The effect of a dynamic geometry learning environment on preservice elementary teachers' performance on similarity tasks

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The effect of a dynamic geometry learning environment on preservice elementary teachers' performance on similarity tasks
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Thesis (Ph.D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 113-123).
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by Helen Gerretson.

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THE EFFECT OF A DYNAMIC GEOMETRY LEARNING ENVIRONMENT
ON PRESERVICE ELEMENTARY TEACHERS' PERFORMANCE ON
SIMILARITY TASKS















By

HELEN GERRETSON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1998

























Copyright 1998

by

Helen Gerretson















TABLE OF CONTENTS


ACKNOWLEDGEMENTS .


ABSTRACT .


CHAPTERS


I INTRODUCTION . . . . .

Rationale . . . . . .
Statement of the Problem . .
Purpose of the Study . . .
Research Questions . . . .
Justification for the Study .
Definitions of Terms .. ......
Organization of the Dissertation


II THEORETICAL FRAMEWORK AND RELEVANT RESEARCH

Constructivism . . . . . . . .
Repetitive Experiences . . . . . .
Phases of Learning . . . . . . .
Erlanger Programm . . . . . . .
Similarity and Proportional Reasoning . .
Computer-based
Geometric Construction Environment ...
Dynamic Geometric Construction Environment
Summary . . . . . . . . .

III RESEARCH DESIGN AND METHODOLOGY . . .


13

* 13
* 14
* 16
S. 18
S. 20

* 23
S. 25
* 31

S. 33


Research Objectives . . .
Population and Sample ....
Instructional Materials . .
Pilot Studies . . . . .
Instrumentation . . . .
Procedure and Data Collection
Statistical Design of the Study
Summary . . . . . .


. . . . . . . . . i i


- V


. . . . 1


pare,









. . . . . 4 3


Research Instrument . .
Research Objectives . .
Goals of the Study . .
Descriptive Statistics
Research Hypotheses . .
Results of Data Analyses
A Reflections Component .
Summary . . . . .


V SUMMARY AND CONCLUSIONS .

Overview of the Study . .
Significance of the Study .
Research questions ...
Results ..............
Discussion of Results . .
Other Observations . .


Limitations


Implications and Recommendations
Summary . . . . . . .


.. . . . . 59


. . . . 58
. . . . 61
. . . . 62
. . . . 62
. . . . 65
. . . . 68
. . . . 69
. . . . 70
. . . . 73


APPENDICES

A THE RESEARCH INSTRUMENT . . .
B CONTENT OF SESSIONS . . . .
C ACTIVITIES WITH RESPONSES . . .
D LETTERS OF INFORMED CONCENT ...
E: DATA SETS . . . . . . .


REFERENCES . . .


BIOGRAPHICAL SKETCH .


S.74
. .86
S.95
* 105
S110


. . . . 113


. . . . . 124


.43
.44
.44
.45
.46
.49
.53
.58


IV RESULTS . .


e O Q Q


D














CHAPTER I
INTRODUCTION


As aspects of the current mathematics reform movement

become reality in the classroom, researchers question the

impact of the restructuring and recontextualizing of the

mathematics classroom. Interest must focus upon reform-

related pedagogy with greater intellectual involvement via

appropriately-chosen experiences in a context of problem

solving and inquiry. In particular, the effects of

implementing technology on the teaching and learning of

mathematics and its impact on mathematics as reasoning must

be addressed. The question is no longer whether technology

belongs in the classroom, but how best to use technology to

enhance learning.


Rationale



Use of technology in mathematics classrooms has

increased dramatically during the past two decades and has

been encouraged by leaders in the field of mathematics

education (Blackwell & Henkin, 1989; English & Halford, 1995;

Leitzel, 1991; National Advisory Committee on Mathematical

Education [NACOME], 1975; National Council of Teachers of

Mathematics [NCTM], 1980, 1989; National Institute of









Education [NIE], 1975; National Research Council [NRC], 1989;

Suydam, 1976). The changing needs of our society demand that

critical issues such as the role of technological advances in
the learning and doing of mathematics be addressed. Computers

are able to aid in visualizing abstract concepts and to
create new environments which extend beyond the physical.
In addition to development of problem solving and

visualization skills, technology offers enormous opportunity

for curriculum reform in mathematics (Abramovich & Brown,
1996). Dynamic software (refer to Definition of Terms) such
as Geometer's Sketchpad (Jackiw, 1995),Cabri Geometre II
(Laborde, 1994), and Mathematica (Wolfram, 1988) provide a
flexibly structured mathematics laboratory: an electronic
micro-world supporting the investigation and exploration of

mathematical concepts at a representational level linking the
concrete and the abstract. Mathematical ideas can be explored

from several different perspectives in an efficient manner,
resulting in deeper conceptual understanding (Kaput &
Thompson, 1994).

Mathematical thinking is now often learned through
problem solving activities that bury the mechanical aspects

of mathematics behind interesting ideas (Vockell & Schwartz,
1992). Dynamic software allows the computer to be a powerful

mathematical tool due to its ability to quickly and
accurately plot, construct, measure, and perform
computations. Through repetitive experiences of exploring and
mathematizing, problem solving skills and one's ability to
construct mathematical ideas are enhanced (Cooper, 1991). In
addition, the interactive mode supports active learning--a









necessary component for effective construction of knowledge

(Lewin, 1991). Furthermore, the dynamic software environment

is motivational for the student because it "promotes

mathematical explorations ... and helps the building of

conceptual understanding" (Fraser, 1988, p. 216).

The Third International Mathematics and Science Study

[TIMSS] gives mathematics educators a benchmark toward

improvement of the teaching and learning of mathematics. The

TIMSS data suggest that when teachers use more challenging

methods to teach more complex mathematical ideas, as called

for in the Standards trilogy: the Curriculum and Evaluation

Standards for School Mathematics (NCTM, 1989), the

Professional Standards for Teaching Mathematics (NCTM, 1991),

and the Assessment Standards for School Mathematics (NCTM,

1995), students' performance is higher (National Center for

Educational Statistics, 1998).



Statement of the Problem



There is reason to change what and how preservice

elementary teachers learn mathematics. Most teachers are

aware of the NCTM Standards, but evidence from the TIMSS

video component suggests that changes in teaching practice
are difficult. In the United States, lessons typically focus

on acquiring mathematical skills rather than conceptual

understanding, with little mathematical reasoning expected.

In contrast, Japanese methodology, requiring students to

think and reason, resemble the recommendations advocated in









the Standards (National Center for Educational Statistics,

1998).

Analyses of data from the TIMSS show, among the content

areas addressed, that United States eighth and twelfth

graders scored significantly below the international average

in geometry and measurement. Moreover, grade twelve students'

performance was relatively lowest in geometry; no country

scored similar to or below the United States (National Center
for Educational Statistics, 1998). This researcher posits

that the lack of exposure to transformational geometry, a

topic often found in other countries' curricula, may be a

leading factor of poor performance. Teachers cannot teach

what they themselves do not know (Fennema & Franke, 1992).

Teachers must be given the opportunity to construct and

thoroughly develop their knowledge of elementary mathematics

in an environment similar to the mathematics classroom they

oversee (Simon & Blume, 1992). Many mathematics teacher

educators advocate that teachers must be taught in ways that

are consistent with how we would like them to teach

(Committee on the Mathematical Education of Teachers, 1991;

Fennema & Franke, 1992; NCTM, 1991).

Implementing new technologies in a reformed curriculum

is not easy and the education of teachers plays a critical

role. The necessity of requiring preservice teachers to learn

mathematics content and methods of teaching in an environment

where computer use is an integral component is clearly

evident. The Mathematical Association of America stresses

this point in their report A call for change: Recommendations

for the mathematical preparation of teachers of mathematics








5
(Leitzel, 1991). The document states:

[Teachers] need experience in using appropriate
technology effectively in solving problems so that
they can learn and adapt strategies and
representations that arise while using that
technology. Because they can reflect on their own
learning and understanding of mathematical ideas
when using technology, teachers will be better
prepared to lead their own students in effective
mathematical learning (p. 7).

The issue is not simply the incorporation of another gadget

in the learning milieu. Rather it encompasses the careful

consideration of the inclusion of technology commensurate

with the current vision for mathematics education reform.

Dynamic software is generally accepted as a fertile

learning environment in which students can be actively

engaged in constructing and exploring mathematical ideas

(Garry, 1997). This researcher postulates that the dynamic

and interactive medium makes gaining an intuitive

understanding more accessible to the [geometry] student (Pea,

1987). Under the psychomotor control of the user, this

"dynamic" capability provides an infinite number of cases to

be explored. For example, once constructed, geometric

figures can be transformed by dragging a point, a vertex, or

an edge. As the figure is transformed, patterns and/or

relationships become visually explicit for further

examination.
The theoretical foundation for using a dynamic learning

environment emphasizing inquiry and exploration is based on

the constructivist theory of learning as well as the role of

repetitive experiences according to Cooper (1991). We have

known since the days of Brownell that mathematical "practice"









often has little to do with developing habits of reasoning

(Thompson, 1994). Long term problem solving is primarily a

function of recognizing well known embedded principles in new

situations. Cooper focuses our attention on the fact that

what we want repeated is a composition of situations in ways
that motivate generalization and support reflection. The

constructivist theory of learning refers to the psychological
level of analysis where emphasis is placed on the

construction of personal knowledge and how a person acquires

this knowledge (Phye, 1997). The philosophical level of

analysis, although a legitimate line of inquiry, will not be
addressed in this thesis.


Purpose of the Study



The purpose of the proposed study was to investigate the

effect of a dynamic geometry learning environment on
preservice elementary teachers' performance in the context of

similarity tasks (refer to Definition of Terms). That is,
this study was concerned with the mathematical evolution of

students majoring in elementary education as a result of

instruction with the dynamic geometry software Geometer's

Sketchpad (Jackiw, 1995). The basic goal of the study was to
detect the extent to which the dynamic environment, given its
ability to efficiently display a plethora of mathematical
situations, assists learning. Attention was also given to the

role of prior knowledge.









Research Questions



Hence, based on the assumptions that teachers need to

learn high quality mathematics and to be taught in a manner

similar to the way they are expected to teach, the specific

questions under investigation were:

1. After controlling for initial performance on
similarity tasks, do preservice elementary
teachers who work on similarity tasks in a
dynamic geometry learning environment exhibit
significantly different posttest performance
than those who study the concept of similarity
in a paper-and-pencil learning environment that
employs traditional tools such as ruler and
protractor?


2. After controlling for prior knowledge as
measured by a standardized achievement test, do
preservice elementary teachers who work on
similarity tasks in a dynamic geometry learning
environment exhibit significantly different
posttest performance than those who study the
concept of similarity in a paper-and-pencil
learning environment that employs traditional
tools such as ruler and protractor?



Justification of the Study



From several perspectives, the problem is important. As

indicated earlier, technology is profoundly changing the

teaching and learning of mathematics. Models of computer uses

in education are needed to show how to incorporate this

technology into daily practice (Garcia-de Galindo, 1994). To

define new roles for teachers in a technologically rich

classroom environment, it is necessary to explore how









computers will modify current ways of teaching and learning.

Carleer and Doornekamp (1990) conclude that "to realize a

valuable integration of computers in education, it will be

necessary to focus the discussion no longer on computers, but

on how to think and make decisions with the improvement of

education as the goal"(p.5).

The acquisition of the concept of similarity is

important for development of proportional reasoning and for

geometrical understanding of one's immediate environment. One

encounters phenomena that require familiarity with

projection, scale factor, and other similarity-related

concepts in everyday situations (Friedlander, 1984; Lappan,
Fitzgerald, Winter, & Phillips, 1986). Similarity-related

concepts are included in many parts of the mathematics
curriculum. Some models for rational number concepts are

based on similarity; an integral part of algebraic thinking

involves ratio and proportion reasoning skills (Lappan &

Even, 1988).

Similar geometric shapes would seem to provide helpful

imagery for analysis of analogous situations. Dynamic

software is relatively new phenomena; in particular, dynamic

geometry software was not widely available a decade ago.

Computerized dynamic geometry is highly visual and efficient
for exploring and discovering properties of similar figures.

However, realization of its potential is stifled by the lack

of research using dynamic geometry software for middle school

mathematics.
At this time, there is no information regarding the

effect of a dynamic geometry learning environment on K-8








9
teachers' learning of the similarity concept. This particular

study adds substance to the growing body of literature

addressing the significance of dynamic geometry software in

the classroom and, in particular, gives evidence whether a

dynamic geometry learning environment enhances efficient

construction of the concept of similarity. Moreover, this

research will help substantiate the need to incorporate

innovative technological tools in teacher education. One can

be taught techniques and tricks, but expertise results from

experience and insight.



Definitions of Terms



The following are some terms that are used throughout

the manuscript; they are defined and collected here for

reference:

A preserviceelementaryteacher is a student, without

prior classroom teaching experience. All participants were

enrolled in a required introductory mathematics methods

course addressing content, methods, and materials for

teaching elementary school mathematics.

The Geometer's Sketchpad (Jackiw, 1995), developed by

Nicholas Jackiw and the Visual Geometry Project at Swarthmore

College and now published by Key Curriculum Press, is an
example of dynamic geometrv software and was used in this

study. Such software transforms the computer interface into a

geometric construction micro-world capable of quickly and

accurately plotting, constructing, measuring, and performing









computations. Once constructed, geometric figures can be

transformed by dragging a point, vertex, or edge. As the

figure is transformed, corresponding measurements and

calculations change accordingly.

A dynamic geometry learning environment is defined as a

learning milieu in which expository teaching is integrated

with laboratory activities. The laboratory activities consist

of explorations and experimentation using dynamic geometry
software that allows for direct construction, manipulation,

and measurement of geometric figures on the plane.

In contrast, the traditional learning environment is

similar to the dynamic learning environment in that
expository teaching is integrated with laboratory activities.

In this case, however, laboratory activities consist of

explorations using manipulatives, ruler and protractor.

A variable tension proportional divider (VTPD) is a

mathematics laboratory device for drawing similar figures.

The VTPD enlarges drawings and pictures in addition to

geometric figures. To build a VTPD, knot together identical

rubber bands. One end is held down by the thumb of one hand,

acting as the projection point, and a pencil is inserted in

the other end. The rubber band is stretched so that a knot

traces the original figure while the pencil is moved on

paper. It is of utmost importance that eyes follow the knot
while the pencil draws. Although the final figures are not

quite accurate, the image clearly conveys the intuitive "same

shape" notion of similarity as one actively creates similar

figures.









A transformation (one-to-one correspondence) on the

Euclidean plane is called a similarity if there exists a

positive number k such that the distance between the images

of any two points is k times the distance between the points.

A dilation about the protection point (center) P of

factor k is a transformation D in which each point A is
mapped to its image D(A) in such a way that the directed
distance between P and D(A) is k times the directed distance
between P and A (Hall, 1973). In particular, the distance

between the images of any two points is k times the distance

between the points. A dilation is also referred to as a

dilatation, a central similarity, or a homothety (King,
1997).


Organization of the Dissertation



In this chapter, the rationale for a study designed to
examine the effect of a dynamic geometry learning environment

on preservice elementary teachers strategies employed and
performance on similarity tasks was described and background

briefly discussed. Chapter II contains theoretical

foundations and a review of related research supporting the
study. Chapter III presents a description of the subjects,
the composition of the instrument, the methodology and design
of the study, and the results of the pilot studies. Analysis,
interpretation of the data, and qualitative aspects for the
present study will be given in Chapter IV. Chapter V will

include a summary of the investigation, a discussion of








12
results, limitations of the study, implications for

instruction, and suggestions for further research.














CHAPTER II
THEORETICAL FRAMEWORK AND RELEVANT RESEARCH


In the previous chapter, the study was introduced and

background was given. In this chapter, the theoretical basis
for the study is presented in detail. This includes a

discussion of constructivist epistemology, the role of

repetitive experience on learning, phases of learning, and an

elucidation of the Erlanger Programm. Also included is a

brief exposition on similarity and proportional reasoning in

addition to relevant research in the area.

The theoretical premise that supports this study is that

a dynamic geometry learning environment, supported by use of

software such as the Geometer's Sketchpad (Jackiw, 1995), can

enhance construction of knowledge and influence learning.

Therefore, an examination of research focusing on software

for geometric constructions follows; this section develops

from static software to dynamic software as research already

conducted on the use of dynamic geometry learning software is

identified.



Constructivism



Using von Glasersfeld's radical theory of constructivism
(1979, 1990, 1995) as a basis, Kilpatrick (1987) describes









the basic tenets of constructivism as an epistemology such
that (1) all knowledge is actively constructed, not passively

received, and (2) coming to know is an adaptive process that

organizes one's experiential world; it is not the discovery

of an independent, pre-existing world outside the mind of the

knower (p. 7). Accepting von Glasersfeld's radical point of

view involves rejecting ontological realism and embracing an

epistemology that makes all knowing active and all knowledge
subjective (Kilpatrick, 1987). This epistemological stance is

in complete harmony with Piaget's theory of learning (1964),

which demonstrates that human beings acquire knowledge by

building it from the inside instead of internalizing it

directly from the environment (von Glasersfeld, 1979; Kamii,

1990). Thus, current research on teaching from a
constructivist perspective follows Piaget's biological
metaphor of development and characterizes mathematical

learning as a process of conceptual reorganization (Cobb,

1995).

The basic tenets of a constructivist epistemology have a
direct implication upon pedagogy. Nel Noddings (1990) posits

that constructivistss in mathematics education contend that
cognitive Constructivism implies pedagogical Constructivism;

that is, acceptance of constructivist premises about

knowledge and knowers implies a way of teaching that
acknowledges learners as active knowers" (p. 10). This is

particularly meaningful in the case of radical
constructivism; it focuses on the individual as a self-

organizing system.









This is not always the case in the many variations of
constructivism; some constructivists embrace Vygotsky's
(1962) epistemological stance that emphasizes the cultural

and social dimensions of development (Cobb, 1995; Confrey,
1995). In contrast, von Glasersfeld (1979, 1990, 1995)

presents constructivism as a model of how an individual's
experience forms the basis for knowing and communication.

Within this model, social interaction constrains (and thus
guides) the processes involved in the construction (von

Glasersfeld, 1990).


Repetitive Experience



The role of experience is to generate or modify the
organization of the relevant cognitive space. As conceived by
Cooper (1991), the role of repetitive experience, repeated

interaction with the environment, is to create, enhance,
and/or reorganize cognitive space. Because the constructed

cognitive space constitutes acquired knowledge, repetitive

experience is of utmost importance. As in practicing typing,
repetitive experience is used to mean repeating a set of

similar and interrelated activities, not doing exactly the

same thing over and over again.
In addition, the construction of such a cognitive space
can serve as the foundation for reflective abstraction from

which new knowledge results. The possibility that repetition
induces reorganizations of knowledge, not just of skills, is
the rationale that Cooper (1991) gives for use of the term









"repetitive experience" rather than "practice" (Thompson,

1994).

Acquisition of knowledge is experiential in that it must

be actively constructed by the learner. Therefore, if

cognitive development is generated by successive interactions

with one's environment, then repetitive activity is necessary
to provide information that facilitates problem solving.

Consider the following task: Two equal-numerosity arrays in

one-to-one correspondence are presented. One is then
screened, and n objects are transferred from the visible to

the screened array. The objective is to predict how many

objects must be added to the visible array to make it equal
to the screened array. Children learn with repeated examples

that a transfer of n produces a difference of 2n (Piaget,

1974/1980; Piaget, Grize, Szeminska, & Vinh-Bang, 1977;
Cooper, Campbell, & Bevins, 1983). Performance on this task

highlights the role of repetitive experience because even
adults frequently err until they have at least one experience

with the task (Cooper, 1991). Robert Cooper's (1991) theory

of repetitive experiences provides a strong foundation for
reflective abstraction, and hence is a critical element for

the development of insightful thought.



Phases of Learning



According to Karplus (1977), conceptual learning

proceeds from an exploration stage to a concept

identification stage to an application stage where new ideas









are used and extended (Simon, 1992). The application stage
triggers a new level of exploration and the cycle
recommences. Karplus' learning cycle, derived from Piaget's

(1964) mental functioning model, provides a pedagogical

framework that complements a constructivist view of learning

(Fleener, 1995). Consequently, it provides a sound foundation

for the middle stage of each learning session with respect to

this research study.

Jurascheck (1983) suggests that learning cycle pedagogy
be applied to mathematics education. Simon (1992) presents a

framework utilized in two research projects for mathematics

teacher learning based on what is understood about students'

mathematics learning. The framework builds on Karplus'
Learning Cycle and identifies a learning cycle that

progresses through the following stages: exploration of a

mathematical situation, discussion leading to concept

identification, and application and extension of new ideas.

Also building on Piaget's work, Dienes (Dienes &

Golding, 1971) advocates a learning cycle in which learners
progress through a series of cyclic patterns, each comprising

activities ranging from concrete to symbolic format. The
earliest learning phase in each cycle begins with free play

to uncover the key features of the structured materials.

Following this, experiences are systematically structured to
facilitate discovery of inherent relations in the material

and in abstracting the concept being represented.
Dienes' learning cycle can be compared to the van
Hiele's phases of learning. Although much has been

communicated regarding the work of the van Hieles in terms of








18
a stratification of human thought (e.g. Senk, 1983), thought

levels comprise only one of the three main components of the

van Hiele model (Hoffer, 1983); the other components are

insight and phases of learning. The van Hieles propose a

sequence of five phases of learning: information, guided

orientation, explication, free orientation, and integration

(Clements & Battista, 1992).
Polya (1965), best known for articulating aspects of
problem solving, posits that any good teaching device must be
correlated somehow with the nature of the learning process

and, therefore, outlines some obvious features in the form of

three principles of learning: (1) the principle of active

learning, (2) the principle of best motivation, and (3) the

principle of consecutive phases. Specific to the third

principle, Three phases are distinguished: exploration,
formalization, and assimilation. Polya (1965) summarizes that

"For efficient learning, an exploratory phase should precede

the phase of verbalization and concept formation and,

eventually, the material learned should be merged in, and

contribute to, the integral mental attitude of the learner"

(p. 104).



The Erlanqer Propramm



In 1872, Felix Klein (1893) established a structure for
the analysis of geometric concepts, now referred to as the

Erlanger Programm, and presented his definition of a
geometry: "A geometry is the study of the properties that are









invariant when the subsets of a set S are mapped by the

transformations of some group of transformations" (Hall,

1973, p.95). Using Klein's definition of geometry, it is
possible to categorize and name various geometries according

to both the invariant properties under that group and the

geometries involved.
The transformations of topology are called
homomorphisms; a one-to-one transformation f is called a
homomorphism if it is continuous and reversibly continuous.
Shapes can be altered by stretching, compressing, bending,
and twisting, but not by tearing or joining (Mansfield,
1985). Projective geometry can be characterized as the study

of properties invariant under the group of collineations;

collineations are special homomorphisms which transform

collinear points into collinear points and, therefore, lines

into lines. Affine geometry is obtained as a subgeometry of
projective geometry by restricting the group of projective
transformations in such a way as to introduce parallelism and

distance. In particular, affine transformations map equal

distances into equal distances on the same or parallel lines,

and midpoints into midpoints.

Whereas an affine transformation multiplies distances in
the same direction by a constant, a similarity transformation

multiplies all distances by the same positive number k, the
ratio of the similarity transformation. Thus, the shape of a

figure is preserved but not its size. The Euclidean
transformation group is a subgroup of the similitude
transformation group as a Euclidean transformation preserves

distance mensurationn).









Similarity and Pronortional Reasoning



Fuson (1978) observed that most of Piaget's (1948/1956,
1960) analyses focus on topological concepts, or on Euclidean
concepts, leaving the projective, affine, and similitude

transformational groups relatively neglected. Nevertheless,
Piaget and Inhelder (1948/1956) describe four experiments
related to the concept of similarity: (i) drawing similar

triangles, (ii) sorting similar cardboard triangles, (iii)
choosing and drawing similar rectangles, and (iv) drawing a
similar configuration of line segments. These experiments
illustrate the gradual procurement of angle and parallelism
concepts. Piaget's categorization indicates that children
begin to use proportions at the age of eleven; perceptual

judgment of similarity is possible two years earlier (p.374).
The failure to perform well in similarity tasks at an earlier
age is attributed to functioning in pre-operational thought

and, later, the inability to use mental constructs (i.e.
proportions) involving second-order relations (Lunzer, 1968).

Freidlander's (1984) study of 675 suburban, middle-

class, midwestern, predominantly white students addressed
four similarity-related topics: (1) basic properties of

similar shapes, (2) proportional reasoning, (3) area
relationships of similar shapes, and (4) applications. Their
teachers had volunteered to teach the Similarity Unit--an
instructional unit developed by the Middle Grades Mathematics

Project (Lappan, Fitzgerald, Winter, & Phillips, 1986) for
grades six, seven and eight. Pre- and post-instructional









performance, at a significance level of 0.05, show

achievement increased as a function of grade level.

In a study involving 119 high school geometry students,
Chazan (1988) reported the results of an investigation into

high school students' understanding of similarity. A unit was
constructed for use with the Geometric Supposers (Schwartz &
Yerushalmy, 1985). Students were observed as they learned
similarity with this unit and were given protests and
posttests on fractions, ratio and proportion, and similarity.
Data from the study show that even those students who show an

understanding of multiplicative ratio may exhibit that
understanding with certain geometric configurations and not
with others. It is important to note that this study did not

compare students who used the Geometric Supposers (Schwartz &
Yerushalmy, 1985) with those who did not; instead this study

focused on students' understanding of aspects of similarity.

Fleener,et al (1993) investigated the following
questions: (1) Does knowing a standard algorithm for solving

proportion problems interfere with the development of

proportional reasoning? (2) Are stronger mathematics students

more flexible or intuitive than weaker students in applying

proportional reasoning strategies to solve problems? and (3)
What is the relationship between a student's general level of
reasoning ability (concrete, transitional, or formal) and the
strategies used for solving proportional reasoning tasks?
Sixteen ninth grade students engaged in proportional
reasoning tasks and computational proportional problem

solving in their science classes made a general positive gain

in Lawson's Classroom Test of Scientific Reasoning scores,









although gains for concrete learners were mixed. Explicit

teaching of the concept of ratios and student engagement in

exploratory studies of the relationships between and among
ratios provoked development of proportional reasoning for

average students. The researchers concluded that students can

benefit from experiences from which the cross-multiply-and-

divide algorithm can be derived.
The understandings involved in proportional reasoning as
embedded in similarity tasks are complex. According to Piaget
(1975), the essential feature of proportional reasoning is

the involvement of a complex quaternary relation, a
relationship between two relationships, rather than a simple
relationship between two directly perceivable quantities

(English, 1995). Tournaire and Pulos (1985) posit one should
expect this type of reasoning to develop slowly over several
years. Research rational numbers and multiplicative

structures suggests that many teachers do not have this

understanding (Cramer & Lesh, 1988; Harel & Behr, 1995;

Lacampagne, Post, Harel, & Behr, 1988).

The aforementioned studies provide a basis for the
discussion of the requirements for an appropriate
instructional environment to maximize the learning of the

concept of similarity and, in general, the development of

proportional reasoning. In view of the eclectic
epistemological framework presented by this researcher, a

thorough investigation of performance on similarity tasks
necessitates an instructional environment where exploration
and inquiry dominate the climate of the classroom.









Computer-based Geometric Construction Environment



Several studies (McCoy, 1991; Yerushalmy, 1991;

Yerushalmy & Chazan, 1990; Yerushalmy & Chazan, 1993) show

that, when used as intended, the Geometric Supposers

(Schwartz & Yerushalmy, 1985) create a powerful learning

environment: geometry lessons are transformed into active

explorations of geometric shapes resulting in stating,

testing, and proving one's own conjectures. Thus, the

conception of mathematics shifts from something the learners

encounter and observe to something they do and invent.

McCoy (1991) compared the geometry achievement of a

tenth grade class (n=29) which used the Geometric Supposers

(Schwartz & Yerushalmy, 1985) periodically during one school

year and a similar class (n=29) which did not use the

software. Analysis of Covariance was used to determine the

effect of the treatment on post geometry treatment scores. To

control for individual differences, the covariate was

mathematics scores from the pretest, the SRA Achievement

Test. The final examination provided by the publisher of the

adopted textbook was used as the posttest. The items were

examined and classified according to Bloom's Taxonomy (Bloom,

1956). Results revealed the experimental group had higher

scores on upper hierarchical (analysis, synthesis, and

evaluation) level problems and there was no statistically

significant difference on lower hierarchical (knowledge and

comprehension) level questions.









Forty-eight eighth graders who worked with the Geometric
Supposers (Schwartz & Yerushalmy, 1985) outperformed a

comparison group of ninety students on a test that measured

knowledge of basic geometry concepts even though the

comparison group was taught the same concepts and topics
during the same amount of time. Yerushalmy (1991) found that

the main difference between the groups was that the

experimental group did not exhibit some of the frequently
observed, persistent misconceptions such as having a
stereotyped image of certain geometric concepts and shapes.

Moreover, Yerushalmy & Chazan (1993) demonstrated that high

school students working with the Geometric Supposers
(Schwartz & Yerushalmy, 1985) seemed to have greater

flexibility in interpreting figures and diagrams; in fact,
the comparison group showed greater difficulty in overcoming

visual obstacles such as the inability to perceive a diagram

in different ways.
To explore which teaching strategies may adapt to a
computer environment that facilitates discovery learning and

encourages cooperative learning, Garcia-de Galindo (1994)

compared the teaching strategies and evaluation methods of

preservice high school geometry teachers observed in

traditional classes and classes implementing the Geometric

Supposers (Schwartz & Yerushalmy, 1985) and/or the Geometer's
Sketchpad (Jackiw, 1995). Comparative analysis was used to

categorize the teaching strategies, teaching styles,
evaluation methods, classroom interactions, and computer uses

of four selected teachers.









The main results were that the preferred teaching

strategies and evaluation methods of the Intuitive-Feeling,

Intuitive-Thinking, and Sensing-Feeling teachers may adapt to

computer environments that encourage discovery and

cooperative learning. The Sensing-Thinking teacher may prefer

the model of teacher-centered instruction when teaching high

school geometry with computers.

The four teachers reported plans to use computers in

different ways. The Intuitive-Feeling teacher related plans

to do calculations; the Intuitive-Thinking teacher, to

generate conjectures; the Sensing-Thinking teacher, to teach

the curriculum; and the Sensing-Feeling teacher, to review

basic skills.

Developed for students to conjecture about Euclidean

geometry, geometric construction software provides an

effective means to collect a wealth of visual and numerical

data to analyze. By observing the relationships revealed by

the data, insightful thinking is exercised. This in turn

generates need for further exploration to collect necessary

data to test the conjectures and the learning cycle begins

anew.


Dynamic Geometric Construction Environment



In comparison to the geometric environment described

above, dynamic construction software affords the added power

of dynamic manipulation of the geometric objects for

efficient exploration of geometric phenomena. The

construction of, for example, a quadrilateral, thus








26
represents the mathematical object in its fullest form due to

its ability to be transformed into, in this case, another

quadrilateral. Thus, the dynamic capability aids in the

imagery process to see the invariance and acquire a richer

global view regarding necessary and sufficient conditions for

construction and analysis of the constructed objects.
Elchuck (1992) explored the effects of the dynamic

capability of the geometric tool software Geometer's

Sketchpad (Jackiw, 1995) on conjecture making ability. One

hundred fifty seven grade nine students were randomly

assigned to either of two treatment groups, based on

capabilities of the tool software. The dynamic group had
access to the full power of the Geometer's Sketchpad (Jackiw,

1995); the static group had access to all but the drag

capabilities of this software. All participants were
administered tests to determine their mathematics achievement

level, spatial visualization skill, locus of control, and van

Hiele level of geometric thought prior to engaging in an

instructional unit. Both treatment groups underwent identical

instruction in the same geometric content. The study

demonstrated that mathematics achievement is a statistically

significant predictor of conjecture making ability. In a

post-hoc regression analysis, the type of software (dynamic

versus static) was found to predict conjecture making ability
when the factor school was also included in the regression
model.

The purpose of Foletta's (1992) case study was to
describe the nature of four high school geometry students'

inquiry by observing how the students used Geometer's









Sketchpad (Jackiw, 1995) and by characterizing their small

group interactions. Factors contributing to the students'

inquiry included the role of the Geometer's Sketchpad

(Jackiw, 1995), the design of the investigations, and the

nature of peer interactions. Foletta (1992) posits that the

students adapt the Geometer's Sketchpad (Jackiw, 1995) as an

extension of the paper-and-pencil medium.

A posttest-only control-group quasi-experimental study
was conducted by Lester (1996) to address the problem of
improving achievement of geometric knowledge through

instructional use of the software program Geometer's
Sketchpad (Jackiw, 1995). An inductive reasoning approach was

the pedagogy of instruction. Forty-seven female high school

geometry students participated in the study. The Geometer's

Sketchpad (Jackiw, 1995) was the tool used by participants in
the experimental group; the control group used traditional

geometry tools: ruler, pencil, protractor, and compass. The

three dependent variables measured on a posttest were:

geometric knowledge, construction, and geometric

conjecturing. Descriptive findings for the dependent

variables geometric knowledge and construction were not

statistically significant at an alpha level of 0.05;

descriptive findings for the dependent variable geometric

conjectures were statistically significant at an alpha level
of 0.05. Results from the study indicated that students

learned geometry skills with greater efficiency and
understood geometry concepts at higher levels as a result of
creating and manipulating dynamic constructions of geometric

objects on the computer screen.









The relationships among 158 high school geometry

students' spatial visualization ability, mathematical
ability, and problem solving strategies with and without the
availability of the Geometer's Sketchpad (Jackiw, 1995) was
explored by Robinson (1994). Following instruction using the
Geometer's Sketchpad (Jackiw, 1995), participants were

randomly assigned to one of two groups: with or without

software access. The availability of the computer was not a
significant factor for performance on problems related to
mathematical locus. Teaching specific skills resulted in

similar strategies used by participants with and without
access to the Geometer's Sketchpad (Jackiw, 1995). Robinson
(1994) posited that strategies learned with the technology

transfer to paper and pencil situations.

Dixon (1995) investigated the effects of a dynamic
instructional environment, English proficiency, and
visualization level on the construction of concepts of rigid
motion transformations. Two hundred forty-one middle school

students were trained on use of the Geometer's Sketchpad

(Jackiw, 1995). The control group was taught using the
traditional textbook approach while the treatment group
worked in a Macintosh computer laboratory. After controlling
for initial differences, Dixon (1995) concluded that students
experiencing the dynamic instructional environment
significantly outperformed students experiencing a

traditional instructional environment on content measures of
rigid motion transformations, as well as on certain measures

of visualization at an alpha level of 0.01.









The purpose of Choi's (1996) study was to investigate

secondary school students' development of geometric thought

during instruction based on a van Hiele model and using

dynamic computer software as a tool. In particular, the

students' learning process was traced in relation to van
Hiele levels of geometric thought with geometric topics using

an interactive computer environment. The clinical interview
procedure was used. It was evidenced that the instruction
based on the van Hiele's five instructional phases was well

integrated with the use of the dynamic nature of the software
since all students showed extensive development of geometric

thought. Also, the use of the dynamic capabilities of the
geometric construction software, Geometer's Sketchpad
(Jackiw, 1995), was found to provide an advantage to students
because it facilitated the movement from symbol to signal and

then to implicatory character.
Research has also been conducted to investigate how
dynamic geometry learning environments qualitatively impact
psychological aspects of geometry learning. Yousef (1997)

studied the effect of using Geometer's Sketchpad (Jackiw,
1994) on high school students' attitude toward geometry. The

sample consisted of two groups with two classes in each
group. All classes involved in the study participated in
exploration activities. The exploration activities for the
students in the experimental group involved the use of the
Geometer's Sketchpad (Jackiw, 1995) and for the control group
involved paper-and-pencil work only. Measurement of attitude
toward geometry was conducted before and after the
implementation of the experiment for both groups. Results








30
indicated that the scores of the pretest and the posttest of

the students in the experimental group were significantly

different and there was a significant difference between the
gain in the scores of the control and experimental group.

Qualitative data were collected from two sources: observation

and interviews. The results supported the quantitative data.

Melzcarek (1996) focused on the effects of problem-

solving activities using dynamic geometry computer software
on readiness for self-directed learning. Attitude towards the

learning of mathematics, the use of computers, and their

mediating effects were also explored. Six high school classes

received the experimental treatment and visited the
mathematics computer laboratory once a week for a period of

six weeks to work on problem-solving activities that were

designed to be used with the Geometer's Sketchpad (Jackiw,

1995). One class served as the control group. The results of

this study indicated a positive relationship between the use

of dynamic geometry computer software and readiness for self-
directed learning through the mediating effects of attitude

towards dynamic geometry computer software.

A task-based clinical interview procedure was used by
Manouchehri (1994) to study the cognitive actions of two

preservice elementary teachers and their interactions with

the Geometer's Sketchpad (Jackiw, 1995) as they worked on
computer-based geometry explorations and problem solving

activities. A final interview session was conducted to
solicit the participants perspectives on their experiences

with the computer-based activities. In the process of active

engagement in exploring with the interactive computer








31
software, the participants moved from a naive intuitive mode

to an analytical mode of thinking. Some difficulties resulted

from subjects' lack of knowledge about how to explore, their

lack of reflection on their operations and findings, and

their inadequate content knowledge base. The collective

impact of these difficulties led the subjects to drawing

false conclusions and overgeneralizations. Both subjects

appreciated being in control of their own learning. They

articulated, however, that such experience was confusing and
frustrating at times.



Summary



The review of the literature presented in this chapter

commenced with a theoretical framework grounded in

constructivist epistemology. According to von Glasersfeld
(1995), new knowledge is constructed and reconstructed from

prior knowledge within an environment of active participation

as one strives to organize one's experiential world. A

discussion of what repetitive experience means to research is
included in addition to theories of instruction focusing on

conceptual development of mathematical ideas. A review of
geometric construction software, including the Geometric

Supposers (Schwartz & Yerushalmy, 1985) and/or the Geometer's

Sketchpad (Jackiw, 1995), revealed that it has been
considerably discussed, tested, and used. The incorporation
of a dynamic aspect to geometric construction software

warranted further discussion.








32
The research provided a strong foundation for
investigation into geometry learning within a dynamic
environment. The research also uncovered the need for further
examination of the impact of a dynamic geometry learning

environment on geometry instruction effectiveness.
Furthermore, the lack of research with preservice elementary
teachers warrants investigation. It is the synthesis of the

information gained from the review that provided the
foundation and justification for this study.














CHAPTER III
RESEARCH DESIGN AND METHODOLOGY


The study methodology is detailed in this chapter which
commences with the research objectives. The population

addressed by this study and the participants of this

particular research study are described next. This is

followed by a discussion of the materials and the measurement
instrument to be used. Then, a summary of the pilot sessions

is described. Finally, the design of the study and the

procedures to be undertaken are detailed.


Research Objectives


This research study was designed to investigate the
effect of the dynamic geometry learning environment

Geometer's Sketchpad (Jackiw, 1995) on preservice K-8

teachers' performance on similarity tasks. In light of the

given eclectic theoretical framework and the literature
review that reports the results of research on aspects of

geometry learning, task performance was identified as a core
category to analyze how preservice teachers respond to

learning in a dynamic environment. The hypotheses for the

study were generated from the questions posed earlier and an

inspection of pilot study data. In order to facilitate









quantitative statistical analyses, following each question is

the list of hypotheses pertaining to that question used for

analyses of data:

1. After controlling for initial performance on
similarity tasks, do preservice K-8 teachers who
work on similarity tasks in a dynamic geometry
learning environment exhibit significantly
different posttest performance than those who study
the concept of similarity in a paper-and-pencil
learning environment that employs traditional tools
such as ruler and protractor?

la. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment, course section, and pretest scores,
using pretest scores as a covariate.

lb. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
course section and pretest scores, using
pretest scores as a covariate.

ic. Hypoothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and pretest scores, using pretest
scores as a covariate.

id. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and course section, using pretest
scores as a covariate.

le. Hypothesis: There is no significant effect on
posttest scores for students grouped by course
section, using pretest scores as a covariate.

if. Hypothesis: There is no significant difference
on posttest scores for students grouped by
treatment, using pretest scores as a
covariate.

2. After controlling for prior knowledge as measured
by a standardized achievement test, do preservice
K-8 teachers who work on similarity tasks in a
dynamic geometry learning environment exhibit
significantly different posttest performance than
those who study the concept of similarity in a
paper-and-pencil learning environment that employs
traditional tools such as ruler and protractor?









2a. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment, course section, and SAT scores,
using SAT scores as a covariate.

2b. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
course section and SAT scores, using SAT
scores as a covariate.

2c. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and SAT scores, using SAT scores as
a covariate.

2d. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and course section, using SAT scores
as a covariate.

2e. Hypothesis: There is no significant effect on
posttest scores for students grouped by course
section, using SAT scores as a covariate.

2f. Hypothesis: There is no significant effect on
posttest scores for students grouped by
treatment, using SAT scores as a covariate.



Population and Sample



The research population consisted of preservice

elementary education teachers attending an American

university. The 52 participants of this study were College of

Education students enrolled in three sections of a

mathematics methods course during a spring semester. This

course, addressing methods for teaching elementary school

mathematics, is a requirement of the teacher preparation

program in elementary education at a major land-grant

university located in the Southeastern United States. The

participants were randomly assigned to two groups within each









section. One group worked in the dynamic geometry learning

environment; another group worked in the traditional paper

and pencil manner (refer to Definition of Terms).



Instructional Materials



The experiential-activity orientation of the Middle

Grades Mathematics Project [MGMP] was used with both
experimental and control groups. These materials follow

recommendations of the Piagetian research on geometrical

development and the instructional implications of the van

Hiele studies (Friedlander, 1984). Each activity utilizes an

instructional model consisting of three stages:
(1) launch introduces new concepts, clarifies
definitions, reviews old concepts, and issues some
challenge;

(2) exploration entails gathering data, looking for
patterns, making conjectures, or developing other
types of problem solving strategies;

(3) summary demonstrates ways to organize data,
discusses the used strategies, and refines these
strategies into efficient problem solving
techniques.

The launch and summary stages are conducted in a whole-class

mode to purposefully generate taken-as-shared knowledge

(Cobb, et al., 1992); while at the exploration phase,
individuals investigate the problem situation and follow
their own path of cognition in analysis of the given task.

The MGMP material presents the concept of similarity at
the van Hiele Level I, properties of shapes, and Level II,

relationships among properties of shapes, of geometric









development (Friedlander, 1984). Consideration of the Van

Hiele levels in the teaching of geometry in the United States
facilitates progress in the learning and understanding of
geometrical concepts (Burger & Shaughnessy, 1986; Carpenter

et al., 1981; Clements & Battista, 1992; Crowley, 1987; Fuys,

Geddes, & Tischler, 1988; Jaime & Gutierrez,1995).



Pilot Studies


In May, 1997, a pilot study was conducted using
undergraduate students as defined by the population sample.
The students participated in four ninety-minute sessions

utilizing the Geometer's Sketchpad (Jackiw, 1995), a dynamic

geometry software package. The first session was simply an
introduction to resources made available to students enrolled
in the class and did not constitute part of the treatment.

Table 3.1 in Appendix B summarizes the academic content of
the learning sessions conducted by the researcher.

As a result of the initial pilot, adjustments were made
regarding the structure of the study. The length of each

session was limited to a fifty-minute session and the number
of learning sessions decreased. Participants considered the
sessions too lengthy and the researcher noted some students
were off task after approximately one hour. In addition, the

"launch" phase for each learning session was viewed by
participants as too in-depth and in need of revision for use
with preservice teachers. Although appropriate for middle
school students, adjustments were made to reflect the
curricular needs of college students.









A second pilot study was conducted during the Fall of

1997. Undergraduate students enrolled in the mandatory

introductory mathematics methods course for the preservice

teacher education program participated in three fifty-minute

learning sessions according to random placement in one of two
groups as described above. Again, adjustments were made

regarding the structure of the study. Content was further

parametrized to linear measurement, therefore eliminating

introduction of the tangential topic of area growth. As a

consequence of the change in curriculum, the research

instrument was appropriately altered, with items addressing

area growth omitted. Additionally, items hypothesized to

address learning attributed to the dynamic component of the

geometric construction software were developed and placed

into the final version of the research instrument. Table 3.2

in Appendix B summarizes the academic content of the learning

sessions conducted by this researcher.



Instrumentation



Performance on the concept of similarity was measured by

an instrument developed by this researcher and based, with

permission, on items used by the Middle Grades Mathematics

Project [MGMP] (1986). The instrument, given in Appendix A,

consisted of 21 multiple choice items with up to five options

for each item. Items were scored by assigning a 1 for a

correct response and a 0 otherwise; no correction is made for

guessing. The total score is considered as a general

indicator of the level of task performance.









Content validity and reliability were considered. To

insure that the instrument reflected the content domain of

transformational geometry regarding the concept of

similarity, a panel of five experts analyzed the twenty-one

items to ascertain validity. The experts, three mathematics

educators and two research mathematicians, reviewed each
problem with consideration to mathematical content.
The instrument was field-tested in Spring, 1998, on

sixty-six preservice elementary teachers enrolled in two

sections the required elementary mathematics methods course

at an American university located in the southeast sector.

Measuring internal consistency of the test, the Cronbach
reliability coefficient calculated from the data is 0.70.

The coefficient of stability, calculated for test-retest

reliability, is 0.70.

Regarding the covariates used in this study, multiple

partial correlations were computed to assess to the strength

of association between the dependent variable and the

covariate while controlling for the other independent

variables treatment and section. The research instrument,

administered as the pretest, served as the covariate to

statistically analyze the hypotheses related to the first

question. For the second question, prior mathematical
knowledge, as measured by the mathematics section of the SAT,

served to adjust for initial random differences in the two

treatment groups. The multiple partial correlation with
respect to the covariate SAT was 0.4281 for this study and,

concerning the covariate pretest, 0.5925 was calculated.









Thus, statistical analyses supported confidence for the

utilization of the specified variables.



Procedure and Data Collection



The procedure consisted of a pretest, three learning
sessions, and a posttest. The pretest was administered eight

weeks prior to the learning sessions. The sessions for the

experimental group were held in a computer laboratory on

campus and involved use of the dynamic geometry software

Geometer's Sketchpad (Jackiw, 1995); sessions for the control
group were held in the traditional mathematics laboratory and

involved use of manipulatives, protractor, and ruler. Written

reflections were collected from all participants at the end

of each learning session in order to gain further insight

regarding the depth and breadth of the learning that had

occurred. During the posttest, administered immediately upon

completion of the research study learning sessions,
participants were given access to whichever tools they had

used during the course of the research study.

All components of the study were administered by this

researcher during scheduled meeting times. The researcher

conducted all lessons to eliminate differences in results due

to instructor variability, but was aware of the possibility

that a threat to validity may be introduced by the researcher

influencing results. This was minimized by the use of

specific lesson plans including printed activities for the

participants of both groups. The contents of the learning








41
sessions are given in Table 3.3 of Appendix B with exemplary

printed activities, showing student work, presented in

Appendix C.

Prior to the study the Institutional Review Board of the

given University granted permission for the investigation to

take place. Students were informed; both researcher as

principal investigator and those who agreed to participate in

the study signed the consent form. An addendum was proposed

and permission granted to access participants SAT scores. In

a similar manner, students were informed and the parties

involved gave written approval. The letters of informed

consent are given in Appendix D.



Statistical Design of the Study



The effect of environmental intervention was examined by

use of a pretest-posttest control-group design with random

assignment. The Borg and Gall (1989) configuration is shown

as

R 0 X 0

RO 0 0
where X represents the experimental treatment, 0 represents

pretest or posttest measurement of the dependent variable,

and R indicates that the experimental and control groups were

formed randomly. The randomization procedure occurred within

each course section.

A literature review of studies focusing on aspects of

geometry learning suggested use of several statistical









procedures (Dixon, 1996; Fleener, et al, 1993; Friedlander,

1984; Lester, 1996; McCoy, 1991; Melzcarek, 1996; Robinson,

1994; Yousef, 1997). To analyze the data collected during the

study, descriptive statistics were obtained for all variables

under consideration and analysis of covariance [ANCOVA] was

conducted. The ANCOVA allows one to test for interactions; in

particular, the interaction of the treatment and course

sections is of particular interest to this study.
Furthermore, ANCOVA does adjust for random differences in the
two groups. In addition to quantitative statistical analyses,

reflections collected in written form were examined to

further elucidate the geometry learning experience.



Summary



This chapter commenced with the research objectives.
Information regarding the population, research sample, and

materials followed. Next, the instrument used in this

particular study was described and the pilots were ,presented.

This chapter ended with the design of the study which

included a description of the procedures utilized in the

study.














CHAPTER IV
RESULTS


This chapter contains the results of the analyses

described in the previous chapter. Descriptive statistics for

the variables under investigation and ANCOVAs are presented

along with the results of the qualitative component detailing

the geometry learning experience. The chapter begins with

descriptions of the research instrument, research objectives,

and the goals of the study. Next, descriptive statistics are

presented and the research hypotheses that reflect the

research questions are articulated. Then, the analyses of the

data are given in light of the hypotheses. Finally, an

exposition of written reflections is presented to elucidate

the geometry learning experience.



The Research Instrument


The research instrument measured performance on

similarity tasks in a multiple choice format with up to five

options for each of the twenty-one items. Appendix A contains

the research instrument. Items were scored by assigning a 1

for a correct response and a 0 otherwise; no correction was

made for guessing. The total score was considered as a

general indicator of the level of task performance.









Research Objectives



To test for significant difference in task performance

between participants grouped by learning environment,

questions were developed to explore the effect of a dynamic

geometry learning environment on preservice elementary

teachers' performance on similarity tasks. Hypotheses were

formulated to identify significant interactions between

various independent variables and to adjust for initial

performance on similarity tasks as measured by the pretest or

by prior mathematics knowledge as measured by a standardized

achievement test.

The comparison of scores on the research instrument that

assessed performance on similarity tasks, administered prior

to and at the end of the study, served as the basis for the

analysis of the first question. The second question focused

upon the effect of a dynamic geometry learning environment on

preservice elementary teachers' performance on similarity

tasks with prior mathematics performance adjusted for by a

standardized achievement test. Hypotheses pertaining to each

question allowed for statistical testing. The results of the

analyses are reported at an alpha level of 0.05 in all cases.



Goals of the Study



Based upon the assumptions that teachers need to learn

high quality mathematics and to be taught in a manner similar

to the way they are expected to teach, two questions of









interest were generated for this particular study, focusing

upon the effect of the dynamic geometry learning environment

on preservice elementary teachers' performance on similarity

tasks as measured by the research instrument:
1. After controlling for initial performance on
similarity tasks, do preservice elementary teachers
who work on similarity tasks in a dynamic geometry
learning environment exhibit significantly different
posttest performance than those who study the concept
of similarity in a paper-and-pencil learning
environment that employs traditional tools such as
ruler and protractor?

2. After controlling for prior knowledge as measured by
a standardized achievement test, do preservice
elementary teachers who work on similarity tasks in a
dynamic geometry learning environment exhibit
significantly different posttest performance than
those who study the concept of similarity in a paper-
and-pencil learning environment that employs
traditional tools such as ruler and protractor?

Thus, the goals of the study included answering the research

questions in light of the given assumptions.



Descriptive Statistics



For the sake of completeness, the data set for the

sample population is given in Appendix E, sorted first by

treatment and then by gain score. This inclusion is to afford

the reader the option to compare the relative performance of

the participants on the posttest. Descriptive statistics for
responses to the pretest, SAT mathematics component, and

posttest for the entire sample, by group, and by group and

course section, are presented in Tables 1, 2, and 3,

respectively. The sample size, mean, and standard deviation

are listed for each response category. The posttest mean








46
scores, adjusted for the covariate, were computed as 14.3699

for the control group and 16.2568 for the experimental group.




Table 1
Descriptive Statistics / Combined


Number Mean Standard Deviation


Pretest 52 12.2885 3.4376
SAT 48 568.3333 80.2213
Posttest 52 15.5769 3.0120





Table 2
Descriptive Statistics / by Group


Group Number Mean Standard Deviation


Pretest control 26 12.2308 3.2287
experimental 26 12.3462 3.6980
SAT control 24 580.0000 73.0098
experimental 24 556.6667 86.8115
Posttest control 26 14.6538 2.6542
experimental 26 16.5000 3.1145


Research Hypotheses


The following hypotheses were generated to test for

significant effect in mathematics achievement regarding

preservice teachers' performance on similarity tasks as

measured by the research instrument. For the first research









Table 3
Descriptive Statistics / by Group and Section


Standard
Section Group Number Mean Deviation


Pretest


SAT








Posttest


a control
experimental

b control
experimental

m control
experimental

a control
experimental

b control
experimental

m control
experimental

a control
experimental

b control
experimental

m control
experimental


10.0000
10.8000

13.8182
13.1667

12.0000
12.1111

585.0000
556.0000

597.0000
563.6363

555.0000
547.5000

13.7143
14.0000

15.0909
17.2500

14.8750
16.4444


question, initial differences in mathematics achievement were

controlled for through the use of the research instrument

administered as a pretest. The latter set of hypotheses

addressed the second question and controlled for initial

differences in mathematics achievement via the mathematics

component of the standardized achievement test SAT.

la. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment, course section, and pretest scores,
using pretest scores as the covariate.


3.7417
4.3818

2.5226
3.7376

2.6726
3.3706

55.0454
99.9000

75.1369
87.4383

83.3238
89.2428

1.4960
3.5637

2.4680
2.7010

3.6425
3.3582









lb. Hvypothesis: There is no significant effect on
posttest scores due to the interaction of course
section and pretest scores, using pretest scores as
the covariate.

ic. Hypothesis: There is no significant effect on
posttest scores due to the interaction of treatment
and pretest scores, using pretest scores as the
covariate.

id. Hypothesis: There is no significant effect on
posttest scores due to the interaction of treatment
and course section, using pretest scores as the
covariate.

le. Hypothesis: There is no significant effect on
posttest scores for students grouped by course
section, using pretest scores as the covariate.

if. Hypothesis: There is no significant effect on
posttest scores for students grouped by treatment,
using pretest scores as the covariate.

2a. Hypothesis: There is no significant effect on
posttest scores due to interaction effect of
treatment, course section, and SAT scores, using
SAT scores as the covariate.

2b. Hvypothesis: There is no significant effect on
posttest scores due to the interaction of course
section and SAT scores, using SAT scores as the
covariate.

2c. Hypothesis: There is no significant effect on
posttest scores due to the interaction of treatment
and SAT scores, using SAT scores as the covariate.
2d. Hypothesis: There is no significant effect on
posttest scores due to the interaction of treatment
and course section, using SAT scores as the
covariate.

2e. Hypothesis: There is no significant effect on
posttest scores for students grouped by course
section, using SAT scores as the covariate.

2f. Hypothesis: There is no significant effect on
posttest scores for students grouped by treatment,
using SAT scores as the covariate.









Results of Data Analyses



The initial analysis of covariance, controlling for

pretest performance, yielded a p-value greater than the set

alpha level of p=0.05 and the null hypothesis was accepted

(Refer to Table 4). That is, no significant effect on

posttest scores due to the interaction of treatment (trt),

course section (scn) and pretest scores (pre), after

controlling for pretest performance, was found (p=0.1017).

Therefore, the following null hypothesis could not be

rejected:

Using pretest scores as the covariate, there is no
significant effect on posttest scores due to the
interaction of treatment, course section and pretest
scores.

Finding no significant three-way interaction, the three-

way interaction term was removed from the model. After
removal of the three-way interaction term, hypotheses lb-ld

were then tested. There were no statistically significant

two-way interactions based on the results of the ANCOVAs. The

p-values for hypotheses Ib-ld were 0.2939, 0.5337, and

0.7871, respectively. Therefore, the following null

hypotheses could not be rejected:

Using pretest scores as the covariate, there is no
significant effect on posttest scores due to the
interaction of course section and pretest scores.

Using pretest scores as the covariate, there is no
significant effect on posttest scores due to the
interaction of treatment and pretest scores.

Using pretest scores as the covariate, there is no
significant effect on posttest scores due to the
interaction of treatment and course section.









Table 4
Initial Analysis Of Covariance
with Pretest as Covariate


Dependent Variable: Posttest

Source DF SS MS F Value Pr > F

trt 1 0.3639 0.3639 0.07 0.7935
pre 1 134.4845 134.4845 25.66 0.0001
scn 2 19.1402 9.5701 1.83 0.1742
pre*trt 1 0.7196 0.7196 0.14 0.7129
trt*scn 2 24.0913 12.0457 2.30 0.1135
pre*scn 2 23.4062 11.7031 2.23 0.1204
pre*trt*scn 2 25.3796 12.6898 2.42 0.1017


Upon finding no significant two-way interactions, the

interaction terms trt*pre and pre*scn were removed from the

model. The term trt*scn was left in the model to guarantee

control of the blocking variable course section. The ANCOVA

information is presented in Table 5. After removal of the

interaction terms, course section showed no statistical

significance (p=0.8332) but treatment was statistically

significant at the alpha level of 0.05 (p=0.0275). Thus, the

following null hypothesis to the initial research question

could not be rejected:

Using pretest scores as the covariate, there is no
significant effect on posttest scores for students
grouped by course section.

However, the following null hypothesis could be rejected:

Using pretest scores as the covariate, there is no
significant effect on posttest scores for students
grouped by treatment.

The analyses relevant to the first question indicated a

statistically significant difference in performance on

similarity tasks between the experimental group and the









Table 5
Analysis Of Covariance
Final Model with Pretest as Covariate


Dependent Variable: Posttest

Source DF SS MS F Value Pr > F

trt 1 29.0884 29.0884 15.19 0.0275*
pre 1 136.3718 136.3718 24.23 0.0001
scn 2 2.0534 1.0262 0.18 0.8332
trt*scn 2 6.7176 3.3588 0.60 0.5534

*statistically significant at alpha level 0.05


control group. LSMeans, with respect to use of the pretest as

covariate, was calculated at 14.74 for the control group and

16.30 for the experimental group. The remaining hypotheses,

addressing the second question, were then analyzed. Again, no

significant difference on posttest scores due to interaction

effect of treatment (trt), course section (scn) and SAT

scores (sat) was found (p=0.7926) after controlling for prior

mathematics performance. That is, the analysis of covariance,

controlling for prior mathematics performance, yielded a p-

value greater than the set alpha level of p<0.05 and the null

hypothesis was accepted (Refer to Table 6). Therefore, the

following null hypothesis could not be rejected:


Using SAT scores as the covariate, there is no
significant effect on posttest scores due to the
interaction effect of treatment, course section and SAT
scores.

The three-way interaction term was removed from the
model upon finding no significant three-way interaction.

After removal of the three-way interaction term, hypotheses

2b-2d were then tested. There were no statistically









Table 6
Initial Analysis Of Covariance
with SAT as Covariate


Dependent Variable: Posttest

Source DF SS MS F Value Pr > F

trt 1 1.5549 1.5549 0.19 0.6681
sat 1 62.8058 62.8058 7.59 0.0091
scn 2 0.9755 0.4877 0.06 0.9428
sat*scn 1 0.2838 0.1419 0.02 0.9830
trt*scn 2 3.2057 1.6028 0.19 0.8247
sat*trt 2 0.2638 0.2638 0.03 0.8592
sat*trt*scn 2 3.8684 1.9342 0.23 0.7926


significant two-way interactions based on the results of the

ANCOVAs. The p-values for hypotheses 2b-2d were 0.9550,

0.7317, and 0.8297, respectively. Therefore, the following

null hypotheses could not be rejected:

Using SAT scores as the covariate, there is no
significant effect on posttest scores due to the
interaction of course section and SAT scores.

Using SAT scores as the covariate, there is no
significant effect on posttest scores due to the
interaction of treatment and SAT scores.

Using SAT scores as the covariate, there is no
significant effect on posttest scores due to the
interaction effect of treatment and course section.

The interaction terms trt*sat and sat*scn were removed from

the model upon finding no significant two-way interactions.

The term trt*scn was left in the model to guarantee control

of the blocking variable course section. After removal of the

interaction terms, course section continued to show no

statistical significance (p=0.2134). However, after

controlling for initial differences on SAT mathematics

scores, a statistically significant difference in means









existed at the alpha level of 0.05 between treatment groups

(p=0.0271). The ANCOVA information is given in Table 7.

LSMeans, with respect to use of the mathematics component of

the SAT as covariate, was calculated at 14.37 for the control

group and 16.26 for the experimental group.


Table 7
Analysis Of Covariance
Final Model with SAT as Covariate


Dependent Variable: Posttest

Source DF SS MS F Value Pr > F


trt 1 38.8572 38.8572 5.26 0.0
sat 1 79.2114 79.2114 10.71 0.0
scn 2 23.7177 11.8588 1.60 0.2
trt*scn 2 2.9500 1.4750 0.20 0.8

*statistically significant at alpha level 0.05

Thus, the following null hypothesis to the second research

question could not be rejected:

Using SAT scores as the covariate, there is no
significant effect on posttest scores for students
grouped by course section.

However, the following null hypothesis could be rejected:

Using SAT scores as the covariate, there is no
significant effect on posttest scores for students
grouped by treatment.


271*
022
134
199


h


A Reflections Component


Written reflections on the learning experience,

collected from the participants at the end of each learning

session, afforded the researcher valuable insights into the

preservice elementary teachers' perception of what they had









learned and the value of this knowledge in their teaching.

During each session, the 52 participants logged responses on

activity worksheets that prompted for description of the

thought processes. After each session, participants were

verbally asked to write what they had learned mathematically,

their reaction to the methodology, and their views as to use

in the mathematics classroom.

The data gathered from responses on the activity
worksheets used in this study were collected to illuminate

the cognitive process occurring within each individual during

active participation. Examples are presented in Appendix C.

The purpose of the collecting of reflections was to gain

further insight by providing opportunities for a richer story

to emerge and for reflective meaning-making to take place.

Critiquing of the written reflections granted the
researcher an opportunity to gain knowledge regarding the

preservice elementary teachers' perspective of the

experience. The participants shared their opinions and

knowledge about the mathematics content, the instruction, and

the technology, in addition to the learning. Affect, as well

as performance, seemed to have improved due to the

correspondence from the preservice teacher to the researcher.

The following excerpts, taken from the written
reflections collected at the end of each session during the

course of the research study, exemplify the effect of the

dynamic geometry learning environment upon cognitive

development. Both treatment groups participated and are

represented.









Experimental group preservice elementary teacher #1:

Day 1: "Today I learned it is easier to see this idea

than if I had to draw inaccurate polygons each time to

compare them. I also learned how the buttons work and how to

make measurements and calculations. I would like to know how

to make calculations such as a2+b2=c2 to prove what I am doing

on the computer. Very cool program!"

Day 2: "I liked today's activity because the student is
guided through a process but the discovery is left to them!

It asks the right questions. It really provides the basis for

pre-geometry study. I don't know if the student will walk

away from this exercise with a grasp on proportion geometry,

but at least it gets them thinking in the right mode."


Control group preservice elementary teacher #2:

Day 1: "We learned about projective geometry mostly

through the use of an instrument made up of a pencil and

rubber bands. The activity was unique to say the least! The

most important thing that I learned from this activity was

the value of practice. I was able to participate in the

activity as a novice. I had no idea what I was doing and I

learned everything as I went along. As a future teacher, this

is a valuable lesson and will help me better understand my

students' learning."

Experimental group preservice elementary teacher #3:

Day 1: "Today I did not LEARN a whole lot. I did,

however, discover quite a bit!! I did not have enough time,

nor enough experience to feel as if I learned. Perhaps

further practice will allow me to learn next time!"








56
Day 2: "Today I became comfortable with the commands on

the computer!! I think because this is such a new concept, I

am enjoying 'playing around' with it! Again, I AM fairly

comfortable w/ dilation, rotation and transformation as we

are learning this in geometry class now!"
Day 3: "I said I was not learning because I already know
this stuff. But I said I discovered because I could make a

conjecture and immediately see the outcome. I'm not very good
on computers but I see that this tool will be great for use

w/ my kids. Being able to see dimensions change is awesome.

Gives one the big picture--quickly and accurately."


Experimental group preservice elementary teacher #4:

Day 3: "I learned a lot because you gave us a different
way to look at the concept. I could pull together old stuff
and ideas and make connections. The dynamics made a

difference, like the old way cartoons were made, it made

things possible to see in a short amount of time."


Control group preservice elementary teacher #5:

Day 3: "I learned I could fail a test and then pass it

with a bit of practice. It was neat to see that the first
test actually had some background to it. When I first took it
I thought it really was a bunch of made-up information. After

the couple of days, I felt like I could ace the test."


The reflections portray the preservice elementary
teachers' perspective regarding the learning experience. It

was interesting to note that, in both groups, many








57
participants regarded the active hands-on approach a form of

practice that was enjoyable in addition to being productive.

Furthermore, connections were established between previous

knowledge and new ideas so that insightful revelations (ah-

ha!) occurred; this was more often evidenced during learning
sessions with the experimental group. For example, teacher #3
quickly mastered the basic skills necessary to explore

Euclidean constructions and was quite enthusiastic regarding
his explorations. He would "conjecture and immediately see
the outcome," evaluate the situation, and repeat the cycle.

Regarding affective aspects, teacher #5 was quite
skeptical regarding participation in mathematical activity.

As a prospective teacher, she believed the course requirement

a waste of time since she intended to teach only in the
primary grades. However, by the end of the study (near the

end of the semester) her participation had increased
dramatically; teacher #5 had constructed a firm foundation of
elementary mathematics that was deemed personally relevant.
In particular, she had developed an informal, intuitive

understanding of the concept of mathematical similarity by
way of a novel approach. Then, teacher #5 reconstructed her

knowledge base regarding the concept to conclude that she had

become mathematically empowered!

Of the participants in the experimental group, most
became comfortable with the software during the launch phase
when participants were given time to experiment with the
tool. Teacher #1 expressed great interest in the capabilities

of the Geometer's Sketchpad (Jackiw, 1995), asking for

assistance to develop a geometric proof of the Pythagorean









Theorem. Able to transcend physical limitations of manual

dexterity, she explored the capabilities of the dynamic

geometry software in order to verify the theorem. The

experimental group participants, for the most part, were

quite impressed by the dynamic capability of the tool. The

participants in the control group, exhibiting enthusiasm in a

similar manner, reported plans to share manipulatives and

techniques with their future students and colleagues.



Summary


Results of the statistical analyses for the study have
been described in detail in this chapter in addition to
presentation of a sampling of collected reflections that

exemplify certain perceptions and themes held by the research

participants. Discussion of the findings, including
implications and recommendations for further research, are

presented in the concluding chapter.














CHAPTER V
SUMMARY AND CONCLUSIONS

This final chapter discusses the results presented
in the previous chapter. An overview of the study is provided
first, including a description of the sample population.

Next, the significance of the study is articulated, followed

by the research questions. Finally, the results and a

discussion of the conclusions based upon the research results
is presented. Implications from the present study are given

next. The chapter concludes with recommendations for possible

future research.


Overview of the Study


The purpose of the study was to explore the role of the
learning environment with respect to the mathematical

evolution of preservice elementary teachers as a result of

active inquiry using the dynamic geometric construction
software Geometer's Sketchpad (Jackiw, 1995). That is, this

study was designed to investigate the effect of a dynamic

geometry learning environment on performance in the context
of similarity tasks. The basic goal of the study was to

detect the extent to which the dynamic environment, given its

ability to efficiently display a plethora of mathematical

situations, assists learning.








60
Complete data were available on 52 preservice elementary

teachers enrolled in three sections of an undergraduate

elementary mathematics methods course in the College of

Education at a major land-grant university located in the

Southeastern United States. The design of the study was a

pretest-posttest control group design with random assignment.

Detail on the design, procedures, and methodology were
presented in Chapter III. The randomization procedure

occurred within each course section. The procedure consisted

of the pretest, three learning sessions, and the posttest.
Sessions for the control group were held in the traditional

mathematics laboratory and involved use of manipulatives,
ruler, and protractor; the sessions for the experimental
group were held in a computer laboratory on campus and

involved use of the dynamic geometric construction software

Geometer's Sketchpad (Jackiw, 1995).

To analyze the data collected during the study,
descriptive statistics were obtained for all variables under

consideration and analysis of covariance [ANCOVA] was

conducted. The research instrument, given to all participants

eight weeks prior to the instructional sessions, served as
the covariate to statistically analyze the hypotheses related

to the first question. For the second question, prior

mathematical knowledge, as measured by a standardized
achievement test, served to adjust for initial differences in

the two treatment groups.

In addition to the quantitative statistical component,

written reflections were collected from all participants at

the end of each session. The purpose of this exercise was to









gain further insight regarding the preservice elementary

teachers' perspective of the instructional environment and to
ascertain what had been learned during the session.



Significance of the Study



Implementing new technologies into the mathematics
curriculum is not easy and the education of teachers plays a
critical role. The necessity of requiring preservice teachers

to learn mathematics content and methods of teaching in an

environment where computer use is an integral component is
advocated by many mathematics teacher educators (Committee on
the Mathematical Education of Teachers, 1991; Curcio, Perez,

& Stewart, 1994; Leitzel, 1991; McNerney, 1994; National

Council of Teachers of Mathematics, 1991) It is, therefore,
imperative to explore how software will modify current ways

of teaching and learning in order to define new roles for
teachers in a technologically rich classroom environment.



Research Questions



Based on the assumptions that teachers need to learn
high quality mathematics and to be taught in a manner similar
to the way they are expected to teach, the specific questions
under investigation were:
1. After controlling for initial performance on
similarity tasks, do preservice elementary
teachers who work on similarity tasks in a
dynamic geometry learning environment exhibit
significantly different posttest performance









than those who study the concept of similarity
in a paper-and-pencil learning environment that
employs traditional tools such as ruler and
protractor?

2. After controlling for prior knowledge as
measured by a standardized achievement test, do
preservice elementary teachers who work on
similarity tasks in a dynamic geometry learning
environment exhibit significantly different
posttest performance than those who study the
concept of similarity in a paper-and-pencil
learning environment that employs traditional
tools such as ruler and protractor?



Results



The hypotheses that were statistically tested, with

consideration of the aforementioned two questions, are listed

below. Following each set of hypotheses, the results of the

analysis are presented for consideration.

la. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment, course section, and pretest scores,
using pretest scores as a covariate.

lb. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
course section and pretest scores, using
pretest scores as a covariate.
ic. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and pretest scores, using pretest
scores as a covariate.

Id. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and course section, using pretest
scores as a covariate.

le. Hypothesis: There is no significant effect on
posttest scores for students grouped by course
section, using pretest scores as a covariate.









if. Hvpothesis: There is no significant effect on
posttest scores for students grouped by
treatment, using pretest scores as a
covariate.

To investigate whether the dynamic geometry
learning environment affects performance on similarity

tasks as measured by the research instrument, ANCOVAs

were conducted with achievement on the research

instrument as a covariate. Initially, ANCOVAs were

applied to assure that interaction effects were not

present and to develop the final model with the

following terms as independent variables: pretest,

treatment, course section, and the interaction of

treatment with course section. The term treatment with

course section was left in the model to guarantee

control of the blocking variable course section.

The initial analyses indicated that there was no
statistically significant effect on posttest scores due

to either three-way or two-way interaction of the

independent variables. Therefore, with the exclusion of

the interaction term of treatment with course section,

the interaction terms were removed.

An ANCOVA was conducted using the final model.

Based on the analysis, there exists supportive evidence

that the difference in learning environment between the

two treatment groups had a significant effect upon the

preservice elementary teachers' performance on

similarity tasks as measured by the research instrument.

Fundamentally, the participants in the experimental

group outperformed the participants in the control group









even though initial variability was taken into

consideration.

Let us now focus upon the hypotheses generated to

address the latter question.

2a. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment, course section, and SAT scores,
using SAT scores as a covariate.

2b. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
course section and SAT scores, using SAT
scores as a covariate.

2c. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and SAT scores, using SAT scores as
a covariate.

2d. Hypothesis: There is no significant effect on
posttest scores due to the interaction of
treatment and course section, using SAT scores
as a covariate.

2e. Hypothesis: There is no significant effect on
posttest scores for students grouped by course
section, using SAT scores as a covariate.

2f. Hypothesis: There is no significant effect on
posttest scores for students grouped by
treatment, using SAT scores as a covariate.


In a similar fashion, to further investigate

whether the dynamic geometry learning environment

affects performance on similarity tasks as measured by

the research instrument, ANCOVAs were again conducted.

For this set of hypotheses, prior knowledge, as measured

by SAT mathematics scores, served as the covariate. Once

more, ANCOVAs were applied to assure that interaction

effects were not present and to develop the final model.

There was no statistically significant effect on









posttest scores due to either three-way or two-way

interaction of the independent variables.

With the following independent variables as terms:

SAT, treatment, course section, and the interaction of

treatment with course section, a final ANCOVA was

conducted. It was concluded that the participants in the
experimental group outperformed the participants in the

control group when consideration was given to prior
knowledge as measured by SAT mathematics scores.

This study focused on the effect of a dynamic
geometry learning environment on preservice elementary

teachers' performance on similarity tasks as measured by

the research instrument. One can be fairly confident

that this effect is attributable to the dynamic learning
environment for three reasons: classes were randomly

assigned to the experimental and control groups, the

observed differences between groups on posttest

performance were statistically significant, and

extraneous variables were effectively controlled.



Discussion of the Results



Constructivism fully agrees with Kuhn's idea
of paradigm shifts (Kuhn, 1970), interpreting
paradigm shifts as "change in the habitual way
of constructing." The more often a construct
is repeated and the greater the number of
larger structures in which it is involved, the
more indispensable it becomes and the more
"given" or "objective" it seems (von
Glasersfeld, 1979, p. 121).








66
The researcher concluded from the data analyses that the

results support the dynamic geometry learning environment as
a viable enhancement to the traditional mathematics

laboratory approach. This study focused on the impact of the

dynamic geometric construction software Geometer's Sketchpad
(Jackiw, 1995) upon preservice elementary teachers'
performance on similarity tasks.

Human beings acquire knowledge by building it from the
inside instead of internalizing it directly from the
environment (Kamii, 1990). Thus, current research from a

constructivist perspective characterizes mathematical
learning as a process of conceptual reorganization (Cobb,
1995). If learning is generated by successive interactions

with one's environment, then repetitive activity is necessary
to provide information that facilitates problem solving
(Cooper, 1991).
Polya (1954, 1957, 1965) advised, when faced with a
problem, to look at examples and special cases, to look at
analogous situations for hints and, whenever possible, to use

specialization and generalization. Polya (1965) claimed that
problem solving is a practical art, like playing the piano.

Such arts can only be learned by imitation and practice.
This type of software allows for direct construction,
manipulation, and measurement of geometric figures. Users are
actively involved in the learning process; the software
supports efficiency in that it allows for dynamic
manipulation of the geometric figures in order to analyze
more cases in a minimal amount of time spent. As in Lester's

(1996) and Yerushalmy's and Chazan's (1993) investigations of









high school students, results from the study indicated

participants learned with greater efficiency as a consequence

of constructing geometric figures on the computer screen.

Graphic support seemed to enhance participants ability to

visualize mathematical relationships and connect geometric
properties to facilitate inquiry.

Calculation of posttest group means presented the

researcher with additional statistics worthy of
consideration. It had been previously mentioned in Chapter
III that the research instrument (refer to Appendix A) was

adjusted in lieu of findings from the pilot studies. Items

hypothesized to address learning attributed to the dynamic

component of the geometric construction software were

developed, reviewed by a panel of experts, and placed into

the final version of the research instrument. Thus, it was

deemed appropriate to investigate individual items. A most

interesting phenomena was that posttest means on parallel

items, items #1 and #12, hypothesized to address learning

attributed to the dynamic component of the geometric

construction software, were at least 0.25 points greater for

the experimental group.

This study suggests that preservice elementary teachers

are able to progress quite well with the use of the

Geometer's Sketchpad (Jackiw, 1995) through the guess-
investigate-conjecture-verify process inherent to the

formalization of the "phases of learning" (Karplus, 1977;
Simon, 1992) and inquiry that are supported by constructivist

epistemology. Polya (1965) summarizes that "for efficient
learning, an exploratory phase should proceed the phase of









verbalization and concept formation and, eventually, the

material learned should be merged in, and contribute to, the

integral mental attitude of the learner" (p. 104).




Other Observations


In addition to the rightful inspection into quantitative
aspects of this study, consideration must also be given to
qualitative aspects intrinsically rich in interpretive value.

The activity worksheets (refer to Appendix C) not only gave
structure to the sessions; participants responses throughout
the guided discovery process provided insight into their
construction and reconstruction of knowledge.

During the first session, participants in either group
could be observed struggling with the novel approach to

understanding a concept many believed they had learned in
middle school. Participants in the control group slowly drew

similar geometric figures using the Variable Tension

Proportional Divider (Friedlander & Lappan, 1987) to develop

an intuitive notion of similarity whereas the participants in
the experimental group used the dynamic capability of the

Geometer's Sketchpad (Jackiw, 1995) to investigate basic
principles of similar figures. By the end of the first
session, the experimental group had measured and analyzed

many different polygonal figures while the control group,
although they had discovered the principles of similarity,
were unsure of whether non-convex polygonal figures would

observe the principles. The control group did, however,








69
convey an enthusiasm to use the Variable Tension Proportional

Divider (Friedlander & Lappan, 1987) with students as often

as the experimental group voiced opinions regarding the

usefulness of the Geometer's Sketchpad (Jackiw, 1995) as a

classroom tool.

Much information can be gained through analysis of
activity worksheets and clinical reflections. The qualitative

component of this study offers a mere sampling of the

preservice elementary teachers' cognitive processes in

action.



Limitations



This study had limitations that affect the
generalizability of the results. Since the study lasted over

a period of several days, variability of attendance was a

problem. Absences were taken into account for the testing

days, but adjustment could not be made to compensate for

instructional time lost.

Concern was present that the small sample size of 52
preservice elementary teachers would have an effect upon the

outcome of this research study. It is a fact that statistical

power increases automatically with sample size. Furthermore,

the larger the sample, the more likely its mean and standard

deviation is representative of the population mean and

standard deviation.

Although statistical analysis supported no statistically

significant effect, course section variability was possible









in the amount of orientation the students had regarding

software use prior to the study. Specifically, the researcher
planned a similar amount of exposure to the software for all

involved in the study, but there was no documentation of

utilization time.

The researcher conducted all lessons to eliminate
differences in results due to instructor variability, but was
aware of the possibility that a threat to validity may be

introduced by the researcher influencing results. This was
minimized by the use of specific presentation plans and

formalized activity worksheets.

In summary, the results of this study are generalizable
only to situations involving similar participants, classroom

environment, and classes of variables.



Implications and Recommendations



Our society has advanced from the industrial age into
the information age where numeracy and technological literacy

are important aspects of effective citizenship. The

mathematics classroom must evolve from a "chalk and talk"

environment into an "active" conceptually-oriented habitat

that models the contemporary workplace.

Classroom teaching is undergoing great change with
innovative dynamic software available. A major challenge

facing mathematics and mathematics education research is the
development of an indepth understanding of the function of
the power technology provides, not only to revise the









mathematics that we teach, but also to transform our

understanding of the teaching and learning processes. To

ensure proper utilization, future teachers must be trained

appropriately in order to bring to the classroom the

experience, confidence, and enthusiasm necessary to

effectively facilitate student learning.

This study added to the body of knowledge that already
exists on technology and its impact upon curriculum and

instruction. However, replication is needed before any

definitive conclusions can be drawn about the effect of a
dynamic geometric construction environment upon learning.
Further investigations can build upon this research and

continue to expand the knowledge base for educational

practices in a technologically rich environment.

Future research should strive for inclusion of other
factors relevant to effective learning. The impact of spatial

factors, such as visualization or orientation, warrant

investigation. This is particularly relevant in light of the

mathematical content addressed. Furthermore, affective

factors should be given consideration in order to suggest

direction for proper reconceptualization of the classroom

environment. In particular, differences in attitudes

regarding computer or software use are rich avenues that have

only recently opened up for investigation. There is also a

need to continue the investigation of grade differences and

learning style differences with respect to the dynamic

environment.

Calculation of posttest score means presented the
researcher with additional statistics worthy of









consideration. It has been previously mentioned in Chapter

III that the research instrument (refer to Appendix A) was

adjusted in lieu of findings of the pilot studies. Items
hypothesized to address learning attributed to the dynamic

component of the geometric construction software were
developed and placed into the final version of the research
instrument. Thus, it was deemed appropriate to investigate

individual items. Research should be undertaken to study
assessment of learning in a dynamic geometric construction
environment in addition to the development of appropriate

test items. Moreover, research should give consideration to

the evaluation of programs that utilize a dynamic learning
environment.

In addition to the rightful inspection into quantitative
aspects of this study, consideration must also be given to

qualitative aspects intrinsically rich in interpretive value.

The activity worksheets (refer to Appendix C) not only gave
structure to the learning sessions; participants responses
throughout the guided discovery process provided insight into
their construction and reconstruction of knowledge.

During the first session, participants in either group
could be observed struggling with the novel approach to
understanding a concept many believed they had learned in
middle school. Participants in the control group slowly drew
similar geometric figures using the VTPD (refer to Definition

of Terms) to develop an intuitive notion of similarity
whereas the participants in the experimental group used the
dynamic capability of the Geometer's Sketchpad (Jackiw, 1995)

to investigate basic principles of similar figures. By the








73
end of the first session, the experimental group had measured

and analyzed many different polygonal figures while the

control group, although they had assimilated the principles

of similarity, were unsure of whether non-convex polygonal
figures would observe the principles. In other words,
participants of the control group were confident regarding
the limited number of examples they were exposed to, but were
unsure when analyzing unknown cases. Further research should

be undertaken to consider possible misconceptions and, also,
levels of geometric thought.




Summary


This final chapter discussed the results presented in
the previous chapter. An overview of the study was provided
first, including a description of the sample population.

Next, the significance of the study was articulated, followed
by the research questions. Then, discussion of the findings
based upon the research analysis was presented. Limitations

followed; the chapter concludes with implications and

recommendations for future research.
















APPENDIX A
RESEARCH INSTRUMENT








1. Point projections are made of the given triangle first
from point P and then from point Q. Each projection uses
a scale factor of 3. Which of the following is true?

A. The triangle image from point P has angles of greater
measure.

B. The triangle image from point Q has angles of greater
measure.

C. The triangle image from point P has a larger area.

D. The triangle images from point P and point Q have the
same area.

E. The triangle image from point Q has a larger area.





o 0
P Q












2. Which of these transformations was used for the given
rectangle and its image:










image
A) (x,y) -> (2x,2y)
B) (x,y) -> (x,2y)
C) (x,y) -> (2x,y)
D) (x,y) -> (2x,4y)
E) (x,y) -> (4x,2y)









3. Which projection point was used to dilate the shaded
rectangle?





0
D


4. What scale
sailboat?


factor has been used to enlarge the small


C) 4 D) 6


A) 2 B) 3


E) 1/4









5. The given rectangles are similar. Find the missing length.


A) 10


B) 11


C) 12


D) 13


E) 14


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


A) 30
B) 21
C) 24
D) 42
E) 48


feet
feet
feet
feet
feet


7. Given rectangles of dimensions 1x6 and 4x24, the area of
the larger triangle is how many times bigger than the
area of the smaller rectangle?

A) 4 times
B) 6 times
C) 8 times
D) 16 times
E) 18 times


1 ~ZIIIIZZ~









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

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



9. A 2 meter stick has a shadow of 1/2 meter at the same time
that a nearby tree has a shadow of 3 meters. How tall is
the tree?

A) 6 meters
B) 12 meters
C) 1.5 meters
D) 3 meters
E) 15 meters



10. The given rectangles are similar. Find the length of the
missing side.

A) 7 B) 9 C) 10 D) 11 E) 15






2- 6


5 ?



11. A 1x5 rectangle grows into a 4x12 rectangle. The area of
the new rectangle is how many times larger than the area
of the small rectangle?

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









12. Point projections are made of the given triangle first
from point P and then from point Q. Each projection uses
a scale factor of 2. Which of the following is true?

A. The triangle image from point P has angles of greater
measure.

B. The triangle image from point Q has angles of greater
measure.

C. The triangle image from point P has a larger area.

D. The triangle images from point P and point Q have the
same area.

E. The triangle image from point Q has a larger area.



0
P










13. Given the bolded segment of the shaded triangle, the
corresponding segment on the projected image is:
A) a
B) b
C) c

Q
i\


/
I
0==


a -


14. Given the bolded segment of the shaded polygon, the
corresponding segment on the projected image is:
A) a
B) b
C) c
D) d







0gd



--b










81
15. Which projection point was used to dilate the shaded
rectangle?

0D D














C
0
0
A









16. The point of perspective is:


A) point A
B) point B
C) point C
D) point D
E) point E








83
17. Choose the scale factor of the similar figures. The
bolded line segment is the pre-image.The scale factor is:

A) 1
B) 2
C) 3
D) 4
E) 5









18. Choose the scale factor of the similar figures. The
shaded circle is the pre-image. The scale factor is:

A) 1
B) 2
C) 3
D) 4
E) 5


19. Which projection point was used to dilate the shaded
rectangle?










20. Which projection point was used to dilate the shaded
trapezoid?









0 C


o0
D




0E












21. Solve the proportionality a/? = w/x.

? = A) a
B) b
C) c
D) d
E) e


0
/!
/1
/ Iy
a ,/ ,
/ I


/ I
/ 0.


"' c


d ---- -
















APPENDIX B
LEARNING SESSIONS








Table 3.1
First Pilot Study



Presession


Information regarding UFIRB form.

Pretest.

Computer lab orientation.


Session 1


Launch Terminology review.

Use of construction tool and measurement tools.

Exploration Properties of similar polygons.

Summary Class discussion of conjectures generated during

individual exploration time

(Corresponding angles congruent. Ratio of 2

sides of a polygon is equal to the ratio of the

corresponding 2 sides of any similar polygon and

they are equivalent fractions)








88


Session 2


Launch




Exploration


Summary


Review of properties of similar polygons.

Introduction to intuitive Projective Geometry.

SCALE FACTOR with DILATE tool.

Properties with respect to (1) point of

perspective and (2)corresponding sides.

Class discussion of conjectures generated during

individual exploration time.

(The ratio of the distance from the perspective

point to a vertex of a similar polygon over the

distance from the perspective point to a vertex

of the given polygon is equal to the scale

factor.

The ratio of the measure of a side of a similar

polygon over the measure of the corresponding

side of the given polygon is equal to the scale

factor.

Properties of similar polygons are invariant as

the position of the point of perspective

varies.)








89


Session 3


Launch


Exploration


Summary


Connection of concept of similarity to ratio and

proportional reasoning.

Given pre-constructed sketches, indirect

measurement tasks.

Class discussion of approaches generated during

individual exploration time.


Session 4


Launch Review of properties with respect to

(1) point of perspective and

(2)corresponding sides.

Use of POLYGON INTERIOR tool.
Exploration Relationship of areas of similar polygons.

Summary Class discussion of conjectures generated during

individual exploration time.

(The ratio of the area of a similar polygon over

the area of the given polygon is equal to the

square of the scale factor.)









Table 3.2
Second Pilot Study


Session 1


Launch








Exploration

Summary


Terminology review.

Use of construction and measurement tools.

Introduction to intuitive Projective Geometry.

Dynamic geometry exposition with animation or

the Variable Tension Proportional Divider.

Properties of similar figures.

Class discussion of conjectures generated during

individual exploration time.

(Corresponding angles congruent.

Ratio of 2 sides of a polygon is equal to the

ratio of the corresponding 2 sides of any

similar polygon and they are equivalent

fractions)








91


Session 2


Launch Define scale factor.

Review properties of similar polygons.

Exploration Properties with respect to

(1) point of perspective and

(2)corresponding sides.

Summary Class discussion of conjectures generated during

individual exploration time.

(The ratio of the distance from the perspective

point to a vertex of a similar polygon over the

distance from the perspective point to a vertex

of the given polygon is equal to the scale

factor.

The ratio of the measure of a side of a similar

polygon over the measure of the corresponding

side of the given polygon is equal to the scale

factor.

Properties of similar polygons are invariant as

the position of the point of perspective

varies.)








92


Session 3


Launch Discussion of how to find the area of a figure.

Exploration Relationship of areas of similar polygons.

Summary Class discussion of conjectures generated during

individual exploration time.

(The ratio of the area of the image over the

area of the given polygon is equal to the square

of the scale factor.)


Launch Review of properties with respect to

(1) point of perspective and

(2)corresponding sides.

Connection of concept of similarity to ratio and

proportional reasoning.

Exploration Given pre-constructed sketches, indirect

measurement tasks.

Summary Class discussion of approaches generated during

individual exploration time.

(The measures of corresponding parts of similar

triangles can be set onto a proportion to find

the unknown measurement.)









Table 3.3
The Study


Session 1


Launch


Exploration




Summary


Terminology review.

Use of construction and measurement tools.

Introduction to intuitive Projective Geometry.

Dynamic geometry exposition with animation or

the Variable Tension Proportional Divider.

Metric properties of similar figures.

Size and position of the image with respect

to the numerosity of the scale factor.

Class discussion of conjectures generated during

individual exploration time.


Session 2


Launch Define scale factor.

Review properties of similar polygons.
Exploration Numeric properties of similar shapes with

respect to

(1) point of perspective and

(2)corresponding sides.
Summary Class discussion of conjectures generated during

individual exploration time.








94


Session 3


Launch Review of properties with respect to

(1) point of perspective and

(2)corresponding sides.

Connection of concept of similarity to ratio

and proportional reasoning.
Exploration Numeric properties of similar shapes with

respect to

(1) scale factor and

(2)corresponding distances.
Summary Class discussion of approaches generated during

individual exploration time.
















APPENDIX C
ACTIVITIES WITH RESPONSES










Sketch
step 1: In the Display menu, choose Line Weight "thick" and Shade "100%."
step 2: Construct any polygon and change the color of each side to be distinct.
step 3: Construct a point outside the polygon and mark it as center in the
Transform menu.
step 4: Select your entire polygon and dilate by a scale factor of 2/1 (Place 2
into top box and 1 into the bottom box). J

Investigation: Corresponding Sides SU ta w j
step 5: Measure the ratio of a side on the dilated polygon with the
corresponding side on the original polygon.
step 6: Repeat step 5 using a different side. What is this ratio?
.50
step 7: Drag a vertex to change the length of a side. What changes? What stays
the same? 0- ',u< 'V A\^Or

ro ^ rwhos sWa~ so\ n ,c]M



Investigation: Projection Point
step 7: Measure an angle of the dilated polygon and the corresponding angle
on the originalpolygon. flQxAL, w-(& 'Wc. ^$4T<-
step 8-.tS rag the point of projection to different locations. Regarding the
e.aO, --figures, what changes (measures, locations,...)? What stays the same?
Make sure to include placing the projection point on a vertex of the
original polygon!
(",, ?- 0 ln! projcc /

/ii o n o o 4U lorTu4
I t b i- ^ . ... ... '*^ -V