THE VAN HIELE THEORY THROUGH THE DISCURSIVE LENS:
PROSPECTIVE TEACEHRS’ GEOMETRIC DISCOURSES
By
Sasha Wang

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mathematics Education
2011

ABSTRACT
THE VAN HIELE THEORY THROUGH THE DISCURSIVE LENS:
PROSPECTIVE TEACHERS’ GEOMETRIC DISCOURSES
By
Sasha Wang
Over the past decade, there has been an increasing trend in the mathematics education
research community to study students’ reasoning in the teaching and learning of mathematics,
and to examine issues emphasizing the use of vocabulary, terminology, and words in the
mathematics classroom. In response, this study investigates changes in prospective elementary
teachers’ levels of geometric thinking, and the development of their geometric discourses in the
classification of quadrilaterals.
In Sfard’s (2008) Thinking as Communicating: Human Development, the Growth of
Discourses, and Mathematizing, she introduces her commognitive framework, a systematic
approach to analyzing the discursive features of mathematical thinking, including word use,
visual mediators, routines, and endorsed narratives. To examine thinking about geometry, this
study connects Sfard’s analytic framework to another, namely the van Hiele theory (see van
Hiele, 1959/1985). The van Hiele theory describes the development of students’ five levels of
thinking in geometry. Levels 1 to 5 are described as visual, descriptive, theoretical, formal logic
and rigor, respectively. This study used the van Hiele Geometry Test from the Cognitive
Development and Achievement in Secondary School Geometry (CDASSG) project (Usiskin,
1982) as the pretest and posttest to determine prospective elementary school teachers’ van Hiele
levels. This study also produces, on the basis of theoretical understandings and of empirical data,
a detailed model, namely, the Development of Geometric Discourse. This model translates the

van Hiele levels into discursive stages of geometric discourses with respect to word use, visual
mediators, routines, and endorsed narratives.
This study reveals discursive similarities and differences in participants’ geometric
discourses at the same van Hiele level, as well as changes in geometric discourse as a result of
changes in levels of geometric thinking. The study also investigates the usefulness of a
discursive framework in providing “rich descriptions” of participants’ thinking processes.

Copyright By
SASHA WANG
2011

To my grandparents, Changmao Wang and Zhongmin Xu for everything.

v

ACKNOWLEGEMENTS

To acknowledge is to direct to the source of my knowledge, and certainly it is the case
here. Many colleagues and friends have shared their wisdom with me. I am grateful particularly
to my advisors, Glenda Lappan and Anna Sfard, who thoughtfully guided me through this
challenging but exciting journey. They inspired me to follow my passions in mathematics
education, and provided uncommon generosity and patience along the way. Without Glenda
Lappan and Anna Sfard, this dissertation would not exist. Thanks to you both for bearing with
me when I was unsure of myself.
I also extend my special thanks to colleagues at the Center For the Study of Mathematics
Curriculum for providing multiple opportunities to learn about the field of mathematics
education. As a doctoral fellow of the center, I was fortunate to benefit from my colleagues and
their knowledge of curriculum development in Mathematics Education. Among those involved, I
am deeply grateful to my teacher, Mike Shaughnessy, for sharing his visions with me in
geometry; and to Zalman Usiskin for his willingness to critique parts of my dissertations. Thank
you for keeping me tuned in geometry.
I also thank the faculty members and graduate students from whom I have been fortunate
to learn and gather advice: my dissertation committee members, Bob Floden, Sharon Senk and
Gerald Luden, who brought expertise to my dissertation and taught me how to conduct
educational research rigorously; my guidance committee members and course faculties, Jack
Smith and Raven McCrory, who in their respective ways, they encouraged and guided me during
my lost-then-found moments; other course faculties, including Susan Wilson, Philip Cusick,

vi

Ralph Putnam, Natasha Speer, Natalie Sinclair and Susan Melnick, who led to the knowledge
about teacher education and research in mathematics education.
It would have been impossible to begin this journey without the inspiration of dear
colleagues at Kapiolani Community College, including Mike Molloy, Maria Bautista and Kyong
Soo Chung, who influenced me with their passions of teaching; and course faculties in the
mathematics department at the University of Hawaii at Manoa, Bill Lampe, Ron Brown and
Mike Hilden, who watered the spark of my ambition and my initial enthusiasm in mathematics. I
thank you for believing in me.
I would be deeply remiss if I did not specifically acknowledge a group of college students
who participated in my study, and they were my primary source of knowledge. I learned
tremendously from their geometric thinking and the way they communicated with me. I thank
you for giving me the opportunity to converse with you and to observe you. I appreciate your
generosity and openness in sharing your thoughts on geometry. It is your participation in the
study, with your bountiful support, that made it possible to complete my dissertation.
Lastly and foremost, my two beautiful children, Rachel and Quincy, are my inspirations.
They made an incredible contribution by giving me the opportunity to watch two exemplary
cases of how kids learn in real time and in close-up. Quincy’s learning of words helped me to
have a better understanding of how a concept of a word is formed by a toddler through his daily
experiences and his interactions with others. Rachel’s early experience as a mathematics learner,
ways in which she juggled with numbers, provided me firsthand scenarios about school kids’
mathematical thinking. Jerry, my most severe critic gave helpful advice on the presentation, and
engaged me in interesting dialogues on issue in mathematics education.
I am indebted to all the people who supported me in pursuing this dream.

vii

TABLE OF CONTENTS

LIST OF TABLES .......................................................................................................................... x
LIST OF FIGURES ..................................................................................................................... xiii
CHAPTER ONE: INTRODUCTION ............................................................................................. 1
CHAPTER TWO: THEORETICAL BACKGROUND ................................................................. 5
Review of Relevant Literature ............................................................................................ 5
The van Hiele Theory ............................................................................................. 5
Research Guided by van Hiele Theory ................................................................... 9
Knowledge of Geometry for Teaching ................................................................. 23
Theoretical Framework ..................................................................................................... 28
Commognition ...................................................................................................... 30
The Levels of Geometric Discourses .................................................................... 34
The General Research Question in Commognitive Terms ................................... 37
CHAPTER THREE: METHODOLOGY ..................................................................................... 40
Participants ........................................................................................................................ 40
The van Hiele Geometry Test ........................................................................................... 41
Interview Tasks ................................................................................................................. 42
Task One: Sorting Geometric Figures .................................................................. 43
Task Two: Investigating Properties of Parallelograms ......................................... 44
Task Three: Prove the Equivalence of Two Definitions ....................................... 46
Data Collection ................................................................................................................. 47
Data Analysis .................................................................................................................... 49
CHAPTER FOUR: FINDINGS .................................................................................................... 52
Changes in van Hiele Levels of Geometric Thinking ....................................................... 52
Changes in van Hiele Geometry Test as a Whole Group ..................................... 52
Changes in van Hiele Geometry Test among Interviewees .................................. 56
Changes in Geometric Discourse ...................................................................................... 58
viii

Case 1: Changes in ATL’s Geometric Discourse ................................................. 60
Case 2: Changes in ANI’s Geometric Discourse .................................................. 89
Case 3: Changes in ALY’s Geometric Discourse ............................................... 123
Case 4: Changes in AYA’s Geometric Discourse .............................................. 165
Case 5: Changes in ARI’s Geometric Discourse ................................................ 197
The van Hiele Levels as Geometric Discourse ............................................................... 226
CHAPTER FIVE: DISCUSSION ............................................................................................... 229
Summary of the Results .................................................................................................. 229
Changes in van Hiele Levels............................................................................... 230
Changes in Geometric Discourses ...................................................................... 231
What Can We Generalize and Why ................................................................................ 235
The van Hiele Levels: Discrete or Continuous? ................................................. 239
What Can Be Asked and Why ........................................................................................ 251
What Do We Learn From Students’ Thinking? .................................................. 251
Practical Information About Teaching and Assessment ..................................... 255
Limitation of the Study ................................................................................................... 257
APPENDICES ............................................................................................................................ 261
Appendix A: Research Study Consent Form .................................................................. 263
Appendix B: Behaviors at Each van Hiele Level ........................................................... 266
Appendix C: van Hiele Geometry Test ........................................................................... 271
Appendix D: Interview Tasks ......................................................................................... 283
Appendix E: Interview Protocols .................................................................................... 286
Appendix F: Development of Geometric Discourse at Each van Hiele Level ............... 293
BIBLIOGRAPHY ....................................................................................................................... 297

ix

LIST OF TABLES

Table 2.1

Responses to Relation Questions about Congruency............................................ 24

Table 2.2

Discursive Translation of van Hiele Levels 1-3 ................................................... 35

Table 2.3

Discursive Translation of van Hiele Levels 4-5 ................................................... 36

Table 3.1

Investigating the Properties of Parallelograms ..................................................... 44

Table 4.1

Distributions of participants’ van Hiele Levels at both tests ................................ 53

Table 4.2

Van Hiele Geometry Test: Item Analysis for Pretest (B) and Posttest (E) ........... 54

Table 4.3

Distributions of interviewees’ van Hiele Levels at both tests............................... 56

Table 4.4

Interviewees’ changes in van Hiele Levels between tests .................................... 57

Table 4.5

A comparison of ATL’s routines of sorting polygons at the two interviews........ 68

Table 4.6

ATL’s routines of verifying on the angles of parallelograms at the Pre-Interview
............................................................................................................................... 70

Table 4.7

ATL’s routines of verifying on the angles of parallelograms at the Post-Interview
............................................................................................................................... 71

Table 4.8

ATL’s routines of verifying on the sides of parallelograms at the Pre-Interview 74

Table 4.8

ATL’s routines of verifying on the sides of parallelograms at the Post-Interview75

Table 4.9

Summary of ATL’s routines in Task Two from both Interviews ......................... 77

Table 4.10

The frequencies of ATL’s use of the names of quadrilaterals at the interviews... 78

Table 4.11

Total frequencies of ATL’s use of names of quadrilaterals at the interviews ...... 78

Table 4.12

The frequencies of ATL’s use of the names of parts of parallelograms at the two
interviews ............................................................................................................. 87

Table 4.13

Total frequencies of ATL’s use of names of parts of parallelograms at the two
interviews…………….………………………………………………………..…88

Table 4.14

ANI’s routine procedure of verification for “diagonals bisect the angles” ........ 115

x

Table 4.15

The frequencies of ANI’s use of the names of quadrilaterals at the two interviews
............................................................................................................................. 117

Table 4.16

Total frequencies of ANI’s use of names of quadrilaterals at the two interviews
............................................................................................................................. 117

Table 4.17

The frequencies of ANI’s use of names of the parts of parallelograms at the two
interviews ............................................................................................................ 122

Table 4.18

Total frequencies of ANI’s use of names of the parts of parallelograms at the two
interviews ............................................................................................................ 122

Table 4.19

ALY’s routine procedures of substantiating that diagonals bisect each other .... 143

Table 4.20

ALY’s written proof that diagonals bisect each other at the Post-Interview ...... 146

Table 4.21

ALY’s proof by constructing a parallelogram with opposite sides equal ........... 150

Table 4.22

ALY’s proof by constructing a parallelogram with opposite angles equal ........ 151

Table 4.23

The frequencies of ALY’s use of the names of quadrilaterals at the two interviews
............................................................................................................................. 156

Table 4.24

Total frequencies of ALY’s use of names of quadrilaterals at the two interviews
............................................................................................................................. 157

Table 4.25

The frequencies of ALY’s use of the names of the parts of parallelograms at the
two interviews. .................................................................................................... 162

Table 4.26

Total frequencies of ALY’s use of names of the parts of parallelograms at the two
interviews ............................................................................................................ 162

Table 4.27

The word distance in geometry and used by ALY ............................................. 165

Table 4.27

AYA’s attempts at proving the equivalence of two definitions .......................... 185

Table 4.28

AYA’s substantiation of two congruent triangles at the Post-Interview ............ 186

Table 4.29

AYA’s newly constructed endorsed narrative at the Post-Interview .................. 187

Table 4.30

The frequencies of AYA’s use of the names of quadrilaterals at the two interviews
............................................................................................................................. 189

Table 4.31

Total frequencies of AYA’s use of names of quadrilaterals at the two interviews
............................................................................................................................. 189

xi

Table 4.32

The frequencies of AYA’s use of the names of the parts of parallelograms at the
two interviews ..................................................................................................... 196

Table 4.33

Total frequencies of AYA’s use of names of the parts of parallelograms at the two
interviews ............................................................................................................ 196

Table 4.34

ARI’s routine procedures of substantiation for “two diagonals are equal” ........ 212

Table 4.35

The frequencies of ARI’s use of the names of quadrilaterals at the two interviews
............................................................................................................................. 218

Table 4.36

Total frequencies of ARI’s use of names of quadrilaterals at the two interviews
............................................................................................................................. 218

Table 4.37

The frequencies of ARI’s use of the names of the parts of parallelograms at the
two interviews ..................................................................................................... 224

Table 4.38

Total frequencies of ARI’s use of names of the parts of parallelograms at the two
interviews ............................................................................................................ 224

Table 4.39

The frequencies of ARI’s use of formal mathematical words at the two interviews
............................................................................................................................. 225

Table 4.40

Total frequencies of ARI’s use of formal mathematical words at the two
interviews ............................................................................................................ 226

Table 4.41

Development of Geometric Discourse at Each van Hiele Level ........................ 293

xii

LIST OF FIGURES

Figure 2.1

Corresponding angles of parallel lines intersected by a transversal. ...................... 8

Figure 2.2

An example of a Level 1 test item with its corresponding van Hiele quotes........ 15

Figure 2.3

Two experimental tasks from the Oregon Project ................................................ 18

Figure 2.4

Bud, Amy and Don’s drawings of different triangles. .......................................... 19

Figure 2.5

An exterior angle of a triangle. ............................................................................. 21

Figure 2.6

The “diagonals” of concave polygons .................................................................. 25

Figure 2.7

A cube (left) and Solid I (right) ............................................................................ 26

Figure 2.8

Responses regarding the altitude of a triangle ...................................................... 27

Figure 3.1

Task One: Sorting Geometric Figures .................................................................. 43

Figure 3.2

Pictures of four parallelograms ............................................................................. 45

Figure 3.3

Summary of Data Collection Phases..................................................................... 47

Figure 4.1

Comparison of correct answers for items at both tests ......................................... 55

Figure 4.2

ATL’s grouping of polygons in the Pre-Interview ............................................... 62

Figure 4.3

ATL’s regrouping of the triangles at the Pre-Interview........................................ 63

Figure 4.4

ATL’s regrouping the quadrilaterals at the Pre-Interview .................................... 63

Figure 4.5

ATL’s routines of sorting polygons at the Pre-Interview ..................................... 64

Figure 4.6

(Previous Page). A comparison of ATL’s grouping of polygons at both interviews
............................................................................................................................... 65

Figure 4.7

ATL’s use of the word parallelogram at the Pre-Interview .................................. 83

Figure 4.8

ATL’s use of the word parallelogram at the Post-Interview................................. 87

Figure 4.9

ANI’s routines of sorting polygons at the Pre-Interview ...................................... 96

xiii

Figure 4.10

ANI’s grouping of parallelograms at the Post-Interview. ..................................... 97

Figure 4.11

ANI’s grouping of the quadrilaterals at the Post-Interview .................................. 99

Figure 4.12

ANI’s routines of sorting polygons at the Post-Interview .................................. 100

Figure 4.13

ANI’s written responses for Task Two at the Post-Interview............................. 104

Figure 4.14.

ANI’s use of the word parallelogram at the Pre-Interview ................................. 120

Figure 4.15

ANI’s use of the word parallelogram at the Post-Interview ............................... 121

Figure 4.16

ALY’s regrouping of polygons at the Pre-Interview .......................................... 125

Figure 4.17

ALY’s subgrouping of the quadrilaterals at the Pre-Interview ........................... 126

Figure 4.18

ALY’s routines of sorting at the Pre-Interview .................................................. 128

Figure 4.19

ALY’s chains subgroupings of the quadrilaterals at the Post-Interview. ........... 129

Figure 4.20

ALY's routines of sorting at the Post-Interview…….…………………………..132

Figure 4.21

ALY’s verification of the first implication at the Post-Interview……………....153

Figure 4.22

ALY’s proof that opposite angles have the same measure ................................. 154

Figure 4.23

(Previous Page) ALY’s use of the word parallelogram at the Pre-Interview ..... 160

Figure 4.24

ALY’s use of the word parallelogram at the Post-Interview .............................. 161

Figure 4.25

(Previuos Page) AYA’s first attempt at sorting polygons at the Pre-Interview .. 168

Figure 4.26

AYA's first attempt of the routine of sorting at the Pre-Interview……………..169

Figure 4.27

Examples of AYA’s regrouping at the Pre-Interview......................................... 169

Figure 4.28

The two subgroups of Group Two at the Pre-Interview ..................................... 171

Figure 4.29

AYA’s second prompt of the routine of sorting at the Pre-Interview................. 172

Figure 4.30

AYA’s use of the word parallelogram at the Pre-Interview ............................... 193

Figure 4.31

AYA’s use of the word parallelogram at the Post-Interview .............................. 195

Figure 4.32

ARI’s routine procedures for sorting polygons at the Pre-Interview .................. 203

xiv

Figure 4.33

ARI’s grouping of the subgroups of the quadrilaterals at the Post-Interview. ... 204

Figure 4.34

ARI’s routines of sorting polygons at the Post-Interview .................................. 205

Figure 4.35

ARI’s use of the word parallelogram at the Pre-Interview ................................. 221

Figure 4.36

ARI’s classifications of parallelograms at the Post-Interview ............................ 223

Figure 5.1

Degree of acquisition of a van Hiele Level ........................................................ 236

Figure 5.2

Characteristics of AYA’s geometrics discourses at Level 2 at the Pre-interview.
............................................................................................................................. 244

Figure 5.3

Characteristics of AYA’s geometrics discourses at Level 2 at the Post-interview.
............................................................................................................................. 245

Figure 5.4

Characteristics of ARI’s geometric discourse at Level 3 at the Pre-interviews. 246

Figure 5.5

Characteristics of ARI’s geometric discourse at Level 3 at the Pre-interviews. 246

Figure 5.6

Characteristics of ALY’s geometric discourse at Level 3 at the Pre-Interview. 248

Figure 5.7

Characteristics of ALY’s geometric discourse at Level 3 at the Post-interview. 248

Figure 5.8

Characteristics of Level 3 geometric discourse. ................................................. 254

Figure Appendix D. 1. Task One: Sorting Geometric Figures………………………………...283

xv

CHAPTER ONE: INTRODUCTION
In a research report prepared for the U.S. Department of Education, Wilson, Floden and
Ferrini-Mundy (2001) reported that research shows a positive connection between teachers’
preparation in their subject matter and their performance and impact in the classroom, and found
that “current results of subject matter preparation are disappointing” (p.35). Darken (2007) also
pointed out that “the weak mathematical preparation of many elementary and middle school (K8) teachers is one of the most serious problems afflicting American education” (p.20). These
conclusions suggest that a teachers’ preparation program needs to emphasize mathematics
content knowledge for teaching. Knowing mathematics for teaching involves knowledge of
mathematical ideas, mathematics reasoning skills, as well as communication skills, fluency with
examples and terms, and thoughtfulness about the nature of mathematical proficiency.
Geometry, considered as a tool for understanding and interacting with the space in which
we live, is perhaps the most intuitive, concrete and reality-linked part of mathematics (ICMI,
1998). It is in the language of geometry that the visual structure of our physical world is
described and communicated between individuals, and the language of geometry helps students
to reason deductively and to think interdependently. “It is written in the language of
mathematics, and its characters are triangles, circles, and other geometrical figures without
which it is humanly impossible to understand a single word of it; without these, one wanders
about in a dark labyrinth (Usiskin, 1996, p.231). Today, the language of geometry is used
without its structure and grammar, and thus it is still a foreign language to many teachers (Pimm,
1987; Usiskin, 1996).
The National Council of Teachers of Mathematics (NCTM, 2000) Principles and
Standards for School Mathematics (PSSM) recommended that students should “analyze

1

characteristics and properties of two- and three dimensional geometric shapes and develop
mathematical arguments about geometric relationships” (p.41). For instance, in the Geometry
Standards for grades 3 to 5 students, it is recommended that all students should identify, compare
and analyze polygons, and develop vocabulary to describe their attributes, as well as to classify
polygons according to their properties, and to develop definitions of classes of shapes. Because
students are expected to learn about geometrical concepts and attributes, as well as relationships
between them, it is important for future teachers to know and be comfortable with the language
of geometry.
In prior research and literature on students’ learning of geometry, and in literature
emphasizing the use of van Hiele theory to categorize students’ levels of thinking, many studies
address the complexity and difficulty of students’ learning of geometry, as well as other
educational and psychological concerns. Based on what others have studied about prospective
teachers’ learning in geometry and their geometric thinking, this study is guided by this
overarching question: What do prospective teachers learn in geometry from their preparation for
the work of teaching geometry?
The mathematics education community has always been interested in the teaching and
learning of mathematics, and we became more aware of the importance of human interaction in
the classrooms, and how it influences the effectiveness of teaching and learning. The notion of
mathematics as discourse and students as being apprenticed into particular ways of doing
mathematics in particular discursive contexts is now gaining prominence in mathematics
education research. This phenomenon prompted the call for the study of teachers’ knowledge in
geometry and of their learning of geometry. This study is informed by prior research and
literature on van Hiele theory (van Hiele, 1959/1985), a framework that describes students’

2

levels of geometric thinking, and by studies using van Hiele theory in the context of
methodology and teacher knowledge in geometry. Additionally, this study is informed by
research in the past that investigates students’ mathematics discourses in their discursive
learning.
While previous work sheds light on prospective teachers’ knowledge and thinking in
geometry, it has not explored how examination of these teachers’ geometric discourses could
help in learning more about their levels of geometric thinking. This study, influenced by the
discursive nature of van Hiele theory, and of discourse analysis in the form of the Commognition
framework described in Thinking as Communicating: Human Development, the Growth of
Discourses, and Mathematizing (Sfard, 2008), seeks to examine prospective teachers’ knowledge
in geometry, and to investigate as well their ways of communicating geometric thinking. The
study revisits van Hiele levels with careful examination of key mathematical features at each
level. These mathematical features include (1) use of mathematical words, (2) use of visual
mediators in the form of geometric figures and their parts, and symbolic artifacts created for the
purpose of communicating about geometry, (3) endorsed narratives such as mathematical
propositions, axioms and definitions, and (4) mathematical routine procedures with which
participants implement well-defined types of tasks. The discursive framework provides a new
lens to investigate students’ geometric thinking.
In Chapter 2, I position this study among studies addressing the teaching and learning of
geometry in mathematics education in general, and studies that examine students’ thinking in
geometry using van Hiele theory, as well as studies emphasizing discursive learning. In addition,
I describe Sfard’s discursive framework in detail, including a description of each of its four key
mathematical features and important phenomenon highlighted in this framework. Chapter 3

3

describes the methodology of the study, including descriptions of van Hiele Geometry Test
instruments, interview tasks and an outline of the design of the study. Chapter 4 contains the
results of the analyses conducted in this study along with interpretations of findings. These
include the van Hiele Geometry pretest and posttest results and analyses of the whole group, and
participants’ in-depth interview results and analyses. Finally, Chapter 5 provides a discussion of
the findings, and Chapter 6 summarizes the study’s contributions to the field, its limitations, and
suggestions for future research.

4

CHAPTER TWO: THEORETICAL BACKGROUND
Review of Relevant Literature
The upcoming sections detail the theoretical framework that will be used in the proposed
study, and present reviews of relevant literature. The first section describes the van Hiele theory
and then summarizes studies guided by the theory in the learning of geometry. In addition, this
section summarizes research that addresses the knowledge of mathematics for teaching,
specifically that related to geometry. The second section describes the commognitive framework
related to discursive learning of mathematics. Included in this section are summaries of studies in
discourse in the mathematics classroom, and the theoretical model of the development of
geometric discourses that aligns the van Hiele theory through the discursive lens. The final
section raises general research questions in discursive terms.
Regarding the teaching and learning of geometry, the van Hieles developed this
influential theory of levels of geometric thinking. In discussing the profound impact of Pierre
van Hiele’s theory in mathematics education, Clements (2003) concludes, “van Hiele theory
gave educators and researchers a model that promoted the understanding of important,
conceptual based level of thinking… It is also a model of synergistic connections among theory,
research, the practice of teaching, and students’ thinking and learning”(p.151). To better describe
the van Hiele Theory and how it has been used in the field of mathematics education, the
following section provides the historical background and a general description of the theory.
The van Hiele Theory
The root of van Hiele theory emanated from the task of improving the teaching of
geometry. A Dutch husband and wife, Pierre Marie van Hiele and Dina van Hiele-Geldof,
developed “the van Hiele Theory” in their doctoral dissertations at the University of Utrecht,

5

Netherlands, in 1957. Dina died shortly after completing her dissertation, and Pierre continued to
develop and disseminate the theory (e.g., van Hiele, 1959/1985, 1986).
When Pierre and Dina worked at Montessori secondary schools as mathematics teachers,
they were very disappointed with “students’ low-level knowledge of geometry”(p.60,
1959/1985). On the other hand, they also realized that teachers and students often fail to
communicate with each other because they “speak a very different language” (p.61). For
example, one of Pierre and Dina’s initial observations was that they seemed to speak about
geometry in a different way than their students. When Pierre and Dina spoke about a square as a
type of rectangle, students were confused because to them a square and a rectangle were quite
different. This led Pierre and Dina to consider the existence of various levels of geometric
thinking and the possibility that those students and teachers at different levels of thinking may
have difficulty communicating with one another. Although Pierre and Dina developed the theory
together, their views were quite different. As a result, Pierre’s dissertation focused on identifying
students’ levels of thinking in learning geometry, while Dina’s dissertation was more about a
teaching experiment designed to investigate how students move from level to level.
The van Hiele theory includes five distinct levels that describe students’ thought levels in
the learning of geometry. However, P. M. van Hiele suggested that mathematics educators
should focus on the first four van Hiele levels, because those are what teachers have to deal with
in school most of the times (van Hiele, 1986). As P. M. van Hiele (1959/1985) described in “the
Children’s thought and geometry”, the five van Hiele levels are as follows (p.62-63): Base Level,
figures are judged by their appearance; First Level, figures are bearers of their properties, and
they are recognized by their properties but not yet ordered; Second Level, properties are ordered,
and they are deduced one from another; at this level, definitions of figure come into play but

6

students did not understand the meaning of deduction; Third Level, thinking is concerned with
the meaning of deduction, with the converse of a theorem, with axioms, with necessary and
sufficient conditions; Fourth Level, thinking is concerned with a variety of axiomatic systems
that are non-Euclidean. Geometry is seen in the abstract.
As described in the levels, students’ levels of thinking attached to the learning of a
particular geometric topic are inductive in nature. At level n-1 certain geometric objects are
studied. Students are able to state some of the relationships explicitly about the objects. At level
n the objects studied are now the statements that were explicitly made at level n-1 as well as
explicit statements that were only implicit at level n-1. Therefore, the objects at level n consist of
extensions of the objects at level n-1. One major purpose of distinguishing the levels is to
recognize obstacles that are presented to students. For example, when a student who is thinking
at level n-1 confronts a problem that requires vocabulary, concepts or thinking at level n, the
student is unable to make progress on the problem, with expected consequences such as
frustration, anxiety and even anger.
The van Hiele levels have several important properties: (1) The levels are discrete and
sequential. Discrete indicates that the levels are qualitatively different from one another.
Sequential refers to the fact that students pass through the levels in the same order, although
varying at different rates, and it is not possible to skip levels. (2) That which was intrinsic at one
level becomes extrinsic at the next level. For example, students operating at Level 1 are able to
name geometric figures only by their appearance as a “whole” – the properties of a figure remain
intrinsic. However, at Level 2, these properties become extrinsic and in fact are the new objects
of study. (3) Each level has its own language and symbols. Van Hiele believed that “In general,
the teacher and the student speak a very different language” (van Hiele, 1986, p. 62). Therefore,

7

teachers and students often have difficulty communicating with one another about geometric
concepts. This linguistic challenge can also extend to communicational difficulties between
students in a classroom when they are functioning at different thought levels. (4) Instructional
methods have a greater influence than either age or grade on a student’s progress through the van
Hiele levels. That is, a teacher’s instructional activities can either foster or impede movement
through the levels.
When assigning students to different van Hiele levels, P. M. van Hiele cautioned that it is
possible to misjudge a student’s level of thinking without careful analysis, because often students
memorize or learn patterns in order to accomplish tasks, but do not really understand the
underlying concepts. An example is when students recognize corresponding angles by finding
the ‘F’ that is formed by parallel lines and the transversal. See Figure 2.1 below.

Figure 2.1

Corresponding angles of parallel lines intersected by a transversal.

This technique simplifies the relation between angles and lines. P. M. van Hiele claimed
that it could be harmful to students if they only seek a quick result and avoid the ‘crisis of
thinking’. In saying ‘crisis of thinking’, P. M. van Hiele meant the difficulties that students need
to transit from one level to a higher level. It is possible for students to derive the answer without
recognizing the relationships between the angles in the figure (e.g., supplementary angles, angles

8

at a point, interior angles). Van Hiele warned that these types of “tricks” might actually prevent
students from moving to the subsequent level of reasoning (van Hiele, 1986, p.42).
The van Hiele theory recognizes the importance of language, which plays a significant
role in communication. According to P. M. van Hiele, students’ levels of thinking are important
not in the sense of the way of their thinking, but in the results of thinking that are revealed in
students’ speaking and writing. For example, the meaning of a statement like, “This figure is a
rhombus.” depends on how one argues about it. For a student who is at Basic level, her answer
could be, “This figure has a shape that looks like what I learned to call ‘rhombus’.” In contrast, if
another student has already obtained the first van Hiele level or higher, her argument could be
quite different. The figure that the student refers to is a collection of properties and those
properties he/she has learned to call “rhombus” (van Hiele, 1986, p.109). By making the same
statement, “This figure is a rhombus”, one could use very different reasoning, and from a very
different level of thinking. This example of students’ responses to a rhombus illustrates how
geometric language can vary among levels.
The van Hiele theory has been influential and extensively studied. In the next section, my
review of the existing literatures focuses on how the van Hiele theory has been used in research
in the years since the theory was developed.
Research Guided by van Hiele Theory
The van Hiele theory was introduced to the United States by the Russian mathematician
Izaak Wirszup in a lecture entitled “Some Breakthroughs in the Psychology of Learning and
Teaching Geometry” at the Closing General Sessions of the National Council of Teachers of
Mathematics in 1974, following its incorporation into a new Soviet geometry curriculum
(Wirszup, 1976; van Hiele, 1959/1985, 1986). After the van Hiele levels were translated into

9

English, they were widely used by many researchers in the United States. During the period of
1980-83, the National Science Foundation funded three major investigations of van Hiele levels
in the United States: one directed by Burger and Shaughnessy at Oregon State University,
another by Fuys, Geddes, and Tischler at Brooklyn College, and a third by Usiskin at the
University of Chicago. Burger and Shaughnessy set out a study using clinical interviews to
determine the usefulness of van Hiele levels for describing children’s geometric thinking in
elementary, middle, and high school grades. Fuys et al. focused their investigation on geometric
thinking in adolescents using instructional models. Usiskin’s project used a large-scale survey to
test whether the van Hiele theory applied to the geometric reasoning of students enrolled in
secondary geometry courses. These three intensive studies have been widely read, discussed, and
cited. After these studies, dozens of other studies using the work of the van Hieles have been
conducted in the United States (e.g., see Mayberry, 1983; Crowley, 1987; Senk, 1983, 1989).
Internationally, Micheal de Villiers in the Netherlands and later in South Africa used van Hiele
theory to develop geometry curricula (de Villiers, 1996), whereas Angel Gutierrez and Adele
Jaime and their students in Spain, and John Pegg and his students in Australia used the theory to
study students’ learning in geometry (e.g., Gutierrez, 1996; Gutierrez, Jaime, & Fortuny, 1991;
Gutierrez, Pegg, & Lawrie, 2000).
In the earlier writing of the van Hieles, the van Hiele levels of geometric thinking mainly
refer to the classification of figures (van Hiele, 1959/1985). At that time, levels were descriptors
and they were not labeled by single words (e.g., “Visual” for Level 1, etc.). Almost thirty years
later, van Hiele (1986) referred to the five levels of thinking as visual, descriptive, theoretical,
formal logic and rigor, and considered such classification to be suitable to a structure of
mathematics (p.53).

10

Over the years, researchers not only used the levels to study students’ levels of geometric
thinking, but also expanded the area of research from classification of the quadrilaterals to the
classification of similar figures, to reasoning and proof, to spatial geometry in three dimension
measurement, etc. In these studies, researchers have proposed various descriptive labels for the
van Hiele levels. As the first to name the van Hiele levels, Hoffer (1981) provided his
descriptors, “levels of mental development in geometry” (p.13), which label Levels 1 through 5
as recognition, analysis, ordering, deduction, and rigor (p.13-14). Besides these five levels,
Hoffer also suggested five basic skills that are expected at each level. These five skills are visual
skills, verbal skills, drawing skills, logical skills, and applied skills. For instance, at Level 1
(recognition), the visual skills only focus on recognizing different figures from a picture, or on
recognizing information labeled on a figure. At Level 2 (analysis), visual skills are developed to
notice properties of a figure as well as to identify a figure as a part of a larger figure. At Level 3
(ordering), visual skills help to recognize interrelationships between different types of figures
and common properties of different types of figures. At Level 4 (deduction), visual skills focus
on using information about a figure to deduce more information. Finally, at Level 5 (rigor),
visual skills are used to recognize unjustified assumptions made by using figures (p.15). Hoffer’s
descriptors suggested that various geometric skills might be expected of students at different
levels of their development in geometry.
Other “level indicators”, suggested by Burger and Shaughnessy (1986), describe the five
levels as visualization, analysis, informal deduction, formal deduction, and rigor, for Levels 0
through 4, respectively. Using Burger and Shaughnessy’s level indicators, Crowley (1987)
provided additional examples of level-specific responses (except for Level 5), concerning how
students would argue a given shape is a rectangle (p. l5).

11

Level 1

“It looks like one.” or “Because it looks like a door.”

Level 2

“Four sides, closed, two long sides, two shorter sides, opposite
sides parallel, four right angles …”

Level 3

“It is a parallelogram with right angles.”

Level 4

“This can be proved if I know this figure is a parallelogram and
that one angle is a right angle.”

Each response assigns to a level. The student at Level 1 gives answers based on a visual
model and is identifying the rectangle by its overall appearance. At Level 2, the student is aware
that the rectangle has properties; however, redundancies (i.e., properties that can be derived from
other properties) are not noticed. A student operating at Level 3 will attempt to give a minimum
number of properties (i.e., a definition), and finally, at Level 4, a student will seek to prove the
fact deductively.
More recently, Battista (2009) elaborates and refines the van Hiele levels with regard to
students’ geometric reasoning. The descriptors of the levels he suggests are visual-holistic
reasoning, descriptive-analytic reasoning, relational-inferential reasoning, formal deductive
proof, and rigor (p.92-94), referring to Levels 1 through 5, respectively. For instance, at Level
1(visual-holistic reasoning), students argue that a square is not a rectangle because a rectangle is
“long”; or claim that two figures have the “same shape” because they “look the same”(p.92). At
this level, students’ justifications of an argument are vague and holistic. At Level 2 (descriptiveanalytic reasoning), students would assert that a square is a rectangle because “it has opposite
sides equal and four right angles.” At this level, students are able to explicitly specify shapes by
their parts and spatial relationships among the parts; however they describe parts and properties
informally and imprecisely using strictly informal language learned from everyday life. At Level

12

3 (relational-inferential reasoning), students start with empirical inference to reason that if a
quadrilateral has four right angles (and this is a rectangle), its opposite sides have to be equal
because by drawing a rectangle with a sequence of perpendiculars, they cannot make the
opposite sides unequal; and then they use logical inference to recognize the classifications of
shapes into a logical hierarchy (p.94).
These descriptors not only provide detailed information about how researchers identify
students’ levels of geometric thinking, but more importantly shed light on the geometric
reasoning and language skills that students need to develop at each van Hiele level. When
conducting studies using van Hiele theory, some researchers use clinical interviews, while others
prefer open-ended survey tests. Among all the van Hiele studies, Usiskin’s Cognitive
Development and Achievement in Secondary School Geometry (CDASSG) project, and Burger
and Shuaghnessy’s “Oregon Project” are two of the most frequently used and cited. In the
following section I summarize the methods used in the van Hiele studies, as well as some
important findings, beginning with these two projects that influence the teaching and learning of
geometry.
Now let me move on to the research methods used in the van Hiele studies. The Usiskin
CDASSG project (Usiskin, 1982; Senk, 1983) used a standard pretest and posttest, involving
four tests, to assess 2699 students in full year geometry classes from 13 public high schools in
five states. The four tests were: Entering Geometry Test, van Hiele Level Test, Comprehensive
Assessment Program Geometry Test and Proof Test. The pretest (i.e., Entering Geometry Test
and Van Hiele Level Test) was conducted during the first week of school, and the posttest (i.e.,
van Hiele Level Test, Comprehensive Assessment Program Geometry Test and Proof Test) was
scheduled three to five weeks before the end of the school year.

13

The van Hiele Geometry Test was designed to predict students’ van Hiele levels at the
beginning and the end of the school year. This test consists of 25 multiple-choice items, with 5
foils per item and 5 items per level, and was designed to capture the key thinking processes
characteristic of each van Hiele level. In order to develop a rigorous test instrument that
describes van Hiele levels in sufficient detail, researchers in the CDASSG project first reviewed
nine original works of the van Hieles, including four originally written in English and five
translated into English from Dutch, German or French. They compiled all the quotes from the
van Hieles’ writings (see appendix B) that describe behaviors of students at a given level. As an
example of the quotes, the following is a selected list of Level 1 behaviors that Usiskin (1982)
provided in the CDASSG project report:
Level 1 (their base level, level 0)
1. “Figures are judged according to their appearance.”
2. “A child recognizes a rectangle by its form, shape”
3. “The rectangle seems different to him from a square.”
4. “A child does not recognize a parallelogram in a rhombus.”
5. “A student was able to produce these figures without error…”
The van Hiele Geometry Test (see Appendix B) instruments were based on the
descriptions of students’ behaviors at each given level. For example, Items 1-3 were derived
from quote number one; Item 4 was derived from quote number eight and Item 5 was derived
from quote number six. Figure 2.2 presents one van Hiele Level 1 item and its corresponding van
Hieles’ quotes.

14

Question 4: Which of these are squares?

H

G

F

I

(A) None of these are squares.
(B) G only
(C) F and G only
(D) G and I only
(E) All are squares.
Figure 2.2

An example of a Level 1 test item with its corresponding van Hiele quotes.

To grade students’ responses to the van Hiele Geometry Test, the project used the 3 of 5
criterion (3 out of 5 correct) and 4 of 5 criterion (4 out of 5 correct), and compared the two
criterions using the analyses of Type I and Type II error. The statistical analysis showed that,
depending on whether one wishes to reduce Type I or Type II error, the 3 of 5 criterion
minimizes the chance of missing a student and yields an optimistic picture of students’ levels,
whereas the 4 of 5 criterion minimizes the chance of a student being at a level by guessing (see
Usiskin, 1982). Based on students’ test responses, the students were assigned a weighted sum
score according to the following:
1 point for meeting criterion on items 1-5 (Level 1)
2 points for meeting criterion on items 6-10 (Level 2)
4 points for meeting criterion on items 11-15 (Level 3)
8 points for meeting criterion on items 16-20 (Level 4)
16 points for meeting criterion on items 21-26 (Level 5) (Usiskin, 1982, p.22)
15

The points were added to give the weighted sum, and the weighted sums were calculated
to allow a person to determine upon which levels the criterion has been reached from the
weighted sum alone. For example, a score of 19 points indicates that the student has reached the
criterion on Levels 1 (1 point), Level 2 (2 points) and Level 5 (16 points). The assigning of
levels, however, was as follows: If a student met the criterion for passing each level up to and
including level n and failed to meet the criterion for all levels above, then the student was
assigned to level n; if the student could not be assigned to any level, then that student was not
said to fit. Thus a student with a weighted sum of 1+2+16 =19 would satisfy the criterion at
Level 1, Level 2 and Level 5 and was assigned to van Hiele Level 2 (p.25). The CDASSG
project used Hoffer’s (1981) descriptors, labeling the levels as recognition, analysis, ordering,
deduction and rigor, from Levels 1 to 5. Additionally, the project reported results using both the
classical theory (i.e., all five van Hiele levels are considered) and the modified theory (i.e., Level
5 is excluded from consideration) to classify students into van Hiele levels (see Usiskin, 1982).
This large-scale research study showed that nearly 40% of students in the United States
finish high school functioning below van Hiele Level 2 (Analysis). Students entering high school
geometry courses with higher van Hiele levels, such as Level 2 or Level 3 (Ordering), were more
likely to succeed in writing proofs by the end of the school year (Senk, 1983, 1989). Of those
studied, students who entered geometry courses functioning at van Hiele Level 1 had a 30%
chance of success in proof writing. Entering geometry at Level 2 provided students with a 56%
chance of success at proof writing, and all students entering at Level 3 experienced success at
proof writing by the end of the school year. These results show that high school students’
achievements in writing proofs are positively related to van Hiele levels of geometric thinking
and to achievement on standard non-proof geometry content (p.318). The study also concluded,

16

“In the form given by the van Hieles, Level 5 either does not exist or is not testable. All other
levels are testable”(Usiskin, 1982, p.79).
It is helpful to use these data to determine the initial status of students’ geometric
backgrounds and to assess their progress. However, there have been questions and doubts about
the feasibility of measuring reasoning by means of items, and about the internal consistency of
the items (Crawley, 1990; Wilson, 1990). Also, one might question what information might be
missed in a paper-pencil test, and how the details of students’ thinking processes might be better
detected. Nevertheless, the main advantage of this method is that it can be administered to many
indiviudals, and it is easy and quick to distinguish between the thought levels of students.
In contrast, Burger and Shaughnessy’s Oregon project (Burger & Shaugnessy, 1986) used
clinical interviews to determine students’ van Hiele levels. They interviewed 45 students from 5
school districts in 3 states, ranging from elementary to middle to high school. The interviews
consisted of eight tasks focusing on geometric shapes, and those tasks were designed to reflect
the descriptions of the van Hiele levels.
The design of the interview tasks involved drawing shapes, identifying and defining
shapes, sorting shapes (e.g., triangles and quadrilaterals); and the interview protocols were
designed to engage participants in both informal and formal reasoning about geometric shapes.
Six of the eight tasks, focusing on drawing, identifying, and sorting, were expected to elicit the
characterizations of van Hiele Levels 0-2 from the protocols. To give an example of the design,
Figure 2.3 shows two tasks that were used, Identifying and defining (2.3a) and Sorting (2.3b).
Identifying and Defining
Students were given a sheet of quadrilaterals (Figure 2.3a), and they were asked to write
an S on each square and R on each rectangle, and if the student was familiar with the terms, a P

17

on each parallelogram and a B on each rhombus. During the interviews, students were asked to
justify their marking. In the defining part of the activity, the student was asked, “What would
you tell someone to look for in order to pick out all the rectangles on a sheet of figures?” Or, an
equivalent question was asked, “Could you make a shorter list? Is No. 2 a rectangle? Is No. 9 a
parallelogram?”(p.34).

2.3a. Quadrilateral to be identified
Figure 2.3

2.3b. Triangles to be sorted

Two experimental tasks from the Oregon Project

Sorting
A set of cut out triangles was spread out on the table (Figure 2.3b). The student was
asked, “ Can you put some of these together that are alike in some way? How are they alike? Can
you put some together that are alike in a different way? How are they alike?” (p.34) This line of

18

questioning was continued as long as the student could come up with new sorting strategies.
These activities sought to explore the student’s definitions and class inclusions.
The project collected and analyzed students’ original written works during the interviews,
and the dialogs between interviewers and the students were analyzed and documented as well.
For example, on the Drawing Triangles task, interviewees were asked to draw “different”
triangles. Based on the interviewees’ drawings during the interviews, Burger and Shaughnessy
found that for Bud, a 5th grade student, “different triangles” meant triangles in different
orientations or positions only. In contrast, for Amy, an 8th grade student, “different triangles”
meant having different angle measures and sizes, and for Don, a 10th grade student, “different
triangles” meant different types of triangles. Figure 2.4 shows the drawings from Bud, Amy and
Don.

Bud

Amy
Figure 2.4

Don

Bud, Amy and Don’s drawings of different triangles.

Recall that the “level indicators” developed by Burger and Shaughnessy (1986) describe
Levels 0 through 4, respectively, as visualization, analysis, informal deduction, formal
deduction, and rigor. Pursuant to students’ responses during the interviews, it turned out that,
even though all three students were to reason about what is meant by “different triangles,” and

19

could provide drawings, all three students were later assigned to three different van Hiele levels:
Bud (Level 0), Amy (Level 1) and Don (Level 2). This example illustrates use of the “same
language” but very different reasoning. Burger and Shaughnessy also documented the original
scripts in which questions were asked for interviewees to complete the task. For example, for the
activity “Drawing Triangles”, interviewees were asked to draw a triangle (called No.1), and then
another triangle (called No.2) that is different in some way from the first one. After the
interviewee had done so, he/she was asked to draw a third triangle that is different from the first
two triangles, and so on. Later, the interviewees were asked questions such as “How is #2
different from #1?” and “How would they be all different from each other?”(p.37).
Burger and Shaughnessy’s project confirmed the hierarchy nature of the levels. They also
found that age is not significantly related to the levels. However, the reviewers of the project had
disagreements and experienced some difficulties of assigning a level to students who appeared to
be in the transition between Levels 0 and 1(p.42).
The interviewees’ written works on one hand, and their verbal responses to the questions
on the other hand, combined to increase the reliability of the data and provide strong evidence on
how the data were analyzed and interpreted by researchers. The great advantage of clinical
interviews is that the information obtained from the interviews results in a deeper knowledge of
the ways students reason. However, this study is clinical with a small sample of students
representing a very broad range of ages (Kindergarten to College). This kind of research is timeconsuming and is unsuitable for assessing many people.
The review of the methods used in van Hiele studies influenced the design of the
proposed study. The van Hiele Geometry Test, used to distinguish students’ van Hiele levels, is
effective in getting initial information about students’ levels of thinking, and the CDSSAG

20

project showed that it is a well-tested and designed test instrument. The Oregon project, on the
other hand, gives an example of how clinical interviews could well detect students’ thinking
processes when engaged in informal and formal reasoning about basic geometric shapes.
There are many other studies using the van Hiele theory that pertain to the learning and
teaching of geometry. One such study by Fuys et al. (1988) focused on clinical interviews with
sequences of instructions known as “Instructional Module Activities” (p.11). In this project, all
subjects were interviewed individually in six to eight 45-minutes sessions as they worked with an
interviewer on the Instructional Modules. The subjects were selected to reflect the diversity of
sixth-grade students from New York City public schools. To categorize the subjects’ levels of
thinking, the interviews focused on their progress (or lack of it) within the levels or to higher
levels, and on learning difficulties as well (p.78). This project was designed to investigate
whether or not instructional modules would help subjects move through the levels. Fuys et al.
(1988) also documented the dialogues between interviewers and subjects. For example, in the
assessment of subjects’ understanding of the exterior angle of a triangle, subjects were given an
open question of finding a possible relationship among the three angles indicated in Figure 2.5.

b
c

a
Figure 2.5

An exterior angle of a triangle.

During the interviews, the interviewer gave several prompts to the subjects such as: “ Is
any part of angle c related to angle a or angle b?” With the help of the interviewer, the subjects

21

would sometimes successfully complete the argument of a relationship between two interior
angles (angles a and b) and their exterior angle (angle c) of the triangle.
In addition to the clinical interviews, Fuys et al. (1988) also analyzed geometric content
of three widely used K-8 textbook series with regard to the van Hiele levels. They reported that
no more than 2% of the lessons contained content that require geometric thinking at Level 3
(formal deduction), and all of Level 3 lessons appeared at Grades 7 and 8. The remaining 98%
represented van Hiele Levels 0, 1, and 2. Analyzing their findings, Fuys et al. concluded that “
average students do not need to think above Level 0 (visual) for almost all of their geometry
experience through grade 8” (p.169). Not surprisingly, the overall results for the van Hiele levels
of students in the United States were discouraging. In their study of the geometric reasoning of
sixth and ninth grade students, Fuy et al. (1988) found the following: 19% of sixth graders
performed consistently at Level 0 (visual), 31% performed sometimes at Level 1 (analysis) and
sometimes at Level 2 (informal deduction), and the remaining 50% performed sometimes at
Level 2 and sometimes at Level 3 (formal deduction). The ninth graders’ corresponding
percentages were 12%, 44%, and 44%.
More recently, Newton (2010) used van Hiele levels to analyze K-8 geometry state
standards. Of the 5,710 Grade Level Expectations (GLEs) contained in the K-8 Geometry and
Measurement strands of 42 states, 1,667 GLEs (approximately 29 %) were labeled as descriptive
geometry. The analysis of the descriptive geometry GLEs indicated that approximately 47% of
the GLEs are Level 1(visualization), 49% are Level 2 (analysis), and 4% are Level 3 (informal
deduction). According to Newton, the absence of Level 3 GLEs in more than 40% of the states
and the near absence in the remaining 60% represent the most compelling result of the analyses,
since formal deduction (Level 4) is generally expected in high school geometry courses.

22

The van Hiele theory has informed and shaped the improvement of the geometry
curriculum (Wirszup, 1976; de Villers, 1999). For example, de Villiers cautioned “no amount of
effort and fancy teaching methods at the secondary school will be successful, unless we embark
on a major revision of the primary school geometry curriculum along van Hiele lines”. In 1999,
Clements et al. encouraged the van Hiele level’s use in guiding curriculum development, and
suggested that “helping children move through these levels may be taken as a critical educational
goal” (p. 193). The following year, in Principles and Standards in School Mathematics, the
NCTM cited van Hiele and others who have studied his theory to develop the importance of “…
building understanding in geometry across the grades, from informal to more formal thinking”
(2000, p. 40).
Knowledge of Geometry for Teaching
“Mathematical knowledge for teaching means the mathematical knowledge used to carry
out the work of teaching mathematics” (Hill, Rowan & Ball, 2005, p.373). When suggesting
what it means to know mathematics for teaching, Ball, Hill and Bass (2005) argue that teaching
involves knowledge of mathematical ideas, mathematics reasoning skills, and communication,
fluency with examples and terms, and thoughtfulness about the nature of mathematical
proficiency. For instance, additional mathematical insight and understanding are required to
explain, listen, and examine students’ work, and more mathematical analysis is required when
correcting students’ errors. In addition to mathematical knowledge for teaching, Ball et al
address the need for teachers to have a specialized fluency with mathematical language, and to
know what counts as a mathematical explanation.
In this section, I summarize research that emphasizes prospective teachers’ knowledge of
geometry. Mayberry (1983) investigated nineteen undergraduate prospective teachers’ geometric

23

understanding when encountering seven geometry concepts: squares, right triangles, isosceles
triangles, circles, parallel lines, similarity and congruence, all common topics to elementary
school textbooks. The study found that two students had difficulty in recognizing a square with a
nonstandard orientation (Basic level), while the properties of figures were often not perceived
(Level I). For example, when asked, “does a right triangle have a longest side?” (p.60), twelve
students responded that they did not think that such a triangle had to have a longest side. With
regard to Level II, the study concluded that class inclusions, relationships, and implications were
not perceived by many of the students, because they answer the questions for particular figures
rather than generalized ones. Responses to questions about congruence (Level III) show that
fifteen out of the nineteen prospective teachers believed that two right triangles with tencentimeter hypotenuses are always congruent. Also, ten of nineteen assert that two circles with
ten-centimeter chords are always congruent (Table 2.1).
Table 2.1

Responses to Relation Questions about Congruency
Always

Sometimes

Never

Don’t know

A square and a triangle

0

1

17

1

Two squares with 10-cm sides

16

2

0

1

Two right triangles with 10-cm hypotenuses

15

3

0

1

Two circles with 10-cm chords

10

8

0

1

Two similar triangles

3

11

3

2

Figure. “ Are these Congruent?”

Findings suggested that the prospective teachers in the study did not yet perceive some of
the properties of basic geometric shapes, and they did not perceive a proof as a logical chain
leading from the “given” to the “conclusion.”
Hershkowitz, Bruckheimer and Vinner conducted in Israel a study of 518 students
(grades 5-8), 142 prospective teachers, and 25 in-service teachers. Findings showed that
24

“teachers have patterns of misconceptions similar to those of students in grades 5-8” (1987,
p.222). More specifically, they assessed 142 prospective elementary teachers’ understanding of
geometry concepts in the context of angles, altitudes of triangles, and diagonals of polygons. For
example, one of the tasks was to assess the understanding of the angle concept by recognizing
the drawing of an angle on a sheet of paper. Results suggested that sixty-eight percent of the
prospective teachers had a proper understanding of the concept of angle (p.223). After assessing
prospective teachers’ understanding of the diagonals of polygons, Hershkowitz, Bruckheimer
and Vinner (1987) found that most of the prospective teachers only drew diagonals that were
inside the polygons and rejected the possibility of an exterior diagonal (see Figure 2.6). This
result suggests that most prospective teachers did not use definitions as their primary tool when
working with these tasks. They tended to use their own individual image of a given concept (e.g,
concept of diagonals of a polygon). These prospective teachers’ individual concept images were
misleading in the case of the diagonals of concave polygons.
l

A

m
A

Figure 2.6

The “diagonals” of concave polygons

Gutierrez, Jaime, and Fornruny (1991) conducted a study to evaluate 32 prospective
teachers’ spatial reasoning in three-dimensional geometry. Nine items of the Spatial Geometry
Test were grouped into five activities. These activities were designed to elicit prospective
teachers’ conceptual understanding of three-dimensional figures, by paying attention either to

25

visual qualities or to the properties of basic geometric shapes such as squares and parallelograms.
Activities that asked prospective teachers to select solids, based on given properties, focused on
how prospective teachers use definitions and properties to identify the polyhedron from a given
set of solids. With regard to the question involving writing the differences and similarities
between a cube and the Solid I (see Figure 2.7), this activity focused on observation and
manipulation of the polyhedran. In response to this question, for instance, one prospective
teacher argued, “In both solids [a cube and Solid I] the faces are parallelograms and both have
six faces. And the differences were, the angles in solid I are not right” (p.246). Another
prospective teacher argued, “Solid I and a cube were alike because both solids have parallel faces
and all edges are the same, and they differed because they don’t have the same shape”(p. 246).

Cube
Figure 2.7

Solid 1
A cube (left) and Solid I (right)

These two responses both reasoned about the differences and similarities between a cube
and a solid, but the arguments provided by the two prospective teachers were quite different. The
first response focused on the properties of the geometric figures; whereas the latter response
mainly paid attention to the visual qualities of figures.
Based on Vinner and Hershkowitz’s (1987) notion of concept image, Gutierrez and Jaime
(1999) conducted a study of prospective elementary teachers’ understanding of the concept of
the altitude of a triangle. They analyzed 190 prospective teachers’ written tests, focusing on

26

concept images, difficulties, and errors related to the concept of the altitude of a triangle.
Analysis showed that there were more correct responses in the test containing the definition of
the altitude than in the test without the definition. This observation suggested that the definition
seemed to provide information helping these prospective teachers to improve their understanding
of the concept of altitude.
From the responses that prospective teachers provided in the study, we learned that there
was confusion and misunderstanding between the concepts of altitudes and medians of a triangle,
and the concepts of altitudes and perpendicular bisectors of the sides of a triangle. These
misunderstandings could be reasons why prospective teachers responded incorrectly to a partial
image that excludes external altitude (Figure 2.8a), and to a partial image that does not take into
account the length of the altitude (Figure 2.8b).
2.8a.

2.8b.

“For interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of the dissertation”
Figure 2.8
Responses regarding the altitude of a triangle

Another issue worth mentioning is the influence of the position of a triangle, as a
consequence of a different rotation of the figure. The easier item appeared to be the prototypical
position with a vertical altitude, which suggests that prospective teachers’ concept images of
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altitudes of triangles were limited to only certain types of triangles, and with certain orientations
of how triangles are positioned.
In the previous section I have presented examples from existing literatures regarding
prospective teachers’ understanding in the context of Euclidean geometry. The review included
teachers from Israel, Spain, and the United States. Some investigations were done through largescale studies; others were done through clinical interviews. These prospective teachers who were
tested all shared some common difficulties and weakness in their learning of geometry.
Theoretical Framework
According to the van Hieles, a learning process is also a process of learning a new
language, because “each level has its own linguistic symbols” (van Hiele, 1959/1985, p.4). The
van Hiele levels reveal the importance of language use, and emphasize that language is a critical
factor in the movement through the levels. The word “language” is not clearly defined in the
broad use of it (see van Hiele, 1986). Moschkovich (2007) argued that the language of
mathematics does not mean a list of vocabulary words or grammar rules, but rather the
communicative competence necessary and sufficient for competent participation in mathematical
discourse.
A real concept is an image of an objective thing in its complexity. Only when we
recognize the thing in all its connections and relations, only when this diversity is
synthesized in a word, in an integral image through the multitude of determinations, do
we develop a concept.

– Vygotsky

In this section, I summarize issues of language and discourse in mathematical learning
from existing literature. Sfard (2008) uses a discursive approach inspired by Vygotsky to make a

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distinction between language and discourse - language is a tool, whereas discourse is an activity
in which the tool is used or mediates. This perspective provides an understanding of mathematics
as a social and discursive accomplishment in which talk, gesture, diagrams, representations, and
objects play an important role; consequently, mathematics learning requires several modes of
communication (Sfard, 2002).
Researchers such as Ball (1993) elaborated the relationship between discourse, content
and community in their research to illustrate how these elements help students to develop what
Schoenfeld (1992) calls a “mathematics point view.” From this perspective, classroom
mathematical discourse is essentially a progress of establishing true claims about mathematics.
In her work, Lampert (1998) advances the case for classroom-based research to consider the
impact of language and discourse on mathematical learning. For example, Lampert illustrates
aspects of “mathematical talk” that includes position taking, question asking, proof and
justification, expanding ideas, use of evidence, conjectures, symbolic reference, and so on.
Kerslake’s (1991) focus on language in mathematics classrooms is an example of work
that attempts to examine the specific language of the content area, and helps to identify how the
use of language becomes a resource for understanding students’ misconceptions. For example,
based on the interviews with students, Kerslake found that students fail to conceive fractions as
numbers because they perceive them as “broken numbers” instead. Moreover, students tend to
rely on the everyday language of “sharing” to describe division, and Kerslake surmises that this
happens because sharing is likely to have been the students’ first experiences of dividing.
Kerslake suggested a closer look at how students think of and talk about fractions in the course
of learning.

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Moschkovich (2010) also acknowledges the significant role of discourse in learning. She
demonstrated this view, through an analysis of a third grade bilingual mathematics classroom, to
illustrate two features of mathematical discourse: situated meaning of words (utterances), and
focus of attention. She suggests that learning mathematics is a discursive activity that involves
participating in a community of practice using multiple materials, linguistic, and social
resources.
Many researchers have attempted to develop frameworks to examine students’ discourses
in learning mathematics. As an example, Sfard’s (2008) communicational approach to
mathematical learning provides a notion of mathematical discourse that distinguishes her
framework from others in several ways. In particular, Sfard (2002) argues that the knowing of
mathematics is synonymous with the ability to participate in mathematics discourse. From this
perspective, conceptualizing mathematical learning as the development of a discourse and
investigating learning means getting to know the ways in which children modify their discursive
actions in these three respects: “its vocabulary, the visual means with which the communication
is mediated, and the meta discursive rules that navigate the flow of communication and tacitly
tell the participants what kind of discursive moves would count as suitable for this particular
discourse, and which would be deemed inappropriate.” (Sfard, 2002, p.4) Therefore, Sfard’s
(2008) commognitive approach is grounded in the assumption that thinking is a form of
communication and that learning mathematics is learning to modify and extend one’s discourse.
Commognition
In Sfard’s (2008) Thinking as communicating: Human development, the growth of
discourses, and mathematizing, she defines discourse as “any act of communication made
distinct by its repertoire of admissible actions and the way these actions are paired with re-

30

actions” (Sfard, 2008, p. 297). A discourse is considered to be mathematical when it features
mathematical vocabulary that relates to numbers and shapes. Geometric discourse, a subcategory
of mathematical discourse, features mathematical vocabulary specifically relating to geometric
shapes, definitions and proofs, etc (p.245). Mathematical discourses are distinguishable by their
vocabularies, visual mediators, routines, and endorsed narratives.
From the commognitve point of view the development of discourse involves modifying
colloquial mathematical discourse into a more precise mathematical discourse, one that follows
meta-discursive rules. For example, in mathematics, geometric shapes are analytically classified
by their properties, not just by how they appear to us holistically. Thus, a stretched out triangle is
still a triangle even if it looks distorted. As long as it has three line segments joined at vertices, it
is a triangle; and because we count those segments and vertices, we engage in a linguistic act
(see Sfard, 2008). In Sfard’s terms, the mathematical discourse develops from colloquial
mathematical discourse; the colloquial mathematical discourse is an important starting point, and
to develop mathematical discourse requires a fundamental change in the discourse practices.
To identify whether a discourse is “mathematical”, four characteristics can be considered
as critical: word use, routines, visual mediators and endorsed narratives. Following is a very brief
description of each of these.
In a mathematical discourse, numbers words and comparison words (e.g., bigger, smaller)
will appear in the conversations discussing numbers and shapes. In this proposed study, the
geometric discourse deals with triangles and quadrilaterals, and their properties; therefore words
will appear such as “angles”, “sides”, “parallel”, “diagonals”, “congruent”, “same”, etc. and
sometimes these words will have multiple meanings depending on how the person uses them.
Word use is an all-important matter because, being tantamount to what others call word meaning,

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it is responsible for how the user sees the world, and it is one of the distinctive characteristics of
discourses (Sfard, 2005, p.245). In particular, a students’ word use reflects his/her levels of
mathematical thinking, which are crucial in this proposed study.
Visual mediators are objects that are operated upon as a part of the process of
communication. In colloquial discourses, visual mediators are images of material things existing
independently of the discourse; whereas in mathematical discourses, visual mediators are often
involved with symbolic artifacts, created specially for the sake of this particular form of
communication. Communication-related operations on visual mediators often become automated
and embodied. For example, “the procedures of scanning the mediator with one’s eyes… With
some experience, this procedure would be remembered, activated, and implemented in the direct
response to certain discursive prompts, as opposed to implementation that requires deliberate
decisions and the explicit recall of a verbal prescription for these operations” (Sfard, 2008,
p.134).
Narrative is defined as “a set of utterances, spoken or written, that is framed as a
description of objects, of relations between objects or processes with or by objects, and which is
subject to endorsement or rejection, that is, to being labeled as true or false” (Sfard, 2008,
p.300). Endorsed narratives are sets of propositions that are accepted and labeled as true by the
given community. In the case of geometric discourse, endorsed narratives are known as
mathematical theories, including definitions, proofs, axioms, and theorems. The statement “a
parallelogram is a quadrilateral with two pairs of parallel sides” is an endorsed narrative of
parallelogram, defining what a parallelogram is mathematically. Mathematical discourse is
conceived as one that should be impervious to any considerations other than purely deductive
relations between narratives. In this proposed study, the narratives will be those utterances

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produced by prospective teachers when identifying and classifying basic geometric shapes,
whereas the endorsed narratives will be the definitions of different quadrilaterals from textbooks
that prospective teachers encounter in their geometry class, and narratives that will be
constructed or endorsed by prospective teachers during the interview.
Routines are repetitive patterns characteristic of the given discourse. Specifically,
mathematical regularities can be noticed whether one is watching the use of mathematical words
and mediators, or follows the process of creating and substantiating narratives about numbers or
geometrical shapes. In fact, such repetitive patterns can be seen in almost any aspect of
mathematical discourses: in mathematical forms of categorizing, in mathematical modes of
attending to the environment, in the ways of viewing situations as “the same” or different, which
is crucial for the interlocutors’ ability to apply mathematical discourse whenever appropriate –
and the list is still long.
When someone is doing mathematics, or to be more specific, is engaging in a
mathematics task in geometry, patterns such as how one is carefully using mathematical words,
or how one is following certain steps when substantiating narratives about geometrical shapes,
can be observed. In fact, those repetitive patterns can be seen in almost any aspect of
mathematical discourses (Sfard, 2005; Sfard 2008). In this proposed study, when prospective
teachers identify and classify basic geometric figures, mathematical regularities will be noticed
through the conversations, to determine whether these prospective teachers are paying attention
to the use of mathematical words and following the process of creating and substantiating
narratives about geometrical shapes.
A Routine is defined as a set of meta-rules defining a discursive pattern that repeats itself
in certain types of situations, and the set can be divided into two subsets – the how of a routine,

33

and the when of a routine (Sfard, 2008, p.208): The how of a routine: a set of meta-rules that
determine the course of the patterned discursive performance (the course of routine, or
procedure); the when of a routine: a collection of meta-rules that determine those situations in
which the discussant would deem this performance as appropriate.
In this proposed study, “the how of a routine” will likely be observed (through
interviews), whereas “the when of a routine” will not be discovered because it requires
observations over a period of time (consistent observations over weeks, months or even years).
The Levels of Geometric Discourses
Many researchers have confirmed the usefulness of van Hiele theory when describing the
development of students’ geometry thinking. However the same researchers often find the van
Hiele theory lacking in depth with respect to the broad description of the levels, and they would
like a more detailed description of students’ levels of thinking. Recall that Hoffer’s (1981)
“Sample Skills and Problems” (p.11) provided a framework that connects the levels of
development with five basic skills (e.g., visual skills, verbal skills, drawing skills, etc) that are
expected at each van Hiele level. This work inspired me to consider the possibility of connecting
the van Hiele theory with a discursive framework, and to translate the van Hiele levels into
discursive terms. I claim that when a student’s geometric thinking develops to a higher level,
simultaneously there is a development of the student’s geometric discourse in discursive terms.
That is, the levels of geometric thinking can be viewed also as the levels of geometric discourses.
If so, the question is: “what additional information do the analyses of geometric discourse
provide about the student’s level of thinking?” To investigate the usefulness of discursive
framework, the study produced a model, on the basis of the theoretical understandings, that

34

describes each van Hiele level as a geometric discourse with respect to word use, routines, visual
mediators and endorsed narratives.
By “the theoretical understandings”, I mean the initial translation of the van Hiele levels
to discursive terms based on the quotes from the van Hieles’ writings, the same quotes used to
design the van Hiele Geometry Test in the CDASSG project. Taking the van Hieles’ descriptions
of students’ behaviors at each level, I analyzed them into the four characteristics described in the
Sfard framework; such translation illustrates student geometric discourse at each van Hiele level.
The translation for five van Hiele levels is shown in Tables 2.2 (Levels 1-3) and 2.3 (Levels 4-5).
Table 2.2

Discursive Translation of van Hiele Levels 1-3

Key terms

Characteristic
Geometric Discourse

Van Hieles’ description of Level 1. Figures are judged by their appearance.
Word use

Naming a figure is matching the figure with its name.

Routines

Direct recognition, a perceptual experience that is self-evident.

Endorsed
narratives

Descriptions of how one perceives. “This one (square) looks different from this
one (a rectangle)”.

Visual
mediators

2-D geometric shapes, the openness of angles, positions of the lines or physical
orientations of a figure are parts of the process of direct recognition.

Van Hieles’ description of Level 2. Figures are bearers of their properties.
Word use

Naming a figure is associated with its properties.

Routines

Direct recognition. Object level routines include checking, measuring and
comparing partial properties of figures.

Endorsed
narratives

Descriptions of visual properties. “Squares are not rectangles because they
have all sides equal, but rectangles are not”.

Visual
mediators

2-D geometric shapes, the openness of angles, positions of segments or
physical orientations of a figure are parts of the process of direct recognition
and identification of visual properties.

Van Hieles’ description of Level 3. Properties are ordered and are deduced one from another.
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Table 2.2 (cont’d)
Word use

Naming a figure signifies the realization of the figure regarding its endorsed
narratives.

Routines

Including routines at Level 2. Object level routines producing endorsed
narratives.

Endorsed
narratives

Descriptions of a definition of a figure and actions on a figure. “Rectangle is a
parallelogram having four right angles”.

Visual
mediators

Figures, lines and angles are parts of the process of identifying necessary and
sufficient condition of a definition.

Table 2.3

Discursive Translation of van Hiele Levels 4-5

Key terms

Characteristic
Geometric Discourse

Van Hieles’ description of Level 4. Thinking is concerned with the meaning of deduction.
Word use

Naming a figure signifies the realization of the figure regarding its endorsed
narratives and its connections with other figures.

Routines

Using abstract symbols. Abstract level of routines producing endorsed
narratives and making connections among them

Endorsed
narratives

Descriptions of abstract relations. Constructions of narratives using deductive
reasoning.

Visual
mediators

All level 3 visual mediators, plus abstract symbols, mathematical diagrams.

Van Hieles’ description of Level 5. Figures are bearers of their properties.
Word use

Naming a figure signifies the realization with its endorsed narratives and
connections with other figures in both Euclidean and non-Euclidean geometry.

Routines

Routines are connected with creativity.

Endorsed
narratives

Descriptions of abstract relations in both Euclidean and non-Euclidean
geometry.

Visual
mediators

Mathematical symbols and artifacts used in both Euclidean and non-Euclidean
geometry.

During this process of translating, the van Hieles’ quotes at each level were reviewed and
analyzed into possible characteristics of a mathematical discourse. For example, the quote No. 2,
“A child recognizes a rectangle by its form, shape”, provides information about how a child
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identifies a figure, what it calls a “rectangle”, based on its physical appearance. When this quote
is translated into discursive terms, the word, “rectangle” signifies a geometric shape (i.e., a shape
that we call a “rectangle”), thus the word use here is a name or a label of the figure. The phrase,
“recognizes… by its form, shape” suggests that the direct recognition triggers the decision
making, and therefore the routine procedure is a perceptual experience and is self-evident (e.g.,
[it is] a rectangle [because I see it] by its form, shape). Narratives are utterances, verbal or
written, that describe objects, and/or relations between objects. The narrative statement is “what
is said” about the object during the interview or observation; a student with behavior described in
the quote is very likely to say, “it is a rectangle because it looks like one”. The visual mediator in
this situation could include a drawing or picture of a four-sided figure looking like a rectangle.
The translation of van Hiele levels into discursive terms provides additional information
about what a student might say (word use and narratives) and do (routines), as well as what
visible objects (visual mediators) are operated upon as a part of the process of communication
through the geometric discourses. Moreover, the translation provides more details about the
development of the levels through the development of geometric discourses. The van Hiele
levels provide a useful framework to distinguish students in different levels, whereas the
geometric discourses at the van Hiele levels provide in-depth analyses of students’ levels of
geometric thinking.
The General Research Question in Commognitive Terms
In the previous section I described Sfard’s (2008) commognitive framework, a systematic
approach to analyzing the discursive features of mathematical thinking. To examine thinking
about geometry, the study connected Sfard’s framework to the van Hiele theory (1959/1985),
and produced a detailed model, namely the Development of Geometric Discourse. This model

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translates the five van Hiele levels into five discursive stages of geometric discourses. I used
empirical data of the study to enrich and refine the model with these questions:
1. At what van Hiele levels do prospective teachers operate?
2. What additional information does the model, the Development of Geometric
Discourse, provide regarding prospective teachers’ levels of geometric thinking?
In view of my interest in investigating prospective teachers’ knowledge in geometry after
taking a university geometry course, I also asked this question:
3. How do prospective elementary teachers’ competencies in geometry change as a
result of their participation in a university geometry course?
According to Sfard (2008), mathematical knowledge in geometry involves two levels of
discursive process: the object level and the abstract level. For example, the geometrical narrative
on geometric shapes, “the sum of the angles in a quadrilateral is 360º”, is considered at the object
level because it expresses a property of quadrilaterals; whereas the abstract level involves a
patterned activity of formulation and substantiation of these object-level narratives. This study
investigates two components when examining prospective teachers’ knowledge in geometry. The
first component is students’ knowledge of the names, properties and classification of geometric
shapes (object level); the second is competence in reasoning. The new knowledge about
geometrical constructs comes from a deductive process (abstract level). The question “How do
prospective elementary teachers’ competencies in geometry change as a result of their
participation in a university geometry course?” pertains to prospective teachers’ familiarities
with basic geometric shapes and their properties, abilities to formulate conjectures, and abilities
to construct geometric proofs.

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In discursive terms, an analysis of “familiarity with basic geometric shapes, their
properties and classification” implies an examination of students’ narratives about geometric
figures utilizing geometric names, where narratives speak about the properties of figures and
relations between them. The analysis of “ability to formulate conjectures about figures, their
properties and relations, and abilities to construct geometric proofs” suggests an examination of
students’ formulation and substantiation of these object level rules about geometric figures.
In the next section, I will discuss the methodology to be used in this study.

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CHAPTER THREE: METHODOLOGY

The purpose of this study is to explore a discursive framework as an analytic tool to
describe students’ geometric thinking through the analysis of their geometric discourses. To this
end, I used three primary data sources, (1) written responses to the van Hiele Geometry Test
(from a Pretest and Posttest), (2) transcripts (from two in-depth interviews, the first interview
conducted right after the pretest, and the second right after the posttest), (3) other written
artifacts (participants’ written statements, answer sheets to the three tasks during the interviews).
I describe these sources in greater detail here followed by the methods used to analyze these data
sources.
Participants
All participants in the study were prospective elementary school teachers. In a certain
mid-west university teacher education program, prospective elementary teachers were required
to complete a sequence of two mathematics content courses designed for elementary school
teachers. The first of these courses deals with on numbers and operations, and the second with
measurement and geometry. The participants of the study were prospective elementary teachers
enrolled in the measurement and geometry course; most of them were juniors and sophomores,
and a few were seniors. All seventy-four students enrolled in the course in the fall of 2010
participated in the pretest, and sixty-three of these participated in the posttest, both tests being
given as class assignments. Twenty-one participants voluntarily participated in the interview part
of the study soon after the pretest, and twenty of these participated in the second part of the
interview soon after the posttest.

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All participants enrolled in the geometry and measurement course for teachers used
Parker and Baldridge’s (2008) textbook, Elementary Geometry for Teachers. Ten chapters are
included in this textbook, all discussing mathematical topics related to geometry and
measurement for prospective elementary school teachers. Most participants had studied
geometry in K-12, therefore the contents of this study related to triangles, quadrilaterals and
proof introduced in Chapter 2 (Geometric Figures) and Chapter 4 (Deductive Geometry) were
partly review to them. For example, in Chapter 2 students were introduced to triangles and
parallelograms. The discussion includes the introduction of angles, perpendicular and parallel
lines, as well as the classification of quadrilaterals. In the classification of quadrilaterals, students
studied parallelograms, rectangles, rhombuses, squares, trapezoids, and kites. In Chapter 4
students learned how to derive new geometric facts from previously known facts using logical
arguments. For instance, in the beginning of Chapter 4, where a problem of finding an unknown
angle in a quadrilateral leads to an unknown angle proof, students learned from a natural
computation to deduce a general fact about a quadrilateral. Later in the chapter, students learned
to construct proofs for congruent triangles, and to use congruent triangles to verify properties of
quadrilaterals. Thus, these participants were introduced to the topics in this study by the textbook
for the course.
The van Hiele Geometry Test
As discussed in the earlier chapter, many mathematics educators have used van Hiele
theory to determine students’ levels of mathematical thinking. In order to identify suitable survey
instruments for the study, literature on the van Hiele levels was reviewed. The van Hiele
Geometry Test (see Appendix C), used in the Cognitive Development and Achievement in
Secondary School Geometry (CDASSG) project, was chosen because it was carefully designed

41

and tested by the researchers of the project (Usiskin, 1982, p.19). This test was used as the
instrument for pretest and posttest to determine the van Hiele level of the students.
The van Hiele Geometry Test contains 25 multiple-choice items, distributed into five van
Hiele levels: Items 1-5 (Level 1), Items 6-10 (Level 2), Items 11-15 (Level 3), Items 16-20
(Level 4) and Items 21-25 (Level 5). These items are designed to identify students’ geometric
thinking at five van Hiele levels. For example, Items 1 to 5 of are designed to identify students’
thinking related to van Hiele Level 1, at which figures are judged according to their appearance.
Items 5 to10 identify students’ behaviors related to van Hiele Level 2, at which figures are the
supports of their properties. The van Hiele Geometry Test was given to provide initial
information on students’ levels of geometric thinking at the two end points: beginning (pretest)
and the end of the semester (posttest). The analyses of the pretest and posttest help to determine
students’ changes in geometric thinking resulting from participating the measurement and
geometry class.
The goals of the interviews were (1) to gather information about students’ knowledge of
with triangles and quadrilaterals, as well as the parts of the triangles and quadrilaterals (e.g.,
angles, sides, etc), (2) students’ abilities in verifying their claims and deriving mathematical
proofs, and (3) to probe further into students’ geometric discourses as revealed through these
mathematical activities.
Interview Tasks
Three tasks (see Appendix D) and corresponding interview protocols were designed for
the interviews. All three tasks were printed individually on four standard 8″× 11″ sheets of white
paper. Task One presents eighteen geometric shapes, labeled with alphabet capital letters shown
in Figure 3.1. Among these eighteen polygons, thirteen are quadrilaterals, four are triangles, and

42

one is a hexagon. Sixteen of these polygons were chosen from the van Hiele Geometry Test
Items 1-5. Two more polygons were added, consisting of Fig. Q, a quadrilateral and Fig. S, a
triangle with no pair of sides equal. These shapes also commonly appear in elementary and
middle school textbooks used by many researchers over the past two decades to categorize
students’ geometric thinking with respect to the van Hiele levels (e.g., Mayberry, 1983; Burger
& Shaughessy, 1986; Gutierrez, Jaime, & Fortuny, 1991).
Task One: Sorting Geometric Figures

Figure 3.1

Task One: Sorting Geometric Figures

Task One was presented to participants at the beginning of the interviews, and each was
asked to sort the eighteen polygons into groups. After the first round of sorting, each participant
was asked to regroup or/ and subgroup the polygons. For example, some participants sorted the
polygons into a group of rectangles (see U, M, F, T, R, G in Figure 3.1), and a group of triangles
(see X, K, W, S in Figure 3.1). The questions “Can you describe each group to me?” and “ Can

43

you find another way to sort these shapes into groups?” allowed participants to produce
narratives about triangles and quadrilaterals based on their knowledge of polygons. Analysis of
the act of grouping gave information on how participants classify triangles and quadrilaterals.
At the end of Task One, I asked each participant to write the definitions of rectangle,
square, parallelogram, rhombus, trapezoid and isosceles triangle, and collected their written
narratives. This information revealed how participants defined these mathematical terms, and
how they made connections between a name and a recognized parallelogram, as well as how
quadrilaterals are related to one another.
Task Two: Investigating Properties of Parallelograms
Task Two of the interview had two components. The first component, divided into Part A
and Part B, was designed to collect participants’ drawings of the parallelograms (see Table 3.1),
and to gather more information on their knowledge of parallelograms.
Table 3.1

Investigating the Properties of Parallelograms

Draw a parallelogram in the space below.
o What can you say about the angles of this parallelogram?
o What can you say about the sides of this parallelogram?
o What can you say about the diagonals of this parallelogram?
Draw a new parallelogram that is different from the one you drew previously.
o What can you say about the angles of this parallelogram?
o What can you say about the sides of this parallelogram?
o What can you say about the diagonals of this parallelogram?

Part A begins with “Draw a parallelogram in the space below”, and next asks
participants to describe the angles, sides and diagonals of the parallelogram. For instance, the
44

question, “What can you say about the angles of this parallelogram?” was to find out about
participants’ familiarity with the angles of parallelograms. Part B starts with “In the space below,
draw a new parallelogram that is different from the one you drew previously”, and asks
participants to describe the angles, sides and diagonals of the new parallelogram. This part of the
task investigated how participants define parallelograms, and their thinking of different
parallelograms.
After participants completed Parts A and B, they were presented pictures of
parallelograms not included in their drawings. Four pictures of parallelograms were prepared for
the interviews, consisting of a parallelogram, a rhombus, a rectangle and a square, each drawn on
a 3″× 5″ white index card. Figure 3.2 shows the four parallelograms.

Figure 3.2

Pictures of four parallelograms

The purpose of these pictures was to encourage discussion of different parallelograms
and their parts. They helped me to explore out why a participant included some parallelograms
but excluded others. For example, after a participant drew a picture of a parallelogram in Part A
and drew a rectangle as a different parallelogram in Part B of Task Two, I presented a picture of
a square, and asked whether it was also parallelogram and why. Thereby, I gathered more
information about participants’ understanding of parallelograms, and was able to gain insights
missed in Task One regarding to participants’ ways of identifying and defining parallelograms. A
set of interview scripts was designed to further aid in analyzing participants’ understanding of
parallelograms (see Appendix E). These scripts were written to help participants make claims
45

about the angles, sides and diagonals of parallelograms. Task Two shed additional light on
participants’ knowledge of parallelograms, and familiarity with their angles, sides and diagonals.
In completing Task Two, participants were engaged in verifying their claims regarding
the properties of parallelograms, constructing informal and/or formal proofs. For example, when
a participant made a statement that the diagonals of a rectangle were equal, she was asked to
justify the claim. In order to convince me, the participant had to engage in a reasoning process.
Such requests of “how do you” or “why” were designed to learn how participants’ verify
mathematical arguments.
Task Three: Prove the Equivalence of Two Definitions
The participants in my study are future teachers, and need to be aware that, although only
one definition of parallelogram dominates books (e.g., a parallelogram is a quadrilateral with two
pairs of parallel sides), other equivalent definitions could be given (e.g., a trapezoid with two
pairs of parallel sides)(Usiskin & Griffin, 2008). The requirement for another definition to be
equivalent to the standard definition is that the defining conditions yield the same figures, and
only such figures. Although Task Three focused on deriving mathematical propositions from
previously known propositions, the choice was made to ask participants to construct
mathematical proofs in order to learn about their understanding of mathematical proof.
Two definitions of parallelogram were presented in Task Three:
1. “A quadrilateral is a parallelogram if and only if both pairs of opposite sides
have the same length”
2. “A quadrilateral is a parallelogram if and only if both pairs of opposite angles
have the same measure”

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Definitions 1 and 2 describe properties of sides and angles, respectively. Before
introducing this task, I first asked what it meant to prove two definitions equivalent
mathematically. I then explained as follows: “To show that two definitions are equivalent, you
must verify that each set of defining conditions implies the other”. That is, to show that the two
definitions of parallelogram are equivalent, one must prove the following implications: (1) “if a
quadrilateral has two pairs of opposite sides of the same length, then the quadrilateral also has
two pairs of opposite angles of the same measure”; and (2) “if a quadrilateral has two pairs of
opposite angles of the same measure, then the quadrilateral also has two pair of opposite sides of
the same length.”
I anticipated that participants would respond to Task Three differently. The task was
designed to learn about their skill in constructing proofs, about their familiarity with “If P, then
Q” statements, and about their use of mathematical symbols. The task also helped measure
abilities to derive a geometric proposition from other ones, more precisely in the context of
quadrilaterals.
Data Collection
Data collection for this study took place in four phases, as summarizes in Figure 3.3.

Phase 1
Administration
of the van Hiele
Geometry Test
at the first week
of class

Phase 3
Administration
of the van Hiele
Geometry Test
at the first week
of November

Phase 2
In-depth
interviews
with participants
after pretest

Last two weeks of Novemebr 2010

First two weeks of the fall 2010

Figure 3.3

Phase 4
In-depth
interviews
with participants
after posttest

Summary of Data Collection Phases
47

The first phase of the data collection was the pretest, a 35-minutes van Hiele Geometry
Test. All students (n=74) enrolled in the measurement and geometry classes took the van Hiele
Geometry Test during the first class of the fall semester of 2010. All the tests were collected and
analyzed, in order to determine participants’ van Hiele levels at the beginning of the semester.
In the second phase, twenty-one students voluntarily participated in a 90-minutes indepth interview with the same researcher a week after the pretest was given. All interviews were
video and audio recorded, and transcribed to serve as the main data recourse for analyzing
interviewees’ geometric discourses relating to triangles and quadrilaterals. All interviews were
completed before the students were introduced to geometric figures in their mathematics content
course.
The third phase of the data collection was the posttest, consisting again of the van Hiele
Geometry Test. Among the seventy-four participants, sixty-three repeated test ten weeks later
during their class time. Again the test responses were collected and analyzed, in order to
determine changes in van Hiele levels between the two tests.
The last phase of the data collection consisted interviews with students who had
participated in the interviews at the beginning of the semester. Among the twenty-one original
interviewees, twenty were interviewed individually for 90-minutes a week after the posttest.
Again the interviews were video and audio recorded, and transcribed and analyzed in order to
observe changes in interviewees’ geometric discourse. All interviews were conducted after
students had finished the chapter introducing deductive reasoning in their mathematics content
course.

48

Data Analysis
In my data analysis I applied Sfard’s discursive framework to analyze interviews. I used
this analysis to investigate participants’ levels of geometric thinking through a discursive lens,
and to gain some perspective on the usefulness of the framework to describe levels of thinking. I
briefly outline the data analyses for this study. First, I compared written responses from the van
Hiele Geometry pretest and posttest, obtaining from the test scores information about the
changes in these prospective elementary school teachers’ van Hiele levels as a whole group. To
determine students’ van Hiele levels, I followed the test grading method used in the CDASSG
project from the University of Chicago in 1982 (Usiskin, 1982). Following the CDASSG
project’s grading method, I used the 4 of 5 criterion1 to determine if a student had reached a
given level. I chose the 4 out of 5 correction criterion because it minimized the chance of a
participant being at that level by guessing (Usiskin, 1982). When assigning a student to a level, I
used the classical van Hiele levels 0-5 introduced by CDASSSG.
The assigning of levels required that the student at level n satisfy the criterion not only at
that level but also at all proceeding levels. For example, if a participant scored 4 of 5 correct for
Levels 1, 2 and 3, 2 of 5 correct for Level 4, and 1of 5 correct for Level 5, this participant was
assigned to van Hiele Level 3 because she not only satisfied the criterion at Level 3, but at all
preceding levels as well. However, in this study there were participants assigned as nofit because
their van Hiele levels could not be determined from the van Hiele Geometry Test. For example, a

1

According to the CDASSG project, the 3 of 5 criterion minimizes the chance of missing a
student and yields an optimistic picture of students’ levels, whereas the 4 of 5 criterion
minimizes the chance of a student being at a level by guessing. I decided to use the 3 of 5
criterion.
49

participant was assigned nofit because her test results satisfied criterions at the levels 1, 3 and 5,
but not at all preceding levels (Usiskin, 1982). I also counted and analyzed students’ overall
performances for each item in the pretest and posttest, looking for changes in answering single
questions as a group (see the results of both analyses in Chapter 4).
My main object of attention in this study is geometric discourse. I try to use the voices of
my participants in describing interview’s whatever possible, so that readers can draw their own
conclusions. In analyzing participants’ geometric discourse, I identified the mathematical
features in interview transcripts using four categories in the framework: (1) Mathematical words,
(2) Visual mediators, (3) Endorsed narratives and (4) Mathematical routines. Mathematical
words and visual mediators utilized mathematical objects of mathematical discourse, whereas
mathematical routines aimed to produce narratives in given situations. To investigate changes in
participants’ geometric discourse, I analyzed (1) participants’ words use regarding to the names
of triangles and quadrilaterals (e.g., rectangle, rhombus, etc), and the hierarchy of classifications
of quadrilaterals, comparing results of the analyses from both interviews, (2) participants’ routine
procedures of verifying claims about properties of parallelograms regarding angles, sides and
diagonals, comparing results of the analyses from both interviews, and (3) participants’ routine
procedures of deriving geometric propositions from other geometric propositions, and in
constructing mathematical proofs (see the descriptions of interviews and results of these analyses
in Chapter 4).
I also investigated the usefulness of a discursive framework as an analytical tool to
describe participants’ behaviors at each van Hiele level in greater detail and depth. The study
produced a theoretical model, the Development of Geometric Discourse, describing participants’
geometric discourse at each van Hiele level. The model includes the descriptions of (1)

50

Geometric Objects, (2) Routines, (3) Endorsed Narratives, and (4) Visual Mediators, at van Hiele
levels 1 to 4. The model provides a new perspective to present levels of geometric thinking as
geometric discourse (see the descriptions of the model in Chapter 4).

51

CHAPTER FOUR: FINDINGS
Changes in van Hiele Levels of Geometric Thinking
During the fall of 2010, the van Hiele Geometry Tests were conducted at the beginning of
the semester (pretest) and ten weeks later (posttest). Sixty-three prospective teachers participated
in both tests; among these sixty-three participants, twenty of them voluntarily participated in the
interview part of the study. In this section I present results of these sixty-three participants’ van
Hiele Geometric Tests as a whole group, as well as the results of the twenty interviewees’ van
Hiele Geometric Tests, in order to give some background information on their changes in van
Hiele levels in the paper-pencil test.
Changes in van Hiele Geometry Test as a Whole Group
The van Hiele Geometry Test contains 25 multiple-choice items, distributed into five van
Hiele levels: Items 1-5 (Level 1), Items 6-10 (Level 2), Items 11-15 (Level 3), Items 16-20
(Level 4) and Items 21-25 (Level 5). These items are designed to identify participants’ geometric
thinking at five van Hiele levels. For example, Items 1 to 5 of are designed to identify students’
thinking related to van Hiele Level 1, at which figures are judged according to their appearance.
Items 5 to10 are designed to identify participants’ thinking related to van Hiele Level 2, at which
figures are the supports of their properties.
Following the Cognitive Development and Achievement in Secondary School Geometry
(CDASSG) project’s grading method, I had a choice of either the 3 out of 5 correction criterion,
or the 4 out of 5 correction criterion, to determine whether a participant has reached a given van
Hiele level. I chose the 4 out of 5 criterion for this study because it minimized the chance of a
participant being at that level by guessing (Usiskin, 1982). When assigning a student to a level,
this study used the classic van Hiele levels introduced by CDASSSG, which included Levels

52

from 0 to 5. The assigning of levels required that the participants at level n satisfied the criterion
not only at that level but also at all proceeding levels. For example, if a participant scored 5 of 5
correct for Levels 1 and 2, 4 of 5 correct for Level 3, 2 of 5 correct for Level 4, and 1of 5 correct
for Level 5, this participant was assigned at van Hiele Level 3 because she not only satisfied the
criterion at Level 3, but all preceding levels as well. However, in this study there were
participants assigned as nofit because their van Hiele levels could not be determined from the
van Hiele Geometric Test. For example, if a participant scored 5 of 5 correct for Level 1, 3 of 5
correct for Level 2, 4 of 5 correct for Level 3, 2 of 5 correct for Level 4, and 4 of 5 correct for
Level 5, she was assigned nofit because her test results satisfied criterions at the levels 1, 3 and 5,
but not all preceding levels (Usiskin, 1982).
To give an overall idea of participants’ van Hiele levels as a whole group, Table 4.1
presents the distributions of levels in both number and percentage from the pretest and the
posttest.
Table 4.1

Distributions of participants’ van Hiele levels at both tests
Level

VHB
N
6
8
9
17
2
0

VHE
%
9.52
12.70
14.29
26.99
3.17
0

N
4
9
8
30
2
1

%
6.35
14.29
12.70
47.62
3.17
1.59

0
1
2
3
4
5
Total
42
66.67
54
85.71
fitting
Nofit
21
33.33
9
14.29
Totals
63
100
63
100
Note: VHB indicates interviewees’ van Hiele levels at the beginning
of the fall semester, whereas VHE indicates their van Hiele levels ten
weeks later.

53

Table 4.1 shows about forty-eight percent of participants at Level 3 (n=30) at the posttest,
and that number almost doubled over the pretest. One participant moved to Level 5 at the
posttest. There was a slight reduction at Level 0 and Level 2, and a slight increase at Level 1
from the pretest to the posttest. There were twenty-one participants assigned to nofit at the
pretest, but only nine such participants at the posttest. These numbers show that a little more than
half (52.4%) of participants were able to reach Level 3 or above ten weeks later after their first
day of class in the fall semester 2010. The reduction of the nofit students suggests that
participants’ responses were more consistent at the posttest than the pretest.
Participants’ responses for each item were counted and analyzed. Table 4.2 presents the
frequencies of each item regarding responses in both the pretest and posttest. The bolded
numbers represent the number of participants having the correct answer for that item.
Table 4.2
Level
1

Van Hiele Geometry Test: Item Analysis for Pretest (B) and Posttest (E)

Choice
A
B
C
D
E

1B
0
53
0
10
0

1E
0
59
0
4
0

2B
0
0
4
59
0

2E
0
0
0
63
0

3B
0
0
63
0
0

3E
0
0
63
0
0

4B
0
41
6
1
6

4E
1
50
5
5
2

5B
0
0
15
0
48

5E
0
0
4
1
58

Item

6B
5
43
9
6
0

6E
5
46
7
5
0

7B
2
1
2
3
55

7E
1
0
1
0
61

8B
39
3
8
2
11

8E
46
0
3
4
10

9B
3
2
54
1
3

9E
1
0
60
0
2

10B
4
4
4
40
1

10E
5
0
3
44
11

Item

2

Item

11B
4
3
54
1
1

11E
3
0
58
0
1

12B
6
47
5
0
5

12E
4
51
4
0
4

13B
45
0
0
0
18

13E
56
0
0
0
7

14B
31
14
10
4
4

14E
46
7
3
2
5

15B
2
29
4
7
21

15E
6
33
1
5
18

A
B
C
D
E
3
A
B
C
D
E

54

Table 4.2 (cont’d)
4
Item
A
B
C
D
E
5

Item
A
B
C
D
E

16B
9
8
31
10
5

16E
10
6
30
13
4

17B
14
9
23
9
8

17E
12
11
29
4
7

18B
14
14
3
27
4

18E
10
15
2
28
8

19B
27
10
14
10
2

19E
37
12
7
5
2

20B
19
2
5
32
5

20E
30
2
1
29
1

21B
33
16
4
1
9

21E
28
16
7
1
10

22B
11
9
6
15
22

22E
12
14
6
19
12

23B
20
8
2
25
8

23E
26
4
5
26
2

24B
2
3
11
16
30

24E
5
5
13
22
18

25B
1
16
2
33
10

25E
2
14
1
37
9

Note the number of students (n=63) who participated in both tests.
Table 4.2 shows an increase in correct answers for Items 1 to 15 from the pretest to the
posttest, indicating that participants did better at the posttest in items related to van Hiele Levels
1, 2 and 3. There was an increase in correct answers at Level 4 and Level 5, but not for every
item individually. Figure 4.1 compares participants’ performance based on the number of correct
answers for each item at the pretest and the posttest.
Comparison of correct answers for items at the pretest and
posttest
70
60
50
40
30
20
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Pretest
Figure 4.1

Posttest

Comparison of correct answers for items at both tests
55

Figure 4.1 shows an increase in correct answers for Items 1 to 15, indicating that the
group of participants (n=63) had a better performance on these items at the posttest than at the
pretest. Thus, participants were better in answering questions relating to van Hiele levels 1 to 3 at
the posttest than at the pretest. However, there was a reduction in correct answers starting at Item
16, and that continued intermittently to Item 25. Items 16 to 25 are designed to identify
participants’ Level 4 and Level 5 thinking, and this inconsistency was no surprise because the
course was not designed to train students at theses levels. In looking at these results, it is clear
that participants were getting more familiar with triangles and quadrilaterals, as well as with
properties related to these polygons. However, it is not clear that participants’ improved in doing
proofs and thinking at an abstract level, mathematical activities related to van Hiele Level 4 and
Level 5.
Twenty students voluntarily participated in the interviews conducted after the pretest and
the posttest. In the next section, I briefly discuss the results of these twenty interviewees’ van
Hiele Geometry Tests before they entered the interview part of the study.
Changes in van Hiele Geometry Test among Interviewees
Twenty-one students voluntarily participated in the interviews shortly after the pretest,
and twenty of these participated in the second interviews ten weeks later after the posttest. Table
4.3 presents distributions of interviewees’ van Hiele levels at the pretest and the posttest.
Table 4.3

Distributions of interviewees’ van Hiele levels at both tests
Level
0
1
2
3
4
5

VHB
N
2
1
2
6
1
0

VHE
%
10
5
10
30
5
0
56

N
1
0
4
10
2
0

%
5
0
20
50
10
0

Table 4.3 (cont’d)
Total
12
fitting
Not fit
8
Totals
20

60

17

85

40
100

3
20

15
100

The table shows that the percentages at each van Hiele level of the twenty interviewees
matched closely with the corresponding percentages of the whole group (n=63). For example,
thirty percent of interviewees were at Level 3 in the pretest, whereas it increased to fifty percent
at the posttest. These numbers are close to those for the whole group, twenty eight percent and
forty eight percent, respectively (see Table 4.2). Thus it appears that the twenty interviewees are
a good sample size for the study. Table 4.4 gives, for these interviewees, frequencies of changes
from one van Hiele level to another between the two tests.
Table 4.4

Interviewees’ changes in van Hiele levels between tests

Levels
Van
0
Hiele
1
levels
2
at the
3
pretest
4
(n=20)
5
Nofit

Van Hiele levels at the posttest (n=20)
0
1
2
3
4
5
2
1
1
1
4
1
3
3
1
-

Nofit
1
1
1

For example, among these interviewees (n=20), one stayed at Level 2, and four stayed at
Level 3 from the pretest to the posttest. There were two interviewees moving three van Hiele
levels from Level 0 to Level 3, while one interviewee moved two van Hiele levels from Level 1
to Level 3, and one interviewee changed from Level 3 to Level 4. A total of seven interviewees
changed from nofit to Level 2 (n=3), Level 3(n=3) or Level 4 (n=1). Note that among the ten

57

interviewees assigned to van Hiele levels at both the pretest and the posttest, five showed no
change, and four showed changes from a lower to a higher van Hiele level.
The van Hiele Geometry Test provided initial information about participants’ van Hiele
levels at the time of the study, but it did not provide rich descriptions about the changes in
participants’ levels of thinking. For a deeper analysis of participants’ thinking, twenty
participants were interviewed soon after the pretest and posttest. In the next section I describe my
interviews with these participants, as well as my findings from these interviews.
Changes in Geometric Discourse
In this section I describe findings about the interviewees who participated in the
interview parts of the study. To narrow the scope, this section focuses on the analyses of five
interviewees’ geometric discourses as examples of various scenarios I encountered during the
interviews. My analyses are organized with regard to the results of their van Hiele levels from
the van Hiele Geometry Tests. These five interviewees have been assigned the names ATL, ANI,
ALY, AYA and ARI.
To analyze interviewees’ geometric discourses in the context of quadrilaterals and
triangles, I devoted my attention to interviewees’ familiarity with polygons in regard to their
word use, including use of the names of polygons (e.g., parallelogram, rectangle, etc), and the
names of the parts of polygons (e.g., angle, side, etc). Also, I analyzed interviewees’ various
routines while engaging in solving geometric tasks during the interviews; these routines included
routines of sorting, identifying, defining, conjecturing and substantiating.
Recall that a routine is a set of meta-rules that describes a repetitive discursive action. As
described in an earlier section, this set of rules is divided into the how of the routine and the
when of the routine. The when of the routine, was influenced by my direct requests during the

58

interviews, as I asked interviewees directly for implementation of given tasks (e.g., sorting the
polygons differently from the way they did before). So for a better analysis of the when of
routines, I needed to observe interviewees’ spontaneously using procedures as a part of more
complex tasks. Therefore, in this study I mainly analyze the how of routines, the routine
procedures that determine the course of patterned discursive performance (Sfard, 2008).
In this study different routines are involved given the nature of the tasks: the routine of
sorting is a set of routine procedures that describes repetitive actions in classifying polygons (e.g,
by their family appearances, by visual properties, etc); the routine of identifying is a set of
routine procedures that describes repetitive actions in identifying polygons (e.g., by visual
recognition, by partial properties check); whereas the routine of defining is a set of repetitive
actions related to how polygons are described or defined (e.g, by visual properties, by
mathematical definition, etc). In endorsed narratives such as mathematical definitions or axioms,
the routine of recalling, a subcategory of the routine of defining, is a set of repetitive actions
using previously endorsed narratives (e.g, I remember this definition because I learned it), and “it
can indicate a lot not just about how the narratives were memorized, but also about how they
were constructed and substantiated originally” (Sfard, 2008, p.236).
With regard to performing mathematical tasks, “guessing and checking” are seen as
common activities. The routine of conjecturing is a set of repetitive actions that describe a
process of how conjecture is formed; and the routine of substantiating is a set of patterns
describing a process of using endorsed narratives to produce new narratives that are true (e.g., an
informal or formal proof using a triangle congruence criterion).
To better understand how learning takes place, and how mathematical concepts are
developed, it is helpful to conceptualize mathematical learning as the development of a

59

discourse, or a change in discourse. Among the twenty interviewees, three showed a change in
their van Hiele levels from lower to higher according to the van Hiele Geometric Test conducted
at the beginning of the semester (the pretest) and ten weeks later (posttest). The three
interviewees include ATL, who moved two van Hiele levels from Level 1 to Level 3; ANI, who
moved three van Hiele levels from Level 0 to Level 3; and ABU who moved one van Hiele level
from Level 3 to Level 4. In the following subsections I describe each interviewee’s geometric
discourse with regard to their routines and word use. From my observations I present evidence to
point out changes in each interviewee’s geometric discourse. I will refer to the interview
conducted at the beginning of the semester as the Pre-Interview, and the interview conducted ten
weeks later as the Post-Interview.
Case 1: Changes in ATL’s Geometric Discourse
ATL was a sophomore at the time of the interviews. ATL took her last geometry class
five years prior to the geometry and measurement class. The van Hiele Geometry Test showed
that she was at Level 1 at the pretest, and ten weeks later she moved two van Hiele levels, to
Level 3 at the posttest. I interviewed ATL after both tests, and analyzed and compared ATL’s
geometric discourses in the context of triangles and quadrilaterals. A summary of ATL’s changes
in geometric discourse as follows:
•

ATL’s routines of sorting changed from grouping polygons according to their
family appearances at the Pre-Interview, to classifying polygons according to
their visual properties and definitions.

•

ATL’s identifying routines changed from self-evident visual recognition at the
Pre-Interview, to identifying partial properties of the polygons (i.e. sides and
angles) and recalling at the Post-Interview.

60

•

ATL’s use of the names of parallelograms changed from describing the
parallelograms as collections of unstructured quadrilaterals that share some
physical appearances at the Pre-Interview, to using the names as collections of
quadrilaterals that share common descriptive narratives at the Post-Interview.

•

In my observations I did not find substantiation routines in the Pre-Interview or
the Post-Interview. That is, ATL did not use measurement tools to prove or
disprove congruent parts of the polygons at the object level; nor did she use
informal or formal mathematical proofs at the abstract level.

Let me begin by introducing Task One, Sorting Geometric Shapes. This Task is used to
analyze interviewees’ routines of sorting, identifying and defining polygons. This task presents
eighteen polygons, including triangles (n=4), quadrilaterals (n=13) and a hexagon. Interviewees
are asked to classify these polygons into groups, without being given measurement information.
One common reaction interviewees might have is to group the polygons based on the number of
their sides (e.g., 3-sided, 4 sided, etc). According to an individual interviewee’s response, I will
ask her to regroup the polygons differently and to subgroup some of the large groups.
Let me begin with argument that there is a change in ATL’s routines of sorting, from
using visual recognition to group quadrilaterals according to their family appearances at the preinterview, to classifying quadrilaterals according to their common descriptive narratives (i.e.,
definitions and properties). I briefly describe my interviews with ATL for Task One.
At the Pre-Interview, ATL stated, “I group them solely on their amount of sides”, and
sorted the polygons into three groups on her first attempt: 1) 3 sides, consisting of Fig. K, W, X,
and S; 2) 4 sides, consisting of Fig. U, M, F, G, P, T, L, J, H, R and Z; and 3) Trapezoid,

61

including Fig. V and Q. See Figure 4.2 for some examples of each group based on ATL’s written
response
3 sides (Triangle)

4 sides (Square/parallelogram)

Trapezoid

Figure 4.2

ATL’s grouping of polygons in the Pre-Interview

ATL included all triangles in the 3-sides group and called it the triangle group. She
included all squares, rectangles and parallelograms in the 4-sides group, and called it the square
and parallelogram group. She grouped Fig. V (a hexagon) and Fig. Q (a quadrilateral) together
as a trapezoid group because to her, a trapezoid was “a figure with five sides, varying in length”,
and “often make these odd shapes”. Fig. N (a right trapezoid) was not included in any of these
groups.
When I asked ATL to find another way to group polygons differently, she regrouped
triangles “base on appearance”, according to attributes of angles and sides in a triangle. ATL
explained by saying, “I know this one has right angle”, and “these ones demonstrate different
length, they are not the same” (see Figure 4.3).

62

Triangle:

right triangle
isosceles triangle
scalene triangles
Figure 4.3
ATL’s regrouping of the triangles at the Pre-Interview

ATL regrouped the triangles into right triangles, isosceles triangles and scalene triangles
according to their visual properties. ATL then regrouped the 4-sided polygons according to their
family appearances, with the names of square, rectangle, parallelogram, and rhombus. For
example, Figure 4.4 shows two of the groups: the rectangles and the squares.
4a. Squares

4b. Rectangles

Figure 4.4

ATL’s regrouping the quadrilaterals at the Pre-Interview

When I asked ATL why she regrouped the polygons in this way, she responded, “I know
this figure (Fig. U, a square) and this figure (Fig. M, a rectangle) are different, but they both
belong to the same quadrilateral group”. Fig. N (a right trapezoid) was not included in any of
these groups again at this second attempt. I asked ATL if I could put Fig. J (a parallelogram) and
Fig. N together, and the following conversation took place:
Interviewer: Can we put these two together?

63

ATL:

I wouldn’t believe so… Just because this [pointing at
Fig. N] shows the angle… it doesn’t have the properties
of a square or a rectangle, the sides…measurement… it
does have four sides, but n
no…. congruent parts.

When ATL sorted polygons into groups, she grouped them by their visual appearances.
ATL’s routines of sorting polygons at the Pre interview are illustrated in Figure 4.5.
Pre-interview
First prompt: “Sort the shapes into groups”
Counting the sides
of shapes

Choosing
intuitively

Grouping by the same
number of sides

Grouping by family name for
quadrilaterals & partial visual
properties of shapes for
triangles

Conclusion

Conclusion

Second prompt: “Find Another way to sort them differently?”
prompt:
Figure 4.5

ATL’s routines of sorting polygons at the Pre Interview
Pre-Interview

ATL mentioned that all triangles and all quadrilaterals “have a broader def
definition of each
other”. When it came to classifying quadrilaterals, ATL’s routine procedures focused on the
.
appearances of the polygons and how their appearances related to their family names. It was
evident that ATL identified polygons with visual recognition. However, as I explained earlier,
explained
ATL regrouped the triangles by their visual properties (e.g., angles and sides). I conclude that
there was no defining routine in ATL’s routines of sorting for Task One. I found identifying
routines such as direct recognition and counting in ATL’s routines of sorting when she sorted
polygons by the numbers of their sides and grouped quadrilaterals by their visual appearances. In
64

the case of triangles, ATL’s identifying routine consisted of identifying visual properties as she
regrouped triangles by the attributes of the angles and sides.
At the Post-Interview, the same task was performed. ATL grouped the polygons by
“looking at the numbers of sides solely” [po1. 2], and she sorted eighteen polygons into three
groups: 1) Triangles (n=4), including all 3-sided polygons; 2) 5-sided (n=1), consisting of Fig.
V; and 3) Quadrilaterals (n=13), including all 4-sided polygons in the task. Figure 4.6 compares
ATL’s first attempts at both interviews with some examples of each group.
Before

Ten Weeks later

3 sides

Triangles (All triangles are included)

4 sides (Fig. N & Q are missing)

Quadrilaterals (All quadrilaterals are
included)

Trapezoid

5-sided

Figure 4.6

A comparison of ATL’s grouping of polygons at both interviews

At the Post-Interview, ATL included both Fig. N (a right trapezoid) and Fig. Q (a 4-sided
figure) into the quadrilateral group with the help of her defining routine. At the Pre-Interview
65

ATL did not include Fig. N in any groups because “it does have four sides, but… not congruent
parts”, but at the Post-Interview she included Fig. N in the quadrilateral group because it is “a
four sided figure with one distinct pair of parallel sides (pointing at Fig. N)”. I asked ATL to
regroup the quadrilaterals, and her response is shown below:
18a ATL: Quadrilaterals, you know that you have your square because …
each forms 90-degree and all the side lengths are equal.
[Pointing at Fig. U]

18c. ATL: these are rectangles because two sides and those two sides are
the same. But again they form 90-degree angles…
[Pointing at Fig. F and Fig. M]

18e. ATL: opposite angles are equal and opposite sides are equal, so these
three would be an example of parallelogram.
[Pointing on Fig. L, J and H]

At the Post-Interview, ATL was able to use her definitions of square, rectangle,
parallelogram and rhombus to identify and to regroup the quadrilateral group. She regrouped
66

quadrilaterals into: 1) squares, including Fig. U, G, and R; 2) rectangles, including Fig. M, F, T;
3) rhombus, consisting of Fig. Z; and 4) parallelogram, including Fig. L, J, H. When I asked
ATL if I could put Fig. U (a square) and Fig. N (a right trapezoid) together, ATL responded:
Interviewer: Can Fig. U and Fig. N group together?

ATL:

They can group together as both being same amount of sides…
but in terms like property…no … they both have two parallel
sides, but a trapezoid cannot be branched off with parallelograms
into rectangles and squares….

This dialogue provides shows ATL’s ability to compare Fig. U and Fig. N, not only
focusing on the “same amount of sides”, but also on visual properties, like “they both have
parallel sides”. To describe what has changed in ATL’s routine of sorting polygons, Table 4.5
summarizes ATL’s course of actions in response to Task One.
Note that there was no change in ATL’s routines of sorting triangles. ATL’s first attempt
of sorting quadrilaterals at both interviews remained the same. However there was a change in
routines of sorting quadrilaterals on the second attempt, when I asked her to regroup the
quadrilaterals differently (see the shaded part in Table 4.5). I found defining routines when ATL
sorted quadrilaterals at the Post-Interview; ATL’s routine of sorting changed from only visual
recognition at the Pre-Interview, to classifying polygons according to their common descriptive
narratives.
67

Table 4.5

A comparison of ATL’s routines of sorting polygons at the two interviews

Before

Ten Weeks Later

First prompt: “Sort the shapes into groups”

First prompt: “Sort the shapes into groups”

1. Counting the sides of shapes (Counting)
2. Grouping by the same number of sides
3. Conclusion

1. Counting the sides of shapes (Counting)
2. Grouping by the same number of sides
3. Conclusion

Second prompt: “Find another way to sort
them differently”

Second prompt: “Find another way to sort
them differently”

1. Direct recognition of possible candidates
(Visual recognition)

1. Direct recognition of possible candidates
(Visual recognition)

2.a Grouping by family appearance of
quadrilaterals and parallelograms (Visual
recognition)
2.b Grouping by properties of angles and sides
in triangles (Defining routine)

2.a Grouping by common descriptions of
quadrilaterals and parallelograms by visual
properties and some mathematical definitions
(Defining routine)
2.b Grouping by properties of angles and sides
in triangles (Defining routine)

3. Conclusion

3. Conclusion

ATL’s responses to the questions in Task Two also revealed changes in her geometric
discourses. Task Two involves two sets of activities about parallelograms, and is designed to
investigate interviewees’ familiarity with the angles, sides and diagonals of a parallelogram.
The first part of Task Two asks interviewees to draw two parallelograms that are different
from each other, and then to discuss the relationship between the angles, sides and diagonals of
the parallelograms. In the second part of Task Two, I present pictures of parallelograms that are
not included in interviewees’ drawings from the first part, and then ask questions about the
parallelograms. Presentation of these pictures of parallelograms is designed to elicit
interviewees’ thinking of “what is a parallelogram” and “what is not a parallelogram”, and to
provide a variety of parallelograms for discussions.

68

Interviewees’ responses to this task vary depending on their familiarities with the
properties and hierarchy of parallelograms, as well as their familiarity with the parts of
parallelograms. For example, when interviewees declare a narrative such as “opposite sides are
equal” regarding the sides of a parallelogram, their substantiation process can be very different.
Depending on their levels of thinking, some interviewees might produce a narrative such as
“because it is a parallelogram” using defining routines, whereas others might conclude, “they
look like they are equal” using identifying routines.
This lead to my argument that ATL’s identifying routines changed from self-evident
visual recognition at the Pre-Interview, to identifying visual properties and using definitions of
parallelograms to draw conclusions about the angles and the sides of parallelograms at the PostInterview. I describe parts of my interviews with ATL for Task Two. As shown in Table 4.6,
ATL drew a parallelogram and then a rectangle as a new parallelogram (see Table 4.6). ATL
declared that the two parallelograms were different because “I would change the sizes of it
[side]”. ATL described the second drawing as, “it’s a rectangle… but it’s not the typical looking
parallelogram”. In response to the questions about the angles of the parallelograms, ATL
expressed her frustrations on the angles, “I am still stuck on the question on what it means by the
angles, … Usually when I’m talking about angles, we have measurements…[pausing] I feel like
the angles would be the same … just based on how it looks”.
Table 4.6 summarizes ATL’s declared narratives of the angles of a parallelogram and a
rectangle, and her verifications of declared narratives.
Table 4.6

ATL’s routines of verifying on the angles of parallelograms at the Pre-

Interview
Q: “What can you say about the angles of this parallelogram?”

69

Rectangle

Parallelogram
Conjecture
(Guessing)

“I would assume that they are
the same for the opposites”

“They would have to be equal…or
add up to a certain amount”

Q: “How do you know?”
Routines

Visual recognition

Visual recognition

Declared
Narrative

“ No. I don’t know.”
“just based on how it looks”

“Just looks more like a stereotypical
parallelogram”

ATL made intuitive claims about the angles of a parallelogram and a rectangle using
direct recognition. For example, ATL assumed that the angles were “the same for the opposites”
in a parallelogram using direct recognition. In this case, the question “How do you know [they
are the same]?” did not lead to any substantiations of the claim, nor lead her to endorse any
narratives using mathematical definitions; instead ATL’s final conclusion was reached by direct
visual recognition which was self-evident. This routine pattern also appeared when ATL was
discussing the diagonals of a parallelogram:
17. Interviewer

What can you say about the diagonals of this
parallelogram?

18. ATL

The diagonals would be equal…

19. Interviewer

How do you know the diagonals are equal?

20. ATL

You have to measure and make sure these were, all
their sides were the same…right here [pointing on the
sides], would all equal… on each side all equaling the

70

same parts.

ATL declared a narrative about the diagonals of the parallelogram stating that, “the
diagonals would be equal”. The diagonals of this parallelogram are not equal, as can be detected
with a ruler. However, ATL did not check because her direct recognition was intuitive and also
self-evident. There was no need to substantiate the narrative, “the diagonals would be equal”, but
instead ATL made her own intuitive conclusion that the “diagonals were equal” because “…all
their sides were …the same”.
Ten weeks later, the same task was performed again. The change in ATL’s identifying
routines was evident. Table 4.7 summarizes ATL’s course of actions in response to the question
“what can you say about the angles of the parallelogram?” at the Post-Interview.
Table 4.7

ATL’s routines of verifying on the angles of parallelograms at the Post-

Interview

Q: “What can you say about the angles of this parallelogram?”

Rectangle

Parallelogram
Declared
Narratives

“Opposite angles equal and …
they don’t form 90-degreee
angle”

“…you could say that the opposite
angles are equal, and in this one
all angles are equal”

Q: “How do you know?”
Table 4.7 (cont’d)

71

Routines

a. Visually identify partial
properties of a parallelogram by
checking the condition of
opposite angles
(Identifying routine)
b. Describe a parallelogram with
no right angles (Defining routinerecalling)

a. Visually identify partial
properties of a rectangle by
checking the condition of opposite
angles (Identifying routine)
b. Describe a rectangle with right
angles (Defining routine –
recalling)

Declared
Narratives

“I would just say the property of
parallelogram” [po2.118]

“It has properties of
parallelogram. It’s a rectangle”
[po2. 4]

Recall that, at the Pre-Interview, ATL did not know how to draw a conclusion about the
angles in a parallelogram without measurements. At the Post-Interview, ATL was able to discuss
the angles of parallelograms using the properties of a parallelogram (defining routine). For
example, when ATL declared the narrative “opposite angles are equal and they don’t form a 90degree angle”, she identified that this 4-sided polygon was a parallelogram (identifying routine)
and described the parallelogram, as it had no right angles using defining routines. Similarly, ATL
was able to identify the differences of the angles between two parallelograms: a parallelogram,
“opposite angles are equal and … they don’t form a 90-degreee angle” and a rectangle, “the
opposite angles are equal, and in this one [rectangle] all angles are equal”.
In this scenario, we begin to see the change in ATL’s routines of identifying, from visual
recognition, to identifying visual properties of the angles in a parallelogram. ATL’s routines of
defining also showed a use of definitions of parallelograms to justify her claims at the PostInterview. However it is important to note that ATL’s routine of defining was more of a
recalling, as it appeared to be memorization of the facts.

72

During the Pre-interview, ATL showed more confidence in discussing the sides of the
parallelogram than the angles of the parallelogram. When I asked ATL about the sides of
parallelogram, the following conversation took place:
9. Interviewer

What can you say about the sides of this
parallelogram?

10. ATL

Opposite sides are equal…

11. Interviewer

How do you know they are equal?

12. ATL

Basically… on just the properties of a parallelogram. If
I measure it out…if I draw it with a ruler, it would
have to be the same for each side

13. Interviewer

Is there a way that you can show me that the opposite
sides are equal?

14. ATL

I would draw it out with two sides having to be the
same measure and these two having to be the same
measure. But for one of the opposite sides, they have
to be longer than others to not to make it the properties
of a square.

In this dialogue, ATL declared a narrative about the sides of the parallelogram, “opposite
sides are equal”. When asked for substantiation, ATL justified her claim by saying “just the
property of a parallelogram” [12]; and provided another explanation, “If I measure it, draw it out
with a ruler...it would have to be the same” [12]. After another prompt, ATL verbally described a
73

set of procedures to justify her claim: “draw on with a ruler”, and “draw it out with two sides
having to be the same measure” [14]. ATL’s routines of verifying the sides of a parallelogram
and a rectangle at the Pre-Interview are summarized in Table 4.8.
Table 4.8

ATL’s routines of verifying on the sides of parallelograms at the Pre-

Interview
Q: “What can you say about the sides of this parallelogram?”

Parallelogram
Conjecture
(Guessing)
Interviewer

Rectangle

“Opposites are equal”
“Opposite sides … one longer
than the other”

“Each opposite side is equal in
length”

“How do you know?”

Routines

a. Visual recognition
b. Identify partial properties of a
parallelogram (defining routinerecalling)

a. Visual recognition
b. Identify partial properties of a
parallelogram (defining routinerecalling)

Declared
Narrative

“just on the properties of a
parallelogram”

“rectangle can still have the
properties of a parallelogram”

At the Pre-Interview, ATL identified the equal sides of a parallelogram and a rectangle
intuitively and verified her claims using properties of parallelograms. However from ATL’s
description of a parallelogram, “opposite sides one longer than the other”, I conclude that ATL
was at the stage of identifying parallelograms by their visual appearances.
In contrast to her responses at the Pre-Interview, ATL declared, “Opposite sides are
parallel and equal” in referring to the sides of a parallelogram at the Post-Interview. To verify
her claims, ATL mentioned only “the properties of a parallelogram”.

74

109. Interviewer

What can you say about the sides of this
parallelogram?

110. ATL

The sides would be the same…or they should be
opposite sides are parallel and they should be equal…

111. Interviewer

Why do you say ‘it should be’?

112. ATL

Because it has the properties of a parallelogram. …by
looking at it, it looks as if they are, so it could be
good.

113. Interviewer

Is there a way that you can show me that the opposite
sides are equal and parallel?

114. ATL

Based on the properties of it [parallelogram].

ATL provided the narrative, “the [opposite] sides would be the same… or should
be…parallel” [110]. After several prompts, I found that ATL’s course of actions consisted of
visual recognition, “by looking at it”, and recalling using what she remembered as, “the
properties of it”. Table 4.8 summarizes ATL’s routine procedures concerning the sides of a
parallelogram and a rectangle at the Post-Interview.
Table 4.8

ATL’s routines of verifying on the sides of parallelograms at the Post-

Interview
Q: “What can you say about the sides of this parallelogram?”

Parallelogram
75

Rectangle

Table 4.8 (cont’d)
Declared
Narratives

“opposite sides are parallel and
they should be equal”

“opposite sides would be
congruent” “They are
parallel to one another”

Q: “How do you know?”
a. Visually identify partial
properties of a parallelogram by
checking the condition of the
sides (Identifying routine)

a. Visually identify partial
properties of a
parallelogram by checking
the condition of the sides
(Identifying routine)

Routines
b. Describe a parallelogram –
opposite sides are equal &
parallel (Defining routinerecalling)

Declared
Narratives

“It has the properties of a
parallelogram”

b. Describe a particular
parallelogram-opposite
sides are equal & parallel
(Defining routinerecalling)
“It’s a basic property of a
parallelogram”
“To be a parallelogram,
opposite sides have to be
parallel, making them
congruent”

ATL described opposite sides as parallel and equal for both parallelograms and rectangles
at the Post-interview, whereas she only mentioned opposite sides as equal at the Pre-Interview.
In verification, ATL’s routine procedures consisted of identifying routines and recalling at the
Post-Interview. ATL’s changes in routines procedures were (1) from visual recognition that was
self-evident, to identifying partial properties and using properties about the angles of a
parallelogram, (2) from identifying partial properties (i.e., equal sides) at the Pre-Interview, to
identifying more properties (i.e., equal and parallel sides) at the Post-Interview as described in
Table 4.9.

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Table 4.9

Summary of ATL’s routines in Task Two from both Interviews
Pre-Interview
Routines

Parts of
Parallelograms

Post-Interview
Routines

Identifying
Routine

Defining
Routine

Identifying
routine

Defining
routine

No

Visual
recognition/
Identifying
partial property

Recalling

Recalling

No

Angles

Visual
recognition
Self-evident

Sides

Visual
recognition/
Identifying
equal sides

Recalling

Visual
recognition/
Identifying
equal and
parallel sides

Diagonals

Visual
recognition
Self-evident

No

Visual
recognition
Self-evident

ATL did not think that squares and rhombi were parallelograms at the Pre-Interview;
therefore my examples of ATL’s identifying and defining routines for Task Two are limited to
the cases of a parallelogram and a rectangle. In both interviews, I did not find a substantiation
routine. ATL did not use measurement tools to verify the congruent angles and sides of
parallelograms, nor did she use definitions and triangle congruency to construct newly endorsed
narratives. ATL’s identifying routines changed from direct recognition, to recalling and
identifying partial properties of parallelograms, when discussing the angles and diagonals of
parallelograms.
When searching for routine patterns, I noticed that ATL’s understanding of the names of
parallelograms, and the names of the parts of the parallelograms, influenced her course of action
in response to the questions related to them. In the following section, I present findings on the
changes in ATL’s use of mathematical words.
77

Analyzing ATL’s use of words helped me to better understand her thinking about
parallelograms, and about the relations among the angles, sides and diagonals of parallelograms.
In this section, I present findings on ATL’s use of mathematical words related to parallelograms.
Recall that, a quadrilateral is defined as “a closed shape in a plane consisting of four line
segments that do not cross each other” (Beckmann, 2008). Among all quadrilaterals, five types
of quadrilaterals are found predominately in school geometry textbooks: parallelograms,
trapezoids, rectangles, squares, and rhombuses (Usiskin, 2008). Interviewees in my study are
also introduced to kite (Parker & Baldridge, 2008) in their course work, and therefore the word
search on quadrilaterals includes also kites. Tables 4.10 and 4.11 summarize the frequencies of
the names of quadrilaterals mentioned by ATL at the Pre-Interview and the Post-Interview:
Table 4.10

The frequencies of ATL’s use of the names of quadrilaterals at the interviews
Name

Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Table 4.11

Pre-T1
2
13
8
12
5
5
0

Pos-T1
10
12
6
14
7
7
1

Frequency
Pre-T2
Pos-T2
0
0
17
12
7
7
9
4
1
6
0
0
0
5

Pre-T3
0
6
0
6
0
0
0

Pos-T3
1
8
1
0
0
0
0

Total frequencies of ATL’s use of names of quadrilaterals at the interviews
Frequency
Pre
Post
2
11
36
32
15
14
27
18
6
13
5
7
0
6

Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

78

Table 4.11 shows that the word parallelogram (n=68) was the most frequently used word
during the interviews and it was mentioned in all three tasks. The word square (n=45) was the
second most frequently used, and the word rectangle (n=29) was third. In contrast, the word kite
(n=6) is the least mentioned during the interviews, used only at the Post-Interview, in Task One
(n=1) and Task Two (n=5). There was an increase in use of the word quadrilateral at the PostInterview, and it was used mostly in Task One (n=10), and a total of eleven times in the entire
interview. Also, there was an increase in use of the word rhombus at the Post-Interview (n=13).
However, the frequencies of the words do not provide details about how and in what way those
words were used. The following findings provide more information regarding ATL’s word
meaning in the use of parallelogram, rectangle, square and rhombus. Let me begin my analyses
of ATL’s word use with this conversation at the Pre-Interview:
15. Interviewer

What is a parallelogram?

16. ATL

A parallelogram is when two sides of each side… all
four are parallel to the opposite one…

17. Interviewer

What is a rectangle?

18. ATL

A rectangle is the two longer sides… the shorter ones
… but in more technical terms, I am sure that they have
congruency on both of those sides too

19. Interviewer

What is a square?

20. ATL

The square is all four of the sides are completely the
same

21.Interviewer

What is a rhombus?

22. ATL

A rhombus… is a square… is just tilted [giggling]

79

ATL’s narratives concerning parallelogram, rectangle, square, and rhombus are
descriptive and visual at the Pre-Interview. ATL gave a descriptive narrative about rectangles
based on physical appearance, “a rectangles is the two longer sides [and two] shorter ones…
have congruency on both…sides”. ATL made connections between squares and rhombi
according to visual appearances, and declared narratives, “a rhombus is a square”, because “they
both have four equal sides”, and “[it] is just a titled [square]”. ATL’s ways of defining
parallelograms triggered the way she classified them. For example, when ATL was asked to
identify all the parallelograms from a set of given figures, her response was as follows (see
earlier analyses about the routine of sorting):
53. Interviewer

Can you identify all the parallelograms on this sheet?
[Pointing to task One]

56a. ATL

Ok. [Marking stars on figures that are parallelograms]

56b. ATL

…Now for these ones, these could be actually… be
considered parallelograms. Based on the side measures …
even though they are rectangles… they could be in the
same category.
80

ATL’s classification was no coincidence. In ATL’s written narratives about the
rectangles, she wrote, “rectangle is when 2 sides are differing from the other 2 sides, however,
opposite sides are equal in length”, and for the parallelograms, she wrote, “parallelogram is
when 2 parallel sides are congruent in length”. To ATL, the word parallelogram was a family
name of figures having opposite sides that were parallel, and having two long sides and two short
sides. For example, when I asked ATL to draw a parallelogram, she provided the following:
1. Interviewer

Draw a parallelogram.

2. ATL

[ATL Draw a figure looking like this]:

3. Interviewer

How do you know this is a parallelogram?

4. ATL

The opposite are equal in length… with the
different sides parallel, they are the same length.

Next, I asked ATL to draw a new parallelogram different from the one she drew:
23. Interviewer Draw a new parallelogram that is different from the one
you drew.
24a. ATL

[ATL Drew a figure looking like this]:

81

24b. ATL

All I know is to change the size of it, but that’s more of
a rectangle…. But it’s not a typical looking
parallelogram…
…….

30a. ATL

I feel it’s a rectangle, but rectangles can still have the
properties of a parallelogram… just a broad term for it.

33. Interviewer Can you say a little more about why this parallelogram
[rectangle] is different from this one [the parallelogram
ATL drew earlier]
34. ATL

They aren’t. Technically, they’re probably not different,
that one just looks more like a stereotypical
parallelogram [Pointing on the parallelogram]. In terms
of properties, there is nothing different.

ATL drew two parallelograms: “a stereotypical parallelogram” and a “not typical looking
parallelogram”. After ATL drew these parallelograms, I presented a picture of a square and a
picture of a rhombus. ATL did not think a square and a rhombus were parallelograms because
“to be a parallelogram, you have to have two long sides and two short ones, here all sides are
equal and it is square”. In the case of a rhombus, ATL responded, “this is similar to the square
that you just showed me, … is a rhombus or just a square”. From these conversations, it is
evident that to ATL the word parallelogram signified two types of polygons, as summarized in
Figure 4.7.
82

Parallelogram

Figure 4.7

ATL’s use of the word parallelogram at the Pre-Interview

ATL’s use of the word parallelogram signified a collection of unstructured polygons by
their family appearances. This family appearance included figures appearing to have opposite
sides equal and parallel, and in particular, two opposite sides longer than the other two opposite
sides. However there was no explicit mention of the necessary condition that these figures be 4sided, nor of any condition on the angles in rectangles.
At the Post-Interview, when I asked ATL to identify all the parallelograms from eighteen
polygons, her response was as follows:
19. Interviewer

What are the parallelograms here? [Pointing to Task One]

20. ATL

L and J and H will be just parallelograms, but all of these
figures [pointing to figures that are squares, rectangles and
rhombus] will be parallelograms, because…they all fit
into the greater property of opposite angles and opposite
sides to be equal.
83

ATL identified two groups of parallelograms: one group contained figures that were “just
parallelograms”, and the other group contained figures that “fit into the greater property of
opposite angles and opposite sides to be equal” . As our conversation continued, ATL provided
the following narratives about the parallelograms:
51. Interviewer What is a square?
52. ATL

A Square is when all the angles form right angles and
they are all the same they are all 90 degrees…and each
side length also has to be the same. [Pointing at Fig. U]

53. Interviewer What is a rectangle?

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54. ATL

A Rectangle, each angle is 90 degrees but these sides are
the same and parallel, and this one is the same and
parallel, but not all 4 of them are the same, necessarily
[Pointing at Fig. M]

55. Interviewer What is a parallelogram?
56. ATL

Um… a parallelogram is when opposite sides are equal
and opposite angles are both equal…
[Pointing at Fig. J]

63. Interviewer What is a rhombus?
64. ATL

Sides are all the same. Does not form 90-degree angle as
rhombus alone.
Pointing at Fig. Z:

85

To better understand her word meaning in the context of parallelograms, I asked ATL if I
could group Fig. J and Fig. Z together, and group Fig. U and Fig. M together. Her response was
“yeah”. The following conversation gives ATL’s responses to these questions ten weeks later:
37. Interviewer: Can I group Fig. J and Fig. Z together?

38. ATL: Mm Hmm.
39. Interviewer: Why is that?
40. ATL: Mm… because they both have opposite sides parallel
and both opposite angle measures are equal.
……
45. Interviewer: Can I group Fig. U and Fig. M together?

46. ATL: Yeah, you can because U has the same property as M.
The only differences is that M does not have all the same sides
length, so M would not have all the properties as U, but U has all
the properties of M…

In these conversations, more dimensions were added to ATL’s use of the word
parallelogram. At the Pre-Interview, the word parallelogram only signified polygons that fit into
86

the physical appearances of parallelograms and rectangles; whereas at the Post-Interview, the
word parallelogram signified a family of polygons that share common descriptive narratives.

Parallelogram

Figure 4.8

ATL’s use of the word parallelogram at the Post-Interview

As shown in Figure 4.8, the word parallelogram signified to ATL a common family
name for all figures that “have opposite sides parallel and opposite angles equal”. This diagram
illustrates how parallelograms were inter-connected. For example, ATL identified that “as a
rhombus alone” [it] does not form a 90-degree angle, and “sides are all the same”. A rhombus
was different from a square with regard to the angles: “all the angles form right angles…and
each side length also has to be the same”. However ATL did not mention how rectangles were
different from parallelograms.
To have a better understanding of ATL’s familiarity of geometric terms, besides names
for quadrilaterals, I searched for words describing the parts of parallelograms. Tables 4.12 and
4.13 present frequencies of the names of parts of parallelograms.

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Table 4.12

The frequencies of ATL’s use of the names of parts of parallelograms at the

two interviews
Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.13

Pre-T1
10
27
2
0
1
0
2
2

Pos-T1
17
17
6
3
6
1
1
5

Frequency
Pre-T2
Pos-T2
10
75
31
33
3
21
1
1
9
17
27
20
0
2
0
13

Pre-T3
14
9
11
0
3
8
0
2

Pos-T3
39
40
3
0
7
10
0
0

Total frequencies of ATL’s use of names of parts of parallelograms at the

two interviews
Frequency
Pre
Post
34
131
67
90
16
30
1
4
13
30
35
31
2
3
4
18

Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.13 shows that the most frequently used words relating to the parts of
parallelograms were angle (n=165) and side (n=157). There was a huge increase in use of the
word angle at the Post-Interview (n=131) over the Pre-Interview (n=34). Besides angle and side,
diagonal was also frequently mentioned at both interviews (n= 66). However parallel sides
(n=5), as one of the most important characteristics of a parallelogram, was the least mentioned.
There was also an increase in use of opposite side (n=30) and opposite angle (n=18) at the PostInterview.
88

These changes of word use associated with parts of parallelograms were not incidental.
Recall that, at the Pre-Interview, ATL showed frustration in discussing the angles without
measurements, and consequently, her responses about parallelograms and parts of parallelograms
focused on the sides (n =67) rather than angles (n=34). At the Post-Interview, she was more
comfortable talking about angles of the different parallelograms and was able to compare the
differences between angles of the parallelograms (e.g. rectangle and parallelogram, square and
rhombus, etc). Moreover, ATL used a narrative, “they have opposite sides and opposite angles
are equal” to classify quadrilaterals into a group of parallelograms. These changes in ATL’s
word use and routines show that she gained more familiarity with the triangles and
quadrilaterals, as well as with the properties of these polygons.

Case 2: Changes in ANI’s Geometric Discourse
ANI was a college sophomore at the time of the interviews, having taken her last
geometry class six years prior to the geometry and measurement class. The van Hiele geometry
Tests showed that she was at Level 0 at the pretest, and moved to Level 3 at the posttest. I
interviewed ANI after both tests, and analyzed her interview responses. ANI’s changes in
geometric discourse are summarized as follows:
•

ANI’s routines of sorting changed from grouping by the names of polygons,
according to their family appearances and visual properties with no order, to
classifying polygons according to their common descriptive narratives, and
structuring quadrilaterals with a hierarchy of classification.

•

ANI’s routines of substantiation changed from visual recognition and recalling at
the Pre-Interview, to using endorsed narratives such as definitions and properties

89

of parallelograms at the Post-Interview; and from comparing parts of
parallelograms visually in the Pre-Interview, to applying various methods (e.g.,
Pythagorean theorem, congruence criterion, algebraic derivations) to verify claims
at an object level in the Post-Interview.
•

ANI’s use of the names of parallelograms changed from visual recognition of
their family appearances at the Pre-Interview, to using these names as collections
of quadrilaterals sharing common descriptive narratives in a hierarchy of
classification.

ANI’s routine procedures for sorting polygons were observed and analyzed in Task One.
She was asked to sort eighteen polygons into groups, consisting of triangles (n=4), quadrilaterals
(n=13) and a hexagon (n=1).
During the Pre-Interview, ANI’s first sorted polygons by their names based on family
appearances, finding triangles (n=5), squares (n=3), rectangles (n=3), parallelograms (n=5) and
trapezoids (n=2). She grouped Fig. V, a hexagon, with the triangles because she identified two
triangles in Fig. V.
2b. ANI

I made a group of triangles, which had three sides.

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2c. ANI

I included V as two triangles, because if you draw
a line here in the middle, it would make two.

drew a dashed line

ANI put Fig. N, a right trapezoid, and Fig. Q, a quadrilateral, together as a trapezoid
group because she was not sure about what to do with these two polygons. She named the group
trapezoid by guessing.
2f. ANI I did one that just had Q and N, which I didn’t
know what to classify. I guess they would be
trapezoids.

When asked for regrouping, ANI combined rectangles and squares together in one group
because she thought that “every square is a rectangle”. She then split the triangle group into right
triangles and isosceles triangles, but did not know what to do with two other triangles.
16. ANI

That’s a right triangle [pointing at Fig. K], and
that’s an isosceles triangle [pointing at Fig. W].

91

17. ANI

I guess X would, I don’t know what the
definition is [pointing at Fig. X], and this one
[pointing at Fig. S] none of the sides on that
looked even to me, so I don’t think it looked like
any of the others

ANI grouped two triangles according to the visual properties of angles (i.e., right
triangle) and sides (i.e, isosceles triangle), and left two other triangles (Fig. X and Fig. S) with no
groups. When subgrouping the parallelograms (n=5), ANI explained as follows:
22a. ANI

L and Z look more like square, I don't know if
there are two different types of
parallelograms…

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22b. ANI

Where P, J, and H look more of a rectangle,
like the opposite side of all of these look the
same. So, I guess you could put it into two
groups that way.

ANI split the parallelograms into a group of squares including Fig. L and Fig. Z, and a
group of rectangles consisting of Fig. P, Fig. J and Fig. H, because they looked like squares and
rectangles, respectively. When I asked for the definitions of parallelogram, rectangle, rhombus
and square, ANI provided her definitions of each:
23. Interviewer

What is a parallelogram?

24. ANI

It has four sides. I think the opposite sides
are the same… I guess it's kind of like a
slanted rectangle.

25. Interviewer

What is a rectangle?

26. ANI

It has four sides with opposite sides being
equal in length.

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ANI made a connection between a parallelogram and a rectangle, as a parallelogram was
a slanted rectangle, based on visual appearances. ANI next gave definitions of square and
rhombus:
29. Interviewer

What is a square?

30. ANI

It has four sides of the same length.

31. Interviewer

What is a rhombus?

32. ANI

A rhombus is like a slanted square and a
parallelogram is a slanted rectangle. They
kind go together like that.

ANI described a square as having four sides of the same length, whereas a rhombus was a
slanted square. She did not mention right angles, a defining condition of rectangles and squares
among the parallelograms. To find out whether ANI considered squares as parallelograms, I
continued:
51. Interviewer

Can I group J and U together?

52. ANI

[Thinking] …You could if you talked about
angles, I guess. You'd say this angle and this
angle are equal [pointing at the opposite
angles of Fig, J], and this angle and this angle
are equal, where the opposite angles are

94

equal here [Fig. U]. So, I guess you could
group it in that way.

ANI agreed that Fig. J, a parallelogram, could group with Fig. U, a square, because
opposite angles were equal in both polygons [52]. Later I found that ANI did not identify a
square as a parallelogram because “a square has all the same length, and a parallelogram has
different sides”. In this case, ANI identified polygons by visual property of their angles.
However, ANI’s response was different when I asked if I could group Fig. J and Fig. M, a
rectangle, together:
53. Interviewer

Can I group J and M together

54. ANI

Yeah. Like J and U, opposite angles are equal…
you also have opposite sides are parallel
[pointing at the sides of Fig. J]…and this side
and this side are both shorter than the other ones
[pointing at the sides of Fig. M], they [Fig. J and
Fig. M] have quite a bit in common.

ANI acted more positive towards the grouping of Fig. J and Fig. M, as she thought, “they
have quite a bit in common” [54]. ANI explained that a rectangle was a parallelogram “because
95

it has two opposite sides equal, a rectangle and a par
parallelogram go together”. In this case, ANI
.
identified polygons by visual property of their sides. ANI did not use any measurement tool to
check the parts of polygons during the Pre Interview in Task One. Her routines of regrouping
Pre-Interview
polygons included direct recognition and identification of polygons by visual properties of angles
identification
and sides. ANI’s routines of sorting polygons at the Pre
Pre-Interview are summarized in Figure 4.9.
rview
4.9
First prompt: “Sort the shapes into groups”
Direct recognition
(Identifying routine)

Grouping by the
names of polygons

Conclusion

Second prompt: “Can you regroup them differently?”
Direct recognition
(Identifying
routine)
Figure 4.9

Grouping by family
appearances & partial visual
properties of angles and sides
of polygons

Conclusion

ANI’s routines of sorting polygons at the Pre Interview
Pre-Interview

Pre-Interview,
In ANI’s routines of sorting triangles and quadrilaterals at the Pre Interview, I did not
find a defining routine because she did not use definitions to group polygons. I found identifying
routines, including direct recognition when she grouped quadrilaterals by their visual
recognition,
appearances, and identifying visual properties when she identified polygons by the attributes of
their angles and sides.
Ten weeks later when I interviewed ANI again, her routines of sorting polygons had
changed from grouping polygons by their visual appearance, to classifying them by common
descriptive narratives with a hierarchy of classifications.
At the Post-Interview, ANI first grouped polygons by their names, finding quadrilaterals
Interview,
(n=13), trapezoids (n=1), parallelograms (n=9), rectangles (n=6), rhombi (n=5) and triangles
96

(n=4). ANI grouped all 4-sided figures into the quadrilaterals group, and all 3-sided figures into
the triangles group. She included parallelograms, squares, and rectangles as parallelograms, but
not the rhombi. The rectangles group consisted of rectangles and squares, and the rhombi group
included squares and rhombi. Note that Fig. V, a hexagon, was not included in any of these
groups. Figure 4.10 presents ANI’s groups of parallelograms, rectangles and rhombi.
Parallelograms group

Rectangles group

Rhombi group

Figure 4.10

ANI’s grouping of parallelograms at the Post-Interview.
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As shown in Figure 4.10, ANI grouped polygons by their common descriptive narratives
(i.e., definitions). For example, ANI explained that she grouped squares and rectangles together
as a rectangles group because “all squares are rectangles, and rectangles have all 90-degree
angles”. She verified that both squares and rhombi were rhombi because “that is what a rhombus
is, four sides of equal length”.
There was a change in ANI’s identifying routine. Recall that at the Pre-Interview ANI
grouped rhombi and squares together, because a rhombus looked like a slanted square. But at the
Post-Interview, she grouped them together because they share a common narrative of having four
sides of the same length. Although ANI grouped rectangles and squares together at both
interviews, there was a difference as her identifying routine changed from recalling that all
squares were rectangles at the Pre-Interview, to identifying common properties of rectangles and
squares at the Post-Interview.
ANI’s identifying routine also changed from grouping the quadrilaterals by their names
as unstructured polygons, to classifying the quadrilaterals with a hierarchy. Figure 4.11
illustrates all of ANI’s the subgroups of quadrilaterals (n=13). ANI classified the quadrilaterals
beginning with the attributes of their sides. This classification included trapezoid, a quadrilateral
with one pair of parallel sides; parallelograms, quadrilaterals with two sets of parallel sides;
rhombus, quadrilaterals with all sides equal; and Fig. Q, a quadrilateral with no two sides equal.
ANI next split parallelograms into parallelograms and rectangles, and split the rhombus group
into rhombi and squares by the characteristics of right angles. ANI did not use rulers or
protractors to check measurements of angles and sides at the Post-Interview, but she did explain
that sides looked like they were parallel, or angles looked like they were right angles, etc. I

98

conclude that ANI used direct recognition as an identifying routine, but it as not self-evident.
ANI’s routines of sorting polygons at the Post-Interview are summarized in Figure 4.12.

Quadrilaterals (n=13)
Trapezoid (n=1)

Parallelograms (n=9)

Rhombus (n=5)

Rectangles ( n=6)

Figure 4.11

Square (n=3)

ANI’s grouping of the quadrilaterals at the Post-Interview

99

First prompt: “Sort the shapes into groups”
Direct recognition
(Identifying
routine)

Grouping by the common
descriptive narratives of
polygons

Conclusion

Second prompt: “Can you regroup them differently?”
Direct
recognition
(Identifying
routine)

Grouping by common descriptive
naratives and properties (i.e., angles &
sides) of polygons with a hierarchy of
classifications of quadrilaterals

Figure 4.12

Conclusion

ANI’s routines of sorting polygons at the Post Interview
Post-Interview

ANI’s routines of substantiation for Task Two were analyzed and compared from both
interviews. I found that ANI’s routines changed from direct recognition and recalling in the PreInterview, to identifying properties of polygons and using triangle congruence criterion to
ce
substantiate her claims. In the Post Interview, she also derived some statements algebraically,
Post-Interview,
and used Pythagorean theorem.
In Task Two, ANI was asked to draw two different parallelograms and to discuss their
properties. At the Pre-Interview she drew a parallelogram and labeled the vertices as A, B, C and
iew
D in clockwise order. She then wrote a statement regarding the angles of the parallelogram, “∠A
= ∠C, ∠B = ∠D”. The following conversation took place when I asked for verification.
he
3. Interviewer

What can you say about the angles of this parallelogram?

ANI’s drawing of a parallelogram

100

4. ANI

Angles A and C are equal, and then B and D are equal
[Pointing to angles A and C, she wrote: ∠A = ∠C, ∠B = ∠D].

5. Interviewer

How do you know?

6. ANI

Mm…I just remember being taught that, I don't actually
know… If you measured them, they would be equal.

ANI referred to her prior knowledge about the angles of a parallelogram to conclude that
the opposite angles were equal. She was able to use mathematical symbols, such as ∠A = ∠C, to
indicate the equivalence of the angles. However ANI remembered the property as a fact without
knowing the explanations. After my prompt, ANI verified that ∠A = ∠C by comparing the space
between the angles:
10a. ANI

If you drew two lines here [adding two perpendicular
lines from angles A and C]
A

B

D

10b. ANI

C

From this line to this line, if you know that’s a 90
degree angle …[pointing at the indicated space]
A

B

D

10c. ANI

C

You know what a 90-degree angle looks like, so it's
easier to go off of that.

101

10d. ANI

…and then you look at the space here and the space
here [pointing at the marked angles], and see if those
are the same.
A

B

D

C

ANI drew two perpendicular lines so that both ∠A and ∠C would be the sum of a right
angle and a smaller angle. [10a]. She started with comparing the right angles because they were
easy to distinguish by their visual appearance [10b; 10c], and then she compared space between
the two smaller angles to check whether they were the same. In this example, ANI’s routines
procedure relied on the visual appearance of the angles to verify her claims.
ANI made two other statements about the angles of the parallelogram, “∠A + ∠D =
180°, ∠C + ∠B = 180°”. When asked for substantiation, she responded as follows:
20a. ANI

If you extend this line out, and draw another straight line
there…
A

D

20b. ANI

A

B

C

D

You can see that this angle equals that angle.

102

B

C

A

B

D

C

ANI pointing at the two marked angles.
21. Interviewer

How do you know they are equal?

22a. ANI

There is a term for it. It’s like a rule that I remember
learning. Maybe parallel angle rule? Or, adjacent angle
rule? Or something.

22b. ANI

And this is angle C [pointing at the angle BCD].

22c. ANI

So if you were to combine them, you would have a
straight line, and that would make a 180-dregee angle.
A

B

D

C

ANI pointing at the two marked angles.

In this example, ANI recognized the structure of alternating interior angles formed by
parallel lines and their transversals [20a]. When ANI explained, “you can see that this angle
equals that angle” [20b], she again relied on the visual appearance of the angles. She did not
know names of the angles, nor the related propositions to support her claim, but referred to a rule
that she had learned. Assuming that the two marked angles were equal [20b], she verified that
∠C + ∠B = 180° because they made a 180-degree angle [22c]. Using the same reasoning, ANI
103

also stated that “∠D + ∠A = 180 I conclude that ANI used visual recognition to compare
A 180°”.
angles, and applied prior knowledge as a fact to verify the statements about the angles of a
parallelogram.
Interview,
In the Post-Interview, ANI drew a parallelogram and labeled the vertices as A, B, C, and
D in a clockwise rotation, and gave the same statements about the angles of the parallelogram as
in the Pre-Interview. Figure 4.13 shows some of ANI responses for Task Two in the Post.
Interview.

Figure 4.13

ANI’s written responses for Task Two at the Post Interview
Post-Interview

In contrast, ANI’s verification of her claims was different than at the Pre-Interview.
PreWhen I asked for substantiation of the statement ∠A = ∠C and ∠B = ∠D, she explained that
“this is a parallelogram because I drew it, so angle A is equal to angle C and angle B is equal to
angle D”. ANI’s verification of ∠ + ∠B = 180° was also different than in the Pre-Interview.
∠A
Pre
16a. ANI

If you were to extend this line…

104

B

C

A

16b. ANI

D

You could look either way, like this angle is equal to
this angle. BC and AD are parallel. They [pointing
at the marked angles] are corresponding angles
because they are on the parallel lines.

B

C

A

16c. ANI

D

Then you could tell that if you add these two angles,
it’s angles on a line. So it’s 180 degrees.
B

C

A

D

pointing at angle B and it’s exterior angle.
16d. ANI

So angle A and angle B add up to 180 degrees.

To verify ∠A + ∠B = 180°, ANI extended side AB so that the structure of the
corresponding angles formed by parallel lines and their transversals was visible [16a]. She
mentioned the corresponding angles were congruent because they were on the parallel lines
[16b], and then concluded that angle A and angle B add up to 180 degrees [16c; 16d]. Although
ANI verified her claim informally, it is important to see the change, as ANI justified her claim
105

that corresponding angles were equal using an endorsed narrative that BC and AD were pa
parallel
at the Post-Interview, whereas she relied on visual appearance of the angles at the Pre
Interview,
Pre-Interview.
ANI provided different narratives about the diagonals of a parallelogram at the two
interviews. At the Pre-Interview, she asserted that the diagonals of a rhombus should be the same
Interview,
us
and the diagonals of a rectangle should intersect at the mi
middle of the rectangle, whereas at the
,
Post-Interview she stated that the diagonals of a rectangle should be equal in length, and the
Interview
diagonals of a parallelogram bisect each other at the Post Interview. Now I describe and compare
sect
Post-Interview.
the changes in ANI’s routine procedures of substantiating these narratives at both interviews.
At the Pre-Interview, ANI drew a rhombus, and stated that it was a different
Interview,
parallelogram because all the sides were the same length. ANI later used the statement, “all the
ause
sides were the same length” to draw conclusions about the angles and diagonals of the rhombus.
73. Interviewer

What can you say about the angles of this
parallelogram

ANI’s drawing of a rhombus
74. ANI

I think they [angles] should all be the same… I
guess if I had drawn it better, all the angles
should be the same.

75. Interviewer

Why do you think they are the same?

106

76. ANI

Because the lengths are the same. [Wrote: ∠A
= ∠B = ∠C = ∠D because lengths of the sides
are the same].
…

93. Interviewer

What can you say about the diagonals of this
parallelogram?

94. ANI

I think the diagonals should be equal in length.

95. Interviewer

How do you know that they should be equal in
length?

96. ANI

Because the sides are all the same length and
the angles are all the same. So, I think the
diagonals should be the same.

In the preceding conversation, ANI claimed that the angles in a rhombus should all be the
same [74], as well as the diagonals [94]. When asked for verification, she explained that all the
angles should be the same because all the sides of a rhombus were the same [76]; and diagonals
should be equal in length because all the sides and angles of a rhombus were all the same [96].
Of course, for ANI’s conclusions about the angles and diagonals of this rhombus to be correct,
the rhombus had to be a square. ANI made incorrect implications from the equivalence of the
sides to the equivalence of the angles, and then suggested that the diagonals must be equal
because of equal sides and angles. There is no routine involved in this verification, other than
making statements based on the fact that all the sides are equal in a rhombus. Obviously, ANI
did not have a correct understanding of a rhombus at the Pre-Interview.

107

ANI stated that the diagonals of a rectangle were longer than its sides, and the
intersection of the diagonals was at the middle of the rectangle.
118b. ANI

The length of this [pointing at the diagonal] is longer
than the length of the longest side [pointing at the
longer side of the rectangle].

diagonal
longest side of rectangle
118c. ANI

They [the diagonals] should intersect in the middle.

middle of the rectangle

Here, ANI’s declared narratives about the diagonals of this rectangle were more like
visual descriptions of what the diagonals appeared to be. She recognized the diagonal as the
hypotenuse of a right triangle, and mentioned, “the Pythagorean theorem, which is how I know
it” to verify the diagonals were longer than the longest side of rectangle. She argued her claim
that diagonals should be at the middle of the rectangle as follows:
130a. ANI

So, if you were to find the midpoint of this
length… [Drew one line passing through the
midpoint of the sides]

108

130b. ANI

…and then if you were to find the midpoint of
this length, and draw a line [Drew another line
passing through the midpoint of the other sides]

130c. ANI

… that the intersection of those two lines should
be the intersection of the diagonals as well.

ANI verified that the diagonals intersect at the middle of the rectangle by locating the
midpoints of the sides of the rectangle, and concluded that the intersection of the two medians
was the same point as the intersection point of the diagonals.
In the preceding examples, ANI’s understanding of the properties of parallelograms was
not clearly demonstrated. To verify the claims, she mostly described what she saw about the
parallelogram.
Ten weeks later I interviewed ANI again, and the same tasks were preformed. ANI first
drew a parallelogram, and then stated that the diagonals of the parallelogram were not equal in
length, but they cross each other at one point. She added that the diagonals create corresponding
triangles.
54. ANI

They [the diagonals] cross at one point.
B

A

C

D

109

56. ANI

They create corresponding triangles. Well, like this
triangle corresponds with this triangle [Shaded the
two corresponding triangles]
B

C

A

D

57. Interviewer What do you mean by “corresponding triangles”?
58a. ANI

This angle and this angle are equal, cause they're
vertical angles…
B

C

A

58b. ANI

D

…And then, this side should equal this side…
B

A

58c. ANI

C

D

…and this side should equal this side. And I know
they're corresponding through Side-Angle-Side.
Like, that would be the rule that…
B

A

C

D

110

59. Interviewer How do you know these sides are equal? [Referring
to 58b & 58c]
60. ANI

Cause diagonals bisect each other.

61. Interviewer How do you know they bisect each other?
62. ANI

I don't really know how I know… I guess it's
because the sides are equal length and they're
parallel, so…

In the preceding conversation ANI started with a descriptive narrative about the diagonals
of a parallelogram, “they cross at one point” [54], and then she asserted that the diagonals
created corresponding triangles [56]. At my request, she verified the corresponding triangles
were a pair of congruent triangles with the Side-Angle-Side (SAS) criterion [58]. Here, ANI used
the endorsed narrative “diagonals bisect each other” to show that the corresponding triangles
were congruent [60]. However, when asked how she knew the diagonals of this parallelogram
bisect each other, she responded, “I don’t really know…I guess, it’s because the sides are equal
length and they’re parallel” [62].
It is clear that ANI remembered how to verify congruent triangles using SAS. She
identified corresponding triangles, as well as the three elements needed for verification of
congruent triangles. She used that fact that “diagonals bisect each other” to justify the
congruency of the sides, and used vertical angles to show the congruence of included angles.
Thus, there was a change in ANI’s routine of substantiating, from no routine at the PreInterview, to using an endorsed narrative to identify three elements for verifying congruent
triangles at the Post-Interview. In this case, ANI’s routine procedure for verifying her claim used

111

the endorsed narrative “diagonals bisect each other”, but she did not know as an endorsed
narrative, but did not know why the narrative was true. It appeared that ANI derived an informal
proof that two triangles were congruent, but she did not clearly demonstrate that she knew what
to substantiate and why.
When discussing the diagonals of a rectangle, ANI provided the narrative “they should be
equal in length”, and tried to verify this claim using the Pythagorean theorem:
128a. ANI

I know the value of this side [pointing at the shorter
side], and I know the value of that side [pointing at the
longer side], so I can find this side [hypotenuse] use
Pythagorean theorem.

128b. ANI

It would be the same over here. I know this and I
know this, so I can find this.

”I can find this”
128c. ANI

…you can see that those two are equal [pointing at the
halves of the diagonal]. And then, if I drew the other
diagonal, you could do it the same way…

112

129. Interviewer

What about this case? If I add the halves of this
diagonal, and add the halves of that diagonal, in my
drawing here, they are not equal.

To verify the diagonals were equal, ANI attempted to show that adding the halves of a
diagonal was equal to adding halves of another diagonal, and use the Pythagorean theorem. I
provided a counterexample to refute her conclusion. ANI replied with another approach by first
labeling the segments a, b, c and d.
134a. ANI

You could find that a is equal to d, they should be equal.

a

c

d

134b. ANI

b

I could do the Pythagorean theorem again, but with this
side and this side. And then I'd find d.

a
d

c
b

Pointing at the two legs of the right triangle

113

134c. ANI

a equal to d because they share this side here [pointing at
the longer leg], and this point is a middle point here, so
these two sides are equal [two shorter legs]. They [a and
d] are equal.

a
d

c
b

ANI described informally her verification of the claim “a = d” by identifying the two
triangles sharing a longer side and having equal shorter sides [134c], and then applying the
Pythagorean theorem to conclude “a = d” [134a; b]. ANI did not argue that the two triangles
were right triangles, an important condition of the Pythagorean theorem, nor did she give details
of the algebraic derivation of “a = d”. I conclude that ANI’s routine procedures changed from
comparing the length of diagonals with the sides visually at the Pre-Interview, to describing a
process of verifying and substantiating her claims at the Post-Interview.
ANI did not think a square was a parallelogram at the Pre-interview, and there was not
much to compare with what she did ten weeks later. However I do want to share ANI’s routine
procedure of using algebra as one way to substantiate her claims at the Post-Interview. We began
with the following conversation:
161. Interviewer

What can you say about the diagonals of the
square?

114

162. ANI

They’re equal… They bisect the angles, split
the angles into two-45 degree angles.

163. Interviewer

How do you know “they are equal”?

164. ANI

The same way I knew with the rectangles.

165. Interviewer

How do you know “they bisect angles”?

166. ANI

It divides the angle into two equal angles.

ANI provided two narratives about the diagonals of the square, “they’re equal”, and “they
bisect the angles”[162]. She applied her knowledge of the diagonals in a rectangle to case of a
square [164]. To verify the diagonals bisect each other, ANI explained that they divide the angle
into two equal angles [166]. The following is ANI’s routine procedure of verification, with
corresponding transcripts.
Table 4.14

ANI’s routine procedure of verification for “diagonals bisect the angles”

Routine Procedures

Transcripts

1. Declare narratives
1.1Draw a diagonal

174a. I guess I’d draw a diagonal

1.2 Identify two right
triangles

174b. It splits the square into two right triangles, because all
of these angles are 90-degrees.

adding right angle sign on each angle of the square
115

Table 4.14 (cont’d)
1.3 Identify the
relation between the
angles and sides of the
right triangle.

174c. By the angle sum rule, all angles add up to 180
degrees. You already have 90 here. So, X plus Y has to equal
90. It’s also an isosceles triangle.

assigning X and Y to the two angles
Q: How do you know it’s an isosceles triangle?
2. Verification of
isosceles triangle

180. These two sides equal.

2.1 Identify congruent
sides of the triangle

Adding two marks on the sides of the triangle
2.2 Verification of
congruent angles

182a. It’s an isosceles triangle. So X is equal Y.

2.3 Finding the angle
measures of X and Y
2.4 Finding other
angles measures

182b. I know that X and Y has to equal 90 degrees. So, I
know that X is 45 degrees and Y is 45 degrees.
190. So, if you know it's 90, and Y is equal 45 degrees, and
this angle is also 45 degrees. Same for X here. So diagonals
splitting into two equal angles and they are 45 degrees each.

3. Conclusion

“this angle is also 45 degrees”
194. Yeah, diagonals bisect each other

It became clear that ANI favored algebraic reasoning in her routine procedures. She
labeled the angles X and Y, and used an endorsed narrative, “all angles are 90 degrees” [174b] to
justify that X and Y were the angles of a right triangle. She used another endorsed narrative,
116

“theses two sides equal” [180] to verify that the triangle is isosceles. Finally, ANI solved X and
Y algebraically, to find that they were 45 degrees each [190]. Using this newly endorsed
narrative, ANI concluded that the diagonals bisect the angles [190; 194]. In this example, ANI
used her knowledge in algebra to help solve a problem in geometry.
ANI’s use of the word parallelogram changed from describing the visual appearances of
the quadrilaterals at the Pre-Interview, to using the word as a common descriptive narrative with
a hierarchy of classifications at the Post-Interview. Here are the frequencies of ANI’s use of the
names of quadrilaterals at two interviews:
Table 4.15

The frequencies of ANI’s use of the names of quadrilaterals at the two

interviews
Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Table 4.16

Pre-T1
0
4
12
10
2
2
0

Pos-T1
3
3
3
5
2
2
1

Frequency
Pre-T2
Pos-T2
0
0
4
0
2
2
1
5
1
3
0
0
0
0

Pre-T3
0
3
3
0
0
0
0

Pos-T3
0
1
0
3
0
0
0

Total frequencies of ANI’s use of names of quadrilaterals at the two

interviews
Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Frequency
Pre
Post
0
3
11
4
17
5
11
13
3
5
2
2
0
1

117

Table 4.16 shows that the word square (n=24) was the most frequently used during the
interviews. The word rectangle (n=22) was the second most frequently used, and parallelogram
(n=15) was third. The names of the parallelograms were mostly mentioned in Task One, and
he
among all the names, rectangle and square were most frequently mentioned at the Pre-Interview
Pre
(see Table 4.15). There was an increase in use of the word square and rhombus in the Post).
Post
Interview. However, there was a reduction in use of the word parallelogram and rectangle in the
Post-Interview. The words kite (n=1) and quadrilateral (n=3) were mentioned only at the PostPost
Interview, whereas the word trapezoid (n=4) was only mentioned in Task One. ANI’s use of the
names of quadrilaterals was much lower than other interviewees’ use of those names.
Recall that in the Pre-Interview, while grouping in Task One, ANI referred to a rhombus
Interview,
as a slanted square, and a parallelogram as a slanted rectangle. Later she drew a rhombus as a
parallelogram, but disagreed that a square was a parallelogram when I showed her a picture of a
parallelogram
square in Task Two:
69. Interviewer

Why is this a parallelogram?

70. ANI

Because AB is parallel to DC, and AD is parallel to BC.

71. Interviewer

Why do you think it’s a different parallelogram?

72. ANI

Because all the sides are the same length.

73. Interviewer

What do you call this shape?

74. ANI

A rhombus.

118

In this conversation, ANI identified a rhombus as a parallelogram because it had two
pairs of parallel sides [70], and recognized that it was a different parallelogram because it had all
sides of the same length. However, in the next conversation, she disqualified a square as a
parallelogram.
99. Interviewer

How about this one? Is this a parallelogram?

100. ANI

No.

101. Interviewer

Why do you think it’s not a parallelogram?

102. ANI

Because all the lengths look like they are the same
sides, and I think that a parallelogram has
different sides.

103. Interviewer

What do you call this shape?

104. ANI

A square.

This inconsistency showed that ANI’s use of the word parallelogram referred to visual
family appearances. From ANI’s grouping of parallelograms in Task One, and her ways of
identifying parallelograms in Task Two, I conclude that the word parallelogram was used to
represent rectangles, parallelograms and rhombi.
Figure 4.14 illustrates ANI’s use of parallelograms at the pre-interview.
Parallelogram (n=8)

Parallelogram

Rhombus

119

Rectangle

Figure 4.14 (cont’d)

Figure 4.14.

ANI’s use of the word parallelogram at the Pre-Interview

ANI’s use of the word parallelogram changed in the Post-Interview, as the word
presented a class of figures sharing a common descriptive narrative, and all parallelograms
presented in the task were connected in a hierarchy. In the Post-Interview, ANI used the word
quadrilateral to extend the family of parallelograms, describing quadrilaterals with a hierarchy
of classifications.
As shown in Figure 4.15, this hierarchy, quadrilaterals have different names depending
on attributes of their angles and sides. This hierarchy has three branches: trapezoid,
parallelogram and quadrilateral. The word parallelogram includes parallelograms, rectangles,
squares and rhombi by definitions. All four-sided polygons with different visual appearances
some with right angles, some have all same sides, but sharing a common descriptive narrative,
“opposite sides parallel and equal”.

120

Parallelograms (n=11)

Rectangles (n=6)

Rhombus (n=5)

Squares (n=3)
Figure 4.15

ANI’s use of the word parallelogram at the Post-Interview

121

In the interviews, ANI mentioned more the parts of parallelogram (e.g., angles, sides, etc)
than the names of parallelograms. Tables 4.17 and 4.18 give the frequencies of these words at the
Pre-Interview and the Post-Interview. Table 4.18 provides total frequencies of each word at the
interviews, whereas Table 4.17 presents the frequencies of each word used at each task in the
interviews.
Table 4.17

The frequencies of ANI’s use of names of the parts of parallelograms at the

two interviews
Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.18

Pre-T1
14
14
3
0
2
0
2
2

Pos-T1
7
12
4
3
0
0
0
0

Frequency
Pre-T2
Pos-T2
32
37
8
27
13
10
0
3
1
1
10
12
1
0
0
0

Pre-T3
5
4
8
0
2
0
0
1

Pos-T3
3
1
2
0
0
0
0
1

Total frequencies of ANI’s use of names of the parts of parallelograms at the

two interviews
Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Frequency
Pre
Post
51
47
26
40
24
16
0
6
5
1
10
12
3
0
3
1

Table 4.18 shows that the most frequently used word relating to the parts of
parallelograms was angle (n= 98), mentioned most frequently in Task Two (see Table 4). The
122

second most frequently used was side (n=66), mentioned most frequently in Task One and Task
Two. The word length (n=40) was third, mentioned in all three tasks. The word diagonal (n=40)
was only mentioned in Task Two. These results were expected, as Task Two asks interviewees
to discuss the angles, sides and diagonals of parallelogram. Right angle (n=3) was least
mentioned, appearing only at the Pre-Interview. There was a slight reduction in the use of the
words opposite angle and opposite side in the Post-Interview. However, ANI mentioned parallel
side (n=6) at the Post-Interview, but did not mention it at all in the Pre-Interview.
The most compelling change in ANI’s word use was her use of the word parallelogram
at the Post-Interview. She had a better understanding of parallelograms and their properties.

Case 3: Changes in ALY’s Geometric Discourse
ALY was a college freshman at the time of the interviews. ALY took her last geometry
class three years prior to the geometry and measurement class. The van Hiele geometry Test
showed that she was at Level 3 at the pretest, and moved to Level 4 on the posttest. I interviewed
ALY after both tests, and analyzed her interview responses. A summary of findings on changes
in ALY’s geometric discourse is as follows:
•

ALY’s routines of sorting changed from grouping polygons by the number of
sides and by their names based on the attribute of their angles, at the PreInterview, to classifying polygons by their common descriptive narratives and
arranging quadrilaterals with a hierarchy of classifications at the Post-Interview.

•

ALY’s routines of substantiation changed from verifying the congruent parts of
parallelograms using recalling, measuring and constructing routines at the Pre-

123

Interview, to formulating mathematical proofs using mathematical axioms and
propositions at the Post-Interview.
•

ALY’s word use changed from representing the word parallelograms as a
collection of polygons sharing common descriptive narratives at the PreInterview, to using words with a hierarchy of classifications of parallelograms at
the Post-Interview. ALY also used more mathematical terms in the Post-interview
than in the Pre-Interview.

ALY’s routines of sorting were analyzed in the interviews for Task One, where ALY was
asked to classify eighteen polygons into groups. These polygons included triangles (n=4),
quadrilaterals (n=13) and one hexagon. The following section details these interviews.
At the Pre-Interview, ALY grouped the eighteen polygons into three groups according to
the number of their sides. She grouped all quadrilaterals into a group, calling it the “four-sided”
group; and she grouped all three-sided polygons (n=4) together, naming the “triangles” group.
ALY called Fig. V (a hexagon) “miscellaneous”, and grouped it as “other”.
When asked to regroup the quadrilateral group, and she regroup them by right angle
versus non-right angle. After regrouping the quadrilaterals, ALY said, “I selected shapes with
right angles… from looking at it. I didn’t measure any of them, but I am assuming that they are
right angles.” ALY’s regrouping of quadrilaterals is shown in Figure 4.16.

Right angle Group (n=8)

124

Figure 4.16 (cont’d)

Non-right angle Group (n=10)

Figure 4.16

ALY’s regrouping of polygons at the Pre-Interview

ALY regrouped the polygons by the attribute of having right angles or not. The right
angle group consisted of polygons having at least one right angle, whereas the non-right angle
group contained all other polygons. When I asked ALY to subgroup the four-sided group (n=13)
that she initially made, she split the group into two: the rectangle group (n=6) and the nonrectangle group (n=7). Figure 4.17 illustrates ALY’s subgrouping of the quadrilaterals.

125

Rectangle Group (n=6)

Non-rectangle Group (n=7)

Figure 4.17

ALY’s subgrouping of the quadrilaterals at the Pre-Interview

ALY used the name rectangle to split the quadrilaterals into two groups. In the rectangle
group, ALY included both rectangles and squares. Consequently, the non-rectangle group
contained all other quadrilaterals such as the parallelograms, the rhombi and a trapezoid. I also
asked ALY to subgroup the non-rectangle group, and she responded that she could split the
group into rhombus and non-rhombus.

126

26. ALY: I would separate them into rhombuses. P is a rhombus,
L is a rhombus, J is a rhombus, Z and H are all
rhombuses. [Pointing at these polygons]

In the subgrouping, ALY identified both parallelograms and rhombi as rhombus. To
investigate further, I asked ALY to identify a parallelogram, and she pointed to Fig. H. I asked
her to identify a rhombus, and she pointed to Fig. J (another parallelogram). ALY’s subgrouping
of the non-rectangle group revealed her confusion between parallelograms and rhombi.
Thus, at the Pre-Interview ALY grouped polygons by the number of sides and by the
characteristics of their angles. ALY also favored dividing polygons into two groups, according to
what they are and what they are not, with names of polygons such as rectangle and rhombus.
ALY identified polygons by direct recognition, but it appeared to me this recognition was not
self-evident. For example, ALY mentioned “just from looking at it, ... I am assuming that they
are right angles” to explain her assumptions. Figure 3 summarizes ALY’s routines of sorting at
the Pre-Interview.

127

Prompt: “Sort the shapes into groups”
Choosing
intuitively

Grouping by the numbers of sides,
and characteristics of the angles of
polygons (i.e., right angles)

Conclusion

Prompt: “Subgroup the group.”

Choosing
intuitively

Grouping by two groups, "what
they are and what they are not"
using rectangle and rhombus
Figure 4.18

Conclusion

ALY’s routines of sorting at the Pre-Interview
Interview

Interview,
At the Post-Interview, at my request, ALY first grouped the eighteen polygons into two
groups, consisting of triangles (n= 4) and non-triangles (n=14) according to the number of their
sides. When I asked ALY to regroup the eighteen polygons differently, she regrouped the
she
polygons into two groups, including quadrilaterals (n=13) and non-quadrilaterals (n=5), again
quadrilaterals
by the numbers of sides. I then asked ALY to subgroup the quadrilaterals, and she split the
,
quadrilaterals into two groups, consisting of parallelograms (n=11) and non-parallelograms
parallelograms
(n=2). I continued to ask ALY to subgroup the parallelograms, and she came up with two
groups, rectangles (n=6) and non
non-rectangles (n=5). This pattern continued with the two
subgroups consisting of squares (n=3) and non-squares (n=3) within the rectangles group.
Figure 4.19 summarizes ALY’s subgroupings of the quadrilaterals at the Post-Interview. As
Interview.
shown in Figure 4.19, ALY’s routines of sorting distinguishing two groups of polygons
sorting,
determined by “what it is and what it is not” proceeding through families of quadrilaterals such
nd
as parallelograms, rectangles and squares.

128

Quadrilaterals (n=13)

Non-parallelograms (n=2)

Parallelograms (n=11)

Non-rectangles (n=5)

Rectangles (n=6)

Non-squares (n=3)

Figure 4.19

Squares (n=3)

ALY’s chains subgroupings of the quadrilaterals at the Post-Interview.

129

During our conversations in the Post-Interview, ALY demonstrated her understandings of
the relations between parallelograms, rhombi and squares by classifying these polygons in a
hierarchy. For example, ALY identified the non-rectangles as parallelograms, a group consisting
of parallelograms and rhombi.
45. Interviewer

What are the non-rectangles?

46. AL

The parallelograms? P, H, Z, J and L. [Pointing at
these polygons]

47. Interviewer

Why are they not the rectangles?

48. ALY

Because they don’t have four right angles.

ALY also identified rhombi as a subgroup of non-rectangles (i.e., parallelograms), and
identified a square, a polygon from a group of rectangles, as a rhombus.
77. Interviewer

Can you identify if there is a rhombus?

78. ALY

A rhombus? I think Z and L are rhombuses. And then U, G,
and R would be rhombuses as well. [Pointing at Fig. Z and L]

130

79. Interviewer

How do you know that L is a rhombus?

80. ALY

Because these sides look equal in length and they are also
parallel to the opposite sides [pointing at the sides of Fig. L]

81. Interviewer

Why is U a rhombus? [Pointing at Fig. U]

82. ALY

Because all sides are equal and opposite sides parallel to each
other.

At the Post-Interview, ALY first grouped the polygons by the numbers of their sides.
When asked to regroup the polygons, ALY classified quadrilaterals by dividing them into two
131

groups each time. During the interview, ALY did not use measurement tools to ve
verify congruent
angles or sides of the polygons. However, ALY did mention that she assumed the angles of the
rectangles to be 90 degrees, and the sides of the parall
,
parallelograms to be parallel. I have concluded
.
that ALY identified polygons intuitively, but agai it was not self-evident. ALY’s routines of
again
evident.
sorting at the Post-Interview are summarized in Figure 4.20
rview
4.20.
Prompt: “Sort the polygons into groups”
Choosing
intuitively

Grouping by the number
of sides of the polygons

Conclusion

Prompt: “Subgroup the group.”
Choosing
intuitively

Grouping into two groups of "what it is
and what it is not" using the names of
parallelograms; and arranging them by a
hierarchy of classificaitons

Figure 4.20

Conclusion

ALY’s routines of sorting at the Post-Interview

We see a similar pattern as ALY grouped the polygons by the number of sides, and sub
subgrouped polygons by dividing them into two groups each time. However, there was a change in
AYL’s subgroupings, as her chains of subgrouping showed a hierarchy of classifi
classifications of
quadrilaterals at the Post-Interview, her subgrouping was limited only to identifying the
Interview,
rectangles and the rhombi without a hierarchy at the Pre
Pre-Interview.
ALY’s change in geometric discourse also appeared in her routines of substantiation. In
the following section, I describe observations from the interviews with ALY in Task Two and
Task Three.
ALY explained and verified her claims about the angles and sides of parallelograms with
sides
recalling and measuring routines at the Pre Interview; whereas she substantiated her statements
Pre-Interview;
132

using endorsed narratives at the Post-Interview. When substantiating the equivalence of two
definitions, ALY constructed two polygons with angle and side measurements that fit the
descriptions of the definitions at the Pre-Interview, whereas she produced a mathematical proof
at the Post-Interview. To illustrate these changes, I will provide the following scenarios.
In Task Two, ALY drew a parallelogram, and stated that it was a parallelogram because
the opposite sides were parallel to each other. When discussing the angles of the parallelogram,
ALY responded that the opposite angles of the parallelogram were equal. When I asked why,
ALY replied, “I think that is just a property of a parallelogram”. The following conversation took
place after I prompted for verification:
9. Interviewer

If I ask you to convince me that the opposite
angles are equal, what would you do?

10. ALY

You mean… prove it to you, that in every case it
would be that way?

11. Interviewer

Yeah.

12. ALY

I could just measure the angles for you, with a
protractor. I've never done a proof before, in this
case. I've done lots of proofs, but not on
something like that, so I don't know.

In this conversation, I noticed that writing a proof about the angles of a parallelogram
was new to ALY, but she was aware of the difference between a verification of a statement
“opposite angles are equal in a parallelogram” in general (i.e., a mathematical proof) and a
verification of an example of a statement (i.e., check the measurements of angles), as evidenced
in her asking, “prove it to you that in every case it would be that way?” and later proposing to

133

measure the angles [12]. In this case, ALY explained the statement that opposite angles are equal
in a parallelogram by remembering it as a property of a parallelogram.
ALY also provided another narrative about the angles of a parallelogram, namely that the
adjacent angles in a parallelogram add up to 180°. When asked for substantiation, ALY replied,
“I have learned it before, but I am not 100% sure…”, and produced her verification as follows:
26a. ALY

This angle looks like it would match up if I was to
extend this line out like this [extending the side]

26b. ALY

… this angle looks equivalent to this angle. And,
actually it is, because I've learned about parallel lines
[pointing at the angles].

26c. ALY

It would, because this angle would be equivalent to
this angle [pointing at the vertical angles]… and is
also equal to that [pointing at alternating interior
angles]

27. Interviewer

Why is that?

28. ALY

That's another property [giggling]... I just remember

134

in high school, we learned a lot of properties about
parallelograms and parallel lines, and things like that.
29. Interviewer

So… What is your conclusion?

30. ALY

That the adjacent angles add up to 180 degrees.

ALY reasoned her way through by drawing extended lines [26a] and by identifying
parallel sides [26b] and congruent angles [26b; 26c]. ALY remembered the properties of a
parallelogram and remembered the propositions of parallel lines, as well as the relation between
the vertical angles [26c], and she applied them in her verification process. It is notable that ALY
identified more congruent angles than she needed for the verification. More specifically, ALY
needed only one of the three congruent pairs of angles, but she identified three pairs [26b; 26c].
ALY neglected to point out how the congruent angles would lead to the proof that the adjacent
angles add up to 180 degrees, an argument crucial in this substantiation. In the preceding
examples ALY applied her prior knowledge about the properties of parallelogram and
propositions of parallel lines to identify the elements needed for verification. However, in this
example, ALY verified her claim only partially and without logical order.
In contrast, at the Post-Interview ALY explained why opposite angles are equal, using the
fact that the adjacent angles in a parallelogram add up to 180° as an endorsed narrative. Here is
ALY’s demonstration that adjacent angles in a parallelogram add up to 180 degrees.
11. Interviewer

How do you know that the adjacent angles add up to 180
degrees?

12. ALY

Because that's one of the properties of a parallelogram, I
can show you if you like?

135

13. Interviewer

Go ahead.

14a. ALY

If you were to rip this in half [drawing a line cut through
the polygon],

14b. ALY

…and then take this top half and put it down here, then it
would line up, like this. Does that make sense?

14c. ALY

Right here… Let's say that this is a and this is b. If we
moved this [b] down here then these would be on the same
line. [Labeling the polygon with a and b representing two
pieces of the polygon]

a

b
a

14d. ALY

b

This angle here [pointing at the vertex angle] would be
right here, and they'd be on the same line. And, angles on a
line add up to 180 degrees.

a

b
a

b

136

To verify that adjacent angles add up to 180 degrees, ALY first referred to this statement
as a property of parallelograms [12]. Then she described an activity of imagining to cut the
polygon horizontally in half [14a], and moving the top half to the bottom [14b] to match the two
adjacent vertex angles side by side in a line [14c; 14d]. ALY used this newly endorsed narrative
that adjacent angles add to 180 degrees to justify her statement that the opposite angles in a
parallelogram are equal.
15. Interviewer

How do you know that the opposite angles are equal?

16a. ALY

Because you have two parallel lines, here, if I was to
extend these lines [Extending two sides of the
parallelogram].

16b. ALY

Then, this would be a transversal.

Transversal
16c. ALY

So, you've got this angle here and this and this angle here
[pointing at the two angles] add up to 180 degrees because
they're adjacent.

16d. ALY

For the same reason, … then this angle and this angle
would also add up to 180 degrees,

137

16e. ALY

So these [pointing at the opposite angles] have to be the
same.

In this example, ALY first identified two parallel lines [16a] and a transversal [16b], and
concluded that adjacent angles add up to 180 degrees [16c]. In order to show that opposite angles
were equal, ALY identified another pair of adjacent angles, with one vertex angle included in the
previous ones [16d], so that two pairs of adjacent angles shared one angle in common. Giving the
same reason [16a; 16b], ALY explained that the second pair of adjacent angles also added up to
180 degrees. Finally, ALY concluded that the opposite angles had to be the same [16e].
This scenario illustrates changes in ALY’s routine procedures. She progressed from
referring to prior knowledge to justify her claims (i.e., recalling routines), to using newly
endorsed narratives to substantiate her claims. ALY also moved from identifying more elements
than needed for verifications, without logical order, to choosing the exact number of elements
necessary to justify statements logically.
In another change, ALY went from measuring the sides of parallelograms with rulers to
verify congruency at the Pre-Interview, to identifying congruent triangles by congruent criterions
to verify the congruent parts of parallelograms at the Post-Interview. In Task Two, I asked ALY
to draw a parallelogram and to discuss its sides. ALY drew a parallelogram and stated that it was
a parallelogram because the opposite sides were parallel to each other and were equal in length.

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The following conversation took place after I asked for substantiation of her claim that “the
opposite sides were equal in length”.
ides
41. Interviewer

Is there a way that you can show me that they are the same length?

42. ALY

In this parallelogram? I can measure it. So this is…4.5
centimeters, this is a little less than 4.5. [Using a ruler to measure
one pair of opposite sides]… Right, this looks about 4.3. Yeah,
about the same. [Measuring another pair of opposite sides]

ALY’s drawing
43. Interviewer

How do you know that for every parallelogram this is true?

44. ALY

You mean prove it? Well, I am not sure…but I know it’s just a
property of a parallelogram.

To verify the claim, ALY first referred to it as a characteristic of a parallelogram, and
characteristic
parallelogram
then used a ruler to measure the sides of the parallelogram to complete her verification. As I
verification.
mentioned earlier, ALY did not have much experience in constructing a mathematical proof in
,
this context, but my observation was that she was aware of the difference between constructing a
proof at an abstract level and checking with a concrete example at an object level. For example,
concrete
her awareness showed when ALY asked, “In this parallelogram? I can measure it” [42].

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Consequently, ALY used a ruler to measure the opposite sides of the parallelogram to check
congruency [42], and referred to this congruency as a property of a parallelogram.
this
This pattern of measuring and checking also appeared when ALY was verifying the
equivalence of diagonals in a rectangle.
95. Interviewer

What can you say about the diagonals of this parallelogram?

96. ALY

They are of equal length
length.

97. Interviewer

How do you know that they are equal?

98. ALY

Because I learned it a long time ago, in a rectangle, the
diagonals are the same.

99. Interviewer

Is there a way that you could convince me?

100a. ALY

I would measure them, is that O.K? [Using a ruler to
measure the diagonals]

100b. ALY

Yeah, they’re both 8.2 centimeters. So, the diagonals have
equal length.

I challenged ALY by asking, “What if you don’t have rulers to measure the diagonals,
what would you do?” ALY replied, “If you look, the diagonals form two triangles”, and
identified two triangles, and explained why the two triangles were congruent.
115. ALY

[Shading the two congruent triangles]
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116a. ALY

This side equal to this side [pointing at the opposite sides
of the paralleogram]

116b. ALY

This side is obviously it’s the same side.

116c. ALY

…Which means this side would have to be equal to this
side [pointing at the diagonals]

ALY intuitively provided a general explanation about the equivalence of the diagonals
using congruent triangles. ALY first identified two triangles [see shaded areas in 116a] where the
diagonals were the hypotenuses of the triangles. ALY chose two congruent elements, using the
congruent
opposite sides of the rectangles as one pair of corresponding sides in the triangles [116a], and
noted a common side [116b]. From there, ALY concluded that the diagonals were equal based on
the equivalence of the two other pairs of sides. Mathematically, to verify that two triangles are
sides.
congruent we need three elements, and in this case ALY only provided two. We need
information about an included angle of the two sides that ALY identified to complete the
verification. Note that, in our earlier conversation, ALY drew this rectangle as a different
141

parallelogram, and she knew that all angles were equal in a rectangle. However there was no
mention of the equivalence of the angles, the third element needed for verification of the
congruent triangles.
Ten weeks later I interviewed ALY again. When ALY discussed the sides of the
parallelogram, she used an argument including the distance between the parallel lines to verify
her claim that the opposite sides were congruent.
22. ALY

Opposite sides are congruent in a parallelogram.

23. Interviewer

How do you know that they are congruent?

24a. ALY

Because the opposite sides are parallel to each
other, they have a fixed distance away from each
other. So, the distance from here to here is the
same as here to here, it's going to be the same all
the way through.

distance
24b. ALY

So, that would mean that this would be equal to
this because they never intersect [pointing at the
opposite sides]

ALY’s used rulers and protractors to measure and check the congruent parts of the
parallelograms at the Pre-Interview. In contrast, ALY used an endorsed narrative, stating that the

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distance between the parallel lines were equal to demonstrate that opposite sides of the
parallelogram were equal at the Post-Interview. The preceding conversation drew my attention to
ALY’s incorrect use of he word “distance” in the context of the distance between parallel lines; I
will discuss this matter later in looking at ALY’s word use.
During the Post-Interview, ALY used triangle congruent criterions to substantiate the
congruent parts of parallelograms. For example, when discussing the diagonals of a
parallelogram, ALY stated that the diagonals bisect each other. When asked for verification,
ALY provided the following justification using the Angle-Side-Angle (ASA) triangle
congruency criterion. Table 4.19 illustrates ALY’s routine procedures of substantiation, with
corresponding transcripts.
Table 4.19

ALY’s routine procedures of substantiating that diagonals bisect each other

Routine Procedures

Transcripts

1. Identify triangles
formed by diagonals

38a. Since I drew two diagonals, we can see that there are
four triangles here.

2. Verification of two
congruent triangles

38b. If you take this angle here and this angle here, they're
equal to each other because they're vertical angles [pointing
at the vertical angles]

2.1 Identify first pair
of corresponding
angles of the triangles
2.2 Identify the second 38c. Because these two lines are parallel, this angle would
pair of corresponding
be equal to this angle here, because they're opposite interior
angles of the triangles angles, …

2.3 Identify the third
pair of corresponding
angles of the triangles

38d. The same goes for this angle and this angle, …because
of the same property.

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Table 4.19 (cont’d)
2.4 Identify the fourth
pair of corresponding
angles of the triangles

38e. …and then this angle and this angle.

2.5 Identify one pair
of corresponding sides
of the triangles

38f. We already know that these two sides are equal because
that's a property of a parallelogram [pointing at the sides of
the triangles]

4. Verify congruent
triangles using A-S-A
correspondence

38g. So, you have Angle-Side-Angle here. And, so, that
shows that this triangle here is congruent to this triangle here

5. Conclusion

38h. When you match up the corresponding sides, this side
would be congruent to this side and then, this would be
congruent to this. So, they are of equal measure, so they
bisect each other.

In this episode, ALY provided a verbal explanation of why the diagonals of this
parallelogram bisect each other. More specifically, ALY used the ASA triangle criterion to
verify that two triangles were congruent [38g], and concluded that the diagonals bisected each
other because the parts of the diagonals were corresponding sides of two congruent triangles.
Note that at each step, ALY was able to provide mathematical justifications for her conclusions.
For example, ALY stated, “because these two lines are parallel” and “because they are vertical
angles” as justifications to demonstrate that two pairs of angles were equal. ALY also used “a
property of parallelogram” to explain the congruent sides.
During this process of verification, ALY identified four pairs of congruent angles [38b38e] and one pair of congruent sides [38f], more elements than needed for verification. In

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particular, she identified one pair of vertical angles [38b] and a pair of alternating interior angles
[38b] that were not needed in her final verification of two congruent triangles.
As our conversations continued, ALY drew a rectangle and stated that the diagonals also
bisected each other in a rectangle. The following conversation took place when I asked for
following
substantiation.
63. Interviewer

How do you know diagonals bisect each other in this case?

ALY’s drawing
64. ALY

For the same reason as last time, do you want me to explain again?

65. Interviewer

When you say, “for the same reason as last time”, what do you mean?

66. ALY

Just, all of it, when you create these triangles and the triangles are
congruent to each other…based on that property, all parallelograms
have diagonals that bisect each other [pointing at the triangles in the
each
parallelogram].

ALY’s drawing

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ALY referred to her newly endorsed narrative about the diagonals bisecting each other in
a parallelogram [64] to justify her claim about the diagonals in a rectangle. Given this
opportunity, I asked for a written proof. ALY’s written proof is presented in Table 4.20.

Table 4.20

ALY’s written proof that diagonals bisect each other at the Post
Post-Interview

p is the intersection of the diagonals

Before ALY started to write the proof, she labeled the vertices of the rectangle with A, B,
C, and D in a clockwise order. ALY used mathematical symbols like ∠, ≅, ∆, ⁄⁄ to replace the
words angle, congruent, triangle and parallel, respectively, in her written proof. For example,
triangle,
itten
ALY wrote “∠APB ≅ ∠DPC”, “
DPC”, “AB = DC” and “∆ABP ≅ ∆CDP” to indicate two congruent
angles, sides and triangles accordingly. I asked ALY for further clarification.
77. Interviewer

Can you explain to me what you wrote?

78a. ALY

I have angle APB, so this angle right here, is equal to DPC,

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because they are vertical angles, And then also angle APD, so
this here, well I guess that's not really important
∠APB ≅ ∠DPC (vertical
angles)
∠APD ≅ ∠BPC (vert. ∠s)

78b. ALY

Angle BAC is congruent to DCP because AB and DC are
parallel to each other, so these two angles are alternate interior
angles, and they're always congruent.
∠BAC ≅ ∠DCP (alt. int ∠s,
AB//DC)

78c. ALY

AB is equal to DC because that is a property of a
parallelogram. So, we have two angles on this side, and that is
enough information to conclude that triangle ABP is congruent
to triangle CDP.
AB = DC (prop. of //ogram)
∆ABP ≅ ∆CDP (ASA)

79. Interviewer

When you say two angles and a side, what do you mean?

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80. ALY

This angle and this angle. And then we have a side.
∠APB ≅ ∠DPC
(vertical angles) –angle
∠BAC ≅ ∠DCP
(alt. int ∠s, AB//DC) –angle
AB = DC (prop. of //ogram) –
side

It is evident that there was a change in ALY’s routines of verifying. She previously
measured and compared the sides of the parallelogram to verify her claims at an object level in
the Pre-Interview, whereas, at the Post-Interview, she constructed a mathematical proof using
mathematical symbols and justifications at an abstract level. However, ALY incorrectly used
Angle-Angle-Side (AAS) to verify congruent triangles. Recall that, in an earlier substantiation,
ALY mentioned “two angles and a side” as ASA to verify the congruent triangles correctly.
However, in this example the same phrase, “two angles and a side” appeared again but with an
indication of AAS. Thus ALY displayed ambiguity in using congruent criterions.
When proving the equivalence of the two definitions, ALY’s routine procedures also
changed. At the Pre-Interview, ALY verified the equivalence of the definitions by constructing
the parts of the parallelograms with measurements that fit the descriptions of the definitions,
whereas she constructed a mathematical proof of the equivalence of the definitions at the PostInterview. The following illustrates ALY’s routine procedure that I observed in Task Three.

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In Task Three, interviewees were asked to construct mathematical proofs from given
definitions. In particular, these interviewees were given two definitions of a parallelogram and
were asked to substantiate that these definitions were equivalent.
•

Definition #1: A quadrilateral is a parallelogram if and only if both pairs of
opposite sides have the same length.

•

Definition #2: A quadrilateral is a parallelogram if and only if both pairs of
opposite angles have the same measure.

In order to verify the equivalence of the definitions mathematically, interviewees needed
to use deduction to prove the following implications:
•

If a quadrilateral has both pairs of opposite sides of the same length, then both
pairs of opposite angles have the same measure; and

•

If a quadrilateral has both pairs of opposite angles of the same measure, then both
pairs of opposite sides have the same length.

Deduction takes place when a newly endorsed narrative is obtained from previously
endorsed narratives with the help of well-defined inferring operations.
At the Pre-Interview, I asked ALY what she would do to prove the two statements were
equivalent. She replied, “I would demonstrate what each definition is saying, then show how it
results in the same thing”. In trying to prove the implication “If a quadrilateral has both pairs of
opposite sides of the same length, then both pairs of opposite angles have the same measure”,
ALY explained, “I would draw a parallelogram where the opposite sides have the same length.
And, because they have the same length, they're going to have the same angles”. ALY used a
ruler to construct a polygon with opposite sides parallel and equal, so that the parallelogram fit
the description in Definition #1. ALY next used a protractor to check all the angles of the

149

parallelogram, and found that the opposite angles had the same measures, matching the
descriptions in Definition #2. ALY finished this part of the substantiation with the conclusion
“it’s going to be that case every time, where the angles will be equal every time.” ALY’s routine
procedure of constructing a parallelogram with both pairs of opposite sides of the same length in
opposite
order to prove the implication, is shown in Table 4.21
4.21.
Table 4.21

ALY’s proof by constructing a parallelogram with opposite sides equal

ALY’s drawing of the parallelogram

1. Constructing a parallelogram with opposite sides equal

2. Verifying the given conditions (Definition #1): Using a ruler to measure the opposite sides of
the parallelogram

3. Verifying the results (Definition #2): Using a protractor to measure the opposite angles of the
#2):
parallelogram

4. Conclusion
ALY: So, it's going to be that case every time, where the angles will be equal every time. I wasn't
measuring the angles when I drew it. I was just focusing on the lengths, but it just came out that
they were the same. And, it will come out that way every time.

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To prove the other implication, “If a quadrilateral has both pairs of opposite angles of the
same measure, then both pairs of opposite sides have the same length.” ALY began by drawing a
quadrilateral with opposite angles equal then she measured the opposite sides. She used a ruler to
check all the sides of the parallelogram, and found that opposite sides had the same lengths
opposite
matching the descriptions in Definition #1. ALY completed her verification with the conclusion
“based on opposite angles being the same. I ended up with sides that were almost identical in
site
length”. ALY’s routine procedure of constructing a quadrilateral with both pairs of opposite
angles of the same measure to prove the implication is shown in Table 4
4.22.
Table 4.22

ALY’s proof by constructing a parallelogram with opposite angles equal
ALY’s drawing of the parallelogram

1. Constructing a parallelogram with opposite angles equal
1.1 Constructing an angle with 65
65°

1.2 Constructing a parallel line to the side and measuring the angle

151

Table 4.22 (cont’d)
1.3 Constructing the fourth side that is parallel to its opposite side and measuring the rest of the
angles

2. Verifying the given conditions (Definition #2): Checking the angle measure of the opposite
angles

3. Verifying the results (Definition #1): Using a ruler to measure the opposite sides of the
parallelogram

4. Conclusion
ALY: This time, when I drew it, I focused on the angles. I drew the lines, then I created the
shape, based on opposite angles being the same. And, I ended up with sides that were almost
identical in length.

At the Pre-Interview, ALY’s routine procedures of proving the equivalence of the two
definitions involved the constructing a parallelogram, measuring the angles or the sides, and
comparing the measurements, all mathematical activities operating at an object level. It is
important to note that ALY did not know how to construct a mathematical proof and her

152

argument was to generalize the result from a particular example, assuming that “it will came out
that way every time.”
Ten weeks later, ALY’s routine procedures for substantiating the equivalence of the two
en
definitions changed. First of all, ALY proved the implication “If a quadrilateral has both pairs of
opposite sides of the same length, then both pairs of opposite angles have the same measure” by
providing an example of the inverse of the implication. That is, ALY drew a polygon that fit the
implication “if a quadrilateral does not have both pairs of opposite sides of the same length, then
both pairs of the opposite angles don’t have the same measure.” ALY’s drawing and her
explanation are shown in Figure 4.21
re 4.21.

ALY’s drawing
ALY: …if they [pointing at the parallel sides] didn't have the same
length, if one of the sides had a different length than the other,
……then they [pointing at the legs of the trapezoid] would not
be a fixed distance away from each other, they could not be
parallel to each other, eventually, they would intersect.
l
ALY: So, opposite angles are not equal.
Figure 4.21 ALY’s verification of the first implication at the Post
Post-Interview

To my request, “Can you prove that opposite angles are equal, knowing that two opposite
sides have the same measure?” ALY drew a parallelogram and constructed a mathematical proof
constructed
by assuming the parallel sides of the polygon as given, and applied the proposition that when
parallel lines are cut by a transversal, the adjacent interior angles add up to 180 degrees. She then
interior
solved the equations algebraically to justify that opposite angles have the same measure. ALY’s
drawing of a parallelogram and her construction of the proof is shown in Figure 4.22.

153

ALY’s drawing

Figure 4.22

ALY’s proof that opposite angles have the same measure

To gain a better understanding of ALY’s thinking, I asked for clarification, and her
thinking,
response to that as follows:
13. Interviewer Can you explain to me what you wrote?
14. ALY

We're given that these opposite sides are parallel to each other. So
it's a parallelogram. We know that adjacent angles add up to 180
degrees.

AB // DC
AD // BC

(given)

15. Interviewer How do you know this is a parallelogram?
Because it's a four sided figure and opposite sides are parallel to
four-sided
16a. ALY

each other and equal to each other, in length.

154

A and D add up to 180 degrees, and also A and B add up to 180
16b. ALY

degrees. So, that must mean that D and B are equal to each other,
because they are both supplementary to A. So, the same is true
for angles C and A, because B and C add up to 180 degrees and B
and A add up to 180 degrees. So, C and A must be equal too.

∠A + ∠B = 180° (prop. of //.ogram)
∠A + ∠D = 180° (prop. of //.ogram)
∠B ≅ ∠D
∠C + ∠B = 180° (prop. of //.orgam)
∠C ≅ ∠A

ALY’s routines of substantiation changed from constructing parallelograms that match
the descriptions in the definitions at the Pre-Interview, to formulating new narratives “angle B
equal to angle D, angle A equal to angle C” using the endorsed narratives “adjacent angles add
up to 180” and “two angles supplement to the same angle are the same” at the Post-Interview.
Although ALY was able to use mathematical symbols in her proofs and she developed some
skills of proving in geometry, she still could not make a clear distinction between what was
given and what was to be proved in the Post-Interview. For example, ALY was to assume a
quadrilateral with opposite sides of the same length as the given, but instead she incorrectly

155

assumed the polygon was a parallelogram and used that to begin her substantiation. It appeared
that ALY was at the beginning stage of constructing mathematical proofs. More evidence of the
change in ALY’s geometric discourse was in her word use. In the following section, I describe
the change in ALY’s use of mathematical terms during the two interviews.
ALY’s use of the word parallelogram changed from indicating both rhombi and
parallelograms in the Pre-Interview, to representing a hierarchy of classifications of
parallelograms in the Post-Interview. ALY also used more mathematical terms between the PostInterview than during the Pre-Interview. Let us look at ALY’s use of the general words
quadrilateral, parallelogram, rectangle, square, rhombus, trapezoid and kite. The total
frequencies of these categories of quadrilaterals at the Pre-interview and the Post-Interview are
listed in Table 4.23 and Table 4.24.
Table 4.23

The frequencies of ALY’s use of the names of quadrilaterals at the two

interviews
Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Pre-T1
0
3
9
1
11
2
0

Pos-T1
1
4
3
3
3
2
0

Frequency
Pre-T2
Pos-T2
0
4
6
14
4
4
3
8
0
3
0
0
0
0

Pre-T3
0
3
0
0
0
0
0

Pos-T3
1
4
1
0
0
0
0

Table 4.24 shows the word parallelogram (n=34) was the most frequently used during
the interviews, being mentioned in all three tasks (see Table 4.24). The word rectangle (n=21)
was the second most frequently used, and rhombus (n=17) the third. The word kite (n=0) was not
mentioned at all in both interviews, and trapezoid (n=4) was the second least mentioned. Table 5
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shows that the words rhombus and trapezoid were mostly mentioned in Task One, where
interviewees were asked to group the polygons. There was an increase in use of the words
quadrilateral, parallelogram and square from the Pre-Interview to the Post-Interview. In
particular, the word quadrilateral was used only during the Post-Interview. The word
parallelogram almost doubled in the Post-Interview, while the word square almost tripled. There
was a reduction in the use of the words rectangle and rhombus at the Post-Interview. The
following section will discuss findings of ALY’s use of the word parallelogram at the
interviews.
Table 4.24

Total frequencies of ALY’s use of names of quadrilaterals at the two

interviews
Frequency
Pre
Post
0
5
12
22
13
8
4
11
11
6
2
2
0
0

Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

In an earlier section, I described ALY’s routine procedures for sorting quadrilaterals in
Task One. In the Pre-Interview, ALY identified all 4-sided polygons with opposite sides parallel
and without right angles, as rhombi. My conversations with ALY revealed that she did not have
clear concepts of a rhombus and a parallelogram.

157

30. ALY: P is a rhombus, L is a rhombus, J is a rhombus, Z and H are
all rhombuses. [Pointing at these polygons]

31. Interviewer: What is a rhombus?
32. ALY: a rhombus would be the opposite sides and the opposite
angles are equal and it’s a four
four-sided figure.
33. Interviewer: What is a parallelo
parallelogram?
34. ALY: A parallelogram with opposite sides and angles are equal. …
This is a parallelogram [pointing at Fig. J].

In this conversation ALY made no distinction between a rhombus and a parallelogram,
parallelogram
as she described both as a four-sided figure with opposite sides and angles equal. In Task Two,
sided
ALY drew a picture of a parallelogram with the properties of a rhombus.
3. Interviewer

Why do you think this is a parallelogram?

158

4. ALY
49. Interviewer

Because opposite sides are equal and parallel.
……
What can you say about the diagonals of this
parallelogram?

50. ALY

They are perpendicular.

ALY’s drawing of a parallelogram looked like a rhombus. Our conversations led ALY to
investigate the sides and the diagonals of the parallelogram. She confirmed that the figure was a
parallelogram with opposite sides equal, and the diagonals of the parallelogram were
perpendicular. Of course, a parallelogram with diagonals perpendicular is a rhombus. These
observations suggest that ALY was not aware of the differences between a parallelogram and a
rhombus, and treated them as the same entity. To ALY at the time of the Pre-Interview, a
parallelogram was a rhombus.

57. Interviewer

Can you identify a rhombus here?

58. ALY

J. [Pointing at Fig. J]

My interview with ALY showed that her use of the word parallelogram applied of
rhombi and rectangles, a family of a four-sided figures that have opposite sides equal and
parallel. Figure 4.23 illustrates ALY’s use of the word parallelogram at the Pre-Interview
Parallelogram

Rhombus/Parallelogram

Rectangle

159

Figure 4.23 (cont’d)

Figure 4.23

(Previous Page) ALY’s use of the word parallelogram at the Pre-Interview

ALY’s use of the words parallelogram and rhombus refers to a family of four-sided
polygons that share a common descriptive narrative: opposite sides are equal and parallel. During
the Pre-Interview, ALY did consider squares as rectangles, but she made no connections between
a rhombus and a square. Her grouping and identification of quadrilaterals suggests that she had
no clear understanding of a hierarchy of classifications of quadrilaterals.
At the Post-Interview, the change in ALY’s use of the names of quadrilaterals showed an
understanding of the word parallelogram, as revealed in her hierarchy of the classifications of
parallelograms. In this hierarchy, the word parallelogram describes a collection of quadrilaterals
with different appearances and names, and arranged by the characteristics of their angles (i.e.,

160

right angle versus non-right angle) and sides (i.e., all sides equal versus opposite sides equal).
Although given different names such as rectangles, parallelograms, rhombi and squares, they
are all called parallelograms because they fit the description of opposite sides being equal and
parallel. This hierarchy of classification is analyzed in Figure 4.24.
Parallelogram (n=11)

Parallelograms

Rectangles

Rhombi

Square

Parallelograms

Figure 4.24

Rectangles

ALY’s use of the word parallelogram at the Post-Interview

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There was also a change in ALY’s use of the names of the parts of parallelograms. She
used more mathematical terms describing the relations between angles and sides in the PostInterview than in the Pre-Interview. A word search of the names of the parts of parallelograms
included angle, sides, length, parallel side, opposite side, opposite angle, right angle and
diagonal. Findings show that ALY used more words describing the parts of parallelograms than
the names of parallelograms. Tables 4.25 and 4.26 provide word usage frequency of word usage
at the two interviews.
Table 4.25

The frequencies of ALY’s use of the names of the parts of parallelograms at

the two interviews.
Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.26

Pre-T1
6
13
0
1
3
0
3
1

Pos-T1
9
10
3
2
3
0
1
0

Frequency
Pre-T2
Pos-T2
23
52
15
27
3
12
2
1
7
9
5
13
0
1
5
1

Pre-T3
5
4
9
1
3
1
0
3

Pos-T3
16
16
7
1
5
0
0
4

Total frequencies of ALY’s use of names of the parts of parallelograms at the

two interviews
Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Frequency
Pre
Post
34
77
32
53
12
22
4
4
13
17
6
13
3
2
9
5

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Table 4.26 shows that the most frequently used word relating to the parts of
parallelograms was angle (n= 111), with its usage of the word doubling in the Post-Interview
over the Pre-interview. The word angle (n=75) was mentioned mostly in Task Two (see Table
4.25). The word side (n=85) was the second most frequently mentioned. The words angle and
side were mentioned in all tasks. After angle and side, the words length (n=34) and opposite side
(n=30) were the next most frequently mentioned at both interviews. The word length was mostly
used in Task Two and Task Three, whereas the term opposite side was mentioned in all three
tasks. Likewise the word diagonal (n=19) was mostly used in Task Two. These results were
expected, as Task Two asks interviewees about the relations of the angles, sides and diagonals of
a parallelogram. The term right angle (n=5) was the least mentioned, and it was used mostly in
Task One. The term parallel side (n=8) was the second least mentioned, and it was mentioned
only one or two times during each task. The term opposite angle was mentioned fourteen times
and it was used mostly in Task Two and Task Three. The words parallel side and opposite angle
were not frequently mentioned during the interviews.
During the Post-Interview, ALY used more mathematical terms to describe the relations
between the angles of the parallelograms than at the Pre-Interview. More specifically, she used
the terms alternating interior angle (n=6) and adjacent angles (n=3) in the Post-Interview,
whereas she used “this angle” and “that angle” to refer to such angles in the Pre-Interview. ALY
also used the word quadrilateral (n=5) in the Post-Interview, whereas she used the term “foursided figures” to describe such polygons in the Pre-Interview.
Lastly, I want to draw attention to ALY’s use of the word distance. The word distance
was mentioned eight times during the Post-Interview, two times for Task Two and six times for

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Task Three. Let us look at one case of ALY’s use of the word distance during the PostInterview.
24a. ALY

Because the opposite sides are parallel to each other,
they have a fixed distance away from each other. So,
the distance from here to here is the same as here to
here, it's going to be the same all the way through.

“fixed distance”
6c. ALY

……
Because they have a fixed distance away from each
other and they always have a fixed distance away from
each other [pointing at the opposite sides of the
parallelograms]

“fixed distance”
ALY’s use of the word distance was ambiguous. She indicated that the distance between
two parallel lines was the length of segments parallel to the other pair of parallel sides of the
parallelogram [24a; 6c]. Mathematically, we define distance differently in this context. Table
4.27 illustrates the mathematical definition of distance and ALY’s use of the word distance.

164

Table 4.27

The word distance in geometry and used by ALY

Distance between two parallel lines:

ALY’s use of “fixed distance”

In this section I have shown evidence of changes in ALY’s geometric discourse. ALY
changed from relying on the measurements of angles and sides to verify their congruency, to
using axioms and propositions to substantiate claims about the congruent parts of the
parallelograms. During the Post-Interview, although ALY’s mathematical proofs were not all
correct, she demonstrated an ability to construct mathematical proofs using symbols and
justifications. ALY’s word use regarding names of parallelograms also changed as we see a
structured hierarchy of classifications of parallelograms at the Post-Interview.
Among the twenty interviewees, five of them showed no change in their van Hiele levels
in the van Hiele Geometry Test conducted at the pretest and posttest. These five interviewees
were AYA (2-2), ARI (3-3), AJA (3-3), ALI (3-3) and ARA (3-3). Therefore, in this section I
describe AYA and ARI’s geometric discourses, and I point out the differences and changes in
their geometric discourses in the context of quadrilaterals and triangles.
Case 4: Changes in AYA’s Geometric Discourse
AYA was a college sophomore at the time of the interviews. AYA took her last geometry
class seven years prior to the geometry and measurement class. The van Hiele Geometry Test
showed that she was at Level 2 at the pretest, and stayed at Level 2 according to the posttest ten
165

weeks later. I interviewed AYA after both tests, and analyzed both her interview responses. A
summary of findings about AYA’s geometric discourse follows:
•

My analyses from the Pre-Interview and the Post-Interview show that AYA’s
routines of sorting polygons remained the same.

•

AYA’s routine of substantiation changed from colloquial mathematical discourse,
where her substantiation was a set process of an activity at object-level, towards a
mathematical discourse using previously endorsed narratives about mathematical
objects at an abstract-level. AYA’s routine procedures were descriptions about the
processes of activities using transformations such as reflection, translation, and
rotation at the Pre-Interview, but were constructions of newly endorsed narratives
using propositions and definitions at the Post-Interview.

•

When verifying congruent figures, AYA chose Side-Side-Angle (SSA) as the
conditions of verification at the Pre-Interview, which was incorrect, whereas at
the Post-Interview AYA chose angle-side-angle and side-angle-side, valid
congruent criterions for verification of congruent triangles.

•

When substantiating the equivalence of two definitions, AYA did not know how
to construct newly endorsed narratives from given definitions at the PreInterview, whereas at the Post-Interview AYA constructed a newly endorsed
narrative using deduction.

•

There were changes in AYA’s use of mathematical terminology such as the
names of polygons and their parts.

AYA’s routine procedures for sorting polygons were observed and analyzed in Task One.
During the Pre-Interview, when AYA was asked to sort polygons into groups, her first question

166

was, “Am I doing it on the assumption that those are right angles [pointing at the angles of a
square], and by itself can I assume anyway?” AYA’s first attempt at sorting geometric shapes “in
terms of the numbers of sides they had” resulted in the following: 1) 3-sided figures (n=4)
consisting of all triangles; 2) 4-sided figures (n=13) consisting of all quadrilaterals; and 3) 6sided (n=1) figures, which is Fig. V (a hexagon). When I asked AYA to subgroup the 4-sided
group, her first reaction was, “If I can assume that the sides appear to be parallel to each other”,
while pointing to the opposite sides of a parallelogram. AYA then rearranged this 4-sided group
into three subgroups, and subsequently, she rearranged the 3-sided group into three subgroups as
well. See Figure 4.25 for details of AYA’s subgrouping of the quadrilaterals and the triangles on
the first attempt.

4-sided shapes (n=13)

Group One

Group Two

167

Group Three

Figure 4.25 (cont’d)
3- sided shapes (n=4)

right triangle
isosceles
scalene
Figure 4.25 AYA’s first attempt at sorting polygons at the Pre-Interview

Figure 4.25 presents three subgroups for quadrilaterals: squares/rectangles,
parallelograms and a group of 4-sided figures that do not fit into the descriptions of the two
previous groups. AYA made it very clear about the characteristics of each group. For example,
AYA talked about the parallelograms group consisting only of the parallelograms that “don’t
have right angles”, and the squares/rectangles group consisting of figures that “have four sides,
all right angles, pairs of sides are parallel and have the same length”. Similarly, AYA sorted
triangles into three groups by the characteristics of their angles or sides. On the first attempt,
AYA’s courses of actions in response to the questions about sorting geometric figures focused on
characteristics of angles (e.g., right angles) and sides (e.g., parallel sides or equal sides). During
the interview, AYA did not use measurement tools such as rulers or protractors to check the
angles and sides of the figures, but instead she chose geometric figures under the assumptions
that “the sides appeared parallel” and “angles are right angles”. Figure 4.26 illustrates AYA’s
routine procedures of sorting geometric shapes into different groups.

168

First prompt: “Sort the shapes into groups”
Counting the sides
of shapes

Grouping by the same
number of sides

Conclusion

First prompt continued: “Can you subgroup 3
3-sided and 4-sided group?”
sided

Choosing
intuitively
Figure 4.26

Grouping by visual properties of
figures such as angles and sides

Conclusion

AYA’s first attempt of the routine of sorting at the Pre-Interview
Interview

When I asked AYA to regroup the figures differently, her first response was, “I want to
separate them into shapes containing right angles and shapes that do not contain right angles”.
angles”
Among the eighteen geometric figures in Task One, AYA included figures (n=8) with at least
one right angle in Group One, and included the figures (n=10) with no right angles in Group
angles
Two. See Figure 4.27 for some examples.
Group One: shapes contain right angles (n=8)

Group Two: shapes do not contain right angles (n=10)

Figure 4.27

Examples of AYA’s regrouping at the Pre-Interview
Interview

169

In Figure 4.27 we see a variety of figures in each group, where AYA simply divided
figures that have right angles from those figures that do not have right angles. I then asked AYA
to subgroup Group One, and she provided the following response:
Interviewer: Can you subgroup Group One?
AYA:

I guess for Group One [subgroup 1]…I could take
squares and non-squares …I could take shapes that
have acute angles…like triangle…like K and N
[pointing at Fig. K and N]

AYA:

I am sure that by defining the second group
[subgroup 2] …I don’t know they kind seem
exclusive.... will follow figures that don’t contain
acute angles [pointing at Fig. U and M]

AYA continued to talk about her strategies of subgrouping Group Two. She divided
Group Two into two subgroups that do not contain right angles: one with figures that have at
least one set of parallel sides, and the other with figures that have no parallel sides. Figure 4.28
illustrates the two subgroups in Group Two.
170

Subgroup 1: with at least one set of parallel sides

Subgroup 2: with no parallel sides

Figure 4.28

The two subgroups of Group Two at the Pre-Interview
Interview

During the regrouping, AYA’s courses of actions for sorting geometric shapes focused
mostly on the characteristics of the angles of figures, not the sides of figures. That is, AYA first
divided the entire group of figures (n=18) into two groups, depending on whether the figures had
ed
a right angle or not; and then divided Group One into two subgroups based on whether the
figures had an acute angle or not. AYA divided Group Two according to whether the figures had
er
parallel sides or not; half of the figures in Group Two are parallelograms. AYA’s routine
procedures in sorting geometric figures at the second attempt are summarized in Figure 4.29.
Second prompt: “Find another way to sort them differently?”
Choosing
intuitively

Grouping by the characteristics
of angles of figures
(i.e., right angles)

171

Conclusion

Figure 4.29 (cont’d)
Second prompt: “Can you subgroup the two groups?”
Choosing
intuitively
Figure 4.29

Grouping by the characteristics of
angles and sides of figures (i.e.,
acute angles and parallel sides)

Conclusion

AYA’s second prompt of the routine of sorting at the Pre
Pre-Interview

Ten weeks later I interviewed AYA again, and found no change in her routine procedures
for sorting geometric shapes when compared to those of the Pre interview. For example, at the
Pre-interview.
Post-Interview, when I asked AYA to group the figures she said, “the first thing I want to do is
Interview,
separate them by numbers of sides, like I did last tim [at the Pre-Interview]”. When I asked
time
AYA to regroup the figures, she replied, “This [group] just assumes that all figures appeared to
have right angles … Group Two could just be all the figures that don’t contain right angles”.
angles”
Therefore AYA’s routine procedures for grouping polygons at the Post-Interview were similar to
rocedures
Interview
what she did at the Pre-Interview.
Interview.
Although I did not find any changes in AYA’s routine procedures in classifying
geometric figures between the time of the Pre
Pre-Interview and the Post-Interview, I noted changes
terview,
in her routine procedures of substantiation of narratives. In the following, I describe changes in
AYA’s substantiation routines in Task Two and Task Three be
between the Pre-interview and the
interview
Recall that a routine of substantiation is a set of patterns describing a process of using
endorsed narratives to produce new narratives that are true. For instance, in the context of this
study, a routine of substantiation describes what an interviewee did, step-by-step, to substantiate
step,
her/his declared statements that opposite sides are equal in a parallelogram. One important
finding in AYA’s geometric discourse was the changes in her routines of substantiation as
observed and analyzed in Task Two and Task Three.
172

Recall that Task Two asks interviewees to draw two parallelograms that are different
from each other, and then to discuss the angles, sides and diagonals of these two parallelograms.
At my request, AYA drew a parallelogram and declared, “in this parallelogram all angles should
add up equal to 360°”. After my prompt for substantiation, “how do you know that all angles add
up to 360°?” AYA produced the following:
14a. AYA

Well, when you have parallel

AYA’s drawing:

sides, you can extend all the
sides…

AYA extended the sides of
parallelogram:

14b. AYA

Pointing at the two angles that

complementary angles …

14c. AYA

…it's 180 degrees and they’re

form straight angles:

…but you can see that this

Pointing at the two angles:

angle really just match this
angle
14d. AYA

…so you know that these two
angles together are gonna
equal 180 degrees

173

Pointing at the two angles:

In the preceding substantiation, AYA first drew extended lines on the sides of the
parallelogram, in saying “you have parallel sides… you can extend…”[14a], and identified a
vertex angle and its corresponding exterior angle forming a “complementary angle”[14b]; and
she then identified an adjacent vertex angle transversal to the same exterior angle and made an
intuitive claim about the two angles, “you …see this angle…matches this angle” [14c]. AYA
concluded that the two adjacent vertices of a parallelogram added up to 180 degrees [14d]. Using
this endorsed narrative, “two angles equal 180 degrees”, AYA continued her substantiation to
the final step:
24a.

From this diagram and the

AYA

Pointing at the two angles:

parallel sides, these two
angles add up to 180
degrees…

24b.

… the fact that it’s just like

AYA

Making an invisible line:

a mirror image, the two 180
sets of angles are just gonna
add up to 360 degrees.

AYA used her previously endorsed narrative, “two angles add up to 180”, and then
endorsed a new narrative, “two sets of 180 degrees angles add up to 360 degrees” because these
are a “mirror image” (i.e., a reflection) of each other, and drew a reflection line (i.e., the dashed
line in 24b). AYA’s substantiation of the narrative, “all angles add up to 360 degrees” was
intuitive and self-evident because the reflection line that AYA drew was not a line of reflection
174

of the parallelogram. Mathematically, this parallelogram only has point symmetry, symmetry
with respect to the center of the parallelogram (i.e., where the diagonals intersect), and not line
symmetry. In this example, AYA used a “mirror image”(i.e., reflection) to draw a conclusion
that all the angles add up to 360 degrees.
During the Pre-Interview, AYA frequently used reflections, rotations and translations in
her substantiations of narratives. For example, when I asked AYA to verify her claim that “two
opposite angles (i.e., ∠ 1 and ∠ 4) are equal”, she provided the following response:
37.Interviewer:

How do you know this angle is equal to this?
[Pointing at ∠1 and ∠4]

3
2

38a. AYA:

4

1

…this angle [pointing at ∠1 ] can just be slid over to this position and
create this angle [pointing at ∠2] …

38b. AYA:

…this line [drawing arrowhead on the line] can be rotated so that this
angle [pointing at ∠2] now becomes this angle [pointing at ∠3].

38c. AYA:

…this angle [pointing at ∠3] at this intersection, can just be slid down
and then be in this angle’s position [pointing at ∠4].

38d. AYA

So these two angles are equal [Pointing at ∠1 and ∠4]

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In this case, AYA used words such as “slid over”, “rotated” and “slid down” to indicate a
sequence of movements preformed to substantiate the claim that “two opposite angles are
equivalent”. Lines and angles are static mathematical objects, but AYA used these sequences of
imaginary movements to complete her substantiation; and through AYA’s description, these
imaginary movements became visible to me. AYA’s substantiation was intuitive and visual.
AYA’s substantiation focused on the processing of the activities of mathematical objects, rather
than on discussions about these mathematical objects. I conclude that AYA’s routine procedures
operated at the object level at the time of the Pre-Interview.
Ten weeks later when asked for substantiation, AYA used mathematical axioms and
propositions to verify her claims. The following brief substantiation was typical at the PostInterview:
16a. AYA

… angles on a straight line add up

Extending one side of the

to 180 degrees…

parallelogram with a dashed
line, and pointing at the two
angles:

16b. AYA

…this angle here is the same as this

Pointing at the angles:

angle… Because parallel lines meet
a third line at the same angle.

16c. AYA

By the same reason [referring to

176

Pointing at the two angles:

16a], this angle added to this angle
equals 180 degrees…

16d. AYA

…these two also add up to 180

Pointing at the two angles:

degrees

16e. AYA

…for a similar reason, these two

Pointing at the two angles:

angles add up to 180 degrees...

16f. AYA

Together they equal 360 degree…

AYA made the same statement as previously about the angles of a parallelogram, “all
added together they equal 360°”. In contrast to AYA’s routines of substantiation at the PreInterview, this example shows two changes that are evident: The first is that in each step of
substantiation, AYA provided endorsed narratives (e.g., mathematical axioms and propositions,
etc) as evidence instead of reasoning intuitively. For example, AYA explained how two
transversal angles are equivalent, not because you “can see it” as in the Pre-Interview, but as a
result of “two parallel lines meet a third line at the same angle” in the Post-Interview. The
second change occurs in AYA’s conclusion that “all angles add up to 360 degree”. At the PreInterview, she argued on the assumption of this “mirror image”, whereas at the Post-Interview
AYA reached her conclusion in a repeat of a similar proof that “two angles add up to 180
degrees” for two adjacent angles in a parallelogram [16e-f]. Thus, one change in AYA’s routine
of substantiation was the shift from descriptions about the processes of activities at the object
177

level towards the abstract level. I will suggest that the maturity of this abstract level of
substantiation is revealed in AYA’s substantiation of congruent triangles.
In the following example, I will describe the changes in AYA’s routines of substantiation
of two congruent triangles that I observed between the Pre-Interview and the Post-Interview. To
describe these changes I looked at two aspects: 1) change from the use of transformations in the
process of substantiation at the object level, to the use of mathematical axioms at the abstract
level; and 2) the change in the choices of elements needed for verification of congruent triangles.
During the interviews, participants were asked to substantiate their declared narratives
about the angles, sides and diagonals of a parallelogram. For example, when asked for
substantiation of the narratives, “opposite sides are equal”, “opposite angles are equal” and /or
“diagonals bisect each other”, some interviewees would support their narratives by using rulers
and protractors to measure the corresponding sides and angles, whereas other interviewees would
try to use mathematical proofs to verify their statements. Using triangle congruency to
substantiate the corresponding sides and angles are congruent in a parallelogram is a common
method students utilize.
During my interviews with AYA, when asked for substantiation of declared narratives
about the sides and angles of a parallelogram, AYA’s first response was, “other than just
measuring them?” AYA expected to substantiate her declared narratives without using the
measurement tools at both the Pre-Interview and the Post-Interview. As an example, the
following are AYA’s routine procedures for the narrative, “diagonals bisect each other in a
parallelogram”, using the triangle congruency method at the Pre-Interview.
When AYA discussed the diagonals of the parallelogram, she talked about diagonals
creating two pairs of congruent triangles. After my prompt for substantiation, AYA identified

178

one pair of such congruent triangles, and then identified two corresponding sides and two
corresponding angles from the two triangles to verify their congruency:
64a. AYA

Because I previously established

AYA added two marks:

that, it is given that these are
parallel sides…
64b. AYA

And, these angles are equal and

AYA added two angle signs:

when lines intersect…

64c. AYA

… it's essentially the same

AYA drew extended lines:

intersection, translated to a new
position…
64d. AYA

… I was suggesting that this angle

AYA added an arrowhead on the

is the same as this angle here.

two extendded sides, and two
angle signs:

64e. AYA

… And that likewise, the

AYA identified two angles that

complementary angles, the smaller

form a striaght angle:

angle that makes it add up to 180
degrees…
64f. AYA

… is the same over here…

AYA identified another two
angles that form a straight angle:

179

64g. AYA

… So, now I know that the angle

Pointing at the alternating

here of this triangle is equivalent

interior angles:

to the angle here of this triangle…

64h. AYA

… and this side length is, the

Referring to the two sides:

same of this side length… So, I've
already shown how a side length
and an angle match of each…
64i. AYA

… And then diagonals bisect

AYA added two marks on the

themselves equally. I can't really

diagonal:

prove that, but I'm suggesting that
this side length is the same as this
side length…
64j. AYA

…this triangle is equivalent to this

The shaded area indicates two

triangle here.

congruent triangles:

AYA’s substantiation included two parts: the first was the substantiation of the
equivalence of alternating interior angles [64b-64g], and the second was the verification of
congruent triangles. The first part of substantiation, “this angle is equivalent to this angle” (i.e.,
alternating interior angles), was intuitive and self-evident. To show that opposite angles are
equivalent in a parallelogram [64b], AYA used an instinctive process of translating the
intersection to a new position [64c], and then “suggested” that the corresponding alternating

180

exterior angles were equivalent [64d], another intuitive act. The second part of substantiation
involved the verification of the two congruent triangles that she identified. During the process of
verification, AYA did not use measurement tools to measure the angles and sides (an object level
of verification) to check equivalence, but instead chose three elements of the triangles to verify
congruent triangles abstractly. However it is important to note that AYA’s choice of these three
elements (angle, side, side) for verification of congruent triangles was incorrect, because this
criterion does not guarantee congruent triangles. I conclude that AYA’s substantiation was a
combination of an objective level of substantiation (e.g., these angles are equal), and an abstract
level of verification (e.g., two triangles are congruent), even though her choice of the elements
for verification was not all correct.
Ten weeks later, I interviewed AYA again, and the same tasks were performed. At the
Post-Interview AYA was able to use triangle congruency to substantiate most of her declarations
stating that “opposite angles are equivalent”, “opposite sides are equivalent” and “diagonals
bisect each other” in a parallelogram. AYA was able to choose three exact elements of the six,
such as Side-Angle-Side and Angle-Side-Angle, to verify congruent triangles, and she was
comfortable using the triangle congruency method. The following response illustrates AYA’s
substantiation that “diagonals bisect each other”:
54a. AYA

… I'm looking at this triangle as

Pointing at the shaded area:

compared to this one here…

54b. AYA

And I know that these two

AYA marked the angle signs on

angles are congruent…

the two angles in the shaded
triangles:

181

54c. AYA

Pointing at the two parallel lines

lines, and now this diagonal,

and one diagonal:

… these angles are also

AYA marked the angle signs on

congruent.

54d. AYA

…And between these parallel

the two angles of shaded
triangles:

54e. AYA

Pointing at the corresponding

angle, side, angle, these two

angles, sides and angles of

triangles are congruent.

shaded triangles:

…which means that this side

Pointing at each half of the

corresponds with this side and

54f. AYA

… So, by the triangle test,

diagonals:

that this side corresponds with
that side. That's probably the
most roundabout way to find
that answer.

In the preceding substantiation AYA first verified that the “two triangles are congruent”
[54e] using the angle-side-angle criterion. She identified the exact three elements (i.e., two
182

angles and their included sides) needed for verification; and used the endorsed narrative “the two
triangles are congruent” to construct a new narrative that “diagonals bisect each other”, by
saying “this side corresponds with this side….”[54f] as a result of congruent triangles. AYA also
made no intuitive claim about the equivalence of alternating interior angles at the Post-Interview,
as she clearly explained:
46d. AYA

…And, we know that between parallel lines, if you take a third
line and cross both lines, then it will have angles that are
congruent. In this case, this angle and this angle.
[AYA extended the two parallel lines, and marked angle signs
on the two alternating interior angles]

During the Post-Interview, AYA applied the same substantiation to other similar
situations. For instance, when I asked AYA why diagonals bisect each other in a rectangle, she
responded, “the same as what I did in parallelogram, I already established that.” When I asked
AYA at the end of Task Two, “is it true that in all parallelograms diagonals bisect each other?”
AYA responded, “Yes, that’s true” and then shared her thinking about this conclusion:

183

148. AYA

…because when you draw the diagonals in a figure, there is an
intersection point and it divides the figure into four triangles.
And, regardless of the figure, if it's a parallelogram, these two
triangles will be congruent and these two triangles will be
congruent [pointing at the two pairs of congruent triangles in the
rectangle] So, it can be found that in congruent triangles,
corresponding sides will be equal [therefore diagonals bisect
each other in all these cases].

In summary I conclude that there was a change in AYA’s routine procedures, from using
translation intuitively in the process of substantiation about the equivalence of the angles at the
object level, to using mathematical axioms to substantiate the same claim at the abstract level. I
am convinced that AYA was more rigorous at the Post-Interview, when she made choices of
three elements needed for verification of congruent triangles, than at the Pre-Interview.
To illustrate AYA’s routines of substantiation, and how that substantiation helped to
produce newly endorsed narratives, the following scenario points out a further change in AYA’s
geometric discourse in the context of her routine procedures of constructing new narratives from
previously endorsed narratives.
Recall that Task Three involves a mathematical proof, in discursive terms, where
interviewees were asked to construct new narratives from endorsed narratives (i.e., definitions of
a parallelogram). In particular, interviewees were given two definitions of a parallelogram and
were asked to substantiate that these definitions are equivalent.

184

•

Definition #1: A quadrilateral is a parallelogram if and only if both pairs of
opposite sides have the same length.

•

Definition #2: A quadrilateral is a parallelogram if and only if both pairs of
parallelogram
opposite angles have the same measure.

My conversations with AYA at both the Pre
Pre-Interview and the Post-Interview show that
Interview
she did not know the mathematical meaning of “two definitions are equivalent”; she thought the
statement meant, “they’re both saying the same thing” about the parallelograms. At the PrePre
Interview AYA did not know how to substantiate the equivalence of two definitions; however
she did attempt an analysis by drawing two quadrilaterals that fit the description in definition #1.
description
AYA’s drawings from the Pre-Interview are shown in Table 4.27
Interview
4.27.
Table 4.27

AYA’s attempts at proving the equivalence of two definitions
a. AYA’ first attempt
AYA’s drawing

Transcripts

AYA: I wanted to draw a figure…
making opposite sides the same
length. I thought I was gonna
measure the angles, except that I
cheated and used a right angle.
So, that's not a very good example.

b. AYA’s second attempt
AYA: Well, this equal this side
length, and that equal this side
length … and pairs of the angles of
the same measure. I think the
definitions are equivalent, but how
do I prove it without any numbers?

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AYA first drew a figure as a concrete example of what is described in Definition #1: a
rectangle has opposite sides equal, and in particular the rectangle that AYA drew has opposite
sides with measurements of 4 and 3 respectively (Table 4.27a). AYA suggested that “it was not a
good example” because the conclusion of “opposite angles are equal” is obvious given that it is a
rectangle. As a second attempt, AYA drew an arbitrary quadrilateral with no specific length
measurements and she assumed that it had opposite sides equal as described in Definition #1
(Table 4.27b); but she could not continue the proof because she did not know how to prove the
angles were equivalent without measurements, as she wondered “what is proof of this?”
In contrast, at the Post-Interview AYA was able to complete the substantiation of “if a
quadrilateral has both pairs of opposite sides of the same length, then both pairs of opposite
angles have the same measure”, using the Side-Side-Side triangle congruence criterion to verify
two congruent triangles. Using this endorsed narrative, AYA then identified the corresponding
angles in the two congruent triangles to construct the newly endorsed narrative that “both pairs
of opposite angles have the same measure”. AYA’s routine procedures are analyzed and the
corresponding transcripts from the Post-Interview are provided in Table 4.28.
Table 4.28

AYA’s substantiation of two congruent triangles at the Post-Interview

Routine Procedures

Transcripts

1. Draw a figure

4a. I guess I would start with a figure…

1.1. Identify the given

4c. …just assume that this is a figure that this side length is
equal to this side length and this side length is equal to this
side length, and that's all we know.
12a. I first would just draw a line from these angles, a
transversal here.

1.2 Draw a transversal

2. Verification of two
congruent triangles

186

Table 4.28 (cont’d)
2.1 Identify corresponding
sides of the triangles

12b. … this is a common side in both of these triangles
common side

2.2 Identifying three elements
needed for verification of
congruent triangles

12b. It's enough to say that two triangles with three sides
congruent to one another, to a corresponding side in
another figure, are congruent figures.

3. Conclusion congruent
triangles using S-S-S
correspondence

two congruent triangles: shaded vs. non-shaded
shaded
AYA’s drawing and writing

AYA wrote: “Two ∆’s are ≅ if they 3 corresponding sides”

AYA’s construction of a new endorsed narrative continued as she was trying to show that
“opposite angles in this quadrilateral of the same measure”:
rilateral
Table 4.29

AYA’s newly constructed endorsed narrative at the Post Interview
Post-Interview

Routine procedure

Transcripts

1. Use previously
endorsed narrative

18a. using the fact that these two triangles are
congruent

2. Identify corresponding
angles of the congruent
triangles
2.1 Identifying one pair of
corresponding angles (i.e.,
the first pair of opposite
angles)

18a. I know that this angle is equal in measure to
this angle…

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Table 4.29 (cont’d)
2.2 Identify second pair of
corresponding angles

18b. … and this angle is equal to measure to this
angle…

2.3 Identify the third pair
of corresponding angles

18c. …And that, this one is equal to this one…

2.4 If all the parts of two
angles are equal then two
angles are equal

18d. …therefore, these two added together is
gonna equal these two added together, which
would make them still the same angle,
corresponding…

3. Conclusion

18d. … a four-sided figure with opposite angles
equal…

It is notable that there was a change in AYA’s routine procedures of constructing new
narratives from previous endorsed narratives. At the Pre-Interview, AYA was not able to finish
the proof because she was unsure how to prove the equivalence of the angles without checking
their measurements. However at the Post-interview, AYA was able to verify congruent triangles
by choosing three elements of six elements needed for verification, and using deduction to
construct a new narrative. Thus, AYA’s geometric discourse made a transition from a geometric
discourse with only a partial properties check at an object level at the Pre-Interview, towards a
geometric discourse with routine procedures capable of constructing new endorsed narratives
using mathematical axioms and definitions.
I have described AYA’s change in routine procedures of substantiation and the changes
in her routine procedures of constructing new narratives; now let me describe the changes in
AYA’s use of mathematical terminologies.

188

I begin with my general analyses of words that AYA used in naming geometric figures in
various conversations; in particular, the words quadrilateral, parallelogram, rectangle, square,
rhombus, trapezoid and kite. The total frequencies of the names of these geometric figures from
the Pre-Interview and the Post-Interview are shown in Table 4.30.
Table 4.30

The frequencies of AYA’s use of the names of quadrilaterals at the two

interviews
Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Table 4.31

Pre-T1
3
7
3
5
6
3
0

Pos-T1
1
5
4
4
0
0
0

Frequency
Pre-T2
Pos-T2
0
0
33
19
0
1
6
7
0
0
0
0
0
0

Pre-T3
2
4
0
0
0
1
0

Pos-T3
1
6
0
0
0
0
0

Total frequencies of AYA’s use of names of quadrilaterals at the two

interviews
Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Frequency
Pre
Post
5
2
44
30
3
5
11
11
6
0
4
0
0
0

Table 4.31 shows that the word parallelogram (n=74) is the most frequently used during
both interviews, being mentioned in all three tasks. The word square (n=22) is the second most
frequently used, and the word rectangle (n=8) is third. Note that the large difference in the
frequency of the words parallelogram and rectangle (n=66), and between the words
189

parallelogram and square (n=52). The w
word kite (n=0) was not mentioned at all in both
interviews. Table 4.30 shows that the word rhombus (n=6) and trapezoid were only mentioned at
the Pre-Interview, and mostly were mentioned at Task One. There was a reduction in use of the
Interview,
words quadrilateral, parallelogram, rhombus and trapezoid at the Post-Interview, and there was
Interview,
a slight increase in use of the word rectangle (n=2) at the Post-Interview. However, the
Interview.
frequencies of the word counts do not provide details about how and in what way these words
were used. The following analyses looks at AYA’s word meaning in the use of parallelogram,
parallelogram
rectangle, square, trapezoid and rhombus.
In an earlier section, I described my observations of AYA’s routine of sorting for Task
One. It is important to note that the natures of the tasks designed for the interviews were limited
the
and pre-constructed. For example, when AYA identified geometric figures among given figures
constructed.
in Task One, the pool of choices was limited to eighteen figures and those figures were pre
predrawn. Consequently, AYA’s misunderstandings about some of the geometric figures were not
detected in Task One. It is in Task Two, when I asked AYA to draw two different parallelograms
at the Pre-Interview, that I began to understand AYA’s misconstrued definition of
Interview,
parallelogram, and it was quite different from what I expected:
2. AYA

AYA’s drawing
drawing:

Note: AYA drew a parallelogram first, and extended sides of the
parallelogram later

190

3. Interviewer

Why is this a parallelogram?

4a. AYA

I believe that this is a parallelogram because I drew it so that this
side… would be parallel to this side [pointing at the two longer
sides of the parallelogram]…

4b. AYA

… and this side would be parallel with this side [pointing at the
two shorter sides of the parallelogram]

Later I asked AYA to draw a new parallelogram different from the one she drew, and she
provided the following responses:
86. AYA

AYA’s drawing of new parallelograms:

Note: AYA drew a hexagon first, and she extended sides of the
hexagon later
87. Interviewer

Why is this a parallelogram?

88. AYA

I think it's a parallelogram… because all the sides are parallel
to another side.

89. Interviewer

Why is it a different parallelogram?

90. AYA

It’s different…because there are more sides and because the
angles are different.

191

The preceding conversations present an interpretive description of parallelogram when
AYA used that word at the Pre-Interview. During our earlier conversation, I asked AYA “what is
a parallelogram”, and she responded, “it [parallelogram] is any figure that has at least one pair of
parallel sides. I think trapezoid [pointing at Fig. N, a right trapezoid] is considered a
parallelogram”. When I asked AYA to write down the definition, she wrote, “A parallelogram is
a figure with all sides being pairs of parallel line segments”, and that was inconsistent with her
verbal statement. Neither AYA’s written narrative nor her verbal narrative about parallelograms
mentioned the necessary condition of a parallelogram being a quadrilateral. Because of this
missing condition, AYA chose a hexagon as an example of a different parallelogram. When
identifying and defining parallelograms, AYA focused on the necessary condition of parallel
sides. At the Pre-Interview, AYA’s concept of a parallelogram was unclear, as she expressed, “I
actually don’t know if parallelograms are strictly four-sided figures… or many shapes should be
parallelograms”.
AYA’s use of the word parallelogram (see Figure 4.30) signifies a collection of figures
that share this visual property of parallel sides. Based on AYA’s definition, this collection of
figures could include figures that have one pair of parallel sides such as trapezoids, two pairs of
parallel sides such as parallelograms, or figures that have more than two pairs of parallel sides
such as hexagons. We notice that rectangles and squares are not included in the family tree of
parallelograms. According to what I observed during the Pre-Interview, AYA did not include
rectangles and squares as parallelograms, but rather considered them as a separate group of
figures that have right angles.

192

Parallelogram

AYA: a parallelogram is a figure with
all sides being a pairs of parallel line
segments.

Figure 4.30

AYA: it is a
figure that has
at least one pair
of parallel sides.
I think a
trapezoid is
considered a
parallelogram.

AYA: I think it’s a
parallelogram because all
sides are parallel to another
side [pr2. 88].
More could fit here based on
AYA’s general definition of
parallelogram.

AYA’s use of the word parallelogram at the Pre-Interview
Interview

At the Post-Interview the most important change in AYA’s word use is her use of the
Interview
word parallelogram. Although AYA showed very similar routine procedures when identifying
.
geometric figures in both interviews, her concept of a parallelogram was different from that of
the Pre-interview. For example, when I asked AYA to draw two different parallelograms in Task
interview.
Two, she drew a parallelogram and a square:
2. AYA:

AYA’s drawing of a parallelogram

193

3. Interviewer:

Why is this a parallelogram?

4. AYA:

Because it has four sides and each opposing side is parallel to one
another.
…

…

60. AYA:

AYA’s drawing of a different parallelogram

61. Interviewer:

Why is this a parallelogram?

62. AYA:

It’s a square… it has four sides of equal measure and all angles are
90 degrees.

63. Interviewer:

Why is this different from the one you drew?

64. AYA

This one is different because all the angles in this figure are equal.

AYA’s use of the word parallelogram changed with regard to this added necessary
condition of “four-sided” figure, and the necessary condition of “parallel sides”. AYA
sided”
considered rectangles and squares as figures with 90 degree angles and as parallelograms.
90-degree
AYA’s use of the word parallelogra now signified a collection of figures sharing this common
parallelogram
descriptive narrative, “a four-sided figure with two sets of parallel sides”.
sided
It is notable that AYA included all quadrilaterals with two sets of parallel sides in this
family tree of parallelograms, relating these quadrilaterals because “they have two sets of parallel
ams,
sides”. AYA did not provide any explicit information about how these figures were related other
than being parallelograms. For example, AYA grouped rhombi together with parallelogra
parallelograms
because all rhombi have two sets of parallel sides; however there were no connections made

194

between rhombi and squares, although AYA defined a rhombus as a “four-sided figure with all
side length equal in measure”. Moreover, AYA did not mention any relations between squares
and rectangles other than that they have four right angles. Therefore, I argue that at the PostInterview AYA had a good grasp of the concept of parallelograms in general, but her
understanding of the hierarchy of parallelograms was missing, or not clearly demonstrated in the
interviews. Figure 4.31 illustrates AYA’s understanding of definition of a parallelogram at the
Post-Interview.
Parallelogram

Figure 4.31

AYA’s use of the word parallelogram at the Post-Interview

According to my observations, AYA focused on the angles and sides of geometric figures
when identifying and defining geometric figures. When asked to group figures AYA’s first
reaction was to group them by the numbers of sides. When asked for subgrouping, AYA looked
at differentiating the figures by their angles, such as by right angles versus acute angles. Table
4.32 provides the frequencies of the names of the parts of parallelograms that AYA mentioned in

195

each task during the two interviews, and Table 4.33 provides the total frequencies of the names
of the parts of parallelograms at the Pre-Interview and the Post-Interview:
Table 4.32

The frequencies of AYA’s use of the names of the parts of parallelograms at

the two interviews
Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.33

Pre-T1
1
21
4
9
0
0
9
0

Pos-T1
4
18
4
6
0
0
10
0

Frequency
Pre-T2 Pos-T2
60
69
41
53
18
34
4
3
1
3
27
13
2
4
1
3

Pre-T3
4
7
7
1
3
4
2
0

Pos-T3
24
13
6
1
0
0
0
3

Total frequencies of AYA’s use of names of the parts of parallelograms at the

two interviews
Frequency
Pre
Post
65
97
69
84
29
44
14
10
4
3
31
13
13
14
1
6

Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.33 shows that the most frequently used words relating to the parts of
parallelograms were angle (n= 162) and side (n=153). The words angle (n=129) and side (n=94)
were mentioned mostly in Task Two during both interviews. After angle and side, the word
length is the next most frequently mentioned at both interviews (n= 73), and it was mostly
mentioned during Task Two (n= 52). Likewise the word diagonal (n=44) was mostly used in task
196

Two. These results are expected, as Task Two asks interviewees about the relations of the angles,
sides and diagonals in a parallelogram. I want to draw a little attention here to words such as
parallel side (n=24), opposite side (n=7) and opposite angle (n=7); these words describe
important characteristics of a parallelogram, but were least mentioned at both interviews.
To support my claims of AYA’s change in geometric discourse, I described her word use
and routine procedures during the interviews, and analyzed the changes in AYA’s routines of
substantiation, routines of constructing new narratives and her use of the word parallelogram.
We see a dynamic change in her geometric discourse, from a colloquial mathematical discourse
towards a mathematical one.
Case 5: Changes in ARI’s Geometric Discourse
ARI was a college sophomore at the time of the interviews. ARI took her last geometry
class in 9th grade, about five years prior to the geometry and measurement class. The van Hiele
Geometry Test showed that ARI was at Level 3 at the pretest, and stayed at Level 3 according to
the posttest ten weeks later. I interviewed ARI after both tests, and analyzed her interview
responses. Based on my observations, findings about ARI’s geometric discourses from the PreInterview and the Post-Interview are as follows:
•

ARI changed her routines of sorting polygons, from focusing on the names and
attributes of the quadrilaterals at the Pre-Interview, to focusing on the hierarchy of
the classifications of the quadrilaterals at the Post-Interview.

•

ARI’s routine of substantiation changed from a combination of recalling and
measuring, a routine procedure using measurement tools to check the results at an
object-level at the Pre-Interview, to routine procedures using previously endorsed
narratives to construct new narratives at an abstract-level at the Post-Interview.

197

•

ARI applied congruence criterions such as angle-side-angle, side-side-side for
verification of congruent triangles; and used the dissection method to verify the
sum of the interior angles of parallelograms.

•

ARI used new mathematical terminology to describe and to justify the congruent
parts of triangles and parallelograms at the Post-Interview, whereas she explained
her claims informally at the Pre-Interview.

In the following sections let me begin with ARI’s routine of sorting. The findings of
ARI’s routine procedures of sorting polygons are mostly observed in Task One. Task One has
eighteen polygons, consisting of triangles, quadrilaterals and a hexagon. During the interviews,
ARI was asked to place these figures into groups, and then to regroup them differently.
At the beginning of the Pre-Interview, when ARI was asked to sort the polygons into
groups, she sorted them based on their names. She grouped eighteen polygons into groups of
triangles, rectangles, squares, parallelograms/rhombi, quadrilaterals, and other. ARI’s method
of grouping was as follows:
2a. ARI

This group is triangles:

2b. ARI

… And then these ones are rectangles [pointing at the Fig. M, F, G]:

198

2c. ARI

U, G, R are squares [pointing at Fig. U, G, R]:

2d. ARI

… and these P, L, J, Z and H are parallelograms… rhombuses…

2e. ARI

And then N and Q just are quadrilaterals…

2f. ARI

And then V is … just a weird shape.

ARI grouped all 3-sided polygons together and called them triangles; she also sorted all
rectangles together, as well as squares. She grouped parallelograms and rhombi together, the
only group with two names. ARI put Fig. N (a right trapezoid) and Fig. Q (a quadrilateral with
no parallel or equal sides) together because both have just four sides. Fig. V, a hexagon, was
grouped by itself. ARI called it “other” because “it is a just weird shape”.

199

I asked ARI to regroup the polygons differently, and she regrouped them by combining
rectangles and squares together. ARI called that group rectangles:
22. ARI These are just rectangles because squares can be also
rectangles…yeah [pointing at Fig. U and M]

ARI continued her regrouping, and she split the parallelograms/rhombuses group into a
group of parallelograms and a group of rhombi
24a. ARI

And then L, Z those are rhombuses

24b. ARI

And then, those are parallelograms

200

ARI sorted the polygons based on the characteristics of their angles. She grouped the
polygons into a right-angled shape group (n= 8), consisting of polygons with at least one right
angle, and an obtuse triangle (n=2) group containing triangles with an obtuse angle.
28. ARI

And these are right-angled shapes.

30a. ARI

..the right angle triangle [pointing at Fig. K]

30b. ARI

…and this has a right angle here [pointing at a right angle in Fig. N]

30c. ARI

All the squares and rectangles [pointing at Fig. U and Fig. M again]

30d. ARI

These ones I put as obtuse triangles

Three polygons were left: Fig. S, Fig. Q and Fig. V. Then the following conversation
took place:

201

33. Interviewer

Why doesn’t Fig. S go with any other groups? [Pointing at Fig. S]

34. ARI

It doesn’t have a right angle, it’s not four-sided…so it’s not a
square, a parallelogram, and then …it’s not obtuse either.

35. Interviewer

How about Fig. Q? Why is Q left out? [Pointing at Fig. Q]

36. ARI

It doesn’t have parallel sides and it doesn’t have a right angle.

37. Interviewer

How about Fig. V? [Pointing at Fig. V]

38. ARI

V? …that’s just a weird shape…

ARI grouped figures according to their common descriptive narratives (i.e., definitions)
by direct recognition. For example, when I asked ARI if I could put Fig. U (a square) and Fig. L
(a rhombus) together, she responded, “yes, because they both can be seen as rhombuses”. When I
followed by asking ARI why she thought the two polygons were rhombuses, she said, “because
all their sides are equal lengths”. ARI did not use any measurement tools to check the angles or
the sides of any polygon for verification while working on Task One. Therefore, ARI’s
judgments about the attributes of the angles and sides were direct recognition and intuition.
Figure 4.32 summarizes ARI’s routine procedures for sorting polygons at the Pre-Interview.

202

Grouping: “Sort the shapes into groups”
Visual recognition
(Identifying routine)

Grouping by family
appearances of the figures
(Identifying routine)

Conclusion

Regrouping: “can you regroup them differently?”
Visual recognition
(Identifying
routine)

Figure 4.32

1. Grouping by the common
descriptive narratives of the names
of the figures (Defining routine)
2. Grouping by the characteristics
of the angles and the sides of the
figures (Identifying routine)

Conclusion

ARI’s routine procedures for sorting polygons at the Pre
Pre-Interview

Ten weeks later I interviewed ARI again. In her routine procedures of sorting
quadrilaterals she arranged them with a hierarchy of classifications. Let me begin with ARI’s
responses in grouping the eighteen polygons. ARI first grouped polygons (n=18) by th numbers
the
of their sides: triangles (n=4), quadrilaterals (n=13) and a six-sided figure (n=1). ARI then
divided the quadrilateral group into subgroups consisting of squares, rectangles,
rectangles
parallelograms, trapezoids and quadrilatera Figure 4.33 illustrates all the subgroups of the
quadrilaterals.
quadrilateral group.
Figure 4.33

ARI’s grouping of the subgroups of the quadrilaterals at the Post-Interview.
Post
Quadrilaterals (n=13)
Trapezoid

203

Figure 4.33 (cont’d)
Parallelograms (n=11)

ARI: “…then P, L, J, Z, H, R, T G, F, M and U are all parallelograms”

ARI: “ M, F T, U, G, R are rectangles”
“ U, G, R are squares”
Figure 4.33

ARI’s grouping of the subgroups of the quadrilaterals at the PostInterview.

204

At the time of the Post-Interview, ARI was able to group the polygons using a hierarchy
Interview,
the
of classifications of parallelograms. In her grouping, a polygon could be placed multiple times
because it was identified in several subgroups by different common descriptive narratives of the
polygons. For example, in the fami of parallelograms, Fig. R, a square, was not only classified
family
as a parallelogram but it was also identified as a rectangle.
When I asked for regrouping, ARI regrouped triangles according to the characteristics of
their angles, splitting the triangle group into three subgroups consisting of obtuse triangles, a
triangles
right triangle and an acute triangle For the parallelograms, the only change ARI made from her
triangle.
,
previous grouping was to put the rhombuses and the squares into the same group, which she
called the rhombuses group. When I asked ARI why she made this change, she responded, “they
are all rhombuses because they have equal sides” In this regrouping process, a square was also
e
sides”.
grouped as a rhombus by this common descriptive narrative, “a rhombus is a four sided figure
with all equal sides”. During the Post Interview, ARI again did not use measurement tools.
.
Post-Interview,
ARI’s routines of sorting polygons at the Post
Post-Interview are summarized in Figure 4.34.
rview
4.34
Grouping: “Sort the shapes into groups”
1. Grouping by the numbers of sides
(Identifying routine)
2. Grouping by the common narratives
on the names of the polygons, and
arranging the polygons by the hierarchy
of classifications (Defining routines )

Visual
recognition
(Identifying
routine)

Conclusion

Regrouping: “can you regroup them differently?”
Visual
recognition
(Identifying
routine)

1. Grouping by the common descriptive
narratives on the names of the figures
(Defining routine)
2.
Grouping by the characteristics of the
angles and the sides of the figures
(Identifying routine)

Figure 4.34

Conclusion

ARI’s routines of sorting polygons at the Post Interview
Post-Interview
205

In ARI’s responses for Task One in the Pre-Interview and the Post-Interview, she was
able to group geometric shapes by their common descriptive narratives. Also, she was able to
classify quadrilaterals into a hierarchy with the help of her defining routines at the PostInterview. At the Pre-Interview, ARI’s defining routines focused on the necessary conditions of
the definitions. Thus figures were categorized by their common names. In contrast, at the PostInterview, ARI’s defining routines focused on both necessary and sufficient conditions of the
definitions, and thus, the quadrilaterals were grouped with a hierarchy of classifications. Change
in ARI’s geometric discourse also occurred in her routines of substantiation. In the following
section I describe ARI’s routine procedures of substantiating her claims.
I observed ARI’s routine patterns of verifying and justifying declared narratives while
working on Task Two. ARI was able to use endorsed narratives to construct new narratives at the
Post-Interview, whereas she depended on recalling and measuring routines at the Pre-Interview.
A recalling routine is a course of action using previously endorsed narratives, and is more about
remembering what one learned previously. In this study, A measurement routine is a set of
repetitive actions where participants measure the parts of polygons, and use these measurements
in their identifying, verifying and substantiating processes. In the next example, ARI tried to
verify that all angles add up to 360 degrees in a parallelogram.
In Task Two all interviewees were asked to draw two parallelograms different from each
other, and to discuss the angles of the parallelograms. ARI drew a parallelogram and wrote that
all of the angles added together equal 360°. When asked for substantiation, ARI first showed that
two adjacent angles added together equal 180°, and next used that result to justify her claim that
“all of the angles added together equal 360°”:

206

26. ARI

If you add this angle and this angle

Pointing at the angle:

together, it’s equal 180…
27. Interviewer

Why is that?

28. ARI

There is a name for it. I forget the
term…

29. ARI

I think this angle is suppose to

Pointing at the two angles:

equal this outside angle here…

30. ARI

…and this line would equal 180

Pointing at the two angles:

31. Interviewer

How do you know these angles are

Pointing at the two angles in

equal?

[29]

32. ARI

…because I learned it in school? I
don’t know how to explain it.

33a. ARI

If that equals 180 added together,

Pointing at the two angles:

33b. ARI

…then this would equal 180 added

Pointing at the other two

together. So all the angles all

angles

together would equal 360.

In this episode, ARI’s routine procedure is a recalling routine, where she remembered
related mathematical rules, without knowing their mathematical terms or why these rules work in
207

a given situation. For example, in looking at the two adjacent vertex angles in a parallelogram,
ARI concluded that they added up to 180 degrees [26]. This conclusion is correct because we
know by definition that a parallelogram has opposite sides parallel, and we can use propositions
about parallel lines and their transversals to conclude that the two angles add up to 180 degrees.
Similarly, ARI was able to recognize that the alternating interior angles were congruent [29], but
she did not know why these two angles were congruent, a consequence of two parallel lines cut
by a transversal. Note that ARI utilized logical thinking in the statement, “if that equals
180…then all the angles…” [33a-b].
In contrast to ARI’s response at the Pre-Interview to the declared narrative that the angles
add up to 180 degrees, her substantiation of this narrative was different at the Post-Interview.
33a. ARI

These are adjacent angles…

Pointing at the angles:

33b. ARI

… and these are alternate interior angles

Pointing at the angles:

and they are equal, because… by the
parallel lines
33c. ARI

… then this plus this angle is 180

Pointing on the angles:

because angles on a line.

33d. ARI

… And so this and this, the adjacent

Referring to the adjacent

angles equal 180.

angles:

208

In this argument ARI began to use mathematical terms to express her ideas. ARI used the
words “adjacent angles” and “alternating interior angles” to replace her informal use of “this
angle” and “that angle”, and “angle inside” and “angle outside” at the Pre-Interview. More
importantly, ARI substantiated her claims using endorsed narratives. For example, when ARI
produced the narrative that alternate interior angles were equal, she used a phrase by … the
parallel lines [33b]; and when ARI declared another narrative that two angles add up to 180
degrees, she used the phrase because angles on a line [33c]. When I asked ARI why she thought
all angles in a parallelogram add up to 360°, the following conversation took place:
34. Interviewer

Why do all the angles add up to 360? What are the angles that
add together to equal 360?

35. ARI

All the interior angles, this plus this plus this plus this equal
360. [Pointing at these angles]:

36. Interviewer

Why is that?

37. ARI

Because if you draw a line, the diagonal, there are two triangles,
and the interior angles of a triangle equal 180. So, two triangles
would equal 180 plus 180. That equals 360. [Draw a diagonal
on the figure]

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I expected ARI to use a newly endorsed narrative, such as two adjacent angles add up to
180° [33d], to make the substantiation. She surprised me with her dissection method, drawing a
diagonal to cut the parallelogram into two triangles. ARI then used the endorsed narrative that
the sum of the interior angles of a triangle equals 180° [37] to complete her proof of the claim
that all interior angles in a parallelogram add up to 360°. ARI’s routine procedures also changed
in verifying the congruent parts of a parallelogram.
ARI was aware of the abstraction of congruent parts of the parallelograms at the PostInterview, whereas she only used measurement routines to check the congruent parts at the PreInterview. That is, when recalling routines did not seem to help, ARI used measurements to
check her claims about the sides and angles of a parallelogram. In the next conversation, I asked
ARI to substantiate her declared narrative about the diagonals of a square:
182. Interviewer

What can you say about the diagonals of this parallelogram?
[Pointing at the square]

183. ARI

They’re gonna be equal lengths

184. Interviewer

How do you know?

185a ARI

Because they are all equal sides and I am pretty sure they
would be all equal diagonals…[Pointing at the sides of the
square]

185b. ARI

I can check…

185c. ARI

Let’s do it in inches…this one is 2.7… that’s also 2.7
[Using a ruler to measure the length of the diagonals of the
210

square].
185d. ARI

Yeah, diagonals are equal lengths.

When asked for substantiation, ARI first made an intuitive claim about why the diagonals
would be equal [185a], and then used a ruler to measure the length of the diagonals, getting
measurements for each diagonal of 2.7 inches [185c]. With this confirmation, ARI concluded
that the diagonals were of equal length [185d], completing substantiation.
Our conversation went on, and ARI declared another narrative about the diagonals of a
square being perpendicular to each other.
189. Interviewer

What can you say about the diagonals of this
parallelogram? [Pointing at the square]

190. ARI

I think they’re perpendicular to each other.

191. Interviewer

What do you mean when you say perpendicular?

192. ARI

At the intersection, they create a 90-degree angle
[Pointing at the intersection of the diagonals]

193. Interviewer

How do you know they are 90-degree angles?

194. ARI

I can measure it… Yeah… it’s 90-degree.
[Using a protractor to measure one of the angles
at the intersection]

195. Interviewer

How do you know they are all 90-degrees?
[Pointing at the other angles at the intersection]
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196. ARI

I am pretty sure that they are all 90-degrees.
Yeah.. this is 90… that’s 90… they are all equal.
[Giggling, and use a protractor to measure two
other angles at the intersection]

When I asked, “how do you know that they are all 90-degrees?”, I was looking for a
routine procedure operating at an abstract level. However, in ARI’s response to this question, she
focused on the concreteness of the congruent angles, in using a protractor to check the angles one
by one [194; 196], thereby using measurement routines to verify her claims at an object level.
As in her responses at the Pre-Interview, ten weeks later ARI declared the same narrative
about the diagonals of a square, stating that the diagonals were equal length. When asked for
substantiation, ARI this time did not measure the length of the diagonals but responded with “I
can prove again that the triangles are congruent”. ARI had just substantiated that the diagonals in
a rectangle were equal, so she applied that argument. Table 4.34 summarizes ARI’s routine
procedures of this verification with corresponding transcripts:
Table 4.34

ARI’s routine procedures of substantiation for “two diagonals are equal”

Routine Procedures

Transcripts

1. Identify two congruent
triangles

111a. ARI: this triangle and this triangle are congruent [pointing
at the shaded triangles]:

2. Declared narrative

111b. ARI: and then that side equals this side [pointing at the
diagonals]:

212

Table 4.34 (cont’d)
Prompt for verification

112. Interviewer: How do you know these two triangles are
congruent? [Referring to 111a]

3. Verification of two
congruent triangles
3.1 Identify
corresponding angles of
the triangles

113a. ARI: well, these are 90 degrees [pointing at the two right
angles]

3.2 Identify
corresponding sides of
the triangles

113b. ARI: they have a common side so that would be the same
for both triangles… [Marked a tally on the common side]

3.3 Identify another
corresponding sides of
the triangles

113c. ARI: opposite sides that are parallel…are equal [Marked
two tallies on the opposite sides of the rectangle]

4. Verify congruent
triangles using S-S-S
correspondence

113d. ARI: that gives you side, angle, side … and makes these
two triangles congurent.

5. Conclusion

113e. ARI: by that, you can conclude that this side and this side
are equal [pointing at the diagonals of the rectagnle].

Table 4.34 shows that ARI first identified a pair of congruent triangles with diagonals as
one set of corresponding sides of the triangles, and drew a conclusion about the equivalence of
the diagonals [111a-b]. After my prompt for substantiation, ARI provided a sequence of steps of
selecting three elements needed for the verification of congruent triangles [113a-c]. During this
selection, ARI did not use a ruler or protractor to check the measurements of the sides and
angles, but instead used identifying routines and defining routines. For example, ARI used the
definition of a rectangle to identify two corresponding angles that “are 90 degrees” [113a], and
two corresponding sides that “are equal” [113c] in the triangles.
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86. Interviewer

Why is this a parallelogram? [Pointing at ARI’s drawing]
[Pointing

87. ARI

Because opposite sides are parallel and the opposite sides are equal.

88. Interviewer

Why is this a different parallelogram from the one you drew?

89a. ARI

Because they all form 90 degree angles, all the angles are equal, not
just the opposite angles.

89b. ARI

This is a rectangle.
his

ARI used an identifying routine to identify the right angles and opposite sides as parts of
the rectangle; and then used a defining routine to confirm her choice of the elements of the
congruent triangles needed for verification. Because the polygon was a rectangle, all angles were
equal and opposite sides were equal. It is legitimate for ARI to apply this proof in the case of a
square as in the next example, because she considered a square as a rectangle (see my earlier
analyses for routines of sorting).
We continued to discuss the diagonals of a square. ARI produced further narratives such
as “diagonals are perpendicular to each other”, “diagonals bisect each other” and “diagonals
“diagonals
bisect the angles”. As noted earlier, ARI made the connection between the diagonals of a square
and the diagonals of a rectangle, as in both cases their “diagonals were equal”. Later ARI made
another connection between the diagonals of a square and the diagonals of a parallelogram, as in
ween
both cases their “diagonals bisect each other”. ARI used the Angle
Angle-Side-Angle criterion to
Angle
substantiate two congruent triangles in a parallelogram, and applied that result to draw t same
the
conclusion in the case of a square. To avoid redundancy, ARI’s substantiation of “the diagonals
214

bisect each other” is not presented here. Instead I share ARI’s substantiation of “diagonals are
perpendicular to each other”:
133. ARI

They are perpendicular to each other. [Pointing at the intersection
of the diagonals]:

134. Interviewer

How do you know?

135a. ARI

[Thinking]… because the angles of the square are 90 degrees
[Pointing at the angle]

135b. ARI

And the diagonal cut this angle in equal halves. [Pointing at the
two angles]

135c. ARI

So this would be 45 degrees and this would be 45 degrees. [Writing
45-degree at each angles]

135d. ARI

So that would equal to 90 degrees … [Adding the two 45-degree
angles]

135e. ARI

For angles a triangle it would be 180 total, so it would have to be
90 degrees. [Pointing at the intersection of the diagonals]

215

To verify that the diagonals are perpendicular to each other, ARI tried to show that the
diagonals form 90-degree angles [135a]. ARI did not use a protractor to measure all the angles at
the intersection, but used endorsed narratives to verify her claims [135e]. In the process of this
verification, ARI made one assumption, that “the diagonal cuts this angle in half” [135b], that
was not mentioned before. Therefore, I asked for substantiation:
142. Interviewer How do you know that the diagonal

Pointing at the angle

cuts the angle in two halves?

143a. ARI

Because, if the diagonals are bisecting

Pointing at the halves of

each other, …. Then the halves are all

one diagonal:

equal lengths too … Because the
diagonals are equal lengths.
143b. ARI

… So from there, if these sides are

Pointing at the base

equal, and it would be an isosceles

angles of the triangle:

triangle…by the definition of an
isosceles triangle, then these angles
would have to be equal.
143c. ARI

Pointing at the adjacent

would also be an isosceles.

143d. ARI

And then the same with this triangle, it

triangle:

So, these are the same isosceles

Pointing at the angles:

triangles. So, the triangles are all

216

congruent, and the angles would all
have to be congruent.
143e. ARI

But this is 90 degrees. So, if these

Pointing at the angle of

angles are equal… then they are all 45

the square:

degrees.

143f. ARI

So the diagonals cut the angles in half.

ARI applied several endorsed narratives to substantiate her declared narrative “the
diagonals cut the angles in half”. ARI first used “diagonals bisect each other” and “diagonals are
equal”, both newly endorsed, to conclude that “the halves are all equal” [143a], and used this
newly endorsed narrative to identify congruent isosceles triangles [143b-d]. By using the
properties of isosceles triangles, ARI showed that all corresponding [base] angles were equal
[143b]. Knowing that the figure was a square, with 90-degree angles, ARI concluded that the
diagonals cut the angles in halves and they were all 45 degrees [143e-f].
In summary, I conclude that ARI changed her routine of substantiation, when she moved
away from an object level of measuring and checking the congruent parts of the parallelogram at
the Pre-Interview, towards an abstract level of substantiation, using endorsed narratives to verify
the congruent parts at the Post-Interviews.
The use of words and language are important when we study ones mathematical
discourse. Different interviewees show different developments in their use of mathematical word
use. In ARI’s case, she was able to use more mathematical terms in the Post-Interview than in
the Pre-Interview; she also developed conceptual understandings of quadrilaterals and

217

parallelograms after the Pre-Interview. Let me begin with some general findings of words that
ARI used in the names of the polygons. More specifically, these words were quadrilateral,
parallelogram, rectangle, square, rhombus, trapezoid and kite. The total frequencies of the
names of quadrilaterals at the two interviews are shown in Table 4.35 and Table 4.36.
Table 4.35

The frequencies of ARI’s use of the names of quadrilaterals at the two

interviews
Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Table 4.36

Pre-T1
3
5
5
6
5
3
0

Pos-T1
4
3
2
1
3
2
0

Frequency
Pre-T2
Pos-T2
0
0
1
4
0
1
1
3
0
1
1
0
0
0

Pre-T3
1
3
0
0
0
0
0

Pos-T3
0
4
7
7
2
0
0

Total frequencies of ARI’s use of names of quadrilaterals at the two

interviews
Frequency
Pre
Post
4
4
9
11
5
10
7
11
5
6
4
2
0
0

Name
Quadrilateral
Parallelogram
Rectangle
Square
Rhombus
Trapezoid
Kite

Table 4.36 shows that the word parallelogram (n=20) was the most frequently used
during the interviews, and being mentioned in all three tasks (see Table 4.35). The word square
(n=18) was the second most frequently used, and rectangle (n=15) was third. The word kite
(n=0) was not mentioned at all in both interviews, and trapezoid (n=6) was the second least
218

mentioned. Table 4.35 shows that the words quadrilateral, rhombus and trapezoid were mostly
mentioned in Task One, where interviewees were asked to group the polygons. There was an
increase in use of the words parallelogram square and rhombus in the Post-Interview, over the
parallelogram,
Interview,
Pre-Interview, and use of the word rectangle doubled in the Post-Interview. ARI’s use of the
Interview,
names of quadrilaterals was much lower than other interviewees’ use of those names.
In an earlier section, I described ARI’s routine procedures for sorting quadrilaterals for
Task One. At the Pre-Interview, ARI first grouped the figures by their names and then by the
Interview,
grouped
characteristics of their angles. However ARI was confused about how a trapezoid and a
parallelogram were related. ARI’s confusion was not detected in Task One, but rather her choice
in drawing a different parallelogram at Task Two shed light on her understanding of the words
llelogram
trapezoid and parallelogram.
1a. ARI

I don’t know if this is right, but I am going to draw it.
[ARI drew a figure]

1b. ARI

I think this is wrong.

3. Interviewer

Why? I just want to know what bothers you.

4. ARI

I know these are parallel sides [pointing at the two
parallel sides], but I don't know if a trapezoid is also a
parallelogram. I am not sure.

5. Interviewer

What is a trapezoid?

6. ARI

I am not sure, cause I think trapezoids need two parallel
sides…

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7. Interviewer

What is a parallelogram?

8. ARI

[Thinking]…A four sided shape with like this [Pointing
four-sided
at her first drawing], opposite sides that are parallel?

The preceding conversation shows that ARI’s concept of trapezoid was unclear, as she
explained, “I don’t know if a trapezoid is also a parallelogram” [4]. ARI produced a verbal
narrative to my request “what is a trapezoid”, stating that a trapezoid needs two parallel sides [6].
two
We see that ARI did know something about trapezoids, four-sided polygons with two parallel
sided
sides, but had no clear understanding that a pair of parallel sides is a necessary condition in the
definition of trapezoid, confusing trapezoid with parallelogram, a polygon requiring two pairs of
,
parallel sides.
In Task One, ARI created the parallelogram group consisting only of parallelograms and
rhombi. The parallelogram group was later split into parallelograms and rhombi as two
subgroups. ARI created the rectangle group consisting of rectangles and squares then divided
them into a rectangle group and a square group. At that time ARI did not group the rectangles
with the parallelograms, and did not provide information abou how rectangles and
about
parallelograms were related. In Task Two, ARI identified a rectangle and a square as
parallelograms, and explained, “they are parallelograms, because they have both opp
opposite sides
parallel”. ARI’s use of the word parallelogram described a collection of figures having parallel
ed
sides. As noted, I included ARI’s drawing of a trapezoid in this diagram because she did consider
at one point that a trapezoid could be a parallelogram, displaying her confusion about what a

220

trapezoid was, and how it related to a parallelogram. Figure 4.35 illustrates ARI’s understanding
that all squares are rectangles, as well as her understanding of the relation between a rhombus
and a parallelogram. However, ARI did not clearly demonstrate the connections between a
rhombus and a square, but she knew that both were a “four-sided figure with all sides equal”. I
conclude that ARI’s uses of the names of parallelograms were limited to their common
descriptive narratives at the local level, while her understanding of the hierarchy of
parallelograms at the global level was missing in the Pre-Interview.
To me, ARI’s use of the word parallelogram referred to a family of parallelograms with
two branches: parallelograms and rectangles, illustrated as follows:

Parallelogram

Parallelograms /rhombi

Figure 4.35

Rectangle

ARI’s use of the word parallelogram at the Pre-Interview

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At the Post-Interview the change in ARI’s word use regarding the names of quadrilaterals
was revealed in the hierarchy of classifications of parallelograms. In ARI’s grouping and
regrouping for Task One in the Post-Interview, the word parallelogram described a collection of
figures that were 4-sided with two pairs of parallel sides. In the family of parallelograms, ARI
also recognized two subgroups: a rectangle group consisting of parallelograms with right angles,
and a rhombus group consisting of parallelograms with all equal sides. Based on the
characteristics of the sides, ARI also split the rectangle group into squares, a group of rectangles
with all equal sides, and rectangles. Lastly, ARI grouped rhombi and squares together because
all their sides were equal.

Parallelogram (n=11)

Rectangle (n=3)

Rhombus (n=2)
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Figure 4.36 (cont’d)

Square (n=3)

Figure 4.36

ARI’s classifications of parallelograms at the Post-Interview

Figure 4.36 shows a hierarchy of classifications of the parallelograms in ARI’s use of the
word parallelogram. In this hierarchy, the word parallelograms denoted a collection of
quadrilaterals with different names, and these names were parallelograms, rectangles, squares
and rhombi. Although these names identify polygons with different physical appearances and
attributes of their angles and sides, they are all parallelograms.
There was also a change in ARI’s use of the names of the parts of parallelograms. In this
study, the names of the parts of parallelograms considered were angle, sides, length, parallel
side, opposite side, opposite angle, right angle and diagonal. Findings show that ARI used more
words describing the parts of the parallelograms than the names of the parallelograms. Tables
4.37 and 4.38 provide information on the frequencies of these words’ usage at the Pre-Interview
and the Post-Interview.

223

Table 4.37

The frequencies of ARI’s use of the names of the parts of parallelograms at

the two interviews
Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.38

Pre-T1
1
10
1
2
2
2
4
0

Pos-T1
2
10
0
1
3
0
1
0

Frequency
Pre-T2
Pos-T2
22
26
11
25
18
14
2
1
6
11
4
12
0
1
4
4

Pre-T3
4
0
5
0
3
0
0
2

Pos-T3
3
7
5
0
7
0
0
4

Total frequencies of ARI’s use of names of the parts of parallelograms at the

two interviews
Frequency
Pre
Post
27
31
22
42
24
19
4
2
11
21
6
12
4
2
6
8

Name
Angle
Side
Length
Parallel side
Opposite side
Diagonal
Right angle
Opposite angle

Table 4.38 shows that the most frequently used words relating to the parts of
parallelograms were side (n= 64) and angle (n= 58). The words angle (n= 48) and side (n=36)
were mentioned mostly in Task Two during both interviews (see Table 7). After angle and side,
the word length was the next most frequently mentioned at both interviews (n= 43), and it was
mostly mentioned at Task Two (n= 32). Likewise the word diagonal (n=16) was mostly used in
task Two. These results were expected, as Task Two asks interviewees about the relations of the
angles, sides and diagonals of a parallelogram. Among the terms opposite sides, parallel sides
224

and opposite angles (n=14) that describe important characteristics of parallelograms, opposite
sides (n=33) was mostly mentioned, and parallel sides (n=6) the least mentioned.
During the interviews, I noticed that ARI used the words sides and lengths
interchangeably. For example, ARI identified Fig. Z as a rhombus because “it has all equal
lengths”[pr1. 50], and later she stated a rhombus “has equal sides”. Also, ARI used the phrases
“same length” and “same side” frequently during the interviews when referring to figures with
the same length measures.
Another important change in ARI’s use of words was in her use of formal mathematical
terminology at the Post-Interview. These formal mathematical terms describe attributes of lines
and angles, as well as the relations between them. In particular, she used the words adjacent
angle, alternating interior angles, complementary, supplementary, transversal and congruent. In
an earlier section, I briefly mentioned that ARI used the terms “adjacent angles” and “alternating
interior angles” to replace her informal use of “this angle” and “that angle”, and “angle inside”
and “angle outside” in her substantiation that angles add up to 360°.
Tables 4.39 and 4.40 provide information on the frequencies of these new mathematical
words used at the interviews.
Table 4.39

The frequencies of ARI’s use of formal mathematical words at the two

interviews
Name
adjacent angle
alt. Interior
complementary
supplementary
vertical angle
transversal
congruent

Pre-T1
0
0
0
0
0
0
0

Frequency
Pre-T2
Pos-T2
0
4
0
8
0
2
0
1
0
2
0
1
0
12

Pos-T1
0
0
0
0
0
0
0

225

Pre-T3
0
0
0
0
0
0
0

Pos-T3
1
0
1
0
0
0
0

Table 4.40

Total frequencies of ARI’s use of formal mathematical words at the two

interviews
Frequency
Pre
Post
0
5
0
8
0
3
0
1
0
2
0
1
0
12

Name
adjacent angle
alt. Interior
complementary
supplementary
vertical angle
transversal
congruent

Table 4.40 shows that all these mathematical words were used only at the Post-Interview,
and most of them were mentioned in Task Two, where interviewees discussed the angles, sides
and diagonals of a parallelograms. The word congruent (n=12) was the most frequently
mentioned, as ARI used congruent triangles to substantiate the congruent parts of the
parallelograms in the Post-Interview. During the process of verification of congruent triangles,
ARI often identified alternating interior angles (n=8) as one of the elements for verification, and
consequently this term was the second most mentioned (see Example Two).
ARI’s transition from colloquial mathematics discourse towards mathematical discourse
also appeared in her use of logical justification. For example, at the Post-Interview, ARI
produced narratives with justifications such as “those sides are equal by corresponding parts in
congruent triangles”, “angles are equal because vertical angles are equal” and “these sides are
equal by the definition of a parallelogram”. To me, ARI’s use of such mathematical language
was a step closer to the language of proof.
The van Hiele Levels as Geometric Discourse
In this section I introduce the model, the Development of Geometric discourse (See
Appendix F) that I developed for this study. In order to have a theoretical basis of the model, I
226

translated each van Hiele level into discursive terms using four mathematical features described
in the commognitive framework. These translations are presented in Chapter 2.
The model consists of four components: (1) Geometric Objects are utilized in the
participants’ use of mathematical words, saming criterions, realizations and systems of objects,
(2) Routines are used in participants’ courses of actions in response to mathematical tasks, (3)
Endorsed Narratives are collected from participants’ written or verbal narratives in this study,
(4) Visual Mediators are collections of symbolic artifacts, geometric figures and their parts, all
involved in the study.
Let me describe Geometric Objects and Routines in a bit more detail. In a mathematical
discourse, a mathematical object constitutes “this thing” that we discuss. In this study, “this
thing” very often is a triangle or a quadrilateral. Perhaps “this thing” also could be parts of a
triangle or a quadrilateral (e.g., sides, angles, etc). It is important to pay attention to the
mathematical objects involved in a given discourse.
In this study, at different van Hiele levels the same geometric figure discussed may not be
the same the geometric object in the corresponding geometric discourse. For example, at Level 1
the word square is used as a label to a picture of a square, just a matching of a word with a shape
(signifier). All squares can be grouped together because they all fit this family appearance of four
sides equal; however, at this level students will not group a rhombus and a square together
because they do not have the same family appearance even though they both have four sides
equal (saming criterion). All figures are grouped by their names only, because each name
represents a family appearance (realization), and there is no hierarchy connecting geometric
figures at Level 1 (system of objects). At Level 3, the word square can also represent a
parallelogram, a rectangle or a rhombus because a square fits the common descriptive narratives

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“opposite sides parallel”, “a parallelogram with four right angles”, and “parallelogram with four
sides equal” (saming criterion). Thus, a square can be grouped with parallelograms, rectangles
and rhombi (realization).
If a student moves from Level 1 to Level 3, the polygon called “square” plays different
roles as a geometric object in the two different geometric discourses. A square in Level 1 is a
picture of “a thing” called a square, whereas a square in Level 3 is an abstract object with
required properties that can have different names. It is important in this study to compare
geometric objects at each van Hiele level.
Routines are discursive patterns that repeat themselves in similar situations. In this study,
routines consist of identifying routines and defining routines, helping me pay attention to the role
of definitions played in van Hiele levels 1 to 3. The model helps me to be more explicit about
how students identify a polygon, and with what evidence, as a repetitive pattern. For example, at
Level 1 a student identified a square because it looked like one (visual recognition), whereas at
Level 2, this student identified a square because it has four equal sides. But the same could also
be rhombus because right angles were not mentioned (identifying partial properties). The
defining routines provide clues to the students’ use of definitions.
This model helped to identify participants’ geometric discourses with regard to their van
Hiele levels. The model was revised during the process of analyzing participants’ geometric
discourses.

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CHAPTER FIVE: DISCUSSION

Efforts in the mathematics education research community toward an understanding of
students’ learning have defined mathematical learning as actively building new knowledge from
experience and prior knowledge (NCTM, 2000), moving to a higher level of thinking (van Hiele,
1959), or as changes in discourse (Sfard, 2008). Other researchers have developed methods to
measure learning quantitatively (Floden, 2002). In this chapter I return to the questions that
motivated this study and guided my analyses of learning. The question that served as the impetus
for the study was: “What do prospective teachers learn in geometry from their preparation for the
work of teaching geometry?” It can be argued that this study does little to answer the question
because of the complexity of participants’ learning, and of the context in which these students
were observed. However, my effort is to conceptualize these participants’ mathematical thinking
through their mathematical discourses as an evidence of their learning, thereby adding some
information to the views of learning as moving to a higher level of thinking, and as changes in
discourse. I will begin with a brief summary of the participants’ changes in levels of thinking as
well as changes in their geometric discourses.
Summary of the Results
To investigate changes in students’ mathematical learning, my study focused on
comparisons of students’ competencies in the topics of triangles and quadrilaterals at the
beginning of a semester and at its end. I conducted the analysis using van Hiele levels (1959) to
investigate changes in students’ geometric thinking, and also used Sfard’s (2008) discursive
framework in which mathematical features of discourse (i.e., word use, visual mediators,
narratives and routines) are analyzed.

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Changes in van Hiele Levels
Changes in participants’ (n=63) competencies were revealed in their overall
performances on the van Hiele geometry pretest and posttest. There were improvements in
answering questions related to van Hiele Levels 1 to 3 at the posttest. In particular, most
participants did better in the following:
•

More than ninety-five percent of the participants correctly named triangles,
squares, rectangles, and parallelograms at the posttest.

•

More than ninety-five percent of the participants at the posttest correctly
identified the properties of isosceles triangles, squares, rectangles, and rhombi
related to their sides, angles and diagonals.

•

About ninety percent of the posttest participants correctly used logical statements
regarding triangles, squares, rectangles, and parallelograms.

These changes show that participants gained familiarity with figures like triangles,
squares, rectangles, rhombi and parallelograms, and with their properties. Participants mentioned
more about the properties of angles and sides in a parallelogram, but less on the properties of
diagonals. The comparisons of van Hiele pretest and posttest levels revealed students’
weaknesses in using deductive reasoning to construct proofs (Level 4) and abstract thinking
(Level 5).
Given these test results, one conclusion is that the geometry course helped students to
move from a lower van Hiele level to Level 3. However, the van Hiele test also showed that a
student entering the class at Level 3 likely would stay at Level 3. But that was expected, as the
course was designed for future elementary and middle school teachers, and the course materials
emphasized activities mostly at Levels 1 to Level 3 of geometric thinking, and included only a

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brief introduction to constructing proofs. The study did not look at how teaching or the use of the
textbook affected these students’ learning, but certainly the textbook and course instructions
contributed in some degree to these prospective teachers’ learning about geometric figures and
their properties.
The van Hiele Geometry pretest and posttest served as a frame to gather general
information about students’ competencies and their thinking as a whole, but it did not provide
details on changes in students’ thinking at an individual level. For this purpose, I also analyzed
changes in participants’ geometric discourses with in-depth interviews.
Changes in Geometric Discourses
I begin with brief remarks on the discursive framework, in order to set up my later
comments. Recall that the discursive framework conceptualizes mathematics as a discourse, and
defines “learning mathematics” as changing of participation in mathematical discourse. The four
key mathematical features highlighted in the discursive framework (along with their definitions)
are:
•

Word use: Mathematical words that signify mathematical objects or process

•

Visual mediators: Symbolic artifacts, created specially for the sake of this
particular communication

•

Narratives: Any text, spoken or written, which is framed as a description of
objects, of relations between processes with or by objects, and which is subject to
endorsement or rejection; that is, to being labeled as true or false.

•

Routines: Repetitive patterns characteristic of the given mathematical discourse
(Sfard, 2008)

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These features interact with one other in a variety of ways. For example, endorsed
narratives contain mathematical words, and provide the context in which mathematical words are
used; mathematical routines are apparent in the use of visual mediators and produce narratives;
visual mediators are used in the construction of endorsed narratives, etc. The most interesting
part of this investigation was the way in which the examination of these mathematical features
contributed to understanding the richness and detail of the participants’ geometric discourse and
their thinking.
In an earlier chapter, I described in detail the uniqueness of each participant’s geometric
discourse at the beginning of the semester, and again at the end of the semester. The main
changes in participants’ geometric discourses occurred in the following two features of
mathematics discourse: word use and routines.
Word use. This study focused on participants’ use of the names of quadrilaterals, and of
the parts of quadrilaterals. Participants’ use of mathematical terminology changed from
describing parallelograms as collections of unstructured quadrilaterals based on family
appearances, to using the names as collections of quadrilaterals sharing common descriptive
narratives. Some participants used of names of quadrilaterals with a hierarchy of classifications
in the Post-Interview.
Word use is a key element in identifying objectification in the discursive framework.
Sfard (2008) defines objectification as “a process in which a noun begins to be used as if it
signifies an extradiscursive, self-sustained entity (object), independent of human agency”
(p.412).
The use of the word parallelogram illustrates the importance of objectification. When a
student states, “this is a parallelogram” based on its family appearance, she uses the word

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parallelogram as a label to match polygons fitting this visual description, rather than a definition.
It is perhaps in course work, through the process of identifying, defining and generalizing
properties of quadrilaterals, that this student has developed a concept of parallelogram, so that
the word is used as a collection of quadrilaterals sharing a common descriptive narrative (i.e.,
definition), such as “a parallelogram is a quadrilateral with opposite sides of the same measure”
That said, the student uses the word parallelogram to include 4-sided polygons such as squares,
rectangles, and rhombi. The change in discourse counting as learning is a transition from nonobjectified speaking to objectified speaking.
Objectification is not straight-forward to detect; however there are clues in the ways
students speak that provide hints about how they are thinking. In the example in the previous
paragraph, “a parallelogram is a quadrilateral with opposite sides of the same measure”, the
word “parallelogram” is used with “is”, “quadrilateral” and “opposite sides of the same
measure”. One clue that “parallelogram” has been objectified is that “is” is used with the object.
That is, “parallelogram” is a noun. Also, in this discourse “quadrilateral” and “opposite sides of
the same measure” are used exclusively with geometric shapes. Thus, “a parallelogram is a
quadrilateral with sides of the same measure” is stated as a mathematical fact in geometry. At the
non-objectified stage of the use of “parallelogram”, we would be more likely to hear something
like “this parallelogram has two long sides equal and two short sides equal”. The use of “has”,
“long sides”, “short sides” and “equal” describes what a student sees about a parallelogram, but
does not have to describe a mathematical fact.
The objectification of parallelogram is perhaps even more complex when we consider
the hierarchy of classifications. I suggest that the objectification of parallelogram is complete

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when students can map out a hierarchy of classifications. Only then is the word parallelogram
recognized in all its connections and relations, and its diversity realized.
Routines. I analyzed changes in participants’ routine procedures, including routines of
classifying, identifying, defining, verifying and substantiating. Briefly stated, participants’
routine procedures changed from identifying polygons using visual recognition, to identifying
them using endorsed narratives. In verifying claims, participants’ routine procedures changed
from recalling, measuring and/or constructing routines, to formulating proofs using mathematical
propositions and axioms. Some participants’ routine procedures also included algebraic
reasoning to verify claims in geometry.
Some participants’ routines of verifying were descriptions of processes of mathematical
activity. For example, one participant verified that diagonals in a rectangle have equal measure
by explaining, “They are the same because I measured it”. The term “I measured it” reveals that
the participant’s routine of verifying relies on comparing and checking measurements, and is a
description of what she did. Another participant verified that two angles were congruent by
asserting, “the angle can slide over to this position and create this angle, and the line can be
rotated so that this angle now becomes this angle”. The use of “angle” with “slide over” and
“create”, and the use of “line” with “be rotated” and “becomes”, indicate that this participant’s
routine of verifying was a description of what the lines and angles did in an imaginary way, but
was not rigorously based on mathematical facts. Changes in discourse that count as learning are
transitions from an object level way of speaking to an abstract level ways of speaking.
In this study, the participants’ routine procedures were less polished than those of
professional mathematicians. However the analyses of their routine procedures shed light on

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participants’ reasoning and problem solving skills, and on their abilities in constructions of
mathematical proofs.
What Can We Generalize and Why
Although, much support exists for van Hiele levels, researchers have questioned and
modified certain aspects of these levels. To answer the question “What additional information
does the discursive framework provide with regard to the levels of geometric thinking?”, I need
to discuss how and in what way my study contributes to mathematics education based on what
we know so far about van Hiele theory. In this section I begin with a brief review of what we
know about so far the van Hiele theory, and follow with descriptions of geometric discourses at
each van Hiele level, and then discuss the usefulness of revisiting the van Hiele levels using a
discursive lens.
To elaborate more on “what do we know so far about the van Hiele theory?”, let me refer
to the following three questions raised by Clements (1992):
1. Are the levels discrete?
2. Do students reason at the same van Hiele levels across topics?
3. Should other characteristics of the levels be considered?
First, according to the van Hieles (1959), the levels are discrete in the sense that they are
qualitatively different from one another, and the “discontinuities are … jumps in the learning
curve, and these jumps reveal the presence of the levels” (p. 76). Research confirmed that the
five van Hiele levels are distinct qualitatively from each other. However, many studies have
questioned whether the van Hiele levels are discrete, because some students’ levels of geometry
thinking are in transition between two levels (e.g., Fuys et al., 1988; Burger & Shuaghnessy,
1986). For example, when assigning students to van Hiele levels in Burger and Shaughnessy’s

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project, reviewers could not resolve disagreements on whether students should be assigned to
van Hiele Level 0 or Level 1 (Burger & Shaughnessy, 1986, p.18). These results encouraged
researchers to provide evidence for a more dynamic and continuous model (e.g., Fuys et al.,
1988; Gutierrez et al., 1991; Usiskin, 1982).
Going in that direction, with the assumption that van Hiele levels are not discrete,
Gutierrez et al. (1991) developed a model, “degree of acquisition of a van Hiele level” (p. 238).
This model assigned a numerical value to indicate one’s acquisition of a van Hiele level. For
instance, when students have no trace of the thinking methods specific to a new level, they have
no acquisition to this level of reasoning. Once the students begin to be aware of the methods of
thinking at a given level, with some attempts to work on this level, they have a low degree of
acquisition of the level. Improvement goes on until students attain complete acquisition of the
new level, when they have complete mastery of this way of thinking and are able to use it
without difficulties. Figure 5.1 shows both the quantitative and qualitative interpretations of the
model (p.238).

Figure 5.1

Degree of acquisition of a van Hiele level

Using this model, Gutierrez et al. found that the possibility that a student can develop two
consecutive levels of reasoning at the same time, and the acquisition of the lower level is more
complete than the acquisition of the upper level. Their study inferred possible continuity within a
van Hiele level.

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More recently, when describing the levels of reasoning, Battista (2009) illustrated
students’ reasoning at each level with different phases. For example, he suggests that “Students
at the beginning of Level 1 [Visual-Holistic reasoning] might identify figures…as squares”
(p.93); “Students at the end of Level 1 might reject… as a square”(p.93); and finally, “Before
reaching the last phase of Level 2 [Descriptive-Analytic reasoning], most students would identify
… as a square”(p.93). Battista’s descriptions of students’ levels of reasoning showed the
continuity of the development within levels. More researchers are convinced intuitively that the
levels are “dynamic rather then static” (Burger & Shaughnessy, 1986, p.45) and refer to
“continuity rather than jumps” (Clements, 1992, p.429) in the process of learning. However, very
little study has been done in this area to verify these claims.
Several studies using van Hiele levels to categorize students’ levels of geometry thinking
across different topics indicate that students may not be working at the same level on all concepts
(e.g., Burger & Shaughnessy, 1986; Mayberry, 1983). For example, Mayberry (1983) assessed
nineteen undergraduate pre-service teachers’ levels of thinking using seven geometry concepts:
squares, right triangles, isosceles triangles, circles, parallel lines, similarity and congruence. The
study found that “the determination of the success criterion for a given topic and level was rather
subjective” (p.68). This conclusion can be understood to mean that pre-service teachers were at
different levels for different concepts. However the study did not provide information concerning
in what way they are different. For instance, one might suspect that a more difficult concept such
as similarity would require a higher van Hiele level of thinking than the classification of a
quadrilateral.
In Burger and Shaughnessy’s project (1986), interview tasks consisted of drawing,
sorting, identifying, and defining geometric shapes such as triangles and quadrilaterals. With

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regard to different tasks, some students operated at different levels. For example, one student
was reasoning at Level 3 (Abstraction) on the sorting task, but was assigned to Level 4
(Deduction) on the identifying and defining tasks because he was able to conjecture and attempt
to verify his conjecture by means of formal proof (p.42).
The ways of identifying students’ levels of geometric thinking suggest that we should
adapt van Hiele levels to the complexity of the human reasoning process because students do not
behave in a simple, linear manner.
The van Hieles argued that a learning process is also a process of learning a new
language, because “each level has its own linguistic symbols” (van Hiele, 1959/1985, p.4). The
van Hiele levels reveal the importance of language use, and language is a critical factor in the
movement through the levels. Van Hiele (1986) provides an explanation of the language use at
each level. For example, at Basic level there is a language, but the use of this language is limited
to the indication of configurations that have been made clear based on observation. At Level 1,
students need to develop the language that belongs to the descriptive level. At Level 2, the
language has a much more abstract character then the descriptive level, and reasoning about
logical relations between theorems begins at this level. At Level 3, students use the language of
proof (p.43-53). However, the word “ language” is not clearly defined in the broad use of it.
Some researchers would consider “language” in the comparisons of informal language versus
formal language, whereas others would refer to it as the different use of mathematical vocabulary
at different van Hiele levels.
In discussing the assessment of students’ reasoning in van Hiele levels, Battista (2007)
argues about the validity of the reasoning, which involves the accuracy and precision of students’
identifications, descriptions, conceptions, explanations and justifications. Researchers must

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determine which, if any, validity or mere uses of proprieties are critical characteristics of van
Hiele levels (p.855). The analysis of van Hiele levels depends largely on illustrating students’
verbal expressions. Many researchers found that activities such as sorting shapes and drawing
pictures of polygons can also provide evidences of students’ levels of thinking (e.g., Burger &
Shaughnessy; 1986; Mayberry, 1983). These activities work well especially with younger
children (Clements et al., 1999), while their language skills are still under development.
Therefore we need to take a greater consideration of what students “say” and “do” when working
on a geometry task in order to better understand their geometric thinking.
My study took a different direction, in examining students’ geometric thinking through
their geometric discourse. The results of these examinations revealed small fractions of the
richness of human thinking, while helping to add a little more data on what we know so far about
the van Hiele Theory.
The van Hiele Levels: Discrete or Continuous?
This ongoing discussion about the continuity and discreteness of van Hiele theory
motivates the study to revisit the van Hiele levels with a different lens. I will use two of the most
revealing characters of geometric discourse, geometric object and routine of substantiation, at
each van Hiele level to argue that my study confirms the discrete nature of the levels, and adds as
well more information on the continuity of the levels. I use the term “geometric object(s)” to
refer to all the mathematical objects involved in a particular geometric discourse, whereas the
term “geometric figure(s)” refers to all the 3 or 4-sided polygons. My discussions about students’
substantiation at each van Hiele level focus on two types of substantiations: the object level and
abstract level substantiations.

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The object level substantiation emphasizes students’ routines of substantiation, looking at
descriptions of how geometric figures are being investigated. Describing static lines, angles and
polygons as movable entities under transformations (i.e., rotation, translation and reflection), as a
way of substantiation, is an example of the object level of substantiation. With regard to
definitions of different quadrilaterals, however, routines of substantiation depending on
measurement routines to check the sides and angles of quadrilaterals, without thinking about how
quadrilaterals are connected, are other examples of the object level of substantiation. Object level
substantiation is a routine of substantiation, where students focus on the concreteness of
geometric figures.
Abstract level substantiation emphasizes students’ routines of substantiation using
endorsed narratives to endorse new narratives. That is, students use mathematical definitions and
axioms to construct mathematical proofs. During my interviews with students, I noticed that
students with an abstract level of substantiations also used object level substantiations to modify
their justifications. For example, a student used the Angle-Side-Angle congruence criterion to
construct a proof at an abstract level that opposite angles of a parallelogram are congruent, and
could also justify why this congruence criterion works using rotations at an object level.
Geometric Discourse at Level 1
At Level 1, students name geometric figures based on their appearance. In geometric
discourse at this level, the word use is passive. That is, the process of naming a polygon is an act
of matching a picture of a polygon with its given name. When a student is asked for verification
of why such polygons are called “rectangles”, or why “opposite sides and angles of a
parallelogram are equal”, the course of actions include direct recognition that are self-evident.
Some students use their prior experiences to draw conclusions, but the course of actions are

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known as rote memorizations, such as “I learned it in school”, or “I know it is a square”. At this
level, grouping quadrilaterals into different groups (i.e., rectangle, rhombus, parallelogram,
square, etc.) is about putting them together by their names. The geometric objects at this level of
discourse are collections of concrete, unstructured, discursive objects, and there is no routine of
substantiation.
Geometric Discourse at Level 2
At Level 2, students are able to identify some properties of geometric figures, but
properties are not yet ordered. In geometric discourse at this level, word use is routine driven,
which means that naming a polygon involves not just matching a polygon with a name, but
referring to it with a common descriptive narrative according to some visual properties. When a
student is asked for an explanation of why a polygon is called a “rectangle”, or why “opposite
sides and angles of a parallelogram are equal”, the courses of actions include direct recognition,
as well as counting the number of sides, or measuring the sides and angles. The student might
respond with, “It looks like it has four right angles”, or “I measured and all the angles are 90
degrees”. At this level, grouping quadrilaterals into different groups involves organizing them by
their names and by some of their visual properties. The geometric objects at this level of
discourse are collections of concrete, unstructured discursive objects which might be placed into
disjoint categories (i.e., they all have right angles, or parallel sides, etc.). The routines of
substantiations in this geometric discourse focus on checking and verifying partial visual
properties of geometric figures.
Geometric Discourse at Level 3
Level 3 is a level where the properties of geometric figures are ordered, and they are
deduced one from another. A student thinking at this level does not understand deduction, but the

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definitions of figures come into play. In geometric discourse at this level, word use is still object
driven, as the naming of a polygon depends on its visual properties, and a common descriptive
narrative accompanying the name of the figure (i.e., a definition of a quadrilateral). When a
student is asked why a polygon is called a “rectangle”, the course of action is to check the
defining conditions of the polygon by counting the number of sides, and measuring and
comparing the sides or angles. A possible response is, “It is a rectangle because it is a
parallelogram, and it has four right angles”. At this level, when the student groups quadrilaterals,
a polygon could belong to multiple groups at the same time by definition. For example, a student
could identify a square as a rectangle, a parallelogram, and a rhombus because it fits the
descriptions of these polygons. Geometric objects at Level 3 are collections of concrete
discursive objects and they begin to connect with joint categories. In the case of quadrilaterals,
all 4-sided polygons begin to fall into a hierarchy of classification.
It is important to note that objectification can be found in geometric discourse at Level 3.
That is, a concrete discursive object, such as a 4-sided polygon labeled as a “square” at a lower
geometric discourse (i.e., a lower van Hiele level), becomes an abstract discursive object at this
geometric discourse, as the word “square” presents this multi-dimensional thing with definitions
and relations to other quadrilaterals. Geometric discourse at Level 3 also reveals the details of
substantiation as a beginning stage of deductive reasoning.
Geometric Discourse at Level 4
At Level 4, students are able to reason deductively. In geometric discourse at this level,
word use is object driven. That is, the naming of a polygon or using a mathematical term (e.g.,
angle bisector, supplement angle, etc.) is guided by common descriptive narratives (i.e.,
definitions). Grouping quadrilaterals into different groups means arranging them by definitions

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with a hierarchy of classification. Routines of substantiations lead to the constructions of new
endorsed narratives. At this level, students are more fluent in using definitions in their
substantiation, and in making connections among endorsed narratives (axioms, propositions, etc.)
to construct new endorsed narratives. Geometric objects at Level 4 are collections of abstract
discursive objects, and the main activity of substantiation is to produce newly endorsed
narratives, or commonly, to construct mathematical proofs.
After the description of geometric discourse at each van Hiele level, the discreteness, in
terms of qualitatively different geometric thinking at each level, is evident. For years, researchers
have examined the possibility that levels are continuous without jumps. Let me move on to my
earlier claim that my study provides evidence of the continuity of the levels; in particular, the
continuity of levels made visible through the variability of participants’ changes in their
geometric discourse at the same van Hiele level, as well as at two consecutive levels.
Continuity becomes more evident once we realize that changes in students’ geometric
discourses are forms of change in thinking and communication, and that thinking is developed
continuously towards, rather than in jumps, to a higher van Hiele level. I wish to discuss in more
detail the development of geometric discourse, highlighting the development of word use and the
changes in routines of substantiation.
Continuity Within a van Hiele Level
AYA and ARI showed no change in van Hiele levels in their responses in the pretest and
posttest, but I found changes in their geometric discourses.
AYA’s van Hiele pretest and posttest responses suggested that her thinking operated at
Level 2 (descriptive). I analyzed AYA’s word use, and found that her use of the word
“parallelogram” changed from the pre-interview to the post-interview. When she spoke the word

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“parallelogram” at the beginning of the semester, she meant any polygon having pairs of parallel
sides, in using a definition of parallelogram with only a necessary condition. Later in the
semester, AYA developed more understanding of parallelograms in the geometry class, and she
was able to use definitions of parallelograms with both necessary and sufficient conditions. Her
thinking at the time fit more towards the descriptions of geometric discourse at Level 3. Figures
5.2 and 5.3 illustrate the characteristics of AYA’s geometric discourse at the pre-interview and at
the post-interview, respectively.

Figure 5.2

Characteristics of AYA’s geometrics discourses at Level 2 at the Preinterview.

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Figure 5.3

Characteristics of AYA’s geometrics discourses at Level 2 at the Postinterview.

Figures 5.2 and 5.3 show two main changes in AYA’s geometric discourse, a change in
word use and a change in reasoning. AYA had developed competence in using definitions to
identify and group polygons with no hierarchy of classification, and had developed some
informal deductive reasoning as her geometric thinking moved towards Level 3. Here, I am not
trying to contradict the findings from AYA’s paper-pencil pretest and posttest with her interview
results, but rather to compile the results and to treat her thinking more dynamically. Her progress
illustrates a student’s geometric thinking developing continuously within Level 2 and in
transition between Level 2 and Level 3, as she was more competent in using definitions to name
polygons, and her routines of substantiation began to operate at an abstract level in using
definitions and axioms to construct mathematical proofs.
ARI’s geometric discourse presents another example of such continuity, but within a
different van Hiele level. Her van Hiele pretest and posttest responses suggested that her thinking
operated at Level 3. ARI came in with the ability to identify and group polygons using
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definitions at the beginning of the semester. By the end of the semester, she began to reason
more abstractly by constructing mathematical proofs using definitions and axioms. Figures 5.4
and 5.5 illustrate characteristics of ARI’s geometric discourse at Level 3, in the pre-interview
and post-interview, respectively.

Figure 5.4

Characteristics of ARI’s geometric discourse at Level 3 at the Preinterviews.

Figure 5.5

Characteristics of ARI’s geometric discourse at Level 3 at the Preinterviews.
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Clearly, there are changes in ARI’s geometric discourse within Level 3. Building on her
familiarity with using definitions of quadrilaterals, ARI later showed understanding of how
quadrilaterals are connected. After ARI was more fluent in using definitions, she developed
routines of substantiations using informal deductive reasoning to substantiate her claims.
In comparing ARI’s geometric discourse at Level 3 with AYA’s at Level 2, we find
similarities between ARI’s geometric discourse at the pre-interview and AYA’s at the postinterview. Both shared familiarity with using definitions in identifying and grouping
quadrilaterals, and were able to reason at an object level. Perhaps this observation indicates the
continuity of learning in transitioning between two consecutive levels, Level 2 to Level 3. I will
next look at ALY’s geometric discourse to make another case that the learning process is
continuous.
Continuity Within Two Consecutive Levels
ALY was one of two students of the study who reached Level 4 at the end of the
semester. At Level 3, ALY demonstrated a typical behavior at this level, as illustrated in Figure
5.6.

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Figure 5.6

Characteristics of ALY’s geometric discourse at Level 3 at the PreInterview.

Similar to other students at Level 3, ALY came in with the ability to use definitions to
identify and group polygons, but she did not show how quadrilaterals were connected, and
performed object level substantiation. At the end of the semester, she was able to draw
connections among quadrilaterals and to use propositions and axioms to construct mathematical
proofs, as illustrated in Figure 5.7.

Figure 5.7

Characteristics of ALY’s geometric discourse at Level 3 at the Postinterview.

Figure 5.7 presents a main characteristic of a Level 4 discourse that is absent in Level 3:
abstract level of substantiations. At this level, ALY showed familiarities with using definitions
and axioms to construct proofs and with using algebraic symbols to write a formal mathematical
proof. However, at Level 4 we also expect to see behavior where students are able to apply
inductive reasoning in an unfamiliar situation, and to connect the knowledge they learned. In
ALY’s case, she was able to apply her knowledge of quadrilaterals to construct mathematical
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proofs in a familiar situation (e.g., to prove opposite angles or sides are congruent using
congruent criterions), having carried out similar proofs in her geometry class. When ALY was
asked to prove two definitions were equivalent, she did not finish the proof because the task was
new to her, and she did not know how to use the same axioms in a new situation. I argue that
ALY was at the beginning stage of Level 4 thinking, starting to gain the skills and languages
needed for mathematical proofs, but needing more practice to move forward to an advanced
abstract level.
AYA, ARI and ALY each made a case that the development of geometric discourse
within a level is continuous, and the development of geometric discourse from one van Hiele
level towards a geometric discourse at the next van Hiele level is also continuous. Using a
discursive lens in this study allowed me to unpack students’ thinking, and to better understand
what students said about geometric figures and what they did when they asked for justifications.
As a result, my study also contributes to answering another question raised by Clements, on
whether there are other characteristics that should be considered at each van Hiele level.
The van Hieles wished to note language differences and different linguistic symbols at
each level, in the study of language in geometric thinking, but were never explicit about it. The
language of mathematics I wish to discuss here does not refer to a list of vocabulary words or
grammar rules, but rather to the communicative competence necessary and sufficient for
competent participation in mathematical discourse.
The van Hiele descriptions of the levels focus largely on how a student reasons about
geometric figures in a language, for instance, in response to what is a rectangle versus what is
not a rectangle, in applying a definition. What is missed or not clearly emphasized is the meaning
of a mathematical term when used by a student. When I consider each van Hiele level as its own

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geometric discourse with characteristics of word use, narratives, routines and visual mediators, I
regard word use as all-important, revealing facts concerning how a concept is formed. In this
study, students’ word use provided significant information about how a concept of a geometric
figure is formed at different van Hiele levels among different students. Moreover, a careful
analysis of students’ mathematical word use in geometric discourse also shed light on how words
are used and whether the words are used correctly for the sake of communications. If any other
characteristics should be considered at van Hiele levels, I recommend adding word use to the list.
Discursive routines do not determine students’ actions, but only constrain what they can
reasonably say or do in a given situation, as negotiated conventions. However, discursive
routines offer valuable information about what students do and say as a course of action to make
conjectures and justifications in a pattern at a geometric discourse. I find it very useful to see the
details of students’ routines of identifying, defining and justifying when working on a task about
geometric figures and their properties, where the roles of definitions are demonstrated at the first
three van Hiele levels. Discursive routines are associated with students’ creativity when they
apply routines in non-routine ways; that is, in applying familiar routines in an unfamiliar
discursive context. In my study, some participants used algebraic reasoning (familiar routines) to
construct geometric proofs (unfamiliar discursive context), without using geometric axioms.
Therefore, if any other characteristics should be considered in van Hiele levels, I also
recommend adding routine, a repetitive discursive action to the list.
I have tried in several ways to explain what additional information this discursive
framework provides with regard to the levels of geometric thinking, as well as how this
additional information adds to what we know about van Hiele levels. There is no closure to what

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we know about the van Hiele levels. Looking at what we do know, we are led to ask what future
studies are needed in this important area of research.
What Can Be Asked and Why
Have you ever had an experience of sitting alone at your desk thinking, lost in thoughts
as if engaged in a conversation with someone? We must concede that thinking is an
individualized form of interpersonal communication, and whatever is created is a product of
collective doing. As a teacher, most of the time, I wish to know what is in my students’ minds
and their thoughts in mathematics. This empirical study in some way gives us an opportunity to
analyze students’ thinking at an individual level. One question natural to ask is, “If anything,
what do we learn from students’ thinking?”
What Do We Learn From Students’ Thinking?
First, I want to discuss the existence of Level 0. Some researchers suspected the existence
of a level prior to the Base Level (Level 1). My empirical study shows that students can reason at
a higher van Hiele level, but their lack of knowledge in geometry, or simply forgetting what they
learned in geometry, has kept them from giving correct answers. In ANI’s case, we learned that
the geometric pretest placed her at Level 0. However, my interview with ANI after the pretest
revealed that she was able to group parallelograms by their names, but did not know the
differences between a rhombus and a square, as well as the differences between a parallelogram
and a rectangle; this geometric discourse fit more to the descriptions at Level 1 than Level 0.
It was quite common during the interviews that a student could not identify a “trapezoid”,
or a “rhombus”, because they did not learn these names, or forgot the names. So, if we consider
the existence of Level 0 (a pre-level to Level 1), then it is likely that we include the possibility of
the kind of reasoning students perform in a domain of knowledge that they have not yet

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explored. This observation led us to consider: “What does van Hiele theory serve to assess?” and
“What do we wish to find out using van Hiele theory?”
Next I want to think about this scenario, when students were prompted to show that two
opposite angles were congruent in a parallelogram. First, visually they were convinced that the
two angles were the same, but further verification was required. One student responded that the
two angles were congruent because she used a protractor to measure the angles and they had the
same measurement. This course of action is typical in geometric discourse at Level 2, where a
student’s reasoning depends on checking and verifying the conditions for being congruent. From
this response we learn that this student has mastered knowing what are “opposite angles” in a
parallelogram, but needs to explore what we call “congruent”, a property of opposite angles, in a
concrete way. For the same task, another student described a sequence of transformations where
a rotation was followed by a translation, to show that the two angles were the same. She stated
that she could rotate one angle, and moved the angle to match the other one, and was sure that
the two angles would match exactly. This course of action is typical in geometric discourse at
Level 3, where a student is familiar with the term “congruent” and tries to explore whether
opposite angles are congruent concretely. From this response, we might infer that students need
to explore the properties of parallelograms through hands-on activities before they reach the
conclusion that “all opposite angles are congruent in any parallelograms”; then inductive
reasoning starts to make sense.
Recall that van Hiele levels are sequential, in that students pass through the levels in the
same order, although varying at different rates, and it is not possible to skip levels. In my study,
we noticed a sequence where a student needs to understand “opposite angles” in the case of a
parallelogram and the meaning of “congruent” first; and then move on to explore the properties

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and relations regarding opposite angles in a parallelogram; and then, perhaps, based on these
concrete experiences, she begins to develop some abstract thinking such as inductive reasoning.
Thus, the object level of substantiations clouds important points van Hiele made that students
must explore domains before describing them, that elaborate descriptions of concrete properties
and relations must be made before abstract relations are explored.
One challenge our students face is the development of abstract relations, because the
abstract relations in geometry may never be fully understood by some students. It takes time for
students to get used to new mathematical terms, as well as to digest the hands-on activities
relating to a particular property, before they can generalize it. When students are introduced to
more advanced thinking in deductive reasoning, some mimic the proofs without fully
understanding them. When we rush to the stage of constructing proofs that a student is not ready
for, it creates obstacles. It is important to give students enough opportunities to explore a
sequence of activities at a level built on other activities at a previous level before abstract
relations are explored.
Next, let us discuss the breath of Level 3. Recall that among the sixty-three students who
took the van Hiele posttest, thirty of them were considered at van Hiele Level 3; and among the
twenty students who participated in the interviews, ten of them were being placed at Level 3 in
the van Hiele posttest, and the interview analyses confirmed that these students were competent
to use definitions to justify their conclusions. It was quite surprising to have more students at
Level 3 than students at the lower levels. However, we also learned that students’ geometric
discourses seemed to develop at different rates, and geometric discourse at Level 3 varied from
person to person and varied in the same person at different times of the semester. Two main
variations of the discourse at Level 3 are 1) how profound a student uses definitions (geometric

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object) and 2) the way she reasons about the geometric figures (substantiation). Figure 5.8
illustrate these two variations in Level 3.

Figure 5.8

Characteristics of Level 3 geometric discourse.

Figure 5.8 highlights possible variations of geometric discourse at Level 3. Having a
geometric discourse at Level 3 indicates that a student has developed competence in applying
definitions in their identifying and justifying routines. In this study, such a student may or may
not make connections among the quadrilaterals, where a hierarchy of classification is presented
and depending on how profoundly the student understands the definitions and uses them
adequately.
ATL’s geometric discourse shows that a student at Level 3 could have competence in
using definitions to identify quadrilaterals, but still needs to develop other stills needed at this
level. In contrast to ATL’s geometric discourse at the same level, AYA and ALY represented a
group of students who could use definitions fluently, as well as reason at the object level. ANI
and ARI represented a group of students who were more advanced at Level 3, when they used
definitions to show a hierarchy of classification among quadrilaterals; and based on their
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experiences of reasoning at object level, they also tried to substantiate their conclusions using
some abstract relations.
In our analysis of the variety of geometric discourses at Level 3, we have learned that
Level 3 thinking is more complicated than previously thought. At this level, students need to be
familiar with and feel comfortable using the definitions fluently; and at the same time, they also
are developing informal reasoning by describing what they observe from exploration of concrete
properties of geometric figures, so that they can see or feel the particularity of figures before
abstract relations take place. All these mathematical activities become students’ prior
experiences in the development of abstract thinking.
There is a challenge for teachers when teaching a group of students whose geometric
discourses vary at different places. What kinds of activities will be suitable to all students so that
they can move toward a higher level of thinking? It is also a challenge for researchers to identify
kinds of activities that will help students in their development of geometric discourse at the same
level and at different levels.
Practical Information About Teaching and Assessment
The empirical data of the study offers a dialogue between a learner and a researcher in a
designed environment. The researcher carefully chooses the tasks, and prompts students’
thinking with well-designed protocols. However, this procedure is not too far from what a
teacher might do in preparing instructional materials. So one question we ask is, “What practical
information does this study offer about teaching and assessment?”
We need more instructional interventions in our classrooms. Students need to explore
“unfamiliar situations” using their existing knowledge. In an instructional sequence, a student
first is introduced to a new concept or new way of presenting her idea in mathematics, such as in

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constructing a proof by imitating what others say and do, and then she is asked to solve similar
problems or construct similar proofs in a similar context. It is a practice of repeating the same
process, the process of exploring the domain of polished tasks and well-designed mathematics
activities, and it is an important part of learning. However, we do want our students to move
beyond this stage and to be more creative. There are opportunities in the classrooms where
students can be creative. Their discursive routines show that sometimes they prefer to use
algebraic reasoning to derive a geometric proof. We need to encourage such thinking and create
more activities to help students make connections between geometry and other domains of
mathematics without losing the goals of introducing definitions and axioms in geometry.
Our concerns about “communication,” “language,” and “discourse” in the mathematics
classroom are not new. Fifty years ago, the same concerns motivated the van Hieles to develop
their theory. Surprisingly, we still don’t have much to recommend what needs to be done in a
geometry classroom for the sake of communication. My study can add some information about
the need to clarify the mathematical terms we use in the classroom, and to be specific about the
context in which they are used.
When a student mentions the word “parallelogram”, she says nothing unless she makes
explicit what she means. My empirical data shows that the word “parallelogram” could mean
quite different things to our prospective teachers. Some thought parallelograms are four sided
figures having two pairs of parallel sides with two sides longer; whereas some thought
parallelograms are tilted rectangles and squares. A few students thought that a parallelogram is a
polygon with pairs of parallel sides, and then of course, hexagons and octagons are
parallelograms. I also found that the use of the word “bisector” was confusing, because of the
several contexts where it was used. Some referred to it as “angle bisector”, whereas others

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thought it “a mid-point of a segment”. A few thought “all bisectors cross each other in the middle
of a parallelogram”. All these are correct ideas of “bisector” in a particular context. Just imagine
a group of students working together, using the same word in their own way, and definitely
creating some miscommunications. Teachers need to be very cautious about the way students use
mathematical terms, in order to make the classroom discussions more understandable. During the
interviews, I found it helpful to get to the bottom of what students meant with a mathematical
term by asking questions such as, “Can you say more about what you mean by…?”, “Will you
give me an example of what you just said about…?” and “Can you show me in this picture where
is (are)…?”.
Every classroom is different. I only can offer what worked in my study with prospective
teachers. But the principle is that classroom teachers need to ask questions and to ask different
questions, and to listen carefully to students’ responses, and give rapid feedback to ensure that
mathematics subjects are communicated well.
Limitation of the Study
This study contributes to the field of mathematics education in several ways. It pilots an
analytic method for investigating students’ geometric thinking using a discursive framework
looking in particular at changes in prospective teachers’ learning about triangles and
quadrilaterals in Euclidean geometry. The discursive framework presents another way to view
van Hiele levels as qualitatively different geometric discourses.
The study illustrates the usefulness of the discursive framework for highlighting the
opportunities for rich description at each van Hiele level through a discursive lens. A detailed
analysis can discover differences in participants’ geometric discourse at each van Hiele level,
which may impact the ways in which students do mathematics, speak about mathematics, and

257

therefore learn mathematics. This study also provides information for thought and discussion to
teacher educators interested in geometry more specifically. For example, should the development
of a concept in geometry begin with a word? Should the introduction of a quadrilateral begin
with the necessary condition of a polygon with four sides? Finally, this study acknowledges
students’ participation by analyzing their thinking processes as the union of the words and
actions observed during the interviews.
The primary limitations of this study have been stated throughout this dissertation, but
will be reiterated here briefly. First, analyzing students’ thinking processes (i.e., in video
recordings and interview transcripts) is challenging. In particular, when participants’ thinking
was not yet consistent and logically ordered, analyzing their geometric discourse was harder.
Hopefully, my descriptions and interpretations are clear. Secondly, the analytic decisions
regarding what to present and to compare for each participant were made to illustrate similarities
and differences. Thus, the primary perspective represented here is mine. I acknowledge that
another individual using the same data (i.e., reading the transcriptions) may see things quite
differently. It is possible that some important responses from participants during the interviews
may have been missed. Finally and most importantly, I interviewed twenty students, and
presented my analyses for five of them. Thus many claims are based on the cases of five
students, or of twenty students. In addition, this study focused only on the “two end points” of
students’ thinking during the time of their course work. I did not consider the individual
aptitudes of students’ learning, and the study may miss some advanced thinking gained after the
course is over. Therefore, there is no closure to my study, but it opens the door for me to pursue
further investigations.

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The study leaves me a long list of looming questions and ideas for future research. First,
there is a group of students whose van Hiele levels could not be determined by their test results,
but who participated in the interviews. I decided not to include their interview responses in my
analyses in order to narrow the scope of the study. To continue the study, one could analyze
these data to find out how the model of development of geometric discourse would help to
identify participants’ levels of thinking. And how would these data help to refine the model of
development of geometric discourse?
This study focused on students’ geometric discourse, and how this discourse helps us to
learn more about their thinking. However, one open question asks what this mathematics
discourse looks like when a student works on different mathematical tasks that include different
content domains of mathematics, and how the subsets of mathematics discourse interact with
each other. Perhaps it will help to gain more information regarding one of Clement’s questions,
“do students reason at the same van Hiele levels across topics?”
As mentioned previously, for those interested in geometry or teaching geometry, an
investigation using a discursive lens into students’ use of mathematical terminology in geometry
would be a next step. This analysis could also be extended to other mathematical topics. More
discussions regarding classroom interactions are needed. What can we do to help students use
mathematical terminology more precisely for the sake of communication and development of a
mathematical concept?
We need to develop frameworks for analyzing activities from both textbooks and
classrooms, and to identify mathematical activities that help students move to higher van Hiele
levels. The rise and popularity of computer software created a new learning environment for
students, and presented an important instructional and learning tool in school curriculum. Many

259

researchers and curriculum developers want to design pre-constructed activities using software,
and they hope that these activities will serve as mediators to help students learn geometry. In
response, there is a need to develop instruments to examine these activities, with the goal of
helping students develop more advanced levels of thinking.
Finally, we need to revisit van Hiele levels with multiple lenses, in order to have a better
picture of human thinking, and to improve communication through classroom interaction. How
can teachers better facilitate classroom discussions at various levels and in various contexts?
In summary, this study provides opportunities for conversations among mathematics
education researchers, curriculum developers, and teacher educators and teachers, on the learning
and teaching of geometry. Such conversations would address students’ levels of thinking in
Euclidean geometry through the lens of a discursive framework, in hopes of improving research
and teaching, and therefore better serving mathematics learners.

260

APPENDICES

261

APPENDIX A

RESEARCH STUDY CONSENT FORM

262

Appendix A: Research Study Consent Form
Dear Participant:
You are invited to participate in a research study (Research title: The van Hiele Theory Through
the Discursive Lens: Prospective Teachers’ Geometric Discourses). Researchers are required to
provide a consent form to inform you about the study, to convey that participation is voluntary,
to explain risks and benefits of participation, and to empower you to make an informed decision.
You should feel free to ask the researcher any questions you may have.
From this study, the researcher hopes to learn about prospective teachers’ knowledge of basic
geometric shapes; their abilities to make conjectures and their abilities to construct mathematical
proofs. Participating in the study will involve discussing your understanding about geometric
shapes and their properties. You will also be asked to explain your reasoning for statements you
make about geometric shapes. The entire study will take you about four hours: the first two-hour
will take place at the beginning of the semester. This part includes a pretest (35 minutes) and an
interview (85 minutes); the second 2-hour will take place at the end of the semester, and it
includes a posttest (35 minutes) and an interview (85 minutes). You will receive10 extra points
for the Math 202 class you are taking after you complete both pretest and posttest. If you are
selected for an interview, you will be asked to work on three geometric tasks, and will discuss
your results with the researcher or other students during the interview. You will receive $20 as
compensation for your time from the researcher one week after you complete the 85-minutes
interview.
In the study, your written work (e.g., pretest, posttest, exercise worksheet, interview tasks etc.)
will be collected, and your activities and conversations with the researcher and with other
students will be audio-recorded and/or video-recorded. The researcher may take notes during the
exercise and interview. The potential benefit for you to participate in this study is that you will
receive the opportunity to learn properties of basic geometric shapes and to learn to formulate
conjectures and geometric proofs, which is a part of the course content of Math 202 you are
taking. Additionally, research indicates that self-reflection on one's thinking aids in increasing
the level of sophistication of that thinking. Therefore, the researcher on this study will encourage
such self-reflection.
The results of this research study might be published in professional publications for teachers
and researchers. Only the research team will use the audio, videotapes or transcripts for analysis.
Your confidentiality will be protected to the maximum extent allowable by law. The data
collected will be coded such that your name and personal information will not show up in or be
linked with any reports and presentations of research project. The data will be stored and locked
in a steel cabin in the office of the researcher in Wells Hall at the Michigan State University.
Only the research term of the project or the Institutional Review Board (IRB) of the Michigan
State University have access to the research records about you and the data collected from you.
All research data will be retained for a minimum of 3 years following closure of project.
Participation in this research study is entirely voluntary. You may choose not to participate at all.
You may also refuse to participate in certain procedures or answer certain questions.
Furthermore, you may decide to discontinue your participation at any time without penalty.
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There are some minimal risks associated with your participation in this research study in that you
may feel tired and embarrassed when you do not know how to justify your thinking. Thinking
out loud may be an unfamiliar process to you and thus you might feel uncomfortable at first.
Please sign your initials if you consent to either of the following:
o
I agree to participate in the pretest and posttest but not the interviews.
Yes
No
Initials____________
o
I agree to participate in the pretest and posttest with the possibility of being chosen for
the interviews.
Yes
No
Initials__________
Please sign your initials if you consent to participate in the videotaping:
o
I agree to allow my image on the videotapes to be included in presentations.
Yes
No
Initials____________
o
I agree to allow audio recording/video recording of the interview.
Yes
No
Initials____________
If you have concerns or questions about this study, such as scientific issues, how to do any part
of it, or to report an injury, please contact Dr. Glenda Lappan at (517) 432-3635, or e-mail
glappan@math.msu.edu or regular mail at A718 Wells Hall, MSU, East Lansing, MI, 48824.
If you have any questions or concerns about your role and rights as a research participant, would
like to obtain information or offer input, or would like to register a complaint about this study,
you may contact, anonymously if you wish, the Michigan State University Human Research
Protection Program at 517-355-2180, Fax 517-432-4503, or e-mail irb@msu.edu or regular mail
at 207 Olds Hall, MSU, East Lansing, MI 48824.
Participant’s Printed Name _________________________________________
Participant’s Signature__________________

Date __________________

Sincerely,
Glenda Lappan, Ph.D.
University Distinguished Professor
Division of Science and Mathematics Education
Michigan State University
A718 Wells Hall
(517) 432-3635, glappan@math.msu.edu
Sasha Wang
Doctoral Student
Michigan State University
A719 Wells Hall
(517) 432-3635, wangsash@msu.edu

264

APPENDIX B

BEHAVIORS AT EACH VAN HIELE LEVEL

265

Appendix B: Behaviors at Each van Hiele Level

Level 1 (van Hieles’ Basic level)
(P.M., 1958-59)
1. “ Figures are judged according to their appearance.”
2. “A child recognizes a rectangle by its form, shape.
3. … and the rectangle seems different to him from a square.”
4. “When one has shown to a child of six, a six year old child, what a rhombus is, what a
rectangle is , what a square is, what a parallelogram is , he is able to produce those
figures without error on a geoboard of Gattegno, even in difficult situations.”
5. “a child does not recognize a parallelogram in a rhombus.”
6. “the rhombus is not a parallelogram. The rhombus appears … as something quite
different.”
(P.M., 1968)
7. “when one says that one calls a quadrilateral whose four sides are equal a rhombus, this
statement will not be enough to convince the beginning student [from which I deduce that
this is his level 0] that the parallelograms which he calls squares are part of the set of
rhombuses.”
(P.M., 1979)
8. (on a question involving recognition of a titled square as a square)
“basic level, because you can see it.”

Level 2 (van Hieles’ first level)
(P.M., 1957)
1. “He is able to associate the name ‘isosceles triangle’ with s specific triangle, knowing
that two of its sides are equal, and draw the subsequent that the two corresponding angles
are equal.”
(P.M., 1957; P.M. and Dina, 1958)
2. “ … a pupil who knows the properties of the rhombus and can name them, will also have
a basic understanding of the isosceles triangle = semirhombus.”
3. “The figures are the supports (lit. ‘supports’ in French) of their properties.”
4. “That a figure is a rectangle signifies that it has four right angles, it is a rectangle, even if
the figure is not traced very carefully.”
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5. “The figures are identified by their properties. (E.g.) If one is told that the figure traced
on the blackboard possesses four right angles, it is a rectangle, even if the figure is not
traced very carefully.”
6. “The properties are not yet organized in such a way that a square is identified as being a
rectangle.”
(P.M., 1959)
7. “The child learns to see the rhombus s an equilateral quadrangle with identical opposed
angles and inter-perpendicular diagonals that bisect both each other and the angles.”
8. (a middle ground between this and the next level) “once the child gets to the stage where
it knows the rhombus and recognizes the isosceles triangle for a semi-rhombus, it will
also be ale to determine of hand a certain number of properties of the equilateral
triangle.”
9. “Once it has been decided that a structure is an ‘isosceles triangle’ the child will also
know that a certain number of governing properties must be present, without having to
memorize them in this special case.”
(P.M., 1976)
10. “The inverse of a function still belongs to the first thought level.”
11. “Resemblance, rules of probability, powers, equations, functions, revelations, sets – with
these you can go from zero to the first thought level.”

Level 3 (van Hieles’ second level)
(Dina, 1957)
1. “Pupils … can understand what is meant by ‘proof’ in geometry. They have arrived at the
second level of thinking.”
(P.M., 1957)
2. “He can manipulate the interrelatedness of the characteristics of geometric patterns.”
3. “e.g., if on the strength of general congruence theorem, he is able to deduce the equality
of angles or linear segments of specific figures.”
(P.M., 1958-59)
4. “The properties are ordered [lit. ‘ordonnent’]. They are deduced from each other: one
property precedes or follows another property.”
5. “The intrinsic significance of deduction is not understood by the student.”
6. “The square is recognized as being a rectangle because at this level definitions of figures
come into play.”
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(P.M., 1959)
7. “the child… [will] recognize the rhombus by means of certain of its properties,…
because , e.g., it is a quadrangle whose diagonals bisect each other perpendicularly.
8. “It [the child] is not capable of studying geometry in the strictest sense of the word.”
9. “The child knows how to reason in accordance with a deductive logical system… this is
not however, identical with reasoning on the strength of formal logical.”
(P.M., 1976)
10. “the question about whether the inverse of a function is a function belongs to the second
thought level.”
11. “The understanding of implication, equivalence, negation of implication belongs to the
second thought level.”
(P.M., 1978)
12. “they are able to understand more advanced thought structure, such as: ‘the parallelism of
the lines implies (according to their signal character) the presence of a saw, and therefore
(according to their symbolic character) equality of the alternate-interior angles’.”
13. “ I [the student] can learn a definition by heart. No level. I can understand that definitions
may be necessary: second level.”
14. “… you know that is meant by it [the use of ‘some’ and ‘all’] second level.

Level 4 (van Hieles’ third level)
(P.M., 1957)
1. “He will reach the third level of thinking when he starts manipulating the intrinsic
characteristics of relations. For example: if he can distinguish between a proposition and
its reverse” [sic. Meaning our converse]
(Dina, 1957)
2. We can start studying a deductive system of propositions, i.e., the way in which the
interdependency of relations is affected. Definitions and propositions now come within
the pupil’s intellectual horizon.”
3. “Parallelism of the lines implies equality of the corresponding angles and vice versa.”
(P.M. and Dina, 1958)
4. “The pupil will be able, e.g., to distinguish between a proposition and its converse.”
5. “it (is) … possible to develop an axiomatic system of geometry.”

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(P.M., 1958-59)
6. “The mind is occupied with the significance of deduction, of the converse of a theorem,
of an axiom, of the conditions necessary and sufficient.”
(P.M., 1968)
7. “ … one could tell him (the student) that in proof it is really a question of knowing
whether these theses are true or not, or rather of the relationship between the truth of
these theses and of some others. Without their understanding such relationships we
cannot explain to the student that one has to have recourse to axioms.” [I include the level
from the first part of the statement; he never identified the level.]

Level 5 (van Hieles’ fourth level)
(Dina, 1957)
1. “A comparative study of the various deductive syatems within the field of geometric
relations is … reserved for those, who have reached the fourth level…”
(P.M. and Dina, 1958)
2. “finally at the fourth level (hardly attainable in secondary teaching) logical thinking itself
can become a subject matter.”
3. “the axiomatic themselves belong to the fourth level.”
(P.M., 1958-59)
4. “one doesn’t ask such question as: what are the points, lines, surfaces, etc.?... figures are
defined only by symbols connected by relationships. To find the specific meaning of the
symbols, one must turn to lower levels where the specific meaning of these symbols can
be seen.”

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APPENDIX C

VAN HIELE GEOMETRY TEST

270

Appendix C: van Hiele Geometry Test

Please print
Name

Section

.

Directions

Do not open this test booklet until you are told to do so.
This test contains 25 questions. It is not expected that you know everything on this test.
When you are told to begin:
1. Read each question carefully.
2. Decide upon the answer you think is correct. There is only one correct answer to each
question. Check the letter corresponding to your answer on the answer sheet.
3. Use the space provided on the answer sheet for figuring or drawing. Do not mark on
this test booklet.
4. If you want to change an answer, completely erase the first answer.
5. If you need a pencil and an eraser, raise your hand.
6. You will have 35 minutes for this test.
Wait until the instructor says that you may begin.

This test is based on the work of P.M. van Hiele.
Copyright ©1980 by the University of Chicago. Reprinted with permission of the University of
Chicago.

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VAN HIELE GEOMETRY TEST
1. Which of these are squares?

(F) K only
(G) L only
(H) M only
(I) L and M only
(J) All are squares
2. Which of these are triangles?

(A) None of these are triangles.
(B) V only
(C) W only
(D) W and X only
(E) V and W only
3. Which of these are rectangles?

(A) S only
(B) T only
(C) S and T only
(D) S and U only
(E) All are rectangles
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4. Which of these are squares?

(A) None of these are squares.
(B) G only
(C) F and G only
(D) G and I only
(E) All are squares.

5. Which of these are parallelograms?

(A) J only
(B) L only
(C) J and M only
(D) None of these are parallelograms.
(E) All are parallelograms.

6. PQRS is a square.
Which relationship is true in all squares?

273

(A) PR and RS have the same length.
(B) QS and PR are perpendicular.
(C) PS and QR are perpendicular.
(D) PS and QS have the same length.
(E) Angle Q is larger than angle R.

7. In a rectangle, GHJK, GJ and HK are the diagonals.

Which of (A) – (D) is not true in every rectangle?
(A) There are four right angles.
(B) There are four sides.
(C) The diagonals have the same length.
(D) The opposite sides have the same length.
(E) All of (A) – (D) are true in every rectangle.

8. A rhombus is a 4-sided figure with all sides of the same length.
Here are three examples.

Which of (A) – (D) is not true in every rhombus?
(A) The two diagonals have the same length.
(B) Each diagonal bisects two angles of the rhombus.
(C) The two diagonals are perpendicular.
(D) The opposite angles have the same measure.
(E) All of (A) – (D) are true in every rhombus.
274

9. An isosceles triangle is a triangle with two sides of equal length.
Here are three examples.

Which of (A) – (D) is true in every isosceles triangle?
(A) The three sides must have the same length.
(B) One side must have twice the length of another side.
(C) There must be at least two angles with the same measure.
(D) The three angles must have the same measure.
(E) None of (A) – (D) is true in every isosceles triangle.

10. Two circles with centers P and Q intersect at R and S to form a 4-sided figure PRQS.
Here are two examples.

Which of (A) – (D) is not always true?
(A) PRQS will have two pairs of sides of equal length.
(B) PRQS will have at least two angles of equal measure.
(C) The lines PQ and RS will be perpendicular.
(D) Angles P and Q will have the same measure.
(E) All of (A) – (D) are true.

11. Here are two statements.
Statement 1: Figure F is a rectangle.
Statement 2: Figure F is a triangle.
275

(A) If 1 is true, then 2 is true.
(B) If 1 is false, then 2 is true.
(C) 1 and 2 cannot both be true.
(D) 1 and 2 cannot both be false.
(E) None of (A) – (D) is correct.

12. Here are two statements.
Statement S: ∆ABC has three sides of the same length.
Statement T: In ∆ABC, ∠B and ∠C have the same measure.
Which is correct?
(A) Statements S and T cannot both be true.
(B) If S is true, then T is true.
(C) If T is true, then S is true.
(D) If S is false, then T is false.
(E) None of (A) – (D) is correct.

13. Which of these can be called rectangles?

(A) All can.
(B) Q only
(C) R only
(D) P and Q only
(E) Q and R only

276

14. Which is true?
(A) All properties of rectangles are properties of all squares.
(B) All properties of squares are properties of all rectangles.
(C) All properties of rectangles are properties of all parallelograms.
(D) All properties of squares are properties of all parallelograms.
(E) None of (A) – (D) is true.

15. What do all rectangles have that some parallelograms do not have?
(A) Opposite sides equal
(B) Diagonals equal
(C) Opposite sides parallel
(D) Opposite angles equal
(E) None of (A) – (D)

16. Here is a right triangle ABC. Equilateral triangles ACE, ABF, and BCD have been
constructed on the sides of ABC.
E

A

F

C

B

D

From this information, one can prove that AD, BE , and CF have a point in common.
What would this proof tell you?
(A) Only this triangle drawn can we be sure that AD, BE , and CF have a point in
common.
(B) In some but not all right triangles, AD, BE , and CF have a point in common.
(C) In any right triangle, AD, BE , and CF have a point in common.
(D) In any triangle, AD, BE , and CF have a point in common.
(E) In any equilateral triangle, AD, BE , and CF have a point in common.
277

17. Here are three properties of a figure.
Property D: It has diagonals of equal length.
Property S: It is a square.
Property R: It is a rectangle.
Which is true?
(A) D implies S which implies R.
(B) D implies R which implies S.
(C) S implies R which implies D.
(D) R implies D which implies S.
(E) R implies S which implies D.

18. Here are two statements.
I:
II:

If a figure is a rectangle, then its diagonals bisect each other.
If the diagonals of a figure bisect each other, the figure is a rectangle.

Which is correct?
(A) To prove I is true, it is enough to prove that II is true.
(B) To prove II is true, it is enough to prove that I is true.
(C) To prove II is true, it is enough to find one rectangle whose diagonal bisect each
other.
(D) To prove II is false, it is enough to find one non-rectangle whose diagonals bisect
each other.
(E) None of (A) – (D) is correct.

19. In geometry:
(A) Every term can be defined and every true statement can be proved true.
(B) Every term can be defined but it is necessary to assume that certain statements are
true.
(C) Some terms must be left undefined but every true statement can be proved true.
(D) Some terms must be left undefined and it is necessary to have some statements,
which are assumed true.
(E) None of (A) – (D) is correct.

20. Examine these three sentences.
(1) Two lines perpendicular to the same line are parallel.
(2) A line that is perpendicular to one of two parallel lines is perpendicular to the
other.
(3) If two lines are equidistant, then they are parallel.
278

In the figure below, it is given that lines m and p are perpendicular and lines n and
p are perpendicular. Which of the above sentences could be the reason that line m
and is parallel to line n?

(A)
(B)
(C)
(D)
(E)

(1) only
(2) only
(3) only
Either (1) or (2)
Either (2) or (3)

21. In F-geometry, one that is different from the one you are used to, there are exactly four
points and six lines. Every line contains exactly two points. If the points are P, Q, R and
S, the lines are {P, Q}, {P, R}, {P, S}, {Q, R}, {Q, S}, and {R, S}

Here are how the words “intersect” and “parallel” are used in F-geometry.
The lines {P, Q} and {P, R} intersect at P because {P, Q} and {P, R} have P in
common.
The lines {P, Q} and {R, S} are parallel because they have no points in common.
From this information, which is correct?
(A) {P, R} and {Q, S} intersect.
(B) {P, R} and {Q, S} are parallel.
(C) {Q, R} and {R, S} are parallel.
(D) {P, S} and {Q, R} intersect.
(E) None of (A) – (D) is correct.
279

22. To trisect an angle means to divide it into three parts of equal measure. In 1847, P.L.
Wantzal proved that, in general, it is impossible to trisect angles using only a compass
and an unmarked ruler. From his proof, what can you conclude?
(A) In general, it is impossible to bisect angles using only a compass and unmarked
ruler.
(B) In general, it is impossible to trisect angles using only a compass and marked
ruler.
(C) In general, it is impossible to trisect angles using any drawing instruments.
(D) It is still possible that in the future someone may find a general way to trisect
angles using only a compass and an unmarked ruler.
(E) No one will ever be able to find a general method for trisecting angles using only
a compass and an unmarked ruler.

23. There is a geometry invented by a mathematician J in which the following is true:
The sum of the measures of the angles of a triangle is less than 180º.
Which is correct?
(A) J made a mistake in measuring the angles of the triangle.
(B) J made a mistake in logical reasoning.
(C) J has a wrong idea of what is meant by “true.”
(D) J started with different assumptions than those in the usual geometry.
(E) None of (A) – (D) is correct.

24. Two geometry books define the word rectangle in different ways.
Which is true?
(A) One of the books has an error.
(B) One of the definitions wrong. There cannot be two different definitions for
rectangle.
(C) The rectangles in one of the books must have different properties from those in
the other book.
(D) The rectangles in one of the books must have the same properties at those in the
other book.
(E) The properties of rectangles in the two books might be different.

280

25. Suppose you have proved statements I and II.
I.
II.

If p, then q.
If s, then not q.

Which statement follows from statements I and II?
(A) If p, then s.
(B) If not p, then not q.
(C) If p or q, then s.
(D) If s, then not p.
(E) If not s, then p.

281

APPENDIX D

INTERVIEW TASKS

282

Appendix D: Interview Tasks

Task One

Figure Appendix D. 1. Task One: Sorting Geometric Figures

283

Task Two
A. Draw a parallelogram in the space below.

1. What can you say about the angles of this parallelogram?

2. What can you say about the sides of this parallelogram?

3. What can you say about the diagonals of this parallelogram?

B. In the space below, draw a new parallelogram that is different from the one you drew
previously.

1. What can you say about the angles of this parallelogram?

2. What can you say about the sides of this parallelogram?

3. What can you say about the diagonals of this parallelogram?

Task Three
Two geometry books define the word parallelogram in different ways.
1: A quadrilateral is a parallelogram if and only if two pairs of opposite sides of the
same length.
2: A quadrilateral is a parallelogram if and only if two pairs of opposite angles of the
same measure.
Show me that these two definitions are equivalent. To verify that two definitions are
equivalent, you must show that each set of defining conditions implies the other.

284

APPENDIX E

INTERVIEW PROTOCOLS

285

Appendix E: Interview Protocols

Before beginning the interview, provide the student with the following materials:
Pencils, ruler, protractor, blank sheets of paper
Turn on both video cameras.

Task One
Present Task One and turn the page to face the student.
1. Say: These are geometric shapes. Sort these shapes into groups. You can sort them any way you want.
Write down your answers at the bottom of the task, and make notes about why you group them in such a
way. Let me know when you are finished.

While the student is working on the task, check the positions of the cameras and see if they are recording
appropriately. Monitor the student while she/he is working on the task, and make notes to prepare
possible questions.

After the student has finished the task, turn on the audiotape.
2. Ask: Can you describe each group to me?
After the student has finished describing her/his results, ask one of the following:
If the student sorts the shapes as all rectangles together, all triangles together, all squares together,
etc, then
•

Ask: Can you find another way to sort these shapes into groups? Try it.

•

Ask: Why?

If the student sorts the shapes as all triangles together, all quadrilaterals together, etc., then
•

Ask: Can you sort these shapes into subgroups? Try it.

•

Ask: Why?
If the student says that he/she doesn’t know any other way to sort the shapes, then

•

Ask: Can “this” (e.g., a rectangle, or a parallelogram) and “this” (e.g, a rhombus, or a
trapezoid) go together?

•

Ask: Why, or why not?

3. Ask: What is a parallelogram?

286

After the student has answered the questions verbally, then give the student a piece of blank paper, and
Say: write it down. Do the same for the following questions.
4. Ask: What is a rectangle?
5. Ask: What is a square?
6. Ask: What is a rhombus?
7. Ask: What is a trapezoid?
8. Ask: What is an isosceles triangle?

Turn off the cameras and audio recorder. Remind the student to write the date and his/her name on all the
worksheets.
Say: I will collect all your worksheets.
Put all Task One materials away, give the student three minutes break and get ready for Task Two.

287

Task Two

Turn on both video cameras and audio recorder.
Present Task Two – “A. Draw a parallelogram …” and turn the page to face the student
Say: Draw a parallelogram in this empty space here.
Once the student has finished drawing, then
1. Ask: What can say about the angles of this parallelogram?
•

If the student says, “the opposite angles are equal”, or “all the vertex angles add up to 360°,
or “the adjacent angles add up to 180°”, then
o

Say: Write down your answer(s), and convince me.

After the student has finished explaining his/her conclusion, then
Ask: Is there any other relationship among the angles of this parallelogram?
•

If the student says, “all the vertex angles add up to 360°”, then
o

•

Say: Write down your answer(s), and convince me.

If the student says, “no, that’s all”, then

2. Ask: What can you say about the sides of this parallelogram?
•

If the student says, “ Opposite sides are equal”, or “opposite sides are parallel”, then
o

Say: Write down your answer(s) and convince me.

After the student has finished explaining his/her conclusion, then
Ask: Is there any other relationship involving the sides of this parallelogram?

Present Task Two – “B. Draw a new parallelogram …” and turn the page face to the student
Say: In the empty space here, draw a new parallelogram that is different from the one you drew
previously.
Once the student finished drawing, then
1. Ask: Why is this a different parallelogram from the first one you drew?
2. Ask: What can you say about the angles of this parallelogram?
•

If the student draws another parallelogram, then his/her answer to this question might be
identical to Task Two A. No need to repeat the process as in Task Two A.

•

If the student draws a rectangle, or a square, or a rhombus, and provides the same answer as
he/she did in Task Two A., then
o

Say: Convince me.

288

3. Ask: What can you say about the sides of this parallelogram?
•

If the student draws another parallelogram, then his/her answer to this question might be
identical to Task Two A. If so, then ask question 4, “what can you say about the diagonals of
this parallelogram?”

•

If the student draws a rectangle, or a square, or a rhombus, and provides the same answer as
he/she did in Task Two A., then
Say: Convince me.

4. What can you say about the diagonals of this parallelogram?
•

If the student draws a parallelogram, after she/he has finished describing the diagonals of the
parallelogram,
o

Ask: Why?

(Present a drawing of a rectangle), and then
o

Ask: What can you say about the diagonals of this one?

o

Ask: Why?

(Present a drawing of a square), and then
o

Ask: What can you say about the diagonals of this one?

o

Ask: Why?

(Present a drawing of a rhombus), and then
o
o
•

Ask: What can you say about the diagonals of this one?
Ask: Why?

If the student draws a rectangle as a new parallelogram, after she/he has finished describing
the diagonals of the rectangle,
o

Ask: Why?

(Present a drawing of a square), and then
o

Ask: What can you say about the diagonals of this one?

o

Ask: Why?

(Present a drawing of a rhombus), and then
o
o
•

Ask: What can you say about the diagonals of this one?
Ask: Why?

If the student draws a square as a new parallelogram, after he/she has finished describing the
diagonals of the square,
o

Ask: Why?

(Present a drawing of a rectangle), and then

289

o

Ask: What can you say about the diagonals of this one?

o

Ask: Why?

(Present a drawing of a rhombus), and then
o
o
•

Ask: What can you say about the diagonals of this one?
Ask: Why?

If the student draws a rhombus as a new parallelogram, after he/she has finished describing
the diagonals of the rhombus,
o

Ask: Why?

(Present a drawing of a square), and then
o

Ask: What can you say about the diagonals of this one?

o

Ask: Why?

(Present a drawing of a rectangle), and then
o

Ask: What can you say about the diagonals of this one?

o

Ask: Why?

5. Is it true that in every parallelogram the diagonals have the same midpoint (bisect each other)?
•

Ask: Why? Or Why not?

After the student has finished describing his/her conclusion, then
•

Say: write it down

Turn off the cameras and audio recorder. Remind the pair to write the date and their names on all the
worksheets.
Say: I will collect all your worksheets.
Put all Task Two materials away, give the pairs three minutes break and get ready for Task Three.

290

Task Three
Turn on both video cameras and audio recorder.

Present Task Three and turn the page to face the student
Say: Read the task carefully, and show your work. Let me know if you have any questions.

If the student shows difficulty understanding the task, and doesn’t know what to do, then
Say: To show that the two definitions are equivalent, you need to show:
1. If in a quadrilateral, two pairs of opposite sides of the same length, then two pairs of opposite
angles of the same measure.
And,
2. If in a quadrilateral, two pairs of opposite angles of the same measure, then two pairs of
opposite sides of the same length.

When the interview is finished, turn off both cameras and audio recorder.
Say: “Thank you” to the student, and let him/her know that you will share the results with them if he/she
is interested.

291

APPENDIX F

THE DEVELOPMENT OF GEOMETRIC DISCOURSES AT EACH VAN HIELE LEVEL

292

Appendix F: Development of Geometric Discourse at Each van Hiele Level
Table 4.41
Level
of discourse

1

2

Development of Geometric Discourse at Each van Hiele Level

Geometric Objects (figures)
Signifier Saming
RealizSystem of
- Word Use criterion
ations
objects
(figures)

proper
name
(passive
use)

according
primary dto family
objects,
appearance
s

unstructured
collection of
concrete dobjects

common
name
(routine
driven
and/or
phrase
driven)

according
to visual
properties
(with no
order)

unstructured
collection of
disjoint
categories of
concrete
d-objects

primary dobjects,

Routines
Examples of
Justifying
Identification
(“Why is this
X?”)

Identifying Routines
Attributed
Examples of
Declared
narratives
(“How do you
know it’s X?”)
visual
“It looks like…” “Because it is”
recognition,
self-evident

Defining Routines
How
When
(“What is
X?”)

--

--

Step 1. visual
recognition

describe
the figure
by visual
properties,
or by
recalling
(no
superordinate);

serves as
necessary
condition for
the use of
word

Step 2.
Substantiation.
Identifying by
partial
properties
check (e.g.
counting,
measuring,
comparing,
etc);

293

“It looks like
they are
parallel…”
“I measured
(sides &
angles)…”

“Because I can
see it”
“Because I
measured...
[some visual
properties, no
superordinate]

Table 4.41 (Cont’d)
3
common
name
(object
driven)

1. common
descriptive
narrative on
the name of
the figures

concrete
d-object
(Objectific
ation
occurred)

May or may
Step 1. visual
not have
recognition
hierarchy of
classifications Step 2.
Substantiation
Identifying by
2*.
definitions
common
*two levels
(check of
descriptive of
defining
narrative on realization
conditions by
the
tree:
counting,
properties
definitions
measuring,
of figures
and
comparing,
(e.g., equal, properties
etc.)
bisector,
of
etc)
geometric
Step 3.
figures,
Construction
and the
of new
relations
narratives
about how
(informal
one
proving
implies the
equality,
other.
congruency,
etc.)

294

“All squares
“If it is a square,
are rectangles” then it has to be
a rectangle, it
fulfills the
definition.”
“All these
figures (e.g.,
“Because they
squares,
all have two
rectangle,
rhombus) are pairs of parallel
parallelograms sides.”
. [critical
conditions are
fulfilled]

describe a
figure or a
mathematic
al term by
definition

serves as a
necessary &
sufficient
condition for
the use of
word

Table 4.41 (Cont’d)
4
1. common
name
(object
driven)

2. common
relations
among
definitions,
axioms,
theorems,
etc.

Common
abstract ddescriptive object
hierarchy of
narrative on
classifications
the name of
figures
Common
descriptive
narrative on
the
properties
of figures
(e.g., equal,
bisector,
etc)

* two-level
realization
tree:
relations
among
definitions,
properties,
axioms,
theorems
about
geometric
figures.

Construction
of new
narratives
(formal
proving using
definitions,
axioms,
theorems, etc)

295

“they are
alternating
interior angles,
and they are
equal.”

“If two parallel
lines are cut by
a transversal,
their alternating
interior angles
are equal.”

describe a
figure or a
mathematic
al term by
definition.

serves as a
necessary &
sufficient
condition for
the use of
word
serves as a
necessity for
the use of
available
axioms,
theorem,
etc.

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296

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