.3, it is clear that
the difference in gains is due to similarity scores (a fact already
revealed in the testing of the four similarity-related topics), but

not to performance in spatial visualization.

3. A second test employing the "7 vs. 8" contrast in grade level revealed
a similar pattern: the test detected an overall Significant
difference (Multivariate F = 5.76, p< .003) between seventh, and
eighth graders, with the seventh graders gaining significantly more
in similarity (Univariate F = 11.15, p< .001), but not in spatial

visualization (Univariate F = 1.22, p< .28).

4. The Stepdown Analysis of Variance for gains in spatial visualization

indicates that after accounting for gains in similarity, there are

 

Percent

A\
70 _.

65 _

60...

55-

50-—

35' .

30-—

25'

20-—

 

95

 

 

._1 1. I 1
BRE POST RE . POST-"‘9

SIMILARITY SPATIAL VISUALIZATION

Figure 4.3 Performance by grade level on the Similarity Test and on the

Spatial Visualization Subtest, prior to, and after instruction
in Similarity.

96

 

 

 

 

Table 4.10 SUMMARY OF ANALYSIS OF REPEATED MEASURES FOR HYPOTHESIS 3.1
Multivariatea Univariate
Source - Stepdown
Variation D.F. F 'P< F P< F P<
Constant 1 2500.58 .0001 Stan Avg.b 14694.83 .0001 4694.83 .0001
sv Avg.e 2580.70 .0001 30.81 .0001
Grade 2
Gr2 (6, 7vs. 8) (1) 106.05 .0001 Sim Avg. 201.13 .0001 201.13 .0001
SV Avg. 105.13 .0001 8.14 .005
Grl (6vs 7) (1) 34.01 .0001 Sim Avg. 66.63 .0001 66.63 .0001
SV Avg. 28.04 .0001 1.34 .25
Sex 1 2.72 07 Sim Avg. 3.00 .09 3.00 .09
SV Avg. 5.02 .03 2.44 .12
Grade by Sex 2 .50 .70 Sim Avg. .45 .64 .45 .64
SV Avg. .99 .38 .65 .52
Between Groups 6
Time 1 514.76 .0001 Sim Diffd 945.00 .0001 945.00 .0001
sv Diffe 187.48 .0001 30.20 .0001
Grade by Time 2
Gr2 (6, 7vs. 8)
by Time (1) 3.99 02 Sim Diff 5.18 .03 5.18 .03
SV Diff 3.97 .05 2.79 .10
Grl (6 vs. 7)
by Time (1) 5.81 .003 Sim Diff. 8.15 .004 8.15 .004
SV Diff. 2.05 .16 3.43 .07
Sex by Time 1 1.48 23 Sim Diff. 1.29 .26 1.29 .26
SV Diff. 2.10 .15 1.68 .20
Grade by Sex
by Time 2 .33 .86 Sim Diff. .37 .70 .37 .70
SV Diff. 22 .80 29 .75

 

Table continued

 

97
Table 4.10 (Cont'd)

 

Within Subjects 508

Univariate MS Error: Sim Avg. 365.22
SV Avg. 714.13
Sim Diff. 107.27
SV Diff. 133.54

 

a Multivariate D.F. = 2 and 507 for main effects
4 and 1014 for interactions

0"

Similarity averaging over time (Pre + Post)

0

Spatial Visualization averaging over time (Pre + Post)

O.

Similarity time effect over subjects (Pre - Post)

e Spatial Visualization time effect over subjects (Pre - Post)

98

no significant grade or sex differences in spatial visualization
gains. The significant time effect shows the existence of gains
in spatial visualization as expressed by the difference of scores
prior and after instruction in Similarity. However, not all of these

gains could be explained by the gains in similarity.

To summarize, there were significant gains in SVST scores measured
before, and after instruction in similarity (a standardized increase
of .85 standard deviation units). After accounting for gains in
Similarity, no grade or sex differences in spatial visualization gains
could be detected. Since the overall gains in spatial visualization
stayed significant even after controlling for gains in similarity, it
is possible that additional factors caused the growth in spatial

visualization.

Effect of Instruction in Spatial Visualization

on Performance in Similarity Tasks

Question 3.2: What effect does instruction in spatial visualization

have on performance in similarity tasks?

Hypothesis 3.2: There will be no Significant effect of instruction
in spatial visualization on performance in SIMT; i.e.,
for grade levels seven and eight, there are no
differences in mean scores and in gain scores on the
SIMT between students who studied the MGM Spatial

Visualization Unit and those who did not.

99

The investigation of this question was limited by a restricted
sample: Sample B consisted of a relatively small number of students
(N = 161) at grade levels seven and eight in one school (see Table
3.3). Moreover, the instruction in spatial visualization occured one
year before the instruction in similarity. Comparisons of results from
the analysis of this question with previously obtained results should

be, therefore, conducted in view of these restrictions.

Hypothesis 3.2 was tested by analyzing pre, and post-instructional
performance on SIMT of seventh and eighth graders who did undergo
instruction in spatial visualization and of students who did not.
Performance in similarity was analyzed in two ways: (1) by employing
a Univariate Analysis of Repeated Measures, and (2) by using the same
analysis as in (l) with the performance in spatial visualization as
a covariate. The second test was designed to detect the extent to
which patterns in similarity scores can be attributed to spatial

visualization.

Table 4.11 presents the design of the data that has been used
to test Hypothesis 3.2, and Table 4.12 is a summary of results Of
the Analyses of Repeated Measures. Further details on the data are
provided in Table E.6 (Appendix E) Of means and standard deviations

on Sample B's performance in similarity and in spatial visualization.

The following conclusions can be drawn from the results of the

analyses, at a significance level of .05:

1. If the level of performance in similarity (as expressed by the average

100

Table 4.11 2 x 2 x 2 UNIVARIATE REPEATED MEASURE DESIGN
FOR HYPOTHESIS 3.2

 

 

 

 

Similarity Test Spatial Visualization
Test
(covariate)
Pre Post Pre Post
sv Unita
Grade 7 Males x1 x2 y1 y2
Females x1 x2 y1 y2
Grade 8 Males x1 x2 y1 y2
Females x1 x2 y1 y2
. No SV Unitb
Grade 7 Males x1 x2 y1 y2
Females x1 x2 y1 y2
Grade 8 Males x1 x2 y1 y2
Females x1 x2 y1 y2

 

a Studied the MGMP Spatial Visualization Unit.

b Did not study the MGMP Spatial Visualization Unit.

101

Table 4.12 SUMMARY OF ANALYSES OF REPEATED MEASURES FOR HYPOTHESIS 3.2

 

 

 

 

 

Source of MANOVAa MANCOVAb
Variation D.F. F P< F P<
Constant 1 1407.88 .0001 42.24 .0001
sv UnitC 1 .75 .39 1.52 .28
Grade 1 .06 .82 .01 .96
Sex 1 4.63 .04 .01 .94
Grade by Sex 1 .69 .41 .27 .61
Grade by SV

Unit 1 .02 .90 2.25 .14
Sex by SV

Unit 1 .11 .75 .15 .71
Grade by Sex

by SV Unit 1 1.53 .22 .47 .50
Between Groups 8 (MS errord: 362.54 268.41)
Time 1 150.31 .0001 119.11 .0001
SV Unit by

Time 1 .06 .81 .04 .85
Grade by Time 1 3.18 .08 4.13 .04
Sex by Time 1 .06 .81 .01 .95
Grade by Sex

by Time 1 .16 .70 .03 .88
Grade by SV

Unit by Time 1 .03 .87 .02 .89
Sex by SV Unit

by Time 1 2.48 .12 2.79 .10
Grade by Sex

by SV Unit

by Time 1 .57 .46 .37 .55
Within Subjects 153 (MS errore: 98.19 86,02)

 

3Analysis of Repeated Measures.fin:SIMT scores

with D.F.=1 and 153.

bAnalysis of Repeated Measure for SIMT scores with SVST scores as
covariate; D.F.=1 and 152.

CEffect of instruction with MGMP Spatial Visualization Unit.

dMean Square error for Similarity averaging over time (Pre + Post).

e . . . . .
Mean Square error for Similarity time effect over subjects (Pre - Post).

102

of pre, and post-instructional scores) is examined, there is no
significant interaction, influence of grade level or effect of
instruction pertaining to spatial visualization. The only marginally
significant effect (p< .04) is the effect of sex. However, when
sex differences in spatial visualization are taken into account,
sex differences in level of performance in similarity become

nonsignificant.

2. 'Lf gain scores in similarity are examined, there is no significant
interaction, and no significant influence of instruction pertaining
to spatial visualization, of grade level, or of sex. However, when
differences in gains in spatial visualization are taken into account,
the gains of the seventh graders in similarity appear significantly
higher than those of the eighth graders--a fact that was detected

also in Sample A.

To summarize, the most relevant finding in this section is that
instruction in Spatial visualization, one year before instruction in
similarity did not have any significant effect on the level of performance
or gains in similarity. The following results seem to contradict previous
findings for sample A: (l) nonsignificant grade differences in level
of performance in similarity, (2) nonsignificant grade differences in
gain scores in similarity, and (3) significant sex differences (i.e.,
male superiority) in level of performance in similarity. The last two
Of these contradictions can be settled by removing the variation that
can be explained by spatial visualization: in that case, the results

Show that, as for sample A, the seventh graders gained significantly

103

more than the eighth graders, and the sex differences in similarity
become nonsignificant. The fact that eighth graders and seventh graders
achieved about the same on the SIMT may be attributed to the use of
a sample restricted to relatively few students from one school, arranged

in a post-hoc blocking according to their experience with instruction

in spatial visualization.

CHAPTER V
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

Significance of the Study

The concept Of similarity is essential in the develOpment of
children's geometrical understanding of their environment. Moreover,
studying similarity provides children with concrete images of
proportional reasoning, and has, therefore, important pedagogical

implications.

One of the four instructional units designed by the Middle Grades
Mathematics Project (MGM) deals with the concept of similarity. The
purpose of the MGM Similarity Unit is to provide a response to the
present deficiencies in the instruction of geometry. By offering a
rich variety of experiences designed to fit the first two levels in
the Van Hiele model for develOpment of geometrical understanding, the
unit attempts to build a solid base for further advance in the
understanding of geometry in general, and of similarity and

proportionality in particular (Lappan, 1983).

The acquisition of similarity concepts in a school setting has
been rarely investigated. One of the goals of this study was to determine
patterns in the performance of middle-grades students in geometrical
similarity tasks both before, and after instruction on similarity via

the MGM unit.

A second goal of this study was to determine the effect, if any,

of instruction with the MGMP Similarity Unit on the spatial visualization

 

of middle-grades students, and conversely, to determine whether

104

105

instruction with the MGMP Spatial Visualization Unit effects the

performance of middle-grades students in similarity tasks.

Design of the Study

The effect of 10-15 days of instruction with the MGM Similarity

 

Unit was determined by comparing pre, and post-instructional performance

on a paper-and-pencil 25-item Similarity Test (Appendix A). Pre-

instructional performance on the Similarity Test was analyzed to

investigate possible grade level and sex differences in the performance

of middle-grade Students in similarity tasks.

1.

The study followed three directions:

Investigation of student performance on four similarity-related
areas: basic properties of similar shapes, proportional reasoning
as expressed in similarity tasks, the area relationship of Similar
shapes, and applications of similarity as expressed in sealed drawings

and indirect measurement.

Comparison of performance in similarity tasks presented in a visual
presentation mode with performance on equivalent tasks presented

verbally (i.e. , without any illustrative aids).

Investigation of the relationship between performance on similarity
tasks and performance in spatial visualization. This was accomplished

by analyzing student performance on a 15-item subtest of the MGM

106

Spatial Visualization Test, in addition to the achievement on the

MGM Similarity Test.

The sample included middle-grades Students from five schools with
a suburban, middle-class, midwestern, predominantly white population and
whose teachers volunteered to the MGM Similarity Unit. The sample
was subdivided in two groups: Sample A consisted of students from four
schools (N=514) that had no instruction in Spatial visualization, and
Sample B (N=161) consisted of students from a fifth school containing
both students who had instruction in spatial visualization the year

before this study, and students who did not have this kind of instruction.

The data Obtained from Sample A was analyzed to determine trends
in performance of middle-grades students in similarity tasks and to
detect any influence of instruction in similarity on spatial
visualization. The data from Sample B was used to analyze the influence
of instruction in Spatial visualization on performance in similarity

tasks.

Before instruction, the teachers participated in a two-hour workshop
on the mathematical content and instructional strategies of the unit,
that was conducted by the MGMP staff, then taught and tested the students

themselves.

Analysis of Variance and Analysis of Repeated Measures served as
the main statistical tools to test the hypotheses that were derived
from the research questions. Group means and correlation coefficients
were additionally examined in order to provide further interpretation

of the data. The analyses were performed by using SPSS programs on a

  

107

CYBER 170, Model 750 computer at Michigan State University.

Main Findings and Conclusions

Performance and Gains in SimilaritLTasks

Two research questions were asked in this study on the performance
of sixth, seventh, and eighth graders in similarity tasks. The first
(Research Question 1.1) was related to grade level and sex differences
in the pre-instructional performance in tasks that require the use of:
(1) basic properties of similar Shapes, (2) proportional reasoning
related to similarity, (3) the principle of area growth, and (4)
applications of the similarity concept in sealed drawings and in indirect
measurement. The second question (Research Question 1.2) referred to
the effect of instruction (i.e., the existence of gains) on achievement

in these four areas.

The data obtained from Sample A on pre-instructional achievement
was analyzed by Multivariate Analysis of Variance. At a significance
level of .05, the test did not Show significant sex differences in
performance on any of the four topics, but indicated significant grade
differences: on three tOpics the level of performance increased
significantly with the grade level. The performance on the topic of
area growth indicated a somewhat different pattern: it was only loosely
correlated (correlation coefficients of less than .25) with the other
three topics, and it was almost uniformly low (average rate of success

of 18-22 percent) for all grade levels.

108

The general pattern of increasing achievement as a function of
grade level (i.e., of age) indicates that cognitive development plays
an important role in the acquistion of the concept of geometrical
similarity. This result confirms the findings of other cognitive studies:
Although researchers may differ on their explanations of factors that
influence the development of the Similarity concept or of proportional
reasoning, they generally agree that starting from about the age Of
11 there is a gradual improvement in the performance of children in
similarity tasks. At earlier ages, perceptual judgments are possible
(Piaget, 1967), but a more general and conceptual use Of prOportionS

could not be detected.

The sequence of topics arranged in an increasing order of difficulty
as indicated by pre-instructional achievement is: (1) basic properties
of similar shapes, (2) applications, (3) proportional reasoning, and
(4) area growth, with overall sample averages of 37.2, 35.0, 33.7, and
19.8 percent respectively. More detailed data is presented in Table
E.1 (Appendix E) and Figure 4.1. The sequence confirms the results
obtained from the second mathematics assessment of NAEP (Carpenter et
al., 1981) that indicate that children exhibit a better understanding
of the basic geometrical concepts than of their applications. The lower
performance in proportional reasoning supports the findings in cognitive
research (e.g., in the Karplus studies) that a systematic use of this
ability is possible about two years later than the first conceptual

judgments in similarity tasks.

109

The very low achievement on tasks that involve the area relationship
of similar shapes also corresponds to findings of other studies. This
phenomenon can be explained in different ways: (1) at this age, only
about half of the children know how to compute the area of a rectangle
(Schnur & Callahan, 1973; Carpenter et al., 1981); (2) children often
confound area with perimeter (e.g., Piaget et al., 1960) confusing
accordingly area growth with linear growth; (3) the separation of the
growth of sides and growth of area in similar shapes requires a formal
reasoning ability that is rarely present at this transitional age

(Montangero, 1976; Lunzer, 1968).

This study indicates a lack of significant sex differences in any
of tin: four similarity-related topics. Other studies showed the same
results for similarity (Cameron, 1925), for informal geometry (Olson,
1970; G.D. Peterson, 1973; Thomas, 1977), and for formal TEuclidean
geometry (Werdelin, 1961; Senk & Usiskin, 1982). On the other hand,
some studies (Shonberger, 1976; Ben-Haim, 1983) detected a nmle
superiority in informal geometry tasks and in spatial visualization.
These sex differences were explained by the boys' richer extra-curricular

experience related to visualization.

Other studies also indicate the existence of a clear male superiority
in proportional reasoning (Chapman, 1975; Brendzel, 1978; ‘Keating .&
Schaefer, 1975; Stage at al., 1980). However, the prOportionality tasks
employed in these studies were not related to geometrical similarity.
The lack of sex differences in similarity-related proportional reasoning

indicated by this study contradicts the findings of other proportionality

110

studies. In view Of more recent studies on sex differences in mathematics
(e.g., Fennema & Sherman, 1978), it may be argued that the experience
required in order to perform in geometrical proportionality tasks is
common to boys and girls, and probably unrelated to extra-curricular

activities.

The data on pre, and post-instructional achievement on the four
similarity-related topics allowed for the analysis of gain scores for
Sample A. The gains could be attributed to the instruction on similarity,
and the possibility of any other confounding influences was considered
low: due to the short period of instruction and to the fact that children
probably did not encounter relevant extra-curricular experiences, the
possibility that confounding events occured between the two tests may
be considered remote. An effect of cognitive maturation may be also
discarded for Similar reasons. In order to avoid a test-retest effect,
teachers were instructed not to discuss the tests after pretesting,
and Student results were available to both teachers and Students only

after posttesting.

At a significance level of .05, the Analysis of Repeated Measures
again did not Show significant sex differences, but indicated significant
grade differences in gains. The seventh graders seem to have profited
the most from instruction: their gains were similar to the eighth
graders' and higher than the gains of the sixth graders on the topics
of area growth and applications; on the other hand, they gained as
much as the sixth graders and more than the eighth graders on the

topics of basic prOperties of similar Shapes and proportional reasoning

(Figure 5.1).

111

Percent

90r-

8O -

70-—

 

 

60-—

 

50-

40'—

 

 

 

30 _.

 

 

20 -

 

10 L.

 

1 1 1 -'LV! I 1 1 I 1 11

6 7 8 6 ‘7 8 6 7 8 6 7 8 ‘5’
BASIC PROPORTIONAL AREA APPLICATIONS
CONCEPTS REASONING GROWTH

 

 

Figure 5.1 Gains in four similarity-related topics--Sample A.

112

Particularly high gains (27-32 percent) were detected in the sixth,
and seventh graders' scores on the topics of basic properties of similar
shapes and of similarity-related proportional reasoning. The relatively
smaller gains (15-17 percent) of the eighth graders in the same topics
may be attributed to a ceiling effect - their average posttest score
reached a mastery level of 80 percent. Although there are no other
studies on similarity to which these results can be compared, these
results are comparable to a study conducted by Kurz and Karplus (1977):
after a three-week-long instruction in proportional reasoning the scores
of the participating high-school prealgebra students on a proportional
reasoning test rose to 62-70 percent. The same results were obtained
by the subjects of this Study at the sixth and the seventh grade level

in proportionality tasks related to geometrical similarity.

Lower gains (5-20 percent) were found for the topics of applications
and of area growth for all grade levels. These results may be attributed
to the cognitive difficulties involving the transfer of the similarity
concept to pmectical situations as required in the application tasks,
and the differentiation between linear growth and area growth of shape
as required in the area growth tasks. The particularly low level of
post-instructional performance in area growth tasks for the sixth.graders
reported in this study (23 percent) and also in Fitzgerald and Shroyer
(1979) may suggest that the cognitive difficulties related to area growth
tasks cannot be overcome by instruction at this age. The fact that

the MGMP Similarity Unit's treatment of area growth is less intensive

 

 

113

than the other topics cannot be the only reason for the low level
of performance in this topic: the instruction in the study conducted
by Fitzgerald and Shroyer concentrated mainly on the growth and shrinking

of figures and objects, but did not produce better results.

The detection of generally significant gains for all similarity-
related topics may be considered an indicator of the effectiveness of
an instruction that employs the MGM Similarity Unit and its corresponding
instructional strategies. Although this study does not compare
alternative instructional interventions, it may be concluded that
teaching the concept of geometrical similarity through informal,
exploratory activities that employ concrete materials at the middle
grades level induced significant improvement in student performance in
similarity tasks. This conclusion adds another empirical basis to
recommendations made by math educators who stress the importance of
teaching informal geometry (e.g., J. C. Peterson, 1973; Wirszup, 1976),
using concrete materials (Fennema, 1972; Wollman & Lawson, 1978) through

a problem-solving approach (Scott et al. , 1971; Hempel, 1981).

Visual versus Verbal Presentation Mode

In order to compare student performance on SIMT items accompanied
by drawings with achievement on items presented in a purely verbal form,
four pairs of equivalent questions were selected to form a visual, and
a corresponding verbal subtest. The obviously visual character of the
instructional unit and pedagogical common sense seem to favor the visual

presentation mode. In other words, it may be expected that students

114

would have a higher performance and would gain more on geometry-related
questions that use also a graphical representation of the concepts

involved.

In this study, two research questions were used to investigate
pre-instructional performance and gains on the verbal and the visual
subtest: The first (Research Question 2.1) referred to grade level
and sex differences in pre-instructional achievement on the items
presented in the two modes, whereas the second question (Research Question
2.2) was related to the effect of instruction on achievement in the

verbal and the visual subtests.

The data obtained from Sample A on pre-instructional achievement
was analyzed by Multivariate Analysis of Variance. At a significance
level of .05, no effects due to presentation mode were detected: all
grade levels and both sexes performed at about the same level on verbal
and on visual items. Furthermore, an examination of the mean scores
(Table E.3, Appendix E) reveals a one-to-two percent higher performance

in items presented in a verbal mode.

An item-analysis reveals a different picture. Two of the four items
in each subtest dealt with area growth-a topic that, as indicated in
the preceeding section, involves many cognitive difficulties. The
achievement on these items is low (3-36 percent) and does not favor
a particular presentation mode. On the item related to indirect
measurement (application of similarity) the performance in the visual
item was by 12-16 percent higher than in its verbal counterpart. The

fourth item required the recognition of similar rectangles by drawings

115

in the visual mode, and by the numerical ratio of their sides in the
verbal form. In this case, the performance in the verbal mode was

superior to the visual presentation by 15-47 percent.

Thus, although no overall difference for a particular presentation
mode could be indicated, for different kinds of Similarity-related tasks

different presentation modes seem to be favored.

The fact that visual representations should not be always favored
is also indicated by Karplus et al. (1974): in this study, performance
on the Mr. Tall/Mr. Short proportionality task improved considerably,
when their subjects were deprived of visual cues (i.e., the drawing

of Mr. Tall) being thus forced to use the numerical data.

The gain scores obtained from the pre, and post-instructional scores
of Sample A were analyzed by conducting an Analysis of Repeated Measures.
At a significance level of .05, no Significant sex differences in gain
scores of the verbal and visual presentation were detected. However,
the test indicated a significant difference in gain pattern as a function
of grade level: seventh, and eighth graders gained more (by 2 and
5 percent respectively) in the visual score, whereas the gains of the
sixth graders in the verbal items were 5 percent higher than in the

visual subtest .

The results indicate that boys and girls have the same level of
performance and the same gain patterns with regard to the two presentation
modes. This finding agrees with the lack of sex differences in geometry
achievement indicated by some studies. However other studies (e.g.,

Werdelin, 1961; Smith & Litman, 1979; Senk & Usiskin, 1982) indicate

 

116

a male superiority on geometry tasks that require visualization, but
equal performance or female superiority on more formal or verbal
geometrical tasks. Therefore, a sex by presentation mode interaction
would be expected. The fact that this phenomenon did not occur in
this study may be a result of unfamiliarity with Similarity tasks for

subjects of both sexes.

The MGM Similarity Unit- emphasizes the visual and the informal
aspect of the concept of similarity. The fact that sixth graders gained
less in their visual tasks than in the verbal ones may be another
indicator of a lower effect of instruction for this group. It is also
possible that during the period of instruction the sixth graders achieved
a higher level in the mastery of the "theoretical" aspects of the
similarity tasks (e.g., numerical ratios, the verbal rule related to
area growth) but were less able to apply it when the data had to be

deduced from drawings.

Relationship with Spatial Visualization

Studies that examined the possibility of a relationship between
geometrical ability and spatial visualization did not Show uniform
results: some studies (Holzinger & Swineford, 1946; Blade & Watson,
1955; Werdelin, 1961) indicate the existence of such a relationship,
others (Weiss, 1954; French, 1964; Bennet et al., 1966) claim that
geometrical ability is independent Of space factors, while additional
studies (e.g., Hanson, 1972) show mixed results. However, there are

strong cognitive and pedagogical reasons (Piaget, 1964; Hoffer, 1977)

117

to assume a close relationship between informal geometry and spatial

ability.

Two research questions were designed to refer to the possibility
of a relationship between achievement in similarity tasks and spatial
visualization. The first (Research Question 3.1) was related to the
effect of instruction in similarity on spatial visualization, whereas
the second (Research Question 3.2) dealt with the effect of instruction

in spatial visualization on performance in similarity tasks.

In order to answer the first question, Sample A scores on the
Similarity Test and on the Spatial Visualization Subtest (Appendices
A and B) prior to, and after instruction in similarity were analyzed
by an Analysis of Repeated Measures. At a significance level of .05,
no grade or sex differences in spatial visualization gains could be
detected. However, there were significant gains in achievement in spatial
visualization (from 37 to 47 percent for the whole sample). Ben-Haim
(1983) indicates that the MGMP Spatial Visualization Test Shows no test-
retest effect. This study also indicates a Strong correlation between
the performance on the similarity, and on the Spatial visualization
test (r=.56 prior to similarity instruction, and r-. 58 after instruction).
Therefore, it may be assumed that the gains in spatial visualization

detected in this study are related to the instruction on similarity.

When similarity achievement was used as a covariate, the results
of the Analysis of Variance for achievement in the MGM Spatial
Visualization Subtest remained basically unchanged. The fact that the

overall gains in spatial visualization stayed significant even after

118

controlling for gains in Similarity, may indicate the existence of

additional factors that caused the growth in Spatial visualization.

Thus, it may be concluded that instruction in similarity had a
positive (although possibly not exclusive) effect on spatial
visualization. This result is consistent with the findings of other
studies (Ekman, 1968; Brinkmann, 1966; Wolfe, 1970; Battista et al.,
1982) that showed significant effects on spatial visualization due to

training in informal geometry.

The data obtained from Sample B was examined to detect any effect

of instruction with the MGM Spatial Visualization Unit (conducted one

 

year before this study) on achievement in similarity.

The Analysis of Repeated Measures for Similarity test scores
indicates that at a significance level of .05, there are no sex or
grade level (seven versus eight) differences in similarity gains.
Moreover, there was no Significant instruction effect: children that
underwent instruction in Spatial Visualization performed at the same
level and gained in similarity scores about the same as children that
did not study that instructional unit. When achievement on the Spatial
Visualization Subtest was controlled, by using spatial visualization
scores as covariates, the same pattern of similarity gains as in Sample
A could be shown: no sex differences, but significantly superior gains
of seventh graders were revealed. The restricted number of subjects,
the use of subjects from a Single school, and the post-hoc blocking
of students according to level of instruction in spatial visualization

posed serious limitations on the data obtained from Sample B.

119

Pedagogical Implication

The results of this study suggest that an informal and activity-
oriented approach in teaching the concept of geometrical similarity,
at the middle-grades level may be effective. Although age appears to

be a considerable factor in children's understanding of similarity, three

 

weeks of instruction with the MGM Similarity Unit helped to improve
the average performance on the Similarity Test by 19, 23, and 17 percent

for Sixth, seventh and eighth graders respectively.

Considerable gains were Obtained in the two topics that were
emphasized the most in the unit: basic properties of similar shapes
(17-27 percent) and prOportional reasoning (15-32 percent). In order
to achieve better results in tasks that involve applications of the
concept of similarity or the principle of area growth, these topics
will have to be accorded an equal amount of attention. However, the
teaching of area growth at the sixth grade level seems to be particularly
unrewarding. Three possible solutions to this difficulty may be
suggested: (1) to postpone the instruction on this topic to an age
when children have the required cognitive skills, (2) to teach first
the principle of area growth in an intensive way (e.g., by the use
of the instructional unit develOped by Shroyer and Fitzgerald, 1979)

and to use the MGM Similarity Unit as a follow-up on the topic, or

 

(3) to consider the teaching of the Similarity Unit as a first step

 

in teaching area growth that will have to be followed at a later Stage
by additional instruction. Since seventh graders were shown to profit

most from instruction, this grade level seems to be particularly favorable

120

for the instruction of the SimilaritLUnit. It is possible that sixth
graders encounter more cognitive difficulties than their older peers,
whereas the concrete-activity orientation of the unit appeals less to

the eighth graders than to younger students.

The lack of sex differences in performance on similarity tasks
found by this study should be a warning against generalizing findings
on male superiority in some visualization tasks to all activities of
this kind. As pointed out by others, the nature of the extra-curricular
experiences required to complete a task seems to be the key to sex
differences in mathematics: tasks that are at the same familiarity
level for both sexes are likely to induce an equal level of performance

for both sexes .

This study questions the assumption that performance in informal
geometry tasks presented in a visual mode is always superior to verbal
tasks. A careful task analysis seems to be required in each case.
Some similarity, and probably other geometrical tasks seem to attract
a higher number of wrong answers when accompanied by drawings. This
finding should not imply that misleading representations should be
avoided. It is recommended that (1) different approaches should be used
for teaching the same tOpic, and (2) the misleading representation should

be used as the final test for understanding a concept.

The fact that instruction in similarity was accompanied by an
improvement in spatial visualization seems to indicate that if the MGMP
Similarity Unit will be followed by instruction with the MGM Spatial

Visualization Unit a better final achievement in spatial visualization

121

may be expected. The first unit will.raise indirectly the achievement
in Spatial visualization, and thus will allow students to start the

second unit at a higher level.

In this study, instruction in Spatial visualization did not have
a Similar effect on achievement in similarity. This result seems to
indicate that placing the two instructional units in a reversed sequence
would not have the same favorable effect for understanding the concept
of similarity. However, since the instruction in spatial visualization
took place a year before this Study, and the sample used in this case
was restricted, further inquires are needed to verify this last assertion.

Recommendations for Further Research

It is recommended that attempts Should be made to overcome some
of the restrictions due to the amount of available time, conditions
of sampling and structure of the evaluation tools. Several studies

that further pursue the directions established in this study are possible:

1. This study may be replicated with samples differing in terms of
student SES, site and ethnic background to verify the generalizability

of the results.

2. A replication of this Study that would also include a delayed testing

would allow for an investigation of retention of the initial gains.

3. A systematic comparison of student responses in Similarity tasks
presented in a verbal and a visual mode may be conducted to confirm

the primary findings of this study. In this case, there is a need

 

4.

5.

6.

122

for a test that would include several pairs of equivalent verbal,
and visual items in each of the four similarity-related tOpicS

identified in this study.

In order to further investigate the relationship between the concept
of similarity and spatial visualization and its pedagogical
implications, a sample of students Should be divided into three

groups: one group will study the MGM Similarity Unit first and

 

the MGMP Spatial Visualization Unit next, another group will study
the two instructional units in reversed order, and the third group
will serve as control. The SIMT and the SVST will be administered
at the beginning and at the end of each instructional unit in order

to draw conclusions about an optimal sequence.

The concept of area growth may be investigated in a separate study.
Results after instruction with the MGM Similarity Unit, The Mouse
and the Elephant (Fitzgerald & Shroyer, 1979), and possibly after
other alternative approaches may be compared. It would be also
interesting to follow the develOpment of the concept of area growth

throughout the high school, and even at the college level.

An in-depth study on the cognitive development of the concept of
similarity may be conducted: by using the SIMT as a primary selection
tool, a limited number of average students will be interviewed

extensively both before, and after instruction with the Similarity

 

Unit. Error-patterns and effect of instruction on students' reasoning

 

1|

123

may be revealed and considered in a revision of the present

instruction.

7. The relationship between the concept of similarity and proportional
reasoning would be another topic for investigations: A test in
proportional reasoning will allow the researcher to compare student
performance on the two related topics and detect any transfer from
proportional reasoning in Similarity tasks to more general

proportionality tasks.

The first four recommendations may extend the primary findings of
this study one step further, whereas the other three may open new
directions in the investigation of the development of children's
geometrical understanding in general, and of the concept of similarity

in particular.

APPENDICES

 

APPENDIX A

MGMP SIMILARITY TEST:

ITEMS INCLUDED IN THE FOUR SUBTESTS

ITEMS INCLUDED IN THE TWO PRESENTATION MODES

124

BASIC PROPERTIES OF SIMILAR SHAPES

1. Given the rectangle

 

Which of the following rectangles is similar to the given rectangle?

(A) (a) (c) (m (5')

      

13. Which of the following rectangles lS similar to a 10 x 15 rectangle?

 

 

 

 

 

 

 

 

 

 

 

 

 

10
7' l a
*.__l L. 5 2:1 °
6 ‘ 5 3 s 10
(A) (B) (C) (D) (E)

14. which of these figures is divided into small shapes each similar to
the big shape?

 

 

 

 

 

 

aflb .g' {fit ' 'fi’
0d " -' ‘7’, ...3 {.71
I 'L i — l 1 l 1 y 7 1
I 8 -__-[-- 1.-“4 o—; : cu: bozo--0
I - -.; D ' 0 1 t #1
I l_ o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

125

15. The ratio of the sides of a rectangle is §5

Which of the following could be the ratio of sides of a similar rectangle?

ung- (8)5;- (mg (mg- (5)-g-

17. Which of these figures :hows the big triangle divided into smaller
triangles each similar to the big triangle?

. . //A\
«A&

19. If two figures are similar, which of the following might be different?

(E)

(A) number of sides (B) lengths of corresponding sides
(C) shape (0) size of angles (E) ratio of corresponding sides

 

126

PROPORTIONAL REASONING

 

3. The given figures are similar. ?//A
5

7 . \\\\E\
.__).

Find the missing length. ' ?
(A) 11 (B) 14 (C) 15 (D) 18 , (E) 21

 

 

 

 

 

 

4. These rectangles are similar. 2 ‘6

 

 

 

01

 

 

 

Find the length of the missing-side.
(A) 7 (B) 9 (c) 10 (D) 11 (E) 15

ON)

These triangles are similar:

(3':
to

Find the missing length.
(A) 10 ,(8) 11 (C) 12 (0) 13 (

 

18. The given figures are similar.

12

 

30 ' ?
Find the missing length.
(A) 4 (B) 5 (C) 6 . {0) :2 (E) l:

 

127

AREA GROWTH

2. A 1x5 rectangle grows into a 4x20 rectangle.

The area of the new rectangle is how many times larger than the area of
the small rectangle?

(A) 3. times (3) 4 times (C) 5 times (D) 15 times (E) 16 times

’-

o. If the lengths of the sides of a triangle are each multiplied by

then the area of the new triangle is?

(A) 3 times larger (8) 5 times larger (C) 9 times larger

(D) 12 times larger (E) 15 times larger

9 The two triangles below are similar and the lengths:ifthe sides of the
larger are 3 times that of the smaller.

/\

.- f \
A f \

How many of the smaller triangles will exactly fit into the larger one?
(A) 4 ' (B) 6 (C) 7 (D) 3 (E) 9

‘5

11. Given rectangles of dimensions 1x6 and 4x24.

 

1E: 41F

p

° 24

. , .. . +
The area of the larger rectangle 15 now many times as big as .he
area of the smaller rectangle?

- -' ’1" 6; ft) 0 *‘ihp
(A) 4 times ’8) 6 times 1C) 3 times 15) .0 ~-“95 1-1 1“ “'E~5
0‘ . \ ‘

 

128

APPLICATIONS

S. A picture of a tall building is taken.

 

/
The following measurements are known. E3
Picture Actual :3
Height of door 10mm 2m
Height of building 40mm ? D
What is the actual height of the building? _ [-1 E3

 

(A) 8m (3) 6m (c) 10m (0) 60m . (a) 80m A W

lg, A man who is 6 feet tall has a shadow which is 8 feet long. At the
same time a nearby tree has a shadow which is 32 feet long. How tall
is the tree?

(A) 30 feet (8) 21 feet (C) 24 feet (D) 42 feet (E) 48 feet

12. Given a picture of a trout.

 

 

 

The scale of the picture to the real trout is 1 to 12.

what is the actual length of the real trout?

(A) 1/2 cm (8) 72 cm (C) 2 cm (D) 144 cm (E) 18 cm

21. Joan estimates the height of a flagpole by uSlng a mirror.

.istances
To eye level 5 rt.
Joan to mirror 2 ft.

Mirror to pole 10 ft.

 

How tall is the pole?

(A) 10 ft. {8) 13 ft. (C) 15

-1»:
d
(I)

V
‘\)
U1
-..‘
fl
0

A

m

.—

O

O

l

129

23. A 2 meter stick has a shadow of 8 m at the same time that a nearby
tree has a shadow of 3m.

How tall is the tree?

 

(A) 6m (8) 12 m (C) 1%") (D) 3:m (E) 15 m

a“, V - 0 -- ~ 0-1 '3
2: What scale factor has been used L0 enlarge the small >alleat.

 

 

(A) 2 (8) 3 (C) 4 (D) 6 (EH/4

 

130

VERBAL PRESENTATION MODE

2. A 1x5 rectangle grows into a 4x20 rectangle.

The area of the new rectangle is how many times larger than the area of
the small rectangle?

(A) 3 times (3) 4 times (C) 5 times (D) 15 times (E) 16 times

6. If the lengths<rfthe sides of a triangle are each multiplied by 3,
then the area of the new triangle is?

(A) 3 times larger (3) 6 times larger (C) 9 times larger

(D) 12 times larger (E) 15 times larger

O

10. A man who is 6 feet tall has a shadow

same time a nearby tree has a shadow
15 the tree?

which is 3 feet long. At the
which 15 32 feet long. How tall

(A) 30 feet (B) 21 feet (C) 24 feet ’

15. The ratio of the sides of a rectangle is

2
‘3'.
Which of the following could be the

ratio of sides of a similar rectangle?

4 4 2 4 _.
(mg (Bl-3- (mg (mg- (a);

 

23-

131

VISUAL PRESENTATION MODE

 

 

 

11. Given rectangles of dimensions 1x6 and 4x24.
1 [:21 4 .
6 l
24

The area of the larger rectangle is how many times_as big as the
area of the smaller rectangle?

(A) 4 times (3) 6 times (C) 8 times (D) 16 times (E) 18 times

9. The two triangles below are similar and the lengths:afthe sides of the
larger are 3 times that of the smaller.

I: /\

How many of the smaller triangles will exactly fit into the larger one?
(A) 4 (B) 5 (C) 7 (D) 3 (E) 9

A 2 meter stick has a shadow of k m at the'same time that a nearby
tree has a shadow of 3m.

 

How tall is the tree?

(A) 6m (B) 12 m (C) 1%”1 (D) 3‘m (E) 15 m

 

132

1. Given the rectangle

 

which of the following rectangles is similar to the given rectangle?

(A) ‘ (B) (C) (0) (E)

      

 

APPENDIX B

MGMP SPATIAL VISUALIZATION SUBTEST:

SAMPLE ITEMS

 

133

26. How many cubes are needed to build this rectangular solid?

 

 

27. You are given a picture of a building drawn from the FRONT-RIGHT corner.
Find the RIGHT VIEH. '

 

 

 

 

 

 

 

 

 

 

 

 

31. Find another view of the first building.

 

 

 

134

3;. Which of these buildings can be made from the two pieces given?

   

' ' ' ii

 

 

33. If a cube were added to the shaded face of the given building, what would the new
building look like?

 

a. a 0 0 ’ ‘ .

     

 

O O O O O t O
0 o O O O 0 O

 

38. How many cubes touch the indicated cube face to face?

4 D
C‘)
U1 {Tl

 

 

-_+_ I...‘ A A

APPENDIX C
THE MGMP SIMILARITY TEST AND

THE MGMP SPATIAL VISUALIZATION SUBTEST:

RELIABILITY COEFFICIENTS, AND

PRETEST - POST TEST CORRELATION COEFFICIENTS

 

135

Table C.1 RELIABILITY COEFFICIENTS--CRONBACH a.FOR THE MGMP
SIMILARITY TEST AND FOR THE MGMP SPATIAL VISUALIZATION
SUBTEST BY TIME, BY GRADE, AND BY SEX.

 

 

MGMP SIMT MGMP SVST
N Pretest Posttest Pretest Posttest
Ci d Oi. 0!.
Grade 6 248 .57 .76 .71 .69
Male 137 .59 .75 .76 .70
Female 111 .53 .76 .59 .65
Grade 7 240 .64 .77 .77 .78
Male 120 .68 .79 .79 .79
Female 120 .60 .74 .75 .76
Grade 8 207 .75 .82 .77 .79
Male 116 .76 .82 .80 .79

Female 91 .74 .82 .71 .77

 

 

136

Table C.2 PEARSON CORRELATION COEFFICIENTS BETWEEN PRETEST AND
POST TEST SCORES ON THE MGMP SIMILARITY TEST AND ON
THE MGMP SPATIAL VISUALIZATION SUBTEST BY GRADE AND

 

 

BY SEX
MGMP SIMT MGMP SVST

N Pre - Post Pre - Post

Correlation* Correlation*
Grade 6 248 .54 .66
Male 137 .49 .69
Female 111 .61 .60
Grade 7 240 .49 .71
Male 120 .52 .72
Female 120 .46 .70
Grade 8 207 .78 .78
Male 116 .76 . .75
Female 91 .79 .80

 

*All significant (P<<.001)

APPENDIX D

THE MGMP SIMILARITY UNIT: OVERVIEW

 

137
OVERVIEW

The unit begins with a very intuitive approach to enlarganent using

 

the rubber band stretcher. While the scale and the shape are not precise,
the basic idea stands out-if we produce an enlargement with two rubber
bands knotted together we get a figure whose sides are twice as long

and whose area is four times as large as the original.

Enlargenents and contractions are also the focus of the second
activity. However, in this activity the use of coordinate system allows
more precise pictures. Tests for similarity can now be made by canparing
angle sizes and corresponding distances. The area of Morris's nose
provides a concrete example of the relationship that area grows as the
square of the scale factor. For exanple if the sides are enlarged
by a factor of three, the area is three squared or nine times as large.
This is an important but subtle idea which will not be learned in

this first exposure.

Activities three and four allow a close look at similarity of
rectangles and develop two tests which can be used to determine whether
rectangles are similar. In activity three, students test two rectangles
by dividing the short side of each rectangle by its long side. If

the resulting ratios are equal, the rectangles are similar.

In activity four, the students find that fmnilies of similar
rectangles will have coinciding diagonals when the rectangles are nested
in their lower left corners. In activity five, first the students apply
what they learned in activities three and four to the idea as similar

right triangles (which are simply halves of the rectangles), then to

 

 

 

 

138
any similar triangles.

The idea of repeating tiles (reptiles) in activity six brings
together in one activity the ideas of similarity, congruence and infinity.
The student will realize that the reptiling process can go on infinitely
in both directions. The tiles can continue to grow as large as one
might want and they can be continually subdivided as small as desired.

Many children are fascinated by the ideas of infinity.

Activity seven provides a means of enlarging figures much more
accrately than the stretcher in activity one. With an unmarked straight
edge the students can construct point enlargements to various scale
factors, both positive and negative. The effects of moving the point
of enlargement determines v_v_h_e_r_e_ the image will lie, but not how large

it will be. The scale factor canpletely detennines size.

Activity eight provides a short respite frcm the more intensive
activities by having the students pause and apply the ideas they have
been studying. They can choose a picture to reproduce by any of the
techniques they have learned so far. They can experiment to see which

of the techniques provides the best results for their purposes.

If you have access to a pantagraph (a mechanical linkage device

for enlarging figures), you might introduce it at this point.

Activity nine engages students in real life applications of
similarity to finding distances or heights that cannot be measured

directly.

 

APPENDIX E

MEAN SCORES, STANDARD DEVIATIONS,
AND CORRELATIONS

RELATED TO THE RESEARCH QUESTIONS

 

139

 

 

 

 

 

Table ELL MEAN SCORES (IN PERCENT ) AND STANDARD DEVIATIONS
ON FOUR SIMILARITY-RELATED AREAS FOR SAMPLE A
BY GRADE LEVEL AND BY SEX
Basic Proportional Area Applications
Properties Reasoning Growth
bi II II
(S.D.) (S.D.) (S.D.) (S.D.)
Pre Post Pre Post Pre Post Pre Post
Grade 6 26.48 53.43 18.25 50.60 17.84 23.19 27.96 38.31
(19.61) (23.82) (26.47) (34.41) (19.66) (24.47) (20.07) (21.83)
Males 29.81 53.89 19.53 54.20 18.43 23.91 29.08 40.15
(20.56) (24.19) (28.07) (33.56) (19.48) (24.04) (21.86) (21.44)
Females 22.37 52.85 16.67 46.17 17.12 22.30 26.58 36.04
(17.62) (23.45) (24.39) (35.06) (19.94) (25.08) (17.61) (22.20)
Grade 7 38.04 65.10 38.09 70.15 22.06 37.65 34.80 53.33
(24.68) (22.15) (35.17) (31.16) (20.86) (29.82) (20.74) (23.89)
Males 37.93 65.90 37.64 ' 70.69 25.00 ' 37.65 33.91 55.71
(25.88) (21.85) (34.71) (31.94) (22.33) (31.64) (20.40) (25.20)
Females 38.15 64.26 38.55 69.58 18.98 37.65 35.74 51.41
(23.50) (22.56) (35.85) (30.51) (18.96) (27.98) (21.17) (22.42)
Grade 8 63.37 79.86 65.63 80.47 10.83 33.33 53.47 69.44
(20.89) (16.74) (31.47) (22.13) (24.78) (30.49) (23.06) (22.25)
Males 63.58 79.01 66.67 80.56 18.06 36.11 56.17 73.46
(22.68) (16.25) (31.85) (22.61) (22.99) (30.20) (24.93) (22.55)
Females 63.10 80.95 64.29 80.36 24.40 29.76 50.00 64.29
(18.60) (17.49) (31.30) (21.76) (26.76) (30.86) (20.16) (21.01)

 

140

Table E.2 CORRELATIONS AMONG PRE-INSTRUCTIONAL
PERFORMANCE ON FOUR SIMILARITY-RELATED AREAS FOR
SAMPLE A BY GRADEa

 

 

Basic Proportional Area
Properties Reasoning Growth
Proportional Total .516
Reasoning Grade 6 .255
Grade 7 .434
Grade 8 .313
Area Total .117 .066*
Growth Grade 6 .183 .062*
Grade 7 -.054* —.028*
Grade 8 .236 .110*
Applications Total .529 .489 .097
Grade 6 .346 .265 .056*
Grade 7 .443 .484 .027*
Grade 8 .461 .348 .210

 

a N = 514,# Grade 6 = 248, # Grade 7 = 170, # Grade 8= 96.

*
Not significant atcx= .05

 

141

Table E.3 MEAN SCORES (IN PERCENT ) AND STANDARD DEVIATIONS

ON TEST ITEMS PRESENTED IN A VERBAL AND A VISUAL
MODE FOR SAMPLE A BY GRADE LEVEL AND BY SEX

 

 

 

 

Verbal Mode Visual Mode
M M
N (S.D.) (S.D.)
Pretest Posttest Pretest Posttest
Grade 6 248 19.76 35.28 18.75 29.44
(21.17) (24.65) (20.89) (25.75)
Males 137 21.72 35.77 19.53 31.20
(22.24) (24.40) (20.94) (25.33)
Females 111 17.34 34.68 17.79 27.25
(19.60) (25.04) (20.89) (26.23)
Grade 7 170 27.35 48.97 24.71 48.53
(20.76) (28.57) (21.58) (27.56)
Males 87 27.59 48.56 26.15 51.15
(23.82) (28.11) (22.53) (28.51)
Females 83 27.11 49.40 23.19 45.78
(17.11) (29.21) (20.58) (26.43)
Grade 8 96 40.36 52.34 38.02 55.21
(22.17) (24.62) (26.65) (26.64)
Males 54 39.35 54.63 38.43 57.87
(20.95) (23.33) (26.91) (26.96)
Females 42 41.67 49.40 37.50 51.79

(23.86) (26.18) (26.62) (26.13)

 

142

Table E.4 CORRELATIONSa AMONG PRE, AND POST-
INSTRUCTIONAL PERFORMANCE ON SIMILARITY AND
SPATIAL VISUALIZATION TASKS FOR SAMPLE A
BY GRADEb

 

SIMT Pre SIMT Post SVST Pre

 

SIMT Post Total .678
Grade 6 .542
Grade 7 .512
Grade 8 .733

SVST Pre Total .562 .509
Grade 6 .297 .270
Grade 7 .437 .387
Grade 8 .551 .608

SVST Post Total .552 .582 .741
Grade 6 .339 .472 .663
Grade 7 .452 .476 .687
Grade 8 .577 .583 .740

 

a All correlation coefficients are significant (P< .001)

b N=514, # Grade 6 = 248, # Grade 7 = 170, # Grade 8 = 96.

143

Table E.5 MEAN SCORES (IN PERCENT) AND STANDARD DEVIATIONS
ON SIMILARITY AND ON SPATIAL VISUALIZATION FOR
SAMPLE A BY GRADE LEVEL AND BY SEX

 

 

 

 

Similarity Test Spatial Visualization Subtest
N - M M
(S.D.) (S.D.)

Pre Post Pre Post
Grade 6 248 22.89 41.76 28.44 39.95
(11.87) (17.28) (18.85) (19.54)
Males 137 24.15 43.36 30.51 43.41
(12.30) (17.04) (20.75) (19.87)
Females 111 21.33 39.78 25.89 35.68
(11.16) (17.45) (15.92) (18.33)
Grade 7 170 31.79 54.82 39.57 48.75
(14.67) (17.52) (22.37) (22.27)
Males 87 31.82 55.45 40.15 49.89
(15.72) (18.86) (22.46) (23.14)
Females 83 31.76 54.17 38.96 47.55
(13.58) (16.08) (22.39) (21.41)
Grade 8 96 50.08 66.88 56.74 63.61
(15.70) (15.03) (20.84) (21.58)
Males 54 50.00 68.44 56.91 64.32
(16.33) (14.67) (22.61) (20.81)
Females 42 50.19 64.86 56.51 62.70

(15.05) (15.42) (18.58) (22.77)

 

 

144

Table E.6 MEAN SCORES (IN PERCENT) AND STANDARD DEVIATIONS ON
SIMILARITY AND ON SPATIAL VISUALIZATION FOR SAMPLE B
BY LEVEL OF INSTRUCTION IN SPATIAL VISUALIZATION, BY

GRADE, AND BY SEX

 

Similarity Test Spatial Visualization Subtest_

 

 

 

N M M
(S.D.) (S.D.)

Pre Post Pre Post

sv Unita 98 33.67 47.43 61.56 67.07
(12.96) (16.13) (21.49) (21.76)
Grade 7 29 31.86 48.83 56.78 60.23
(11.98) (15.06) (22.51) (22.71)

Males 15 34.93 52.00 60.44 68.89
(8.34) (11.71) (22.88) (17.58)

Females 14 28.57 45.43 52.86 50.95
(14.54) (17.79) (22.26) (24.48)

Grade 8 69 34.43 46.84 63.57 69.95
(13.36) (16.64) (20.88) (20.85)
Males 43 36.28 47.35 68.68 74.88
(14.59) (18.33) (20.60) (20.57)

Females 26 31.38 46.00 55.13 61.79
(10.59) (13.68) (18.81) (19.00)
No sv Unitb 63 32.06 45.27 40.74 45.50
(14.34) (17.42) (20 88) (20.27)

Grade 7 29 30.62 45.79 45.52 48.05
(13.66) (18.16) (19.93) (23.19)
Males 13 33.85 50.77 55.90 60.00
(15.02) (19 76) (18.77) (24.65)

Females 16 28.00 41.75 37.08 38.33
(12.31) (16.26) (17.03) (17.13)

 

Table continued

145

Table E.6 (COnt'd.)

 

Similarity Test Spatial Visualization Subtest

 

 

 

N M M
(S.D.) (S.D.)

Pre Post Pre Post
Grade 8 34 33.29 44.82 36.67 43.33
(14.99) (17.02) (21.10) (17.47)

Males 16 33.50 48.50 39.17 46.25
(17.34) (15.79) (25.05) (18.77)
Females 18 33.11 41.56 34.44 40.74
(13.07) (17.84) (17.30) (16.31)

 

a Studied the MGMP Spatial Visualization Unit.

b Did not study the MGMP Spatial Visualization Unit.

 

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