AN INVESTIGATION OF STUDENT CONJECTURES IN STATIC AND DYNAMIC
GEOMETRY ENVIRONMENTS
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. The
dissertation does not include proprietary or classified information.
_________________________________________
John M. Gillis
Certificate of Approval:
__________________________ ________________________
Marilyn E. Strutchens W. Gary Martin, Chair
Associate Professor Associate Professor
Mathematics Education Mathematics Education
__________________________ ________________________
Margaret Ross Chris Rodger
Associate Professor Professor
Educational Foundations Mathematics
___________________________
Stephen L. McFarland
Acting Dean
Graduate School
AN INVESTIGATION OF STUDENT CONJECTURES IN STATIC AND DYNAMIC
GEOMETRY ENVIRONMENTS
John M. Gillis
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
In Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 13, 2005
iii
DISSERTATION ABSTRACT
AN INVESTIGATION OF STUDENT CONJECTURES IN STATIC AND DYNAMIC
GEOMETRY ENVIRONMENTS
John M. Gillis
Doctor of Philosophy, May 13, 2005
(M.S., Auburn University, 2002)
(Ed.S., Columbus State University, 1998)
(M.Ed., Columbus State University, 1997)
(B.S., University of Florida, 1989)
171 Typed Pages
Directed by W. Gary Martin
This study was designed to investigate the mathematical conjectures formed by
high school geometry students when given identical geometric figures in two different
types of geometric environments. Student conjectures formed in a static geometry
environment were compared with those formed in a dynamic geometry environment
generated by dynamic geometry software. These conjectures and the environments in
which they were formed were examined both quantitatively and qualitatively.
Results indicate that students who used dynamic geometry software made more
relevant conjectures, fewer false conjectures, and the conviction in the correctness of
their conjectures was higher when compared to students working in a static geometry
iv
environment. These differences were found to be statistically significant using linear
regression analysis.
Qualitative data was collected by means of participant observations, a survey
instrument, selected participant interviews, and a qualitative analysis of the conjectures
made by the students in each environment. Qualitative analysis focused on the following
themes: Student concepts of conjecture and proof, student preferences concerning each
environment, the kind of language used in the conjectures formed in each environment,
the ability to find counterexamples using dynamic geometry software, the ?dragging?
techniques used by the participants using dynamic geometry software, and the students
conviction in the output generated by dynamic geometry software.
Results indicated a strong preference for the dynamic environment and a high
conviction in the output generated by dynamic geometry software. The language used in
forming conjectures in the dynamic environment was noticeably different and reflected
the environment itself. The participants? concept of proof included both inductive and
deductive frames when dynamic geometry software was available, and many of the
students had difficulty with forming and finding of counterexamples using dynamic
geometry software when confronted with a false conjecture.
v
ACKNOWLEDGEMENTS
The author would like to thank the participants of this study, the administers of
the school used in this study, the teachers and educators that served as outside graders,
the advisory committee and chair for their support, and my wife and daughter for their
patience and understanding.
vi
Style Manual Used:
Publication Manual of the American Psychological Association, Fifth Edition
Computer Software Used:
Microsoft Word 2002
Geometers? Sketchpad version 4.02
vii
TABLE OF CONTENTS
CHAPTER I: INTRODUCTION????????????...?????????1
CHAPTER II: LITERATURE REVIEW ???.........................................................?.7
CHAPTER III: METHODOLOGY ????????????.................................40
CHAPTER IV: QUANTITATIVE RESULTS ????????????????55
CHAPTER V: QUALITATIVE RESULTS ?????????????????72
CHAPTER VI: DISCUSSION ??..?????????????..????..?102
REFERENCES ????????????????????.???????..122
APPENDIX A: LAB ACTIVITIES WITH RUBRICS ?.............................................127
APPENDIX B: VAN HIELE INSTRUMENT WITH RUBRIC ????????...153
APPENDIX C: PARTICIPANT SURVEY ????????????????...165
APPENDIX D: INTERVIEW PROTOCAL ????????????????.166
APPENDIX E: QUANTITATIVE DATA ?????????????????169
viii
LIST OF FIGURES
1. Dragging a Point to Form a Continuum of Parallelograms in a Dynamic
Geometry Environment ????????????????????????..2
2. A Flow Chart of Student Research in Geometry Showing the Role of
Conjecture and Counterexample in the Proof Process ????????????.23
3. A Continuum of Quadrilaterals under the Drag Mode Using
Geometers Sketchpad ????????????????????????...26
4. The Exploration of the Midsegment Quadrilateral ?????????????..27
5. Shipwreck Problem Used by Mudaly and de Villiers (1999)???????...........35
6. Constructions of Triangles with One Congruent Side and Two Congruent Angles ?38
7. Participant Distribution of 2003 PSAT Mathematics Scores ?????????..41
8. First Situation Posed to the Interview Participants in the Dynamic Environment
Along with the Given Conjecture ??????????????????...?.77
9. A General Quadrilateral with a False Conjecture and True Theorem ??????.85
10. A Concave Quadrilateral with Angle Measures Shown ???????????91
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LIST OF TABLES
1. Participants? Gender and Racial Demographics ??????????????..42
2. Means and Standard Deviations for Both Assignment Cases for the Three
Dependent Variables in Both Environments ????????????????56
3. Means in Terms of Environment for Each of the Dependent Variables ?????..57
4. Means and Standard Deviations for the Covariate ACHIEVEMENT ??????.57
5. Means and Standard Deviations for the Covariate VH ????????????58
6. Zero Order Pearson Correlation Coefficients for the Odd Assignment Case ???..59
7. Zero Order Pearson Correlation Coefficients for the Even Assignment Case ???.60
8. Regression Analysis for the Variable RELEVANT in the Odd Assignment Case ?..62
9. Regression Analysis for the Variable RELEVANT in the Even Assignment Case ?63
10. Regression Analysis for the Variable FALSE in the Odd Assignment Case ???64
11. Regression Analysis for the Variable FALSE in the Even Assignment Case ??...65
12. Regression Analysis for the Variable CONVICTION in the Odd Assignment
Case ???????????????????????????????66
13. Regression Analysis for the Variable CONVICTION in the Even Assignment
Case ???????????????????????????????67
14. Conviction Scores for Relevant and False Conjectures for Each Environment ??68
15. Collinearity Results for the Independent Variables for Each Sequential
Regression Model ?????????????????????????..69
1
CHAPTER I: INTRODUCTION
Reform measures in mathematics education have called for an increased emphasis
on the inductive processes of exploration and conjecture. Mathematical conjectures are
formed by observing data, recognizing patterns and making generalizations. These
generalizations are unproven statements based on inductive reasoning (Serra, 1997). The
use of conjecture as a means of instruction contrasts with the traditional pedagogy of
memorizing and proving already known geometric theorems. The National Council of
Teachers of Mathematics (NCTM) recommends that the practice of conjecture be an
integral part of instruction regarding mathematical reasoning and proof across the grades.
In particular, the subject of geometry is particularly well suited for exploration and
conjecture (NCTM, 2000).
Over the last decade, a number of dynamic geometry software packages have
been developed that allow students to construct, measure, distort, and explore geometric
figures. Two of the most popular geometry software packages used today are Cabri
Geometre (Laborde, 1990) which is marketed by Texas Instruments, and Geometer?s
Sketchpad (Jackiw, 1991) which is marketed by Key Curriculum Press. Geometer?s
Sketchpad version 4.02 (Jackiw, 2001) was used to create the dynamic geometry
environment used in all of the activities and instruments in this study.
Dynamic geometry software enables students to examine many cases without
having to reconstruct the figure. With dynamic geometry software students are able to
2
select any vertex, segment, or any other part of a geometric figure and move it using the
computer?s mouse to ?drag? that part causing a distortion of the figure. Figures can be
constructed so that the defining characteristics of a specific type of geometric figure are
maintained. For example, a constructed parallelogram in a dynamic geometry
environment will always keep the opposite sides of the quadrilateral parallel. Therefore,
the students are able to distort a parallelogram into another parallelogram by the
?dragging? process. Any measured sides or angles will automatically change
accordingly. By continuing this process the user essentially has a continuum of
parallelograms to investigate rather than one specific static figure. Figure 1 illustrates
dragging a constructed parallelogram in a dynamic geometry environment. Dynamic
geometry software was used to construct this figure.
Figure 1. Dragging point A to form a continuum of parallelograms in a dynamic
geometry environment.
3
Deforming the figure by dragging allows students to directly observe how various
components of geometric figures and their measures are affected by dynamic changes.
By generalizing the patterns that emerge during these explorations and observing changes
in the figures and their measures, students may be able to form their own mathematical
conjectures (Glass, Deckert, Edwards, & Graham, 2001). The use of dynamic software
enables students to examine many cases, thus extending their ability to formulate and
explore conjectures.
The challenge for teachers of geometry is to integrate dynamic geometry
environments in their teaching as a way of encouraging students to explore ideas and
develop conjectures while continuing to help them understand the need for proofs or
counterexamples of conjectures (NCTM, 2000). The classical approach to proof can be
enriched by the advent of dynamic geometry software. Haddas and Hershkowitz (1999)
claimed, ?A main pedagogical feature of many dynamic geometry based learning
environments is that the discovery and conviction of geometrical facts is greatly
enhanced by means of dynamic processes? (p. 25). With the use of dynamic geometry
software, students are no longer solely reliant on stated theorems and formal proofs of
those theorems to verify geometric principles. They now have a tool that will facilitate
conjecturing and aid in exploration of geometric principles.
Purpose and Objective
If developing conjectures is to be an integral part of secondary geometry courses,
then the tools and environments presented during instruction should enhance the students'
ability to form conjectures. This study will investigate differences in students? geometric
4
conjectures in both static and dynamic geometry environments. This study will explore
whether the use of a dynamic geometry environment significantly increases the students'
ability to form worthwhile conjectures.
In order to address this question, high school geometry students were allowed to
independently conjecture in both static and dynamic geometry environments. The static
environment did not allow students to distort the figures by dragging. In the dynamic
environment, students were able to drag the figures and observe changes in the figures
and the measurements provided. Note that the figures presented in both environments
were identical with the exception of the dragging capabilities.
Research Questions
This study addressed the following questions:
1. Does the number of relevant conjectures formed by students significantly
differ in static and dynamic geometry environments?
2. Does the number of false conjectures formed by students significantly
differ in static and dynamic geometry environments?
3. Does students? conviction in their conjectures significantly differ in these
environments?
4. How do the variables of gender, mathematics achievement, and geometric
reasoning level relate with the students' ability to conjecture in these
different environments?
It was hypothesized that in the dynamic geometry environment, students will
produce significantly more relevant conjectures, significantly fewer false conjectures, and
5
that student?s conviction in all conjectures will be significantly greater when compared to
the conjectures formed in the static environment. The variables of gender, mathematics
achievement, and geometric reasoning level were used as covariates and as exploratory
variables.
This study also included several qualitative themes addressing other aspects of the
students? experience during the course of the study. Four a priori themes were developed
as a result of the initial literature review, prior to the collection of data. The four a priori
themes were: Student preferences concerning each environment, the students? concepts
of conjecture and proof, the students? ability to form and find counterexamples in the
dynamic environment, and the students? conviction in the output generated by dynamic
geometry software. Specific instruments designed to address these themes will be
discussed in Chapter III. During the analysis of data, two more emergent themes were
developed. These emergent themes were: The use of dynamic language in the
conjectures formed in the dynamic environment and a description of different dragging
techniques used by the students. The selection and analysis of all six of these themes will
be discussed further in Chapters III and V.
Qualitative methods and studies of mixed design are increasingly common in the
field of mathematics education (Schoenfeld, 2000). By including both quantitative
methods to answer specific research questions and qualitative methods to explore the
above themes, this study was designed, not only to compare, but also to describe student
conjecturing in two different geometric environments.
6
Summary
This study investigated students? ability to form geometric conjectures in both
static and dynamic geometry environments. All participants were exposed to both
environments and participated in up to eight lab activities that allowed them to conjecture
independently in each of the geometric environments. These activities were aligned to
the curriculum of a secondary geometry course throughout the semester and were used as
part of the instruction during regular class hours.
In order to lay the foundation for this study, the next chapter will review literature
concerning traditional geometric proof instruction, different levels of geometric
reasoning, the role of student conjecturing in mathematics instruction, and the uses of
dynamic geometry software. Chapter III discusses the methodology used to collect and
analyze both the quantitative and qualitative data obtained in this study. Chapter IV
reports all of the quantitative results while Chapter V reports the results of the qualitative
themes addressed in this study. The final chapter will discuss the research questions and
the qualitative themes in terms of implications and suggestions for further research.
7
CHAPTER II: LITERATURE REVIEW
This chapter outlines research in several areas that have important consequences
for understanding the role of dynamic geometry environments on student conjecturing.
Since a primary motivation for introducing student conjecturing in the mathematics
classroom is to enhance the instruction of proof, literature concerning traditional proof
instruction in geometry is discussed. The van Hiele theory of geometric reasoning also
plays a significant role in understanding the deductive process of proof; therefore,
literature concerning the van Hiele theory of geometric reasoning will also be reviewed in
this chapter.
Studies that focus on the role of conjecture in the classroom are included in this
chapter as well as literature that links the process of conjecturing with the use of dynamic
software and to the instruction of proof. Several recent studies have investigated the uses
of dynamic geometry software and its applications in the geometry classroom. The
debate over the role of dynamic geometry software with regards to the instruction of
proof is a common theme in recent studies and several of those studies will be discussed
in this chapter as well. Although most of the studies describe research conducted in the
secondary classroom, some studies using preservice teachers and practicing teachers as
participants are included in this chapter. These studies provide insight on these
8
participants? attitude toward proof and the use of dynamic geometry software in the
classroom.
Traditional Proof Instruction
The traditional axiomatic approach to proof gives little or no place for
conjecturing. Theorems are simply given to the students by the teacher or by the
textbook. These theorems are assumed true, and students are asked to verify them by
using deductive reasoning, often in a formal two-column proof format. This traditional
approach to proof has, in the past, left many students disenchanted with the proof process
(Battista & Clements, 1995). Senk (1989) has reported that geometric proof was among
the most difficult and disliked mathematical topics for college-bound students in the
United States. Senk (1985) also stated the typical high school mathematics program
provides virtually no opportunity for students to practice proof writing outside the context
of geometry class. Difficulty with proof is not merely a problem for students in the
United States. Data from the Third International Mathematics and Science Study
(TIMSS) indicate that, in general, students worldwide have particular difficulties
organizing arguments (National Center for Education Statistics, 1998).
Findings suggest that the transition to proof is too abrupt in the traditional
mathematics curriculum and that this transition is often difficult for even those who have
done superior work in preceding courses. Within this traditional environment, tasks
concerning proofs are presented in the form ?prove that ??, where the statement to be
proved is already provided to students (Furinghetti, Olivero, & Paola, 2001). Battista and
Clements (1995) claimed that most mathematics instruction and textbooks lead us to
believe that mathematicians make use only of formal proof and deductive reasoning
9
based on axioms. However, in reality, mathematicians pose problems, analyze examples,
make conjectures, look for counter examples, and revise conjectures all as part of
creating mathematics. Deductive proof is seen as the final step of this creative process.
In a study consisting of 2699 students in 99 geometry classes from five states,
Senk (1985) reported that only about 30% of the students in a full year geometry course
that taught proof were able to master proofs that were similar to those presented in
standard secondary geometry textbooks. In this study, a wide variety of schools and
students were used to achieve a realistic cross section of students with regards to
achievement and socioeconomic conditions.
Senk (1985) found three specific instructional issues that should be addressed as a
result of this study. First, many students could not get started after listing the given
statements, suggesting that teachers need to pay special attention to helping students
begin a chain of deductive reasoning. Second, many students cited the theorem to be
proved within the proof itself, suggesting that teachers should place greater emphasis on
the meaning of proof. Finally, many students had difficulty with embedded figures and
auxiliary lines showing the need to instruct students how, why, and when they can
transform a figure in a proof.
Mingus and Grassl (1999) described the ?attitude barrier? concerning formal
proof, which prevents students from taking the risk of justifying or explaining their
reasoning to others in the classroom. This attitude is realized by frustration and a disdain
for proofs causing students to quit before even attempting to write a proof. This attitude
toward and inability to do proof is a serious deficiency in a student?s mathematical
training.
10
Mingus and Grassl (1999) reported on two different studies concerning proof
frames and abilities. In a study of elementary and secondary preservice teachers, Mingus
and Grassl (1999) examined the participants? experience with proof and their beliefs
concerning proof. The participants included 30 preservice elementary teachers enrolled
in a mathematics content course and 21 preservice secondary mathematics teachers
enrolled in an abstract algebra course. The study found that most of the elementary
preservice teachers (80%) and most of the secondary math preservice teachers (55%) had
either no proof instruction in secondary school or just the traditional axiomatic proof
instruction in a high school geometry class. The majority of both elementary and
secondary preservice teachers felt uncomfortable with proof and felt unprepared to tackle
formal proofs in college level mathematics. They felt that their secondary mathematics
curriculum had not prepared them for this task, and most of the participants felt that some
kind of proof instruction should be incorporated into the earlier grades to help prepare
and nurture students toward formal proof.
Mingus and Grassl (1999) also reported on a study using 215 middle and high
school students that attempted to judge students? ability to produce a convincing
mathematical argument. The students were asked to show that there are just as many
even numbers as there are odd numbers. The problem was presented in written form
during school hours, and 170 of the students provided written justification of their
reasoning. Responses included elegant arguments involving one-to-one correspondence,
the role of digits in the unit?s position, and proofs by contradiction. The study found that
most of the students were able to provide a proof at some level and, surprisingly, the
11
younger students in the sixth to eighth grades often showed the most creativity in their
justifications.
The implications of these two studies led to the conclusion that proofs should be
encountered as early as possible in the curriculum and that a broad view of proof should
be adopted to encourage a wide variety of mathematical ideas. More precision and
breadth should be expected as students advance through the grades. Teachers should also
demonstrate appropriate proofs as they reinforce the students? attempts at proof at all
levels (Mingus & Grassl, 1999).
The way in which students perceive the role of figures used in geometric proofs
was the subject of a study by Martin and Harel (1989a). The study asked 410 university
students enrolled in a lower division mathematics course to judge the correctness of the
following geometric statement: ?A segment connecting the midpoints of two sides of a
triangle is one half the length of the third side? (p. 266). This statement is sometimes
referred to as the midsegment theorem.
Three different instruments were used in this study. The first two instruments
used general proofs of the statement accompanied by combinations of general figures and
specific ?non-generic? figures of triangles. For each figure, participants were asked if the
given proof was valid for that figure. The third instrument used an argument tailored to a
specific triangle. It was also accompanied with both general and particular figures of
triangles. Like the other two instruments, the participants were asked to judge the
validity of the given argument for each of the figures provided. The results of this study
provided two interesting findings concerning the students? concepts of proof with regards
to geometric figures. First, the use of particular ?non-generic? figures did not appear to
12
influence the students? judgment about the correctness of a proof. Second, students
indicated that new proofs would have to be done if the figures were changed. In other
words, students conceived that the proof was only valid for the figure that was provided
and not necessarily valid for all figures of that ?type?. This inability to generalize to
other figures is indeed a problem for students? concept of proof, and the authors
suggested further research on student interpretations of figures in geometric proof.
Martin and Harel (1989b) also reported on a study of proof frames of preservice
elementary teachers. In this study 101 preservice elementary teachers enrolled in a
sophomore level mathematics course were given a proof instrument in which both a
familiar generalization and an unfamiliar generalization were given. Along with these
generalizations, the instrument provided inductive examples and patterns as well as a
general proof, a false proof, and a particular proof.
The results of this study showed that students accepted both inductive and
deductive arguments as proofs. The authors postulated that inductive and deductive
arguments represent two different proof frames constructed by students as the result of
experiences outside and inside the mathematics classroom. Martin and Harel (1989b)
concluded that the inductive frame, which is formed earlier, is not deleted when students
acquire the deductive frame. This finding is relevant to reform recommendations
concerning the instruction of proof and to this study. Having students form conjectures
in geometry class may indeed help them develop their ability to use inductive reasoning,
although it must be emphasized that inductive arguments are not valid mathematical
?proofs?. Teachers must recognize the students? ?inductive frame? and how to
incorporate this frame into formal proof instruction.
13
All of these studies address different aspects of proof instruction and offer
suggestions to improve the instruction of proof. Students should be exposed to methods
of justification in both inductive and deductive frames at an earlier age. The act of
conjecturing and justifying using both inductive and deductive methods may indeed
better prepare students for formal proof instruction in secondary school mathematics.
The role of figure in geometric proofs and how students perceive the figures is an
important element in the students? concept of proof. Teachers in elementary and middle
grades should be trained to recognize and encourage creative ways for students to justify
their conclusions. The National Council of Teachers of Mathematics recommends that
instructional programs from kindergarten through secondary school incorporate
investigation and conjecturing. Such activities should help students develop and evaluate
mathematical argument and proof, select and use various types of arguments, and
recognize proof as a fundamental aspect of mathematics in general (NCTM, 2000).
The van Hiele theory of Geometric Reasoning
The van Hiele theory of geometric reasoning originated in the respective
dissertations of Dina van Hiele-Geldof and her husband Pierre van Hiele in 1957 (Fuys,
Geddes, & Tischler, R., 1988). The van Hieles described stages of geometric reasoning
that students pass through as they acquire geometric knowledge and cognitive
sophistication with regards to geometric reasoning. The ?four level? van Hiele model of
geometric reasoning is summarized as follows (de Villiers, 1996; Gutierrez & Jaime,
1998):
14
? Level 1 - Recognition: Students recognize figures by appearance and
recognize squares, rectangles, triangles, etc. by their shape. They do not
explicitly identify the properties of these figures.
? Level 2 ? Analysis: Students begin analyzing the properties of figures and
use proper terminology to define and describe figures. They do not yet
inter-relate the figures and their properties.
? Level 3 ? Ordering: Students are able to classify and inter-relate figures
by their properties. Students can order properties of figures by short
chains of deductions.
? Level 4 ? Deduction: Students start developing longer sequences of
statements and begin to understand the concept of deduction, the role of
axioms, theorems, and proof.
To further examine the effect of the van Hiele levels on proof writing ability,
Senk (1989) used a sample of 241 secondary school students enrolled in full year
geometry classes. She found that proof writing ability correlated significantly with van
Hiele level even when entering knowledge of geometry and geometry achievement
throughout the course were used as covariates. She concluded that a student?s entering
van Hiele level could serve as a strong predictor of the students? ability to master proof in
a secondary geometry course.
In order to measure the students? proof writing achievement, a six item test was
given to each student with a 35 minute time limit. This test contained two short answer
items and four full proofs. This test was given to the students during the last month of
the school year and contained proof items similar to those found in typical secondary
15
geometry texts. A Cronbach?s alpha reliability coefficient of .85 was reported for the
students taking the test. Students who were able to correctly prove at least three out of
the four given proofs were considered to have ?mastered? proof writing.
This study also supported the notion that formal deductive proof in geometry
requires at least thinking at level three in the van Hiele hierarchy. Senk (1989) used a
?five level? van Hiele scale in which level five represents the ability to perform rigorous
mathematical proofs. At the onset of the study only 15 of the 241 participants were at
level three and none at levels four and five. By the end of the study 68 students had
reached level three, 13 had reached level four, and four had reached level five. Only 22%
of the students who reached level two were able to master proof whereas 57%, 85%, and
100% of the students who reached levels three, four, and five respectively were able to
master proof. It should be noted, however, that the majority of the students (65%) were
still at level two or below at the end of the course.
Students must pass through lower levels of geometric thought before they can
attain higher levels, and this process does take a considerable amount of time. The van
Hiele theory suggests that instruction should gradually progress through lower levels of
geometric thought before students begin a proof-oriented study of geometry. Because
students cannot bypass levels and achieve understanding, prematurely dealing with
formal proof can cause students to rely on memorization without understanding. The
traditional axiomatic approach to formal proof then is unlikely to be productive for the
vast majority of students in high school geometry (Battista & Clements, 1995).
16
The Role of Conjecture
An alternate approach to formal proof instruction in geometry would involve
including attention to the inductive step of conjecture as a prelude to formal deductive
proof. The discovery of geometric facts by conjecturing may lead to formal and informal
justifications of why the conjectures are true, thus laying the groundwork for deductive
proof. Students can be given an environment in which to explore geometric situations
and derive their own conjectures rather than relying on the teacher or the textbook to tell
them the mathematical truths that can be found in any particular geometric situation.
This kind of instruction is consistent with recommendations from the National
Council of Teachers of Mathematics (NCTM). The NCTM supports instruction in which
students are able to learn with understanding rather than memorize mathematical facts
and procedures. Learning with understanding includes proposing ideas and conjectures
as well as evaluating those ideas and conjectures (NCTM, 2000).
The NCTM?s ?Reasoning and Proof? standard for all students from pre-
kindergarten through high school includes the following four elements:
? Recognize reasoning and proof as fundamental aspects of mathematics.
? Make and investigate mathematical conjectures.
? Develop and evaluate mathematical arguments and proofs.
? Select and use various types of reasoning and methods of proof. (NCTM, 2000, p.
56)
As seen from this list, conjecturing and various types of reasoning are coupled with
formal proof. As students move through the grades, conjecturing and informal
17
justifications should prepare students for the more rigorous act of deductive proof in
secondary school mathematics, but they are certainly never abandoned.
The mathematical environment provided for students when they form conjectures
may indeed determine the quality of their conjectures and the conviction in the
conjectures that they form. de Villiers (1992) reported on a study concerning students?
conviction on given mathematical conjectures. The study hypothesized that the majority
of children would base their conviction of the truth of the given statements on the
authority of the teacher and/or textbook rather than on personal conviction. de Villiers
(1992) also hypothesized that the majority of the children would not easily distinguish
false statements on their own, but would be dependent on the authority of the teacher
and/or textbook for this distinction.
To test his hypothesis, de Villiers (1992) studied 40 grade 10 students from two
different schools and 99 grade 11 students from five different schools in order to
determine what mathematical statements the students were convinced or doubtful about
and the reasons for their conviction. All students were given a series of 15 geometric
statements in 35 minutes and asked to respond with one of four different codes. The
codes are as follows:
Code 1: Believe it is true from own conviction.
Code 2: Believe it is true because it appears in the textbook or because the
teacher said so.
Code 3: Do not know whether it is true or not.
Code 4: Do not think it is true.
18
It was found that the majority of pupils based their conviction on authoritarian
grounds rather than personal conviction, as hypothesized. The students in this study
found formal proof to be less convincing than verification from an authority and very few
students (2%) could accurately identify false statements. This study implies that students
may not be adequately prepared to conjecture on their own when simply handed a paper
of mathematical statements (de Villiers, 1992). Students may indeed need a proper
environment that promotes the act of conjecture through experimentation and
manipulation.
Furinghetti, Olivero, and Paola (2001) supported presenting students with tasks
that use dynamic exploration to help foster the ability to conjecture. Such explorations
and the resulting conjectures might support students in the process towards proof and
make the way of doing mathematics in the classroom closer to the way of
mathematicians. Furinghetti, Olivero, and Paola (2001) used the exploration of short,
easily-understood geometric statements that foster discovery, conjecture, and do not
suggest a method of proof. They reported on a classroom experiment that used the
following geometric statement given to a small group of adolescent geometry students:
You are given a right-angled triangle ABC, AB being the hypotenuse.
Take a point P on AB. Draw the parallel lines to AC and BC through P.
Name H and K, the points of intersection with AC and BC respectively.
For which position of P does the line HK have minimum length?
(Furinghetti, Olivero, & Paola, 2001, p. 324)
The students were grouped in threes and required to form conjectures, which then needed
to be justified or proved to the entire class. The use of an open problem helped students
19
produce strategies and because of the need for communicating them to their colleagues in
the classroom, these strategies were explicit.
Furinghetti, Olivero, and Paola (2001) reported that almost all students were
actively engaged in the activity, and a variety of different strategies were explored and
discussed among the groups. The use of videotape allowed the students, teacher, and
researchers to capture important learning moments and statements that would normally
be overlooked or forgotten by regular observation. Although the use of dynamic
geometry software was not part of the original classroom experiment, a follow up activity
did use dynamic geometry software for demonstration purposes.
Boero, Garuti, and Mariotti (1996) reported on a teaching experiment that used
conjecture as a stepping-stone toward proof. Once again the setting of an open-ended
problem set the stage for the experiment:
In the past years we observed that the shadows of two vertical sticks on
the horizontal ground are always parallel. What can be said of the
parallelism of shadows in the case of a vertical stick and an oblique stick?
Can shadows be parallel? At times? When? Always? Never? (Boero,
Garuti, & Mariotti, 1996, p. 3)
This problem was posed to thirty-six eighth grade Italian students in two separate
mathematics classes. Many of the students started working with thin sticks or pencils.
The absences of sunlight caused students to use dynamic mental processes to formulate
conjectures. The conjectures were discussed, with the help of the teacher, until two
collective statements resulted:
20
1. If sun rays belong to the vertical plane of the oblique stick, shadows are
parallel. Shadows are parallel only if sun rays belong to the vertical plane
of the oblique stick.
2. If the oblique stick is on a vertical plane containing sun rays, shadows
are parallel. Shadows are parallel only if the oblique stick is on a vertical
plane containing sun rays. (Boero, Garuti, & Mariotti, 1996, p. 5)
Although these statements could be combined into one conjecture in a compact ?if and
only if? form, they were left as is for purposes of proof. The proof exercises for this
activity involved both working in pairs and teacher guided discussions. Activities
focused on what it means to prove something in mathematics and the concept of
?necessary and sufficient? conditions.
In this study, all of the participants actively took part in developing the initial
conjecture and 29 of the 36 were able to complete all of the follow-up proof exercises in
a productive way. Videotape analysis of the classroom activities showed more than half
of the students exploring the problem in various dynamic ways by using props or hands
to simulate movement. During these explorations conjectures were often abandoned or
modified. The authors of this report claimed that because of the relative success of the
use of ?dynamic? mental processes educators should seek to find ways to incorporate
dynamic environments into traditionally ?static? mathematical situations (Boero, Garuti,
& Mariotti, 1996).
21
The Role of Dynamic Geometry Software
Throughout the last two decades a number of dynamic geometry software
packages have been marketed that allow users to construct, measure, distort, and explore
geometric figures (Battista & Clements, 1995; de Villiers, 1996). Because of the
dynamic nature of the environment created by these packages, students are able to
explore multiple figures without having to reconstruct them. The Geometric Supposer
software series (Schwartz & Yerushalmy, 1986) was a precursor to modern dynamic
geometry software packages. It allowed students to choose a primitive shape, such as a
triangle or a quadrilateral, and perform measurements and constructions on it. The
program would then repeat the same operations on other shapes allowing students to
explore the generality of the consequences of their constructions (Battista & Clements,
1995).
One of the first true ?dynamic? geometry programs produced was Cabri-
Geometre (Laborde, 1990), a French program that was introduced to the international
mathematics education community at a conference in Budapest in 1988 (de Villiers,
1996). Unlike the Geometric Supposer, this ?state of the art? package had dragging
capabilities that gave the user instant control over the dynamic processes. Other
packages soon followed, including the Geometer?s Sketchpad (Jackiw, 1991) which is
widely used in the United States as the premier dynamic geometry software package
(Battista & Clements, 1995).
22
As stated in de Villiers (1996):
The development of dynamic geometry software in recent years is
certainly the most exciting development in geometry since Euclid.
Besides rekindling interest in some basic research in geometry, it has
revitalized the teaching of geometry in many countries where Euclidean
geometry was in danger of being thrown into the trashcan of history
(p. 25).
The Role of Conjecture using Dynamic Geometry Software
de Villiers (1996) provided a framework in which dynamic software can be used
to verify true conjectures and construct counterexamples for false conjectures. He
suggested that teachers and educators in mathematics should focus more on teaching and
developing the process aspects of mathematics. Teachers should allow students to
actively construct their knowledge in the class rather than being presented with
preplanned content. The following figure shows de Villiers? (1996) illustration of the
role of conjecture and counterexample in the proof process.
23
Figure 2. A flow chart of student research in geometry showing the role of conjecture
and counterexample in the proof process (de Villiers, 1996. p. 27).
In figure 2, the step referred to as ?testing? would use some kind of dynamic
geometry software to test the students? conjectures. Students can test to see if their
conjectures hold true under the dragging process. By examining a continuum of figures
in the dynamic environment, students can confirm or reject their conjectures. If students
discover counterexamples to their conjectures during this dynamic process, they can
adapt their conjecture to accommodate the counterexample or start afresh with a new
conjecture. Eventually this process of exploration, discovery, and testing leads to an
Conjecture
Testing
Confirmation
Proof
Successful
STOP
Counter-example
Reformulation or
Rejection
Reformulation or
Rejection
Unsuccessful
24
attempt to prove the conjecture deductively. An unsuccessful attempt at the proof leads
the student back to the conjecturing phase where as a successful attempt will end the
process unless the student wishes to make further conjectures.
Implications of Dynamic Geometry Software
Arcavi and Hadas (1996) outlined some of the educational implications brought
about by the use of dynamic geometry technologies:
? In the dynamic geometry environment, students are led to explore and to
play with many particular cases. As a result these observations may add
insight and provide a basis for proving and further exploration.
? Students can conduct explorations and conjecture independently without
the need for a teacher to confirm or judge the outcome. The role of the
teacher can then be that of a guide that forces students to take a stance on
conjectures and asks the all-important question ?why??
? Making sense of a situation while playing with the situation itself first
enhances both the understanding of the situation and the representations
used to analyze the situation such as measures, graphs, and symbols.
? Traditional boundaries between mathematical subdisciplines are blurred
and connections are enhanced. One subdiscipline may serve as the model
for another enhancing the student?s sense of consistency among various
branches of mathematics.
Arcavi and Hadas (1996) described several important components of the dynamic
geometry environment that contribute to its success, including the attributes of
25
visualization, experimentation, surprise, feedback, and need for proof. Dynamic
geometry environments allow students to construct visual images with certain properties
and then transform them continuously in real time thus adding to the visual experience of
the user. Dynamic environments enhance the student?s ability to experiment, to look for
extreme cases and negative examples. This experimentation is the basis of stating
generalizations and forming and dismissing conjectures (Arcavi & Hadas, 1996). When
the element of surprise is incorporated into dynamic geometry activities, students may
become more thoughtfully engaged in the problem at hand. The impact of puzzlement or
curiosity is not a negative or judgmental result but rather causes students to question
themselves and enhances meaningful learning. This element of surprise may also
motivate the students? need for proof. In this way, the dynamic environment supports
deductive proof and helps the students ?close the circle? by engaging in proof (Arcavi &
Hadas, 1996). By allowing multiple representations such as Cartesian graphs,
measurements, and animations, mathematical connections are enhanced. While seeing
objects in a dynamic state, students can observe the act of variation that is betrayed in
static representations. Dynamic geometry environments support the notion of increasing
and decreasing measures as well as the concept of variation and extrema (Arcavi &
Hadas, 1996).
Goldenburg and Cuoco (1998) raised issues concerning student perceptions of the
dynamic geometry environment. Exactly what do students perceive on the computer
screen using dynamic software? Do they see a continuum of figures? Or do they perceive
several discrete cases? Do students have to re-examine their existing definitions to suit
the dynamic environment?
26
Figure 3. A continuum of quadrilaterals under the drag mode using Geometers
Sketchpad (Jackiw, 1991).
To illustrate this last question, consider students investigating a construction in
which the midpoints of the four sides of a quadrilateral are connected in order. The
resulting figure referred to as the ?midsegment quadrilateral? appears to be a
parallelogram. The students may wish to explore and conjecture that this will be true for
all quadrilaterals. While exploring by ?dragging?, students are likely to create figures
that they do not consider to be quadrilaterals, such as the degenerate triangle that is
formed as the quadrilateral transforms from convex to concave or the ?crossed bowtie?
figure that is not even considered to be a polygon. Notice that the conjecture, however,
still holds for these ?monster? polygons. How do students resolve these cases? Do they
ignore them, redefine their existing notion of quadrilateral, or treat them as separate cases
(Goldenberg & Cuoco, 1998)?
27
Figure 4. The exploration of the midsegment quadrilateral.
Possible negative consequences
The use of dynamic geometry software as an instructional tool is not without
controversy. Jones (1999) reported on some data from a longitudinal study designed to
examine how using the dynamic geometry package Cabri-Geometre (Laborde, 1990)
mediates the learning of certain geometrical concepts. He conducted case studies of five
pairs of 12 year old pupils working through a series of specially designed tasks that
involved the construction of various quadrilaterals. Jones (1999) concluded that children
working with computers might focus on the screen product at the expense of reflection
upon its construction. Students modified the figure ?to make it look right? rather than
debugging any problems in the construction process. Students did not necessarily
appreciate how the computer tools they used constrained their behavior. After making
inductive generalizations, students frequently failed to apply them to a new situation
(Jones, 1999).
Lange (2002) described a classroom episode in which students used dynamic
geometry software to explore perpendicular bisectors. The instructor was expecting the
students to discover that any point on the perpendicular bisectors is equidistant to the
endpoints of the segment being bisected; this was the ?targeted? conjecture of the
28
exercise. The author described how many students failed to observe this property and
instead focused on non-conjectures that were already given or were irrelevant. The
instructor?s goal in this classroom exercise was to link three main components of proof
writing: observation, conjecture, and then deduction. Students struggled with what
observations should be made in the dynamic geometry environment which made the next
step of conjecture difficult.
Both Jones (1999) and Lange (2002) pointed out difficulties that students have
adapting to the dynamic geometry environment. It may indeed take some students a
considerable amount of time and exposure to realize the potential of the dynamic
geometry environment in terms of proper construction techniques and conjecturing in
geometry.
Dynamic Language
Mariotti (2001) analyzed a long-term teaching experiment carried out in the ninth
and tenth grades aimed at introducing pupils to theoretical thinking by using dynamic
geometry software. The analysis of the teaching experiment was aimed at discussing the
specific role played by dynamic geometry software. It was found that the use of dynamic
geometry software in the generation of a conjecture is based on the internalization of the
dragging function as a logic control, which is able to transform perceptual data into a
conditional relationship between hypothesis and thesis (if?then). The use of dynamic
geometry software on the computer created a channel of communication between the
teacher and the pupil based on a shared language (Mariotti, 2001).
29
Jones (2000) also looked at students? interaction with the dynamic geometry
environment and how it affected the students reasoning and language used in their
explanations. Like Mariotti (2001), students developed a language that reflected the
dynamic environment and used that language to communicate their mathematical
explanations. Jones (2000) also reported on a longitudinal study of 12-year old students
who completed a 30-week unit of geometry that used dynamic geometry software as a
means of delivery. Students passed through three phases during this unit that focused
primarily on the properties and classification of quadrilaterals.
In the first stage students relied on description rather than actual explanation.
Mathematical language and reasoning were not yet present. In the second stage,
explanations became more mathematically precise and were influenced by the dynamic
environment itself. For example, words like ?dragging? were used in the explanations.
In the third stage of the teaching unit students were providing explanations that were
entirely in the context of the dynamic geometry environment.
Both of these studies show how the dynamic geometry environment becomes part
of the actual mathematics being addressed over time. Language that reflects the
environment is used in explanations and as a means of communication between pupil and
teacher. The environment promotes action leading to the if-then form of a conjecture and
this action becomes part of the reasoning behind the explanations (Jones, 2000).
Dynamic Software and Proof
Dynamic software has the potential to encourage both explanation and proof,
because it makes it so easy to pose and test conjectures. ?Unfortunately, the successful
30
use of this software in exploration has lent support to a view among many educators that
deductive proof in geometry should be downplayed or abandoned in favor of an entirely
experimental approach to mathematical justification? (Hanna, 2001, p. 13).
Pandiscio (2002) reported on a study that examined preservice teachers?
conception of proof and the use of dynamic software. In this case study, four preservice
teachers who were enrolled in a semester long course centered on effective secondary
mathematics pedagogy were used as the participants. These participants were given two
geometric situations to explore using a dynamic software package name Geometers?
Sketchpad. Pandiscio (2002) used the following problems:
? Are the six small triangles that are formed by the intersection of the
medians of a given triangle congruent? Do they have equal area?
Investigate and then prove.
? If two secant segments are drawn from a point outside a circle, then the
product of the measures of one secant segment and its external part is
equal to the product of the measures of the other secant segment and its
external part. Explore these relationships, and then prove the theorem
(p. 215).
Surveys, observations, and interviews of the participants revealed that after using
dynamic software, preservice mathematics teachers expressed the concern that high
school students will believe proofs are unnecessary. They still believed that a formal
proof is different from ?proof by many examples,? but after repeated use of dynamic
software, they questioned the value of the formal proof for high school students.
31
Through observation it was noted that the participants explored the problems
more deeply with the software than without. The participants claimed that the greatest
value of dynamic software is in helping students understand key relationships that are
embedded in the figures being explored rather than the proofs of these relationships. This
study suggests that the use of dynamic geometry software does not strengthen the
perception of preservice teachers that proof is critical to high school geometry. In fact,
this study suggested that such a powerful inductive tool may actually decrease the
perceived need for students to write deductive formal proofs (Pandiscio, 2002).
Laborde (2001) reported on a case study of four secondary geometry teachers?
attitudes related to the use of dynamic geometry software in their classes. The study
examined two veteran teachers with experience in dynamic geometry software, a novice
teacher who was experienced in computer science and finally a veteran teacher who had
little familiarity in using technology in mathematics teaching. Specifically, the study
focused on the tasks that were explored by these teachers? students using dynamic
geometry software.
For the novice teacher, technology was used to show and confirm that certain
geometric theorems and postulates were indeed true and that the figures did behave as
expected. The veteran teacher who had little experience with dynamic geometry used the
technology mainly for observation and for construction, but pencil and paper
environments were provided in addition to the dynamic environment. The two teachers
who were experienced in the uses of dynamic geometry software tended to have more
open explorations to make use of the drag mode to enhance student conjecture and to
pose problems for further exploration and discussion at the end of the activities. This
32
case study illustrated how different types of teachers may use dynamic geometry
software. Teachers that are familiar and comfortable with the technology may tend to use
tasks that lead to student exploration and discovery. They may allow students to work
independently using the technology to form conjectures that would perhaps lead toward
deductive proof or counter-examples (Laborde, 2001).
A study by Marrades and Gutierrez (2000) specifically examined the use of
dynamic geometry software to facilitate students? transition to deductive proof in
geometry. These authors considered two case studies of pairs of students working in a
unit that used Cabri-Geometre (Laborde, 1990). Each pair of students was part of a class
of sixteen secondary students of age fifteen or sixteen. All students in this class used
dynamic geometry software twice a week throughout the unit of 30 activities. The
typical activity in this unit would first ask the students to construct geometric figures and
then explore them using the Cabri software. Students were then asked to make
conjectures about the figures, and finally they were asked to justify their conjectures.
Marrades and Gutierrez (2000) classified the justifications as empirical or
deductive. Empirical justifications were inductive in nature and relied on observation of
the figures. These observations were classified as ?na?ve empiricism?, ?crucial
experiment?, or ?generic example?. Na?ve empiricism referred to justifying the
conjecture by showing that it is true in one or more examples, usually selected without a
specific criterion. Crucial experiment justifications used a specific, carefully selected
example. Students were aware of the need for generalization so they would choose an
example that was as non-particular as possible although it was not considered as a
representative of any other example. Students assumed that the conjecture was always
33
true if it was true in this example. The final type of empirical justification, generic
example, was based on a specific example seen as a characteristic representative of its
class. This kind of justification also used abstract reasons for the truth of the conjecture
by means of operations or transformations on the chosen example.
Two types of deductive justifications were also classified in this study. ?Thought
experiment? referred to the use of a specific example to help the student organize their
justifications; however, the use of axioms, definitions, and accepted theorems was used to
form deductions. ?Formal deduction? referred to the use mental operations without the
help of specific examples. In this kind of justification only generic aspects of the
problem are discussed. This kind of justification is akin to formal proof.
The researchers collected data by collecting handed in worksheets, downloaded
files of the sketches constructed by the participants, and informal videotaped interviews.
The results showed that even though the students did not reach the stage of formal
deduction, the use of the dynamic geometry software increased the quality of students?
conjectures over time and justifications moved toward deduction by reaching the
empirical stage of generic example. Therefore, the students had looked beyond specific
shapes and moved toward making conjectures for general figures.
Marrades and Gutierrez (2001) reached several conclusions based on the results
of this study:
1) Dynamic geometry software may well help secondary school students
understand the need for abstract justifications and formal proofs.
34
2) The types of justifications and the phases in the process of producing
justifications are complementary elements and allow us to make a detailed analysis of the
solutions of proof problems.
3) By stating carefully organized sequences of problems, and giving students free
time to explore these problems, it is possible to have students progress toward more
elaborated types of justifications.
4) Students need a considerable amount of time devoted to experimentation using
dynamic geometry software before they become confident with formal deduction.
This study is an example of how dynamic geometry software can prepare students
to tackle the task of deductive proof. The authors claim that ?we need to know students?
conception of mathematical proof in order to understand their attempts to solve proof
problems, that is, what kinds of arguments convince students that a statement is true??
(Marrades & Gutierrez, 2001, p. 88).
Mudaly and de Villiers (1999) investigated students? use of dynamic software in
order to explore student conviction in discovered conjectures and their need for further
explanation concerning those conjectures. The study asked the following questions: Are
students convinced about the truth of their discovered geometric conjecture and what is
their level of conviction? Do they require further conviction? Do they exhibit a desire
for an explanation for why the result is true? Can they construct a logical explanation for
themselves with guidance and do they find it meaningful? The following figure
illustrates a question that was used to prompt investigation.
35
Sarah, a shipwreck survivor manages to swim to a desert island.
As it happens, the island closely approximates the shape of an equilateral
triangle. She soon discovers that the surfing is outstanding on all three of
the island?s coasts and crafts a surfboard from a fallen tree and surfs
everyday. Where should Sarah build her house so that the total sum of the
distances from the house to all three beaches is a minimum?
m PI+m HP+m PF = 5.40 cm
m PF = 1.00 cm
m HP = 2.47 cm
m PI = 1.93 cm
m CA = 6.24 cm
m BC = 6.24 cm
m AB = 6.24 cm
H
F
I
C
B
A
P
Figure 5. Shipwreck problem used by Mudaly and de Villiers (1999, p. 2).
Students were able to use this dynamic figure to discover that the sum of the
lengths of the three segments PI, HP, and PF remains constant while dragging point P
within the confines of the equilateral triangle.
The equilateral triangle figure, already constructed in a dynamic geometry
environment, was presented to a sample class of fourteen year old students. After
exploring the problem using the dynamic features of the software, most of the students
36
were completely convinced that, indeed, the sum of the lengths of the three segments PI,
HP, and PF remains constant while dragging point P, and most of them wanted to know
why. The research indicated that the learners displayed a need for further explanation for
a result, independent of their need for conviction. Given such high levels of conviction
one might expect that it should have made no difference to the students whether there
was some logical explanation for the result. Yet they found the result surprising and
expressed a strong desire for an explanation that was effectively utilized to introduce
them to proof as a means of explanation rather than verification (Mudaly & de Villiers,
1999).
Hadas, Hershkowitz, and Schwarz (2000) found similar results in their study that
introduced the elements of contradiction and uncertainty with the aid of dynamic
geometry software. Two activities were developed for a sample of eighth grade students
and designed to cause contradiction or uncertainty in the students? initial intuitive
conjectures. The first activity was concerned with the interior angle sum and the exterior
angle sum of convex polygons. Students initially conjectured about the sum of the
interior angles of various convex polygons with the aid of dynamic geometry software.
Many students conjectured that indeed the sum of the interior angles increased by 180
degrees with each additional side or angle. Many of them even expressed the relationship
algebraically. This result may have caused the students to then assume that the sum of
the exterior angles would also increase as the number of sides increased. In fact 37 out of
49 responses indicated just that. The fact that the sum of the exterior angles is always
360 degrees for convex polygons was a surprising result that contradicted the students?
conjectures. This result was also discovered by means of dynamic geometry software;
37
however, the explanation of why this sum is constant motivated students to go further
into interesting inductive, deductive, and visual arguments (Hadas, Hershkowitz, &
Schwarz, 2000).
In the next activity, students used dynamic geometry to explore the conditions for
congruent triangles, to investigate if and when two triangles having several congruent
parts are congruent. In one of the tasks, the students were asked if it is possible to
construct a triangle with one side and two angles congruent to another triangle but that is
not congruent to that triangle. This task was designed to cause uncertainty among the
students. Some of the students were able to make the construction using dynamic
geometry software while others claimed that the triangles must be congruent. Those
students who claimed that the triangles must stay congruent had constructions in which
the corresponding sides were the included sides of the two congruent angles in both of
the triangles. Those who assigned the congruent side as the included side in one triangle
and a non-included side in the other triangle realized through the dragging process that
these triangles were not congruent. In fact in this case the triangles are similar (Hadas,
Hershkowitz, & Schwarz, 2000). Figure 6 illustrates these two possibilities.
Once again students were driven toward deductive explanations to resolve this
uncertainty. Results of this study show that most of the students resolved this problem
using deductive means and discovered the triangle congruence propositions in the process
(Hadas, Hershkowitz, & Schwarz, 2000).
38
m?DFE = 67.02?
m?DE F = 43.74?
EF = 4.38 cm
YZ = 4.38 cm
BC = 4.38 cm
m?XZY = 67?
m?ACB = 67?
m?ZXY = 44?
m?ABC = 44?
Z
E
F
B
A
C
D
Y
X
Figure 6. Constructions of triangles with one congruent side and two congruent
angles, showing that the positioning of the congruent parts determines the congruence of
the triangles.
These last two studies contrast with the findings in Pandiscio (2002) which
suggested that preservice teachers may find the use of dynamic software a hindrance to
proof instruction and that their students would not be motivated to use deductive proof
after using dynamic software. With proper guidance students were motivated and were
able to construct deductive arguments after exploring and conjecturing in the dynamic
geometry environment.
39
These dynamic geometry studies explore a variety of issues concerning the use of
the dynamic geometry environment in the classroom. Topics include student perceptions
and misconceptions, student conjectures and conviction, student need and ability to use
dynamic geometry software to communicate and foster proof, and teacher as well as pre-
service teachers? attitude and uses of dynamic geometry software. The recurring themes
of conjecture and proof in all of these studies make it evident that these topics are of
utmost importance in studies concerning the use of dynamic geometry software.
Summary
This chapter has reviewed literature concerning student?s abilities and attitudes
concerning proof. A brief discussion of van Hiele theory was provided to show the
relation to proof and possible causes for the failure of traditional proof instruction.
Literature concerning the role of conjecture in the proof process was then followed by a
discussion of the dynamic geometry environment created by computer software packages
and the role of dynamic geometry software in terms of conjecture and proof. This
chapter serves to lay the groundwork for this study. The following chapter details the
methodology used to collect both quantitative and qualitative data, a description of the
context of the research, a detailed explanation of the instruments that were used, the
variables that were measured, and the methods used to analyze the data.
40
CHAPTER III: METHODOLOGY
This chapter provides a detailed outline of the methods employed to answer the
quantitative research questions and to address the qualitative themes regarding student
conjecture in geometry and the use of dynamic geometry software. A discussion of the
participants used in the study and the instruments used on those participants will be
followed by the procedures for collecting data and analyzing that data. Note that the
researcher of this study was also the instructor of the students used in this study and is a
full-time employee of the school and the district in which this study took place.
Participants
Participants were recruited from two secondary school geometry classes taught by
the researcher. These two classes were designated as class A and class B for the purposes
of this study. All students in these classes were asked, but not required, to participate in
the study. Parental permission was obtained for each participant before the study began.
All students who volunteered and returned a written permission form signed by a parent
or legal guardian were included in the study. Forty-two out of a possible fifty-five
students returned written permission forms. The remaining students still participated in
all of the conjecturing lab activities; however, their collected data was not analyzed as
part of this study.
41
The participants were enrolled in a public high school in a southern city of the
United States. The total school population for grades 9-12 is approximately 1100.
Approximately 70% of the school population is African American, 20% White, 7%
Latino, and 3% Asian. The geometry classes in which the subjects were enrolled were
neither remedial or honors classes and reflected the general school population in terms of
race, ethnicity, and achievement. Most of the students in these geometry classes were
sophomores or juniors with one freshman and four seniors included.
A wide range of mathematical abilities was represented; however, data obtained
on mathematics achievement indicated that most of the participants were in the lower
percentiles in mathematics achievement when compared with national data. The measure
of mathematics achievement used in this study was the students? most recent score on the
mathematics section of the Preliminary Scholastic Aptitude Test (PSAT) already on file
prior to the study. Figure 7 shows the distribution of student scores on the 2003 PSAT
math section. The national mean for this test was 44.5 for tenth grade students and 48.8
for eleventh grade students for the year 2003 (College Entrance Examination Board,
2003).
0
2
4
6
8
10
12
14
20-24 25-29 30-34 35-39 40-44 45-49
Figure 7. Participant distribution of 2003 PSAT mathematics scores
42
Forty-two students began the study, and one student was dropped because of lack
of attendance. All of the students participating in the study were taking the course for the
first time. The following table provides demographic data for the participants used in
both classes. Note that the racial percentages are relatively close to the school averages
and there is a balance of gender for the overall sample of students.
Table 1
Participants? gender and racial demographics
Race
Class A
Male Female
Class B Total
Male Female
African American
6
6
6
11 29
White 3 2 0 2 7
Latino 3 0 1 0 4
Asian 0 0 0 1 1
Total
12
8
7
14 41
Instruments
Lab activities were developed to measure the students? conjecturing abilities in
each environment, as well as their conviction in the correctness of their conjectures. In
addition, two instruments were used to measure statistical covariates of the study. The
mathematics section of the Preliminary Scholastic Aptitude Test (PSAT) was used to
43
measure mathematics achievement and a Van Hiele geometric reasoning instrument was
used to measure the students? geometric reasoning level. Two qualitative instruments
were developed to explore qualitative themes: a student survey with open-ended
questions and an interview protocol. Each of the instruments used are discussed in turn.
Conjecturing Lab Instruments
Eight different interactive lab instruments were developed with parallel versions
for both the static and the dynamic geometry environments. Each of the lab instruments
contained geometric figure(s) with appropriate measurements given. These figures were
constructed using Geometer?s Sketchpad (GSP) version 4.02 (Jackiw, 2001), a dynamic
geometry software package. In the dynamic environment, students were able to deform
the figures by using a dragging feature. This dragging utility gave the students the ability
to observe how the components of the figure and the given measurements shown
reflected the deformations. The static environment used the exact same figures and
measurements; however, the dragging capabilities of the software were disabled.
Therefore, the only notable difference between the two environments on any particular
lab activity was the ability to drag. On each instrument, students were asked to state, in
their own words, as many conjectures as they could about the geometric situations the
figures represent. Students were then asked to rank their personal conviction of the truth
of their conjectures on a scale of 1 to 10 for each conjecture, with 10 being the highest
confidence in the truth of their conjectures.
The labs were designed to serve as introductory lessons before the actual
theorems and postulates were presented to the class. For example, the parallelogram lab
was completed prior to or on the day in which parallelograms were introduced in the
44
course curriculum. This gave the students an opportunity to explore and conjecture about
the properties of this figure before being presented with the theorems concerning
parallelograms. Topics were selected to align with the course curriculum. The lab topics
included:
1. Parallel Lines and Transversals
2. Angles of Triangles
3. Triangle Midsegments
4. Parallelograms
5. Diagonals of Rectangles and Squares
6. Diagonals of Rhombi and Kites
7. Trapezoids
8. Inscribed Angles and Circles
These instruments are presented in Appendix A.
The quality of the conjectures formed by the students was assessed using the
following general rubric following Lange (2002):
? An R was assigned to a conjecture that was true and relevant to the lab
activity. Relevant conjectures include the lab activities? ?targeted?
conjectures and other true conjectures that pertain to the figure(s) and had
not yet been introduced in the course curriculum at the time of the activity.
? An I was assigned to conjectures that were true but irrelevant to the lab
activity. These conjectures had either been introduced earlier in the course
curriculum or they pertain to a more general figure than the figure(s)
involved in the lab activity. For example, consider a student making the
45
following conjecture about a figure showing a parallelogram: ?The sum of
the interior angle measures of a parallelogram is 360 degrees.? This
conjecture is true but irrelevant, since it is true for all quadrilaterals and
was discussed earlier in the course.
? An A was assigned to a conjecture that was ambiguous and could not be
reasonably interpreted as true or false. Conjectures of this kind were often
poorly worded, and it was difficult to determine what the student is trying
to communicate mathematically.
? An F was assigned to conjectures that were false. These conjectures may
have been true for some cases but if a counterexample exists,
mathematically speaking, the conjecture is false. For example, consider
the following false conjecture: ?A parallelogram has two acute angles and
two obtuse angles.? Although this statement is true for many
parallelograms, it is false if the parallelogram is a rectangle or a square.
For each lab instrument, item-specific rubrics that specify what kinds of
conjectures are considered relevant, irrelevant, ambiguous, or false were developed; see
appendix A. The validity of the instruments and their rubrics was validated by a
mathematician and by a mathematics educator, both employed by a local university.
Their suggestions were used in modifying the instruments to their final form.
All responses to the lab instruments were assessed by the researcher as well as an
outside grader experienced in teaching secondary geometry. While different outside
graders participated in the validation, only a single outside grader was used to assess each
individual lab activity. For example, one outside grader was used to assess lab activity
46
one where a different outside grader was used for lab activity two. Outside graders used
the general rubric and item-specific rubrics to assign one of the four letters mentioned
above to each of the subjects? conjectures. Interrater reliability percentage of agreement
data is presented in chapter four of this report.
Geometric Reasoning Instrument
Participants completed an instrument to assess their entering level of geometric
reasoning following the work of van Hiele (Gutierrez & Jaime, 1998) during the second
week of class instruction. The instrument consisted of six open ended items with
multiple parts. It was patterned after instruments suggested by Gutierrez and Jaime
(1998) who have done extensive work with van Hiele levels of geometric reasoning.
Gutierrez and Jaime (1998) state two sources of validity for the content used in these
items. The first source of validity was a series of pilot studies conducted by the authors
and the second source of validity was based on analysis made by several researchers with
expertise in van Hiele theory (Gutierrez & Jaime, 1998). Gutierrez and Jaime (1998) also
reported Guttman Coefficients ranging from .98 to 1.00. No additional information
concerning validity and reliability is provided by Gutierrez and Jaime (1998). This
instrument was graded by the researcher and an outside grader with knowledge of van
Hiele theory using a general rubric suggested by the authors. This instrument ranked the
students? geometric reasoning level (1 ? 4) and is presented along with the rubric used in
appendix B.
47
Mathematics Achievement Instrument
Information about mathematics achievement was obtained from student records
already on file in the school?s guidance department. Thirty-nine of the forty-one subjects
had taken the Preliminary Scholastic Aptitude Test (PSAT) and the results were part of
their school file. The score from the mathematics sections of this assessment was used to
measure the students? mathematics achievement. The mathematics portion of this test
consists of two timed 25 minute mathematics sections containing arithmetic, algebra, and
geometry items. Comprehensive reviews and analyses are conducted to ensure that
questions are a valid measure of mathematics problem solving ability and are fair for
different groups of students (College Entrance Examination Board, 2003). Item types
range from multiple choice, quantitative comparison, and free response. Reported scores
for the PSAT mathematics sections may range from 20 to 80 points, with one point
awarded for each correct response, 0 points for no answer, and a ? point deduction for
each wrong answer on a multiple choice item. Percentile scores were also reported which
compared the students nationally. A reliability coefficient of .87 was reported for the
2003 version of the mathematics portion of the PSAT (College Entrance Examination
Board, 2003).
Survey Instrument
Student surveys were given to all participants after all lab activities had been
concluded to assess their reactions to the experience. Questions on this instrument were
open ended. They addressed the students? concept of conjecture, environment
preferences, and the purpose of dragging in the dynamic environment. The purpose of
this instrument was to address several of the a priori qualitative themes of this study:
48
Which environment do the students prefer and why? What is the students? concept of
conjecture? How much do students use the dragging utility while in the dynamic
environment? This survey instrument is presented in appendix C.
Interview Protocol
Interviews took place with ten selected participants representing a cross section of
students by level of mathematics achievement. These interviews took place after all lab
activities and the survey instrument were completed. Individual interviews were
conducted with the students, who were seated at a computer station for the interview.
Students were presented with geometric figures in the dynamic software environment,
and conjectures associated with them. The students were asked to evaluate the truth of
these given conjectures. Students were allowed, and sometimes encouraged, to drag the
figures while the interview was taking place. The questions focused on the concept of
proof versus conjecture, finding counterexamples to disprove false conjectures, and
student conviction in the computer output of the dynamic environment. These questions
coincided with three of the a priori qualitative themes addressed in this study. Particular
attention was placed on the students? dragging technique during the interviews and the
manner in which the students? used the dragging utility was noted in the researcher?s
journal. All interviews were audiotaped and transcribed. The interview protocol and
accompanying figures are presented in appendix D.
49
Procedures
An informed consent form was distributed to all students of both classes during
the first week of instruction. All of the students were invited to participate in the study,
and those that returned the form with a parental signature were included in the study. No
additional recruiting was necessary, and all students participated in the lab activities as a
requirement of the course regardless of their participation in the study.
The van Hiele geometric reasoning instrument was administered during the
second week of instruction. The PSAT scores for each participant were also obtained
from the guidance department of the high school during the second week of instruction.
The eight conjecturing labs, both static and dynamic, consisted of 45-minute
increments of time at the beginning of a 90-minute block of instruction. These labs were
spread over eight weeks of instruction beginning on the fourth week. All lab activities
took place in one of the school?s computer labs. Students were seated at a computer
station in which the file containing the lab activity was already open. The class that was
assigned the static environment had their dragging tool hidden and only their text tool
was functioning to allow them to write their conjectures. The class that was assigned the
dynamic environment also had the dragging tool operational.
All students were shown how to use the dragging tool but no other training on the
capabilities of the dynamic geometry software package was presented to the students
until the conclusion of the study, since all of the figures presented in the activities were
constructed by the researcher. Thus, the students did not have to construct, hide, or
measure anything on the screen in order to make relevant conjectures. In the directions,
students were asked to state any ?new? conjectures not already discussed in the
50
classroom concerning the figures in the lab activities. Students were also asked to rate
their conviction in the truth of their conjectures on a 1-10 scale for each of their
conjectures with 10 being the most confident.
The researcher?s task during the lab activities was to ensure that the students were
on task and were properly using the computer software and hardware. At no time did the
researcher confirm the validity of any conjecture or help any student with their
conjectures. All students received full credit for participating in each lab activity
regardless of the quality of their conjectures or their participation in the study. Students
saved their final output on a floppy disc provided by the researcher. The researcher
collected each disc at the end of the conjecturing lab and reissued them before the next
lab. When each lab activity was concluded, all students met in the classroom where the
researcher led a discussion on the students? findings.
Class B participated in the static environment on lab activities 1, 3, 5, and 7 while
class A participated in the dynamic environment during these lab activities. During
activities 2, 4, 6, and 8 these assignments were reversed. The purpose of this assignment
design was to ensure that all students were equally exposed to both environments so as to
not to deprive any student from any educational benefits offered by the static or dynamic
environments. Because of this design, it was necessary to run separate data analyses on
the two different assignment cases (odd and even).
The qualitative data collection occurred following the completion of the final lab
activity. The survey instrument was administered one week after the completion of the
final lab activity. Students were selected for interviews based on their reported
mathematics achievement scores to ensure that subjects represented various levels of
51
mathematical achievement. Each interview lasted approximately fifteen minutes and was
conducted privately with the researcher during the school day.
In addition, the researcher kept a journal throughout the course of the study to
incorporate field notes recorded during the research activities. Field notes were taken
after the completion of each lab activity and after each participant interview. These notes
were immediately entered into an electronic journal in chronological order. Entries
following the lab activities typically were a paragraph in length and included general
observations of the classes, their use of the software, how quickly they finished the
activity, and how well they participated during the discussion following the lab. Entries
following the interviews made note of the participants? dragging technique during the
interview as well as their general behavior during the interview process. The interviews
themselves were audio taped and later transcribed.
Data Analysis
This study analyzed both quantitative and qualitative data. Both of these analyses
are described in the following sections.
Quantitative Analysis
Descriptive statistics, correlations, and sequential linear regression analysis were
used to answer the quantitative research questions. The following independent variables
were collected for each participant: Gender (GENDER), mathematics achievement
(ACHIEVEMENT), and van Hiele level (VH). The independent variable for
environment (ENVIRONMENT) was recorded for each activity and for each student.
52
The following dependent variables were collected from each participant in each
environment: mean score for the number of relevant conjectures (RELEVANT), mean
score for the number of false conjectures (FALSE), and the mean conviction score for
each conjecture (CONVICTION). Therefore, each student had a score for the number of
relevant conjectures per lab activity and for the number of false conjectures per lab
activity. These means were calculated separately for each environment since all
participants conjectured in both environments.
To compute RELEVANT, the total number of relevant conjectures was summed
across the activities in a particular environment for each participant and then divided by
the number of labs completed by that participant in that environment. This yielded a
mean score for the variable RELEVANT for each participant in each environment. False
conjectures were treated in a similar manner to calculate the variable FALSE for each
participant in each environment. The conviction score for each participant was obtained
by calculating the mean of all convictions for each conjecture in a particular environment.
When differences between the researcher?s rating and the outside grader?s rating
were present within a lab activity, the mean of those ratings was used. For example, if
the researcher had found three relevant conjectures for a student on a particular lab
activity, and the outside grader had found only two relevant conjectures for that same
student on that same lab activity, a score of 2.5 was used for that lab activity.
A correlation matrix was created using all variables to determine zero order
correlation coefficients. Three different sequential linear regressions were performed on
the three dependent variables for each of the two assignment cases. The independent
variables, ACHIEVEMENT, GENDER, and VH, were used as covariates in the
53
sequential linear regression. Therefore, this analysis encompassed six sequential linear
regressions with ENVIRONMENT being the independent variable of interest. Sequential
regression analysis was used to test the following null hypotheses:
? The opportunity to alter a figure in the dynamic geometry environment by
dragging does not significantly increase a student?s ability to make relevant
conjectures.
? The opportunity to alter a figure in the dynamic geometry environment by
dragging does not significantly decrease a student?s likelihood of making false
conjectures.
? The opportunity to alter a figure in the dynamic geometry environment by
dragging does not significantly increase a student?s conviction in the conjectures
they make.
Statistical analyses were also used to explore what role gender, mathematics
achievement, and van Hiele level play in the conjecture process. All statistical analyses
were conducted using the Statistical Package for the Social Sciences (SPSS) (Shannon
and Davenport, 2001).
Qualitative Analysis
The qualitative analysis of this study used four sources of data: The survey
instrument, the participant interviews, the researcher?s journal, and the conjecture labs
themselves. The survey instrument and the interview protocol were constructed to
address the four a priori qualitative themes. These themes were developed prior to the
collection of data, motivated by the literature reviewed for this study. They included:
Student preferences concerning the conjecturing environments, students? concepts of
54
conjecture and proof, students? ability to form and find counterexamples using dynamic
geometry software, and students? conviction in the output generated by dynamic
geometry software. The specific literature sources used to develop these themes are
discussed in greater detail in Chapter V.
The qualitative analysis concentrated on coded key words and phrases found in all
of the mentioned sources of data. The researcher?s journal as well as all surveys,
interviews, and conjectures were read thoroughly. On a second reading, key words and
phrases were identified and placed on index cards along with their location on the
instrument. These key words and phrases were assigned as ?codes? that helped organize
the data into several core categories. All codes and text associated with those codes were
then transferred from index cards to a computer data base for easy access.
Bogdan and Biklen (1998) refer to codes as ?words, phrases, patterns of behavior,
subjects? ways of thinking, and events that repeat or stand out (p. 171).? The act of
coding involves several steps: Searching your data for patterns and topics of interest,
identifying words and phrases that identify those patterns and topics, placing these words
and phrases into core categories, and using these categories to sort the data (Bogden &
Biklen, 1998).
After the coding process, many of the core categories applied directly to the four
a priori themes developed for this study. However, several other categories gleaned from
the conjecturing labs and the researcher?s journal suggested the development of two
additional qualitative themes. Thus, emergent themes concerning dynamic language and
dragging tendencies were added to the analysis.
55
CHAPTER IV: QUANTITATIVE RESULTS
This chapter reports the results of the quantitative data analysis conducted in order
to answer the quantitative questions of this study. The descriptive statistics of each of the
variables are reported first, followed by the interrater reliability percentage of agreement
concerning the scores obtained from the instruments designed for this study. The results
of zero-order correlations of all of the variables are reported followed by the results of the
sequential regression analysis. This chapter concludes by summarizing how these results
address the research questions as well as the exploratory findings.
Descriptive Statistics
The following tables provide descriptive statistics for the three dependent
variables: RELEVANT, FALSE, and CONVICTION. Tables reporting results for the
covariates ACHIEVEMENT and VH are then provided. Table 2 provides means and
standard deviations for all three dependent variables in both assignment cases regardless
of the environment used. Recall that two different assignment cases were used in this
study. Class A received the instruments in the dynamic environment in the odd
assignment case and Class B received the instruments in the dynamic environment during
the even assignment case.
56
Table 2
Means and standard deviations for both assignment cases for the three dependent
variables in both environments
Dependent
Variable
Odd Assignment Case
Mean S.D.
Even Assignment Case
Mean S.D.
RELEVANT 1.3374 0.95512 1.5469 0.81262
FALSE 0.1315 0.22820 0.1964 0.30440
CONVICTION 8.1667 1.53885 8.5231 1.54195
Note that in this table, there is no distinction between the environments used,
therefore the means represent an average over both environments. The means in the odd
assignment case represent both the means of Class A, conjecturing in the dynamic
environment, and Class B, conjecturing in the static environment. The roles are reversed
with the even assignment case. It should not be surprising that the scores are quite
comparable, since they include both cases.
Table 3 provides means for each variable within a specific environment.
Differences in means are apparent when considering the two different environments.
RELEVENT and CONVICTION is higher in the dynamic environment and FALSE is
lower in the dynamic environment. This is true in both assignment cases. Further
analysis will determine the statistical significance of these differences.
57
Table 3
Means in terms of environment for each of the dependent variables
Dependent Variable Odd Assignment Case
Class A* Class B
Even Assignment Case
Class A Class B*
RELEVANT 1.80 0.92 1.21 1.93
FALSE 0.02 0.238 0.31 0.06
CONVICTION 8.8 7.6 7.90 9.20
* Group using the dynamic geometry environment
Tables 4 and 5 provide means and standard deviations for the covariate variables
ACHIEVEMENT and VH. The mean achievement of 33.3 in Class A corresponds with
the 16th percentile when compared with national data and the mean score of 36.8
corresponds with the 26th percentile when compared with national data (College
Entrance Examination Board, 2004).
Table 4
Means and standard deviations for the covariate ACHIEVEMENT
ACHIEVEMENT Class A Class B
Mean 33.3 36.8
Standard Deviation 6.3 7.5
58
In Class A, 19 of the 20 subjects were assessed at van Hiele Level 1 with one subject at
Level 2. In Class B, 17 of the 21 subjects were assessed at van Hiele Level 1 with four
subjects at Level 2.
Table 5
Means and standard deviations for the covariate VH
VH Class A Class B
Mean 1.05 1.19
Standard Deviation 0.22 0.42
Interrater Reliability
Each of the lab instruments used in this study as well as the van Hiele geometric
reasoning instrument was assessed by the researcher and by an outside grader
experienced in the teaching of secondary school geometry. There were no differences in
the assigned scores for the van Hiele instrument and only small differences in the lab
instruments. Percentage of agreement was calculated for each instrument for both
RELEVANT and FALSE, and they ranged from 92% to 100%. When a score for any of
these variables was found to differ, the mean score was used in the analysis.
Correlations of Variables
The following tables show the zero order Pearson correlations for all variables in
each of the two different assignment cases. Statistically significant correlations are
indicated for the .05 and .01 levels. These zero order correlations do not take into
59
account collinearity aspects of the data; therefore, they should not be interpreted as
?unique effects? (Shannon & Davenport, 2001).
Table 6
Zero order Pearson correlation coefficients for the odd assignment case
RELEVANT
FALSE
CONVICTION
ENVIRONMENT
GENDER
ACHIEVEMENT
VH
RELEVENT
-.191 .695** .474** .154 .332* .161
FALSE
-.191 -.184 -.501** -.253 .382* .259
CONVICTION
.695** -.184 .445** .333* .111 .151
ENVIRONMENT
.474** -.501** .445** .267 -.255 -.215
GENDER
.154 -.253 .333* .267 -.093 .102
ACHIEVEMENT
.332* .382* .111 -.255 -.093 .518**
VH
.161 .259 .151 -.215 .102 .518**
*p<.05 **p<.01
This correlation matrix indicates a significant correlation of the independent
variable ENVIRONMENT with the three dependent variables: RELEVANT, FALSE,
and CONVICTION. In a later section, sequential regression analysis will be used to
explore the unique contribution of the independent variable ENVIRONMENT on each of
the dependent variables as well as the unique effects of the covariate variables on each
dependent variable.
There is also a strong positive correlation between the dependent variables
CONVICTION and RELEVANT indicating that those students that have more relevant
conjectures are more confident in those conjectures. The results also indicate that higher
achieving students tend to make more relevant conjectures and somewhat surprisingly,
60
more false conjectures as well. Note the strong correlation between van Hiele level and
mathematics achievement, as well as the correlation between GENDER and
CONVICTION, indicating that males have a higher confidence in their conjectures.
Table 7
Zero order Pearson correlation coefficients for the even assignment case
RELEVANT
FALSE
CONVICTION
ENVIRONMENT
GENDER
ACHIEVEMENT
VH
RELEVANT
-.246 .455** .419** -.053 .412** .147
FALSE
-.246 -.517** -.427** -.036 -.048 -.026
CONVICTION
.455** -.517** .415** -.022 .262 .116
ENVIRONMENT
.419** -.427** .415** -.267 .255 .215
GENDER
-.053 -.036 -.022 -.267 -.093 .102
ACHIEVEMENT
.412** -.048 .262 .255 -.093 .518**
VH
.147 -.026 .116 .215 .102 .518**
*p<.05 **p<.01
This matrix shows many of the same results hold for the odd assignment case.
The independent variable ENVIRONMENT correlates strongly with the three dependent
variables; RELEVANT, FALSE, and CONVICTION. CONVICTION correlates
strongly with RELEVANT as in the other case. In this case, however, ACHIEVEMENT
correlates significantly with RELEVANT, but not significantly with FALSE, and there is
not a significant correlation between GENDER and CONVICTION. In the next section,
sequential regression analysis will be used to explore unique contributions of the
geometric environment as well as the covariates.
61
Regression Analysis
A sequential regression analysis was used. In step one of the regression analysis
the variables of ACHIEVEMENT, GENDER, and VH were entered. In step two of the
analysis the ENVIRONMENT variable was added as a predictor. This analysis will show
the unique effects of the ENVIRONMENT variable after the other variables have been
accounted for. Since there are three dependent variables to consider and two separate
assignment cases for each of the dependent variables, the results of six sequential
multiple regressions are presented.
Relevant Conjectures
The variable RELEVANT measured the average number of relevant conjectures
made per lab activity for each subject. The results of this variable in the odd assignment
case yielded an overall mean of 1.34 relevant conjectures per lab activity with a standard
deviation of .955. Table 8 presents the results of the sequential regression analysis. Step
one of the regression analysis yielded R = .377 which was not statistically significant,
F(3, 35) = 1.928, p = .143. The R
2
= .142 indicated that approximately 14% of the
variance in RELEVANT could be explained by a combination of mathematics
achievement, van Hiele level, and gender. In step two of the analysis, the
ENVIRONMENT variable was added yielding R = .698 which was statistically
significant, F(4, 34) = 8.072, p <.001. The change statistics produced by the addition of
ENVIRONMENT were statistically significant with R = .588, F(1, 34) = 22.886, and
p <.001. The R
2
change of .345 explained the variance of the dependent variable
RELEVANT over and above the covariates. Thus the unique contribution of
62
ENVIRONMENT is responsible approximately 35% of this variance, which is
statistically significant.
Table 8
Regression analysis for the variable RELEVANT in the odd assignment case
Variable
B
Unstandardized
Coefficients
Beta
Standardized
Coefficients
R
Semipartial
Correlation
R
2
Sig.
ACHIEVEMENT .060 .440 .368 .135 .005**
VH .295 .105 .086 .007 .491
GENDER -.102 -.054 -.049 .002 .694
ENVIRONMENT 1.242 .649 .588 .346 <.001**
*p<.05 **p<.01
The results of the RELEVANT variable in the even assignment case yielded an
average of 1.55 relevant conjectures per lab activity with a standard deviation of .813.
Table 9 presents the results of the sequential regression analysis. Step one of the
regression analysis yielded R = .393 which was not statistically significant, F(3, 35) =
2.129, p = .114. The R
2
of .154 indicated that approximately 15% of the variance in
RELEVANT could be explained by a combination of mathematics achievement, van
Hiele level, and gender. In step two of the analysis, the ENVIRONMENT variable was
added in yielding R = .509 which was statistically significant, F(4, 34) = 2.966, p = .033.
The change statistics produced by the addition of ENVIRONMENT were statistically
significant with R = .323, F(1, 34) = 4.787, and p < .036. The R
2
change = .104
63
explained the variance of the dependent variable RELEVANT over and above the
covariates. Thus the unique contribution of ENVIRONMENT is responsible
approximately 10% of this variance, which is statistically significant.
Table 9
Regression analysis for the variable RELEVANT in the even assignment case
Variable
B
Unstandardized
Coefficients
Beta
Standardized
Coefficients
R
Semipartial
Correlation
R
2
Sig.
ACHIEVEMENT .036 .312 .261 .068 .087
VH -.169 -.071 -.058 .003 .698
GENDER -.015 -.009 -.008 <.001 .955
ENVIRONMENT .581 .362 .323 .104 .036*
*p<.05 **p<.01
False Conjectures
The variable FALSE measured the average number of false conjectures made per
lab activity for each subject. The results of this variable in the odd assignment case
yielded an average of 0.1315 false conjectures per lab activity with a standard deviation
of .228. Table 10 presents the results of the sequential regression analysis. Step one of
the regression analysis yielded an R = .461 which was statistically significant, F(3, 35) =
3.149, p = .037. The R
2
= .213 indicated that approximately 21% of the variance in
FALSE could be explained by a combination of mathematics achievement, van Hiele
level, and gender. In step two of the analysis, the ENVIRONMENT variable was added
64
yielding R = .588 which was statistically significant, F(4, 34) = 4.494, p =.005. The
change statistics produced by the addition of ENVIRONMENT were statistically
significant with R = .365, F(1, 34) = 6.928, and
p <.013. The R
2
change of .133 explained the variance of the dependent variable
FALSE over and above the covariates. Thus the unique contribution of ENVIRONMENT
is responsible approximately 13% of this variance, which is statistically significant.
Table 10
Regression analysis for the variable FALSE in the odd assignment case
Variable
B
Unstandardized
Coefficients
Beta
Standardized
Coefficients
R
Semipartial
Correlation
R
2
Sig.
ACHIEVEMENT .008 .241 .201 .040 .156
VH .035 .052 .043 .002 .761
GENDER -.049 -.107 -.097 .009 .490
ENVIRONMENT -.184 -.409 -.365 .133 .013*
*p<.05 **p<.01
The analysis of the FALSE variable in the even assignment case yielded an
average of 0.1964 false conjectures per lab activity with a standard deviation of .304.
Table 11 presents the results of the sequential regression analysis. Step one of the
regression analysis yielded R = .050 which was not statistically significant, F(3, 35) =
0.030, p = .993. The R
2
= .003 indicated that less than 1% of the variance in FALSE
could be explained by a combination of mathematics achievement, van Hiele level, and
65
gender. In step two of the analysis, the ENVIRONMENT variable was added yielding R
= .448 which was not statistically significant, F(4, 34) = 2.129, p =.099. The change
statistics produced by the addition of ENVIRONMENT were statistically significant with
R = .445, F(1, 34) = 8.407, and p =.007. The R
2
change of .198 explained the variance of
the dependent variable FALSE over and above the covariates. Thus the unique
contribution of ENVIRONMENT is responsible approximately 20% of this variance,
which is statistically significant.
Table 11
Regression analysis for the variable FALSE in the even assignment case
Variable
B
Unstandardized
Coefficients
Beta
Standardized
Coefficients
R
Semipartial
Correlation
R
2
Sig.
ACHIEVEMENT .001 .012 .010 <.001 .947
VH .088 .098 .080 .006 .606
GENDER -.110 -.182 -.165 .027 .289
ENVIRONMENT -.300 -.498 -.445 .198 .007**
*p<.05 **p<.01
Conviction
The variable CONVICTION measured the average self-rated conviction in all
conjectures made in every lab activity for each subject. The results of this variable in the
odd assignment case yielded an average conviction of 8.1667 with a standard deviation of
1.54. Table 12 presents the results of the sequential regression analysis. Step one of the
66
regression analysis yielded R = .386 which was not statistically significant, F(3, 35) =
2.044, p = .126. The R
2
= .149 indicated that approximately 15% of the variance in
CONVICTION could be explained by a combination of mathematics achievement, van
Hiele level, and gender. In step two of the analysis, the ENVIRONMENT variable was
added in yielding R = .566 which was statistically significant, F(4, 34) = 4.009, p =.009.
The change statistics produced by the addition of ENVIRONMENT were statistically
significant with R = .414, F(1, 34) = 8.578, and p =.006. The R
2
change of .171
explained the variance of the dependent variable CONVICTION over and above the
covariates. Thus the unique contribution of ENVIRONMENT is responsible
approximately 17% of this variance, which is statistically significant.
Table 12
Regression analysis for the variable CONVICTION in the odd assignment case
Variable
B
Unstandardized
Coefficients
Beta
Standardized
Coefficients
R
Semipartial
Correlation
R
2
Sig.
ACHIEVEMENT .038 .172 .144 .021 .317
VH .670 .147 .121 .015 .400
GENDER .600 .196 .178 .032 .217
ENVIRONMENT 1.410 .464 .414 .171 .006**
*p<.05 **p<.01
The results of the CONVICTION variable in the even assignment case yielded an
average conviction of 8.5231 with a standard deviation of 1.54. Table 13 presents the
67
results of the sequential regression analysis. Step one of the regression analysis yielded R
= .265 which was not statistically significant, F(3, 35) = 0.880, p = .461. The R
2
= .070
indicated that approximately 7% of the variance in CONVICTION could be explained by
a combination of mathematics achievement, van Hiele level, and gender. In step two of
the analysis, the ENVIRONMENT variable was added in yielding R = .444 which was
not statistically significant, F(4, 34) = 2.084, p =.105. The change statistics produced by
the addition of ENVIRONMENT were statistically significant with R = .356, F(1, 34) =
5.366, and p =.027. The R
2
change of .127 explained the variance of the dependent
variable CONVICTION over and above the covariates. Thus the unique contribution of
ENVIRONMENT is responsible approximately 13% of this variance, which is
statistically significant.
Table 13
Regression analysis for the variable CONVICTION in the even assignment case
Variable
B
Unstandardized
Coefficients
Beta
Standardized
Coefficients
R
Semipartial
Correlation
R
2
Sig.
ACHIEVEMENT .048 .218 .144 .021 .244
VH -.419 -.092 -.075 .006 .628
GENDER .327 .106 .097 .009 .534
ENVIRONMENT 1.215 .399 .356 .127 .027*
*p<.05 **p<.01
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Conviction of Relevant versus False Conjectures
In the previous section, student conviction in the correctness of their conjectures
was averaged for all of a participant?s conjectures, including the false conjectures.
Having a higher conviction on false conjectures is not desirable; therefore, a further
analysis of CONVICTION comparing the averages of relevant versus false conjectures
was conducted using a repeated measure ANOVA. In this analysis, the students?
conviction in their relevant conjectures was compared with their conviction in their false
conjectures. This analysis was grouped by the type of environment the students were
using. Students that had made no false conjectures could not be included in this analysis.
Table 14 summarizes the conviction means that were compared.
Table 14
Conviction scores for relevant and false conjectures for each environment
Dynamic
Environment
Static
Environment
Relevant Conjectures
9.5 8.6
False Conjectures
6.7 6.9
The effect within subjects was significant, with F(1, 27) = 4.561, p = .042
indicating that students? conviction in relevant conjectures is significantly higher than
their conviction in false conjectures. The effect between subjects was not significant,
with, F(1, 27) = .843, p = .367. However, a post-hoc t-test on relevant conjectures by
environment resulted in t = 2.629, p = .014, indicating that conviction in relevant
conjectures is significantly higher in the dynamic environment. The t-test on false
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conjectures by environment resulted in t = -0.350, p = 0.729, showing no significant
difference in the conviction of false conjectures within the two environments.
Collinearity of Predictors
Table 15 shows the tolerance and variance inflation factor (VIF). The tolerance
shows the percentage of each variable that is not related to the other variables and the
VIF is merely the reciprocal of the tolerance. With relatively high tolerance levels and
VIF levels close to one there is not a problem with collinearity among predictors in this
study (Shannon and Davenport, 2001).
Table 15
Collinearity results for the independent variables for each sequential regression model
Model ACHIEVEMENT GENDER VH ENVIRONMENT
1 Tolerance
VIF
.706
1.416
.950
1.053
.701
1.427
.796
1.256
2 Tolerance
VIF
.700
1.429
.823
1.216
.669
1.495
.796
1.256
Summary
In all six multiple regressions the variable ENVIRONMENT was a significant
predictor of the dependent variable of concern. As a result, the null hypotheses were
rejected since all results show that students made significantly more relevant conjectures
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in the dynamic environment, significantly fewer false conjectures in the dynamic
environment, and have significantly greater conviction in their conjectures in the dynamic
environment. In this study, the only difference between the static and dynamic
environments provided was the opportunity to alter the figures by dragging in the
dynamic environment. Therefore, it was concluded that the opportunity to alter figures
by dragging results in students making more relevant conjectures, fewer false
conjectures, and also results in a greater conviction in their conjectures. These
differences were shown to be statistically significant by sequential regression analysis.
Zero-order Pearson correlation coefficients between the covariates and the
dependent variables found some significant correlations, however, the regression analysis
which examined unique effects of the covariates as predictors of the dependent variables
rendered some of these relationships insignificant. The zero order correlation between
GENDER and CONVICTION was significant in the odd assignment case but the
subsequent linear regression found that the unique contribution of a student?s gender was
not a significant predictor of conviction in that assignment case. A similar result
occurred between the variables ACHIEVEMENT and FALSE in the odd assignment
case. Once again the regression analysis showed the unique contribution of a student?s
mathematics achievement not to be a significant predictor of the number of false
conjectures that student will form, even though the zero order correlations were
significant.
One covariate was found to be significant. The unique contribution of the
variable ACHIEVEMENT was found to be highly correlated with RELEVANT, but only
in the odd assignment case. Therefore, there is some evidence to support the claim that
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students with higher mathematics achievement levels tend to make more relevant
conjectures regardless of the environment.
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CHAPTER V: QUALITATIVE RESULTS
This chapter will analyze six qualitative themes that were developed prior to and
during the course of the study. The instruments used to collect qualitative data include:
? The student conjectures created during the eight lab activities.
? A short participant survey with open ended items that was administered
to all participants after the conclusion of the lab activities.
? A series of ten individual interviews also performed after all of the
conjecturing labs had been concluded.
? The researcher?s field notes taken during the lab activities and student
interviews. These notes were recorded in journal form by the researcher.
Examination of these sources of data by the researcher yielded results for six
qualitative themes concerning student conjectures and the use of dynamic geometry
software. Four of these themes were developed before the study and were motivated by
the literature sited in Chapter II. The survey and interview instruments were developed
to address these a priori themes. The two additional themes concerning dynamic
language and student dragging tendencies emerged during the course of the study. The
theme concerning dynamic language led to an additional review of the literature, which
was added to Chapter II. This chapter will report on the findings of each of four a priori
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themes as well as the two emergent themes. The a priori qualitative themes discussed
are:
1. Student concepts of conjecture and proof
2. Student preferences concerning the geometry environment
3. Students? ability to form and find counterexamples in the dynamic environment
4. Students? conviction in computer generated output
The Emergent themes are:
5. Student dragging techniques
6. The use of dynamic language
Concept of Conjecture and Proof
Senk (1985) questions if students really understand the meaning of proof while
Mingus and Grassl (1999) contend that even younger students are capable of forming
proofs at some level given freedom to do so. Martin and Herel (1989b) found that
students often use both inductive and deductive frames of reasoning when asked to form
proofs, and Pandescia (2002) questioned the need for formal proof instruction when the
use of dynamic geometry software can easily verify conjectures. This led to the
questions: ?What do students consider to be a proof??, ?What is their concept of
conjecture??, ?How do they differ??
This theme was explored using two primary data sources. First, the survey
instrument asked the subjects to ?describe what a conjecture is in your own words.?
Larson, Boswell, and Stiff (2001) define a conjecture as ?an unproven statement that is
based on observations? (p. 4). The participants had been exposed to this definition of
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conjecture during classroom instruction prior to the beginning of the study; however,
students were expected to reply with a definition given in their own words.
The coding of this survey item resulted into two core categories: Students that
expressed a concept of conjecture and those that expressed little or no concept of
conjecture. The data was further analyzed by comparing the students? core category with
the students? ability to form relevant conjectures demonstrated by the lab activities used
in the study. The following key words and phrases emerged as codes to organize the
students? responses.
1. Guess
2. Opinion
3. Observation
4. Hypothesis
5. Statement of Truth
6. Lack of Concept
The first five codes were used to classify responses that showed some concept of the
working definition of conjecture. The sixth code was reserved for those responses that
showed little or no concept of what a mathematical conjecture is. Approximately 80% of
the subjects were classified as having a reasonable concept of conjecture. Consider some
of the responses associated with the codes used to classify them.
Several of the subjects defined conjecture as some kind of guess: ?A conjecture
is a guess that you may or may not be able to prove.? ?A conjecture is a guess about a
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given item on a figure.? ?A conjecture is an educated guess.? Others saw a conjecture as
a personal opinion: ?A conjecture is an opinion of what is true.? ?A conjecture is your
own opinion made about a particular figure.? ?A conjecture is a fact or opinion about a
given geometric figure.?
Another common word used in subject responses was ?observation? which is also
present in the working definition. Consider the following responses: ?A conjecture is a
statement made by observing and testing.? ?A conjecture is made by observing and
thinking about what you see looking at a figure.? ?A conjecture is your observation of
what is in a figure.? The word hypothesis also occurred several times in subject
responses: ?A conjecture is a hypothesis or judgment based on evidence.? ?A
conjecture is your hypothesis about the facts of the figure?. Once again these statements
were accepted by the researcher as evidence of student understanding of conjecture.
The most common code indicated some kind of belief of truth about the figures.
These responses were varied but did show an understanding of the concept of conjecture.
Consider the following examples: ?A conjecture is what you think is true about a
problem.? ?A conjecture is what you conclude to be true.? ?A conjecture is your own
words about what you think are true about the figure.? ?A conjecture is a fact that you
think is true mathematically.?
In total there were 33 responses similar to those mentioned that were categorized
by these five codes and showed evidence of some concept of conjecture. Eight students
that did not show evidence of the concept of conjecture on the survey instrument:
?A conjecture is a figure that a person could make a statement about and answer it.?
?A conjecture is a theorem proven by statements.? ?A conjecture is a theorem of many
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angles.? ?A conjecture is a figure with congruent sides.? However, when averaging
these students number of relevant conjectures across both environments it was found that
these eight participants averaged 1.75 relevant conjectures per lab activity, which is
higher than the overall average in both assignment cases. This is a surprising result and
indicates that students can find relevant conjectures independent of their stated concept of
conjecture.
Additional core categories related to this theme emerged from student responses
to the interview. Interview participants were given conjectures about given figures and
asked if they agreed or disagreed with those conjectures and how they would prove or
disprove those conjectures. The purpose of these questions was to see if the participants
made a distinction between inductive conjectures based on observation and facts proven
by deduction.
To examine the concept of proof or conjecture, figure 8 was presented in the
dynamic environment to ten participants interviewed individually.
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Figure 1: The midsegment quadrilateral
A midsegment quadrilateral is formed by connecting the midpoints of the sides of a quadrilateral
Conjecture: The midsegment quadrilateral will always be a parallelogram.
Figure 8. First situation posed to the interview participants in the dynamic environment
along with the given conjecture.
Subjects were asked if they agreed with the stated conjecture. After some
dragging and observation, all of the subjects agreed with the stated conjecture, which is in
fact true. The subjects were then asked how they would prove the conjecture. Of the ten
subjects none of them mentioned the idea of a purely deductive proof even though most
of the targeted conjectures used in the lab activities were proven deductively as part of
the course curriculum. Two of the participants did not indicate how they would prove the
conjecture. Eight of the ten subjects indicated that they would use a combination of
deduction and observation to prove the conjecture. The following excerpts from the
interviews demonstrate this kind of ?mixed? reasoning.
Mark is a relatively high-achieving male student when compared to the entire
sample. He claimed that he would prove that the midsegment quadrilateral is always a
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parallelogram by observing that the opposite angles of the midsegment quadrilateral stay
congruent throughout dragging in the dynamic environment.
Researcher: What you see on this first figure is called a midsegment
quadrilateral; it is formed by connecting the midpoints of any general
quadrilateral. A conjecture from a former student says that the midsegment
quadrilateral will always be a parallelogram. Do you think this conjecture is true?
Mark: It looks like it here.
Researcher: How would you test your conclusion? (Student begins to drag)
Mark: To see if the opposite sides stay parallel as I drag around. Yes, I think it is
true.
Researcher: Would you like any measures?
Mark: Yes
Researcher: What measures?
Mark: How about the angles? (Pause while researcher measures and displays the
interior angles of the midsegment quadrilateral)
Researcher: How does this help with your conclusion?
Mark: Opposite angles are congruent as I drag around so I am convinced.
Researcher: How can you prove that the conjecture is true?
Mark: I don?t know exactly how on paper but this proves it because for a
parallelogram the opposite angles are always congruent.
Note the combination of deductive reasoning and inductive reasoning used by this
student. Deductive reasoning is used by claiming since the opposite angles are
congruent, the quadrilateral is a parallelogram. This is an accurate deduction. Inductive
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reasoning is used by observing that during the dragging process, the opposite angles
remained congruent. Thus, this student drew no distinction between inductive and
deductive reasoning and combined them into what he considered to be a ?proof?.
Another example of this kind of reasoning was provided by Tasha. Tasha is a
relatively low achiever when compared to the rest of the sample. She was somewhat
reluctant to use the dragging utility during the interview. However, she demonstrated the
same kind of inductive and deductive combination in her reasoning.
Researcher: What you see on this first figure is called a midsegment
quadrilateral; it is formed by connecting the midpoints of any general
quadrilateral. A conjecture from a former student says that the midsegment
quadrilateral will always be a parallelogram. Do you think this conjecture is true?
Tasha: Yes
Researcher: How would you test your conclusion?
Tasha: To see if the opposite sides equal up.
Researcher: Would you like to see the measures of those sides?
Tasha: Yes, (Pause while researcher measures and displays the measures of the
sides of the midsegment quadrilateral)
Researcher: OK, what do you notice?
Tasha: That the opposite sides are equal.
Researcher: How can you prove that the conjecture is true?
Tasha: I don?t know? Maybe just keep dragging. (The student begins to drag
slightly). These opposite sides stay equal and so this must be a parallelogram.
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Six other interview participants used a similar kind of reasoning by stating the
observation that the opposite angles or sides remained congruent while using the
dragging utility. They then claimed that these observations proved that a midsegment
quadrilateral is a parallelogram. They used the inductive process of observation through
the dragging process to conclude that these properties hold for all cases and followed that
by a true deduction that if these properties are true for all cases then the midsegment
quadrilateral is indeed a parallelogram. This combination of inductive and deductive
reasoning did not pose a conflict for the students and gave rise to a discussion of what
students understand to be proof.
To summarize the results of this theme, approximately 80% of the subjects
demonstrated a reasonable concept of conjecture when compared with the working
definition; however, those who did not respond with a reasonable definition of conjecture
averaged more relevant conjectures than the sample as a whole. This result indicates that
the ?skill? of finding relevant conjectures is independent of the subjects? ability to
articulate a reasonable concept of conjecture. Students tend to use a combination of
deductive and inductive reasoning when asked to prove conjectures if the dynamic
environment is available. Although using inductive observation by the dragging process
is not considered a valid proof, the subjects see it as proof and shy away from a purely
deductive path to proof when the dynamic environment is available. The results indicate
that although most of the participants were able to articulate a reasonable concept of
conjecture, they lacked an understanding how this relates to mathematical proof.
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Student Preferences
Furinghetti, Olivero, and Paulo, (2001) suggest presenting students with dynamic
explorations to foster conjecturing. Arcavi and Hadas (1996) list several positive
implications of the use of dynamic geometry software including the enhancing of
understanding and insight. Haddas and Hershkowitz (1999) claim that discovery and
conviction are greatly enhanced by dynamic geometry software. However, do students
feel that dynamic geometry software is aiding them in the forming of conjectures? What
features of the software do the students claim aids them the most in the forming of
conjectures?
The catagories for the theme of student preferences emerged from two primary
data sources: the student survey and the researcher?s journal. On the student survey
subjects were asked the questions: ?What are the differences between the static and
dynamic environments?? and ?Which environment, static or dynamic, is better for
conjecturing? Why?? As in the preceding theme a number of key words and phrases
were identified as codes to sort and analyze the responses to these questions.
All of the participants correctly identified the differences in the two environments
as the ability to drag, cause movement, or change. Consider the following excerpts: ?In
the dynamic environment you can move the figure and see changes while in the static
environment you can?t move it.? ?The static environment has shapes that can?t be moved
but in the dynamic environment you can drag segments and points to see the
measurements change.? ?In the static environment you just have to look at it and figure it
out, but in the dynamic environment you can do hands on movements on the figure.?
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All of the participants stated that the dynamic environment was better for
conjecturing. Coding the responses yielded the following categories used to classify the
reasons why the students felt the dynamic environment was better for conjecturing.
1. Interactive Qualities
2. Facilitate Conjecture
3. Improve Conviction
The category of Interactive Qualities referred to responses that indicated that the
dynamic environment was superior because of the ability to interact with the figure by
dragging and to observe change within the figures and their measures. ?The dynamic
environment is better because when you drag you can see what changes and what does
not.? ?The dynamic is better because you can play with it and see how the shape
changes.? Of the forty-one participants surveyed, eighteen of the responses fell in this
coded category making it the most common of the three.
The category of Facilitate Conjecture referred to responses in which the subjects felt
the dynamic environment was better because it was easier to find relevant conjectures in
this environment. In many of these responses, the subject expressed the concept of
having to do less to get more. Consider the following responses: ?The dynamic is easier
because you don?t have to think so hard, and you can observe the patterns formed when
dragging instead of figuring out just one shape.? ?It?s a lot easier when you can move to
make new shapes instead of studying one still shape.? ?You can figure out more when
you drag to get new figures, and you don?t have to rack your brain on one figure.?
Twelve of the responses fit into this category and interestingly, ten of those twelve were
male subjects.
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Ten of the 41 subjects indicated that the dynamic environment was better because
it improved their confidence in their conjectures. This improvement in conviction is the
last coded category used for this analysis. ?Getting hands on experience increases your
confidence level in the conjectures.? ?In the dynamic environment you can make sure
that your conjectures are right.? ?In the dynamic environment you have the ability to
move shapes around to prove your conjectures.? Notice this participant considered the
testing of a conjecture by dragging equivalent to proving conjectures. Gender also had an
effect but this time nine of the ten subjects that were classified under this category were
female students.
To summarize, about half of the participants considered the dynamic environment
superior to the static environment simply because of the ability to interact with the figures
by the dragging process. Many male students tend to favor the dynamic environment
because it facilitates the conjecturing process whereas many females favor the dynamic
because it increases their conviction in their conjectures.
Student preferences toward the dynamic environment were also apparent during
the course of the study and were noted by the researcher in field notes written in the
researcher?s journal. The subjects typically spent more time on the figures while in the
dynamic environment. The static environment seemed to bore many of the students and
some of the figures were left blank with no conjectures even attempted. During the
discussions after the lab activities, the class who used the dynamic environment usually
had a more lively discussion and all of the targeted conjectures were found by at least one
of the participants.
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Forming and Finding Counterexamples
Arcavi and Hadas (1996) claim that dynamic geometry aids in the finding of
counterexamples while de Villiers (1996) considers the finding of counterexamples an
important component of the overall proof process. NCTM (2000) also claims that
students should learn to understand the need for counterexamples to false conjectures.
Does the use of dynamic geometry software aid in the finding of counterexamples?
In this theme students? ability to use dynamic geometry software to find
counterexamples in order to disprove false conjectures was examined. The second figure
shown to the students during the interviews was that of a general quadrilateral with angle
measures shown. This figure was accompanied with a false conjecture: ?A quadrilateral
will have at most two obtuse angles.? Students were asked if they thought the conjecture
was true or false. The results provide insight on the subjects? ability to form a negation,
find a counterexample, and use the dragging utility to find that counterexample.
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Theorem: The interior angles of a quadrilateral will always add up to 360?
m?AB C+ m ?BCD+m?CDA+m?DAB = 360.00?
m?CDA = 61.51?
m?DAB = 115.32?
m?BCD = 85.29?
m?AB C = 97 .89 ?
Figure 2: Genera l qua drilateal
Conjecture: A quadrilateral will have at most two obtuse angles
D
A
B
C
Figure 9. A general quadrilateral with a false conjecture and true theorem
Of the ten individual interviews only four of the participants were able to form the
negation and find a counterexample to disprove the false conjecture even though logical
statements and their negations had been covered in the curriculum of the course. The six
participants who claimed that the conjecture was true either used flawed logic, were not
able to form a proper negation to the conjecture, or demonstrated a weak dragging
technique which convinced them that a counterexample did not exist. The following
excerpts from three different interviews will demonstrate these three phenomena.
Chris is an average achiever who demonstrated flawed reasoning which led to the
incorrect conclusion that the conjecture was true. Consider the following excerpt.
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Researcher: On this next figure we have a general quadrilateral with all four
interior angles measured. A former student conjectured that a quadrilateral will
have at most two obtuse angles. Do you agree with this conjecture?
Chris: (Pause) Yea, I agree.
Researcher: You don?t want to drag at all?
Chris: No.
Researcher: OK, How would you prove or disprove this conjecture.
Chris: Well the most it could have would be two because if it had three it would
be over 360. I don?t think it is possible to have more than two because each
obtuse angle would have to have an acute angle to cancel it out to 180 degrees, so
two obtuse and two acute angles will give you 360.
Note that this student did not even attempt to form or find a counterexample. He
was convinced in the truth of this conjecture by using the same logic that leads to the
conclusion that a triangle must have at most one obtuse angle. However, this student
failed to realize that the quadrilateral has one more degree of freedom thus allowing for
three obtuse angles with room to spare for a forth acute angle. Two other participants
used the same kind of flawed logic when asked how they would prove the conjecture.
However, they at least attempted to form or find a counterexample before concluding the
conjecture was true.
Two of the participants had difficulty forming a negation of the conjecture and
thus were not able to find a proper counterexample because they did not know what to
look for. Beth is an average achiever who had difficulty understanding the concept of
counterexample. Consider the following excerpt from the interview.
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Researcher: On this next figure we have a general quadrilateral with all four
interior angles measured. A former student conjectured that a quadrilateral will
have at most two obtuse angles. Do you agree with this conjecture?
Beth: I don?t know. There is two here but you could have right angles that would
make it false.
Researcher: Would that make it false?
Beth: Well they would not be obtuse.
Researcher: How could you prove your conclusion?
Beth: I will drag some more?
Researcher: So what are you looking for?
Beth: Angles that are greater than 90. (Student stops dragging when the
quadrilateral has only one obtuse angle and two right angles.)
Researcher: So what about the conjecture?
Beth: Is it false then?
This student reached the proper conclusion but did not actually form a proper
negation and thus did not find a valid counterexample. One other student failed to
understand the concept of counterexample and simply gave up on the conjecture without
ever saying if it was true or false.
Three of the participants had no problem forming the negation of the conjecture
and knew what to look for when they began dragging; however, they failed to actually
find a counterexample. Consider this excerpt with an average-achieving student named
Joe.
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Researcher: On this next figure we have a general quadrilateral with all four
interior angles measured. A former student conjectured that a quadrilateral will
have at most two obtuse angles. Do you agree with this conjecture?
Joe: I agree but I?m not positive.
Researcher: How would you test that conjecture?
Joe: Well if three angles are obtuse then the conjecture is false but I don?t think
that is possible.
Researcher: You don?t think it is possible to have three obtuse angles in a
quadrilateral?
Joe: Can I drag?
Researcher: Sure.
Joe: (Does some dragging). I don?t think you can have three because the sides
won?t connect so I think the conjecture is true.
Joe reached the wrong conclusion because of a weak dragging technique which
was a common observation by the researcher. He simply gave up too soon and did not
make the fine adjustments to the figure to find a counterexample. The next section of this
chapter will discuss student dragging techniques in detail.
The four students who formed and found a counterexample to this conjecture all
used the dragging utility to find the counterexample instead of logically deducing a
counterexample. In other words, no student started off by saying that the conjecture was
false by citing a counterexample. They all had to convince themselves by finding a
counterexample through the use of the dragging utility.
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In the following excerpt Juan disproved the conjecture by exploring using the
dragging utility, even though he intuitively thought the conjecture was true at the onset.
Researcher: On this next figure we have a general quadrilateral with all four
interior angles measured. A former student conjectured that a quadrilateral will
have at most two obtuse angles. Do you agree with this conjecture?
Juan: I think so because two obtuse and two acute angles will give you 360 like
on the figure.
Researcher: How would you test your conclusion?
Juan: By forming different quadrilaterals and measuring their angles.
Researcher: So what are you looking for?
Juan: A shape that does not have two obtuse angles but has three obtuse angles.
(Student begins dragging and stops when the figure shows three obtuse angles.)
Researcher: So what does this show?
Juan: This shows that the conjecture is not true.
Researcher: Does this disprove the conjecture?
Juan: Yes, by showing a quadrilateral with more than two obtuse angles.
Researcher: What if the shape only had one obtuse angle or none?
Juan: That would not disprove it because the conjecture said at most two obtuse
angles.
This student overcame his initial incorrect reasoning that each obtuse angle has to
be paired with an acute angle. He formed a proper negation and used the dragging utility
to find a counterexample, thus proving the conjecture false.
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To summarize this theme, it was found that less than half of the interviewed
participants were able to disprove a false conjecture. Some students who used flawed
logic, were not able to form the proper negation to the false conjecture, or simply had
weak dragging skills and did not find a counterexample even after searching for one. Of
those that did disprove the conjecture, the dragging utility was essential to their arrival at
a counterexample.
Conviction in Computer Output
Mudely and de Villiers (1999) examined students? conviction in the output
generated by dynamic geometry software and found that students were very confident in
the output even though the result was surprising. They used the surprising result as a
motivation to prove. However, consider the case where students are confronted with a
surprising result generated by dynamic geometry software that was in conflict with a
known theorem stated by the teacher or the textbook. De Villiers (1992) showed that
students tend to hold more conviction in the authority of the teacher and the textbook
than they do on their own conjectures. However, what level of conviction will they hold
in dynamic geometry software, relative to other sources of information? If students see
something on the screen that contradicts what they have learned from the instructor or the
text book, how will they rationalize that contradiction?
The second situation presented to the interview participants also stated the true
theorem, ?The interior angles of a quadrilateral will always add up to 360 degrees.? This
theorem is true for all quadrilaterals and had been proven by the instructor and accepted
by the students as part of previous classroom instruction. However, Geometer?s
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Sketchpad gives angle measures only between 0 and 180 degrees. Therefore, if a
quadrilateral is dragged so that it is concave, forcing an interior angle greater than 180
degrees, the software package will default to measuring the angle that measures less than
180 degrees on the exterior of the quadrilateral and produce the false result that the sum
of the interior angles of a concave quadrilateral is less than 360 degrees. Figure 10
illustrates this deceiving result.
Theorem: The interior angles of a quadrilateral will always add up to 360?
m?AB C+ m ?BCD+m?CDA+m?DAB = 268.53?
m?CDA = 61.51?
m?DAB = 47.49?
m?BCD = 25.27?
m?ABC = 13 4.27?
Figure 2: General quadrilateal
Conjecture: A quadrilateral will have at most two obtuse angles
D
A
B
C
Figure 10. A concave quadrilateral with angle measures shown, leaving the impression
that the sum of its interior angles are less than 360 degrees.
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This figure illustrates that the output shows angle ABC as an angle measuring
134.27 degrees rather than the interior angle that would measure 225.73 degrees (360-
134.27). Of the ten interview participants, nine of them believed that the sum of the
interior angles was indeed not equal to 360 degrees when the quadrilateral was concave.
The following excerpts from participant interviews are examples of modifying a given
theorem falsely in order to rationalize the computer?s visual output. The first excerpt is
from the interview of Tamika, a relatively high achiever and the only freshman in the
class.
Researcher: OK, here is a theorem from the text book. The sum of the interior
angles for any quadrilateral is 360 degrees. Do you think this is true for all
quadrilaterals?
Tamika: I?ll drag to see. (Student drags until a concave quadrilateral is formed.)
Researcher: What just happened?
Tamika: It got too far in.
Researcher: What do you mean?
Tamika: It?s like not convex but concave.
Researcher: So should the theorem be changed?
Tamika: Yes
Researcher: How?
Tamika: By using a biconditional.
Researcher: What do you mean?
Tamika: Like saying the interior angles of a quadrilateral add up to be 360 if and
only if the quadrilateral is convex.
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Note that this participant observed under what conditions the output was not
producing a sum of 360 degrees. She then questioned the theorem rather than the output
and made a false modification to the theorem. However, this modification was consistent
with the computer?s visual output. The next excerpt is from Juan.
Researcher: OK, on the same figure scroll down to see that the sum of all the
interior angles is equal to 360. This is actually a theorem from the book. Do you
think that this will always be true? (Student drags and the shape goes concave)
Juan: It?s concave and not 360.
Researcher: So is the theorem wrong?
Juan: It?s not wrong, however it only works on a normal quadrilateral not a
concave one.
Researcher: Should the theorem be changed?
Juan: Yes, I would specify the kind of quadrilateral. The concave ones add up to
be less than 360.
Once again the subject failed to examine the output carefully and instead
questioned the theorem. Note that the modification once again is false relative to the
definitions used in the course but consistent with the generated output. The sum will
always appear to be less than 360 degrees when the quadrilateral is concave. Only one of
the interview participants noticed the inconsistency in the output and was able to figure
out why the output was reading less than a 360 degree sum. Consider this excerpt from
Mark?s interview.
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Researcher: OK, on the same figure scroll down to see that the sum of all the
interior angles is equal to 360. This is actually a theorem from the book. Do you
agree with this theorem?
Mark: Yes.
Researcher: Can we confirm it with the computer? (Student begins dragging until
shape goes concave.)
Mark: Here it goes less than 360.
Researcher: What happened?
Mark: Oh, it?s measuring the outside angle.
Researcher: Do you think the theorem is wrong?
Mark: No, because it is measuring the wrong angle and if you add up the angle
that is greater than 180 it will probably still work.
Mark was the only student that was able to rationalize the inconsistent output and
noticed right away what angle measure was really being displayed.
The results of these interviews show that students tend not to question computer
output even when it contradicts what the class has proven deductively and what the
textbook has stated as fact. They also failed to closely observe the interior angles that
were formed in the concave quadrilateral. One of these angles is indeed greater than 180
degrees. Instead, students may falsely alter the facts in order to rationalize the output
generated by the computer even though they were aware of the fact that angles can
measure more than 180 degrees. In this experiment, close examination of the angle
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measures themselves reveals the inconsistency. However, only one student closely
examined these measurements.
Dragging Tendencies
The theme of dragging tendencies was the also an additional emergent theme
developed during the course of data collection. This theme was drawn from the
researcher?s journal, based on careful observation of students during the eight lab
activities and the ten individual interviews. The researcher identified three different
kinds of dragging. They were designated as Random Dragging, Dragging for Pattern
Recognition, and Dragging for a Specific Purpose. This section will define and discuss
these three kinds of dragging.
The category of Random Dragging was especially prevalent during the first two
lab activities. This category is characterized by a considerable amount of random
dragging where students moved the figures around without really looking at how the
measures changed. This dragging technique is characterized by sporadic and
unpredictable motions by the user. At first glance it may seem like this kind of dragging
served no useful purpose and was employed simply to amuse the user; however, the
students made comments that suggested that they were using the dragging utility to test
the limits of the figure and how different points and segments behaved during the
dragging process. Students would often ask why different points moved in a different
manner. The researcher always replied that these different motions were due to the
construction of the figures. Therefore, random dragging did serve the purpose of testing
the robustness of the figure during the dragging process.
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The category of Random Dragging was frequently followed by Dragging for
Pattern Recognition within a lab session. Students would drag in less sporadic motions
with pauses to observe the figure and the measures of its components. During this kind
of dragging students were looking for some kind of pattern. For example, a student may
deform the parallelogram to a new parallelogram and observe that the opposite sides are
congruent. Doing this several times and stopping to observe that the opposite sides are
always congruent would lead the student to conjecture that the opposite sides of a
parallelogram are always congruent. This dragging technique is used to find conjectures
and is characterized by smaller deformations with pauses for observation.
The third category of dragging was Dragging for a Specific Purpose. For
example, several of the subjects were observed dragging the parallelogram into a
rectangle or dragging the transversal of the parallel lines so that all angles would be right
angles. This kind of dragging sometimes was evidenced in the conjectures themselves.
?If two parallel lines are intersected by a transversal then there will be four acute angles
and four obtuse angles unless all of the angles are right angles.? This conjecture shows
evidence that the user attempted to see what would happen when the angles were right.
Without this last qualifying statement about right angle the conjecture would have been
false. This third level of dragging was also observed during the interviews where
students were searching for a counterexample. Those who found the counterexample
displayed this technique. The technique is characterized by careful meticulous
adjustments to the figure by slight drags until the figure displays what the user intends.
In some sense, these categories might be considered to be levels of dragging,
representing increasing levels of sophistication in their use and understanding of
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dragging. Most of the participants in this study achieved and operated on the second
level of dragging, Dragging for Pattern Recognition. Many of the participants did not use
dragging characterized by Dragging for Specific Purpose; only four of the ten interview
participants were able to find a counterexample for a given false conjecture. It must be
noted that the participants of this study did not receive any particular instruction on
dragging techniques and developed their own personal style of dragging independently.
Dynamic Language
The theme of dynamic language was developed during the course of data
collection as an ?emergent? theme. However, this theme also suggested in studies by
Jones (2000) and Mariotti (2001), both of which observed the formation of dynamic
language among students using dynamic geometry software, although these studies were
not identified until after the fact.
The categories in this theme address how students phrase their conjectures in the
two different environments. It was noted during the study that many of the subjects used
language that reflected the dynamic environment itself. By examining the conjectures
produced for all of the eight lab activities it was determined that the conjectures made in
the dynamic environment tended to be more general in their phrasing while the
conjectures made in the static environment tended to be more specific to the figure that
was presented. The more general conjectures tended to have the word ?always?
embedded in the phrasing of the conjecture: ?If a triangle is isosceles then there will
always be two congruent angles.? ?If a parallelogram is a rectangle then the diagonals
are always the same length.? ?If two parallel lines are intersected by a transversal then
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corresponding angles will always be congruent.? The word ?always? used in this context
was much more frequent in the responses made in the dynamic environment.
In the static environment it was common for the subjects to give responses that
referred directly to the figure often by using the labeled parts of the figure. For example:
?If a quadrilateral is a parallelogram then angles DAB and DCB are congruent, and so are
angles ADC and ABC.? This is a relevant conjecture indicating that the opposite angles
of a parallelogram are congruent but using a particular figure. These kinds of responses
were much more common in the static environment.
After examining the responses from all lab activities, two categories were
identified that reflected the dynamic environment itself: Move or Drag, and Change or
Stay the Same. These codes used to develop these categories were only found embedded
in the responses from the dynamic environment and reflect the environment itself.
The ?Move or Drag? category referred to conjectures which incorporate the
student?s action during the formation of the conjecture. Consider the following
responses: ?If a triangle is equilateral then all the angles will be congruent anyway you
move it.? ?If a triangle is a midsegment triangle then it will divide the area by four and
the perimeter by two no matter how you move the points.? ?If a parallelogram is a
rectangle then the diagonals will be congruent as you drag different points.? Note that
these conjectures make more sense if you consider the dynamic environment in which
they arose.
The ?Change or Stay the Same? category refers to the figure itself transforming
into a different shape. Embedded within the conjectures is what the subjects see
changing or not changing within the figure during deformations. Consider the following
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responses: ?If a parallelogram is a rhombus then the diagonals will change size but they
will stay perpendicular.? ?If a figure is a triangle then the angles may change but they
will always add up to 180.? ?If an angle is an inscribed angle then when you change the
measure of the angle the arc will stay twice as big.? Once again these conjectures refer to
the dynamic environment in which they were discovered. The words ?change? or ?stay
the same? never occurred in the responses found in the dynamic environment.
Several conjectures were classified by both categories. These conjectures referred
to the subject?s action and the subject?s observation. Consider the following responses:
?If a parallelogram is a square then when you drag the square to change its size the
diagonals stay congruent and all the angles stay the same.? ?If a trapezoid is isosceles
then when you move the points the angles change but the top angles stay congruent and
the bottom angles stay congruent.?
The presence of dynamic language was not consistent throughout the lab
activities. Lab 1 (parallel lines and transversals) and Lab 4 (parallelograms) were found
to have very few conjectures that contained dynamic language while Lab 5 (rectangles
and squares) and Lab 3 (midsegments) had numerous examples of dynamic language. In
the examples sited, all of the conjectures were relevant; however, dynamic language also
occurred in irrelevant conjectures. For example consider the following irrelevant
conjectures: ?If a segment is a trapezoid midsegment then when you drag the segment
the measurement changes.? ?If a quadrilateral is inscribed in a circle then when you
change its shape and size the angles still add up to be 360 degrees always.?
The analysis of this theme shows that the environment not only has an effect on
the number of relevant conjectures, false conjectures, and the conviction in those
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conjectures, but also has an effect on the phrasing of the conjectures themselves.
Conjectures formed in the dynamic environment tend to be more general often using the
word ?always?. These conjectures often use language that reflects the environment itself
referring to the action taken by the participant, the changes observed, or both. The
conjectures formed in the static environment tend to be more specific to the figure used
and contain no language of action or change.
Summary
The qualitative results of this study provide a description of the subjects
experience over the six different themes explored. It was observed that most of the
participants had a reasonably accurate concept of conjecture but that the ability to
articulate a definition of conjecture was independent of the ability to form relevant
conjectures about geometric figures. The interview subjects displayed a concept of proof
that included a mixture of both inductive and deductive reasoning.
All of the participants stated that the dynamic environment was better for finding
conjectures and an interesting gender effect was noted where many of the males preferred
this environment because it facilitated the conjecture process; many of the females
preferred the dynamic environment because it strengthened their conviction in their
conjectures.
The results from the quantitative portion of this study indicated that the dynamic
environment had a statistically significant effect on the students? ability to form relevant
conjectures and false conjectures as well as a statistically significant effect on the
students? conviction in their conjectures. A close qualitative examination of these
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conjectures indicates that the environment has an effect on the way in which students
phrase conjectures. Conjectures formed in the static environment tend to be specific to
the figure given while those formed in the dynamic environment are more general often
using language of action and change.
Most of the interviewed subjects in this study had difficulty finding a
counterexample to a false conjecture even when using dynamic software. Some used
flawed logic to deduce that the conjecture was true. Others had difficulty articulating the
negation of the conjecture, while some simply did not find a counterexample because of
weak dragging technique. Dragging techniques themselves were classified into three
distinct categories, each with its own characteristics and purposes.
The final theme examined students? conviction in the output generated by the
computer software package. The results indicate that the students tend to question the
authority of the instructor and the text book rather than question the output of the
computer. A discussion of these themes in terms of meaning, implications, and
recommendations for further research will be presented in chapter six of this report.
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CHAPTER VI: DISCUSSION
This study focused on both quantitative and qualitative data concerning student
conjectures using two different geometric environments: The static geometry
environment in which a specific static figure was presented and the dynamic geometry
environment which allowed students to deform the figure. The quantitative results of this
study focused on statistical differences between predetermined and measurable variables
in each of the geometric environments. These results were used to answer the specific
research questions concerning differences in the participants? conjectures formed in the
two environments. The qualitative results explored data categorized by six themes that
delved deeper into the students? experiences, tendencies, and concepts developed while
participating in this study. The only notable difference between the dynamic and static
environments used in this study was the ability for the participants to employ the
dragging utility in the dynamic environment. Because the ability to drag is a
distinguishing factor of the dynamic environment, considerable attention was given to the
act of dragging during this study.
This final chapter will discuss the implications of this study by reflecting upon the
results from the two previous chapters. This chapter will also discuss the limitations of
the research and suggestions for further research.
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Implications
This study has yielded several results with possible implications and benefits for
students as well as professionals in mathematics education. A discussion of the four
research questions and six qualitative themes will be put into the context of potential
benefits for students and implications for professionals in the field of mathematics
education, including teachers of mathematics and educators in mathematics education.
Potential benefits for students
The quantitative results of this study support the use of dynamic geometry
software as a classroom tool for student conjecturing in geometry. These results indicate
that the use of dynamic geometry software aids in students making a significantly greater
number of relevant conjectures about a given figure when compared to the use of a static
representation of the same figure. The results also showed that the use of dynamic
geometry software significantly decreases the students? likelihood of making false
conjectures about geometric figures.
In this study, the researcher had already constructed the figures posed in both
environments. The same measurements were also provided in each environment.
Therefore, the ability to use the dragging utility in the dynamic environment was the only
notable difference between the dynamic and static environments. This ability proved to
be an important factor in making correct and relevant conjectures. The participants were
not ?coached? on dragging techniques and were left on their own to develop their own
dragging techniques. Therefore, it is reasonable to assume, based on the results of this
study, that simply given the ability to use the dragging utility in a dynamic environment,
a students? ability to make relevant conjectures concerning geometric figures should
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increase, and the likelihood of the student making false conjectures about those same
figures should decrease when compared to a traditional static setting.
Arcavi and Hadas (1996) explained, ?Making sense of a situation while playing
with the situation itself first, and then interpreting its representations, enhances both the
understanding of the situation and of the representations? (p. 8). Students are able to
form conjectures in geometry by recognizing and generalizing patterns during
exploration. Pattern recognition is stimulated by the use of dynamic software which
allows them to manipulate diagrams easily and focus on the patterns and relationships
displayed on the screen (Glass, Deckert, Edwards, & Graham, 2001). It is not surprising
that students were able to find significantly more relevant conjectures and stated
significantly fewer false conjectures when given the ability to manipulate the figures in
the dynamic environment.
Another benefit for the users of dynamic geometry software is the increase in the
conviction of their conjectures. Results showed that when subjects were asked to rate
their own conviction in the conjectures that they made in both the static and dynamic
environments, those ratings were significantly higher in the dynamic environment. This
result implies that students are more confident in their conjectures when they are able to
use the dragging utility. Students have a continuum of figures to test their conjectures
resulting in greater confidence in those conjectures. This view is shared by Mudaly and
de Villiers (1999) who claimed that higher levels of conviction are attained when
dynamic geometry software is used. They claim that learners would not likely achieve
comparable levels of conviction using only pencil and paper methods.
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In terms of the research questions posed by this study, the results of this study do
support the hypotheses that students will make significantly more relevant conjectures
and significantly fewer false conjectures in the dynamic geometry environment when
compared to the static geometry environment. The results also support the hypothesis
that the conviction of conjectures made in the dynamic geometry environment would be
significantly greater than the conviction of those conjectures made in the static
environment.
Further analysis indicated that students in the dynamic environment were
significantly more confident in their relevant conjectures but not in their false
conjectures. Thus, the reported increase in conviction occurred primarily with true
conjectures. This is, of course, a desirable outcome. Although significantly fewer false
conjectures are formed in the dynamic environment, we do not want the students to have
a higher conviction in those false conjectures that were formed.
The forth exploratory question asked how the variables of gender, mathematics
achievement, and entering van Hiele level of geometric reasoning are related to the
students? ability to conjecture. It was found by regression analysis that mathematics
achievement is a significant predictor of a student?s ability to form relevant conjectures.
This is not a surprising result; however, it was also found that in one of the assignment
cases, the higher achieving students tended to make more false conjectures. This result
was unexpected but can be somewhat explained by the fact that the higher achievers
tended to make more conjectures in total. Therefore, their likelihood of making false
conjectures may increase because of the higher number of total conjectures made.
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It was shown that mathematics achievement was not a significant contributor to
the students? conviction; however, there was a strong positive correlation between the
number of relevant conjectures and the subjects? conviction in their conjectures. This
correlation suggests that subjects tend to be more confident with conjectures that are
relevant, which is also is not a surprising result.
Even though the gender variable was significantly correlated with conviction in
the zero-order correlations, this correlation did not hold up in the regression analysis and
thus it was concluded that gender did not significantly affect any of the dependent
variables. The participants? van Hiele level of geometric reasoning also did not
significantly affect any of the dependent variables. Therefore, the environment itself was
shown to be the greatest predictor of a students? ability to form relevant conjectures,
make fewer false conjectures, and increase the conviction of their conjectures.
This finding is important because it implies that the use of dynamic geometry
software contributes to the students? ability to make good conjectures more than the
students? level of mathematics achievement does. This implies that lower achieving
students benefit by the use of dynamic geometry software. The use of dynamic geometry
software may give them the opportunity to achieve results similar to higher achieving
students.
The participants of this study were predominately African American, many from
low socio-economic backgrounds, and most were relatively low achievers in mathematics
with weak backgrounds in algebra and basic geometry skills. This finding suggests that
the use of dynamic geometry software may make an important contribution to achieving
equity among students of different backgrounds and achievement levels. The NCTM?s
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?Equity Principle? states that ?technology can assist in achieving equity and must be
accessible to all students? (NCTM, 2000, p. 12). The results of this study support that
claim.
Implications for Professionals
The results of the six qualitative themes developed for this study yielded some
interesting points for professionals in mathematics education and teachers of
mathematics. Those themes included: Student concepts of conjecture and proof, student
preferences concerning the geometry environment, language that reflects the dynamic
environment, students? ability to form and find counterexamples in the dynamic
environment, student dragging tendencies, and students? conviction in computer
generated output.
Concept of Conjecture. Results from the survey and the lab instruments indicated
that the ability to form relevant conjectures is independent of the subject?s ability to
articulate a concept or definition of conjecture. It was an unexpected result that those
students who could not articulate a sensible concept of what a conjecture ?is? actually
averaged a greater number of relevant conjectures across the eight lab activities when
compared to the entire sample mean of relevant conjectures. This result supports the
notion that students may possess a mental concept of a mathematical process without
knowing or understanding a definition of that process.
Tall and Vinner (1981) claim that a student?s concept definition of a mathematical
process and their concept image of that process are not the same. The concept image
consists of all of the cognitive structures in the individual?s mind related to that process,
many of which cannot be expressed verbally. The concept definition is the individual?s
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way of defining that concept verbally. The concept definition may include both formal
and informal mathematical language. The concept definition then is merely a component
of the total concept image.
When students were asked to explain what a conjecture was in their own words,
they were expressing their concept definition. Some of those students who lacked the
ability to articulate a definition of conjecture may still have had a strong concept image
of conjecturing since they were able to form relevant conjectures unassisted.
Concept of proof. To address the concept of proof in terms of proving
conjectures, selected students were interviewed and presented a true conjecture in the
dynamic environment. These students were not told that the conjecture was true and
were asked to indicate the truth of the conjecture. All of the participants indicated that
the midsegment quadrilateral was indeed a parallelogram, as conjectured. When asked to
tell how they would ?prove? that conjecture, the participants overwhelmingly responded
that they would use the dynamic dragging tool as part of their ?proof?. Participants
indicated that they would use the dragging utility to show that either the opposite sides or
the opposite angles remained congruent while distorting the figure by dragging. They
then concluded that since these parts remained congruent, the quadrilateral must be a
parallelogram. This mix of inductive observation and deductive reasoning was common
for the interviewed participants of this study.
Martin and Harel (1989b) indicate that in everyday life, people consider ?proof?
to be ?what convinces me.? Indeed, the combination of inductive and deductive
reasoning used by the participants may form a convincing argument. However, it is not
mathematical proof. The question is, ?Does the use of dynamic geometry software
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enhance or substitute the need for deductive proof?? Pandiscio (2002) raised this same
question in his study of preservice teachers? conception of proof and the use of dynamic
geometry. In his study, the sample of preservice teachers questioned the need for formal
proof when dynamic geometry software was available. They were also concerned that
the students would not see the need for proofs after being exposed to dynamic software.
The results of this study support the idea that the use of dynamic geometry may substitute
for deductive proof in the minds of the student. However, if the purpose for proof is to
merely confirm the truth or validity of an argument, then perhaps deductive proof can be
replaced with dynamic geometry software. de Villiers (1999) agreed that when dynamic
geometry software is available for students, they have little need for further conviction or
verification. However, de Villiers (1999) also argued that the use of dynamic geometry
software can serve to enhance rather than substitute deductive proof.
de Villiers (1999) claimed that students may be motivated toward deductive proof
by being asked the question, ?Why?? The use of dynamic geometry software can
certainly convince students that a particular statement is true or false but it does not
necessarily shed any illumination on why the statement is true or false. de Villiers (1999)
contended that beside verification, deductive proof serves the purposes of explanation,
discovery, systematization, and intellectual challenge. Students may find that deductive
proof does serve to explain why statements are true and also serves to systemize
mathematical theorems by linking them together deductively.
The results of this qualitative theme concerning students? concepts of conjecture
and proof indicated that many students do not have a strong concept of what
mathematical proof really is and how inductive and deductive reasoning differ. They did
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not see a distinction between inductive and deductive reasoning and mixed the two
together to form what they see as proof. Students should be aware of what proof is and
how it differs from conjecture. Classroom teachers should emphasize this difference and
point out that the verifications and discoveries found using dynamic geometry software
are not proofs but rather conjectures not yet proven. The results of this study support the
claim that dynamic geometry software may indeed be seen as a substitute for proof by the
students. This implies that de Villiers? (1999) suggestions concerning further explanation
and challenge be taken into account for teachers using dynamic geometry software as a
classroom tool.
Hadas, Hershkowitz, and Schwarz (2000) suggested using dynamic geometry
activities that lead to surprise or uncertainty and motivate students to question further and
motivate them to explore deductive proof as a means of explanation. The study
conducted by Marrades and Gutierrez (2000), where students were asked for
justifications of a variety of conjectures may also be of help in understanding how
inductive explorations with dynamic software may lead towards deduction. Recall that
they found that even the best students would choose to justify using empirical means
rather than deductive means, but that the use of dynamic geometry software helped
prepare students for the transition to deductive proof. Marrades and Gutierrez (2000)
claimed that this transition is slow and in fact the progress from van Hiele level two to
van Hiele level four may take several years.
Student preferences. The theme that explored the student preferences in terms of
the environment showed that all of the students preferred the dynamic environment
mainly because of the ability to interact with the figure. Students would spend more time
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examining the figures and experimenting with the figures by using the dragging utility.
The dynamic environment tended to be much more engaging to the students and much
more enjoyable. Discussions after the lab activities were richer when dynamic geometry
software was used when compared with the static environment.
The fact that the students prefer to work with dynamic geometry software and
spend more time engaged in conjecturing activities when using dynamic geometry
software implies that the teacher?s role may indeed be changed from instructor to
facilitator while dynamic geometry software is being used. Students may enjoy the
freedom given by the use of dynamic geometry software without being told what to do by
a teacher. They may work harder and show greater interest in the subject matter
(Hannafin, 2001).
There was somewhat of a gender effect in terms of the reasons why the students
preferred the dynamic environment. Results show that many of the male students
preferred the dynamic environment because it facilitated the conjecturing process. They
did not have to ?think as much? or ?figure it out.? The dynamic environment provides a
continuum of like figures that allow students to find patterns by observation while
dragging. Many of the female students preferred the dynamic environment because it
improved their conviction in the conjectures they made. It is reasonable to assume from
the results of this study that the use of dynamic geometry facilitates the conjecturing
process and results in greater conviction in conjectures. It is interesting that these two
reasons for the preference of the dynamic environment fell into gender categories where
males leaned toward the facilitation aspect and females leaned toward the conviction
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aspect. This may suggest that there is a qualitative gender effect concerning students?
attitude toward the use of dynamic geometry software.
Dynamic language. In the theme concerning the language used in forming
conjectures in the two different environments, it was found that many of the conjectures
formed in the dynamic environment reflected the environment itself. Often the action of
dragging or change was written into the conjectures formed from the dynamic
environment. The conjectures in the dynamic environment were much more general as
stated theorems are. This raises an important point raised by Martin and Harel (1989a).
In their study of the role of figure in students? concepts of geometric proof, it was found
that students often did not accept a proof for a general figure and would think that a proof
had to be redone for each specific figure.
Dynamic software may aid with the understanding of the general figure rather
than the specific. Dynamic geometry software transforms specific figures into general
figures by the dragging process. Conjectures that are general may lead toward the
acceptance of theorems and toward proof. Conjectures that were worded in terms of the
specific figure shown in the static environment lead a reader of that conjecture to believe
that the conjecture only applies to that ?particular? figure and not necessarily to a general
conclusion about figures of that ?type?.
Many of the conjectures formed in the dynamic environment reflected the
environment itself. This dynamic language was also found by Jones (2001) who
observed a group of students evolve their language around the use of dynamic geometry
software. In his observations terms like ?moving? and ?dragging? became more frequent
as students progressed through a unit that used dynamic geometry software as the key
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component of instruction. He found that by the end of the unit, explanations related
entirely to the context of dynamic geometry software.
The fact that student?s use of language that is more attuned to the general rather
the specific is an important finding and implies that the use of dynamic geometry
software may indeed help students toward a concept of the ?generic? or general figure
and help them along the road to deductive proof of general theorems. Teachers who
incorporate dynamic geometry software in their instruction should foster the dynamic
language used by the students into the classroom semantics.
Finding Counterexamples. In the theme that examined the use of dynamic
geometry software to find counterexamples and disprove false conjectures, the results
showed that many of the students struggled with the concept of the negation and forming
a proper counterexample. Other students had difficulty finding the counterexample
because of poor dragging technique. They were able to form the proper negation to the
false conjecture but because of their weakness in using the dragging utility, they did not
actually find a counterexample and concluded that the false conjecture was indeed true.
The results of this theme imply that greater attention should be given to the logic
of negation and disproving conjectures. Many of the interviewed students could not form
the proper negation that would disprove the false conjecture and claimed that the false
conjecture was indeed true because it appeared true on the figure provided and remained
true during their dragging process. It must be noted that the students in this study were
not ?coached? in terms of dragging techniques and developed their dragging styles
independently. Some of the students were able to form a proper counterexample to the
false conjecture but because of inadequate dragging techniques they claimed that the
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counterexample did not exist. This result implies that proper modeling or coaching in the
uses of dynamic geometry might benefit students? abilities to get a maximum benefit
from the tool. Class discussions on the use of the software, with student input, may be
worthwhile for all of the students in the class.
Reliance on dynamic software may have substituted for the logical thought
process in which the students could have arrived with a counterexample without the use
of dynamic software or even without a figure being displayed. The conjecture stated that
?A quadrilateral may not have more than two obtuse angles.? With some thought,
students can produce specific counterexamples even without a figure. For example,
consider a quadrilateral with three angles measuring 100 degrees each and one sixty
degree angle. This quadrilateral has three obtuse angles thus negating the conjecture.
This quadrilateral does exist because the angle sum of 360 degrees is satisfied. Teachers
should recognize that the use of dynamic geometry software is a powerful and useful tool
but it does not replace logical thought.
Dragging categories. During this study, the researcher noted the way in which
students used the dragging utility. During observation throughout the eight lab activities
and the ten interviews three distinct types of dragging were noted, each with their own
characteristics and purposes. The first is referred to as ?random dragging?. Random
dragging was sporadic and fluid and used to test the robustness of the figure and how
different components of the figure behaved during the drag. ?Dragging for pattern
recognition? was choppy with pauses to observe the measurements. This kind of
dragging may indeed be necessary to arrive with conjectures based on the changes shown
on the figure and its measures, pausing to observe and contemplate these changes.
115
?Dragging for a specific purpose? was characterized by small and careful motion used to
produce a specific outcome. This kind of dragging may be necessary to find
counterexamples to false conjectures and to test for specific properties of figures.
This classification of dragging tendencies should be useful to researchers
observing students using dynamic geometry software and to teachers who are coaching
their students in the use of the dragging utility. It must be noted that these different kinds
of dragging tendencies were observed while the participants were finding and confirming
conjectures. Other dragging techniques may exist when dynamic geometry software is
used for other purposes such as construction.
Conviction in the computer output. The final qualitative theme explored the
students? conviction in the computers? output using dynamic geometry software. All but
one of the interviewed participants believed the output of the computer without question
when confronted with output that violated a known theorem. The dynamic geometry
software package used in this study was Geometers Sketchpad. By default, Geometers
Sketchpad measures angles between 0 to 180 degrees and does not consider angles
greater than 180 degrees. When confronted with a concave quadrilateral in which an
interior angle was greater than 180 degrees, nine out of ten students stated that the sum of
the interior angles was less than 360 degrees. In reality, the software program was
measuring an exterior angle by default and thus indicating a misleading result concerning
the sum of the interior angles of the concave quadrilateral. It is interesting that the
students made modifications to a known theorem to accommodate this anomaly.
de Villiers (1992) found that the majority of students based their conviction on
authoritarian grounds rather than personal conviction when confronted with conjectures.
116
It is interesting to observe that in this study, the participants held conviction in the
observed computer output over the authoritarian sources of teacher and textbook. It is
important that instructors who use dynamic geometry software become acquainted with
any anomalies created by the software and help their students see that they should not
always accept the output generated by the software at face value. Students should
question the computer output and learn to examine why these kinds of inconsistencies
occur. Teachers may also turn these inconsistencies into learning opportunities for their
students, asking the students to resolve these inconsistencies through discussion, close
examination of the figures, and use of mathematical principles.
Limitations
Although this study yielded a variety of significant results, it was far from a
purely experimental study conducted in a laboratory setting. Because of the ?real world?
setting of this study, there are some limitations that should be mentioned concerning the
results.
One of the prime limitations to the results of this study was the fact that almost all
of the participants entered the study with a van Hiele level of geometric reasoning at level
one. Because of the low variability of the entering van Hiele levels of geometric
reasoning it is difficult to state that this variable has no impact in the inductive process of
conjecture even though none was found by the results of this study.
The samples used in this study were not random and were selected from the
participating schools? registration process. A more controlled selection of participants
using a random sample may produce different results. Likewise the setting of the
117
classroom and computer lab was certainly less than ideal with both classes being
overcrowded and sometimes difficult to accommodate.
The fact that the researcher was also the full time instructor of the classes in the
study may limit the objective nature of study. In a sample of students not taught by the
researcher, the researcher would not have been influenced by the daily classroom
exposure to the participants during the entire semester. The researcher would not have
been the full time instructor influencing the content knowledge of the participants and
their conceptual understanding of geometry. Therefore, results may have been different
if the researcher conducted this study on students taught by another instructor.
Suggestions for Further Research
This section offers some suggestions for further research concerning topics related
to this study and topics that may stem from the results of this study. While examining the
variables across the different lab activities, it was noted that there was no trend over time.
Conjecturing abilities did not increase or decrease. Instead, some kinds of figures
favored the dynamic environment while other figures showed no significant differences
in the measured variables. Some of the lab activities produced good results in both
environments while others did not. It would be interesting to examine what kinds of
situations are most enhanced by the dynamic environment and what kinds of situations
facilitate conjecture in general.
In this study, the dragging utility was highlighted, however, would the use of
construction in the dynamic environment improve the students? ability to conjecture?
The figures in this study were already constructed by the researcher who designed the
118
instruments. It stands to reason that a greater appreciation of the tool and a better
conceptual understanding of the figures could be attained by allowing the students to
construct the figures by themselves with guidance by the instructor. Research concerning
construction in the dynamic geometry environment as a means to improve conjecture and
conceptual understanding of the definition of figure would be worthwhile.
Reproducing this study or a study similar to it using a sample of students with
different demographical backgrounds would also be worthwhile. It would serve to
strengthen and to extend the results of this study. It would be interesting to use a sample
of higher achieving students from higher socioeconomic backgrounds and run a duplicate
study using similar instruments and methodology. Comparing the results of both studies
would be insightful. It would be interesting to see what aspects of both studies remained
constant and what aspects differed.
Concluding Remarks
Participants of this study had the opportunity to explore and discover geometric
truths in both static and dynamic environments without the aid of an instructor or text
book. Allowing students to explore independently allowed them to build inductive
reasoning skills and uncover truths or untruths on their own. The instruments used for
this study provide a series of dynamic lab activities that are closely aligned with typical
secondary geometry curricula and may be used in practical geometry instruction or in
research studies such as this one.
This study yielded results supporting the use of dynamic geometry software in the
classroom as well as interesting results concerning the interaction of gender, mathematics
119
achievement, and van Hiele level with student ability to conjecture in geometry. Results
indicate that even with little or no training, students may use the dragging utility available
in a dynamic geometry environment to significantly improve their ability to conjecture in
geometry and significantly increase the conviction in their conjectures, even without the
ability to construct the figures on their own.
This study uncovered some of the potential weaknesses and misconceptions of
students in the conjecture process and their use of dynamic software. It pointed out how
students may blur the distinction between conjecture and proof or between inductive and
deductive reasoning. It demonstrated how students misunderstand the act of
counterexample and what it means to ?disprove?. The results of this study stress the
importance of good dragging technique and the pitfalls of trusting output without
question. This study also explored an important question concerning the role of proof
with the availability of dynamic geometry software. Results indicate how students may
see the use of dynamic geometry software as a replacement to deductive proof.
Goldenburg and Cuoco (1996) stated the following concerning the use of dynamic
geometry software:
If the potential hazards and pitfalls seem larger and more numerous than
one might have imagined, it is also clear that the opportunities to make
new and important mathematical connections between classical geometric
content and big ideas from other mathematical areas are too intriguing to
ignore. So is the evidence in students? eyes and on their faces. We must
come to understand the new terrain well, so that its roughness becomes a
part of, and not a detraction from, its beauty. (p. 30)
120
This study has not only provided support for the use of dynamic software as an
effective conjecturing tool, but has explored some of these mentioned ?pitfalls?,
classified distinct dragging techniques used by students while finding and testing
conjectures, and explored interesting themes concerning the students experiences and
perceptions using dynamic geometry software as a conjecturing tool.
de Villiers (1996) claimed that many of the teachers in the past have simply
avoided the informal exploration of geometric relationships by construction and
measurement with paper and pencil since they are so time consuming. de Villiers (1996)
also claims that dynamic geometry software has revitalized the teaching of geometry in
many countries where Euclidean geometry was in danger of being ?thrown into the
trashcan of history.? (p. 25).
The results and implications of this study help strengthen the claim that dynamic
geometry software is a valuable tool that facilitates student conjecturing in geometry and
promotes equity among students of different achievement levels. However, teachers who
use dynamic software must be aware that the students may see the tool as a substitution
for proof and must strive to use dynamic software to enhance the need for proof rather
than replace it. The results of this study indicate that the use of dynamic software aids
with the students? concept of the general figure which is an important concept in the
context of deductive proof. Finally, this study classified different techniques used with
the dragging utility and uncovered several pitfalls that students may encounter when
conjecturing using dynamic geometry software. It is important that these pitfalls be
uncovered and addressed in order to make the use of dynamic geometry software in the
121
classroom a worthwhile and productive experience for both instructors and learners of
mathematics.
122
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127
APPENDIX A
LAB ACTIVITIES WITH RUBRICS
128
Lab 1
In the following diagram lines AC and DF are parallel to each other.
Segment GH is a transversal of these parallel lines. Using the measurements provided
make as many conjectures as you can about the angles formed by transversals and
parallel lines.
We already know that vertical angles are congruent and linear pairs are supplementary.
Now consider the relationships of the corresponding, alternate interior, alternate exterior
and consecutive angles when you make your conjectures. After each conjecture indicate
how sure you are that your conjecture is true for all transversals and parallel lines. Use a
1 ? 10 scale with 10 being the most confident.
m?GEF = 120.0?m?DEG = 60.0?
m?FEB = 60.0?m?DEB = 120.0?
m?CBE = 120.0?m?ABE = 60.0?
m?HB C = 60. 0?m?ABH = 120.0?
E
BA
C
H
G
D F
Conjecture(s): If two parallel lines are intersected by a transversal then:
129
Rubric Lab 1
This lab focuses on the angle pairs formed by transversals and parallel lines. Four
conjectures are targeted: If parallel lines are intersected by a transversal then?
1) Corresponding angles are congruent.
2) Alternate Interior angles are congruent.
3) Alternate Exterior angles are congruent.
4) Consecutive (same-side interior) angles are supplementary.
True statements concerning the perpendicular case such as ?If a transversal is
perpendicular to one line then it will be perpendicular to the other line.? will be counted
as a relevant conjecture. Conjectures that are poorly worded may still be judged relevant
if you determine that the statement is reasonable enough to be interpreted in proper
mathematical language. Any true conjectures concerning vertical angles or linear pairs
will be counted as irrelevant since these are theorems already covered in previous class
lessens. The following chart has some examples taken from previous student responses.
Code
Conjectures
Relevant R Consecutive interior angles have the same measure if and only if they
are right angles.
Alternate interior angles will always be the same.
Corresponding angles are equal.
For same side interior angles they equal up to 180.
Irrelevant I Vertical angles are congruent.
Lines AC and DF are parallel.
Linear pairs are supplementary.
Ambiguous A All alternate interior angles are two angles that lie between two opposite
sides and intersect two or more coplanar lines on different points.
If you take one alternate interior and one alternate exterior then they add
up to be 180 degrees.
False F You always get four acute angles and four obtuse angles.
One consecutive angle is twice as big as the other.
Same side interior angles are never congruent.
130
Lab 2
The measurements of the three interior angles of triangle ABC are shown.
The measurement of the exterior angle ACD is also shown.
Make as many conjectures as you can about the interior and exterior angles of triangles.
After each conjecture rate how convinced you are that your conjecture is true for all
triangles. Use a 1-10 scale with 10 being the most confident.
m
?
ABC+m
?
BAC = 150.0
?
m
?
ABC+m
?
BAC+m
?
ACB = 180.0
?
m
?
ACD = 150.0
?
m
?
ACB = 30.0
?
m
?
BAC = 70.0
?
m
?
ABC = 80.0
?
B D
A
C
Conjecture(s): If a figure is a triangle then ?
131
An isosceles triangle has at least two congruent sides. Triangle EFG is an isosceles
triangle.
Make as many conjectures as you can about isosceles triangles.
After each conjecture rate how convinced you are that your conjecture is true for all
isosceles triangles. Use a 1-10 scale with 10 being the most confident.
m
?
FGE = 67.3
?
m
?
FEG = 67.3
?
m
?
EFG = 45.4
?
m GE = 3.0 cm
m FG = 4.0 cm
m EF = 4.0 cm
F
G
E
Conjecture(s): If a triangle is isosceles then ?
132
Equilateral triangles have all three sides congruent. Triangle HIJ is equilateral.
Make as many conjectures as you can about equilateral triangles.
After each conjecture rate how convinced you are that your conjecture is true for all
equilateral triangles. Use a 1-10 scale with 10 being the most confident.
m
?
JIH = 60.00
?
m
?
HJI = 60.00
?
m
?
IHJ = 60.00
?
m JH = 3.9 cm
m IJ = 3.9 cm
m HI = 3.9 cm
I
H J
Conjecture(s): If a triangle is equilateral then ?
133
Rubric Lab 2
This lab focuses on interior and exterior angles of triangles. Four conjectures are targeted
using three different figures:
Figure 1: If a figure is a triangles then ?
5) The sum of the measures of the interior angles is 180 degrees.
6) The measure of an exterior angle is equal to the sum of the measures of the two
remote interior angles.
Figure 2: If a triangle is isosceles then ?
3) The base angles are congruent.
Figure 3: If a triangle is equilateral then ?
4) The triangle is equiangular.
True statements concerning the number of interior angles that are obtuse or right will be
counted as relevant. For example ?A triangle can only have one obtuse (right) interior
angle.? is a relevant conjecture. Conjectures that are poorly worded may still be judged
relevant if you determine that the statement is reasonable enough to be interpreted in
proper mathematical language. Any statements about the exterior angle and its adjacent
interior angle being supplementary will be counted as irrelevant. The following chart has
some examples taken from previous student responses. If you are unsure of what the
student is trying to convey, score the conjecture as ambiguous.
Code
Conjectures
Relevant R The angles of the triangle add up to be 180.
Two angles are the same for the isosceles.
The equilateral has all angles 60 degrees.
If you add up two of the triangle angles you get the outside angle ACD.
If you get one obtuse angle inside then the other angles inside are
smaller than 90.
The other angles of a right triangle add up to 90 degrees.
Irrelevant I ACB and ACD are linear pairs.
The isosceles has two sides the same.
ABC is a right triangle.
Ambiguous A Angle ABC has been measures it will not change as much as the others.
If you split angle ACD the measures would be close to each other.
False F The exterior angles are obtuse.
Isosceles triangles are acute triangles.
The base angles will be greater than the angle at the top.
134
Lab 3
A triangle midsegment joins the midpoints of two sides of a triangle.
In the following figure segment RS is a triangle midsegment. Write as many conjectures
as you can about triangles midsegments. Some measurements are provided with the
figure. After each conjecture rate how confident you are that your conjecture is true for
all triangle midsegments. Use a 1 - 10 scale with 10 being the most confident.
m
?
ROQ = 41.5
?
m
?
PRS = 41.5
?
m RS = 1.5 in.
m QO = 3.0 in.
S
R
O
P
Q
Conjecture(s): If a segment is a triangle midsegment then ?
135
Triangle DEF is formed by joining the midpoints of the sides of triangle ABC.
Triangle DEF is called a midsegment triangle.
Write as many conjectures as you can about midsegment triangles using the diagram and
the measures provided.
For each conjecture rate how convinced you are that your conjecture will be true for all
midsegment triangles. Use a 1-10 scale with 10 being the most confident.
Perimeter ABC
()
Perimeter DE F
= 2.00
Area ABC
()
Area DE F
()
= 4.00
Perimeter DEF = 3.3 in.
Perimeter ABC = 6.6 in.
Area DEF = 0.5 in
2
Area ABC = 2.0 in
2
F
ED
A
B
C
Conjecture(s): If a triangle is a midsegment triangle then ?
136
Rubric Lab 3
This lab focuses on the triangle midsegments. Four conjectures are targeted using two
figures.
Figure 1: If a segment is a triangle midsegment then ?
1) It is parallel to the remaining side.
2) Its measure is one half the measure of the remaining side.
Figure 2: If a triangle is a midsegment triangle then ?
3) The area of the midsegment triangle is one forth the area of the original
triangle.
4) The perimeter of the midsegment triangle is one half the perimeter of the
original triangle.
True statements concerning the congruence of triangles found within figure 2 will be
counted as relevant; Any statement concerning the similarity of the midsegment triangle
with the original triangle will also be counted as relevant Conjectures that are poorly
worded may still be judged relevant if you determine that the statement is reasonable
enough to be interpreted in proper mathematical language. If a student states that the
midsegment is simply ?smaller than? the remaining side, count it as irrelevant they
should be more specific. The following chart has some examples taken from previous
student responses. If you are unsure of what the student is trying to convey, score the
conjecture as ambiguous.
Code
Conjectures
Relevant R The bottom side is twice the size of the midsegment.
There are three other triangles just like the middle triangle.
The middle triangle is the same shape as the big triangle but one forth of
the size.
The midsegment has the same slope as OQ.
Irrelevant I The points D, E, and F are midpoints.
Triangle DEF is smaller than triangle ABC
Ambiguous A You find the perimeter of both triangles by dividing them by each other.
The midpoints of A, B, and C equal the perimeter of the inner triangle.
The measurement of the area is more than the perimeter.
False F The midsegment triangle is equilateral.
All segments are congruent and all angles are the same.
137
Lab 4
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Quadrilateral ABCD is a parallelogram.
Use the diagram below and the measurements provided to make as many conjectures you
can about parallelograms.
Do not use theorems or postulates about quadrilaterals already discussed in class.
After each conjecture rate how convinced you are about your conjecture for all
parallelograms. Use a 1 to 10 scale with 10 being the most confident.
m
?
CDA = 60.0
?
m
?
BCD = 120.0
?
m
?
ABC = 60.0
?
m
?
DAB = 120.0
?
BE = 1.95 in.
DE = 1.95 in.
CE = 1.18 in.
AE = 1.18 in.
BC = 1.79 in.
AD = 1.79 in.
CD = 2.68 in.
AB = 2.68 in.
E
BA
CD
Conjecture(s): If a quadrilateral is a parallelogram then?
138
Rubric Lab 4
This lab focuses on the properties of parallelograms. Four conjectures are targeted: If a
quadrilateral is a parallelogram then ?
3) Opposite sides of a parallelogram are congruent.
4) Opposite angles of a parallelogram are congruent.
5) Adjacent sides of a parallelogram are supplementary.
6) The diagonals of a parallelogram bisect each other.
True statements concerning the congruence of triangles found within the figure will be
counted as relevant; however, multiple statements concerning the congruence of triangles
will count only as one relevant conjecture. For example, there are four different pairs of
congruent triangles found within the figure. If a student states four different conjectures
listing these pairs only one relevant conjecture should be awarded. Conjectures that are
poorly worded may still be judged relevant if you determine that the statement is
reasonable enough to be interpreted in proper mathematical language. Any statements
about the congruence of the vertical angles resulting by the intersection of the diagonals
will be counted as irrelevant. Any statements that hold true for a general quadrilateral
such as ?The sum of the measures of the interior angles is 360 degrees? or ?The
parallelogram has two diagonals? will be counted as irrelevant. The following chart has
some examples taken from previous student responses. Remember that the conjectures
should reflect the properties of parallelograms and not just quadrilaterals in general. If
you are unsure of what the student is trying to convey, score the conjecture as ambiguous.
Code
Conjectures
Relevant R The measure of AB and DC will stay the same.
The angles across the diagonal are congruent.
Line segment AEC bisects line segment DEB.
BE and DE are equal and AE and CE are equal.
The diagonals make congruent triangles.
When a ?perfect rectangle? the diagonals are congruent.
Irrelevant I All angles add up to be 360 degrees.
The opposite sides are parallel.
Both of the diagonals intersect at E
Ambiguous A The wider the shape the more it increases.
The corresponding angles are congruent.
The corresponding lines are congruent.
False F The diagonals are congruent.
The diagonals bisect the angles of the parallelogram.
The diagonals are perpendicular.
All sides are congruent.
139
Lab 5
A rectangle is a parallelogram with four right angles. ABCD is a rectangle with
diagonals shown dashed.
Write all of your conjectures about the diagonals of rectangles.
After each conjecture rate how confident you are that your conjecture is true for all
rectangles. Use a 1- 10 scale with 10 being the most confident.
m?DE C = 46.6?
m?CDE = 66.7?
m?ADE = 23.3?
m BD = 10.0 cm
m AC = 10.0 cm
E
D
B
C
A
Conjecture(s): If a parallelogram is a rectangle then ?
140
A square is a parallelogram with four congruent sides and four right angles. ABCD is a
square with diagonals shown dashed.
Write as many conjectures as you can about the diagonals of a square.
After each conjecture rate how confident you are that your conjecture is true for all
squares. Use a 1 - 10 scale with 10 being the most confident.
m
?
ABC = 90.00
?
m
?
CBE = 45.00
?
m
?
ABE = 45.00
?
m DB = 7.63 cm
m AC = 7.63 cm
E
B
A
CD
Conjecture(s): If a parallelogram is a square then ?
141
Rubric Lab 5
This lab focuses on the diagonals of rectangles and squares. Four conjectures are
targeted using two figures
Figure 1) If a parallelogram is a rectangle then ?
1) Diagonals are congruent.
Figure 2) If a parallelogram is a square then ?
2) Diagonals are congruent.
3) Diagonals are perpendicular.
4) Diagonals bisect the interior angles of the square.
The conjectures in this lab should focus on the diagonals of rectangles and squares and
not other properties that are inherent in these shapes. Any general properties of
parallelograms or quadrilaterals should be counted as irrelevant. Conjectures that are
poorly worded may still be judged relevant if you determine that the statement is
reasonable enough to be interpreted in proper mathematical language. Any statements
about the congruence of the vertical angles resulting by the intersection of the diagonals
will be counted as irrelevant. Any statement concerning the congruence of the triangle
pairs formed by the diagonals will be counted as irrelevant since this is a property of the
general parallelogram. However, any statement that targets the fact that the diagonals
form four congruent triangles will count as relevant since this is not a property of the
general parallelogram but rather the rhombus and square. The following chart has some
examples taken from previous student responses. Remember that the conjectures should
reflect the properties of the diagonals of these shapes. If you are unsure of what the
student is trying to convey, score the conjecture as ambiguous.
Code
Conjectures
Relevant R For a rectangle the diagonal lengths are the same.
The diagonals of the square are 90 degrees.
The diagonals are cutting the angles of the square in half 45 degrees.
The diagonals of a square are the same.
Irrelevant I Rectangles and squares have 90 degree angles
The opposite sides are parallel.
The diagonals of a square cut the square in half.
Ambiguous A I believe that all the diagonal angles are equal in a rectangle.
The line bisects at the same point.
False F The diagonals of a square form equilateral triangles.
The diagonals of a rectangle bisect the angles of the rectangle.
142
Lab 6
A rhombus is a parallelogram with four congruent sides. ABCD is a rhombus with
diagonals shown dashed.
Write as many conjectures as you can about the diagonals of a rhombus.
After each conjecture rate how confident you are that your conjecture is true for all
rhombi. Use a 1 - 10 scale with 10 being the most confident.
m
?
BCE = 32.42
?
m
?
DCE = 32.42
?
m
?
DEC = 90.0
?
m
?
CDE = 57.6
?
m
?
ADE = 57.6
?
m BD = 5.0 cm
m AC = 7.9 cm
E
A
C
B
D
Conjecture(s): If a parallelogram is a rhombus then ?
143
A kite is a quadrilateral with two pairs of adjacent sides congruent but opposite sides not
congruent.
DEFG is a kite with diagonals shown dashed.
Write as many conjectures as you can about the diagonals of a kite.
After each conjecture rate how confident you are that your conjecture is true for all kites.
Use a 1 - 10 scale with 10 being the most confident.
m?DHE = 90.0?
m?GDH = 50.3?
m?EDH = 50.3?
m?FEH = 67.7?
m?DE H = 39.7?
m GE = 5.2 cm
m DF = 8.5 cm
H
D
E
G
F
Conjecture(s): If a quadrilateral is a kite then ?
144
Rubric Lab 6
This lab focuses on the diagonals of rhombi and kites. Four conjectures are targeted
using two figures
Figure 1) If a parallelogram is a rhombus then ?
1) Diagonals are perpendicular.
2) Diagonals bisect the interior angles of the rhombus.
Figure 2) If a quadrilateral is a kite then ?
3) Diagonals are perpendicular.
4) Diagonals bisect one pair of interior angles of the kite.
The conjectures in this lab should focus on the diagonals of rhombi and kites and not
other properties that are inherent in these shapes. Any general properties of
parallelograms or quadrilaterals should be counted as irrelevant. Conjectures that are
poorly worded may still be judged relevant if you determine that the statement is
reasonable enough to be interpreted in proper mathematical language. Any statements
about the congruence of the vertical angles resulting by the intersection of the diagonals
will be counted as irrelevant. Any statement concerning the congruence of the triangle
pairs formed by the diagonals of a rhombus will be counted as irrelevant since this is a
property of the general parallelogram. However, any statement that targets the fact that
the diagonals form four congruent triangles will count as relevant since this is not a
property of the general parallelogram but rather the rhombus and square. Statements
concerning the congruence of triangle pairs for the kite will be counted as relevant since a
kite is not a parallelogram. The following chart has some examples taken from previous
student responses. Remember that the conjectures should reflect the properties of the
diagonals of these shapes. If you are unsure of what the student is trying to convey, score
the conjecture as ambiguous.
Code
Conjectures
Relevant R The diagonals of a rhombus make 90 degrees.
The diagonals of a kite can be outside the shape when the kite is
concave.
The diagonals of a kite make right angles.
One diagonal bisects the kite angles but the other diagonal does not.
One diagonal is bisected but not the other for a kite.
Irrelevant I Kites have one pair of congruent angles.
The opposite sides of a rhombus are parallel and all sides are congruent.
The diagonals of a rhombus cut the shape in half.
For a rhombus diagonals bisect each other.
Ambiguous A The angles of the kite are as not to be a parallelogram.
Opposite points are congruent.
False F The diagonals of a rhombus are not congruent.
The diagonals of a kite are not congruent.
145
Lab 7
Segment EF is a trapezoid midsegment. It connects the midpoints of the legs of
trapezoid ABCD.
Write as many conjectures as you can about trapezoid midsegments.
After each conjecture rate how confident you are that your conjecture is true for all
trapezoid midsegments. Use a 1 - 10 scale with 10 being the most confident.
m DC = 4.0 in.
m EF = 3.0 in.
m AB = 2.0 in.
Slope DC = 0.20
Slope EF = 0.20
Slope AB = 0.20
F
E
D
C
A
B
Conjecture(s): If a segment is a trapezoid midsegment then ?
146
An isosceles trapezoid has congruent legs. WXYZ is an isosceles trapezoid.
Write as many conjectures as you can about isosceles trapezoids.
After each conjecture rate how confident you are that your conjecture is true for all
isosceles trapezoids. Use a 1 - 10 scale with 10 being the most confident.
m?WXY = 45.5? m
?XYZ = 45.5?
m?XWZ = 134.5?
m?WZY = 134.5?
m ZX = 2.8 in. m YW = 2.8 in.
X
Z
W
Y
Conjecture(s): If a trapezoid is isosceles then ?
147
Rubric Lab 7
This lab focuses on trapezoid midsegments and isosceles trapezoids. Four conjectures
are targeted using two figures
Figure 1) If a segment is a trapezoid midsegment then ?
1) It is parallel to the bases.
2) Its measure is one half the sum of the measure of the two bases.
Figure 2) If a trapezoid is isosceles then ?
3) Diagonals are congruent.
4) Base angles are congruent.
The conjectures from figure one should focus only on the midsegment. Any statement
concerning the adjacent angles being supplementary will be scored irrelevant because this
is a property of the general trapezoid not the isosceles trapezoid. A statement concerning
the opposite angles being supplementary for an isosceles trapezoid will be scored
relevant. Conjectures that are poorly worded may still be judged relevant if you
determine that the statement is reasonable enough to be interpreted in proper
mathematical language. Any statements about the congruence of the vertical angles
resulting by the intersection of the diagonals will be counted as irrelevant. The following
chart has some examples taken from previous student responses. If you are unsure of
what the student is trying to convey, score the conjecture as ambiguous.
Code
Conjectures
Relevant R The trapezoid midsegment is parallel to AB and DC.
AB + DC divided by 2 is the midsegment.
The diagonals will be equal in the isosceles trapezoid.
The opposite angles will add up to 180 in the isosceles trapezoid.
The isosceles trapezoid has two pairs of congruent angles.
An isosceles trapezoid has two acute and two obtuse angles.
Irrelevant I The legs are not parallel.
The angles on the side are supplementary.
Top and bottom sides are parallel.
Ambiguous A Angle BAD is not congruent to any of the sides.
AD is in a right angle.
False F The trapezoid has two acute and two obtuse angles.
The diagonals bisect each other.
A trapezoid has all different angles unless it is isosceles.
148
Lab 8
An inscribed angle is an angle whose vertex is on a circle and whose sides contain
chords of that circle.
In the figure below Angle ACB is an inscribed angle in Circle D and Angle ACD has an
intercepted arc AB. The measurements of the inscribed angle and its intercepted arc are
shown. Write as many conjectures as you can about inscribed angles and their
intercepted arcs. After each conjecture rate how convinced you are that your conjecture
is true for all inscribed angles and their intercepted arcs. Use a 1 - 10 scale with 10 being
the most confident.
m
?
ACB = 74.0
?
m AB = 148.0
?
D
B
A
C
Conjecture(s): If an angle is an inscribed angle then ?
149
In the figure below Angle EFG and Angle EHG are both inscribed angles who share the
intercepted arc EG.
Write as many conjectures as you can about inscribed angles that have the same
intercepted arc.
After each conjecture rate how convinced you are that your conjecture is true for all
inscribed angles that have the same intercepted arc. Use a 1 - 10 scale with 10 being the
most confident.
m?EHG = 45.0?
m?EFG = 45.0?
E
H
F
G
Conjecture(s): If inscribed angles have the same intercepted arc then ?
150
In the figure below a right triangle is inscribed in a circle. Make is many conjectures as
you can about inscribed right triangles. After each conjecture rate how convinced you are
that your conjecture is true for all inscribed right triangles. Use a 1 - 10 scale with 10
being the most confident.
m
?
JKI = 29.7
?
m
?
IJK = 60.3
?
m
?
KIJ = 90.0
?
K
L
J
I
Conjecture(s): If a right triangle is inscribed in a circle then ?
151
In the figure below a quadrilateral is inscribed in a circle. Make is many conjectures as
you can about inscribed quadrilaterals. After each conjecture rate how convinced you are
that your conjecture is true for all inscribed quadrilaterals. Use a 1 - 10 scale with 10
being the most confident.
m
?
PMN = 80.0
?
m
?
OPM = 60.0
?
m
?
NOP = 100.0
?
m
?
MNO = 120.0
?
Q
O
M
N
P
Conjecture(s): If a quadrilateral is inscribed in a circle then ?
152
Rubric Lab 8
This lab focuses on inscribed angles. Four conjectures are targeted using four figures
Figure 1) If an angle is an inscribed angle then ?
1) Its measure is half the measure of its intercepted arc.
Figure 2) If inscribed angles intercept the same arc then ?
2) They are congruent.
Figure 3) If a right triangle is inscribed in a circle then ?
3) The hypotenuse is a diameter of the circle.
Figure 4) If a quadrilateral is inscribed in a circle then ?
4) The opposite angles are supplementary.
All relevant conjectures should be equivalent to the targeted conjectures above.
Conjectures that are poorly worded may still be judged relevant if you determine that the
statement is reasonable enough to be interpreted in proper mathematical language. The
following chart has some examples taken from previous student responses. If you are
unsure of what the student is trying to convey, score the conjecture as ambiguous.
Code
Conjectures
Relevant R The inscribed angle is half of the arc.
The opposite angles of the quadrilateral add up to 180 degrees.
The right triangle takes exactly half of the circle.
Angles of the same arc are congruent.
Irrelevant I Smaller arcs mean smaller angles.
The inscribed angle is inside the circle.
The right triangle has two angles that add up to 90.
Ambiguous A The intersection of the inscribed angles makes the lines vertical.
The thick arc is the most taken by the inscribed angle.
False F Inscribed angles are acute angles.
Inscribed angles intersect to make similar triangles.
153
APPENDIX B
VAN HIELE INSTRUMENT WITH RUBRIC
154
155
Rubric Item 1
The correct answers for shapes 1 through 9 are as follows:
1) T, P
2) Q, P
3) Q, P
4) N
5) N
6) P
7) Q, P
8) N
9) T, P
This item can be answered with a level one response. For instance, a student
wrote that shape 2 is not a polygon because ?It does not follow any rule?, and shape 7 is a
quadrilateral because ?It is a rhombus with its sides diagonal and parallel.? Item one can
also be answered with a level two response, when the reasons for classification are based
on the number of sides and, in shapes 4, 5, and 6, on their openness or curvature
(Gutierrez & Jaime, 1998 p. 35).
156
157
Rubric Item 2
The correct answers for shapes 1 through 7 are as follows:
1) I, X
2) I, V
3) R, X
4) R, X
5) I, X
6) I, V
7) I, V
This item can be answered in level one or two. Some level one students may
classify regular polygons as those that are ?familiar? to them (1, 3, 4, and 5), and the
irregular polygons as ?estrange? shapes. Level two students should base their
classification on the (in)equity of the angles and sides (Gutierrez & Jaime, 1998 p. 36).
158
159
Rubric Item 3
The correct answers for shapes 1 through 8 are as follows:
1) R, P
2) P
3) H, P
4) Blank
5) S, R, H, P
6) Blank
7) Blank
8) Blank
Students may answer with a level three response if the multiple combinations
shown above are answered correctly indicating hieratical classification. Students with the
correct one letter responses with 4, 6, 7, 8 left blank demonstrate level 2 understanding of
definition.
160
161
Rubric Item 4
Level one students will find the number of sides the only property common to
squares and rhombuses, a difference among these shapes may include a description such
as ?rhombuses are pointy but squares are not?. Students at level two will not differentiate
properties that are shared by the two shapes from those that belong to only one. For
example, the property of squares and rhombi having four sides may be mentioned as both
a shared property and a differentiating property. Level two students may use exclusive
classifications using lists of properties of angles, sides, and diagonals. The students in
level three are able to justify either inclusive or exclusive classifications. For example a
differentiating property would be ?Squares have four right angles but rhombuses have
two acute and two obtuse angles.? (Gutierrez & Jaime, 1998 p. 36).
162
Rubric Item 5
Students who successfully complete a proof in any form are assigned a level three since
this item uses a ?hint? as part of the item. (Gutierrez & Jaime, 1998 p. 37).
163
164
Rubric Item 6
The correct answers for questions 1 through 4 are as follows:
1) 1 diagonal from each vertex, 2 total diagonals
2) 2 diagonals from each vertex, 5 total diagonals
3) 3 diagonals from each vertex, 9 total diagonals
4) n-3 diagonals from each vertex, n(n-3)/2 total diagonals
Level two may draw the polygons with their diagonals and count the number of
diagonals answering the first three questions but are unsuccessful at question 4. Level
three students are successful on question 4 with the answer above or an equivalent form
of it. Level four students are able to justify the correct response in question 4 with a valid
deductive proof. (Gutierrez & Jaime, 1998 p. 38).
165
APPENDIX C
PARTICIPANT SURVEY
1) Describe what a conjecture is in your own words?
2) Which environment, dynamic or static, is better for conjecturing? Why?
3) How do you prove a conjecture?
4) Do you do a lot of ?dragging? when you are in the dynamic environment?
5) What is the purpose of dragging?
6) What is a ?generic? figure?
7) Are the figures in Sketchpad generic figures?
8) Are the figures in the static environment generic?
166
APPENDIX D
INTERVIEW PROTOCAL
Student is seated in front of a computer and Figure 1 is opened.
Researcher: In this interview I will be asking you some questions about the figures
shown on the computer screen. Feel free to use the mouse to drag at any time during the
interview. This interview is being recorded and may be used in a published report but at
no time will your real identity be published or made public.
This figure shows a midsegment quadrilateral that is formed by joining the
midpoints of any quadrilateral. A former student conjectured that the midsegment
quadrilateral will always be a parallelogram.
Do you agree with this conjecture?
How could you test your conclusion?
Would you like to have any measurements shown? Which measurements?
How could you prove or disprove this conjecture?
The figure of a general quadrilateral is then opened.
Researcher: A former student conjectured that there will be at most two obtuse interior
angles for any quadrilateral.
Do you agree with this conjecture?
How could you test your conclusion?
How could you prove or disprove this conjecture?
167
Using the same figure and the sum of the interior angles measured. A theorem
states that the sum of the interior angles for any quadrilateral is 360 degrees.
Do you agree with this theorem? (Encourage the student to drag until the quadrilateral is
concave)
How do you resolve this contradiction?
168
Figure 1: The midsegment quadrilateral
A midsegment quadrilateral is formed by connecting the midpoints of the sides of a quadrilateral
Conjecture: The midsegment quadrilateral will always be a parallelogram.
Theorem: The interior angles of a quadrilateral will always add up to 360
?
m
?
ABC+m
?
BCD+m
?
CDA+m
?
DAB = 360.00
?
m
?
CDA = 61.51
?
m
?
DAB = 115.32
?
m
?
BCD = 85.29
?
m
?
ABC = 97 .89
?
Figure 2: General quadrilateal
C onjecture: A q uadrilateral will ha ve a t most two o btu s e ang les
D
A
B
C
169
APPENDIX E
QUANTITATIVE DATA
The following tables show the results of all of the variables used in this study on
both assignment cases. The first table shows the results of the odd lab activities where
class A with 20 subjects conjectured in the dynamic environment and class B with 21
subjects conjectured in the static environment. The second table shows the results of the
even lab activities with the environments reversed.
170
Input Data for the Odd Assignment Case
Subject environment relevant false conviction gender achievement vh
A1 1 2.25 0 9.4 1 38 1
A2 1 2.75 0 10.0 0 37 1
A3 1 1.00 0 9.0 0 34 1
A4 1 1.00 0 5.0 0 31 1
A5 1 1.33 0 9.2 1 28 1
A6 1 1.33 0 6.7 0 32 1
A7 1 3.00 0 9.7 0 41 1
A8 1 2.50 0 9.1 1 32 1
A9 1 1.75 0 9.6 1 28 1
A10 1 3.50 0 9.3 0 36 1
A11 1 2.50 0 9.1 1 40 1
A12 1 1.25 0 8.6 1 24 1
A13 1 1.33 0 10.0 1 38 1
A14 1 2.67 0 9.4 1 46 2
A15 1 0.50 0 9.1 1 30 1
A16 1 0.33 0.33 6.8 1 32 1
A17 1 2.00 0 9.6 1 40 1
A18 1 2.25 0 9.8 0 26 1
A19 1 1.50 0 9.3 1 21 1
A20 1 1.25 0 8.0 0 32 1
B1 0 2.00 0 10.0 0 35 1
B2 0 1.75 .25 8.8 1 1
B3 0 1.75 0 9.2 1 45 2
B4 0 1.50 0 7.0 1 1
B5 0 0.00 0 6.6 0 22 1
B6 0 0.00 0 6.8 0 22 1
B7 0 2.00 0.5 7.5 0 41 1
B8 0 1.67 0.67 7.8 0 44 1
B9 0 1.33 0.67 7.8 0 45 2
B10 0 1.75 0.5 9.6 0 43 1
B11 0 0.00 0 5.5 0 42 1
B12 0 0.00 0 5.6 0 38 1
B13 0 0.00 0.67 7.6 0 36 1
B14 0 0.00 0.25 6.6 0 40 1
B15 0 1.00 0.5 8.2 0 42 2
B16 0 0.25 0 6.3 0 27 1
B17 0 2.00 0.25 9.2 1 44 2
B18 0 0.67 0 9.7 1 31 1
B19 0 0.75 0.25 5.8 0 37 1
B20 0 0.00 0.5 6.0 1 28 1
B21 0 1.00 0 6.0 1 38 1
171
Input Data for the Even Assignment Case
Subject environment relevant false conviction gender achievement vh
A1 0 0.25 0.25 8.4 1 38 1
A2 0 1.00 0 8.8 0 37 1
A3 0 0.75 0 9.3 0 34 1
A4 0 0.67 0.67 4.5 0 31 1
A5 0 1.33 0 7.8 1 28 1
A6 0 1.50 0 7.5 0 32 1
A7 0 1.50 0 10.0 0 41 1
A8 0 1.33 0 9.1 1 32 1
A9 0 2.00 0 9.7 1 28 1
A10 0 2.67 0.33 7.2 0 36 1
A11 0 1.25 1.00 8.3 1 40 1
A12 0 0.75 0.25 8.2 1 24 1
A13 0 2.00 0 10.0 1 38 1
A14 0 1.50 0.50 10.0 1 46 2
A15 0 1.00 0 8.9 1 30 1
A16 0 0.75 0.50 6.1 1 32 1
A17 0 1.00 0.50 4.2 1 40 1
A18 0 1.50 1.00 9.6 0 26 1
A19 0 1.00 0.33 6.5 1 21 1
A20 0 0.50 1.00 4.6 0 32 1
B1 1 2.50 0 10.0 0 35 1
B2 1 2.67 0 9.6 1 1
B3 1 3.00 0 9.4 1 45 2
B4 1 2.75 0 9.5 1 1
B5 1 1.75 0 9.2 0 22 1
B6 1 1.00 0 8.2 0 22 1
B7 1 4.00 0 9.4 0 41 1
B8 1 1.50 0 10.0 0 44 1
B9 1 2.33 0.33 8.9 0 45 2
B10 1 2.75 0 10.0 0 43 1
B11 1 0.50 0 10.0 0 42 1
B12 1 2.00 0.25 8.6 0 38 1
B13 1 2.00 0..5 9.3 0 36 1
B14 1 2.00 0.25 8.7 0 40 1
B15 1 1.00 0 7.8 0 42 2
B16 1 0..50 0 8.0 0 27 1
B17 1 1.75 0 9.1 1 44 2
B18 1 2.00 0 10.0 1 31 1
B19 1 2.75 0 9.7 0 37 1
B20 1 1.00 0.5 8.8 1 28 1
B21 1 1.75 0 8.6 1 38 1