English language proficiency and spatial visualization in middle school students' construction of the concepts of reflec...

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English language proficiency and spatial visualization in middle school students' construction of the concepts of reflection and rotation using The Geometer's Sketchpad
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Thesis (Ph. D.)--University of Florida, 1995.
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ENGLISH LANGUAGE PROFICIENCY AND SPATIAL VISUALIZATION IN
MIDDLE SCHOOL STUDENTS' CONSTRUCTION OF THE CONCEPTS OF
REFLECTION AND ROTATION USING THE GEOMETER'S SKETCHPAD











B/

JULI KIM DIXON











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


1995

























Copyright 1995
by
Juli Kim Dixon




















This dissertation is dedicated to three generations of life and love.
In loving memory of my grandfather,
Albert P. Harap,
whose belief in me was invaluable,
to my parents,
Harvey and Joy Inventasch,
who instilled in me a respect for education,
and to my husband,
Marc Todd Dixon,
who respected my dream and made it his own.












ACKNOWLEDGEMENTS
My gratitude is extended to the many people who have provided
academic and emotional support throughout the dissertation process.
To Professor Mary Grace Kantowski, my chair and mentor, who
provided educational challenges and support, I owe my future
contributions as a mathematics educator. I am thankful to my
committee members, each of whom provided guidance throughout the
dissertation process. Professor David Miller shared his statistical
expertise. Professor Clemens Hallman led me to a deeper
understanding of the needs of students with limited English
proficiency and provided financial support through the Bilingual
Education and Minority Languages Affairs Fellowship. Professor
Donald Bernard contributed to my growth through his experiences as
an educator. Professor Charles Nelson enhanced my understanding of
mathematics from the viewpoint of a mathematics educator.
I am grateful to Susan Lee and the mathematics faculty and
students of the central Florida middle school for making the
research for my dissertation possible.
I am especially indebted to Nick Jackiw of Key Curriculum
Press for providing me with the experimental version of The
Geometer's Sketchpad used in the study and providing participating
teachers with The Sketchpad for the continued use with their
students. Nick also provided ongoing and greatly appreciated
technical support.









I am grateful for the financial assistance provided by Dr.
Charlie T. Council and the Florida Educational Research Council in
support of this study.
I am forever grateful to my family, especially to my parents,
Harvey and Joy Inventasch, my husband, Marc Todd Dixon, and my
friend, Thomasenia Lott Adams whom I consider family. These very
special people have provided me with the love and support that only
family can provide. I would never have become the person and
mathematics educator I am today without each and every one of
them.











TABLE OF CONTENTS

page

ACKNOWLEDGEMENTS .................................iv

ABSTRACT ..................................... .viii

CHAPTERS

I DESCRIPTION OF THE STUDY

Introduction ...... ... .. ... .. .. .. ..... .... .... 1
Statement of the Problem. ........................ 5
Significance of the Study ......................... 8
Organization of the Study ......................... 9

II LITERATURE REVIEW

Overview .. ....................... ........ 11
Theorietical Background ......................... 11
Studies on Transformations ...................... 17
Visualization ................................ 33
Limited English Proficient Students. ................. 46
Sum m ary ..................................49

III METHODOLOGY

Research Objective ............................ 52
M measures .......................... .........53
Pilot Study ....................... ..........56
Sampling Procedures ................... ........ 61
Experimental Treatment and Procedures. .............. 62
Research Design ..............................67










IV RESULTS


Descriptive Statistics .................
Limitations of the Study ...............

V QUALITATIVE RESULTS

M ethodology ........................
Computer Projects ...................
Observations .......................
Data from Reflection/Rotation Instruments .
Icon Questionnaire. ....................


.... 70
. .. .... .. 78




. . 81
.. .... ... 83
. .... .... 90
......... 105
......... 108


VI CONCLUSION


Sum m ary ......... ................
Discussion ........................
Implications........................
Recommendations ...................
Suggestions for Future Research .........


. . 110
. . 116
... ... .. .. 123
. .. .... 124
... ....... 127


APPENDICES


A PAPER AND PENCIL REFLECTION/ROTATION INSTRUMENT. .. .129
B PAPER AND PENCIL REFLECTION/ROTATION INSTRUMENT
OVERLAYS ..................................140
C TRAINING LESSONS ............... .............147
D ANGLE AND DISTANCE MEASURE ACTIVITIES. ........... .150
E EXPERIMENTAL TREATMENT LESSONS ................. 158
F REFLECTION AND ROTATION KEYS .................. .165
G REFLECTIONGAME SCREEN ........................ 170

REFERENCES ................. ...... ............. 171

BIOGRAPHICAL SKETCH ............................. .180











Abstract of Dissertation Presented to the Graduate School of the
University of Florida in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy

ENGLISH LANGUAGE PROFICIENCY AND SPATIAL VISUALIZATION IN
MIDDLE SCHOOL STUDENTS' CONSTRUCTION OF THE CONCEPTS OF
REFLECTION AND ROTATION USING THE GEOMETER'S SKETCHPAD

By

JULI KIM DIXON

May 1995

Chairman: Eleanore L. Kantowski
Major Department: Instruction and Curriculum

This study was designed to investigate the effects of a dynamic
instructional environment, English proficiency, and visualization
level, independently and interactively, on middle school students'
construction of the concepts of reflection and rotation. Also
examined were the effects of a dynamic instructional environment
on students' two- and three-dimensional visualization.
The dynamic instructional environment involved exclusive use
of The Geometer's Sketchpad with lessons created for the purpose
of allowing for student discovery of properties of reflection and
rotation. Students were also provided extensive opportunities to
perform the transformations.
After controlling for initial differences, it was concluded that
students experiencing the dynamic instructional environment








significantly outperformed students experiencing a traditional
instructional environment on content measures of the concepts of
reflection and rotation as well as on measures of two-dimensional
visualization. The students' instructional environment did not
significantly affect their three-dimensional visualization. There
was no statistically significant differences on any of the dependent
variables between the performances of limited English proficient
students and those of their English proficient peers when the same
instructional environment was experienced.
Observations provided examples of how students communicate
about mathematics, visualize movement, and construct concepts of
reflection and rotation while experiencing a dynamic instructional
environment. Concrete information was gained regarding the depth
of students' knowledge about concepts involved with reflecting and
rotating. Insight was obtained into the class attitudes about and
understanding of transformations.
















CHAPTER I
DESCRIPTION OF THE STUDY


Introduction
The Curriculum and Evaluation Standards for School
Mathematics (National Council of Teachers of Mathematics (NCTM),
1989) include the following objectives in the standard for middle
level geometry: "In grades 5 8, the mathematics curriculum should
include the study of the geometry of one, two, and three dimensions
in a variety of situations so that students can visualize and
represent geometric figures with special attention to developing
spatial sense; [and] explore transformations of geometric figures"
(p. 112). This standard is not being implemented in many middle
schools in the United States. The Second International Mathematics
Study: Report for the United States (SIMS) (Crosswhite, 1985)
identified topics taught in other countries which were not covered
by the majority of eighth-grade classes in the United States. "For
example, topics in transformational geometry, taught in some
countries, were reported taught by only 12% of United States eighth
grade teachers" (p. 20). Furthermore, the results of the study
indicated that teachers primarily taught according to their textbook
with very little use of manipulatives or other materials not included









with the text. This finding was corroborated by data collected in a
comparative study with seventh- and eighth-grade students in the
United States and Japan (Iben, 1988). Reasons for the lack of
emphasis on transformation geometry may include the need for an
effective and accessible instructional strategy.
For students to develop spatial sense, they need many and
varying experiences with drawing, measuring, transforming,
visualizing, comparing, and classifying geometric shapes (NCTM,
1989). "Spatial sense is an intuitive feel for one's surroundings and
the objects in them. To develop spatial sense, children must have
many experiences that focus on geometric relationships; the
direction, orientation, and perspectives of objects in space; the
relative shapes and sizes of figures and objects; and how a change in
shape relates to a change in size" (NCTM, 1989, p. 49). The term
"spatial sense" is related to what has also been labeled spatial
visualization, spatial perception, visual imagery, spatial ability,
visual skill, spatial reasoning, mental rotations, and visual
processes (Bishop, 1983; Davey & Holliday, 1992; Stanic & Owens,
1990; Wheatley, 1990). The curriculum in geometry and instruction
practices must include appropriate experiences for students in the
area of visualization as it reflects this important aspect of
mathematics (National Council of Supervisors of Mathematics
[NCSM], 1989; National Research Council [NRC], 1989).
The recent availability of computers in classrooms has
provided the tool for such instruction. The dynamic graphic
capabilities of the microcomputer allows for geometry to be









introduced to students through transformations (Kantowski, 1987).
NCSM (1989) supports the use of computers throughout the
mathematics curriculum as well as instruction in visualization and
transformations.
Effective instructional strategies are also needed for use with
language minority students. According to Mathematics for Language
Minority Students, the NCTM's position statement, "cultural
background or difficulties with the English language must not
exclude any student from full participation in the school's
mathematics program" (NCTM, 1987, n. p. ). The Curriculum and
Evaluation Standards for School Mathematics states, "students
whose primary language is not standard English may require support
to facilitate their learning of mathematics" (NCTM, 1989, p. 80).
These students tend to spend most of their time on the prerequisite
basic skills and rarely have exposure to higher order mathematics
skills (Schwartz, 1991; Secada & Carey, 1990; Stoloff, 1989). The
position of NCTM (1989) in general and this researcher in particular
is that this current trend is unacceptable.
The Curriculum and Evaluation Standards for School
Mathematics (NCTM, 1989) and the Professional Standards for
Teaching Mathematics (NCTM, 1991), both commonly referred to as
the Standards, stress communication as an important part of
teaching and learning mathematics. Mathematics can be taught and
learned visually; communication does not necessarily refer
exclusively to verbal exchanges. Visual communication in
mathematics education is especially important to students having










limited proficiency in English (Cummins, 1984; Dawe, 1983;
Presmeg, 1989). The Standards also stress the need to empower all
students through mathematics.

Since the assumption persists that mathematics depends
only marginally on expertise in English, mathematics
typically serves as the first academic course beyond
English as a Second Language (ESL) into which counselors
place students from foreign countries. However,
counselors and bewildered students soon discover that
limited English proficiency is a discouraging obstacle to
learning mathematics. (Kimball, 1990, p. 604)

Limited English proficient (LEP) students can make progress in
learning mathematics when the teacher recognizes their special
needs and uses communication, manipulatives, hands-on experiences,
the first language of the students when possible, and when the
teacher systematically addresses the necessary vocabulary (Kober,
1991). Mathematics taught with an emphasis on visual clues could
make the mathematics traditionally taught in a highly verbal
instructional mode more attainable to students having limited
proficiency in English.
Use of The Geometer's Sketchpad (Jackiw, 1991) for the
dynamic presentation of rotations and reflections is an appropriate
vehicle to facilitate full participation of limited English proficient
students while encouraging the development of all students'
visualization abilities in the middle school mathematics program.
The Geometer's Sketchpad is a dynamic program for the Macintosh
and IBM computers. The Sketchpad ". has great potential for









teachers and students to use in investigations of ideas in geometry"
(Wilson, 1992, p. 157). This tool allows mathematics to be taught
visually to the class as a whole, to small groups, or to individuals.
Furthermore, the experimental version of the Sketchpad used in this
study incorporated icons, or pictorial representations, as clues to
appropriate menu selections displayed in English. These menu
selections were coupled with symbols designed to provide context
clues for the associated vocabulary.


Statement of the Problem
The study of the rigid motion transformations and experience
with visualizing in two- and three-dimensions are integral parts of
the Standards for eighth-grade students in mathematics. These
topics are not currently addressed adequately in the middle school
curriculum in the United States.
Achievement in mathematics in a given language seems to be
related to the degree of proficiency in that language (Secada,
1992a). Contrary to what many people believe, the study of
mathematics does not transcend language barriers. The changing
demographics of the United States dictates that appropriate
mathematical content, teaching strategies, learning tools, and
classroom environments must be incorporated in all schools to close
the gap of achievement in mathematics between English speaking
and LEP students. Current programs designed with the pretense of
closing this gap are ". evaluated based on their ability to curtail
student drop out and to improve student scores on standardized









tests. As a consequence, compensatory programs mimic the tests by
which they are evaluated and which are focused on lower level and
computational skills" (Secada, 1989, p. 38). This focus on lower
level skills is not consistent with NCTM's Standards and is not an
acceptable solution to the problem of the achievement gap.
The purposes of this study included: (1) to explore the effects
of a dynamic instructional environment and students' visualization
ability on the identification of and ability to perform the rigid
motion transformations of reflection and rotation; and (2) to explore
the effects of the relationship between the students' level of
visualization and the instructional environment on the identification
of and ability to perform these rigid motion transformations.
Another purpose of this study was to investigate the effects of
students' English proficiency on the identification of and ability to
perform reflections and rotations and the effects of the relationship
between students' English proficiency and a dynamic instructional
environment on the identification of and ability to perform the rigid
motion transformations of reflection and rotation. Finally, the
study was designed to ascertain the effects of a dynamic
instructional environment and students' English proficiency on
students' visualization. The treatment in the dynamic instructional
environment included exploring the rigid motion transformations of
reflection and rotation through a dynamic presentation and
subsequent activities using The Geometer's Sketchpad (Jackiw,
1991). The rigid motion transformations of reflection and rotation
were addressed based on the view of the researcher that the motion









involved with reflection and rotation are very similar with the
exception that reflection involves three dimensions and rotation
involves two dimensions. The rigid motion transformation of
translation was not addressed in this study due to the view of the
researcher that the movement involved with translation is not
closely related to that of reflection and rotation.
The theoretical bases for using a dynamic instructional
environment emphasizing student collaboration and exploration as
the instructional strategy was based on the constructivist theory of
teaching and learning mathematics as well as Vygotsky's (1978,
1986) zone of proximal development. Vygotsky believed that
students are provided more opportunities to learn when they are able
to collaborate with more capable peers and adults. The
constructivist theory promotes collaboration among peers during the
construction of knowledge. The choice of the computer program and
design of associated student activities was based on Cummins'
(1984) theory of context embedded versus context reduced
instruction. The Geometer's Sketchpad provides context clues
through the dynamic, visual presentation of geometric properties.
Together with the activities, the software was used to aid in
students' acquisition of the concepts and vocabulary involved with
rigid motion transformations in a context embedded rather than a
context reduced instructional environment.









The objectives of the study were

1. to explore whether students' level of visualization will
interact with their instructional environment to affect
their ability to identify and perform reflections and
rotations in a (a) dynamic testing environment or a (b)
static testing environment.

2. to explore whether students' English proficiency will
interact with their instructional environment to affect
their ability to identify and perform reflections and
rotations in a (a) dynamic testing environment or a (b)
static testing environment.

3. to explore whether students' (a) instructional environment,
(b) visualization level, and (c) English proficiency will
affect their ability to identify and perform reflections and
rotations in a (a) dynamic testing environment or a (b)
static testing environment.

4. to explore whether students' (a) instructional environment
and (b) English proficiency will affect their (a) two
dimensional and/or (b) three-dimensional visualization.


Significance of the Study
If the transformations are to be studied in their own
right, it would seem pointless to do this in a didactic,
expository manner. The fact that the transformations
can be defined in terms of actions (folding and turning)
and their results represented in a very direct manner by
drawings means that the topic is ideally suited to a
practical and investigative approach. (Hart, 1981)

The study of transformations provides an excellent opportunity
for a dynamic instructional environment for all students regardless
of their proficiency with the English language. Throughout the









recent history of mathematics education, individuals and influential
reform groups have devoted chapters and entire textbooks to the
study of transformations.

Transformations often are used to represent physical
motions such as slides [translations], flips [reflections],
turns [rotations], and stretches dilationss]. Students
should use computer software based on this dynamic
view of transformations to explore properties of
translations, line reflections, rotations, and dilations, as
well as compositions of these transformations. These
graphic experiences not only help students develop an
understanding of the effects of various transformations
but also contribute to the development of their skills in
visualizing congruent and similar figures. (NCTM, 1989,
p. 162)

The Geometer's Sketchpad was first published in 1991 by Key
Curriculum Press. The Sketchpad is a highly visual and dynamic
tool for exploring and discovering geometric properties, but because
of the lack of research using the Sketchpad in middle school
mathematics its potential is not being realized. This study
established the effectiveness of the Sketchpad as a tool for the
instruction of all students, including LEP students, to successfully
recognize and perform rotations and reflections. The Sketchpad
was also useful for improving students' visualization ability.


Organization of the Study
This chapter included the statement of the problem and its
significance in the field of mathematics education. The review of
the relevant literature presented in chapter II includes the





10



theoretical bases in which this study has been framed, studies
involving instruction with transformation geometry, studies of
visualization, and studies involving mathematics and science
instruction for limited English proficient students. The design and
methodology of the study is reported in Chapter III. In Chapter IV,
results of the quantitative analysis and limitations of the study are
provided. Qualitative results of the study are reported in Chapter V.
A summary of the results, implications, and recommendations for
future research are presented in Chapter VI.
















CHAPTER II
LITERATURE REVIEW


Overview
A review of the relevant literature is presented in this chapter.
The areas under discussion are the theoretical bases in which this
study has been framed, studies involving instruction with
transformation geometry, studies of visualization, and studies
involving mathematics and science instruction for limited English
proficient students.


Theoretical Backaround
This study was designed to explore whether an environment in
which English proficient and limited English proficient students
work together using computers will result in their construction of
knowledge of reflections and rotations while simultaneously
improving visualization skills.
Research must be supported by educational theory. The
constructivist theory of teaching and learning explains how teachers
can create environments in which students can construct knowledge.
Vygotsky's (1978, 1986) zone of proximal development addresses
how students working together, with adult guidance as needed, and









using proper tools could construct this knowledge socially.
Cummins' theory of context embedded language versus context
reduced language justifies how English proficient and limited
English proficient students could work together in the same
instructional setting.
Although the constructivist theory of teaching and learning
exists in a variety of expressions, there is general consensus on the
following principles (Confrey, 1990; Davis, Maher, & Noddings, 1990;
Goldin, 1990; Kamii and Lewis, 1990; Noddings, 1990; von
Glasersfeld, 1990; Wheatley, 1991):

1. Students build their own knowledge; they do not
receive knowledge prepackaged from others.
2. Knowledge is not built passively but through physical
and mental action.
3. Truths are not found; rather, interpretations are built
to explain experiences.
4. Learning takes place through social interaction.

Inherent in these principles is the need for an environment
conducive to student exploration and interaction. "Constructivists
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"
(Noddings, 1990, p. 10).
The constructivist view of learning mathematics is consistent
with the belief that students come to the classroom with differing
understandings and ways of conceptualizing mathematics (Kamii &









Lewis, 1990; Steffe, 1990; Wheatley, 1991). Different students
construct concepts in different ways depending on many variables,
such as the students' native and learned abilities, aptitudes,
dispositions, learning styles, native language, and past experiences.
The constructivist view of teaching allows for these differences by
providing experiences for children to make sense of mathematics
through varying instructional strategies. These strategies include
meaningful mathematical exploration and experiences, use of
multiple representations, multiple modes of instruction, and, as in
the case of this study, a dynamic medium for learning ( Kamii &
Lewis, 1990; Noddings, 1990; Wheatley, 1991). "The teacher's main
function is to establish a mathematical environment" (Noddings,
1990, p. 13).

Viewing mathematical knowledge as a learner activity
rather than an independent body of "knowns" leads to
quite different educational considerations. Rather than
identifying the set of skills to be gotten in children's
heads, attention shifts to establishing learning
environments conducive to children constructing their
mathematics in social settings. Since there is a social
dimension of knowledge, this learning environment
necessarily includes children talking mathematics with
each other. Such learning environments provide
opportunities for children to share ideas with peers, both
in small groups and within the society of the classroom.
(Wheatley, 1991, p. 12)

Vygotsky (1978) believed that the pedagogical and social needs
of the student could be met through the use of appropriate tools in
an environment that promoted collaborative inquiry about concepts.
In so doing, the students' zone of proximal development would be










formed and learning legitimized. According to Vygotsky (1978,
1986), the zone of proximal development is the distance in
developmental level between problems the student can solve
independently and problems the student can solve in collaboration
with more capable peers or under guidance from an adult. Students'
potential to learn is then judged by the problems that they cannot
yet solve alone but can accomplish in collaboration with others
rather than the problems that they can already perform alone.
Vygotsky believed that learning within this zone is thus the only
worthwhile learning. He proposed

that an essential feature of learning is that it creates
the zone of proximal development; that is, learning
awakens a variety of internal developmental processes
that are able to operate only when the child is
interacting with people in his environment and in
cooperation with his peers. Once these processes are
internalized, they become part of the child's independent
development achievement. (Vygotsky, 1978, p. 90)

Vygotsky believed that the student was working within his zone
of proximal development when he was able to solve given problems
only with assistance from more capable peers or adult guidance and
was not able to solve the problems independently. This researcher
believes that students working in collaboration are able to help each
other within their zones of proximal development because, according
to the constructivist view of learning, each student brings different
experiences to the learning situation and hence may be more capable
on different aspects of the same problem. In this situation, the









"more capable peer" may alternate from one student to the other
while working on the same problem. And, in the event that the
students cannot provide the proper catalyst to solve the problem,
the teacher, acting as facilitator, can supply the necessary guidance.
This role of the teacher is in accordance with both the
constructivist view of teaching and learning mathematics and
Vygotsky's zone of proximal development (Vygotsky, 1978, 1986;
Wright, 1990).
For LEP students in particular and all students in general,
collaborative inquiry should be placed in a context embedded
situation. A context-embedded situation is one that provides
comprehensible input so that the students do not need to guess about
the teacher's intentions but may gather an understanding of the
intentions based on linguistic as well as situational cues. This
context-embedded language may include pictures, interpersonal
interactions, and nonverbal information (Fradd, 1987). According to
Cummins (1984) and Fradd (1987), this environment is more
compatible to LEP students than is a context reduced situation (the
traditional classroom lecture style of environment) where students
must rely solely on linguistic cues in a language with which the LEP
student is not yet proficient.
The degree of cognitive involvement in activities where
communication is used can be explained by viewing these activities
in a continuum. At one end there exist ". .communicative tasks and
activities in which the linguistic tools have become largely
automatized (mastered) and thus require little active involvement









for appropriate performance" (Cummins, 1984, p. 139); at the
opposite ". end of the continuum are tasks and activities in which
the communicative tools have not become automatized and thus
require active cognitive involvement" (Cummins, 1984, p. 139).
Contextual support is most important when the degree of cognitive
involvement is the greatest (Cummins, 1981, 1984; Fradd, 1987).
When exploring new mathematical concepts with unfamiliar
mathematical vocabulary, students, especially LEP students, will
not have automatized the associated communicative tools. They may
be experiencing just this kind of situation when working within
their zones of proximal development. It may seem that the limited
English proficient students need help with the mathematics when in
actuality the problem is with the language. And, according to Fradd
(1987), "comprehensible input is the foundation of effective
instruction" (p. 143).

Comprehensible input shares both a point of commonality
and a point of difference with the zone of proximal
development. The two constructs are similar in that
advancement is possible to the extent that language
input is made comprehensible to the learning by an adult
or more capable peer. Moreover, both constructs
underscore the importance of being responsive to the
learner's perceived needs. The difference lies in the
focus each construct places on the role of language
learning. Whereas comprehensible input stresses the
importance of adjusting speech to the learner's
linguistic level of competence, the zone of proximal
development focuses on adjusting speech to the learner's
interactional level of competence. (DeVillar & Faltis,
1991, pp. 23-24)










Studies on Transformations
Several studies have addressed geometric transformations.
Some have involved extensive instruction in performing
transformations (Edwards, 1990, 1991; Ernest, 1986; Johnson-
Gentile, 1990; Pleet, 1990; Williford, 1972); most have included
only a brief introduction to transformations, if any at all, before
testing or interviewing students on their ability to perform
transformations (Edwards & Zazkis, 1993; Hart, 1981; Kidder, 1976;
Law, 1991; Moyer, 1978; Schultz & Austin, 1983; Soon, 1989). A few
studies have incorporated the use of computers in the presentation
of transformations (Edwards, 1991; Edwards & Zazkis, 1993; Ernest,
1986; Johnson-Gentile, 1990; Pleet, 1990).
An investigation involving students' actual construction of
transformation concepts involved the use of the computer software
Logo with small groups of students (Edwards, 1990, 1991). Edwards
investigated "the learning of a small group of children who
interacted over a short period of time with a computer microworld
dealing with transformation geometry" (Edwards, 1991, pp. 122-
123). The microworld, referred to as the TGEO computer
microworld, was based on Logo, which all of the students had
previously learned. Prior to the study, the students did not have
formal instruction in transformation geometry. Twelve students
from grades six through eight were chosen from a small, private
school based on their interest in participating in the study and their
ability to meet after school. The students were introduced to









transformations in their regular computer classes during two 50-
minute sessions. After this introduction, the students met in pairs
after school once a week for five weeks. The pairs of students
worked together on tasks at the computer or on worksheets. The
investigator introduced the activities then answered questions
while videotaping the session. A pretest was administered after the
first small group session and administered again as a posttest at
the conclusion of the study. Twelve items of the 24 items on the
posttest were identical to those used in a large scale British study
(Hart, 1981). As this was a qualitative dissertation, most of the
data were gathered via videotape.

The results of the study suggest that the microworld and
the associated activities were effective in assisting the
students to construct a working knowledge of the
transformations. The average performance of the
students on the written final examination was 70%
correct. The subjects in [Edward's] study performed
better than the British students on 10 of the 12 items
common to both tests. Furthermore, by the time of the
final session, all the students were able to carry out any
transformation on the computer without error or
hesitation. (Edwards, 1991, p. 129)

Edward's (1991) study, which was conducted at a small private
school, included the components for the necessary environment. Her
microworld necessitated facility with Logo, a computer
programming language. It is unrealistic to assume that the already
overburdened and underfunded public schools will develop curriculum
that will be altered to include a computer programming course in the
middle school before students can learn to perform transformations.









Transformations have been performed on microcomputers using
programs other than Logo such as student and teacher generated
programs in Basic using Apple II Graphics (Orton, 1990; Shilgalis,
1982).
Orton (1990) described a short program in Basic that uses
students' knowledge about algebra, transformations, and computer
programming to perform reflection and compositions of reflections
over user generated mirror lines. This program was presented by
Orton in response to a conference of NCTM at which short programs
were called for in order to teach and learn mathematics. Shilgalis
(1982) also described a program for Apple II graphics that performs
reflections as well as rotations, translations, glide reflections,
dilations, and compositions of the transformations. Both programs
rely on the coordinate axes as references for points, lines, and
vectors. The user enters the appropriate information as a response
to program generated prompts. Hence, to use the programs, students
must have some knowledge of coordinate axes and vectors as they
apply to transformations.
The large scale British study was part of Concepts in Secondary
Mathematics and Science (CSMS). This research program (Hart,
1981) conducted at Chelsea College, University of London, included
widescale testing of British 13- to 15-year-old students' ability to
perform rotations and reflections. The results of the test were
reported in Children's Understanding of Mathematics: 11-16 (Hart,
1981). Two hundred ninety-three 13-year-old students, 449 14-
year-old students, and 284 15-year-old students (1026 subjects









total) were administered a 52-item paper and pencil test lasting one
hour. The test consisted of three parts: single reflections, single
rotations, and combinations of reflections and rotations. The
students practiced and received answers to examples of reflections
and rotations immediately prior to the test. No other formal
instruction was provided by the investigators because reflections
and rotations were included in the British curriculum for ages 13 to
15. Nearly all students experienced some success performing single
reflections and rotations; however, most students had a difficult
time performing combinations of transformations.
Ernest (1986), who also used items from Hart's (1981) test,
studied the effects of computer gaming on students' performance in
transformational geometry. Eighteen 15-year-old students of below
average mathematics ability received six hours of instruction on
geometric transformations. The students were divided into two
comparable groups of nine subjects each. Prior to the first
computer session, each group was given a pretest involving specific
and general transformations. The general transformations involved
items from Hart's "Reflection and Rotation Test." Both groups
worked in pairs or trios on computer games for two half-hour
sessions. The experimental group played transformation games
using the computer program Triangles. The control group played
computer games without transformations. A posttest of an
equivalent form to the pretest was administered immediately after
the final computer session. The experimental group performed
significantly better on the transformations specifically related to










the game. There was no significant difference on the performance
on the test of general transformations.
Unlike Edwards' (1990) study, the computer games used in
Ernest's (1986) study did not require knowledge or experience with
programming. While the students had access to computers, it could
be concluded that they were not allowed sufficient time to explore
transformations through the use of this technology. The students
were exposed to the software for two 30-minute sessions. Perhaps
this brief duration of time explains the successful performance of
the experimental group only on transformations directly related to
those experienced in the computer game. The results might have
been different if students were allowed more time at the computer
and/or if there was a longer delay between the computer session and
the final.
Pleet (1990) compared the use of the computer program
Motions to the use of the manipulative Mira on eighth-grade
students' ability to perform transformations and mental rotations.
Eight teachers at eight different schools in Los Angeles taught one
class using Motions and one class using Mira for a three week
period. Seven additional teachers at seven additional schools in Los
Angeles administered pre- and posttests with no transformation
instruction to control for test effects. Complete data were
collected on 560 subjects. The Card Rotation Test from the Kit of
Factor-Referenced Tests (Ekstrom, French, Harman, & Derman, 1976)
and a researcher-made transformation geometry test were given as









pre- and posttests at the beginning and end of the three-week
period. Each lesson during the three-week period was structured
with a 20-minute review-explanation-discussion period followed by
a 30-minute activity session. The Motions groups transferred to an
Apple computer lab for the activity period while the Mira groups
stayed in their classrooms. There was no significant difference
between the Motions computer program group and the Mira
manipulative group on either acquisition of transformation geometry
concepts or mental rotation ability. Pleet recommended that
teachers who would be teaching transformations using either a
computer graphics program or hands-on manipulatives receive
sufficient training in both use of and teaching strategies for the
appropriate program or manipulative.
Johnson-Gentile (1990) also investigated the effects of
computer and non-computer environments on students' achievement
with transformation geometry. The researcher examined the effects
of a Logo version and a non-Logo version of a "motions" unit
developed by Professors Clements and Battista based on the van
Hiele levels. Two hundred twenty-three fifth and sixth grade
students from two different schools participated in the study. One
school was a city school with 50% of its students labeled as gifted
and talented; the other was an upper middle class suburban school.
The students were divided by class into two treatment groups and
one control group. The two treatment groups consisted of a Logo
based motions curriculum and a non-Logo based motions curriculum.
The non-Logo based curriculum was identical to that of the Logo









group's in every way except when the students worked on the
computer. The non-Logo groups worked with paper and pencil,
transparencies, and the Mira while the Logo groups worked with
Logo on the computer. The control group did not participate in any
motions activities. The Logo groups were part of a larger study
involving use of the Clements and Battista curriculum. A pretest of
achievement in geometry was given to all groups in the Fall of 1988;
the study was conducted during the Spring of 1989. The motions
unit lasted two weeks. A posttest consisting of objectives related
to the rigid motion transformations was administered to the two
treatment groups and the control group immediately following the
unit. A subsample of 36 students evenly distributed according to
sex and ability was chosen by stratified random sampling for
interviews. Each interview was 30 minutes long, the process began
within four days of the posttest. The interviews were designed to
collect information pertaining to precision of language with respect
to transformations and levels of thinking based on the van Hiele
levels. Finally, a retention test (identical to the posttest) was
administered one month after the posttest. According to the
pretest, the groups were equivalent with respect to geometric
ability at the time it was given. Both treatment groups scored
significantly higher than the control group on the posttest and the
retention test. There was no significant difference between the
Logo and non-Logo groups on the posttest but there was a significant
difference between the groups on the retention test. The Logo group
scored higher on the retention test compared to the posttest and the









non-Logo group scored lower. There was no significant difference
between the Logo and non-Logo groups on interview measures but
both treatment groups scored significantly higher than the control
group on the same measures. It should be noted that the Logo group
had been using a different curriculum (Clements/Battista
curriculum) previous to the beginning of the study and after the
administration of the pretest. The pretest was given in the Fall of
1988 and the study began in the Spring of 1989. The groups may not
have been equivalent with respect to experience with geometric
concepts at the beginning of the study.
The following studies involved little or no instructional
intervention. Soon (1989) examined whether the hierarchical levels
a student follows while learning concepts in transformation
geometry are consistent with the van Hiele theory for geometrical
understanding. Textbook material related to transformation
geometry in Singapore were also examined to determine whether
they were consistent with van Hiele based levels. Twenty female
students in their final year of secondary school were randomly
selected from one school in Singapore. The students all used the
same textbook but were taught by four different teachers. Two
interview sessions of 2 1/2 hours were conducted with each
student. The first session was videotaped and the second session
was audiotaped. Subjects were instructed to think aloud as they
solved problems using paper, pencil, and concrete objects. The
subjects were given 31 problems. The problems were separated into
levels based on the van Hiele levels of learning geometry as they









relate to concepts in transformation geometry. The hierarchy of
understanding of concepts in transformation geometry the students
followed were consistent with the van Hiele theory. Forty-two and
one-half percent of the students responded according to criteria for
the Basic Level which "required students to visually identify and
discriminate the following transformations: reflection, rotation,
translation, and enlargement" (Soon, 1989, p. 78). Thirty-six and
one-quarter percent of the students were at Level 1. "At this level
questions focused on a knowledge of the properties of each
transformation, and what happens to each figure after each
transformation" (Soon, 1989, p. 79). Six and one-quarter percent
were at Level 2 in which "students were required to interrelate the
properties of the different transformations. They were expected to
deal with composition of transformation and the use of matrices in
transformations" (Soon, 1989, p. 93). Finally, 12.5% were at Level 3
which required the use of transformations in proofs. Soon also
found that the textbooks did not provide students with opportunities
to explore and conjecture about transformations. The levels of the
textbooks were consistent with the levels of a majority of the
students.
Law (1991) also looked at the hierarchy of the acquisition of
the concept of transformation in an attempt to determine how
preservice elementary school teachers construct the concepts of
translation, reflection, and rotation. Eighteen preservice
elementary school teachers in a college geometry class volunteered
to participate. This class was the second college level mathematics










class in the preservice program of study. There were 15 students
from the Spring 1989 semester and 3 from the Fall 1990 semester.
The three transformations were covered in class through lectures.
Each student was interviewed following the coverage of the
material. "Based on his understanding of the concept, the researcher
conjectured that students learn the concept in the order of learning
definition of transformations first, and then single-point movement,
then figure movement, and finally identification of transformations"
(pp. 72-73). Furthermore, "the researcher implicitly assumed that a
student cannot understand any concept if he or she did not
understand the previous concept" (p. 73) in the given order. The
examples that Law used during his interview were limiting in
number (eight questions) and the fact that each subject received one
interview does not seem sufficient to develop a hierarchy.
Preservice teachers' understandings of transformations were
also examined by Edwards and Zazkis (1993) in a study of student
teachers' initial understandings of the concepts of reflection and
rotation. Edwards and Zazkis hypothesized that the student
teachers, having no previous instruction in transformation geometry,
would use primitive definitions of reflection and rotation even if
such examples were not readily available. Results based on initial
interviews with the 14 student teachers involved in the study as
well as written assignments completed by the student teachers
after computer experience with transformations supported the
previously stated hypothesis. The interviews included situations









where the subject was asked to rotate or reflect a figure on a table
such as a number nine or a rectangle. The students were not given
any parameters. For the most part, the students responded by
carrying out the request without asking for more direction. In all
responses, the subjects rotated the object around a center that was
located on the object or reflected the object over a mirror line that
ran through or adjacent to the object. After the interviews, the
subjects spent 2 1/2 hours using the TGEO computer microworld
described by Edwards (1990). The subjects worked individually
during this time playing a game that involved the rotation and/or
reflection of a figure so that the image of the figure completely
covered a congruent figure. The written assignment that followed
the computer experience involved the subject in a similar game
using paper and pencil. The written assignment also included
problems in which the student was asked to perform reflections and
rotations given the mirror line or the center and angle of rotation.
Edwards and Zazkis (1993) found that student solutions to the
games both on the computer and with paper and pencil showed
strategies based on using sequences of transformations to reach a
target rather than a single reflection or rotation with the mirror
line or center of rotation not located on the object. The solutions to
28% of the reflection and rotation problems were based on incorrect
concepts of transformation. The misconceptions seemed to be based
on the belief that rotations and reflections affect the object alone
rather that the plane containing the object. For example, if the
center of rotation was not located on the object, the student would









translate the object so that it was located on the center before
executing the rotation. Similarly, if the mirror line was not
adjacent to the object, the subject would translate the object so
that it was adjacent to the mirror line before reflecting the object
about the mirror line. Edwards and Zazkis concluded that these
prevalent misconceptions should not be ignored but should be
addressed so that students may build upon their prior knowledge to
understand new situations.
Kidder (1976) investigated fourth-, sixth-, and eighth-grade
students' ability to perform single transformations, compositions of
transformation, and inverse transformations. Fifteen students
identified by their teacher as belonging to the upper three-fourths
of the class in general ability, were administered a pretest dealing
with spatial analogies. Following the pretest, the investigator
spent 10 to 15 minutes with each student instructing them in an
"operational" definition of each of the three transformations
through demonstrations of the transformations and discussion of
student attempts at performing the transformations. Rigid figures
were transformed according to operations indicated by wire models
(lines and or arrows) for each transformation by both investigator
and student. The transformation test immediately followed the
session. Students were asked to use triangles made of toothpicks to
illustrate the indicated individual, composite, and inverse
transformations. "The data did not support the experimental
hypotheses that adolescents could perform Euclidean
transformations at the representational level" (Kidder, 1976, p. 49).









It seems reasonable to conjecture that 15 minutes was not ample
time to learn a new concept and be able to perform applications of
this concept on command, especially for the age group used in
Kidder's study. This lack of support might also be due to the choice
of model (for both object and operation).
Schultz and Austin (1983) studied the effect of transformation
direction on the difficulty of performing the transformations.
Slides, flips, and turns were investigated in horizontal, vertical, and
diagonal directions (for the purpose of consistency, slides, flips, and
turns will be referred to as translations, reflections, and rotations
respectively). Fifteen tasks involving each of these transformations
and directions were performed by 105 first-, third-, and fifth-grade
students. The students had no formal experience with
transformations. Prior to collection of the data, the researchers
insured that each student exhibited proficiency with copying the
task object (a sailboat) in several different orientations. Two
transparent platforms, one placed on top of the other, were
positioned in front of each student. The bottom platform contained a
flat model of a sailboat. The top platform was empty. After
watching the investigator perform a transformation of the top
platform, the students were to place a duplicate sailboat in the
appropriate location and orientation on the top (image) platform.
The students' responses to the tasks were scored on a scale from
zero to four. The scorer reliability coefficient was 1.0. The 15
tasks consisted of five translations, five reflections, and five
rotations. "[Translations] seem to be the easiest transformations









for students this age to visualize. However, the direction of the
transformation influences the relative difficulty of [rotations] or
[reflections]" (Schultz & Austin, 1983, p. 101).
Moyer (1978) investigated the connection between the
mathematical organization of transformations and the cognitive
structure of the learner. The following questions were addressed:
Is dynamic presentation more appropriate than static presentation?
Are reflections easier to perform than rotations and translations?
Under what structure do different age students perform topological-
like or Euclidean-like transformations? Nine tasks involving
translation, reflection, and rotation (referred to as slides, flips, and
turns respectively) of marked circles were administered to the
students. The tasks were divided into three categories: with color
clues and with motion, without color clues and with motion, and
with color clues and without motion. The students' responses on the
tasks were scored on a scale from zero to three. There were 24
randomly selected students from each of the following grade levels:
preschool, kindergarten, first, second, and third (120 subjects
total). Analysis of the data indicated that there was a statistically
significant difference between rotations with motion and without
motion, but motion was not a significant factor for the translation
and reflection tasks. The students performed best on the translation
task followed by the reflection task. The rotation task was the
most difficult for the students. Analysis of the data indicated a
direct correlation between grade level and Euclidean-like
structures. "The results of this study confirm that mathematical










and cognitive structures are not always in total accord" (Moyer,
1978, p. 90).
Williford (1972) also focused on primary grade students in his
investigation involving transformation geometry. The purpose of
Williford's study was to ascertain information regarding second-
and third-grade students' ability to perform transformations after
being taught about transformations through a specific teaching
strategy as well as the effects this instruction had on the students'
spatial ability. Sixty-three second- and third-grade students of
average or above average general classroom performance ability
(according to their teachers) were chosen for the study. The pre-
and posttest consisted of a spatial ability test and an achievement
test on congruence and transformations. After administration of the
pretest, the students were randomly divided into experimental and
control groups. The experimental groups were removed from class
for 12 sessions each lasting 25 to 30 minutes over the duration of
four to five weeks. During these sessions, the control groups
remained in class working on subjects unrelated to transformation
geometry. The experimental groups were involved in demonstrations
and activities related to congruence and transformations. The
control groups received a lesson including an overview of the
experimental groups' lessons. The lesson consisted of appropriate
vocabulary and examples dealing with each of the three rigid motion
transformations, reflection, rotation, and translation. These
lessons were designed to compensate for the lack of experience with









the terminology of the test. The decision to include the control
group treatment was based on information from a pilot study. The
researcher hypothesized that control group subjects may gain
knowledge of transformations through everyday experiences but may
not have the vocabulary to demonstrate this knowledge on the
posttest. Following the completion of the lessons, the posttest was
administered. The experimental groups performed significantly
better then the control groups on the achievement test; however,
there was no statistically significant difference between group
performance on the spatial ability test. The experimental groups did
not show success on the application questions on the achievement
test of transformations. This may explain the lack of significance
between groups on the spatial ability test, assuming that spatial
ability is actually an application of transformations. This inability
may also be explained in part by the age of the subjects (primary) or
by the lack of exploration involved in the use of technology.
Studies dealing with the rigid motion transformations that
have included substantial instruction on transformations have used
Logo or Motions (Edwards, 1990; Johnson-Gentile, 1990; Pleet,
1990). Logo involves a language for the computer that must be
taught to children in order for them to perform the transformations
on the computer. Children do not typically find the program to be
complicated; however, Logo takes time to learn. Motions is a
transformation program based primarily on Logo. The majority of
studies dealing with transformations have included little or no
instruction on transformations (Hart, 1981; Kidder, 1976; Law,









1991; Moyer, 1978; Schultz & Austin, 1983; Soon, 1989). After
reviewing the literature relevant to instruction with rigid motion
transformations, it became evident that there is a lack of consensus
between conclusions based on studies involving this topic.
The studies previously reviewed have provided a basis for the
discussion of the requirements for an appropriate instructional
environment for successful construction of the concepts of
reflection and rotation. According to the constructivist view of
teaching and learning mathematics, a thorough investigation of
students' ability to perform transformations must allow for an
environment which includes student opportunities for
communication with each other and the teacher as well as time to
explore the mathematical concepts so knowledge may be built. The
appropriate technology and a well organized instructional strategy
will help to create this environment.


Visualization
Visualization studies tend to be involved with either the
interaction between students' spatial ability and performance in
specified areas of mathematics (Battista, Wheatley, & Talsma,
1982; Connor & Serbin, 1985; Ferrini-Mundy, 1987; Kiser, 1990;
Moses, 1977; Perunko, 1982; Tillotson, 1984) or students'
trainability with visualization (Baker, 1990; Battista, Wheatley, &
Talsma, 1982; Ben-Chaim, Lappan, & Houang, 1988, 1989; Brinkmann,
1966; Connor, Schackman, & Serbin,1978; Connor & Serbin, 1985;
Ferrini-Mundy, 1987; Lord, 1985; Mendicino, 1958; Miller & Miller,









1977; Moses, 1977; Pallrand & Seeber, 1984; Smith & Schroeder,
1979; Tillotson, 1984). The current research involved both issues:
hence, a review of related literature in both areas is presented
below.
The relationship between aptitude with spatial visualization
and success in a computer enhanced instructional environment was
examined by Kiser (1990). Fifty-two students in two intact college
algebra classes taught by the same instructor at an aeronautical
university made up the subjects in the study. The Paper Folding Test
and Form Board Test (Ekstrom, French, Harman, & Derman, 1976)
were used to measure students spatial visualization ability. Both
classes in Kiser's study experienced the same course objectives and
criterion measures. The class instruction differed in mode of
presentation for one week during which both classes were to learn
the procedures for solving linear absolute-value inequalities. The
treatment group was exposed to a highly-visual, computer-enhanced
instruction (CEI) version of the lesson based on a discovery approach
to learning. The students were able to make conjectures about
solutions and test them using the graphic visual feedback of the
computer software while the teacher verified the algebraic
solutions on the chalkboard. The control group experienced a more
traditional presentation of the topic with the instructor working out
algebraic solutions on the chalkboard while using the overhead
projector to display graphic solutions. Kiser concluded that there
was a significant interaction between spatial ability and the
treatment. Furthermore, he found that "spatial ability is a









significant predictor of achievement of the CEI group but not the
traditional group" (Kiser, 1990, p. 95).
The relationship between spatial ability and mathematical
problem solving performance as well as the effect of instruction in
perceptual tasks on spatial ability were explored by Moses (1977).
The Card Rotation Test and Punched Holes Test were administered in
conjunction with the Figure Rotation Test, Form Board Test, and
Cube Comparison Test (Ekstrom, French, Harman, & Derman, 1976) to
acquire a spatial ability score. Moses created a problem solving
instrument to test mathematical problem solving performance. The
subjects were fifth-grade students in four classes at one
elementary school, two classes in the control group and two in the
treatment group. All classes were administered all tests as
protests. The control group received no training. The treatment
group received training with perceptual tasks during one class
period per week for nine weeks. The perceptual training consisted
of three instructional phases. The first phase involved hands on
experience with three-dimensional objects. The second phase
involved drawing of two- and three-dimensional objects. The third
and final phase involved the translation of mathematical word
problems into two- and three-dimensional representations. The
researcher taught all lessons to each class in the treatment group
and administered all tests to both treatment and control groups. The
tests were given again directly following the training to all classes
as posttests. Moses found that the Card Rotation Test and the
Punched Holes Test were good measures of visualization ability for










fifth-grade students. Moses found a positive correlation between
spatial ability and mathematical problem solving performance.
Training in perceptual tasks positively effected spatial ability but
not mathematical problem solving performance. The experimental
group experienced gains in the ability to solve spatial mathematical
problems whereas the control group did not experience such gains.
Among several relationships investigated by Perunko (1982)
was the relationship between spatial ability and mathematical
problem solving. Ninety-three community college students were
given tests including measures of students' ability to rotate visual
material (a measure of spatial ability) and students' ability to solve
algebra and geometry problems. The students had not experienced
success in high school mathematics courses and were enrolled in
developmental algebra II and geometry classes at the community
college. Perunko found a positive correlation between spatial
ability and performance on visual mathematics problems but not
verbal mathematics problems. It was also noted that students
approached mathematics problems that combined visual and verbal
components with visual techniques.
Baker (1990) also investigated issues of spatial visualization
with college students. Results from two related studies with
undergraduate students revealed that students preparing for spatial
related fields outperformed students preparing for less spatial
related fields on tests of spatial reasoning. These same spatially
oriented students reported more academic and nonacademic spatially









enriching experiences then their less spatially oriented peers. Baker
concluded that spatial reasoning could be improved through training.
However, this conclusion was based on instruction in four scheduled
mathematics classes; according to Baker, there was no control group
and there was high experimental mortality.
Tillotson (1984) and Connor and Serbin (1985) were interested
in both spatial visualization ability as a predictor of success in
mathematics and as a trainable skill. Tillotson worked with sixth-
grade students and Connor and Serbin worked with seventh- and
tenth-grade students in a series of three studies. All used the Card
Rotation Test and Punched Holes Test among other instruments to
measure spatial ability. Tillotson added the Cube Comparison Test
to determine the subjects' spatial index.
The specific purposes of Tillotson's research were to
determine if spatial visualization ability was a predictor of problem
solving performance and if training in spatial visualization would
improve spatial visualization ability as well as problem solving
performance. Students from two elementary schools all in the sixth
grade made up the subjects in the study. Students at one school
acted as the treatment group while students at the second school
made up the control group (102 subjects total). The treatment
lasted eight weeks and consisted of one 45-minute session per week
taught by the researcher. The pre- and posttest were administered
the week preceding and following the treatment respectively. The
visualization training for the treatment group consisted of
manipulating and imagining the manipulation of three dimensional









models, performing transformations of two dimensional drawings,
and solving problems. The control group received no visualization
training. Based on the collected data, Tillotson concluded that
spatial visualization is a trainable skill and a predictor of problem
solving performance. However, instruction in visualization skills
did not affect high visualization students differently from low
visualization students.
After testing seventh- and tenth-grade students with a wide
variety of both spatial orientation and visualization instruments and
measures of mathematical achievement in two separate studies,
Connor and Serbin (1985) concluded that spatial ability is a
predictor of mathematical achievement for both boys and girls. A
third study with eighth-grade students involved in a series of half
hour spatial ability training sessions resulted in significant
increases in spatial ability.
Ben-Chaim, Lappan, and Houang (1988, 1989) conducted a study
with sixth- through eighth-grade students that was motivated by
the researchers' observation that middle school students tended to
have difficulty communicating information about three dimensional
solids. Building description tasks were created "to determine
whether students' preference for representation mode and rate of
success on the task would be affected by instruction in spatial
visualization activities" (Ben-Chaim, Lappan, & Houang, 1989, p.
123). The tasks required the students to describe a three-
dimensional solid constructed out of cubes. The students were not
instructed how to communicate the information and the mode of









representation was not implied in the directions for the tasks.
Seven classes in each of the three grade levels participated in the
study. All subjects were administered a building description task as
the pre- and posttest with three weeks of spatial visualization
activities in the interim. A wide variety of representations for
building descriptions were present in both pre- and posttests,
however, out of the three modes of representation (verbal, graphic,
and mixed), the verbal mode was nearly eliminated after the
visualization activity intervention. The rate of success was also
dramatically improved following the visualization activities. It is
clear that Ben-Chaim, Lappan, and Houang's (1988) study supports
the claim that training can improve spatial visualization skills. "It
should be noted that a significant positive training effect was also
evident in students' performance on the Middle Grades Mathematics
Project (MGMP) Spatial Visualization Test. In addition,
a persistence over time of the effects was demonstrated" (Ben-
Chaim, Lappan, & Houang, 1989, p. 142).
As with the previous study, Brinkmann (1966) improved eighth-
grade students' spatial visualization through training. The training
in Brinkmann's study was in the form of programmed instruction in
elementary geometry including topics about points, lines, line
segments, rays, angles, planes, plane figures, and solids. The
program stressed problem solving rather than mathematical proof.
Sixty students from two eighth-grade classes were assigned to
treatment and control groups so that the groups were approximately
matched according to visualization pretest score and sex. The









treatment group followed the described programmed instruction
during mathematics class for three weeks. The control group
received no training. The treatment group performed significantly
better than the control group on a geometry inventory based on the
instruction as well as on spatial visualization.
Success in visualization training was also experienced in
studies as shown by Miller and Miller (1977) and Connor, Schackman,
and Serbin (1978) with first-grade students. Miller and Miller
illustrated this success in three similar experiments involving
visual perspective. Each study compared the students' ability to
coordinate projective space after receiving different levels of
training in perspectives from teacher directed visualization
activities to no visualization activities. In all but the first study,
the students experiencing the visualization training performed
significantly better on a test of perspective ability then students
not exposed to such training. The first study showed significant
improvement for boys receiving the training but not for girls.
Connor, Schackman, and Serbin (1978) found test-retest
improvement with the control group as well as effects from training
in spatial visualization with the treatment group. The test was
similar to the training; when given a different test of spatial
ability, the same gains in performance was not realized.
Smith and Schroeder (1979) investigated but did not find
differences between boys' and girls' spatial ability. In this study,
fourth-grade students were randomly assigned to two groups within
three intact classes. One group received the spatial visualization










test prior to training in tasks similar to those tested, while the
other group was tested after the training. The training and test
were both based on Tangram activities. Results indicated that
fourth-grade boys and girls were equally trainable in spatial
visualization and significantly outperformed students who were not
similarly instructed.
Findings from Ferrini-Mundy (1987) and Mendicino (1958) were
not consistent with the success in visualization training illustrated
in the studies previously discussed. Ferrini-Mundy investigated the
effects of a training program in spatial skills on college students
spatial ability and achievement in calculus. A random sample of 334
students was selected from 1054 preregistered Calculus I students
and randomly assigned to seven classes making up two treatment
groups and one control group. The treatment was made up of six
lessons presented via audiotapes and slides with students
completing a worksheet with each lesson. One treatment group did
not use accompanying manipulatives the other treatment group used
the manipulatives along with the lessons. The lessons involved two-
and three-dimensional tasks including rotations and translations of
two-dimensional representations to three-dimensional images. The
control group received no spatial training. Based on pre- and
posttest scores, Ferrini-Mundy found no treatment effect on the
calculus or spatial measures. "The case for enhancing college
students' spatial scores through the type of training in this study
was not supported by the results" (Ferrini-Mundy, 1987, p. 136).









Students' scores might have been different had the training involved
teacher-student and/or student-student interaction.
Mendicino (1958) compared the spatial perception scores of
tenth grade students enrolled in a vocational machine shop class to
students not enrolled in a vocational class of that nature. Pretests
of mechanical reasoning and space perception were given to both
groups at the beginning of the year and posttests were given at the
end of the year. No difference was observed between the group
receiving vocational machine shop instruction and the group not
receiving the instruction.
Lord (1985) worked with students on a program similar to
Ferrini-Mundy's but he experienced success in positively effecting
college students' spatial ability through training. He worked with
undergraduate biology students, randomly assigning 84 students to
treatment and control groups. The treatment group received training
with two- and three-dimensional figures; the control group received
no training. The treatment group met thirty minutes per week for
twelve weeks in the intervention. During the lessons, the students
worked with both manual and mental operations dealing with three-
dimensional figures. Most of the lessons involved imagining the
cross-sections of sliced three-dimensional objects. The
intervention was effective in enhancing the scores of the treatment
group on tasks similar to those practiced during the lesson.
However, differences in pre- and posttest scores on spatial
visualization and spatial orientation were also significantly in favor
of the treatment group.









Another study centered around improving the spatial ability of
undergraduate science students illustrated success in increasing
spatial ability (Pallrand & Seeber, 1984). Four groups of students
from a community college made up the subjects for this study: three
groups from an introductory physics course and one group from a
liberal arts course whose students had not taken and were not
planning to take physics. One physics group received the treatment,
one physics group received the placebo treatment, and one physics
group received no treatment. The liberal arts group was tested to
compare the spatial ability of physics students to non-physics
students. All students were given the same pre- and posttests
consisting of perception, spatial orientation, and spatial
visualization measures. Included in the tests were the Paper Folding
Test and the Card Rotation Test. The treatment intervention lasted
65 minutes per week for ten weeks and consisted of perspective
drawings, geometry of lines, planes, angles, solid figures, and
transformations, and relative position activities. The placebo
intervention was also 65 minutes per week for ten weeks but
consisted of lessons on the history of science. The control and
liberal arts groups received no intervention. All groups experienced
gains in spatial ability, however, the liberal arts group received the
smallest gain leading Pallrand and Seeber to believe that taking
physics may improve spatial ability of the students involved.
Additionally, "Visual-spatial scores of the liberal arts group were
lower than those of the physics sections, suggesting that visual-









spatial ability influences course selection" (Pallrand & Seeber,
1984, p. 507).
The following account of educational research with
undergraduate students focused on preservice elementary teachers.
Battista, Wheatley, and Talsma (1982) investigated the importance
of spatial visualization and cognitive development for learning of
geometry. They further investigated whether the geometry learning
would enhance spatial visualization of the same preservice
elementary teachers. Data were collected on 82 students enrolled in
a geometry course. Pre- and posttests were given to the students at
the beginning and end of the semester respectively. During the
semester, the students experienced lessons in symmetry, paper
folding, tracing, and Mira math as well as transformation geometry
through hands-on activities. Not only were spatial visualization, as
measured by the Purdue Spatial Visualization Test: Rotations, and
cognitive development, as measured by a modified version of the
Longeot test of cognitive development, directly related to
performance in the geometry class, but students spatial
visualization significantly increased by the end of the semester.
This latter finding suggests that the type of geometry lessons used
in the course may actually improve students' spatial visualization
ability.
Further proof of students' trainability with respect to
visualization skills was also exemplified in Iben's (1988)
comparative study between seventh- and eighth-grade students in
Japan and the United States. Iben investigated students'










development of spatial relations and found that Japanese students
had significantly more developed spatial relations than American
students of the same grade. Four explanations were offered for this
observed difference. Japanese students have spent approximately
2.4 school years more in school by the time they are in seventh- or
eighth-grade than U. S. students of the same grade. The Socratic
method is used to teach Japanese students whereas students in the
U. S. are primarily taught using direct instruction. Students in the
U. S. are generally given computational spatial relations activities
while Japanese students have hands-on spatial relations activities.
And, spatial relations is a large part of the early Japanese
curriculum through paper folding activities such as origami.
Students in the United States do not tend to be exposed to spatial
relations activities until seventh- or eighth-grade. Based on Iben's
observations, Japanese students' superior performance on spatial
relations activities compared to the performance on the same
activities by U. S. students may be due in part to the substantially
greater amount of time spent in such training as well as the type of
training received. As with the SIMS (1985) study, students in the
United States tend to experience much less hands-on activity in the
classroom then students in other countries.
Many of the reviewed visualization studies concluded that
students' visualization ability could be improved and that
visualization ability was a predictor of achievement with
mathematics. Study of the rigid motion transformation seems to be









an appropriate setting for visualization ability to be improved as
transforming geometric figures can involve the visualization of the
figures as they are flipped, turned and slid. This improvement of
visualization ability is important to increase students' achievement
in mathematics.


Limited English Proficient Students
Bilingual-education-program research and evaluation
have been driven by concerns for the development of
English and of academics among LEP students. These
studies have taken for granted the school mathematics
curriculum that LEP students are exposed to and, even
when problems in instruction are noted, those concerns
get cast in terms of language development. (Secada,
1992b, p. 218)

This same dilemma does not currently exist in school science
(Sutman & Guzman, 1992). Cheche Konnen (Rosebery, Warren, &
Conant, 1992; Warren & Rosebery, 1993; and Warren & Rosebery,
1992) is a program conducted by the Teacher Education Research
Center in Cambridge. Working with bilingual teachers and LEP
students, Rosebery and Warren ". are attempting to elaborate an
approach to science teaching and learning that supports the
development of scientific sense-making communities in the
classroom" (1992). Data have been collected describing the training
of teachers and the resulting experiences of the students. The
teachers in the program include bilingual teachers, English as a
second language teachers all with no training in teaching science,
and a science specialist.










In pilot projects, the researchers found that the training of the
teachers must include both science content and science pedagogy to
make effective facilitators, co-investigators, and mentors to the
students. Teachers needed to experience scientific sense-making in
a setting conducive to collaborative inquiry themselves. The
teachers are trained by the researchers in a setting similar to that
in which the students are placed. The teachers and the researchers
maintain contact throughout the program.
The emphasis on collaborative inquiry in the training of the
teachers and the experiences of the students build on Vygotsky's
(1978) belief, in that "robust knowledge and understandings are
socially constructed through talk, activity, and interaction around
meaningful problems and tools" (Rosebery, personal conversation,
2/17/94; Warren & Rosebery, 1992, p. 280). The collaborative
inquiry approach allows for shared responsibility in learning. This
is especially helpful for the LEP students as they are contending
with learning the language as well as the science content. Rosebery
and Warren might argue that language and science cannot be
separated.
Data have been collected with students in kindergarten, seventh
and eighth grade, and tenth grade. Exploration was based on diverse
content areas including weather patterns, water taste tests and
bacteria content, and swamp life. The science that the students
experience is partly controlled by the inquiries the students
generate. Results from data collection gathered from the









aforementioned settings suggest that students' knowledge of the
topic explored and their ability to reason about hypotheses and
experiments in an organized manner increased between September
and June.

A major goal of Cheche Konnen is to forge links between
learning science and doing science, and among science,
mathematics, and language. This is in large part what
makes it a powerful model for language minority
students, in particular, and perhaps for all students.
(Rosebery, Warren, & Conant, 1992, p. 3)

Sutman and Guzman (1992) reported a thematic based program
similar to Cheche Konnen for teaching science to LEP students. The
primary emphasis of this program is illustrated through its
assumption that instruction in science and the English language can
coexist effectively without placing excessive emphasis on the
students' first language. The authors believe that use of the
students' first language by the teacher may enhance instruction
when used judiciously, however, it is not entirely necessary.
Thematic lessons are used in this approach to scientific inquiry
and discovery to provide students the opportunity to become
accustomed to the vocabulary and syntax associated with one
concept. The authors described lessons called IALS (Integrated
Activity (based) Learning Sequences) that have been developed for
both the elementary and secondary curriculum. The lessons include
preparation and materials for teachers, objectives, hands-on
activities for students, and questions that students might ask.
Sutman and Guzman contend that even though a discovery/inquiry









lesson has been outlined, it is up to the teacher to maintain the
environment in which students are free to discover and inquire about
the concept.
Existing research in science education focused on providing
meaningful science instruction for LEP students is related to
meaningful mathematics instruction for LEP students. The
instruction of LEP students in both science and mathematics has as
its focus the language development of the students in a content
oriented situation or the students' development of conceptual
knowledge of the content while the associated language is
developed. The research in science education included in this review
had the latter as its focus as does this research.


Summary
Numerous studies have provided evidence that students can
learn to identify and perform the rigid motion transformations of
reflection and rotation and improve their spatial visualization.
Results from research have provided evidence that spatial
visualization is linked to achievement in mathematics. Projects
have set the framework for environments in which LEP students may
work within their zones of proximal development constructing
meaningful mathematics while simultaneously building their
associated English vocabulary. It is the synthesis of the information
gained from these valuable studies and the theoretical framework
previously described that provides the foundation and justification
for the current study.









This study extends the research of Edwards (1990, 1991) by
placing the students within an entire class during class time rather
than after school in groups of two or three, using computer software
that is much less restricting and much easier to learn than Logo, and
including LEP students in the sample. The duration of this study
which included three weeks of instruction using the Sketchpad to
construct the concepts of reflection and rotation was influenced by
the lack of generalizability of knowledge reported by Ernest (1986).
The students in Ernest's study received only six hours of instruction
with transformations.
Pleet's (1990) recommendation combined with information
gained from programs in the science education of LEP students lead
the researcher to make the decision to teach the experimental
treatment lessons due to experience using the Sketchpad in
computer laboratory settings.
The instruments used for this study were chosen based on
information gained from a review of the related literature. Hart's
(1981) test of students' ability to identify and perform reflections
and rotations, the basis of many previous studies, influenced the
types of problems and grading keys used in this study. The Card
Rotation Test and Paper Folding Test used as measures of spatial
visualization for this study were used for similar purposes in many
of the studies reviewed in this chapter.
The choice of instructional environment was predominately
influenced by the theoretical bases for the study. The constructivist
view of teaching and learning mathematics combined with the zone









of proximal development requires that the instructional environment
of the learner be such that collaborative inquiry is emphasized and
accommodated. Students working in pairs at computers to discover
the properties of reflection and rotation using computer software
that allows for student conjectures and provides visual feedback is
consistent with these requirements. Results from research
conducted by Soon (1989), Edwards and Zazkis (1993), and Kiser
(1990) reinforced the instructional environment used in this study
and Moyer (1978) provided evidence that the motion involved with
use of the Sketchpad might aid students' acquisition of the
concepts involved with rotation.
Many studies provided evidence that spatial visualization is a
trainable ability, however they also lead this researcher to believe
that the type of lesson used to increase students' spatial
visualization effects the extent of the increase in spatial
visualization. Instructional environments providing student-student
and/or student-teacher interaction and hands-on activities seemed
most effective. This study attempted to provide an environment
incorporating such interaction and activities.
















CHAPTER III
METHODOLOGY


Research Obiective
The purposes of this study included: (1) to explore the effects
of a dynamic instructional environment and students' visualization
ability on the identification of and ability to perform the rigid
motion transformations of reflection and rotation; and (2) to explore
the effects of the relationship between the students' level of
visualization and the instructional environment on the identification
of and ability to perform these rigid motion transformations.
Another purpose of this study was to investigate the effects of
students' English proficiency on identification and performance of
reflections and rotations and the effects of the relationship
between students' English proficiency and a dynamic instructional
environment on the identification of and ability to perform the rigid
motion transformations of reflection and rotation. Finally, the
study was designed to ascertain the effects of a dynamic
instructional environment and students' English proficiency on
students' visualization. The treatment in the dynamic instructional
environment included exploring the rigid motion transformations of
reflection and rotation through a dynamic presentation and










subsequent activities using The Geometer's Sketchpad (Jackiw,
1991). The following null hypotheses were tested:

1. Students' level of visualization will not interact with their
instructional environment to affect their ability to identify
and perform reflections and rotations in a (a) dynamic
testing environment or a (b) static testing environment.

2. Students' English proficiency will not interact with their
instructional environment to affect their ability to identify
and perform reflections and rotations in a (a) dynamic
testing environment or a (b) static testing environment.

3. Students' (a) instructional environment, (b) visualization
level, and (c) English proficiency will not affect their ability
to identify and perform reflections and rotations in a (a)
dynamic testing environment or a (b) static testing
environment.

4. Students' (a) instructional environment and (b) English
proficiency will not affect their (a) two-dimensional or (b)
three-dimensional visualization.


Measures
The measures for the study consisted of three covariates (one
continuous and two discontinuous) and four posttests. The
covariates consisted of the Card Rotation Test and Paper Folding
Test of the Kit of Factor-Referenced Cognitive Tests (Ekstrom et
al., 1976) as well as the Language Assessment Battery (LAB). Two
posttests were designed by the researcher and were versions of a
Rotation/Reflection Instrument. The Card Rotation and Paper
Folding Tests were also used as posttests.









Paoer Folding and Card Rotation Tests
The Card Rotation Test, S-1 of the Spatial Orientation Factor
and the Paper Folding Test, VZ-2 of the Visualization Factor were
used to measure students' visualization. The Card Rotation Test was
used to control for initial differences between the subjects and was
used as a continuous measure of students' two-dimensional
visualization. The students were divided into thirds according to
their scores on the Paper Folding Test. Based on their scores,
students were categorized as having high, medium, and low
visualization levels. It was also used as a continuous measure when
investigating students' change in three-dimensional visualization.
The Card Rotation Test required that the student differentiate
between figures that were equivalent in every way but orientation
from cards that had been reflected. The Paper Folding Test requires
that the student imagine a piece of paper being folded and having
holes punched through all thicknesses according to drawings. The
student was then to choose the appropriate result once the imagined
paper had been unfolded. Both tests contain two parts and each part
has a time limit of three minutes. Each of the tests calls for spatial
orientation and visualization skills required to identify and perform
rotations and reflections. The Card Rotation Test involves the
ability to visualize movement in two-dimensional space and the
Paper Folding Test involves the ability to visualize movement in
three-dimensional space. The Card Rotation Test and Paper Folding
Test were each used by Edwards (1991) in her investigation of
students' construction of the concepts of transformations.









Language Assessment Battery
The Language Assessment Battery was used as a measure of
English proficiency. Scores were provided by the students' middle
school and were used to categorize students as either English
proficient or limited English proficient according to district
mandated levels. Students scoring at or above the sixtieth
percentile were assumed to be English proficient, students scoring
below the sixtieth percentile were assigned to the limited English
proficient category.


Reflection/Rotation Instruments
A paper and pencil version and a computer version of the
Reflection/Rotation Instrument were designed by the researcher,
used in the pilot study, and validated for content by four experts.
The experts were two mathematics education professors, a
mathematician, and an educational technology professor. The
instruments were adapted, with permission, from the examination
designed for a large scale British study which was part of Concepts
in Secondary Mathematics and Science (Hart, 1981).
The computer and paper and pencil versions of the
Reflection/Rotation Instrument are similar in content and difficulty.
However, the computer version was more dynamic in nature than the
paper and pencil version as the computer version involved motion.
The paper and pencil version was more static in nature compared to
the computer version as the students were not exposed to motion
while taking the examination. The paper and pencil









Reflection/Rotation Instrument can be found in Appendix A. The
acetate overlays used to grade the paper and pencil version of the
instrument can be found in Appendix B. The objectives of both
versions of the Reflection/Rotation Instruments are given in Table
1. A description of the motions for the computer version of the
Reflection/Rotation Instrument is given in Table 2. The computer
and paper and pencil versions of the instrument were not equivalent
forms of the same test, therefore, the students' performance on the
two tests could not be statistically compared.


Pilot Study
A pilot study was conducted during May 1994. The study
included three classes taught by the researcher. The researcher
trained the classroom teacher in use of The Geometer's Sketchpad
during the six weeks preceding the pilot study. The pilot study
began with the administration of a personal inventory designed by
the researcher to determine students' language background and
proficiency with English. All classes were then given the Card
Rotation Test and Paper Folding Test. Following the administration
of the covariates, the classroom teacher and researcher instructed
the students from all three classes in use of The Geometer's
Sketchpad. Lessons on angle measure and distance were used to
train the students After the students became familiar with the
A
computer program, the paper and pencil and computer versions of the
Reflection/Rotation Instrument were administered to all classes as
protests. Following the tests, two classes were given the reflection









Table 1
Objectives of Reflection/Rotation Instruments


Items Objective: Test the student's ability to .


1-4 recognize examples and nonexamples of reflections.
5-8 perform reflections given a figure, a mirror line, and a
background grid.
9-12 perform reflections given a figure and a mirror line.
13-16 recognize examples and nonexamples of reflections when
the mirror line is hidden and draw mirror lines where
appropriate.
17-20 recognize examples and nonexamples of rotations.
21-24 perform rotations given a figure, a center point, and a
reference circle.
25-32 perform rotations given a figure and a center point.
33-36 draw the center of rotation given a figure and its image.
37-40 determine the correct composition of transformations
given the figures and their images without mirror lines or
center points.










Table 2
Motion in Computer Version of Reflection/Rotation Instruments


Items Description of motion: The figure moves from the
preimage ...


1-4 to the image following the properties of a reflection when
the example portrays a reflection.
5-12 to the mirror line then disappears. The movement follows
the properties of a reflection.
13-16 to the image following the properties of a reflection when
the example portrays a reflection.
17-20 to the image following the properties of a rotation. The
center of rotation may or may not be correctly drawn in
each example.
21-32 approximately 200 in the direction of the given reflection
then disappears. The movement follows the properties of a
rotation.
33-36 to the image following the properties of a rotation. The
center point is hidden in each example.
37-40 to the image to the image after composition following the
properties of rotations and reflections as appropriate. All
mirror lines and center points are hidden.









and rotation unit using the computer. The third class was given the
reflection and rotation unit without the computer. Upon completion
of the units, all students were given the paper and computer
versions of the Reflection/Rotation Instrument as posttests.
The Reflection/Rotation Instruments were scored by the
researcher using acetate overlays similar to those used to score the
CSMS exams (Hart, 1981). Reliability estimates were calculated for
the paper and computer versions of the Reflection/Rotation
Instrument based on the scores from the protests using the Kuder
Richardson (KR-20) formula. The estimates were found to be .87 and
.84 respectively.
There were no statistically significant differences between the
three classes on the posttests after controlling for initial
differences between students using the protests and the
visualization tests. Several observations by the researcher may
explain this lack of treatment effect.
All students were exposed to aspects of the treatment in
various forms from the pretest and teaching. The computer version
of the Reflection/Rotation Instrument was used as a pretest. This
version of the Instrument involves motion similar to that
experienced by the treatment group in the dynamic instructional
environment. Hence, while learning in a traditional instructional
environment, the control group may have been envisioning the
movement experienced during the administration of the pretest and
in this way, experiencing the treatment.









The researcher taught all classes. Due to the researcher's
extensive experience using The Geometer's Sketchpad to teach
transformations, the teaching of the control group was contaminated
with the treatment. This observation was dually noted by the
classroom teacher who indicated that the static lessons were not
taught using traditional means of explanation.
Lastly, the dynamic unit experienced by the treatment groups
did not include a sufficient number of lessons to highlight the
properties of reflections and rotations. This lack was due to the
belief held by the researcher at that time that the students would
not be capable of using the Sketchpad for lessons of such
complexity.
The following changes were based on previously described
observations made during and directly after the pilot study. The
paper and pencil and computer versions of the Reflection/Rotation
Instrument were not used as protests, only as posttests. The Card
Rotation Test was found to be highly correlated with the pretest and
was therefore sufficient for the continuous covariate. The Paper
Folding Test was to be used as a discontinuous measure of students'
level of visualization to determine whether or not this level
affected performance on the dependent measures.
Classroom teachers with no previous experience using The
Geometer's Sketchpad taught the control classes. Lastly, more
advanced lessons designed to address the properties of reflection
and rotation using The Geometer's Sketchpad were created to be
used with the treatment groups.









Sampling Procedures
Letters were sent to district mathematics supervisors of all
counties in Florida believed to have high percentages of LEP
students. The supervisors were asked for assistance in locating
middle schools having Macintosh computer laboratories and LEP
students enrolled in English speaking, heterogeneously grouped
mathematics classes. Based on the responses, a central Florida
public school district was chosen for the study. This district most
closely met the requirements of the study. The supervisor of that
county directed the researcher to a middle school having the
necessary attributes for the purposes of the study. The researcher
met with the chairperson of the mathematics department of the
school to request her participation in the study. The teacher and her
principal agreed to participate in the study and a proposal to conduct
research was submitted to the school district and subsequently
accepted. Both the pilot study and the study were conducted in the
same school. The pilot study was conducted during the school year
preceding the school year in which the data were collected for the
study. The pilot study was conducted with the chairperson present
using three heterogeneously grouped mathematics classes usually
taught by the chairperson. The remaining two eighth-grade teachers
were contacted during the summer between the pilot study and the
study and agreed to participate in the study.
This middle school had three eighth-grade teams, each eighth-
grade team had one mathematics teacher who taught four
mathematics classes. One mathematics class on each team was










made up of "advanced" students chosen at the end of their sixth-
grade school year based on mathematics grades. The other three
classes consisted of the remaining students, heterogeneously
grouped. All classes contained LEP students.
The researcher began the school year as a long-term substitute
teacher replacing the chairperson for the first five weeks of the
school year. Three classes normally taught by the chairperson and
one class taught by the remaining teachers were used as the
Treatment Group and taught by the researcher. The first three
classes made up all of the heterogeneously grouped classes taught
by the chairperson and the fourth class was chosen based on
heterogeneity of grouping and ease of scheduling. The remaining five
classes, two taught by one of the remaining teachers and three by
the other, made up the remaining heterogeneously grouped eighth
grade mathematics classes in the school. These five classes made
up the Control Group.


Experimental Treatment and Procedures


Training of Teachers
The teachers of the Control Group did not undergo training in
the use of The Geometer's Sketchpad This was necessary to insure
that the study would not be contaminated in the manner observed in
the pilot study. The teachers were briefed about the study and the
need for the teachers to avoid use of the Sketchpad until the









completion of the study. The teachers from the Control Group were
promised training in use of The Geometer's Sketchpad after
completion of the study.


Training of Students
The students from the Treatment and Control Groups were
taught how to use The Geometer's Sketchpad prior to the collection
of the data. All training was conducted by the researcher and lasted
one class period each. The middle school followed block scheduling,
therefore the students spent approximately two consecutive hours in
the mathematics class period every other day. The training lesson
was designed by the researcher to require students to use all of the
tools necessary to successfully complete the computer
Reflection/Rotation Instrument. At no time during the training did
the students witness or perform reflections or rotations using The
Geometer's Sketchpad. A detailed account of the lesson is in
Appendix C. Captured computer screens from the activities
associated with the lesson were printed and can be found in
Appendix D. The teachers for the Control Group administered the
Card Rotation and Paper Folding Tests to the Treatment Groups while
their classes were trained by the researcher.


Data Collection
Following the training on use of The Geometer's Sketchpad, all
classes were given the Card Rotation Test and Paper Folding Test.
After completion of the above measures, the Control Group was









taught about the concepts of reflection and rotation using the
traditional textbook approach. The control group teachers used
Chapter Eight of Transition Mathematics (Usiskin et al., 1990), the
adopted textbook for eighth-grade at this school. This chapter
addresses the rigid motion transformations.
The researcher taught the lessons in the computer lab using
The Geometer's Sketchpad. The treatment classes were held in the
Macintosh computer lab throughout the unit on reflections and
rotations. Since there were 15 computers and from 28 to 32
students in each mathematics class, the students usually worked in
pairs at the computers. For the most part, the students chose their
own partners. Several pairs included students whose first language
was Spanish. These students tended to converse in their first
language when discussing properties of the geometric
transformations as well as problems posed by the researcher.
The researcher assumed the role of facilitator and problem
poser for the Treatment Group. An experimental version of The
Geometer's Sketchpad was used throughout the study. The
transformation commands; translate, rotate, reflect, mark center,
and mark mirror were accompanied by descriptive icons (pictorial
representations) for each command. The adjusted menu of the
experimental version was requested to aid in the students'
acquisition of vocabulary related to transformations. The
researcher did not call attention to the icons at any time during the
study. A subsample of the Treatment Group was given a short









questionnaire about the usefulness of the icons approximately two
weeks after the administration of the posttests.
The reflection and rotation units for the Treatment and Control
Groups each lasted approximately eight class periods over three to
four weeks. The Card Rotation Test and Paper Folding Test as well
as the paper and computer versions of the Reflection/Rotation
Instrument were administered to both groups at the end of the units.


Treatment Lessons
All experimental treatment lessons were preplanned so that
any teacher with adequate knowledge of geometry, middle school
pedagogy, the computer software, and computer lab pedagogy could
teach using the lesson plan. Lesson plans for all experimental
treatment lessons have been included in Appendix E. A brief outline
follows:


Lesson 1
Objective: to have students learn how to execute reflections
and rotations using the Sketchpad.
Procedure: students worked in pairs on exercises requiring
students to conjecture about positions of mirror lines and centers of
rotation so that the images of figures after the transformations
would rest entirely within the interior of given squares. Students
were given keys to help with the steps to perform reflections and
rotations (see Appendix F). A typical screen is located in Appendix
G









Lesson 2
Objective: to have students make conjectures about positions
of the images of figures after a reflection or a rotation.
Procedure: students worked in pairs conjecturing about the
position images after reflections or rotations given a figure and
either a mirror line or a center and angle of rotation. The students
checked each other's conjectures by performing the given
transformation using the Sketchpad. The class then played a game
involving more practice with similar conjectures.


Lesson 3
Objective: to allow for more practice in conjecturing with
rotations and to introduce the computer project.
Procedure: The students practiced conjecturing using the same
format used during the previous lesson. The instructor then
introduced the computer project. The students spent the remaining
class time working on the computer projects. The computer project
is described in detail in the description of the lessons in Appendix E.


Lesson 4
Objective: to have students work on computer project.
Procedure: students worked in pairs on computer project
introduced during lesson 3.










Lesson 5
Objective: to have students complete and present computer
projects.
Procedure: Students spent first part of class time completing
projects and remaining class time presenting projects to
classmates.


Lesson 6
Objective: to have students present remaining projects and to
review properties of reflections and rotations.
Procedure: students spent first part of class time presenting
projects to classmates and remaining class time reviewing the
properties of reflections and rotations with the instructor.


Research Design


Quantitative Methodology
A three factor, nonequivalent control-group design was used for
the study. This quasi-experimental research design involved a
2X2X3 matrix to examine three factors. The three factors were the
level of computer use, the level of English proficiency, and the level
of visualization of the students. There were two groups, one
Treatment Group and one Control Group, with four classes in the
Treatment Group and five classes in the Control Group. The
Treatment Group used the computer unit on reflections and









rotations. The Control Group followed the lessons included in the
text used by the middle school.
An analysis of covariance (ANCOVA) was used to control for
initial differences between groups. Two tests from the Kit of
Factor-Referenced Cognitive Tests (Ekstrom, et. al., 1976) and the
Language Assessment Battery (LAB) served as covariates. The Paper
Folding Test was the discontinuous measure used to assign students
to visualization levels. The Card Rotation Test was the continuous
covariate used to control for initial differences between groups.
The objective of the design was to determine the effects of the
independent variables (computer use, English proficiency, and
visualization level), individually and interactively, on the dependent
variables (posttest scores). Of interest were the interaction
between computer use and English proficiency and the interaction
between computer use and visualization level.
The performance on two dependent variables was investigated,
the computer version and the paper version of the
Reflection/Rotation Instrument. Two separate ANCOVAs were run to
determine the effects of the independent variables on each of the
dependent variables. The first ANCOVA described the effect of the
independent variables individually and interactively on the
performance of the sample on the computer version of the posttest.
The second ANCOVA described the effect of the independent
variables individually and interactively on the performance of the
sample on the paper version of the posttest.






69


Additional hypotheses were tested using a 2X2 matrix to
examine the effects of a dynamic instructional environment and
English proficiency on students' visualization. The last two
ANCOVAs were run with the purpose of describing the effects of the
two independent variables on the performance of the sample on the
Card Rotation Test and Paper Folding Test.















CHAPTER IV
QUANTITATIVE RESULTS


Descriptive Statistics


Sample
The sample consisted of all students enrolled in
heterogeneously grouped, English language, eighth-grade
mathematics classes in a central Florida middle school. Data were
collected from 241 students, 109 in the Treatment Group and 132 in
the Control Group. Table 3 describes the sample according to race
and Table 4 describes the sample according to sex.


Table 3
Frequency and Percentage of Race by Group
Race Treatment Group Control Group Sample
African American 8 (7.3%) 9 (6.8%) 17 (7.1%)
American Indian 0 (0%) 1 (0.8%) 1 (0.4%)
Asian American 3 (2.8%) 1 (0.8%) 4 (1.7%)
Caucasian 63 (57.8%) 79 (59.8%) 142 (58.9%)
Hispanic American 35 (32.1%) 42 (31.8%) 77 (32%)









Table 4
Frequency and Percentage of Sex by Group
Sex Treatment Group Control Group Sample
Female 50 (45.9%) 67 (50.8%) 117(48.5%)
Male 59 (54.1%) 65 (49.2%) 124 (51.5%)



Instruments
Descriptive statistics for the Paper and Pencil
Reflection/Rotation Posttest, Computer Reflection/Rotation
Posttest, Card Rotation Test, and Paper Folding Test for the entire
sample, Experimental Group, and Control Group are given in Tables 5,
6, and 7 respectively. The mean, standard deviation, and range have
been calculated for each instrument.


Table 5
Entire Sample
Posttest Means, Standard Deviations, and Ranges for All Instruments
Instrument Mean S. D. Range Total Points
Paper and Pencil Test 17.7 6.9 5 36.5 40
Computer Test 18.2 7.9 2-36 40
Card Rotation Test 98.7 39.4 -5-158 160
Paper Folding Test 7.1 4.5 -3.8-20 20









Table 6
Experimental Group
Posttest Means, Standard Deviations, and Ranges for All Instruments
Instrument Mean S. D. Range Total Points
Paper and Pencil Test 20.9 7 5 36.5 40
Computer Test 21.5 7.6 3.5 36 40
Card Rotation Test 102.5 38.9 -3-154 160
Paper Folding Test 7.5 4.2 -1.3 16.3 20



Table 7
Control Group
Posttest Means, Standard Deviations, and Ranges for All Instruments
Instrument Mean S. D. Range Total Points
Paper and Pencil Test 14.9 5.3 5.5 31 40
Computer Test 15.4 6.9 2-35 40
Card Rotation Test 95.6 39.5 -5 158 160
Paper Folding Test 6.7 4.6 -3.8 20 20




Statistical Analysis
Initial differences were controlled for through the use of the
Card Rotation Test (correlates strongly with Reflection/Rotation
Instruments), the Paper Folding Test (determines visualization
level), and the Language Assessment Battery (distinguishes between
limited English proficient (LEP) and English proficient (EP)
students). There were no statistically significant interactions
between the students' level of visualization and the treatment based
on the results of both the Dynamic and Static Reflection/Rotation
Instruments (see Tables 8 and 9). There was no statistically









significant interaction between students' English proficiency and
the treatment based on the results of both the Dynamic and Static
Reflection/Rotation Instruments (see Tables 8 and 9). Therefore,
the following null hypotheses could not be rejected:

Students' level of visualization will not interact with their
instructional environment to affect their ability to identify
and perform reflections and rotations in a (a) dynamic
testing environment or a (b) static testing environment.

Students' English proficiency will not interact with their
instructional environment to affect their ability to identify
and perform reflections and rotations in a (a) dynamic
testing environment or a (b) static testing environment.

Finding no significant interaction, the interaction terms were
removed from the model. After removal of the interaction terms,
students' instructional environment was a statistically significant
variable on performance on both the dynamic and static
Reflection/Rotation Instruments (see Tables 10 and 11).
Visualization level was also a significant predictor of students'
performance on the dependent measures (see Tables 10 and 11) with
students at the high visualization level performing better than
students at the medium visualization level, as expected. Students at
the medium visualization level performed better than students at
the low visualization level (see Table 12). Furthermore, it was
determined that LEP students did not perform statistically
significantly differently on either the dynamic Reflection/Rotation










Table 8
Analysis of Covariance: Dynamic Reflection/Rotation Instrument

Source DF SS F Ratio p

Card Rotation Test 1 1385.75 35.12* 0.00
Visualization Level 2 550.41 6.97* 0.00
English Proficiency 1 65.00 1.65 0.20
Treatment 1 80.92 2.05 0.15
Treatment x Viz. Level 2 43.08 0.55 0.58
Treatment x Eng. Prof. 1 35.47 0.90 0.34
Model 8 5520.53 17.49* 0.00
Error 212 8365.73

Note: *significant for p=.01



Table 9
Analysis of Covariance: Static Reflection/Rotation Instrument

Source DF SS F Ratio p

Card Rotation Test 1 883.50 29.68* 0.00
Visualization Level 2 466.52 7.84* 0.00
English Proficiency 1 27.81 0.93 0.33
Treatment 1 103.44 3.47 0.06
Treatment x Viz. Level 2 111.83 1.88 0.16
Treatment x Eng. Prof. 1 6.78 0.23 0.63
Model 8 4412.75 18.53* 0.00
Error 212 6490.14

Note: *significant for p=.01










Instrument or the static Reflection/Rotation Instrument when
compared to their EP peers (see Tables 10 and 11).
Therefore, the following null hypothesis could be rejected:

Students' (a) instructional environment and (b) visualization
level will not affect their ability to identify and perform
reflections and rotations in a (a) dynamic testing
environment or a (b) static testing environment.

The following null hypothesis could not be rejected:

Students' English proficiency will not affect their ability to
identify and perform reflections and rotations in a (a)
dynamic testing environment or a (b) static testing
environment.



Table 10
Analysis of Covariance: Dynamic Reflection/Rotation Instrument

Source DF SS F Ratio p


Card Rotation Test 1 1425.00 36.26* 0.00
Visualization Level 2 1104.64 14.05* 0.00
English Proficiency 1 34.39 0.88 0.35
Treatment 1 1858.14 47.28* 0.00
Model 5 5437.17 27.67* 0.00
Error 215 8449.09

Note: *significant for p=.01










Table 11
Analysis of Covariance: Static Reflection/Rotation Instrument

Source DF SS F Ratio p

Card Rotation Test 1 870.03 29.07* 0.00
Visualization Level 2 607.92 10.16* 0.00
English Proficiency 1 32.64 1.09 0.30
Treatment 1 2021.52 67.55* 0.00
Model 5 4289.21 28.67* 0.00
Error 221 10902.89

Note: *significant for p-.01



Table 12
Means by Visualization Level on Reflection/Rotation Instruments
Visualization Level Mean for Dynamic Test Mean for Static Test
High Visualization 21.65 20.22
Medium Visualization 18.59 18.57
Low Visualization 14.49 14.76




After the researcher controlled for initial differences on
pretest scores, the students who received the dynamic treatment
performed significantly better on the Card Rotation Test at the .05
level than their peers who did not receive the dynamic treatment
(see Table 13). There was no statistically significant difference
between the Treatment and Control Groups on the Paper Folding Test
(see Table 14). Once again, LEP students did not perform









statistically significantly differently than their EP peers on either
of the dependent measures (see Tables 13 and 14). Therefore, the
following null hypothesis could be rejected:

Students' instructional environment will not affect their
two-dimensional visualization.

The following null hypotheses could not be rejected:

Students' instructional environment will not affect their
three-dimensional visualization.

Students' English proficiency will not affect their (a) two-
dimensional visualization or (b) three-dimensional
visualization.



Table 13
Analysis of Covariance: Card Rotation Test

Source DF SS F Ratio p

Card Rotation Pretest 1 213366.74 403.00* 0.00
English Proficiency 1 368.71 0.70 0.40
Treatment 1 3078.71 5.81* 0.02
Model 3 227300.83 143.10* 0.00
Error 227 120185.47

Note: *significant for p=.05









Table 14
Analysis of Covariance: Paper Folding Test

Source DF SS F Ratio p

Paper Folding Pretest 1 2117.31 192.32* 0.00
English Proficiency 1 0.25 0.02 0.88
Treatment 1 26.38 2.40 0.12
Model 3 2166.02 65.58* 0.00
Error 224 2466.08

Note: 'significant for p=.01



Limitations of the Study
The present study had several limitations including the fact
that the researcher instructed the experimental group throughout
the study. The decision to teach the treatment group was based on
previous studies (Pleet, 1990; Rosebery, Warren, & Conant, 1992;
Warren & Rosebery, 1993; and Warren & Rosebery, 1992) in which
recommendations were made to insure that teachers using
computers in lab settings have excellent command of the software
as well as the pedagogical methodology required in computer labs.
The student scores on the Language Assessment Battery (LAB)
were supplied by the students' middle school. The students were not
tested simultaneously but were tested throughout the previous
school year and during the first week of the current school year.
Students who were tested during the previous school year may have
increased their English language proficiency to the extent that they









would not have fallen below the sixtieth percentile on the LAB. This
may have altered the classification of some students as limited
English proficient.
The students in the experimental group were videotaped by an
individual other than their teacher during the first class period
spent in the computer lab. This limitation was removed directly
following that session due to the belief held by the researcher that
the video taping was effecting the students' performance. Students
seemed to be distracted by the camera when they believed it was
focused on them. A stationary, unmanned camcorder was used for all
subsequent sessions in the computer lab. The stationary camcorder
did not seem to elicit a similar reaction however some students
were aware of its presence.
The computer and paper and pencil instruments were similar in
objectives and content. However, they were not equivalent forms so
they could not be compared statistically. Because of the nature of
the computer test, it was helpful if students were unsure of
directions of positive and negative angles. The movement involved
with the computer test did not assist the students to proper
solutions if they did not already possess substantial knowledge of
the concepts of reflection and rotation. The paper and pencil test
was not any more helpful than the computer test if students did not
understand the content* it did, however, allow students to turn and
fold the paper to aid in visualization of solutions. Equivalent forms
of the computer and paper and pencil tests would be necessary for an
objective analysis of the tests' comparative instructional value.









The sample consisted of all nine heterogeneously grouped
eighth-grade mathematics classes in the middle school. In order to
include four classes in the treatment group, the researcher was
required to teach one class that was not originally on the
researchers' schedule. The other three classes had been taught by
the researcher from the first day of school, whereas the fourth
class was added after the students had been in school for four days.
This class was the least cooperative of the four classes possibly
due to the students' belief that they were being taught by a
"substitute".
The students labeled as advanced in mathematics were taught
in homogeneously grouped classes and were thus excluded from the
study. The advanced group were those in the top quartile of the
mathematics students in the middle school. These students may
have benefitted from the treatment differently from the non-
advanced students.
Finally, the classes were intact and nonrandomly assigned to
treatment and control groups. This is not uncommon among studies
conducted in school settings.
















CHAPTER V
QUALITATIVE RESULTS


Methodology
The students' computer projects, observations of students
working on activities or their computer projects, and the data
gathered from student responses on the computer and paper and
pencil Reflection/Rotation Instruments afforded the researcher
valuable insights into students' understandings and
misunderstandings of reflection and rotation and related concepts.
The students' computer projects were helpful in assessing students'
ability to measure distance and angle, negotiate the meaning of
positive and negative directionality, and use the properties of a
circle as they apply to measurement.
Observations were conducted throughout the study of students
in the Treatment Group. The observations were audiotaped and
videotaped through the use of a stationary audio/video camera set up
in the corner of the lab and left running for the duration of each
lesson. The tapes were viewed and portions involving students
working on activities or their computer projects that might reveal
insight were transcribed and interpreted. The purposes of the
observations were to reach a better understanding of how students










interacted together using The Geometer's Sketchpad to construct
the concepts of reflection and rotation and to determine whether the
dynamic computer environment is an effective instructional setting
for this construction to take place. An additional purpose of the
observations was to gather insight into visualization strategies
used by students as they performed rotations and reflections using
the computer. Of particular interest was LEP students'
collaboration as appropriate English vocabulary was developed for
reflections and rotations.
The data gathered from student responses on the
Reflection/Rotation Instruments provided evidence of student's
actual success or lack of success with recognizing and performing
reflections and rotations. The activities and the computer project
were part of the treatment therefore the Control Group did not take
part in either. The Reflection/Rotation Instruments were
administered to both Treatment and Control Groups.
A short questionnaire was administered to a subsample of the
treatment group in order to determine whether or not the
experimental version of The Geometer's Sketchpad was helpful in
students' acquisition of the mathematical vocabulary involved in
reflections and rotations. The experimental version of the
Sketchpad used for this study incorporated icons, or pictorial
representations, as clues to appropriate menu commands. The
commands under transformations were coupled with symbols
designed to provide context clues for the associated vocabulary.









Computer Projects
A major goal of the computer project was to have the students
explore and discover the properties of reflection and rotation. The
students, working in pairs, were to use up to four screens (on The
Geometer's Sketchpad) to demonstrate properties of reflection and
rotation. The students were required to use drawings, color, at
least one circle, measurement, and written explanation to illustrate
the properties they had discovered. The project requirements were
designed to influence the students' use of multiple representations
to communicate about various properties of reflection and rotation.
The multiple representations used by the students supplied evidence
of the depth of the students' conceptual knowledge of the properties
of reflections and rotation. In figure 1, the property that an object
and its image after reflection will be equidistant from the mirror
line was represented using measurement, drawing, and writing.
Various uses of the circle were also observed. Figure 2
illustrates two students' use of circles to verify the relationship
between dimensions of a flag and its images after rotation, the
circles also show that the flags are all the same distance from the
center point, however, the students do not call attention to this
fact. Figure 3 exhibits the circles' use as a measure of distance
from the center to any point on the circle as equidistant. Figure 4
displays the circle as part of a drawing.
Some students defined the same term in various ways, for
instance, the direction of positive and negative angles in figure 5.








Other students discovered properties that far exceeded the
expectations of the researcher as in figure 6.




REFLECTIONS


THE FLAGS
ARE ALWAYS
THE SAME
DISTANCE
AWAY FROM
EACH OTHER.


Distance(J to E1) = 4.00 ca


Distance(J' to El) = 4.90 cm


Distance(L to N) = 3.32 cr Distance(L' to N) = 3.32 cm


Distance(B1 to K) 4.83 cm Distance(K' to BI) = 4.83 cm
Distance(B1 to K) = 4.83 cm Distance(K' to B1) = 4.83 cm


THE FLAGS
ARE ALWAYS
THE SAME
SIZE,SHAPE,
COLOR,AND
FACE THE
OPPOSITE
DIRECTION
FROM EACH
OTHER


Figure 1
Equidistance Including Students' Written Comments










ROTATIONS


-450 w


i-135o


ALWAYS THE
SAME DISTANCE
FROM THE
CENTER POINT.

Distance(D1 to Fl) 1.98 1800


THE CIRCLE SHOWS
THAT THE FLAGS ARE
THE SAME DISTANCE
FROM TOP TO
BOTTOM.

90*



Figure 2
Multiple Uses of Circles Including Students' Written Comments






























THE CIRCLES ARE SHOWING THAT THE OBJECTS
ARE T'i-IE SAME DISTANCE FROM THE CENTER
POINT ALL AROUND.



Figure 3
Circle as Measure of Distance Including Students' Written Comments





Este dibujo es otro ejemplo de REFLECION.


joy


Ao


Figure 4
Circle as Part of Drawing Including Students' Written Comments


t -af


t:












This is a good way to explain
rotationrbecause it makes iteasier to find
the results to your problem and will also
help you leam rotations faster and better.
MAIN POINT The drawing is showing you the rotations of
the MAIN flag point It shows that the
4 negative stations go clockwise and that the
positive rotations go counter dock wise. This
drawing also shows that if you draw a circle
with center at rotation the circle will touch
the flag at the same spot as the other flags.
900 .900


When you go this
way it equals the
negative rotation!

180* When you go this
-180o way t equals the
positive rotation!





Figure 5
Positive and Negative Angles Including Students' Written Comments














These quadralatrals are 45 degrees apart.
We fit 8 quadralatrals in 360 degrees---


Figure 6
Repeated Rotations Including Students' Written Comments









Observations
The Treatment Group was audiotaped and videotaped through the
use of a stationary audio/video camera set up in the corner of the
lab and left running for the duration of each lesson. The tapes were
viewed and portions were transcribed and interpreted. The purpose
of the observations was to reach a better understanding of how
students interacted together using The Geometer's Sketchpad to
construct the concepts of reflection and rotation in order to
determine whether the dynamic computer environment is an
effective instructional setting for this construction to take place.
An additional purpose of the observations was to gather insight into
visualization strategies used by students as they performed
rotations and reflections using the computer. Of particular interest
was LEP students' collaboration as appropriate English vocabulary
was developed for reflections and rotations.
Each of the following observations was based on the
aforementioned audio/video recordings taken while students were
working in pairs on activities or their computer projects using The
Geometer's Sketchpad. Student conversations dealt with
conjecturing about reflections and rotations or meeting the
requirements of the assigned computer project. The requirements of
the computer project included the use of four screens or less to
illustrate the properties of reflection and rotation. The students
were to illustrate the properties both pictorially as well as
verbally. Line segments, circles, measures, and color were all
necessary components of the project. A brief description of the









setting for each observation precedes the associated transcription.
The captured computer screen resulting from the students'
collaboration follows the transcription of each conversation leading
to the completion of a computer project.


Observations from Activities
Shortly after the class had begun to practice conjecturing with
reflections, two students called the teacher to their computer.
According to the students, they kept choosing incorrect locations of
images after reflection and they did not understand why their
conjectures were wrong. Together, the teacher and the students
determined that the conjectures were actually correct but the
procedures to test the conjectures were incorrect. The students
were not marking the predetermined mirror line before reflecting
the figure. In this case, the students were questioning what had
previously been considered as the "authority" when they could not
make sense of the "solutions".
Once students had learned how to perform reflections and
rotations using the Sketchpad and had played the reflection and
rotation games, the students were directed to work together
conjecturing about positions of images after reflection and rotation.
When conjecturing about reflections, one student was to choose the
location of a figure and construct a mirror line, the other student
was to draw an outline of the figure in the location he/she believed
the image would lie after the transformation had been performed.
The following observation took place while two students were




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