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Going Public:
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Pedagogical Content Knowledge

presented by

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GOING PUBLIC:
THE DEVELOPMENT OF A TEACHER EDUCATOR’S
PEDAGOGICAL CONTENT KNOWLEDGE
By

Jeffrey Joe Wanko

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

2000

ABSTRACT
GOING PUBLIC:
THE DEVELOPMENT OF A TEACHER EDUCATOR’S
PEDAGOGICAL CONTENT KNOWLEDGE
By

Jeffrey Joe Wanko

When Lee Shulman and his colleagues introduced pedagogical content knowledge
(PCK) to the education lexicon in the 19805, they gave teachers and teacher educators
some technical language that could be used for talking about the knowledge needed for
work that they do in classrooms, thereby helping to establish teaching as a profession.
Since that time, the PCK of classroom teachers has been studied and documented across
various content areas. But the PCK of teacher educators has remained a largely
unexamined area of research, especially in the providing experiences in helping
preservice teachers develop their own PCK. This study examines this issue more fully.
Specifically, “Can pedagogical content knowledge be a useful framework for a teacher
educator in designing and teaching a mathematics content course for preservice teachers
and if so in what ways?”

In this study, I use my own teaching and classroom of prospective elementary
teachers as the site for investigation. I examine the ways in which my own PCK as a
teacher educator influenced and was influenced by my work with students. Data for the
study are provided by my teaching journal, lesson and units plans, student work, and

audiotapes of class proceedings.

In conclusion, I present three major findings of this study. First, this study
highlights and problematizes Shulman’s notion of representation that is used in defining
pedagogical content knowledge. In mathematics there are mathematical and empirical
representations—classifications which do not map easily onto Shulman’s use of
representation. This study exposes some of those inherent distinctions and seeks to make
Shulman’s work more applicable to the field of mathematics. Second, this study describes
the importance of task design—a process that is particularly essential in teaching
mathematics—and finds that Shulman’s notions of PCK and the pedagogical reasoning
and action cycle miss or obscure its significance. And third, this study introduces the
notion of shared reflection to Shulman’s model for pedagogical reasoning and action
when it is applied to teacher education. It also finds that the act of going public with
one’s ideas through shared reflection can be a useful tool for teacher educators in the

development of their pedagogical content knowledge.

ACKNOWLEDGEMENTS

Conceptualizing, conducting, analyzing, and writing this research and dissertation
have been part of a long and difficult process. Without the help and support of many
colleagues, family members, and friends, it would have been next to impossible.

I wish to express my sincerest gratitude to Dr. Sandra K. Wilcox. You have
served as a mentor and a guiding force through all of my work as a doctoral student. Your
constant assistance and advice have been invaluable. Thank you also to Dr. Glenda
Lappan, Dr. Daniel Chazan, and Dr. Helen Featherstone for your help in reviewing my
work and providing feedback.

To the staff of the Connected Mathematics Project, I thank you for all of your
support over the past six years.

To my parents and the rest of my family, I thank you for encouraging me through
my professional life and for providing the support structure that I have always relied on.

And to my wife, Molly, I thank you for coming into my life and for your undying

patience and love.

TABLE OF CONTENTS

LIST OF TABLES ....................................................................................................... viii

LIST OF FIGURES ........................................................................................................ ix
CHAPTER 1

INTRODUCTION TO THE STUDY .............................................................................. 1

Introduction ............................................................................................................... l

Pedagogical Content Knowledge: A Brief Historical Perspective ......................... 3

What Do Preservice Teachers Need to Know for Teaching? ................................ 5
CHAPTER 2

METHODOLOGY .......................................................................................................... 9

Introduction: Designing the Course and Study ........................................................... 9

Forming An Initial Question .................................................................................... 10

Background: Why Did This Interest Me? ................................................................. 11

Learning About and Teaching Preservice Teachers ............................................ 11

Teaching in the Teacher Education Department ........................................... 12

Teaching in the Mathematics Department with the Connected Mathematics

Project ...................................................................................................... 14

Learning About and Teaching Middle School Students ...................................... 16

Learning About Sources for Pedagogical Content Knowledge ........................... 20

The Course .............................................................................................................. 24

Mathematics Content ......................................................................................... 25

Model for Learning and Classroom Environment ............................................... 26

Mathematical Investigations ......................................................................... 26

Writing and Talking About Mathematics ..................................................... 27

Using Manipulatives for Learning ................................................................ 27

Instructional Model ...................................................................................... 28

Pedagogical Content and PCK ........................................................................... 29

The Participants ................................................................................................. 31

Assessment and Grading .................................................................................... 31

Influences from Research Literature .................................................................. 32

The Study ................................................................................................................ 34

Studying One’s Own Practice ............................................................................ 35

Informing Students ............................................................................................ 39

Documentation .................................................................................................. 40

Lesson and Unit Plans .................................................................................. 41

Teaching Journal .......................................................................................... 42

Textbooks .................................................................................................... 42

Students’ Written Work ............................................................................... 46

Audiotapes of Class Proceedings ................................................................. 48

Other Data ................................................................................................... 49

Data Analysis .......................................................................................................... 50

The Emergence of Two Themes ........................................................................ 50

Formulas and Other Representations ............................................................ 50

Nature of Tasks and Issues of Task Design .................................................. 54

Revisiting Shulman’s Pedagogical Reasoning .................................................... 56

Refocusing My Study ........................................................................................ 59

The Story ................................................................................................................ 60
CHAPTER 3 .

REPRESENTATIONS .................................................................................................. 61

Introduction ............................................................................................................. 61

Episode 1: Trapezoid Area ...................................................................................... 63

Background: Working with Triangles ................................................................ 63

Using Two Copies of a Figure ...................................................................... 63

Dissecting and Rearranging .......................................................................... 65

Area of a Trapezoid—Connecting Symbolic and Spatial Representations .......... 68

Area of a Trapezoid—Introduction of Mathematical Language Representation. 72
Reflecting on the Use of Mathematical Language, Spatial Manipulation, and

Symbolic Representations ............................................................................... 74
Reflecting on Mathematical Assumptions .......................................................... 75
Dissecting and Rearranging .......................................................................... 76
Method B ..................................................................................................... 78
Method A ..................................................................................................... 80
Method B ..................................................................................................... 81
Method C ..................................................................................................... 81
Method D ..................................................................................................... 82
Comparing Representations through Shared Reflection ...................................... 82
The Perceptual Proof Scheme of Dissecting and Rearranging ............................ 85
Episode 2: Circle Area ............................................................................................. 88
Representations and Their Limitations ............................................................... 88
Method A ..................................................................................................... 89
Method B ..................................................................................................... 92
Reflecting and Shared Reflection on Multiple Representations .......................... 96
Perceptual and Ritual Proof Schemes ................................................................. 98
Episode 3: Volume of Prisms, Cylinders, Cones, and Spheres ............................... 100
Background—Understanding Layers ............................................................... 100
Areas of Regular Polygons—My Comprehension of Subject Matter ................ 103
Shared Reflection—Modeling Good Teaching Practice ................................... 109
Rice and Clay—Selection of Appropriate Tools and Representations ............... 110
Using Shared Reflection to Move Learners to Thinking as Future Teachers ..... 116
Using a New “Blended” Representation ........................................................... 1 18
Making Sense of Formulas: An Informal Proof Scheme ................................... 120
Conclusion ............................................................................................................ 122

vi

CHAPTER 4

TASK DEVELOPMENT ............................................................................................ 123
Introduction ........................................................................................................... 123
Posing a Task ........................................................................................................ 124
Using Student Work to Analyze the Design of the Task ......................................... 127

Analyzing Themes in Student Work ................................................................ 132
Trivial Observations ................................................................................... 132
Applications of Mathematical Ideas Previously Covered in MTH 202 ........ 134
Applications of Other Basic or Advanced Mathematics .............................. 134
Unproven Hypotheses ................................................................................ 136
Incorrect Statements ................................................................................... 138

Helping Students Learn from Others—Supplementing a Task .......................... 14]

Comparing to the Original Task ....................................................................... 147

Conclusion ............................................................................................................ 1 50

CHAPTER 5

IMPLICATIONS AND LIMITATIONS ..................................................................... 152
Implications ........................................................................................................... l 52

Complex Ways of Thinking About Representation .......................................... 152

Growing Appreciation of Task Design ............................................................. 155

Shared Reflection: Going Public ...................................................................... 158

Discussion ............................................................................................................. 160
Pedagogically Useful Mathematical Understanding ......................................... 161
Using PCK As the Lens ................................................................................... 163
Limitations ............................................................................................................ 165

APPENDICES
Appendix A: Class Meetings with Mathematical and Pedagogical Content ............ 169
Appendix B: MTH 202 Grading Policy ................................................................. 174
Appendix C: Initial E-Mail to Students .................................................................. 176
Appendix D: Documentation of MTH 202 ............................................................. 178
Appendix E: Informed Consent for Study of MTH 202, Section 1 ......................... 180
Appendix F: Learning Log Prompts ....................................................................... 182

BIBLIOGRAPHY ....................................................................................................... 1 85

vii

Table 1:

Table 2:

Table 3:

Table 4:

Table 5:

Table 6:

LIST OF TABLES

A Model of Pedagogical Reasoning and Action ............................................. 4
Senior and Intern Methods Courses with Classroom Placements .................. 12
Important Geometry and Measurement Topics and Alignment in CMP ........ 26

Five Methods for Understanding Area of a Trapezoid with

Three Types of Representations ................................................................... 83
Methods for Understanding Area of a Circle ................................................ 89
Student Responses to Prism Volume Problem ............................................ 104

viii

Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure 7:
Figure 8:

Figure 9:

Figure 10:
Figure 11:
Figure 12:
Figure 13':
Figure 14:
Figure 15:
Figure 16:
Figure 17:
Figure 18:
Figure 19:
Figure 20:

Figure 21:

LIST OF FIGURES

Shulman’s Cycle of Pedagogical Reasoning and Action ................................. 5
Sample Problem from EDT 335 ................................................................... 17
Sources for a Teacher’s PCK Development .................................................. 23
Sources for a Teacher’s PCK with Focus on Content ................................... 24
A Teacher Educator’s Pedagogical Reasoning and Action ............................ 58
Proof Schemes ............................................................................................. 62
Triangles A and D on Grid Paper ................................................................. 63
Two Copies of Triangle A ............................................................................ 64
Two Copies of Triangle D ............................................................................ 64
Triangle A as Rectangle with Half of the Height .......................................... 66
Triangle A as Rectangle with Half of the Base ............................................. 66
Triangle D with Half of the Height in One Cut ............................................. 67
Triangle D with Half of the Height in Two Cuts ........................................... 67
Trapezoid Method A .................................................................................... 68
Trapezoid Quiz Task .................................................................................... 70
Trapezoid Method B .................................................................................... 71
Trapezoid Method C .................................................................................... 71
Trapezoid Method D .................................................................................... 72
Trapezoid Method B .................................................................................... 75
Dissection Puzzle ......................................................................................... 77
Proof of Trapezoid Method E ....................................................................... 79

Figure 22:
Figure 23:
Figure 24:
Figure 25:
Figure 26:
Figure 27:
Figure 28:
Figure 29:
Figure 30:
Figure 31:
Figure 32:
Figure 33:
Figure 34:
Figure 35:
Figure 36:
Figure 37:
Figure 38:
Figure 39:
Figure 40:
Figure 41:
Figure 42:
Figure 43:

Figure 44:

Proof of Trapezoid Method A ...................................................................... 80
Circle Divided into Sectors .......................................................................... 90
Circle Sectors Rearranged as an “Almost Rectangle” ................................... 90
Circle with a Radius Square ......................................................................... 93
Circle with Four Radius Squares .................................................................. 94
Circle with Three Radius Squares ................................................................ 94
Box with Layers of Cubes .......................................................................... 101
Construction of Triangular and Pentagonal Prisms ..................................... 103
Base of Triangular Prism ........................................................................... 106
Base of Hexagonal Prism ........................................................................... 107
Base of Pentagonal Prism ........................................................................... 107
Cone and Corresponding Cylinder ............................................................. 111
Cylinder Containing Volume of One Corresponding Cone ......................... 112
Clay Sphere in a Cylinder .......................................................................... 113
F lattened Sphere in a Cylinder ................................................................... 114
Combined Flattened Sphere and Rice in Cylinder ...................................... 116
Archimedes’ Ratios of Volumes ................................................................. 118
Trapezoid Task Version A ......................................................................... 125
Trapezoid Venn Diagram ........................................................................... 126
Dana’s Work on Trapezoid Task ................................................................ 128
Rachel’s Work on Trapezoid Task ............................................................. 129
Margie’s Work on Trapezoid Task ............................................................. 130
Patti’s Work on Trapezoid Task ................................................................. 131

Figure 45:
Figure 46:
Figure 47:
Figure 48:
Figure 49:
Figure 50:

Figure 51:

Margie’s Special Isosceles Trapezoid 1 ...................................................... 137

Margie’s Special Isosceles Trapezoid 2 ...................................................... 137
Margie’s Special Isosceles Trapezoid 3 ...................................................... 139
Special Isosceles Trapezoid Labsheet, Page 1 ............................................ 143
Special Isosceles Trapezoid Labsheet, Page 2 ............................................ 144
Trapezoid Task Version B .......................................................................... 148
Learning Log Trapezmd ................................. 176

xi

CHAPTER 1

INTRODUCTION TO THE STUDY

Introduction

The preparation of elementary school teachers of mathematics has traditionally
been viewed as having two distinctly separate components-pedagogy and content.
Preservice teachers are a part of teacher education programs in which students might take
courses on child development, educational psychology, teaching methods, and other
matters that are pedagogical in nature. The focus in these courses is learning how to
teach—sometimes with respect to specific subject matter, but often from a generalist
standpoint. On the other hand, prospective teachers are expected to learn and understand
mathematics by taking courses often taught by instructors in a mathematics department.
These courses vary in both content and rigor, but invariably the focus remains on learning
mathematics.

A problem arises, however, when preservice teachers become practicing teachers
and they attempt to cross this divide——when they search for ways to make their
experiences in understanding mathematics useful tools for teaching children and, in turn,
helping them to understand some more mathematics themselves (Brown & Borko, 1992;
Copes, 1996). It is one thing for a teacher to learn some mathematics him- or herself, but
it is quite another for a teacher to take that knowledge and apply it to ways for helping
students learn and understand some mathematics. Replicating the teachers’ experiences is

obviously not the answer for a number of reasons: the content being addressed can be

quite different, children may approach the mathematics in very different ways, the
children’s experiences are likely to be quite different from the teacher’s and fi'om those of
other students, and the teacher has to think about the mathematics from a teaching
perspective rather than from a learning perspective. With such a large curriculum of
study—in both content and pedagogy—it is not surprising to hear stories of novice
teachers having limited knowledge in mathematics. For example, while preservice
teachers may get to learn some mathematics in their content courses, that mathematics
typically lacks connectedness, a look at multiple perspectives, work beyond studying
basic ideas, and attention to what content is taught across grade levels (Ma, 1999).
Suffice it to say, the separation of content and pedagogy in a teacher preparation program
has often not served preservice teachers well when they begin teaching (Becher, 1989).
An important skill for teachers is the ability to transform one’s own knowledge
into something that can be understood by one’s students. Successful teaching is much
more than demonstrating one’s own knowledge; it involves leading others to know—and
believe—things that they didn’t before (Hansen, 1995). This happens when a teacher
recognizes how students can best experience the acquisition of that knowledge——using
various strategies, representations, and problems to move students toward a desired goal.
An understanding of the mathematics is central to the transformation process, but it
involves much more than processing content—it recognizes that children learn in
different ways and that teachers need a multitude of strategies for“ helping their students
to learn mathematics. Thus it is unclear where preservice teachers can and should begin

to attend to transforming their mathematical knowledge—as a part of a course on

pedagogy or in content.

The transformative process became recognized as an essential element of a
teacher’s knowledge through the work of Lee Shulman and his colleagues in the mid-

1980s. Here I describe the essence of their work as well as how it is applied to my study.

Pedagogical Content Knowledge: A Brief Historical Perspective

Within cognitive psychological research, schema theory developed as a model for
ways in which people organize abstract thoughts into various collections and
representations of those ideas (Anderson, 1984). When Shulman and others applied
schema theory to teaching (Shulman & Grossman, 1988; Wilson, Shulman, & Richert,
1987), they theorized that teachers draw upon seven different domains of knowledge as
they teach: knowledge of subject matter, pedagogical content knowledge, knowledge of
other content, knowledge of the curriculum, knowledge of learners, knowledge of
educational aims, and general pedagogical knowledge.

When pedagogical content knowledge (PCK) was first introduced to the
educational community (Shulman, 1986), it was described as a domain of knowledge that
was different from both knowledge of the content and general knowledge of teaching.
PCK was portrayed as a specialized knowledge for teachers, one which recognizes the
subject matter needed for teachers as it differs from that as understood by other content
specialists. Shulman’s work was actually a continuation of some of the work done by
progressive educators (Dewey, 1902) and theorists (Bruner, 1960), but he was the first to

label and define PCK as such.1

 

’ Some have argued that by giving a name to pedagogical content knowledge, Shulman was also working
to establish his domain of educational research as a legitimate area of study when compared with other
long-standing areas of research.

In analyzing successful teaching, Shulman (1987) described the process as one in
which the teacher transforms content knowledge into forms that are pedagogically
powerful and adaptive to particular groups of students—ones which draw upon a
teacher’s pedagogical content knowledge. To this end, he employed the framework of
pedagogical reasoning and action, detailing a cycle of comprehension, transformation,

instruction, assessmentz, reflection, and new comprehension (Table l and Figure 1).

Table 1: A Model of Pedagogical Reasoning and Action

 

 

Comprehension
Of purposes, subject matter structures, ideas within and outside the discipline

Transformation

Preparation: critical interpretation and analysis of texts, structuring and segmenting, development of a
curricular repertoire, and clarification of purposes

Representation: use of a representational repertoire which includes analogies, metaphors, examples,
demonstrations, explanations, etc.

Selection: choice from among an instructional repertoire which includes modes of teaching, organizing,
managing, and arranging

Adaptation and Tailoring to Student Characteristics: consideration of conceptions, preconceptions,
misconceptions, and difficulties, language, culture, and motivations, social class, gender, age, ability,
aptitude, interests, self concepts, and attention

Instruction
Management, presentations, interactions, group work, discipline, humor, questioning, and other aspects
of active teaching, discovery or inquiry instruction, and the observable forms of classroom teaching

Assessment
Checking for student understanding during interactive teaching
Testing student understanding at the end of lessons or units
Assessing one’s own performance, and adjusting for experiences

Reflection
Reviewing, reconstructing, reenacting and critically analyzing one’s own and the class’s performance,
and grounding explanations in evidence

New Comprehension
Of purposes, subject matter, students, teaching, and self
Consolidation of new understandings, and learning from experiences

 

2 Shulman used evaluation instead of assessment. I have chosen to replace Shulman’s terminology with
one that is more accepted today to indicate Shulman’s original intent. Lesh and Lamon (1992) made this
distinction by defining evaluating as “assigning a value to testing” and assessing as “describing a current
state—probably with reference to some conceptual, or procedural or developing landmarks” (p. 7).

Comprehension

  

Transformation (of subject matter)

1. Preparation

2. Representation
3. Selection

4. Adaptation

Reflection

instruction

Assessment (ofstudcnts' understanding and teacher's performance)
Figure 1: Shulman’s Cycle of Pedagogical Reasoning and Action

Each phase of the process is an important one and while the phases can occur in a
slightly different sequence, one fact remains constant—the process begins and ends with
comprehension. As Shulman states, “To teach is first to understand” (Shulman, 1987, p.
14). Research evidence indicates that pedagogical reasoning is underdeveloped in novice
teachers (Brown & Borko, 1992). At any number of places in the pedagogical reasoning
and action cycle, novice teachers experience difficulties. The transformation phase of the
process is perhaps one of the most critical—for ideas that are comprehended by the
teacher must be transformed in some way if they are to be taught to and understood by

the students—and one of the most difficult for novice teachers (Geddis & Wood, 1997).

What Do Preservice Teachers Need to Know for Teaching?

The preparation of elementary school teachers needs to address their content
knowledge of the subject matter as well as how to teach it. But there also needs to be
some focus on preservice teachers’ pedagogical content knowledge. The dilemma comes

in finding an appropriate place in the teacher preparation program for it. PCK enables

teachers to make connections between their knowledge of teaching and their knowledge
of subject matter—connections that are critical for teaching effectiveness (Cochran,
DeRuiter, & King, 1993). With PCK such an essential part of a teacher’s knowledge
base, it makes sense that they begin to develop their own PCK even as preservice
teachers. It also makes sense that PCK development occurs not only in teaching methods
courses, but in content courses as well. Since PCK involves understanding ways to
transform one’s own understanding of subject matter, a sensible place to learn about this

process would be within the context of studying that subject matter itself.

In this study, I suggest an approach to the development of preservice teachers’
PCK within a mathematics content course. In Chapter 2, I detail the evolution of my
research question and study. Initially, I was interested in what a mathematics content
course might look like when the instructor is also attending to issues around teaching the
mathematics to elementary school students. As the study progressed, my focus shifted to
how, as an instructor, a closer look at my own PCK development is helpful in designing
and teaching the course. Little research has been done around the subject of a teacher
educator’s PCK development (for an example, see Femandez-Balboa & Stiehl, 1995),
much less around teaching such a course designed around a content area. In this study, I
look at some specific ways in which PCK can be considered an area of study for teacher
educators as well as their students within the domain of mathematics—by looking at the
role of representations (Chapter 3) and the process of task design (Chapter 4).

This is a study in which the researcher and the subject are one and the same. In

the field of qualitative educational research, the first-person perspective has emerged as a

relatively new genre and one that is useful in some very specific cases—where the
researcher is interested in studying in-depth some element of his or her practice as it
pertains to the larger research community.

This study is done from the first-person perspective by necessity as well as by
design. Since mathematics content courses are often taught by professors, graduate
students, and instructors well-versed in the subject matter but lacking a background in the
study of pedagogy, it was a matter of necessity that I had to study my own practice
around this question. I was the only mathematics instructor in the department at the time
who was interested in the question of looking at what PCK preservice teachers could
learn in a content course, much less researching what PCK was important for a teacher of
preservice teachers. But there are also issues of design which necessitated that this be a
self study. Much of what I was interested in centered around the decisions that a teacher
educator makes when attending to issues of PCK. Since I was the course instructor, I
have particular insight to the decision making process I used in the design of the course.

1 These decisions were based on my cumulative experiences in classrooms and all of these
experience became an important part of my story and my study. As the researcher, I had
access to my own experiences in ways that would have been much more difficult—and
potentially less enlightening—if I had studied someone else’s practice. In Chapter 2, I
describe some of these experiences and how they contributed to the development of both
the course and the study.

Recently, questions have been raised within the mathematics education
community as to how useful the idea of pedagogical content knowledge is when applied

specifically to mathematics. In this study, I set out to look at ways in which Shulman’s

construct of PCK was useful to me in teaching content to preservice teachers. I found
places where the model was insufficient in describing aspects of mathematics
teaching—places which were revealed as critical when I reflected on my own teaching
and when I shared my reflections with my students. These deficiencies are described in

the findings of my study in Chapter 5.

CHAPTER 2

METHODOLOGY

Introduction: Designing the Course and Study

In Chapter 1, I described this study as one in which I look more closely at my own
teaching. In particular, I look at the decisions I made in teaching a mathematics course
for preservice teachers where issues of pedagogical content knowledge are also
addressed. This is, then, a qualitative study in the field of “inquiry in teaching,” but also
one in which I am both researcher and subject. Thus I am using what Lampert called “the
perspective of practice” (Lampert, 1998; Ball, 2000) to examine teaching from the inside.

The MTH 202 course that I taught evolved over a three-year period and was based
on ideas I learned from teaching previous sections as well as discussions I had about the
course design with some of my colleagues in mathematics education. For the purposes of
this study, it culminated in the course that I taught in the spring of 1999. In this chapter, I
describe some of the aspects of the course in its final iteration and the design of the study
I conducted around my teaching of the course.

In this chapter, I also describe the evolution of my study—from one that centered
around identifying and including PCK ideas in a course for preservice teachers to one that

focused on the development of my own PCK as a teacher educator.

Forming An Initial Question

As a teacher educator, I was both aware of and intrigued by the interplay between
teaching mathematics content and teaching pedagogy. There is much to cover in each
domain, and the instructors of teacher education and instructors of mathematics can be
extremely proprietary of their subject matter. At times, the situation plays itself out as a
choice between extremes—forsake the pedagogy to focus solely on the mathematics or
give too little attention to the content so that students could learn about teaching.
Between these two extremes, however, lies an expansive continuum of possibilities for
design of courses in teacher preparation programs.

When given the chance to teach a mathematics course for preservice elementary
teachers, I looked forward to creating a content course designed to meet the needs of a
specific population of students. Because the students were all future teachers, I saw it as a
unique opportunity to incorporate some issues of pedagogy within the study of
mathematics. In particular, I wondered what a mathematics content course might look
like when the instructor wants to attend to issues that relate to the teaching of that content
to K-8 students?

As I considered this question, I realized there were a number of issues that needed
to be addressed. For example: What would the mathematics content be of such a course?
What pedagogical issues would be addressed and in what ways? How would decisions
about the content and design of the course be made and on what experiences would they
be based? How could pedagogical concerns be addressed without taking away from the

mathematics experiences expected of the course?

10

Background: Why Did This Interest Me?

My reasons for wanting to design and study my teaching of such a course were
threefold. First, it grew out of my experiences teaching teacher education courses and
mathematics courses for preservice teachers. I was frustrated by time constraints as well
as the convention of separating the learning of subject matter from learning how to teach
that subject matter. Second, I drew upon my own experiences teaching children. As a
novice middle school teacher, I was often unable to replicate in my classroom for my
students the kinds of rich learning experiences I had as a preservice teacher. I felt as if
there was something missing in my undergraduate experience in moving from my
understanding the mathematics to helping students understand the content. I wanted to
find out how to make that transition smoother and more explicit for future teachers.
Third, there was a growing recognition in the field that developing a teacher’s PCK was
critical in the shaping of his or her practice. The vast majority of the research, however,
was centered around the development of PCK in teacher education courses. I began to
consider whether attention to preservice teachers’ PCK development in a mathematics

content course might be possible and even desirable.

Learning About and Teaching Preservice Teachers
From the fall of 1994 through the spring of 1999, I taught several different
courses for students in the elementary education program at Michigan State University. I

began teaching a course for interns1 and eventually I shifted my focus to teaching a

 

' At Michigan State University, teacher education is a five-year program. After four years of study,
students graduate from the university with a baccalaureate degree. The program requires a fifth year
internship for certification. During this fifth year, interns are placed in a public school for two semesters
with a collaborating teacher and over time take the lead teaching role. As a part of the internship,

11

content course in the mathematics department for preservice teachers—the site for this
research. What follows is a description of how I grew to reconceptualize the content
course through my experiences as an instructor in teacher education with a growing
awareness of the theory of pedagogical reasoning, and as an instructor in the mathematics
department.

Teaching in the Teacher Education Department. For two years, I taught courses in
the “Reflection and Inquiry in Teaching Practice” sequence——TE 802 and TB 804 (Table
2). These courses were designed to work in support of the internship placement as a
second round of methods courses. In their senior years, students were placed in
elementary classrooms for 4 hours per week. As interns, they spent most of a school year
in a classroom, working closely with a collaborating teacher and taking increased
responsibility for the daily and long-terrn planning, teaching, and assessment. The course
I taught was designed to take up issues that students weren’t ready to grapple with as
seniors. As they took more of the lead teaching responsibility in the elementary
classrooms, the students were faced with needing to think about curriculum and unit
planning, learning about what sense their elementary students were making of the subject

matter, and making instructional decisions based on assessment of student learning.

Table 2: Senior and Intern Methods Courses with Classroom Placements

 

 

 

 

 

 

 

 

Course Year Elementary Classroom Placement
TE 401 Senior 2 hours, 2 days/week
TB 802 Intern, lst semester Typically 4 days/week
TB 804 Intern, 2nd semester Typically 4 days/week

 

 

students also take two master’s level courses each semester. These four courses can count toward the
university’s Master of Arts in Curriculum and Teaching.

12

In the first semester course, I taught mathematics methods to groups of about 20
interns at a time for five-week periods, using measurement as the mathematical context
for the pedagogical seminar. For the second semester course, I worked with the two
dozen interns who selected mathematics as a focus area, and I used geometry as the site
for further developing their practice as mathematics teachers. In each course, the
mathematics served as a launching point for discussing important pedagogical issues that
were critical for interns to be thinking about as novice teachers. For example, interns used
their schools’ curricular guidelines for their grade level placement to develop plans for
teaching geometry and measurement units. They worked with various curriculum
materials and made informed decisions about the appropriateness of the materials based
on district and state curriculum guidelines and national standards. In addition, interns
assessed not only their student’s understanding of the content, but they assessed their own
knowledge about and teaching of the content as well. These issues were central to the
courses and its main goal of developing reflective educators who use inquiry of their own
practice to grow as professionals.

I enjoyed working with the interns as they began their teaching careers. But at the
same time, I experienced some frustration with not having the opportunity to focus more
directly on establishing a strong subject matter base for these novice teachers. By the
time they were interns, they were moving beyond the mathematics content and were
focusing on applying that knowledge to their teaching. But I was bothered by how weak
that content knowledge was—particularly in geometry and measurement.

One of the goals of teaching interns in the teacher education department was to

use their classroom placements as sites for studying and developing their teaching

13

practices. This involved helping preservice teachers work through the transformation
process—selecting appropriate lessons and adapting those lessons to meet the needs of
their students. And yet it is impossible for teachers to move to the transformation phase
effectively if comprehension is weak, as was the case with some of the interns I taught.
Their lack of mathematical understanding became evident as they struggled to create
lesson and unit plans, and they typically had to rely on lessons presented in a textbook
which did not draw upon any understanding of what their students knew and didn’t know.

This served to underscore the importance of focusing on preservice teachers’
understanding of the mathematics content.

Teaching in the Mathematics Department with me ConnectedMathemJaths
PLjo'ggt After teaching in the teacher education department for two years, I shifted my
focus to teach a course in the mathematics department. I eventually concentrated my
efforts primarily on MTH 202, the geometry course for preservice elementary educators.

In late 1996, I was asked by Glenda Lappan if I would be interested in co-
teaching MTH 202 with her in the spring of 1997. Not only was I intrigued by teaching a
mathematics content course, but Glenda was using some of the Connected Mathematics
Project materials in the course as well and I was interested in seeing what preservice
teachers could learn from these materials.

The Connected Mathematics Project (CMP) is a middle school (grades 6-8)
mathematics curriculum project funded by the National Science Foundation and was

based at Michigan State University. Glenda was one of the authors of the project and I

14

was in my second year as a graduate assistant for the project.2 Glenda explained that she
was planning to use some of the CMP units as a part of the content for the section of
MTH 202 that she was teaching.

I was surprised to hear that Glenda wanted to use middle school materials in a
course for preservice teachers. These would be twenty-year-olds, not the early teenagers
for whom the units were written. I wondered if the students would feel as if they were
being taught a watered-down version of the MTH 202 course material and if it would be
challenging enough to them. Glenda assured me that the course that we were going to
teach would be thought-provoking and mathematically challenging for the students. She
also suggested that there were other ways that the CMP materials could be useful in a
teacher preparation course. For example, she described a course in which the
mathematics would be the focus, but in which the cuniculum materials (specifically the
teacher editions) would give students a chance to experience some of the complexities of
making instructional decisions about what students need to know. She also explained that
she viewed CMP as a curriculum that was being written to help teach mathematics to
kids, but even more she viewed it as a curriculum to help teach some mathematics to
teachers. If that mathematics could be taught to teachers before they entered the teaching
profession, then all the better.

As I taught MTH 202 over the next few years, I tried to capitalize on some of

these ways in which the materials might be useful and I began to develop a different kind

 

2 The CMP materials had gone through five years of development, writing, field-testing, revision, and
subsequent publication. In the spring of 1997, the project focus was shifting from the initial development
phase to an implementation phase, but there were still a few eighth grade units to be finalized.

15

of course, one that kept the mathematics front and center, but also included some rich,

embedded pedagogical discussions.

Learning About and Teaching Middle School Students

Before teaching preservice teachers, I graduated with a teaching degree in
secondary mathematics education and taught middle school mathematics in southwest
Ohio for six years. As a part of my undergraduate program, I took two methods courses
in teacher education and over a dozen content courses in mathematics and statistics. All
of the content courses were taught by mathematics professors and those of us majoring in
education were mixed in with students of other majors. Understandably, these courses
focused solely on mathematics content.

The first methods course, “Instructional Procedures in Secondary School
Mathematics” (EDT 335), was taught by a professor in mathematics teacher education.
The course focused on preparing preservice mathematics teachers for teaching middle
school; the second methods course in the sequence concentrated on issues of teaching
high school. Although EDT 335 was based in the department of teacher education and
was centered around learning to teach mathematics, content was an integral part of our
work. Pedagogical ideas were typically framed around mathematical investigations, and
the mathematical topics studied—rational numbers, number theory, computation,
informal geometry, measurement, and algebra—were linked to some of the core ideas in
upper elementary and middle school mathematics curricula.

Although I don’t remember it being stated as such, some of the mathematical

content of EDT 335 was centered around several big mathematical problems (Figure 2).

16

These problems were often open-ended tasks designed to build on preservice teachers’
existing mathematical knowledge and although I was fairly adept at a wide range of

mathematics, I was invariably challenged by the content of these problems.

 

 

Construct an organized chart showing all of the proper fractions (one-half, one-third, two-
thirds, one-fourth, two-fourths, etc.) through twelfths. Include an exact decimal for each.
By studying these and, other examples as necessary, develop hypotheses to answer these
questions:

0 What about a fraction determines the nature of its decimal? [i.e., What determines
whether it repeats or terminates, how long its repetend is, how many places it has,
when is it a decimal that repeats but not at first, ?]

0 How can we predict the nature of a fraction’s decimal?

0 What fractions turn out to produce cyclic permutations of the same digits for their
decimals?

- In a/b, what is the worst possible decimal?

 

Figure 2: Sample Problem from EDT 335
In retrospect, I think that all of the problems posed in EDT 335 were designed to
push at preservice teachers’ mathematics deeper understanding of topics taught in middle
school. The problems also provided access to understanding some of the underlying
structure of the mathematics beyond what middle schoolers need to know. For example,
the problem in Figure l asks preservice teachers to investigate patterns of decimal
equivalents of fractions. Middle school students typically work with a lot of fractions and

decimals. And although middle schoolers do not usually need to understand a great deal

17

 

behind the patterns that emerge when looking at decimal equivalents for groups of
fractions, it can be helpful for the middle school teacher to have an understanding of the
structure of our number system that causes those patterns to emerge.

When I started teaching middle school, I wanted to recapture some of the
investigative nature of the problems I worked on in EDT 335. For me, the problems were
challenging and interesting, and they provided access to exciting mathematical
discoveries. I thought that middle schoolers would be able to learn in much the same
ways I did from these problems and I wanted to find ways to incorporate some of the
problems into the classes I was teaching. But while I understood that many of the
problems would need to be altered in ways to make them accessible to middle school
students, I found that what I lacked was an understanding of ways to do this.

As a novice teacher, I tried to recreate some of the lessons and experiences that I
had as a preservice teacher, only to find that I was unable to initiate the same learning
situations that I remembered as a student. For example, I recall working as a preservice
teacher on the well-known Locker Problem3 and having a lively, informative
conversation about what we, as students, learned from the problem. However, when I
tried to use the problem with my seventh grade students, I was unable to provide the

kinds of learning experiences for my students I thought were possible. With my seventh

 

3 The Locker Problem is used in many curricula today, but it is first attributed to George Polya, often
thought of as the father of today’s problem solving heuristics. The Locker Problem (Pelya, 1957) lends
itself to using a number of problem solving techniques and the solution is surprising until some simple
number theory is applied to its understanding.

One version of the Locker Problem is as follows: In a school of 1,000 students there are lockers
numbered from 1 to 1,000. One the first day of school, students line up outside of the building with all of
the lockers starting out closed. Student #1 goes through and opens every locker. Student #2 goes through
and closed every second locker, starting with locker #2. Student #3 goes through and reverses every third
locker, starting with locker #3—that is, if a locker is closed, he opens it, if a locker is open, he closes it.
Student #4 goes through and reverses every fourth locker, starting with locker #4. Student #5 goes
through and reverses every fifth locker, starting with locker #5. This continues until all 1,000 students

18

graders, the problem degenerated into an exercise in looking for patterns4 and the
students failed to make the connections between important mathematical conceptss It
was a lack of understanding of this set of necessary prerequisite skills that kept me from
moving my students to making an important mathematical discovery.

After thinking about examples like this, I decided that one important piece was
missing. When I was an undergraduate student, my professor presented the problem and
lead us through a discussion of the mathematics, but we were not given any insight about
what he did when presenting the problem or leading the discussion. I viewed his work
from a student’s limited perspective and I was unable to play the part of the teacher in the
lesson when I tried to do so. I gave minimal consideration to the fact that I would be
working with seventh graders who had more limited mathematical backgrounds. I did not
provide the right mathematical scaffolding for their learning, leaving out some important
prior knowledge—like square numbers and factor pairs—that are integral to

understanding the solution.

 

have gone through the building, reversing their appropriate lockers. After all students are done, which
lockers are left open and why?

Upon looking at a simpler version of the problem (with 20 to 30 lockers), students quickly found an
interesting pattern of open lockers emerging: lockers 1, 4, 9, 16, and 25 are left open. Some students
noticed that the open locker numbers increase by consecutive odd numbers (I :3 =4, 4 :5 =9,

9 :1 = 16, 16 +_2 = 25, etc.), other students noticed that the lockers that are left open are the square
numbers. Either way, students may end up focusing more on the patterns than on the underlying
mathematical reason for why these are the lockers that are left open. The patterns are important, but they
can actually serve to illuminate the deeper reasons for the patterns and give students access to higher-
powered mathematics.

After finding the pattern of lockers that are left open, the question of why they are the square numbers
was raised, challenging the students to access the underlying mathematics. An important idea comes
from realizing that each locker is touched by the students whose numbers are the factors of the locker
number. For example, locker 20 is touched by students 1, 2, 4, 5, 10, and 20—all of the factors of 20. The
lockers that are left open, then, are those that are touched by an odd number of students (since the first
student to touch the locker opens it, the second one closes it, the third opens it, etc.). Once this is
established, students think about which numbers have an odd number of factors. From their work with
factor pairs, they recall that the square numbers have an odd number of factors and are therefore the
lockers that are left open. With my students, however, I didn’t assess for any of the necessary
background knowledge, and my students were not ready to talk about factors, factor pairs, or special
properties of square numbers.

19

When planning for teaching a mathematics content course, I recalled the
experiences of trying to use problems from EDT 335 in teaching middle school students.
Preservice teachers need rich mathematical experiences as students, but I also felt that
learning about these problems should also include learning about how to create
appropriate learning experiences around these or other challenging problems for

elementary and middle school students.

Learning About Sources for Pedagogical Content Knowledge

While I was teaching MTH 202, I became aware of a field of educational research
that received a fair amount of attention in the late 1980s and early 1990s which also
looked at the ways that pedagogy and subject matter were linked—the study of
pedagogical content knowledge.

Shulman writes about the development of a teacher’s pedagogical content
knowledge (PCK)—an area of knowledge that is actually different from both the content
and pedagogy. In fact, PCK is a knowledge base that is truly unique to the teaching
profession in that it involves the teacher’s developing understanding of the content and
the ways in which the teacher transforms his or her understanding of the content to make
it understandable to students (Shulman, 1986, 1987).

Pedagogical content knowledge builds on the content knowledge but looks at how
it is affected by what the students learn and how they learn it. This is what makes
pedagogical content knowledge a unique category of knowledge for teachers—it is one in
which the focus begins with what the teacher knows and how that knowledge is

transformed into ways in which is it learnable for students. In short, it is for teachers, a

20

“special form of professional understanding” (Shulman, 1987). Educational researchers
have come to realize that students learn concepts in a variety of ways and that students’
understandings are built upon previous knowledge. Pedagogical content knowledge

- consists of learning how to represent specific topics in ways that are appropriate for and
make sense to the diverse abilities and interests of the students (Borko et al., 1993). In
describing the importance of pedagogical content knowledge, Grossman (1991) wrote, “If
teachers are to guide students in their journey in to unfamiliar territories, they will need to
know the terrain well. Both knowledge of the content and knowledge of the best way to
teach that content to students help teachers construct meaningful representations,
representations that reflect both the nature of the subject matter and the realities of
students’ prior knowledge and skills” (p. 203).

The transformation of subject matter for teaching occurs when the teacher reflects
on and interprets the subject matter (Cochran, DeRuiter, & King, 1993), adapting it to
meet the students’ needs and tailoring the material to those students. Gudmundsdottir
(1987) recognized this constant restructuring of the subject matter to meet the students’
constantly changing needs and ’Buchmann (1984) discussed this need of a flexible
understanding of the subject matter knowledge as crucial to the success of the instruction,
thus validating the importance of connected subject matter knowledge for teachers.

For preservice teachers, pedagogical content knowledge is of particular
importance. While it may be argued that knowledge of how to teach may best be learned
through experience, teacher education programs have long held that meaningful
pedagogical knowledge can be learned as a preservice teacher. How that pedagogy can be

played out within a specific subject matter is an area ripe for study as perspective

21

teachers learn about the subject matter. Brown and Borko (1992) reported that novice
teachers are especially weak in their pedagogical content knowledge and that university
courses should make pedagogical content knowledge a priority. Grossman (1988, 1991)
also believed that pedagogical content knowledge can and should be taught to preservice
teachers, stating that teachers must have pedagogical maps of content as well as an
understanding of the subject matter to help students toward new conceptions of
mathematics.

I wondered if my MTH 202 course could be a source for looking at preservice
teachers’ PCK development, for here was a place in which I wanted to introduce
pedagogical concerns in a content course. Teachers develop PCK through teaching
content to students, but I wondered how PCK could begin to be developed during the
preservice period—when teachers’ ideas and impressions of the profession are being
fostered in a college setting.

Grossman (1990) suggested that there are four possible sources for PCK
development—apprenticeship of observation, subject matter knowledge, teacher
education, and classroom (teaching) experience (Figure 3). In Grossman’s model, the
most important and influential source is a teacher’s teaching experience (the bottom
triangle which appears separately from the other three sources). I, on the other hand, was

interested in how teachers develop PCK through their teacher education experiences.

22

 

 

/ / .3:'.°:::. \ \
l

PEDAGOGICAL CONTENT KNOWLEDGE

 

 

Conceptions of Purposes for Teaching Subject Matter

 

Knowledge of C 'cular Knowledge of
Students' Instructional
led
Understanding Know ge Strategies

 

 

 

 

Experience

 

 

Figure 3: Sources for a Teacher’s PCK Development
But most of what Grossman and other researchers refer to as teacher education
sources for PCK are methods courses taught in departments of teacher education. My
interest lay in considering how PCK might be developed in the context of a content
course in which pedagogical ideas were also addressed and studied. What I was

beginning to think about was how a mathematics content course—such as MTH

23

202—might provide a particularly useful site for PCK development, especially when

taught by an instructor with a background in teacher education (Figure 4).

 

 

 

/ / Teacher \‘ ~\: ‘‘‘‘‘
Education ‘ * T ‘

Content

Courses

   
 

  
  

Figure 4: Sources for a Teacher’s PCK with Focus on Content
This appeared as a relatively untapped area of research—the study of PCK
development in a content course for preservice teachers. It also addressed many of the
concerns I experienced working with preservice teachers in trying to find an effective
way to combine teaching content with teaching pedagogy and concerns as a middle
school teacher working with children in understanding ways to transform meaningful

learning experiences for me into ones that carried much the same impact for students.

The Course
When making plans for teaching MTH 202, I was given the opportunity to
develop my own curriculum—as long as it focused primarily on geometry and

measurement ideas for preservice teachers. An implicit expectation of the mathematics

24

department was that the course needed to remain focused on the mathematics content.6
The mathematics department chair had full knowledge of my interest in using the CMP
materials to teach MTH 202 and he expressed his confidence in my ability to design a

course that was appropriate and challenging for students.

Mathematics Content

Good teaching involves an understanding of the major topics in a domain of
mathematics and having the ability to make connections between these ideas (Schifier,
1999). To model this thinking for students, I decided to construct my sections of MTH
202 around what I perceived as some of the important mathematical topics in geometry
and measurement. By looking at the recommended text for the course and by talking to
my colleagues who were also teaching sections of the course, I identified those ideas as
perimeter and area, similarity, volume and surface area, the Pythagorean Theorem,
symmetry, and transformational geometry. These ideas were chosen based on my work
with CMP and what I had observed as critical areas of misconceptions and partial
understandings held by preservice teachers.

The first semester that I taught MTH 202, I attempted to cover all of these topics,
but I found that I was not able to give adequate time to each one. In each iteration of the
course I taught over the next two years, I chose some subset of the mathematical topics
listed above, based on the work done by students in previous semesters. These topics

aligned with the CMP curriculum as indicated in Table 3.

 

6 This is a salient point, in that I was a doctoral student in teacher education—unlike almost every other
graduate student teaching MTH 201 and MTH 202 who were graduate students in mathematics.

25

Table 3: Important Geometry and Measurement Topics and Alignment in CMP

 

 

 

 

 

 

 

 

Mathematical Topic(s) CMP Unit(s)
Perimeter and Area Covering and Surrounding
Similarity Stretching and Shrinking
Volume and Surface Area Filling and Wrapping
The Pythagorean Theorem Looking for Pythagoras

 

Model for Learning and Classroom Environment

I wanted students to experience the mathematics content by investigating
questions that I posed and that grew out of class discussions. To that end, I worked to
create a classroom environment that encouraged group and individual investigations,
writing and talking about mathematics, and using manipulatives to enhance learning. I
also made use of a teaching model that enabled many of these ideas to be incorporated
into the daily classroom routine.

Mathematical Investigations. The mathematics of the course was problem-
centered—that is, the mathematics was developed through the introduction of sequenced
problems that encouraged students to employ appropriate mathematics and problem
solving strategies. This approach differs from other, more traditional models in that the
problems provide the impetus for learning and the mathematics is utilized as a tool for
solving the problems. This is in contrast to a “traditional” classroom, in which skills are
taught and then problem are (sometimes) introduced as applications of the skills.

Students were assigned to small groups (2 or 3 students) and were typically given
time to work individually and in their groups on problems. This recognizes that there is

often more to be learned working with others, utilizing each individual’s respective

strengths, than by oneself.

26

Writing and Talking About Mathematics. Writing has become an important part
of every subject matter in schools. Children are often given opportunities to write about
what they have learned and the process of writing is recognized as a critical element of
the learning process (Rose, 1989). Since the publication of the NCTM Standards (1989),
there has been a renewed interest in helping students communicate about
mathematics—and writing is one important aspect of communication.

In MTH 202, I regularly asked students to write about the mathematics—directing
them to think as much about the why behind the answer as the answer itself. On formal
assessments, students were asked to explain their answers, an indication that writing
about the mathematics was necessary. In addition, students were regularly given writing
prompts as a part of their student learning logs. These are described in more detail on
page 42.

Closely connected to group investigation and writing is the subject of meaningful
student discourse. The NCTM Professional Standards (1991) describe the importance of
establishing good classroom discourse, comparing it to a piece of music as it “has themes
that pull together to create a whole that has meaning” (NCTM, 1991, p. 35). In
establishing good classroom discourse, the teacher plays the pivotal role, establishing
norms for discussion, agreement, and disagreement; valuing various opinions and ideas;
and conveying messages about who is able to contribute and who has status in the
classroom (Cazden, 1988).

Using Manipulatives for Learning. Most of the problems and investigations that
were a part of MTH 202 utilized some kind of manipulative to enhance the learning.

When mathematical manipulatives became a popular component of many elementary

27

curricula in the 19808, many teachers embraced using manipulatives as a panacea for
alleviating students’ difficulties in understanding mathematics. Instead, manipulatives
should be used when developmentally appropriate and when they can effectively
contribute to the learning process. Manipulatives need to be used when they offer
something new and relevant to the learning, not used just for the sake of giving students
something to play with. Nor should it be assumed that the use of manipulatives will
automatically provide insight to mathematical knowledge (Ball, 1992).

In MTH 202, students investigated problems using a variety of concrete materials.
In understanding the concepts of perimeter, area, surface area, and volume, they utilized
square grid paper, square dot paper, isometric dot paper, transparency paper, square tiles,
cubes, string, clay, and rice. The square dot paper also proved to be particularly useful in
understanding notions of similarity and the Pythagorean Theorem. Students also used a
variety of tools in working with these manipulatives throughout the course, including
scissors, calculators, rulers, and angle rulers. For specific problems, I provided students
with labsheets that had drawings, figures, diagrams, and/or data that I found useful.

Mtiopal Model. In the CMP curriculum, problems and lessons are built
around a specific instructional model that I have also found useful in my teaching MTH
202. With this model, lessons are organized around three phases of instruction—the
launch, explore, and summarize phases. In the launch phase, the teacher introduces the
problem—establishing baseline knowledge for the entire class, introducing new terms or
ideas, and issuing the problem’s challenge to the students. In the explore phase, students
work alone, in pairs, or in small groups on the problem, employing new ideas or utilizing

mathematics or strategies from previous lessons. Also during the explore phase, the

28

teacher works as a facilitator as he or she works between individuals and
groups—answering questions, keeping students on-task and on-track, posing questions
that can help students get unstuck, and providing additional questions and extra
challenges for students. During the final phase, the teacher typically brings the class
together again and the class works together to summarize the mathematics that has been
learned, the strategies that have been used, and the answer(s) to the problem(s). During
this summary phase, the teacher keeps classroom discourse and activity focused on the
problem, its solution, and the underlying mathematics.

Through my work with CMP, I had a reasonable amount of experience in
designing lessons around this instructional model and in facilitating lessons using it.
Many of the other aspects of the course——mathematical investigations, group and
individual discovery, writing about mathematics, student discourse, and using
manipulatives for learning—are embedded in the instructional model. Students responded
well to this model and midway through the course, they were able to describe every

aspect of it in detail, before it was ever made explicit to them.

Pedagogical Content and PCK

The knowledge base for teaching is far from being fixed and final. In fact, the
systematic study of teaching as an enterprise is relatively new and largely uncharted
(Shulman, 1987). The transmission of knowledge to preservice teachers thus becomes a
grossly inefficient model for teacher education, and one that is frustrating for novice
teachers as they encounter situations for which they feel unprepared. As argued by

Fensterrnacher (1978, 1986), the goal for teacher education is “not to indoctrinate or train

29

teachers to behave in prescribed ways, but to educate teachers to reason soundly about
their teaching as well as to perform skillfully.” In this model, the teacher’s knowledge
base must be flexible and under constant development. Teachers must also be able to use
their knowledge base to make reasoned decisions. In this view, teaching becomes a
process, rather than an endpoint. A teacher takes an idea and works to place it in his or
her existing framework. Then the teacher begins to mold and shape the idea in new ways
that will enable his or her students to better understand the idea. In this respect, teaching
focuses on the teacher’s interaction with ideas and how they can be transformed in ways
that students can understand them—more simply, by giving attention to the teacher’s
pedagogical content knowledge.

When I began the course, I wasn’t completely sure what the pedagogical content
of the course would be, nor to what extent the course would focus on matters of teaching.
In general, I hoped that I could use the mathematical conversations and investigations of
the course to address some pedagogical issues, such as assessing student learning and
making day-to-day and long-terrn curriculum decisions.

I was also not exactly clear what I thought pedagogical content knowledge was.
When I was planning for the course, I turned to Shulman’s definition and description of

PCK for some initial guidance:

Pedagogical content knowledge embodies the aspects of content most germane to its teachability.
Within the category of pedagogical content knowledge I include, for the most regularly taught
topics in one’s subject area, the most useful forms of representation of those ideas, the most
powerful analogies, illustrations, examples, explanations, and demonstrations—in a word, the
ways of representing and formulating the subject that make it comprehensible to others. . .. [It] also
includes an understanding of what makes the learning of specific concepts easy or diflicult: the
conceptions and preconceptions that students of different ages and backgrounds bring with them to
the learning. (Shulman, 1986, p. 9)

My initial understanding of PCK centered around the fact that it differed from

subject matter knowledge and knowledge about teaching. I viewed it as “more than just a

30

marriage of pedagogy and content. [I]t is another type of knowledge and it is one which
teachers must draw upon extensively in order to teach” (Personal Teaching Journal,
December 1998). As I taught the course and tried to identify the places in which
pedagogical content and PCK were appropriate (see Appendix A), I found that I was
mostly focusing on ideas of representation and task development (which provided the
basis for the analytic framework in my study), discourse, writing, and teacher
professionalism—all of which are part of a knowledge base that is unique to the teaching

field.

The Participants

The class met two days a week for an hour and twenty minutes a class session.
The class was comprised of thirty-four students—thirty-one females and three males—all
majoring in elementary education. One female student dropped the course about two-
thirds of the way into the semester due to a family crisis. The other thirty-three students

completed the course in its entirety.

Assessment and Grading

F ormal assessment of the students consisted of the following: five quizzes (one of
which was a group quiz), three tests, responses to ten learning log prompts (see p. 42 and
Appendix F), a project and a final cumulative exam. A description of the grading policy
that was given to the students is included as Appendix B. Students were regularly
assigned homework problems which provided extra practice and further investigation of

content that was covered that day in class and which extended the discussions from class.

31

Influences from Research Literature

The course was focused on students learning meaningful mathematics using a
problem-centered curriculum with an emphasis on students’ constructing their
mathematics, situated cognition, and connections. In addition to the mathematics content,
students were exposed to ideas about transforming the content they learned into
appropriate mathematics for elementary and middle school students. I approached the
teaching of the course primarily as a reflective practitioner—reflecting on my teaching
and the students’ learning, and involving students in the process.

The current movement of using problems as the framework for a mathematics
curriculum can be traced back to John Dewey in The Child and the Curriculum and
Jerome Brunet in The Process of Education—both of whom recognized the ways
students learn by applying preexisting knowledge to new problem situations. The value
of building a classroom and a curriculum around the investigation of problems has been
documented by educators and mathematicians alike (Lo, Gaddis, & Henderson 1996,
Schoenfeld, 1996, Fellows, 1992). In describing a mathematics classroom where
investigations grow out of interesting problems, Larry Copes advocated that students do
“real mathematics”—-—the kinds of work that mathematicians do (Copes, 1996). In

preparing teachers to teach in a problem-centered classroom, he stressed that:

Teachers should know ways that they can teach mathematical investigations. When the emphasis
in on discovering processes rather than reading answers from Tablets in the Sky, teachers need not
be able to read those tablets themselves. They need not know all of the answers to engage students
in investigating. (Copes 1996 pg. 273)

The bottom-up approach to cuniculum design (in which problems form the
foundation) was recommended in Everybody Counts (National Research Council, 1989)

and the Curriculum and Evaluation Standards for School Mathematics (N CTM, 1989). In

32

the Standards, NCTM envisioned “classrooms as places where interesting problems are
regularly explored using important mathematical ideas. ...What a student learns depends
to a great degree on how he or she has learned it” (NCTM, 1989, p. 5). Brown, Collins,
and Duguid (1989) took it even further, describing this type of learning as social
construction, where the learning outcomes become inseparably associated with the
learning process. The current push by some curriculum developers for using “authentic
problem situations” recognizes the influence of situated cognition. When students interact
with problems that have meaning for them, they learn concepts within problem solving
contexts. This helps to create a web of “connected mathematics” for each learner in
which the knowledge can be easily accessed in new problem situations. This act of
situated cognition also conceives mathematics learning as an “essentially constructive
activity instead of an absorptive one” (Schoenfeld, 1992). This theory has yielded the
current wave of mathematics literature around constructivism, which focuses education
on the learner and his or her efforts to construct meaning around problems (Davis, Maher,
& Noddings, 1990; Hiebert & Carpenter, 1992).

The theory of constructivism applies to teaching as well. The teacher’s role is to
find and adapt tasks which set up learning situations for students. Just as social
interactions between students are critical to knowledge construction, so are the
interactions between students and the teacher (Koehler & Grouws, 1992). The teacher
plays a critical role in what students learn and understand—not only in task design, but in
choosing representations, interacting with students, assessing their understanding, etc.
Shulman’s (1987) description of successful teaching using the model of pedagogical

reasoning and action demonstrates how important a role the teacher plays in developing

33

student understanding. One critical phase of Shulman’s model is reflection—the process
of critically analyzing one’s own efforts—as well as those of the students. Yet teachers
often see themselves as sources of knowledge—confident in their understanding of the
content as well as the pedagogy. Unless challenged, teaching is therefore not a practice
that naturally engenders self-reflection (Schbn, 1987). But reflection is the very action
that enables teachers to learn from their experience and become better teachers (Brown &

Borko, 1992).

The Study
In this study, I set out to find ways to construct a course for preservice teachers
that covered both mathematics content and pedagogical content knowledge. I was
interested in finding ways in which PCK could be an embedded, natural part of the
curriculum, much as constructivist teachers want mathematical understanding to grow out
of experiences their students have. To increase my understanding of this goal, a focal
question and several sub-questions were developed to guide the study and the analysis.
The focal research question was:
0 What does a mathematics content course look like when the instructor also
wants to attend to issues related to teaching that content to K—8 students?
Sub-questions included:
0 What do I, as an instructor of my course, do to address what preservice
teachers need to know about geometry and measurement to teach it to

elementary level students?

34

0 What do preservice teachers need to know about ways of transforming what
they know about geometry and measurement for the purposes of teaching it to
elementary students and how do I attend to those needs as an instructor?

0 In what ways and to what extent is the course contributing to the development
of preservice teachers’ content knowledge and pedagogical content

knowledge?

Studying One’s Own Practice

This study is an example of a relatively new field of research—one in which the
researcher studies his or her own practice. There is a long tradition of teachers writing
about their own practice (e.g., Parker, 1993; Tsuruda, 1994), but the first-person
perspective has only become accepted as a viable form of educational research in the past
fifteen years (Richardson, 1996). Through the work done by some of the early advocates
of first-person educational research (e.g., Lampert, 1986; Lensmire, 1991; Heaton, 1994;
Simon, 1995), this method of inquiry has developed into one which is distinctive from
other case study research in two critical ways—through design and subject (Ball, 2000).

The first issue of design is critical in that with first-person research, it begins with
the identification of an issue and then a context is designed in which the issue can be
examined. That is, the context grows out of the study, instead of the other way around.
This places the emphasis more on the study itself, for without it, the phenomenon that is
being studied wouldn’t necessarily exist (Ball, 2000).

It could be argued that the version of MTH 202 I was studying was simply the

result of a natural progression of previous sections I taught. To some degree, this is true.

35

But in many ways, the course was designed not only because of what I learned from
teaching other sections, but because I was studying this iteration of the course.

For example, I was aware that geometric formulas were becoming a larger focus
of the MTH 202 course each time I taught it. I saw how preservice teachers struggled
with formulas and ways to validate them and I wondered if there were some “geometric
proofs” which were more helpful, more convincing, and generally more powerful than
others in establishing this understanding. I tried various ways to engage students in
looking at these geometric proofs and it is clear to me that whether I was conducting a
study of my teaching this version of MTH 202 or not, formulas would have continued to
play an important role. However, in thinking about what I wanted to study, I began to
focus on the development of PCK in preservice teachers. When I saw that Shulman’s
definition of PCK includes the phrase “the most useful forms of representation of those
ideas” (Shulman, 1986), I was struck with the realization that formulas and their
geometric proofs are two forms of representation, and that there was the potential for
using these representations as the basis for part of my study (in fact, it provided the
framework for my analysis in Chapter 3). So because the study was an integral part of the
design of the course, the language of “representations” became a regular part of my
vocabulary.

Not only did I develop better ways of talking about what I wanted to look at, but I
also began to focus more on representations in my teaching journal. For example, in
reflecting on a class discussion on the development of formulas for volume and surface

area of rectangular solids, I wrote:

I explained that in middle school, we “don’t stop until we get to variables”... and I pushed them to
write down a representation for the surface area of that rectangular solid. Three different forms
were presented and we were able to have a rich conversation about these forms and their

36

equivalence. . . . I feel a lot better about taking the time to talk about the flat patterns and the
development of the surface area formulas than when I did this last year and didn’t focus on the flat
patterns. They really are a critical step in [my students’] understanding surface area. (Personal
Teaching Journal, 3/16/99)

In thinking about the design of the study and reflecting on my teaching, important issues,
such as the role that representations play in the development of PCK, were raised and
attended to. The course that I taught—and which became an important part of my
study—would not have existed in this form had I not been conducting this research.

The second critical difference between first-person research and other forms of
case study is the subject and the relationship between the subject and the researcher.
Since, in first-person research, these are one and the same, it is apparent that the
researcher has access to special insight on his or her subject. Ideas, memories, attitudes,
impressions—all of these can be easier to address in a self-study than in even the closest
third-person participant observation.

But this also raises a valid concern about maintaining an appropriate distance
between researcher and subject. When analyzing data, the researcher needs to stay
outside of the experience, remaining an objective observer of events and phenomena. But
when the researcher is himself an integral part of these events, that separation can be
difficult. On the other hand, distancing himself too far from the events can negate some
of the unique insight that can be gained from doing first-person research. It’s important to
find a middle ground within these two extremes.

In teaching MTH 202, I was constantly struggling with posing good questions. I
often had an idea that I was trying to move students toward, much like Martin Simon’s
“hypothetical learning trajectory” (Simon, 1995), but I experienced difficulties in

wording my questions to get at the ideas I wanted to address. This issue of task writing

37

was often on my mind as I prepared for class—as when writing good test questions that
would cover all of the important mathematical concepts in a unit but would be concise
enough that students could complete in one class period, or when asking questions in the
course of a lesson that would move students toward an intended goal without giving away
too much of the discovery process. These concerns over the development of good
questions, problems, and tasks surfaced frequently in my personal reflections as in the
following:

I think I need to look harder at the questions that I ask and how I can open up the floor for better
discussion. It seems like some of my questions are too directed and not open enough (“Anyone
know anything about the history of the standard measurement system?”) and I need to pay better
attention to how I ask these questions. (Personal Teaching Journal, 1/26/99).

Yet many of these concerns that I had would not have been apparent to an outside
observer. These were issues that I thought about in practically every stage of course
design and development—in planning lessons, in writing assessment questions, in
interacting with students in class, through e-mail, and in my office, and in reflecting on
the class and my teaching. This theme emerged because the study was about my own
teaching and I had much broader insight to the issues I was struggling with as a teacher.
The developing attention to task design—an issue that was not an explicit part of my
original research design—thus grew to be an important aspect of my study and provided
the framework for my analysis in Chapter 4.

I was able to keep an objective perspective on my task analysis because much of
the analysis was taking place long after the fact. As I explained to the students in MTH
202, I worked to make sure that the study would not add work to or detract my attention
from the course in which they were enrolled. The actual analysis of classroom

interactions, teacher reflections, and student work (for the purpose of the study) would

38

not occur until after the course was over. This was to ensure students that they would
have my undivided attention during the semester as students—not as participants in a

study. It was also my attempt to separate the subject from the researcher.7

Informing Students

As this was a study of my own teaching involving college students, I wanted to
make sure that they understood the separation between the course and the study I was
conducting. And although I was the actual subject of the study, I thought it was important
to inform the students throughout the process since they would be participants in the
study and their work would be a part of my data. I obtained a preliminary class list and on
January 5, l999—one week before the first class meeting—I sent the students an e-mail
to describe the course and the study I was conducting (Appendix C). I wanted students to
know how this section of MTH 202 would be different from others and to allow them the
opportunity to choose another section if they desired. In the e-mail, I explained that
students in the class could choose to be a participant in the study or not. This decision
would be made at the end of the course and I would not know who granted permission
until after the grades were submitted to the university.

On the first day of class, students received the course syllabus which outlined the
mathematics content, expectations, and grading policy. Students also received copies of

the Consent for Documentation Form (Appendix D) which described the documentation

 

7 Heaton took an even more novel approach when attending to issues of distance (Heaton 1994). In her
doctoral dissertation, she referred to herself as Ruth 1, Ruth 2, and Ruth 3—identities that situated her in
different time periods with relation to the data (Ruth 1 was the teacher teaching a class, Ruth 2 was the
teacher reflecting during the same year but some time later, and Ruth 3 was the teacher looking back 3
years later on her work). Not only did I want to avoid talking about myself in the third person, but since
some of my data actually goes back to work 1 did as an undergraduate student, I wanted to avoid
referring to myself with non-positive subscripts.

39

process I was using and the Informed Consent Form (Appendix E). I explained that the
documentation I would be doing (audiotaping, collecting student work, etc.) would be a
part of the study, but it was also part of the practice of good teaching. Students were
asked to read and sign the Consent for Docurnentation—-either that day or as soon as they
wished. I explained that I was giving them a copy of the Informed Consent Form to them
on the first day so that they would know what they were going to agree or not agree to at
the end of the course.

On the final day of class, students were given another copy of the Informed
Consent Form. I again explained the process and that their decisions whether to grant
consent would not have any repercussions on their grades. While I was out of the room,
students filled out their forms and handed them to one classmate who kept the completed
forms until after I submitted the course grades to the university. At that time, I obtained
the completed forms from the student and found that all granted permission for their work

to be included in the study.

Documentation

There were two types of documentation used for the study—artifacts from the
course (data that was a regular part of the course, but collected for the study), and data
that was designed and/or collected specifically for the purposes of the study. Artifacts
included the lesson and unit plans I made for the course, my teaching journal, the texts
that students and I used, and students’ written work on both instructional and assessment
tasks—including students’ “learning logs.” For the purposes of the study, I collected

audiotapes of all class proceedings, pre- and post-course concept maps showing students’

40

understanding of geometry, and pre- and post-course surveys of students’ attitudes
toward geometry.

Lesson and Unit Plans. Since this study is centered around my planning and
teaching of MTH 202, it was clear that my own thoughts and decisions about the makeup
of the course would be important data. Before the course began and as it progressed, I
kept written records of what I planned to teach. My plans included outlines and
mathematical work that I did to prepare for class (working out homework problems,
planning for assessments, investigating ideas that had been raised in class, etc.). These
plans always included notes about mathematical content and tasks I planned to use,
problems that arose in other students’ work that I wanted to use in class discussion, and
ideas for homework assignments. At times, the plans contained notes about aspects of
PCK I wanted to include in class. The lesson and unit plans were kept in a binder and
were helpful in maintaining an ongoing written account of howl envisioned the content
of the course.

The unit plans focused on the mathematical topics I identified in geometry and
measurement and that I planned to use as the basis for the course. My mathematical goals
for the course included in-depth investigation of these mathematical topics (outlined for
the semester in Appendix A) as well as:

0 Giving students opportunities to read and write about the mathematics and to

engage in thoughtful discourse about the mathematical content.

0 Making students aware of additional resources that are available to teachers,

such as professional journals, the NCTM Standards, Internet resources, and

multiple elementary and middle school curricula.

41

0 Highlighting and encouraging students to find connections between different
mathematical ideas.
0 Connecting the mathematics of the course to other subjects taught in
elementary school, such as literature, social studies, and science.
0 Identifying preconceptions about what geometry is and giving students a set
of positive experiences around the mathematics content of the course.
Teaching Journal. Each day after class, I immediately returned to my oflice where
I sat down for two or three hours, thinking about what had happened in class that day and
recording my thoughts into a teaching journal. The teaching journal afforded me the
opportunity to reflect on my teaching and what had just transpired in the classroom.
When teaching in a middle school or two back-to-back college courses, my teaching
schedule often did not afford me the time to reflect—especially this soon after the
event—and this resulted in some missed opportunities for me and my students. But when
I was able to block out time immediately after teaching MTH 202, I was able to recall
important, but small, events which might have been forgotten, I was able to attribute
particular insights to students who had made them, and I was able to make better plans on
where to pick up for the next class session. The teaching journal was also helpful in that it
provided me with a record of my thoughts and ideas which was a rich source of data.
Textbooks. The texts for the course were student editions (SE’s) and teacher
editions (TE’s) of the following CMP units: Covering and Surrounding (SE only),
Stretching and Shrinking (SE and TE), Filling and Wrapping (SE and TE), and Looking

for Pythagoras (SE and TE).

42

The other instructors of MTH 202 used a college geometry text in their sections
(Geometry: An Investigative Approach by O’Daffer and Clemens). The O’Daffer text did
a nice job presenting problems for preservice teachers to solve and making some
connections fi'om the mathematics to teaching the mathematics. But the text fell short in
preparing teachers to make curricular decisions—which topics to cover, what topics are
appropriate for students at various levels, how to engage students in the mathematics, etc.
Russell (1997) suggested that elementary and middle school curriculum materials might
be more appropriate for teaching preservice teachers the mathematics content and ways to
teach it. Bright (1999) reviewed some current reform curricula with this purpose in
mind—of teaching preservice teachers content using student and teacher texts. He

”8 to have the most

suggested that those teacher texts which contained “dialogue boxes
potential in teaching preservice teachers.

For two years prior to this study, students in my sections of MTH 202 purchased
and used various student and teacher editions of geometry and measurement CMP texts.
Initially, I had students use SE’s for all of the units and TE’s for only one of the units. I
did this for several reasons. First, I thought that if students had the answer keys to some
of the homework questions, it would decrease the amount of time spent in class going
over answers to problems. This would leave more time for discussion of the mathematics
embedded in the problems, applications of these ideas to teaching elementary school, and

investigations of new ideas. Second, I reasoned that since many of the problems and their

solutions involved geometric diagrams, it would be easier if students could refer to an

 

a The dialogue boxes appeared in teacher editions and were used to give evidence of student reasoning on
problems, strategies that students often utilized, and suggestions for teaching the mathematical content to
students.

43

answer key with figures drawn more accurately than those that I might be able to
reproduce in class.

As I taught the course, however, I found that by giving students access to more of
the TE’s we began to talk about other issues as well. The CMP TE’s contain more than
just correct answers to student problems. They are developed through interacting with
dozens of experienced teachers who used the materials as they were being developed and
so they reflect lessons learned by those teachers and suggestions that they made. The
TE’s contain multiple answers (when appropriate) and various examples of strategies that
middle school students may use when working on problems. They also push teachers to
understand the content at a much deeper level than that of their students, encouraging
them to know more about the why behind the mathematics they encounter.

By having access to the TE’s, the preservice teachers were being exposed to a
number of concepts that are often talked about in teacher education courses but may be
difficult to comprehend when not placed in a mathematics context—things like
discussions on various strategies that middle school students could use in solving
problems and possible ways to engage students in the mathematics that is being studied.
As a result of this exposure, students initiated class conversations about lesson planning,
connections between mathematical ideas, assessment, curriculum design, and other
pedagogical concerns. I

For example, a large part of one class session midway through the course centered
around my asking the students to “describe the typical mathematics lesson” from MTH
202. The students were writing their own grade-level specific unit plans or lesson plans

for a topic in elementary school geometry. In preparing to write their own lessons, I

44

wanted them to think about the normal mathematics class they remembered from their

school experiences as compared to what I typically did in MTH 202. Some of the

students’ responses were:

“You start by bringing out our background knowledge for a certain
mathematical idea.”

“Without telling us what we are going to find, you let us explore first.”
“You use ‘guided experimenting’... You set up the situations for us to go in
specific directions.”

It’s very hands-on. I learn from touching something—to be able to move the
tiles or to wrap the yarn around my foot.”

“(After we get back together as a class), you go back and talk about each
group’s answers and how they are different, (with a goal of) finding the most
accurate or the most efficient method.”

“At the end, you tie everything into the big picture, how it all relates to each

other.”

Some students simply picked up on these ideas after being a student in the class

for eight weeks while other students explained that the TE’s were helpful in seeing what I '

was doing when teaching the class. At this point, we talked about phases of instruction

and an instructional model (see p. 28) that are explicit in the TE’s, and how this model

for learning differed from what was experienced in other mathematics classes. Students

commented that this conversation served as a helpful analysis of teaching and learning

mathematics as they prepared their own lesson plans. I think that fewer students would

45

have been able to participate in this classroom analysis had they not used TE’s on a
regular basis.

Students’ Written Work. I collected and photocopied many pieces of student
work. These included quizzes, tests, the final exam, and responses to learning log
prompts. All students consented to the use of their work in the study.

This is primarily a study that looks at what I did to plan for and teach a
mathematics content course for preservice teachers that gave attention to issues related to
teaching that content to children. However, I found it helpful to use evidence of student
work in looking at the impact of what I did on what students learned. Students were often
encouraged to supply reasons and show work for their answers. This evidence supports
not only what they learned, but it also provides support of claims I make about how the
design of the course has an impact on the learners.

One of the richest sources of data around student work was the student learning
logs. As part of the course, I required each student to supply a small notebook that I
called a learning log. The learning logs were used for students to reply to designated
prompts throughout the semester (a total of ten different prompts). I regularly collected,
read, evaluated and orchestrated class discussions about their responses. The learning log
prompts were designed, in part, to give students some experience in writing about
mathematics. Many elementary teachers are asking students to keep mathematics journals
to record their thoughts, insights, methods, and attitudes. I wanted preservice teachers to
have some experience in writing about mathematics as well, just as they might ask their
students to do. In addition, the learning log prompts were written in conjunction with the

learning goals I had for the class and were designed to:

46

Establish a base-level understanding of what students understood about a
concept before it was addressed in class. In the first prompt, I asked students
to “write down the first three geometry formulas you think of and tell what
they are the formulas for.... Describe some of the pros and cons for using
mathematical formulas. What do you think about using formulas with
elementary school students? ”
Address some of their preconceived ideas and attitudes about geometry. It has
been my experience that some preservice elementary teachers view geometry
only through the lens of their high school geometry course—positively and/or
negatively. In one prompt I asked students to “describe in detail an event fiom
your high school geometry course which is particularly memorable to you.
Why do you think this event sticks with you? ”
Give students a mathematical task in addition to what was done in class. I
used this opportunity to pose more open-ended tasks. This gave me some rich
data on how students could write about mathematics and on their
mathematical thinking. In one prompt I asked, “When the midpoints of the
sides of a quadrilateral are connected in order, another quadrilateral is
formed on the inside. What kinds of quadrilaterals can be formed on the
inside? Which original quadrilaterals produce which types of quadrilaterals
on the inside?... ”
Raise other pedagogical issues. After the class discussion on how some of the

geometry ideas in the course connect to other areas of mathematics (number,

algebra, data analysis, etc.), I included a learning log prompt asking students

47

to “...give examples of some of the ways in which geometry and measurement
connect to other school subjects aoossibly including science, social studies,
language arts, music, etc. ). ”

0 Have students reflect on the MTH 202 course. Since I was interested in how
the design of the course had an impact on my students’ understanding and
attitudes, I included one prompt in which I asked them to “describe in detail
one event fiom your MT H 202 course which is particularly memorable to you.
Why do you think this event sticks with you? ”

A complete list of learning log prompts that I used in the course are given in

Appendix F.

Audiotapes of Cgss Proceeding; At the start of each class session, I turned on a
small cassette recorder that was placed at the front of the classroom and recorded the
entire class session. All students were made aware that this type of data would be
collected and they granted consent for their voices to be recorded. The tape recorder was
near me and it did a good job picking up my voice throughout. Some parts of student
conversations were harder to hear.

Although I did not use the recordings while teaching the course (finding the time
to listen to class sessions and review what was said might have been helpful but
extremely time-consurning), they provided a relatively thorough record of class
proceedings and discussions. I used the teaching journal and the lesson and unit plans to
identify critical events in the course and was then able to turn to the audiotapes for a

more detailed account which I transcribed.

48

Other Data. Two additional pieces of data were collected in the course and were
designed to help me look at student growth in knowledge and disposition. The first was a
set of students’ concept maps that were drawn at two different times in the course. After a
brief discussion on making concept maps early in the course, I asked students to place the
word “geometry” in the center of a blank piece of paper. Students then constructed their
own concept maps of how they thought about the content of geometry, naming big and
small ideas and connecting them as a representation of their thinking. These maps were
collected and redistributed on the last day of class. Students were then given the same
task—to construct a concept map of how they thought about the content of
geometry—and asked to make those maps on the backside of their initial ones. Students
were then asked to compare their two maps with the hope of their seeing how much their
understanding of the domain grew.

The second was a set of student responses to an attitude survey about geometry
and measurement. As with the concept maps, these surveys were administered at both the
beginning and at the end of the course. In the case of the surveys, however, students were
not given access to their earlier responses and so could not use them to make a self-
assessment about any changes in attitude or disposition.

As the analysis of the data progressed, I didn’t find the concept maps very helpful
in the study, so they were not used. A few of the student responses to the attitude survey
helped shed some light on their dispositions toward learning geometry and thus became a

part of the study.

49

Data Analysis
This was a study of what I did as a teacher educator to attend to issues of
pedagogical content knowledge in a content course. To that end, I used coding methods
to find themes that surfaced in my teaching of MTH 202—reading my personal accounts
of the course in my teaching journal and lesson and units plans, listening to and
transcribing episodes of class discussions, and analyzing student work. Data is drawn
from thirty class sessions, my writings in preparing for those sessions, and my writings in

reflecting after those sessions and after the course was completed.

The Emergence of Two Themes

After the course was completed and all students agreed to be participants in the
study, I began the work of analysis by reading—and rereading—my teaching journal. I
had almost four months of reflections and observations that were recorded and I wanted
to refamiliarize myself with what I wrote over the course of the semester. In particular, I
was looking for trends or recurrent themes in my writing—ideas which could potentially
develop as themes for analysis.

Formulas and Other Representations. One of the most prevalent themes that
emerged came as no surprise to me—that of geometric formulas. As my MTH 202 course
evolved over a three-year period, I found myself constantly coming back to issues
concerning formulas——understanding where they come from, helping students make
sense of them, finding ways to remember them, etc. Many of the problems and
investigations I developed and that I used from the CMP curriculum involved areas and

perimeters of shapes and volumes and surface areas of solids, topics that often led to

50

discussions around understanding formulas used to calculate those measures. I
recognized early on that formulas would be a possible theme for analysis and I made a
conscious effort to pinpoint places in my teaching journal where discussions around
formulas occurred.

The importance of understanding representations has not been lost on the
mathematics education community. Mathematics has long concerned itself with various
categories of representations (e.g., symbolic, spatial, and language) as well as specific
types of representations that are particularly useful in mathematics (e.g., graphs, tables,
diagrams, and formulas). These mathematical representations are established models in
the world of mathematics. Recently, the National Council of Teachers of Mathematics
raised mathematics educators’ level of consciousness about representations by adding it
to its list of essential process standards for school mathematics—a list that now includes
problem solving, reasoning and proof, communication, connections, and representations
(N CTM, 2000).

Mathematical formulas are one example of a type of representation. A formula is
a symbolic representation of a mathematical relationship. For example, A = s2 is a
formula for the area of a square. This formula shows—concisely and eloquently—that the
area (A) of a square can be determined by multiplying the side (s) by itself, (squaring it).
With this type of representation, a great deal can typically be said in a small space.
Another example is a pictorial or spatial representation. As I taught different sections of
MTH 202 over the years, I became increasingly interested in ways in which various

spatial representations could be used to explain or validate the geometric formulas that

51

were a part of the course. I began to wonder if different forms of representation were
helpful for students in understanding a concept.

But the word representation refers not only to the product (the formula, the
picture, etc.) but also to the process—the “act of capturing a mathematical concept or
relationship in some form” (N CTM, 2000, p. 67). The research around the role of
representations gives credence to both aspects, the process and the product. For not only
is it important to look at which types of representations can be used with students and
have an impact on student learning, but it is also important to recognize the development
of modes of representation as they occur in the classroom.

But representations also play an important role in the development of pedagogical
content knowledge as well—although the representations that are mentioned in PCK
research literature do not necessarily map easily onto the representations in the
mathematics literature. Ball identified “‘forms of representation’ . . . [as] the crucial
substance of pedagogical content knowledge” (Ball, 1988). Borko et a1. claimed that
making subject matter understandable for students hinges on the teacher’s ability to
“represent specific t0pics in ways appropriate to the diverse abilities and interests of
learners” (Borko et al., 1993). And in summarizing the literature on PCK, Van Driel et al.
suggested that “all scholars agree on Shulman’s two key elements—that is, knowledge of
representations of subject matter and understanding of specific learning difficulties and
student conceptions” (Van Driel, Verloop, & De Vos, 1998). Indeed, representation
appears as a critical component of PCK not only in Shulman’s introduction of PCK to the
field (Shulman, 1986), but also as a key element of the transformation phase in his model

for pedagogical reasoning and action (Shulman, 1987).

52

It seemed to me that the mathematical representations (like spatial, symbolic, and
language) could be examples of some of the representations mentioned in the literature
on PCK, as in Ball’s “substance of pedagogical content knowledge,” but that much of the
writing about PCK didn’t involve physical representations as much as it did
representations of knowledge and understanding. With the word representation being
used in many different ways, I found that it was easy to get confused about its meaning
when applying ideas of PCK specifically to learning mathematics—a content area in
which representations already play an important role.

I decided to focus some of my analysis around the important mathematical
representations as they appeared in my data. Because I identified formulas and their
accompanying spatial representations as a potential theme before collecting data, I
encountered numerous references to these representations in my unit and lesson plans,
my teaching journal, the learning log prompts and assessment questions I wrote as well as
the student work done around them, and in the audiotapes of the class proceedings—any
source with evidence of my thinking and/or my words. The CMP textbooks (specifically,
the TE’s) also provided some evidence of identifying various representations and the
ways that they can be used in helping students understand a mathematical concept.

A useful method of analysis in thinking about types and roles of representations
was in developing and using coding categories. In Chapter 3, I give evidence of my work
in forming various categories of representations and how this process revealed a type of
representation that I previously disregarded—the representation of mathematical
language. I also discuss how the process of shared reflection was an important tool in

developing my understanding of the various ways in which representations were used, as

53

I shared some of struggles and frustrations with preservice teachers and made these
conversations a part of my learning about teaching.

Nature of Tasks and Issues of Task Design. After looking more carefully at what I
wrote in my lesson plans and teaching journal, I uncovered a second theme. I noticed that
one particular problem—one of the learning log prompts—stayed a part of our class
discussions for parts of four different sessions. This was considerably longer than the one
day I planned to investigate this task. In thinking about why this task embedded itself so
deeply in my (if not the class’) conscious, I conjectured that if the task were worded
differently, it might not create the difficulties that it did for my students and me.

This supposition was gradually generalized to include basic ideas of task design.
That is, what goes into the development and construction of a good task—one that does
not mislead students but provides an avenue for investigating the problem as the teacher
intended.

The NCTM Professional Standards described worthwhile mathematical tasks as
ones “that are likely to promote the development of students’ understandings of concepts
and procedures in a way that also fosters their ability to solve problems and to reason and
communicate mathematically” (N CTM, 1991 , p. 25). While it goes on to explain that
tasks may be selected or adapted from other resources, or generated by teachers
themselves, it provides little detail on the process that teachers go through in creating
meaningful tasks.

Since task design clearly depends on a flexible understanding of the underlying
mathematics around which the task is developed as well as identifying ways to help

students understand the mathematics themselves, it seems that a teacher education course

54

with an interest in developing preservice teachers’ PCK should include some attention to
its development.

In my data, I found several sources for data around task development. In my
lesson and unit plans and in my teaching journal, I found a number of entries around this
particular learning log prompt alone. I also had copies of all of my students’ work on this
task—data which I used to triangulate my findings and theories on how elements of the
task I designed produced unexpected results. In addition to this, I had historical records of
this task when it was given to me as a student in a preservice teacher education course.
This was helpful in doing a critical analysis of the two versions of the task and looking at
the differences in expectations that the tasks make of students. In Chapter 4, I give
evidence of my using student work to investigate the design of a task and of my
comparing two different versions of the same mathematical task. I also discuss how the
process of shared reflection was a useful tool in helping me learn more about the ways in
which the task could be revised to better reflect the potential for learning that I was
hoping to capture.

In each of these themes—representations and task design—I started by looking at
ways in which they exemplified preservice teachers’ PCK development. But in thinking
about each case, I kept wanting to turn from what my students needed to know to what I
needed to know to teach preservice teachers and what I learned through the use of shared
reflection. In light of this, I wondered if this was really a story about developing my own

PCK as a teacher educator in the context of attending to students’ PCK development.

55

Revisiting Shulman’s Pedagogical Reasoning

As described from the start, this study was built around my wanting to look at
what I did to design and teach a mathematics content course that also attended to issues
of teaching that content to K—8 students. The students in the course were not intended to
be the subjects, but rather participants in a study of my own practice. So while I initially
thought the themes that I was developing centered on looking at developing students’
PCK, I began to recognize that another important aspect was the development of my own
pedagogical content knowledge.

Little had been written about the development of a teacher educator’s PCK.
Bennett and Carré (1993) applied Shulman’s model to a study of a postgraduate teacher
education program and Cochran, DeRuiter and King (1993) analyzed a constructivist
teacher education program using Shulman’s model. But both of these studies were done
on a general level, looking at an entire program instead of a specific case. These studies
also focused more on looking for PCK in the final end product, rather than on examining
the complexities of its development.

Geddis and Wood (1997) described a case study of a teacher educator in which
they documented a specific instance of teaching as a transformation of knowledge about
mathematics pedagogy for preservice teachers. Although rich in detail, their study looks
only at part of the transformation process of teacher education and does not address the
process the teacher educator went through himself to develop his own PCK about the
subject matter.

It appears that a critical missing element in these studies is understanding the

teacher educator’s reflection on his or her learning and understanding—about what the

56

students know about the content and pedagogy, and about what the teacher educator
knows about the content and the pedagogy. Because the teaching is taking place in a
teacher education context, the preservice teachers and the teacher educator can all benefit
when this reflection becomes a part of the classroom conversation.

In reflecting on my own experiences as an undergraduate student and a novice
middle school teacher, I recalled instances where I struggled to recreate a rich learning
situation for my students that I had as a preservice teacher. As with the example of using
the locker problem (page 14), I found that I was lacking some important understanding
about the problem in making it meaningful for my middle schoolers. Some of my
shortcomings centered around an understanding of the mathematics——the necessary
prerequisite skills that were needed and the sequencing of problems that could lead to a
deeper understanding. Other deficiencies were around the pedagogy—not knowing how
to organize my students and classroom to work on the problem, introducing a strategy too
early and not giving students an opportunity to construct their own, etc. But after further
reflection, I realized that all of these problems fell under the umbrella of my lacking
PCK, that is, ways to transform what I understood about the problem as a teacher and
making the mathematical knowledge accessible to my students. In the case of the locker
problem, I decided that what I was missing was some insight on what ‘my college
professor thought about when he chose the task, adapted it for our class, made decisions
about introducing representations, and other elements of the pedagogical reasoning
process. With some of that pedagogical content knowledge, my experience of using the

locker problem with middle school students might have been quite different.

57

Teaching teachers and giving some attention to their PCK could involve taking an
additional step—providing for them some ideas for making curricular and pedagogical
decisions. These are skills that are often developed through years of teaching, but novice
teachers can also begin to have access to this decision-making process when the teacher
educator shares with them the process that he or she went through for using that
particular lesson or problem.

In fact, this process makes the pedagogical reasoning and action model explicit.
Instead of the teacher educator just working through the phases, the process itself is
highlighted and the preservice teachers have exemplary teaching modeled for them. This
is one way in which shared reflection can be utilized—where the teacher educator shares
his or her thinking process with preservice teachers. The subject of shared reflection can
be any phase in the pedagogical reasoning and action cycle but by definition, it needs to
occur within the teacher educator’s reflection cycle. Figure 5 shows an adaptation of my
model of Shulman’s pedagogical reasoning and action cycle to describe the process of a

teacher educator employing shared reflection.

Comprehension

    
 

Transformation (of subject matter)

1. Preparation
2. Representation
3. Selection

fl 4. Adaptation

Reflection
Shared Reflection
(with students)
Instruction

Assessment (of students’ understanding and teacher's performance)

Figure 5: A Teacher Educator’s Pedagogical Reasoning and Action

58

For example, the teacher educator may choose to wait until a complete lesson has
been planned, taught, and assessed before he or she begins to share with the preservice
teachers some of the decisions that were made around the choice of the problem, the
representations that were used, the way in which the lesson was presented, or the
assessment has been completed. In other instances, the teacher educator may take the
time to address some of the phases either before, during, or after those phases present
themselves in the lesson, but still after the teacher educator has reflected on the phase
him- or herself. These are decisions that the teacher educator needs to address at all
times, deciding whether the students should be let in on the decision-making process or if

they should be allowed to “muck about” in the mathematics or the pedagogy for awhile.

Refocusing My Study

Looking back at my data, I found many instances of how I used aspects of shared
reflection in helping my students develop their PCK. In each case, the journey begins and
ends with my looking at my own PCK of the problem and thinking about how giving
attention to the role of representations or task design using shared reflection may have
helped students with their PCK development. To that end, I clarified my central question.
I was now interested in tackling the issue of: “Can pedagogical content knowledge be a
useful framework for a teacher educator in designing and teaching a mathematics content

course for preservice teachers and if so in what ways?”

59

The Story

This study is a story about my efforts to give attention to the pedagogical content
knowledge development of preservice teachers through a process of examining my own
PCK development as a teacher educator. Although I started the study with an idea of
focusing on students’ PCK development, it wasn’t until midway through the course that I
began thinking about focusing on the development of my PCK for teaching prospective
teachers and using that experience as a lens for thinking about what the subject matter of
PCK might be for preservice teachers. Reflecting on my experiences as an undergraduate
student helped me to formulate a model for teacher education in which an emphasis on
shared reflection gives preservice teachers some insight to teaching content to K-8
students. This study details my some of my efforts to use shared reflection in lessons

about representations and task design.

60

CHAPTER 3

REPRESENTATIONS

Introduction

This chapter takes the form of three episodes. Each episode describes: (a) a
mathematical topic that was covered in the course; (b) the ways in which the mathematics
was presented with respect to the development and use of representations; and (c) how
shared reflection played an important role in the course.

The three episodes center around trapezoid area, circle area, and volumes of
prisms, cylinders, cones, and spheres. The first episode deals with my gaining a deeper
understanding of different types of representations—symbolic, spatial, and mathematical
language. The second episode deals with my looking at limitations of some
representations and the ways in which multiple representations are linked with each
other. The third episode deals with my struggling to reconstruct some of my own prior
mathematical knowledge and my sharing those struggles with the students. In the third
episode, I also look at the use of a blended representation that proved to be a particularly
powerful one for preservice teachers.

Throughout each episode, there is also an underlying tension concerning the
importance of providing justification. Students come to believe that certain mathematical
ideas are true. But as described in the NCTM’S Curriculum and Evaluation Standards for
School Mathematics (1989) and Principals and Standards for School Mathematics

(2000), students need to consider what it takes to prove the validity of these assertions. In

61

a course for preservice teachers, some argue that attention should be given to
understanding what it takes to validate a mathematical statement and various
methods—or proof schemes—for providing this justification (Harel & Sowder, 1998).
The reasoning goes that just as school students are being expected to provide reasoning
about their mathematics, so must their future teachers work on understanding what it
takes to justify a mathematical assertion.

In looking at justification in this chapter, I use the classifications developed by
Harel and Sowder in which proof schemes are described as externally based, empirical,
or analytic. Each scheme has several sub-categories (Figure 6) which are helpful for
looking at various representations that are used (Sowder & Harel, 1998). In each of the
episodes, I examine what proof schemes are implied by the inherent assumptions that

were made by me as the teacher educator.

 

I Externally based proof schemes I
I i
J

 

 

 

IAuthoritarian II Ritual I I Symbolic I

 

 

 

Empirical proof schemes I

l

 

 

 

 

I Perceptual J I Examples-based I

 

 

 

Analytic proof schemes
(mathematical proofs)

 

 

 

 

I Transformational I I Axiomatic I

 

 

Figure 6: Proof Schemes

62

Episode 1: Trapezoid Area

Background—Working with Triangles

Students’ work with triangles earlier in the course involved investigating ways to
generalize the area of a triangle and developing a symbolic formula for triangle area. A
series of triangles were presented on grid paper and students were given time to develop
strategies for finding area. The goal was to find one or more methods that could be used
with as many different triangles as possible. (1" wo triangles from this labsheet are presented
in Figure 7.) Two methods were common: pairing a triangle with a congruent figure or

dissecting the triangle and rearranging the pieces to form a new figure.

 

Figure 7: Triangles A and D on Grid Paper

Using Two Copies of a Figure. When investigating a right triangle like A, students
often paired it with a congruent copy, rotated 180° and placed so that the hypotenuses
matched, creating a rectangle with an area twice that of the original shape (Figure 8). The
rectangle’s area was easily determined (by counting squares or by using the base multiplied
by the height) and the area of the triangle was commonly stated as “half of the area of the

rectangle.”

63

 

Figure 8: Two Copies of Triangle A
Although non-right triangles, like D, were less likely to be paired with a congruent
copy, some students continued the reasoning from the right triangles—using the resulting

parallelogram instead of a rectangle (Figure 9).1

 

Figure 9: Two Copies of Triangle D

With both right triangles and non-right triangles, students used two copies of the
original triangle to make a new shape for which they could find the area. And since two

copies had been used, the area of the resulting figure was double that of the original shape.

Students then concluded that the area of the original triangle was only % the area of the

resulting figure, leading to the general formula A = Ibh.

 

’ In a previous class, students discovered that a parallelogram has an area equal to a rectangle with the same
base and height as those of the parallelogram, leading to the general formula for the area of a parallelogram
(rectangular or not) as A = bh.

64

Dissecting and Rearranging. Many of our earlier investigations with area centered
on ways to take a shape, subdivide it, and then put the pieces back together in a new shape
whose area could be determined. This strategy—which I call dissection and
rearrangement—is important in mathematics both historically and as an example of a proof
scheme.

Some simple dissections were well-known to the ancient Greeks, but the first
formal collection of dissection problems was most likely written by the tenth-century
Persian astronomer, Abul Wefa (Gardner, 1969). The problems were typically presented as
interesting mathematical diversions, showing for example how three congruent squares can
be cut into nine pieces to make one single square. In the early twentieth century,
mathematicians (like the puzzle creator Henry Ernest Dudeney) took on geometric
dissections with a renewed recreational fervor, working to set records based on the fewest
number of cuts needed to dissect a given figure and rearrange the pieces into a specific
shape.

In MT H 202, I referred to some of the ways in which methods of dissection and
rearrangement were used by mathematicians to determine the areas of unknown regions,
like how Leonardo da Vinci calculated the areas of many curved figures by cutting them
apart and rearranging the pieces to create figures with known areas. I referred to this
method not as a mathematical recreation, but as a viable way to justify a geometric idea.
Many students employed this technique in a number of investigations on area, making it a
common strategy for looking at the triangles.

When working with right triangles like A, the most typical approach to dissecting

and rearranging was to make one cut—horizontal or vertical—and reduce either the height

65

or the base to half its original length. These two pieces could then be formed into a
rectangle with half of the height and the same base as the original triangle (Figure 10), or

the half of the base and the same height (Figure 11).

   

Figure 10: Triangle A as Figure 11: Triangle A as
Rectangle with Half of the Height Rectangle with Half of the Base
The notion of making one cut to create two pieces which could be rearranged to
make a known shape was used with non-right triangles as well. However with these
triangles students discovered that the cut had to be made parallel to the “base” (the side that
lay on one of the grid lines)2. For example, on Triangle D, one cut could be made parallel
to the base that would intersect each of the other sides at their midpoints (Figure 12). The
top piece could be rotated 180° and placed on either side to create a non-rectangular
parallelogram with a base equal in length to the base of the triangle, but with a height half

of the triangle’s height.

 

2 Because this triangle activity was focused on developing and understanding the basic triangle area formula
A = i bh, all of the triangles presented in class had at least one side drawn on a grid line. The idea of finding
the area of a triangle whose endpoints all lie on the intersection of grid lines but none of whose sides lie on
grid lines was also presented and was played out in the context of surrounding a shape with a rectangle and
subtracting off the areas of the unwanted pieces. It was important that students saw and worked with
triangles of many different types, but for the purposes of the lesson on developing a general formula for
areas of triangles, only triangles with at least one side on a grid line were considered.

66

 

Figure 12: Triangle D with Half of the Height in One Git
Some students made two cuts with the non-right triangles (Figure 13), still
recognizing the fact that the resulting figure would have a height half of that of the original
triangle. These students thought of the top triangle as being composed of two smaller right
triangles, each of which fit nicely with the larger bottom piece to create a rectangle. These
students were likely guided by the existing grid lines to consider this second cut and to

create a final shape with all sides lying on the grid lines.

 

Figure 13: Triangle D with Half of the Height in Two Cuts

In each of the above examples (Figures 5-8), the shape resulting from dissection

III a I
9 s7

and rearrangement was a . or not. And each of these resulting
parallelograms had one dimension (either the base or the height) the same length as the

original triangle. The other dimension was half the length of the original triangle. Students
reasoned that the parallelograrns they made from the triangles had areas equal to either (a)

the original triangle’s base multiplied by half of its height, or (b) the original triangle’s

67

height multiplied by half of its base. Symbolically, this led to the triangle area formula of

_i
A—pn

Area of a Trapezoid—Connecting Symbolic and Spatial Representations

The course was designed so that one subject typically led to the next, helping
students to see not only how different mathematical ideas were connected, but also how a
strategy that was useful with one topic might be employed with another. For this reason, I
chose to follow triangles with some work on trapezoids.3

Since many students had used the dissection and rearrangement approach with
triangles, I decided that it would be a useful strategy to use with the class when reasoning
about the trapezoid formula. When I introduced the trapezoid area formula, I presented the

diagram in Figure 14.

 

 

 

Figure 14: Trapezoid Method A

I expected students to reason through the problem as they had done with other
shapes before, using the figure to justify the parts of the formula. In particular, I expected
that students would recognize that the resulting figure (OQLN) was a parallelogram with an

area equal to the area of the original trapezoid (JKLM). Furthermore, the parallelogram had

 

3 A trapezoid is sometimes defined as a quadrilateral with at least one pair of parallel sides. Using this
definition, the set of parallelograrns (quadrilaterals with two sets of parallel sides) is a subset of the set of
trapezoids. I chose to use the more limiting definition of a trapezoid as a quadrilateral with exactly one pair
of parallel sides. This allowed me to introduce parallelograrns first, giving students some knowledge on a
figure which was useful when reasoning about the formula for the area of a trapezoid. The alternate
definition has parallelograrns as a special case of trapezoids, rendering parallelograrns much less useful in
understanding trapezoids.

68

a base equal to the sum of the trapezoid’s two bases (b1 + b2) and the height of the resulting

parallelogram was equal to half the height of the original trapezoid. Substituting these

values into the basic parallelogram area formula A = bh, we would get A = (b + b2) x %h

which is equivalent to the more standard form: A = %(b1 + b2)h.

I was intrigued with the idea of using a spatial representation to justify a known
area formula. I thought that it was a good way to show preservice teachers that geometric
figures and their manipulation were legitimate means of proving a mathematical
statement (here, an area formula) and that there were multiple ways in which a figure
could be manipulated to yield similar results. I knew that these students, as teachers,
would need to be open to different approaches presented by their students and so I
decided to present three other methods for reasoning about the trapezoid area formula. I
did this in the context of an assessment task (Figure 15), given after students had worked
with the trapezoid area formula and after they had been shown the first justification of the
area formula using spatial reasoning.

The three spatial representations in Figure 15 had not been discussed in class. I
asked students to choose one and explain how it could be used to justify the known
formula. All three of the methods were in some way related to the parallelogram and
triangle work we had done earlier and I expected students would use this prior knowledge

to give good mathematical justifications.

69

 

 

We learned in class that the formula for the area of a trapezoid is A = %(b1 + b2)h.

 

Each of the three diagrams below is a different verification of the formula for the

area of a trapezoid. Choose one (by circling it) and explain in detail how that

picture shows that the area of the original trapezoid is fi-(b; + b2)h.

 

 

 

 

 

Figure 15: Trapezoid Quiz Task
A majority of the students chose the first spatial representation in the task
(reproduced in Figure 16) to write about. Students recalled that we had used two copies
of a triangle to create a parallelogram and they followed that same reasoning with the

trapezoid. Thus, parallelogram RKLN had an area double that of trapezoid JKLM. They

70

knew that the area of RKLN was (b1 + b2)h, so they reasoned that the area of JKLM was

%(br + b2)h-

 

Figure 16: Trapezoid Method B
Students who chose the second representation in the task (Figure 17) saw that the
trapezoid was divided into two triangles and that the necessary dimensions of these
triangles (base and height) corresponded to known dimensions on the trapezoid. They
reasoned that since the area of each triangle could be represented symbolically as one-half

of its base length multiplied by the height, the area of the trapezoid could be written as the

sum of the areas of the two triangles, or % blh + % bzh. To make this representation look like

a trapezoid area formula, they then had to factor and rearrange factors, or as some put it,

“combine like terms.” Recognizing this, students were able to show that:

§b1h+§b2h=§mbl+b9

 

 

Figure 17: Trapezoid Method C
The students who chose to write about the third spatial representation in the task
(Figure 18), also used something they had learned about triangles, but with this diagram

they had to recognize that the trapezoid JKLM was dissected and rearranged as triangle

7]

RNL. They observed that the base of the triangle was the sum of the trapezoid’s bases
(b1+ b2) and that the height of RNL was the same as the height of JKLM. And with their

knowledge of the triangle area formula, students reasoned that the area of RNL (and thus

JKLM) was g-(b, + b2)h.

 

 

Figure 18: Trapezoid Method D

Area of a Trapezoid—Introduction of Mathematical Language Representation

As I planned for our work with trapezoids, I approached the task of making sense
of the symbolic rule for area only through spatial reasoning—primarily through various
forms of geometric manipulation. So when I introduced the trapezoid discussion by
asking students what they knew about the area of a trapezoid, I thought I would handle
any interpretation of the formula with an appropriate spatial representation. What I got,

however, was a statement that I was not at all prepared to hear, much less think about.

JW: ...What is the formula for the area of a trapezoid? Anybody
remember this? (Drawing picture and labeling it.) This is b1 this is
b2 and this is h. OK? Anybody remember the formula for the area
of a trapezoid? (Pause)

Suzanne: The average of b1 and b2 times the height.

’5

72

JW: (Repeating for class.) The average of b; and b2 times the height.

Another way to think about that... (Writing) Area equals % of b1

plus b2. .. the average of base 1 and base 2... and then times the

height. OK? And I want you think about why that might be true.

In my thinking, the i— in the formula represented something spatially. The figures for

Methods A, B, C, and D were firmly established in my mind and I was planning on having

a discussion on where the % comes from—the parallelogram with half the height of the

original trapezoid, for example. But in the context of the spatial representations that we had
been using in class, I struggled to make some sense of how this student’s interpretation
could be modeled with a drawing. In my attempt to transform the subject matter into
something that students could better understand, I had not considered the possibility that
another form of representation—mathematical language4—might be presented and used by
the students. And so I was unprepared to recognize it, much less handle its appearance in
class.

It was possible that Suzanne’s use of “average of the two bases” was a restatement
of a rule she remembered from a high school geometry teacher. Or she may have applied
some knowledge from her previous mathematics course for elementary education
maj ors—one with a focus on statistics and probability and in which the concept of

averaging was covered. It was also possible that Suzanne adapted a method that we had

 

4 My use of “mathematical language” is not meant to imply that the other representations do not involve
ways of communicating mathematical ideas. Here, mathematical language indicates the language (words)
that are used to communicate ideas in mathematics—the language of mathematics. I previously referred to
this as “natural language” to indicate that Suzanne‘s language employed words that were fairly common in
everyday usage. It became elem, however, that the words that Suzanne was using had mathematical
meanings that were far from “natural” (see Layzer, 1989).

73

discussed in class the previous week for estimating the area of an irregular shape on grid
paper-5

But at the time, I was focused on how a geometric manipulation like that in Method
A could justify Suzanne’s work, and I couldn’t immediately see how the concept of the
average of the two base lengths could be represented in this diagram. As time was running
short that day, I worked to steer the students back into thinking about a spatial
representation with manipulation, and I demonstrated Method A. I essentially forgot about

Suzanne’s method.

Reflecting on the Use of Mathematical Language, Spatial Manipulation, and
Symbolic Representations

While I was caught off guard by Suzanne’s statement, it wasn’t until the semester
was over and I went back over my course records that I began to think more about the
implications of her method. While I wish I could say that I immediately recognized how
it might influence the way I was thinking about representations, the reality was that I got
caught up in some other students’ concerns after class and I forgot about Suzanne’s
introduction of some mathematical language that did not fit with any of the methods I had
thought about.

My first impulse was to go back to her statement and think about whether a good

spatial representation of the average of the two bases exists. After some trial and error

 

5 With this method, students drew a number of horizontal and vertical cross sections on a shape and the
average of the horizontal lengths was multiplied by the average of the vertical lengths, giving the area of a
rectangle that approximates the area of the original figure. In the case of the trapezoid, there were two
different horizontal lengths indicated (the two bases) and one vertical length (the height), so the average of
the two bases would make some sense in estimating—but not necessarily finding an exact measurement
of—the area of the trapezoid.

74

and talking with some colleagues, I developed the diagram shown in Figure 19 and the

accompanying reasoning.

 

 

 

VM

Figure 19: Trapezoid Method E
Q and P are the midpoints of KL and JM respectively. Segment OP is the

midsegment6 of the trapezoid and by its definition, has a length that is the average of the

two bases, or % (b; + b2). Triangles QUL and PVM are rotated 180° to create the rectangle

S I U V, which has a base the same length as QP and a height the same as trapezoid
JKLM. The area of S I U V (and thus, JKLM) is % (b1 + b2)h.

After developing this spatial representation, I felt as if I had finally made some
sense of Suzanne’s statement in a way that was satisfying to me. While it was readily
apparent to me that the mathematical language that she was using—the “average of the
two bases multiplied by the height”—was mathematically equivalent to the symbolic
representation, I was not interested in recognizing her statement as a legitimate

representation until I developed an appropriate spatial manipulation to accompany it.

Reflecting on Mathematical Assumptions
In thinking through this reasoning, however, I realized that a number of
mathematical assrunptions are embedded in it. In fact, my subsequent analysis of the

spatial representations for Methods A, B, C, and D shows that each method carries with it

 

6 Some sources refer to the midsegment as the median line.

75

a set of mathematical ideas that would have to be taken for granted if the figures are to be
accepted as geometric proofs of the formula. These are not ideas that I had originally
considered in presenting these representations, but they grew out of my continued
reflection on the methods and my wondering what it would really take to understand the
spatial manipulations involved. By investigating these underlying assumptions, I hoped to
push myself to reason about my own mathematical knowledge and what it might take to
transform that understanding into something I could use in teaching preservice teachers.

I begin by looking at a few general assumptions that are common to all of the
methods and then look more carefully at the mathematical assumptions that are specific
to each of the methods.

Dissecting and Rearranging, The first assumption comes from this underlying
principle—that the area of a shape stays constant when it is cut into pieces and those
pieces are moved around. From a psychological perspective, this is Piaget’s conservation
of area, an indicator of the concrete operational stage of child development (Piaget,
1952/1963; Piaget & Inhelder, 1969). While this is not a concern for preservice teachers
as students, it is a relevant concern for them to consider as future teachers. They would
need to be aware that not every child in the elementary grades would consider this a
plausible approach—that some of the original figure’s area may be lost when the shape is
subdivided.

There is another underlying issue that comes up in most every case of dissection
and rearranging—whether the pieces actually fit together as nicely as the drawing
suggests. Mathematical issues such as parallelism, congruency, and similarity are

embedded within this issue and they are detailed in the discussion of the five methods

76

which follows. This mathematics involved in looking at how pieces fit together when
they are rearranged is far from trivial. A classic puzzle first credited to W. W. Rouse Ball
in his Mathematical Recreations and Essays but popularized by Sam Lloyd in his
Cyclopedia of 5 000 Puzzles illustrates an inherent problem to this issue (Gardner, 1956;
De Villiers, 1998). The problem involves taking a square that has been cut into four

pieces and rearranging them into a rectangle (Figure 20).

 

Figure 20: Dissection Puzzle

In looking at these two figures, it would appear that since the two shapes are made
up of the same four pieces, they would have the same area, but the square has area 64 and
the rectangle has area 65! How could the pieces of one shape be rearranged to create
another shape with a different area?

This is a case where a visual observation leads to an incorrect mathematical
statement. The assumption is that the four pieces of the square actually do fit together as
neatly as they appear, when in actuality, a small gap exists between the pieces when they
are used to create the rectangle. This gap has an area of one, thus enabling the four pieces
(plus the gap) to have an area of 65.

Preservice teachers made many similar assumptions about the ways pieces fit
together and. created new figures. In each of these cases, though, their assumptions were

correct—although neither I nor the students accounted for the what it would take to prove

77

their assumptions true. My investigation of Method B during the analysis of the data first
raised this issue of mathematical assumptions, so I present an investigation of that
method first.

Method B. The first issue concerns the midpoints Q and P and whether the
segment that connects them is parallel to the bases JK and LM—an important factor in
establishing the fact that the resulting figure is truly a parallelogram. To accept this,
students need to have an understanding about ratios of segment lengths that are formed
by parallel lines. Specifically, we can start by working backwards through the problem.

First, we take Q as the midpoint of KL. The must be a line through Q that is
parallel to KJ and LM. This line will intersect JM at some point (say, point A). Since we
have three parallel lines (KJ, QA, and LM) with two transversals (KL and JM), the ratio
of KQ to QL must be the same as the ratio of JA to AM. Since Q is the midpoint of KL,
that ratio is 1:1, which means that the ratio of JA to AM is also 1:1, indicating that A is
the midpoint of JM. A is therefore concurrent to the existing point P, so QA and QP are
the same line segment. QP is therefore parallel to JK and LM.

The next assumption that must be addressed in this dissection and rearrangement
method is that the midsegment QP is indeed the average of the two bases KJ and LM.
The definition of a midsegment addresses this concern, but not everyone is familiar with
this definition. To understand this idea, students would need to observe that when
perpendiculars are dropped from J, K, P, and Q, a number of rectangles are formed
(Figure 21). These rectangles are very helpful in setting up a set of equations which,
when combined, lead to the desired result.

(1)QP=KJ+UW+ZV.

78

Then, it must be noted that AKQX z- AQLU, giving LU = QX = UW. A similar
argument is made to show VM = YP = ZV. From these two statements, it is shown that U

is the midpoint of LW and V is the midpoint of ZM. So:
(2)UW=%LWandZV=%ZM.
Combining (1) and the equations in (2), you get
QP=KJ+§LW+§ZM
QP=%[KJ+KJ] +§LW+§zM
__ r 1 1 r
QP— 5KJ+3KJ+5LW+EZM
QP=§KJ+-2‘-[KJ+LW+ZM]
_1 r
QP—TKJ+-2-LM
QP=-;-[KJ+LM]

The midsegment QP is thus shown to be the average of the two bases KJ and LM.

TK b2 JS

 

l. U W b, 2 V M
Figure 21: Proof of Trapezoid Method E
All that remains to be shown is whether trapezoid JKLM and rectangle STUV
have equivalent areas. This would be true if triangle LUQ could be cut out and moved to
fill in the space of triangle KTQ. Consider the fact that JK and LM are parallel. With KL
as a transversal, AQLU E AQKT making AQKT and AQKJ supplementary angles. And

since Q is the midpoint of KL, KQ E QL and the right triangles KTQ and LUQ are

79

congruent. Similar reasoning can be employed to prove that triangles J SP and MVP are
also congruent and so trapezoid JKLM and rectangle STUV have equivalent areas.
Method A. As with Method E, it is important to establish that the midsegment QP
is parallel to the bases JK and LM, but there are several additional assumptions that need
to be addressed. The first is that in drawing the midsegment QP, the height of PQLM is
half the height of JKLM. This is simply another case of the argument that was previously
made. If a perpendicular is dropped from J to intersect LM at Z and QP at Y (Figure 22),
.12 can also be thought of as a transversal through parallel lines KJ, QP, and LM. The
ratio of JY to YZ is also 1:1, so the height of PQLM (which is YZ) is halfthe height of

JKLM (which is JZ).

 

 

Figure 22: Proof of Trapezoid Method A

The final assumption is that when JKQP is rotated to MN OP, the pieces actually
form a parallelogram. The easy part of this is observing that since P is the midpoint of
JM, JP = MP and so one piece is not longer than the other when placed together. The
more detailed piece is seeing that OQ and LN are indeed straight line segments. Since
vertical angles AJPQ and AMPO are congruent and since AJPQ and AQPM are
supplementary, then [MPG and AQPM are also supplementary, indicating that OQ is a
straight line segment. To show that LN is a straight line segment, students need to

observe that with JM as a transversal to JK and ML, the adjacent interior angles AKJP

80

and APML are supplementary. Thus when JKQP is placed on MNOP, APML and
[NMP are also supplementary.

We have shown that Q0 and LN are straight line segments and that they are
parallel. To show that OQLN is a parallelogram, all that remains to be shown is that QL
and ON are parallel. Consider KL as the transversal to JK and LN. AJKQ and AQLM are
supplementary. When JKQP is rotated to MNOP, K coincides with N, so AONP and
AQLM are supplementary. This shows that LN is a transversal to parallel line segments
QL and ON.

Method B. As with Method A, showing that final figure (here it’s RKLN) is
indeed a' parallelogram is critical. RK and LN can be shown to be a straight line segments
by noting that AJML 5 AMJR and that AJML and AMJK are supplementary. Therefore,
AJML and [NM] are supplementary (as are AMJK and ARJM). Thus RK and LN are
straight line segments and are also parallel. Using an argument similar to the one
described in Method A above, it can easily be shown that KL and NR are also parallel,
making RKLN a parallelogram.

Method C. By drawing the diagonal KM and subdividing JKLM into two
triangles, the only geometrical assumption that needs to be made is that triangles KLM
and MJK have the same height. But since JK and LM are parallel, the height remains

perpendicular to each of the bases, even when it falls outside of the triangle MJK.

Seeing that the symbolic representation i— blh + i—bzh is equivalent to the form
%(b1 + b2)h requires students to understand some symbol manipulation, factoring the

original expression as % h(b1 + b2) and rearranging the factors.

81

Method D. This dissection and rearrangement method requires students to
understand two things—that the pieces fit together as neatly as they appear to make a
triangle, and that the height of the triangle is the same as the height of the original
trapezoid. To show that RLN is a triangle, it must be understood that the midpoint P
creates two congruent segments, JP and MP. Then to show that LN is a straight line
segment, students need to observe that with JM as a transversal to JK and ML, the
adjacent interior angles AKJP and APML are supplementary. Thus when JKLP is placed
on MNRP, APML and ANMP are also supplementary. And since vertical angles AJPL and
AMPR are congruent and since AJPL and ALPM are supplementary, then AMPL and
ARPM are also supplementary, indicating that RL is a straight line segment.

The simplest way to show that the height of LNR is equal to the height of JKLM

is to observe that R, J, and K are collinear and so the height remains constant.

Comparing Representations through Shared Reflection

With these analyses and a developing appreciation for mathematical language as
an additional representation, I decided to investigate the ways in which mathematical
language might also be used with the other methods for understanding trapezoid area
(Table 4). Not surprisingly, the other four methods didn’t seem to lend themselves as
nicely to mathematical language representations. In fact, the mathematical language
representations that I came up with were either literal translations of the symbolic
representations (e.g., “Half of one base multiplied by the height plus half of the other
base multiplied by the height” for Method C) or a description of the spatial reasoning

(e.g., “The sum of the areas of two triangles”).

82

 

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83

The mathematical language of Method E that Suzanne shared with the class was
the one example of language representation that students used regularly. It is
fimdamentally different fi'om the others in that it employs some language (“average
of. . .”) that is neither geometric in origin nor in use. It is a case where mathematical
language helps to explain the accompanying spatial reasoning, instead of describing it

Preservice teachers were most at case using the mathematical language of Method
B in remembering and using a way to understand the area of a trapezoid. In subsequent
class sessions, the phrase “the average of the two bases multiplied by the height” was
almost exclusively used by students—the “one-half” that I had been so concerned about
helping them to understand in a geometric sense was folded into the averaging of two
lengths. Students seemed to like the mnemonics of the representation and no one was
bothered by the fact that we had not come up with an appropriate spatial representation to
describe it.

But the fact that this mathematical language emerged as the representation of
choice does not indicate that it should be presented as the sole method for preservice
teachers’ understanding of trapezoid area. In fact, it underscores the importance of
helping them to develop multiple representations for understanding. Shulman wrote,
“Since there are no single most powerful forms of representation, the teacher must have
at hand a veritable armamentarimn of alternative forms of representation, some of which
derive from research whereas others originate in the wisdom of practice” (Shuhnan,
1986, p. 9). I was originally focused on helping preservice teachers justify symbolic
representations with multiple. forms of spatial reasonings, and I was surprised to find that

many students were more inclined to focus on a mathematical language representation.

84

By the same token, these students—as future teachers—need to be prepared to think
beyond the use of the one method or representation with which they may be most
comfortable, in meeting the varying needs and levels of understanding of their future
students. Sharing this experience with preservice teachers is one way in which a teacher
educator’s attention to his or her own pedagogical content knowledge growth can
contribute to the developing knowledge bases of his or her students. In this case, this
reflection occurred in retrospect and so was not as enlightening for these students. But I
envision using this experience as a central piece of a future course that I or others may

teach for teacher educators—one with attention to shared reflections.

The Perceptual Proof Scheme of Dissecting and Rearranging

When preparing for MTH 202, I did not emphasize students’ use of justifications.
From my past experience in teaching preservice teachers, I knew that many come into
MTH 202 with a dislike of geometry—often stemming from their high school
experiences in geometry classes where proofs were emphasized. In the attitude survey
that was given on the first day of class, I asked students: “What is geometry? What comes

to your mind when you hear the word ‘geometry’?” One student wrote:

When I hear the word geometry I think of shapes and measurements. I also think of my high
school geometry class and doing endless amounts of proofs!

Another student wrote:

A bunch of lines and angles where we have to find measures. Also, proofs and theorems to be
defined which I do not like.

While many students mentioned the importance of shape and measurement in
geometry, most of the students included some reference to proofs, theorems, or

postulates—and these were rarely mentioned in a very positive way. While I thought that

85

justifications were important to consider when making geometric hypotheses, I felt it was
more important that students view geometry as more than a study of proofs and theorems,
and that the study of geometry can be less daunting than they may remember from high
school. This was my main purpose for de-emphasizing proofs in MTH 202.

The first (and only) reference in my teaching journal concerning proofs comes
when I was thinking about what content I wanted to teach during the last week of the

course. I wrote:

The only other thing that I had been thinking about is the notion of proof. It’s an issue that I have
not really addressed with my students, yet it’s one that is getting a fair amount of attention right
now, especially in conjunction with the ways in which the [Principals and Standards for School
Mathematics] is looking at proof and reasoning together across the grade levels. I know that we
have a nice geometric proof in Looking for Pythagoras and that there are some good algebraic
proofs that make a lot of sense with respect to using similar triangles, so I’m not sure how (or if) I
will work this in. If I only had another three or four weeks with them. .. (Personal Teaching
Journal, 4/20/99)

In my thinking, proof was a topic that would be covered—time permitting—at the
end of the course and not something that was important to think about in the context of
various mathematical situations during the course. And throughout my teaching journal, I
only mentioned justification as a synonym for proof, focusing on the mathematical
construct of a formal proof rather than as justifying the reasoning behind one’s
mathematical thinking.

But I did think about the spatial reasoning as a kind of proof of the symbolic
representation. Referring to my learning about the spatial reasoning of Method D, I

wrote:

Actually, this very idea of a new method for me was what had pushed me to consider making the
topic of formulas and geometric justifications an emphasis of my course this semester. (Personal
Teaching Journal, 1/29/99)

In this way, I was thinking of the dissection and rearrangement method as an

example of an empirical proof scheme. In these schemes, justification is based solely on

86

the use of examples. More specifically, since the examples were of spatial reasoning, I
was relying on a ideas that were highly perceptual. In the perceptual proof scheme,
conclusions are often based on one or several drawings that are used to argue a
mathematical hypothesis. When teaching MTH 202, I was bound by the idea that there
had to be a geometric representation (typically using some form of dissection and
rearrangement) of a formula. I first classified the five methods of understanding the
trapezoid area formula in Table 1 by the spatial representations—thinking that the
accompanying figures were the most important representation.

In doing so, I was also giving students the idea that a perceptual proof scheme
was a sufficient way to justify mathematical thinking. Students were not challenged to
consider the possibility that the perceptions implied by the drawings were
erroneous—that the pieces might not fit together as neatly as they appear. In retrospect,
there were a number of places in the course where this issue could have been raised—like
asking students how they knew that two copies of a trapezoid would make a
parallelogram or even whether each of the methods would work on different trapezoids. I
have little doubt that preservice teachers need more experiences developing mathematical
justifications for observations and hypotheses—to enrich and deepen their own
mathematical understanding as well as to understand what it may take in helping kids do
the same. However, I decided before the course and throughout the semester that proof

and justification would not be a critical element of what I was teaching.

87

Episode 2: Circle Area

Representations and Their Limitations

When investigating the area of a circle, the representations used have an impact
on the ways in which students might make sense of the problem. In addition, each
representation bring with it different mathematical assumptions that must be addressed
within the context of each method. But not all representations convey the mathematical
ideas they symbolize in the same way nor with the same potential for providing access to
understanding. And some representations even carry some inherent problems to
understanding the underlying mathematics. Preservice teachers need to understand how
these representations relate to one another and be aware of the potential difficulties that

are inherent with each one (Wilson et al., 1987).

Both of the methods that we used for understanding the formula for the area of a
circle (Table 5) bring new problems into the conversation and reveal some of the
limitations of relying on the accompanying representations. The first method requires
students to make a mathematical assumption that, while seemingly obvious, truly requires
an understanding of integral calculus for its complete justification. Mathematical
language plays an important role in the less commonly used second method that the
students and I investigated in class. But the introduction of this second method also raises

an important, and somewhat problematic idea—that of mathematical precision.

88

Table 5: Methods for Understanding Area of a Circle

 

 

 

 

Mathematical language Symbolic Spatial Reasoning
A An “almost rectangle” with an area A = l C r
equivalent to that of the original circle. 2
B Pi radius squares. A = 1t 12

 

 

 

 

 

By comparing these two methods and looking at the limitations that are embedded
in each one, I found that another case for using multiple representations is in using the
strengths of one representation to help cover theweaknesses of another. By juxtaposing
the first method with the second, I found that preservice teachers were able to gain some

knowledge about the area of a circle that they may not have otherwise had.

Method A. The first method that we looked at is one that is typically used to help
middle school students understand the formula for the area of a circle. This method
employs the strategy of dissection and rearrangement that was used in many other area
examples in the class. With the Piagetian idea of conservation of area firmly in place, the
dissection and rearrangement strategy can be a powerful one for preservice teachers.
Lengths and angles from one geometric representation remain constant as they reappear
in another spatial representation, enabling students to use a rearrangement of pieces that

resembles another (already accepted) shape.

With a circle, though, the curved outside of the shape presents a problem. A circle

is typically the first shape with a curved edge presented to students and no amount of

89

 

dissecting and rearranging can transform the curves into the straight lines of triangles and
parallelograrns.

Historically, this conundrum caused problems for ancient peoples as well. Our
best guess as to how people first discovered the formula for the area of a circle is the
same way in which I first introduced the concept to my preservice teachers—using a
method of dissection and rearrangement and connecting the calculation of the area to that
of the circumference. The Babylonians and Chinese calculated the area of a circle using
the formula A = Cd/4 = (C/2)(d/2) where C is the circumference and d is the diameter, a
formula which was likely derived fi'om a dissection and rearrangement (Katz, 1993).
Japanese documents from the 1698 show this method being used and Leonardo da Vinci
himself also employed this idea in the 16th century (Beckmann, 1971).

With this method, a circle is out along any number of radii (Figure 23), and the

pieces are rearranged to create a figure that resembles an “almost rectangle”7 (Figure 24).

 

 

Figure 23: Circle Figure 24: Circle Sectors Rearranged
Divided into Sectors as an “Almost Rectangle”
The reasoning goes that as the circle is cut into more and more smaller sectors, the

curved edges of the resulting figure become less pronounced and appear more like a

 

7 I refer to this shape as an “almost rectangle” because as the size of the sectors becomes smaller, the shape
looks more like a rectangle. Also, it’s important to think of the limiting case 3 a rectangle rather than a
parallelogram because only then is the height of the figure equal to the radius of the circle.

90

straight line, creating a figure that is more rectangular in each iteration. After a while,
those curves become so small that the resulting figure might as well be taken as a
rectangle and the rectangle area formula (A = base x height) can be used. Here, however,
the base length of the reCtangle can be thought of as half of the original circle’s

circumference and the height of the rectangle is equal to the radius of the original circle,
yielding the formula A = i— C r.8 Once the constant, 1:, was accepted, this gave us a

formula for determining the circumference of a circle, C = 1t (1 or C = 2 1r r. Substituting

this last expression into the circle area formula above, we got A = %(2 1: r) r and after

some simplification, A = 1: r2.

This proved to be a nice way for students to make some sense of how the formula
could be derived from a geometric representation of the problem. However, it required at
least one leap of faith. I have always been bothered by the mathematical hand-waving
that must be done in assuming that arranging smaller and smaller sectors of the circle will
result in a figure that, for all intents and purposes, is a rectangle. Granted, the students
were never bothered much by it for it seemed to make sense to them that this could be
true. And since the resulting formula was the one that students already knew for circle

area, then this seemed to be an ideal time to let the ends justify the means.9

With this method, the symbolic representation appears to be a powerful one. For

students who have dissected and rearranged pieces to make sense of parallelogram,

 

' Prior to the class session in which this method was presented, we had investigated the circumference of
circles and had looked at the development of 1: as the ratio of a circle’s circumference to its diameter, or

n=§.

9 In actuality, some higher level mathematics can easily simplify the problem, employing integral calculus
and the theory of limits to the example. But this doesn’t typically help those preservice elementary
teachers who will never study calculus, and it certainly did not help da Vinci whose work predated the
discovery of integral calculus by about a century.

91

triangle, and trapezoid areas, this strategy makes a great deal of sense. But unlike the
polygons, the circle presents a unique twist as the resulting figure does not appear to be a
true rectangle. The curved edge of the circle forces students to take an additional step in
reasoning about how this method might be useful—noticing that with a greater number of
sectors, the resulting shape looks more like a rectangle and assuming that it will
eventually be one. For the first time, students are confronted with the possibility that the
dissection and rearrangement method cannot completely solve the problem.

Method B. The second method for looking at the area of a circle comes from the
Connected Mathematics Project (CMP), a middle school mathematics curriculum that
was developed at Michigan State from 1991-1997. Having been a part of the writing of
many CMP units and having had the opportunity to talk with the authors of the
curriculum as well as with a number of teachers who have used the materials, I found that
many of the middle school ideas which are presented in CMP are extremely appropriate
for preservice elementary teachers. Not only do they give us the opportunity to expand on
the preservice teachers’ knowledge base, but they provide us with a forum to discuss a
number of pedagogical issues and a set of problems to access the pedagogical content
knowledge that I am trying to address.

The sixth grade CMP unit that investigates the ideas of perimeter and area is
called Covering and Surrounding. In the final set of problems in Covering and
Surrounding, middle school students are exposed to several big ideas around circles—the
relationship between the circumference and the diameter of any circle as well as ways to
estimate and calculate the area of a circle. The work with circumference results in the

students’ seeing that the ratio of a circle’s circumference to its diameter is a constant

92

(although, due to measurement error, this constant ratio may not initially be apparent) and
that this constant ratio is a number with a value a little greater than three. For middle
school students, this is often the first time that they are exposed to the irrational number
it, or even the notion that numbers exist that cannot be expressed as the ratio of two
integers. The topic of irrationality is not intended to be investigated at this point in the
curriculum and the subject is only mentioned in describing the existence of TC. Preservice
teachers, however, already have some knowledge of it and irrational numbers, so I chose
to use these problems more as a review and an example of a way that a concept can be
introduced in a more investigative way—where students learn through doing and not just
rote memorization.

After reviewing the idea of rt and methods for estimating areas of shapes drawn
on grid paper, we moved to the problem of making sense of the area formula for circles.
The question was embedded in an investigation of the relationship between two measures
of the circle, just as the circumference and diameter lengths had been compared. Students
were given a number of circles drawn on grid paper. Each circle included a square drawn

on the radius of the circle (Figure 25). This square was referred to as the radius square.

 

Figure 25: Circle with a Radius Square
Students were asked to find a relationship between the area of a circle’s radius

square and the (estimated) area of the circle. In Covering and Surrounding, this question

93

is posed using language that is consistent with the language of the unit, instructing
students to “find out how many radius squares it takes to cover each circle.”

At this point, I observed a variety of techniques being used. Some students
focused on the language of the question and immediately began covering the circle with
radius squares (Figure 26). This helped us to establish an upper limit of the relationship,
that four radius squares could completely cover the circle, but that there was a fair amount

of material left over.

 

Figure 26: Circle with Four Radius Squares
Other students started dissecting and rearranging radius squares, covering the
circle with three radius squares as best as possible (one way is shown in Figure 27) and
then cutting the leftover pieces apart and fitting them into the remaining space. This
method typically yielded the intriguingly familiar statement that the area of the circle was

“a little more than three” radius squares.

 

Figure 27 : Circle with Three Radius Squares

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And other students stuck with their initial numerical estimates of the areas of the
circles and compared them with the numerical values for the areas of the corresponding
radius squares. Students employed a variety of techniques for estimating the circle areas
and these area estimates varied, but they were often within several square units of each
other. These students then gave answers that represented the ratio between the circles’
(estimated) areas and their corresponding radius square’s area, with answers such as 3.1,
3.15, and 3.25.

After all of the students had a chance to work on the problem, I brought the class
back together and asked for different students to share their findings. Students began to
notice that most of the answers that they came up with for the number of radius squares it
would take to cover a circle was “a little more than three” or actual numbers a little
greater than 3. I asked where else we had already looked at answers similar to these, and
they referred to the work we had done with relating a circle’s circumference to its
diameter.

At this point I led the students to take that “leap of faith.” I pointed out that it
would be quite a coincidence that two different problems involving circles would lead us
to two different numbers, each one “a little more than three.” I went on to state that, in
fact, this ratio of the circle’s area to the number of radius squares needed to cover the
circle is exactly the same ratio as the circle’s circumference to its diameter—the irrational
number it. I went on to point out how helpful this representation of the area of the circle
is in looking at the corresponding formula, for the circle can be thought of as having the
same area as “pi radius squares” which gives way to A = 1r r2. Here, the mathematical

language of the problem fits with the symbolic representation quite nicely, with the

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notation for square matching with the geometric picture of the square. The three
representations—mathematical language, symbolic, and spatial—coalesce in an almost
ideal example of how one representation connects to another.

The method itself can be open to criticism for its apparent lack of mathematical
rigor. There is an assumption that, because the work is being done with circles (which
had previously resulted in an investigation on the number 1:), the students’ answers for
the number of radius squares it takes to cover a circle must also be approximations for 1:.
Students can argue, however, that their answers may actually be approximations for
another number—a number close (but not exactly equal) to 1:. With middle school
students, this method alone may be convincing proof that a circle can be covered with “pi
radius squares,” but for preservice teachers, it is important to understand that there is a
level of mathematical rigor that is being glossed over with this method, no matter how

helpful the mnemonics of the representations may be.

Reflecting and Shared Reflection on Multiple Representations

In fact, this investigation is helpful in looking at the ways in which these
representations provide substance for pedagogical content knowledge. In talking with
other instructors of MTH 202, I found that the example using radius squares to cover a
circle (Method B) was not used to show why the formula A = 1: r2 makes sense for the
area of a circle—in fact, most instructors didn’t give any attention to understanding the
formula at all. In the course that I developed, representations and understanding how they
can be used to make sense of formulas became a central focus of the class. In the case of

circles, there are some mathematical assumptions being made about how precise one

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must be to use this model for understanding the circle area formula. But in a course
where pedagogical content knowledge is valued and given some attention, the
investigation with radius squares—when paired with the previous one using sectors of a
circle—can help to move preservice teachers into thinking about ways to help their
students make better sense of the mathematical ideas and to move between the
representations.

After both methods were introduced and discussed, I talked to the students about
my reasons for presenting both methods. I explained that for the purposes of the class, I
felt that neither one was complete enough to stand alone—that as I was thinking about
the models, I realized that the representations used in each model had their limitations. I
asked the students to explain what they liked and disliked about using each model and
what each model helped them to understand about circle area.

About Method A, students explained that they liked the idea of cutting a shape
apart and fitting the pieces together to make a new shape that is like one that was already

studied. Several also described how using algebraic manipulation was

satisfying—transforming what looked like a new formula for the area of a circle (A = i C

r) into a known formula (A = 1: r2) using some simple algebraic substitutions. They liked
using strategies that had been utilized with other area problems, making the circle
investigation more similar to, than different from, the ones done with other shapes. But
students also wondered how elementary school students would be able to make sense of
this method, since the symbol manipulation would not be a tool that elementary students
would have at their disposal. Several students also expressed some concern about how

cutting the circle into only a few sectors would make a shape that had noticeably curved

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bases but none expressly described any hesitation to accept that a larger number of circle
sectors would actually create a rectangle.

With Method B, students described the use of radius squares as “interesting,”
“totally new,” and “something [they] had never thought about before.” A number of
students liked seeing a method that can be used with middle school and upper elementary
school students. Students also explained that they liked how this method reinforces the
verbalization of the formula (“pi radius squares” for “pi times the radius squared”) and
how this method helps to explain the use of “square units in covering something with a
curved edge.” However, a number of students explained that it was difficult to accept
using approximately 3.14 radius squares to cover the circle when their work resulted in a
different number.

By centering the discussion around the representations instead of just the
outcomes, I wanted students to be more aware of the power behind using different and
multiple representations and to be thinking about the ways in which these representations

can have an impact on what they contribute to other understanding.

Perceptual and Ritual Proof Schemes

In many ways, the first method for understanding circle area used in MTH 202
was highly perceptual—it involved dissection and rearrangement of pieces to make a
figure look like a figure with a known area. It also implied that subsequent iterations (i.e.,
more sectors) would produce a final figure that more closely resembled a rectangle,

leaving students to reason about the argument based on only a small number of examples.

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But since it required students to accept that the rearranged pieces form an “almost
rectangle,” my use of the method also constituted the use of a ritual proof scheme.

A ritual proof scheme is one in which students judge “the correctness of an
argument solely by the form of the argument rather than by the correctness of the
reasoning involved” (Martin & Hare], 1989). A typical example of a ritual proof scheme
is a students’ reliance—based on prior experience in a traditional geometry class——on the
two-column format as the only acceptable method of proof. To these students,
mathematical truth lies in the structure that is used in making the argument, rather than in
the reasoning itself. With Method A for understanding the area of the circle, students
were likely led to believe in the results because the method of subdividing and
rearranging pieces was established in earlier examples (parallelograrns, triangles, and
trapezoids), and so they were willing to accept the method as justification for the results.

My use of Method B does not follow a proof scheme. In that method, I did not ask
students to accept the process as absolute mathematical truth, but that the end-product
(seeing the area of a circle as “approximately pi radius squares” and relating it to the area
formula, A = 1: r2) was the important element. My plans in introducing students to this
method centered around their using it more as a mnemonic device for remembering the
formula rather than for understanding the irrational number n and for making sense of the
method. I placed the emphasis on providing students with multiple representations for the
same idea, and not on trying to validate one method with the other. In my mind, I viewed
Method A as the justification of the area formula and Method B more as a spatial

representation of the symbols that are used.

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Episode 3: Volume of Prisms, Cylinders, Cones, and Spheres
Background—Understanding Layers

After completing some additional work in two-dimensional geometry, I decided
to push on some of the concepts that we developed in 2-space (like relationships between
perimeter and area) and see how students could apply these ideas to 3-space
(investigating surface area and volume relationships).'The titles of the Connected
Mathematics Project units that we used (first Covering and Surrounding and then Filling
and Wrapping) were helpful in signaling this shift as well as in attaching some
mathematical language to the important mathematical ideas.

In previous times that I taught MTH 202, I spent very little time on the idea of
surface area—thinking that it was less important (especially in understanding some of the
more common formulas used in solid geometry) and easier to understand. However, I
learned that preservice teachers had a fair amount of difficulty in visualizing how various
flat patterns (or nets) could be folded to wrap a shape and that these flat patterns were
helpful in understanding basic formulas for surface area of rectangular prisms. So this
semester, I included several investigations on working with flat patterns—visualizing the
resulting solid that would be created and using the flat pattern to determine the surface
area as well as seeing how to generalize with a formula. This worked very well with the
students this semester, and it helped them to get a better handle on the concept of surface
area without spending too much time on it. But with volumes of prisms, I have always
used some of the same approaches with preservice teachers that the CMP curriculum has

found to be powerful for middle school students—particularly the notion of layers.

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When moving from flat figures to solids, it’s helpful to fall back upon the Med-
and-true method of counting. Kids find counting reassuring, and I have found that
preservice teachers do as well. Just as we had first counted squares to determine areas of
rectangles, students began by counting cubes to determine the volume of solids. When I
asked them to think about ways in which kids might start to find shortcuts for counting a
large number of cubes in a rectangular solid, students suggested that they could count the
number of cubes in one layer of the shape and multiply by the number of layers. For
example, students found that the box that measured 3 cm by 2 cm by 5 cm had 6 cubes in

the bottom layer and 5 layers of cubes, for 6 x 5 or 30 cubes (Figure 28).

 

5cm

 

 

 

 

 

2cm
30m

Figure 28: Box with Layers of Cubes
With a rectangular prism, this naturally leads to a generalization that the number
of cubes in each layer can be found by multiplying the two dimensions of the base and
that the volume can be determined by multiplying the three dimensions of the solid, or
V = l x w x h. This is a formula with which most preservice teachers have some

familiarity and as such, they are eager to encounter it and use it. However, it is common

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to see some students over-generalize and try to use this formula to calculate the volume
of any solid (especially ones in which three different dimensions are given).10

I found that more preservice teachers were able to avoid this misunderstanding
when the concept of layers was used and that language was applied to the situation. And
instead of moving to the well-known volume formula, V = l x w x h (which is only useful
with some prisms), it was more powerful to use the concept of layers which can be
applied to all prisms. By using layering, students develop a firmer understanding of
volume as the number of cubes in one layer (typically thought of as the bottom layer or
the base of the solid) multiplied by the number of layers (or the height of the solid). With
this idea, the result is a slightly different, yet more generalized formula for volume:
V = area of base x height.11

The power behind this formula and its development is that it can be applied to any
prism and it makes sense to students why this would be true. It also makes for a nice
application of—and connection to—areas of polygons, and it provides a useful tool in
reasoning about cylinder volumes as well. But before I get to the cylinders and how
representations provided some interesting avenues for discoveries about spheres and
cones with preservice teachers, I first turn to the investigation of volumes of prisms with
various bases as an example of my own initial comprehension of knowledge, the
beginning of its transformation, and how this was a powerful example of shared

reflection for preservice teachers.

 

’0 Similarly, students who have little understanding of how formulas can be justified with spatial reasoning
often use the formula for the area of a rectangle when determining the areas of triangles, trapezoids, and
other shapes.

" As it is the accepted notation in mathematics, this came to be represented as V = B x h where the capital B
stands for the area of the base (instead of a lower-case b which is used for the length of a side labeled as a
base of a parallelogram).

102

Areas of Regular Polygons—My Comprehension of Subject Matter

In our class work on perimeter and area, we constantly came back to the ideas of
how perimeter and area are related. In particular, questions kept coming up about whether
a shape with a greater perimeter necessarily indicated a shape with a greater area and
students wrestled with these ideas throughout the course. They quickly reasoned that in
the case of rectangles with a fixed perimeter, many possible areas could result. ’2 Students
eventually discovered that the rectangle with a maximum area and a fixed perimeter is a
square.

But I also wanted them to think about what shape would have a maximum area
for a given perimeter if the rectangle constraint were removed from the task—a problem
that was not investigated when we were working in two dimensions, so I decided to
revisit the issue when working with cylinders.

A problem in Filling and Wrapping set the stage for this investigation quite
nicely. As part of a larger problem, students were asked to find the volumes of triangular,
square, pentagonal, and hexagonal prisms formed by taking identical sheets of paper and
folding them into 3, 4, 5, or 6 equal-sized rectangles that would represent the lateral faces

of the prisms (two examples are shown in Figure 29).

Aver-1.1:."
A U u r

I

C

I

 

 

 

------1
.-.---
------

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 29: Construction of Triangular and Pentagonal Prisms

 

'2 For example, rectangles with a fixed perimeter of 12 and whole number lengths include a 1 x 5, a 2 x 4, and
a 3 x 3, with areas 5, 8, and 9 respectively.

103

Students were given this task to work on in class and results were then compiled

for the entire group. My primary goal had been to set up the situation where students

would see that as the number of lateral sides increased, the volume would also increase.

And since the height of the prisms is constant, the problem is actually analogous to the

problem of keeping a perimeter of a polygon constant (represented as the base) and

maximizing the area.

It turned out that the preservice teachers in my class had very few problems with

this concept. Some students quickly reasoned that since the prism bases were looking

more and more like a circle, then for a fixed lateral surface area, a cylinder (a “circular

prism”) would have the greatest volume. From our previous work, these students had

possibly realized that a circle would be the maximal two-dimensional shape for a given

perimeter and they were ready to apply that knowledge to this new situation. Other

students might not have realized this fact prior to this investigation but were able to work

back to that conclusion based on their experiences with the cylinders. Either way, it was

an important lesson for the class—either in validating a prior belief with an extension

problem or in tying together a number of previously-seen but disconnected ideas.

But what caused a great deal of concern for the class was the fact that there was

little agreement on what the volumes for the various solids should be (Table 6).

Table 6: Student Responses to Prism Volume Problem

 

 

 

Triangular Prism Square Prism Pentagpnal Prism Heggonal Prism
817 cm3 1058.4 cm3 1242 cm? 1144.8 cm3
825 cm3 1053.5 cm3 933.12 cm3 1196.5 cm3
823 cm3 1043.6 cm3 2364 cm3
820 cm3 1053.5 cm3 906 cm3
940.5 cm3 1043.6 cm3

756 cm3 1078 cm3

1051 cm3

 

 

 

 

104

 

Students were encouraged to measure, estimate, and/or calculate the volumes of
their prisms in any way they could. Some students took precise measurements of their
prisms and did the calculations with a high level of accuracy. Other students did a fair
amount of estimation—tracing the base of each prism on grid paper and counting the
number of square units in the bottom layer as a part of their calculations. One student
suggested subdividing an unknown base shape into other shapes with areas that could be
calculated—drawing one diagonal in the pentagon and turning it into a triangle and a
trapezoid. Some students worked with what they could recall fi'om their previous

mathematics experiences to figure out what the “right” answers were supposed to be.

Since all students were working with standard sheets of 8% in by 11 in paper, I thought

that their answers would have less variance, but something was obviously not working
right. The class session ended with a discussion about maximizing volume and area, but
the students and I were troubled by the messiness of their answers (when Cara shared her
answer of 933.12 cm3 for the volume of the pentagonal prism, she saw how it was smaller
than any of the square prism volumes and added “but it’s probably off”).

As part of my preparation for the next class, I decided that I would take a look at
the problem again myself. I decided that for ease of calculation, I would convert the
measurements fi'om inches to centimeters (using the conversion 2.54 em = 1 inch and
rounding to the nearest tenth, making the original paper 21.6 cm by 27.9 cm). The square
base was the easiest to calculate, giving an approximate volume of 1051 cu cm. Next, I
tackled the triangular base (Figure 30). The triangle has a base of 9.3 cm. To determine

the height, I recognized the fact that the small triangle (AABD) is a 30-60—90 triangle, and

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therefore the ratio of the long leg to the short leg is J3 :1 , giving a height of about 8.1

cm. Using this, the volume of the triangular prism would be approximately 814 cu cm.

B

    

9.3 cm

 

A D C
Figure 30: Base of Triangular Prism
Having already tackled the equilateral triangle, I decided to move on to the
hexagon, recognizing that some of my equilateral triangle work might be applicable to
this problem as well. First I determined that one side of the hexagon would have a length
of 4.65 cm (I decided to leave the extra decimal place in since it was exactly half of the
equilateral triangle’s side length). I drew in the diagonal BE (see Figure 31) and
recognized that this creates two congruent “special isosceles trapezoids” (see page 125 in
Chapter 4), each of which is made up of three equilateral triangles. Since one of these
triangles has a side that is half as long as the triangle in Figure 30, the height of one of
these triangles is also half the height of the triangle in Figure 30, or about 4.0 cm. This

gives a volume of about 1213 cu cm for the hexagonal prism.

106

A 4.65 cm F
Figure 31: Base of Hexagonal Prism
But when I looked at the pentagonal prism base, I had no idea where to start. I

knew that I was looking at a regular pentagon with sides of about 5.58 cm, but I was at a
loss for figuring out the area. I recalled doing something like this in high school
geometry, but 17 years is a long time and it was not a technique that I remembered using
very much since then. I started to doodle around on my paper, to see if anything would
come to me. The equilateral triangles in the hexagon had been so helpful, so I decided to

subdivide the pentagon into five congruent triangles as well (see Figure 32).

 

 

A

5.58 cm E

Figure 32: Base of Pentagonal Prism
These triangles were obviously not equilateral, so I had no quick way of figuring
out the height of one of them. I remembered that this was something that could be solved

with some trigonometry, so I used what I relearned from working with CMP to figure out

107

one of the interior angles of the pentagon13 and I recalled sohcahtoal4 and calculated the
height of the triangle, only to get a number that was much smaller than I was anticipating.
After doing the same process on the hexagon and comparing the result with what I knew
had to be the answer, I realized that I was doing something wrong—my calculator was in
radian mode. It had been almost 15 years since the last time I worked with radians, but
here it was again, staring me in the face. I switched to degrees and quickly found the
pentagonal prism to have a volume of approximately 1157 cu cm.

None of this mathematics was particularly difficult. In high school, I would not
have had any problem pulling these pieces together to solve the problem. But I was
clearly rusty in some of my mathematical content knowledge. As I wrote in my teaching
journal:

This exercise has helped me to think about my own mathematical knowledge and how I need to
constantly exercise some of those skills that have been dormant for too long. There are some
things in higher level mathematics that I haven’t done for a longtime (like calculus and
trigonometry) that are going to be important to me in my future as a mathematics educator that I
need to be able to recall more efficiently. I think that by doing the puzzles on the calendar page in
the Mathematics Teacher and some other recreational mathematics, I can get some of what I need,
but also having engaging, mathematical discussions with teachers and others here at the university,
we can push each other to think harder about the mathematics that lies beneath what we talk about
every day. (Personal Teaching Journal, 3/30/99)

In addition to working with colleagues on additional mathematics tasks however,
I have come to realize that my own teaching has also become a site for challenging my
mathematical thinking and for revisiting ideas that I encountered some time ago. Through

continued reflection on the challenges that arise in the preparation of teaching as well as

 

’3 In the CMP unit Covering and Surrounding, students learn that the sum of the interior angles of a polygon
can be determined by multiplying two less than the number of sides by 180°. This builds from the idea that
any triangle has an angle sum of 180° and that the number of triangles (with all vertices on the vertices of
the polygon) that a polygon can be divided into is two less than the number of polygonal sides.

’4 Schcahtoa is a mnanonic device introduced in many high school trigonometry classes. It is helpful for
remembering the basic trigonometric ratios by relating them to their corresponding angle lengths in a right
triangle (sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent).

108

from the classroom itself, I have been able to expand my own content knowledge. I have
recognized that my knowledge base is indeed in a constant state of reconstruction and

development through reflection on practice.

Shared Reflection—Modeling Good Teaching Practice

I decided not to stop simply with my own reflection, but to share my experiences
with the students. The next day in class, I decided to recreate for them the process I went
through in determining the volumes of the four prisms. Utilizing the same mathematics
that I made use of, I recreated through the problem and the process again—engaging
students once again in the investigation while pointing out where I made mistakes and
where I struggled to find an appropriate mathematical approach to solving the problem.

In Shulman's model for pedagogical reasoning I, as a teacher, was moving from
what I had originally understood about this problem to a new level of comprehension.
This was occurring as I was beginning the transformation process—specifically, I was in
the preparation phase of my pedagogical reasoning cycle. I had to think through the
mathematics of the problem on my own, just so that I could begin to talk about it with the
students.

Drawing on my model which includes shared reflection, I replicated my thinking
for the class. These students had all taken several years of mathematics in high
school—experiences which were much more recent for them than my high school days
were to me. With this background, most of the students were able to follow the logic I
used as evidenced by their ability to participate in the discussion and suggest other

possible approaches. The accompanying conversation allowed us to cover more

109

mathematics than I thought would be possible with the class and in a short amount of
time. By asking the students for their suggestions in tackling my questions and then
walking them through my reasoning on this problem, I was able to touch on ideas of
similarity, trigonometry, and the Pythagorean Theorem. I demonstrated some of the ways
in which these mathematical ideas are connected and how one strategy may have an
impact on other approaches.

Even for those students who may not have seen the trigonometry before, the
experience of watching and listening to a teacher explain how he struggled with these
ideas and refilsed to give up until an answer had been reached was, I feel, an important
one for them. The discussion also continued to demonstrate to preservice teachers the
importance of being a reflective practitioner and that their developing knowledge was
constantly growing and making new connections. When I thought about how many of the
students were able to make the connections between maximizing area with a given
perimeter and maximizing volume with a given lateral surface area without my making
these connections explicit, I realized that the task of helping preservice teachers
recognize and build their content knowledge is a complex one—yet one that I was

making some headway on. From my own personal reflections:

Last year, my students had a hard time extending this idea to why the cylinder has a maximum
volume for a given lateral surface area—this year that question was solved before I even asked the
question. My students now have such a flexible, connected knowledge of maximizing and
minimizing perimeters, areas, surfaces areas, and volumes that they are hypothesizing about new
situations with little difficulty. It’s nice to see them making sense of the formulas they had
learned but didn’t understand and reasoning about new problem situations in a variety of ways.
(Personal Teaching Journal, 3/30/99)

Rice and Clay—Selection of Appropriate Tools and Representations
With a useful formula for the volume of a prism in hand and an informal

understanding of a cylinder as a “circular prism”, students easily tackled the task of

110

finding the volume of a cylinder. This problem, however, only served as a brief prelude
to the investigation that was cited by more students than any other as the “most
memorable” of the semester. Filling and Wrapping contains a numbers of problems that I
find extremely impressive when I see middle school students, teachers, and preservice
teachers investigate them. But there are two problems which really stand out for me—the
development of formulas for the volume of a sphere and a cone.

1 had often seen this particular cone investigation before, even as an eighth grader
and as a preservice teacher in a methods course. It involves the comparison of a cone and

a cylinder with equal heights and congruent bases (Figure 33).

 

Figure 33: Cone and Corresponding Cylinder
The cone is filled (with rice, popcorn, water, or some other medium) and dumped
into the cylinder (Figure 34). The cone is filled and emptied into the cylinder a total of

three times.

111

 

Figure 34: Cylinder Containing Volume of One Corresponding Cone

This demonstrates that the volume of the cone appears to be ;— that of the

cylinder. If a numerical value has been determined for the cylinder’s volume, then the
cone’s volume is easily determined. Or, if you are dealing with the cylinder’s volume

symbolically as V = 1r 12 h, then the volume of the corresponding cone could be expressed

asV=§1r12h

But similar to some of our work with circle area, the exactness of the answer
comes into question. While it may take a little higher level of acceptance to agree that it
takes exactly 11: radius squares to cover a circle, students must take at least a small leap of
faith to buy into the fact that it takes exactly 3 cones to fill the cylinder. Perhaps it takes
just a little more than 3 (maybe even it!) cones to fill the cylinder and the rice that the
teacher uses fills in the space of the cylinder better than the space of the cone. Or maybe
the teacher has chosen a cone and a cylinder of a specific size to make it come out
evenly. As a teacher, I always felt that too many questions were raised by this
investigation to irrefutably prove the desired relationship, but my middle school students
seemed convinced by it and my preservice teachers were always impressed with it

(especially since it helped to validate a formula they may have already seen). So I

112

continued to use this problem, not realizing its full potential until I saw how it was
embedded in a set of problems in the CMP Filling and Wrapping unit.

In some of the recent CMP workshops that are done at Michigan State, a new
credo has arisen to help teachers understand the CMP philosophy of curriculum
development—nothing is arbitrary. That is, there is a reason to the order that problems
are presented and they are developed and written the way they are for some specific
reasons. The cone/cylinder activity is used in other resources, but in the CMP curriculum
the problem is immediately preceded by one that sheds some new light on the task.

After students are introduced to some of the basic vocabulary that they need for
the problems (sphere, cone, height of a sphere, radius of the cone base, and radius of the
sphere), they are given some clay or modeling dough and are instructed to “make a
sphere with a diameter between 2 inches and 3.5 inches.” Then, they are given some
transparent plastic and instructed to create a cylinder that fits snugly around the sphere,

trimming the height of the cylinder so it matches the height of the sphere (Figure 35).

 

Figure 35: Clay Sphere in a Cylinder
Next, the students are instructed to “flatten the sphere so that it fits snugly in the

bottom of the cylinder” and to “mark the height of the flattened sphere on the cylinder”

(Figure 36).

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a:

    

Figure 36: Flattened Sphere in a Cylinder
Following this, students are asked to measure the height of the flattened sphere,
the height of the cylinder, and the height of the empty space for the purpose of looking
for a relationship between the volume of the sphere and of the cylinder.
At this point most middle school teachers and preservice teachers find that the

height of their flattened sphere is about double that of the empty space. This indicates that
the original sphere had a volume of about § that of the cylinder. This typically becomes
even more powerful when data are compared for the entire class and students see that the

ratio holds true for whatever size sphere they started with. With what they know about

cylinder volume, students can then reason that to find the volume of the sphere, they

would take 3 of the volume of the corresponding cylinder. Symbolically, this leads to a
3

sphere volume formula V = _ 1r rzh. The only part of this process that students typically
3

find difficult is realizing that the height of the sphere is not given and that since it is the
same as the diameter of the sphere, it is double the radius.
For the mathematically-savvy preservice teacher, this formula often comes as a bit

of a surprise, for the formula that is usually presented for the volume of a sphere is

114

v = g 1! r3. But with a little symbolic manipulation (and the understanding that with a

sphere, h = 2r), it can be shown that the formulas are equivalent:
V= an2h= Zar2(2r)= i rtr3
3 3 3

But this problem also raises the same difficulties as the one involving the cone
and the cylinder—that some amount of hand-waving must be used to gather an agreement
that the 2:3 ratio between the sphere and the cylinder volmnes is indeed correct. But here
I return to the fact that these two problems are placed one after the other. The students
even use the same cylinder that was created by wrapping the plastic around the sphere to
investigate the cone and cylinder relationship. It is the combination of these two
investigations that lends some additional credence to the outcome. There may still be
some question about the exactitude of the relationship, but the fact that two related
problems produce such seemingly related results is harder to dismiss.

In these two problems, the interaction of the representations is particularly helpfirl
for understanding the relationships between the volumes. The spatial reasoning that is
used here is not initially done with a picture, but with actual obj ects—the clay sphere, the
paper cone of rice, and the plastic cylinder. Students transform these objects to
investigate the relationships——creating the cylinder by wrapping the plastic around the
sphere and trimming it to fit, flattening the sphere so that it fits tightly into the cylinder,
and filling the cone with rice and dumping it repeatedly into the cylinder. These physical
transformations themselves become important and powerful ways of building an
understanding of the relationship. Students can recall the image of how high the flattened

sphere appears in relation to the cylinder, but the physical process of pressing down the

115

clay becomes intricately embedded in their understanding. When Janet poured the rice
from the cone on top of the flattened sphere (Figure 37), she observed that the combined
volumes of the sphere and cone were equal to the volume of the cylinder. By doing this,
she added a new dimension to the spatial reasoning that she was developing—one that

demonstrated to her more about the interplay of the various relationships.

 

Figure 37: Combined Flattened Sphere and Rice in Cylinder

Using Shared Reflection to Move Learners to Thinking as Future Teachers

As a teacher educator, I observed a grong deepness of my own understanding
as well as my students’ with these problems. In Shulman’s model for pedagogical
reasoning, I look primarily at the transformational phase of the teaching process, where
the selection of subject matter and the adaptation of lessons results not only in important
representations of the subject matter, but also in the accompanying instructional strategies
(the ways I set up the tasks, the language, drawings, and models I used to aid in
understanding). The choice of these models and the order of their introduction was far
from arbitrary—these decisions were made based on the work done around CMP, my
presentation of these lessons with practicing teachers in inservice sessions, and my

previous use of these investigations with preservice teachers.

116

But what set this presentation of these problems apart from the other times that I
had tried them was my use of shared reflection throughout the lessons. I constantly asked
the preservice teachers to change their hats—to move from being a student of
mathematics to a future teacher of children. I shared my reflections with the students on
the decisions that I had made about teaching these problems and I shared with them the
successes and failures that I had experienced. I asked the students to compare the
combination of these two lessons (finding the volume of the sphere and the volume of the
cone) with using the lesson on the volume of the cone by itself. I asked the students to
think about whether they learned anything new about the cone by doing it in conjunction
with the sphere activity. Students explained that each lesson helped to validate (in their
minds) the other. They described the ways in which combining the two lessons helped

them to recall the formulas. In her learning log, Tammy wrote:

I liked this lesson not necessarily just because we played with clay, but I liked it because of the
way it sticks in my head. During this lesson, we learned the formulas for a sphere and a cone.
These formulas are very easy for me to remember surprisingly because they are actually tough
formulas. The reason I am able to remember them so easily is because in my mind I remember
what happened with the rice and clay. I can see how the sphere made of clay filled up two-thirds
of the cylinder, and also how the cone of rice filled up one-third of the cylinder. (4/20/99)

I asked the students to think about the kinds of things they learned as students in
the class and to reflect on how they might use their experiences as learners when they are

teachers. Tammy went on to write:

This lesson was a favorite of mine, because you found a good way for us to easily remember a
tough idea. My goal when I become a teacher is to do the same thing. (4/20/99)

These comments are particularly insightful when considering that Tammy was
one of the brightest students in the class. She had a rich mathematical background and
was often the first student to recognize a problem’s underlying mathematics. In her

learning log, Tammy shows that she has been thinking about how powerful these

117

representations might be for her students—explaining that manipulatives involved more
than just “play[ing] with clay,” but that it makes the mathematics more accessible to

students at all ability levels and with various learning styles.

Using a New “Blended” Representation

In preparing to teach these lessons to my preservice teachers, I recalled reading
about Archimedes and how he had discovered this fascinating set of volume
relationships. I turned to some mathematical history books and found what I was looking

for, a succinct visual representation of Archimedes’ discovery” (Figure 38).

A

.V
if v. ......... .

1 :2 :3

Figure 38: Archimedes’ Ratios of Volumes

 

 

I decided that while this representation captures much of the beauty of the volume
relationships, I wouldn’t show it to the students right away. For even though it describes
the ratio between the volumes of any two solids quite nicely (just cover up one of the

figures and the number beneath it, leaving behind the ratio relating the volumes of the

 

’5 Archimedes’ original representation in of the volume relationships actually involved a cone enclosed in a
hemisphere which was enclosed in a cylinder. To discover this relationship, Archimedes clearly had to
know the formula for the volume of sphere—a formula that was unknown at the time. In 1906, the Danish
philologist John Ludvig Heiberg discovered The Method, a lost work by Archimedes in which he described
his technique for finding the formula for the volume of a sphere. Archimedes employed principles of
mathematics and physics to derive the formula. Today, most mathematicians use calculus to derive the
volume formula for a sphere—a tool that was not available to Archimedes as it wasn’t discovered until
almost 2,000 years after Archimedes’ death (Swetz, 1994, pp. 180-181).

118

two remaining solids), it circumvents the discovery nature of the volume experiment, a
powerful lesson for preservice teachers and an experience that I want them to have.
But after the preservice teachers had the opportunity to work on the problems, I
drew the three solids and the accompanying compound ratio on the board without an
explanation. I asked the students to describe what this representation meant to them. A
few students immediately pointed out the ratio 2:3. I asked them to elaborate and they

explained that the ratio corresponds with the volumes of the two figures above it, that the

sphere has a volume that is % the cylinder’s volume. Several other students quickly

added that the 1:3 ratio is also contained in the representation, indicating that the cone has

a volume that is g. the cylinder’s volume. This representation also indicated to the

students the additional ratio of 1:2, showing that the sphere’s volume is double the cone’s
volume—a fact that some students discovered in manipulating the rice and clay but a fact
that became easier to comprehend with this new representation.

Many students chose to use the representation (Figure 38) in later work—I found
it drawn on many tests and quizzes as students worked to recall the relationships—while
others relied solely on what they learned from the lessons. Students used this
representation to help them reconstruct the formulas for the volmne of a cone and a
sphere from the formula for the volume of a cylinder. This blended representation is a
powerful abbreviation of the spatial, symbolic, and mathematical language
representations of the volume relationships and it serves as an ideal example of how
important representations can be in the development of preservice teachers’ deeper

mathematical understanding.

119

Making Sense of Formulas: An Informal Proof Scheme

As with the circle area, these investigations on the volumes of the sphere and the
cone were less about developing an understanding of the mathematical reasoning
involved and more about remembering and being able to reconstruct formulas for
calculating those volumes. This focus was largely a result of my experience with
preservice teachers who had misunderstandings about formulas coming into MTH 202.

On the first day of class I posed a learning log prompt asking students, in part, to
“Write down the first three geometry formulas you think of and tell what they are the
formulas for.” A few were not able to come up with three formulas and many of the other
students either stated incorrect formulas or did not write down what the formulas
measured. For example, Tina wrote:

I x w = area %bh = triangle area a2 + b2 = c2

Tina’s second formula and description of what it measured were fine, but her .first
formula lacks the specificity of the shape it measures and she shows no evidence of
understanding what the third formula represents or can be used for.

Maryanne’s entries were:

A = 21 - 2w P = 21 + 2w perimeter ofa square = 4s

Maryanne’s third formula was both correct and specific about what it represented,
but her second one gave no indication of what kind of shape the formula can be applied
to and her first one was incorrect, although it may show Maryanne’s generalization of the
perimeter formula for a rectangle (an additive process) to an area formula (a

multiplicative process).

120

I found that these students were fairly representative of the preservice teachers I
worked with in MTH 202. For this reason, I decided that I would focus the course around
helping students remember formulas used in geometry correctly, what the formulas are
used for, and how to make sense of the formulas.

The third goal—making sense of formulas—is the only one that alluded to any
level of proof schemes. Even then, I truly thought about it as “making sense,” and not
anything as formal as a kind of proof, much less the all-encompassing “justification.”

To me, “making sense” implied a less formal level of justification. It indicated that
students reasoned about the mathematics of the problem and by making connections to
other ideas, using multiple approaches or representations, or thinking about the
underlying mathematics of the problem were able to be more confident about their
results. I wanted students to feel that the formulas made sense, in much the same way
teachers help students gain number sense (e.g., by reasoning that an estimate is in the
ballpark and by understanding the magnitude of numbers and the effects that operations
have on those numbers). Geometric formula sense for preservice teachers can be thought
of as a specific type of mathematical PCK that encompasses ideas of using symbolic,
spatial, and mathematical language representations to think about the reasonableness of a
formula, learning about ways in which various representations help students to reason
about formulas in words and in symbols, and pushing at a teacher’s own mathematical

content knowledge.

121

Conclusion

In this chapter, I used representation as the organizing tool of analysis in looking
at howl attended to issues of PCK while focusing on the development of my own
pedagogical content knowledge. Although I began the course focusing on ways to make
sense of geometric formulas using spatial reasoning with various manipulations, my
understanding of the mathematics and ways to help students deepen their knowledge
grew as I expanded my perceptions to include other types of representations that were
used in class. The development of a blended representation that encompassed ideas
inherent to symbolic, spatial, and mathematical language was a particularly powerful tool
for students in understanding relationships between figures and it provided students with
a way to remember formulas and relationships that might otherwise be difficult to recall.

I also used Harel and Sowder’s proof scheme to investigate further my approach
to using justification in teaching MTH 202. The notion of geometric formula sense
doesn’t exactly classify as a proof scheme, but I found it to be a helpful way to describe
the ways in which I encouraged preservice teachers to “make sense” of the formulas they

encountered.

122

CHAPTER 4

TASK DEVELOPMENT

Introduction

Effective teaching involves not only finding and creating tasks that are engaging
and mathematically challenging, but also recognizing the complexity of using those tasks
in interactions with students (N CTM, 2000). There is a great deal that goes into decisions
about the tasks that are chosen—understanding students’ abilities and potentials, thinking
about how students might interpret the task in the way it appears to them, and deciding
whether the task truly reflects the mathematical goals the teacher has for the students.
These decisions involve knowing students, knowing mathematics, and knowing how
students might best learn the content—aspects of teaching that are difficult for
experienced teachers, much less novices.

Yet descriptions about the process of designing tasks give little detail about how
these decisions are made. Are there words or phrases that are more useful than others in
conveying specific desired mathematical processes? Are there instances when a spatial
representation might be helpful and times when it might be misleading? What
assumptions are being made when a task introduces terminology that may be new to
some students and how might this affect students’ performance on the task?

In this chapter, I don’t look to establish definitive answers to these and other
similar questions about task design. Instead, I examine how thinking about questions like

these can help a teacher, or in this case a teacher educator, create tasks that better reflect

123

what he or she intends. This chapter gives a detailed account of the evolution of a task,
following it from its creation for MTH 202, examining how students interpreted and
answered task, comparing it to another version of the task, and charting its development
through several subsequent iterations.

In analyzing student work and the evolution of the task in each of its iterations,
one theme that kept bubbling up to the surface was the issue of generality. It is not
uncommon for teachers to find that students are interacting with a specific case in a
problem and not generalizing their results to a larger domain, as the teacher may have
expected (Schwartz, 1986; Chazan, 1993; Goldenberg & Cuoco, 1998). Geometry classes
are particularly susceptible to the phenomenon, as problems are often posed with an
accompanying drawing, intended to exemplify one possibility or case, and students treat
hypotheses based on the example as also holding for the more general.

Parzysz made a distinction between the diagrams that indicated cases of the
particular and the general when he introduced the words drawing and figure to the
research literature in geometric learning (Parzysz, 1988). A drawing was one specific
picture and a figure was meant to represent a larger class of shapes which were related by
some shared traits. I use this terminology in describing the task being analyzed in this
chapter and to underscore the importance of making students aware of the difference

between the two types of diagrams and the tasks that accompany them.

Posing a Task

Following a series of learning log prompts that asked students to write about some

of their previous geometry experiences, I posed a question that I thought would link some

124

of the topics we were studying in class and would provide access to some additional
geometry topics. Since I was still getting them used to writing in a mathematics class, I
posed an open-ended question that would encourage the students to write as much as they
could about a problem. I recalled a problem that seemed to meet all of these needs—a
problem that was given to me as an undergraduate student in mathematics education that
centered around investigating a unique kind of trapezoid, so I recreated the problem as I
remembered it.

As students walked into class one day, I distributed slips of paper with a problem

and figure (Figure 39).

 

 

Learning Log Prompt #4

 

Here is a special isosceles
trapezoid. Write down all of the

interesting things you can find

 

about this isosceles trapezoid.

 

Figure 39: Trapezoid Task Version A
I chose to introduce the term “special isosceles trapezoid” to encourage students
to wonder what was special about this figure. In my mind, I envisioned a classification of
trapezoids (Figure 40) in which special isosceles trapezoids make up a subset of the set of
isosceles trapezoids (trapezoids with congruent non-parallel sides). The special isosceles
trapezoids are those in which the two congruent non-parallel sides are also congruent to

the smaller base and in which the larger base is twice the smaller base.

125

 

 

 

 

   
 

 

*Special
Isosceles
Trapezoids

 

Figure 40: Trapezoid Venn Diagram

At this point, I expected students to at least notice some interesting properties
about the figure—that three sides are one length and the fourth side is double that length,
that there are two 60° angles and two 120° angles, that the figure has a line of symmetry,
etc. I expected students then to branch out and look at other interesting properties of the
figure involving diagonals, subdividing and rearranging the shape, segments connecting
midpoints of sides, and any other geometric ideas they could think of.

I also planned on using this task to launch a unit centered around similarity. I
wanted students to begin looking at what it means for two different shapes to be similar,
since the figure I included with the task was meant to be one example of the set of
different-sized special isosceles trapezoids. However, the design of the task led most of
my students to assume that the only trapezoid that they were supposed to consider was
the single one drawn with the prompt and thus gave them a more restricting view of the
question than I planned. In much of their work, they treated the diagram as a drawing and

not as a figure as I intended.

126

Using Student Work to Analyze the Design of the Task

When students turned in their learning legs, I was surprised at their limited
interpretation of the question I posed—until I reread what I had asked them to do and
realized that they had followed my instructions. Instead of looking at special isosceles
trapezoids in general, many students focused on the one that they were given and treated
it as the one special isosceles trapezoid in existence. For them, what was special was this
drawing, not the set of special isosceles trapezoids which was a subset of the set of
isosceles trapezoids. I did an inadequate job expressing the question I planned, which
made it nearly impossible for students to approach the problem in the same ways that I
anticipated.

Four sets of student work are presented here as examples of how preservice
teachers answered the question based on their interpretation of the task (Figures 41, 42,
43, and 44). I look at some of the general themes present in the work of the four
students—Dana, Rachel, Margie, and Patti—and how the design of the task may have led
students to respond in the some of the ways that they did. The samples of student work
that are provided also include my comments, corrections, and evaluation of the student’s

work, giving each student a grade based on a 10-point scale.

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131

Analyzing Themes in Student Work

The four pieces of student work on this task are fairly representative of what was
done by students in the class. All students made at least one kind of error in their work,
although I was not assessing students on the accuracy of their findings alone. Almost all
students made at least one statement that grew out of an understanding of the definition
of trapezoid—these statements could be viewed as trivial. And almost all students applied
some ideas that were introduced earlier in the course to this task.

I grouped the responses of these four students into five different categories:

1. Trivial observations

2. Applications of mathematical ideas previously covered in MTH 202

3. Applications of other basic or advanced mathematics

4. Unproven hypotheses

5. Incorrect statements

While there is some overlap between categories (e.g., a statement might be an
unproven hypothesis that is also an application of a previously covered idea from MTH
202), I tried to make some distinctions between the kinds of statements I placed in each
category.

Ig'v_ia:l Observations. A word of clarification before these are presented: these
observations may or may not have seemed trivial to the students who made them. I am
classifying them as trivial with respect to the task that was given, to find “all of the
interesting things” students could about the figure. The statements that fall into this
category are generally a result of the shape being a trapezoid and have nothing to do with

the uniqueness of the shape.

132

Margie made several trivial observations about the special isosceles trapezoid,
noting that has four sides, its perimeter can be found by adding up the lengths of the four
sides (“P = A + B + C + D”), and its area can be found using a formula that was

+
discussed at length in class (“A = 1%; h”). Unlike the other students, Margie didn’t

use either of the formulas to determine the trapezoid’s perimeter or area, she simply
restated a formula that was used in class. Both Margie and Dana listed at least one of the
spatial representations for finding the area of the trapezoid, but did not determine the area
using that method. And three students (Rachel, Margie, and Dana) stated that the two
bases of the figure are parallel—a reiteration of a part of the definition of a trapezoid.

It is possible that each of these trivial observations is a result of some confusion
that arose from the wording of the task. Although I wanted students to discover aspects of
the figure that made it “special,” I instructed the students to find “interesting things”
about the figure, a phrase that is somewhat vague and open to interpretation. In addition, I
ofien asked students to make connections back to prior class work. We spent several
previous class periods discussing general trapezoids. The trapezoid in the task likely
indicated to some students that the task was also an assessment of what they already
knew about trapezoids, and so they reproduced some facts that were not unique to the
special isosceles trapezoid. Finally, students knew that their responses to the learning log
prompt were evaluated as a part of their course grade. By asking for students to find “all
of the interesting things” they could find, I may have inadvertently indicated to some
students that the more things they wrote, the higher their grade would be. These students
may have decided to include everything they could come up with about the figure—no

matter how trivial.

133

minions of Mathematicial Ideas Previously Covered in MTH 202. In addition
to some of the trivial observations presented in the previous section, there was a set of
statements made by students that related to things discussed in class. I classified these
statements in a different category from the trivial observations because to me, these
observations involved an application (instead of a simple restatement) of an idea that was
previously discussed.

Dana, Rachel, and Patti measured the dimensions of the drawing and calculated
its perimeter and its area. This involved making some decisions about the trapezoid’s
measurements—how accurate to be, what measurement system to use, etc. Afier
measuring in centimeters (the drawing was made so that its sides were whole number
centimeter lengths), Dana went to the trouble of measuring in inches and calculating its
perimeter and area. Two students, Dana and Margie, also applied ideas about subdividing
a shape to find its area with Dana listing no fewer than four different ways that this could
be done.

As with the trivial observations, it is possible that these applications of previously
covered topics drew from my common insistence on connecting back to other ideas.
Students may have assumed that I wanted them to apply concepts from the course to this
task.

Applications of Other Basic or Advanced Mathematics. I was pleased to see the
introduction of ideas that I wasn’t planning to cover in MTH 202 or had not yet covered
in the course. These ideas included some of what I was hoping to see in students’
work—either some ideas of what made the figure “special” or thinking about some of the

relationships between the figure’s attributes. Some of the more advanced ideas that

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students gave involving terminology or ideas that are not part of student’s daily lives
were likely carried over from the their middle school mathematics or high school
geometry course.

Dana, Rachel, and Patti observed some relationships between side lengths (three
sides were the same, one base was half the length of the other base, etc.) and all four
students discussed at least one of the angle relationships (two sets of two different sized
angles, larger angles measured twice the smaller angles, etc.). All students also made an
observation about the symmetry of the shape, showing how it “folds over and matches”
(Margie), that “if you divide the trapezoid vertically down the middle and fold it over
onto itself, it’ll reflect” (Patti), or that the diagonals are of the same length (Rachel).

Patti recalled some relationships she learned before but I was planning to
introduce later in the course—relationships of side length in a 30-60-90 triangle. She
subdivided the trapezoid into two 30-60-90 triangles and rectangle and used the
relationships she knew to find the exact height of the special isosceles trapezoid. This
was a clever approach to use and it showed some nice geometric insight on Patti’s part,
especially when compared to the other students who took a measurement of the height
(3.5 cm) to be exact.

Rachel also made a number of unique observations about the special isosceles
trapezoid, finding that “if you connect the midpoints of the two non-parallel sides it will
form a line parallel to the two parallel sides” (a restatement of the definition of a I
midsegment, which was not explicitly discussed in class) and that “if you connect the

midpoints of the two parallel sides it will form the perpendicular bisector [to the bases].

Rachel also turned to an outside source to write a defrrrition of an isosceles

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trapezoid—one in which “two base angles are equal” (assuming that one accepts the fact
that the shape is trapezoid).

In responses such as these, students felt free to begin exploring relationships
within the figure. This is the type of response I was anticipating when I included the word
interesting in the design of the task. I wanted students to feel free to experiment with the
figure, drawing diagonals, connecting midpoints, and basically wondering, “what if I did
this...” when playing with the shape. When designing a task, it’s helpful to find some
way to express to students when the desired outcome involves trying out something new.
Asking students to find “interesting things” about a situation is one possible way, but as
was previously discussed, that choice of words may connote other unintended ideas as
well.

Unproven Hypotheses. When I posed a learning log prompt asking students to
“Describe in detail an event from your high school geometry course which is particularly
memorable. . .”, a number of them included a discussion about working with proofs. As
mentioned in Chapter 3, I wasn’t interested in making proof a formal part of the course.
However, I did want to find some place in the MTH 202 curriculum to allow for some
discussion of issues of justification and I thought that this task might provide that kind of
access.

Some of the students’ observations about the special isosceles trapezoid were
quite interesting and yet they left me wanting to know if students could make a formal or
informal argument about whether the statements were true. For example, Rachel stated
that “if you extend the two non-parallel sides until they meet, you will form an isosceles

triangle. . .and the new [smaller] triangle [on top of the trapezoid] will have 3 equal

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sides.” Aside from the fact that Rachel could have used to word equilateral to describe
both the large and small triangle that were created, Rachel’s two statements seemed like
great examples of hypotheses that preservice teachers might be able to prove.

Margie was the queen of unproved hypotheses—statements that I felt almost
begged to be justified in a proof—two-column or otherwise. She wrote that by drawing
the two diagonals, the trapezoid “can be divided into 4 triangles—two of the triangles
appear to be equal to each other [or congruent].” Although she was not very eloquent in
describing her hypothesis, in her accompanying diagram, Margie clearly indicated the

two triangles she was talking about (Figure 45).

 

 

 

Figure 45: Margie’s Special Isosceles Trapezoid 1
In the next statement, Margie looked at the other triangles formed by two
diagonals and wrote, “I am guessing that triangle A is half the area of triangle B” (Figure
46). Margie’s statement was not correct—the area of triangle A is actually one-fourth the
area of triangle B—but I was excited about the kind of statement that she made. I also
appreciated that Margie admitted that this observation included some “guessing,” just as
in her previous one, she noted that the two triangles “appear” to be equal. Unlike most

other students, Margie didn’t make unfounded claims, she gave unproven hypotheses.

 

A

B

 

Figure 46: Margie’s Special Isosceles Trapezoid 2

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This is what initially bothered me about many of the statements that students
made. Not only did they offer any justification for why their statements were true, but
most students didn’t seem to recognize that some of the observations they were making
were actually hypotheses until they were proven. Then they could be considered
statements of truth.

But to be fair, students had no indication that this was something I was concerned
about. We had not really talked about what it means to prove something and I was not
ready to tackle that subject yet in class. There was never any expectation made of
students to recognize their statements as hypotheses, and nothing in the design of the task
would indicate that. I simply asked for “interesting things.” While I, personally, found
hypotheses to be interesting, there was no reason for me to think that students would do
the same.

Incorrect Statements. As previously indicated, some of the students’ observations
that were listed in other categories also contained incorrect statements. These categories
are not mutually exclusive and I coded these statements as incorrect because they
specifically indicated to me some mathematical misunderstandings that were relevant to
the course.

The first statement was one that was made by Margie concerning the areas of two
parts of the figure after it was subdivided. She wrote that you get “equal areas when you
cut the trapezoid in half.” Without the drawing she provided, I would have thought she
was writing about cutting the trapezoid in half along the vertical line of symmetry. Her
statement would then have been correct. However, in Margie’s drawing that accompanied

this statement (Figure 47), she drew the midsegment of the trapezoid—how she was

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likely thinking about cutting it in half. Since she gave no indication of what shapes she
claimed have equal areas, she could be talking about the two trapezoids (which clearly do
not have equal areas), or the two triangles at the left-hand side of the figure (which do
have equal areas). Since we just finished several weeks of understanding areas of shapes,

I was surprised that Margie might still have this misconception about areas of shapes.

 

 

 

 

Figure 47 : Margie’s Special Isosceles Trapezoid 3

An incorrect statement that Patti made also looked at the isosceles trapezoid after
the median line was drawn in. In Patti’s statement, she claimed that “if you divide the
trapezoid horizontally so that the B2 would become 6 cm, you end up with two similar
trapezoids. The angles are similar.” Patti’s mistake was in using the notion of
similarity—an important mathematical topic that we were in the process of investigating
when this task was assigned. Before this task, we established the fact that similar shapes
have congruent corresponding angles—the idea that Patti seemed to be getting at in
saying that the “angles are similar.” But while we weren’t to the point of understanding
about the proportionality of corresponding sides in similar figures, we had established the
fact that the similar figures had to have the same basic shape. By looking at her response
to this task, Patti may not have completely understood what was meant by two trapezoids
that are mathematically similar having the “same shape,” since the bottom trapezoid is
visibly stretched from side to side when compared to the top trapezoid. Patti had

problems with using the word similar and/or the idea of similarity in her statement. These

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misconceptions led to an incorrect statement that a preservice teacher at this level should
not have made.

The final example of an incorrect statement was made by Dana as she noticed
something interesting about the perimeter and area of the special isosceles trapezoid
drawing. Dana wrote, “Area is 21 cm2, which is very close to the perimeter (20 cm).” In
my notes to Dana, 1 corrected her to make her statement read that they are “numerically
very close.” I wanted her to realize that while the measurements and calculations she
found may be interesting, her statement is not quite mathematically correct. Dana focused
on the fact that the numerical parts of the two measurements were close while ignoring
the fact that the units were different (cm2 versus cm).

Her statements should also have raised some questions when in her later

observation, she calculated the area to be “ .(square) inches,” which is quite different

from what she would have found if she calculated the perimeter of the trapezoid when

measured in inches (8', by her measurements). In light of this, her statement about the

closeness of the two measurements might read “when measured in centimeters and square
centimeters, the perimeter and area are numerically very close.”

Dana was mathematically incorrect in the language she used in describing
something she noticed about measurements she made of the drawing. We discussed this
distinction between units in an earlier class session, and I was surprised to find Dana

confusing the issue once again.

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Helping Students Learn from Others—Supplementing a Task
After considering the work that some students did on the task as it was posed to
them, I decided that there was more that could be learned fiom investigating the special
isosceles trapezoid even further. I felt that other students could benefit from some of the
individual observations that were made—both right and wrong—and I looked at the
students’ work to pull out some of the items I felt would be most beneficial. I decided to
create a new labsheet of questions that I pulled from some of the students’ observations.
In Shulman's model for pedagogical reasoning I was taking what I learned from
assessing students and transforming that knowledge into appropriate questions and new
tasks that would prompt more investigation of various ideas in geometry. As part of the
transformative process, I was thinking about the original task that I created and what may
have caused students to respond to it in the ways that they did. I considered the language
that was used, the spatial representation that was presented, and the introduction of the
new term special isosceles trapezoid. I looked at ways students responded to the learning
log prompt and considered questions such as:
o What level of observations should be expected of preservice teachers and how
can I encourage students to reach that level?
0 How could I get students to consider the properties of the family of special
isosceles trapezoids instead of one specific example?
0 Should I introduce the term special isosceles trapezoid in the task? Does the
term mislead students or take away from some of the investigative nature of

the task?

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0 Should students simply state conjectures or should they also include some
proof of their hypotheses? What role should proof play in MTH 202?

Thinking about questions like these helped me decide which individual student
observations I would include on the labsheet. I compiled a series of observations and
conjectures that were representative of varying types and levels (Figures 48 and 49). I
also decided to reformulate their observations as questions. Rather than presenting facts ‘
(which may prove true or false), I wrote questions to encourage further inquiry into the
task and the observations that students made.

With regards to my PCK development, I was taking what I learned about what
students understood about the special isosceles trapezoid, and designed a new level of the
task to address the specific misunderstandings and insights that were made by individual
students of the class. I was becoming more aware of the ways I designed the initial task
that didn’t satisfy the goals I had for my students’ understanding, and I used their work to
help me design a new task comprised of a series of specific questions to reach those
goals.

The shared reflection in this stage of the task design involved my giving students
some feedback on the work that was done by the class as a whole. I explained that the
initial learning log prompt that I wrote was, in some ways, misleading and did not
indicate some of what I wanted them to do with the problem. I explained how I decided
to revisit some of the more intriguing observations that were made by creating a new
labsheet of problems that would address some of the issues I hoped students would

observe.

142

 

 

Most people found it helpful to name

 

A
the vertices of the trapezoid in some
way like those shown here. This helps
when refening to vertices and line
segments later.
A

1. By drawing one diagonal as

 

shown, two triangles are created.
Describe these two triangles as
much as possible. How are their
areas related?

2. By drawing both diagonals as A

 

shown, four triangles are created.

Describe the relationship

between AADM and ABCM. M
3. Describe the relationship

between AABM and ACDM. D

 

4. The line segment that connects
the midpoints of the sides is

 

called the midsegment. How is
the midsegment length related to
the lengths of the two bases?

 

D

5. One student said that ABFE ~ EFCD. Is this true? Why or why not?

 

Figure 48: Special Isosceles Trapezoid Labsheet, Page 1

143

 

 

 

10.

ll.

12.

l3.

14.

If perpendicular line segments
are drawn from A and B, what
are the three shapes that result?

Can you find the length of AG
only knowing the lengths of the

sides of the original trapezoid?

One student said that the height is equal to the square root of the sum of the two bases. Is

this true? Why or why not?

 

 

 

 

D G H

Fold ABCH over BH. How does this help to find the area of the original trapezoid?

If line segments are drawn fi'om
A and B to the midpoint of CD,
what three shapes result?

If the sides are extended to meet,

what shape results?

In general, what is the sum of

[DAB and (ADC?

What is the angle sum of the
original trapezoid? Are there any
other polygons with the same

angle sum?

What shape(s) can be made by
putting together two copies of
the original trapezoid?

A B

 

 

D

 

Figure 49: Special Isosceles Trapezoid Labsheet, Page 2

144

 

I separated the class into groups of three students each and I distributed the

labsheet. I explained that the questions were pulled from students’ responses to the

learning log prompt. I asked them to work on the questions—some of which were

designed to push them to think about aspects of the problem they might not have

considered. Students worked on the questions for about 15 minutes and I pulled the class

together to go over some of their work.

When I asked students to name one of the questions they thought was interesting

or that they wanted to talk about in class, Terry requested question 10.

JW:

Terry:

JW:

Terry:

Terry:

Terry:

So what did you find out about question 10?

It makes three equilateral triangles.

How do you know that they are equilateral?

I measured the sides and they came out the same. They looked like
they would, and when I measured them, I saw that they are.

Is that what you found interesting about this question?

Yeah, I was surprised by this. I mean, I knew that drawing these
two lines [AI and BI] would make three triangles, but I didn’t think
that they would be equilateral. I figured that they would be
isosceles, but I wasn’t sure about equilateral.

Why did you think that they would be isosceles triangles?

Well... I knew that I is the middle of DC, so I figured that the
middle triangle [ABI] would be isosceles. Then I figured that the
other two triangles would be the same...

As the middle one or as each other?

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Terry:

Paula:

Patti:

Each other. Since there’s a line of symmetry down the middle of
the trapezoid, I figured that the left and right sides would be the
same.

This is good. Well, let me show you something else that’s neat
about this problem in particular. (Takes out pattern blocks.) A lot
of elementary school teachers use pattern blocks to teach different
topics in math class. As a couple of you pointed out earlier, there is
a pattern block that is the same shape as the special isosceles
trapezoid (places red transparent pattern block on overhead). There
is also a pattern block that is one of these equilateral triangles
(places green transparent pattern block on overhead). By playing
with these shapes, kids learn early on that they can place cover the
trapezoid pattern block with three of these equilateral triangles.
What mathematical ideas does this help you think about teaching
elementary students?

Area.

Fractions.

What started as a discussion around an observation about the special isosceles

trapezoid moved into an introduction to using pattern blocks in teaching elementary

school. This is an example of my trying to teach students content, and also trying to move

them beyond thinking about the mathematics solely as students but also as how this

knowledge pertains to the teaching of elementary students. Terry’s initial interest in this

property of the special isosceles trapezoid triggered an idea in my head that I wasn’t sure

146

I would cover. I had the transparent pattern blocks with me to use the red pattern block as
an example of the special isosceles trapezoid, but when the situation presented itself, I
decided to talk with the students about how the pattern blocks are used to teach other
mathematical topics as well.

I was pleased with how the questions on the labsheet initiated student
conversations and investigations like the one described above. But I was bothered by the
fact that I wasn’t able to elicit this level of inquiry from all of my students with the
learning log prompt I wrote. I knew that I needed to do better at writing questions that
indicate the kinds of work I wanted students to do, but I wasn’t sure how to go about it on

a regular basis. From my personal reflections:

One other thing that keeps rearing its ugly head in my class is the issue of my asking a different
question than what I am trying to get at. This happened on the first test and it happened to a much
lesser degree on the most recent quiz and the learning log prompt about the special isosceles
trapezoid. On quiz #2, I asked students to compare areas of similar rectangles numerically and
pictorially—at least, that’s what I thought I had asked. Only four students drew the picture to
“validate (their) answer(s)” with the sketch. I knew what I wanted, but I didn’t ask it very clearly.
(Personal Teaching Journal, 2/18/99).

Comparing to the Original Task
After the course was over, I turned back to my notes from my undergraduate
course and found the “special isosceles trapezoid question” as it was originally posed to

me (Figure 50) in the EDT 335 methods course.

147

 

 

 

Make a list of hypotheses about trapezoids that A B
have three congruent sides and a fourth side / \
twice as long as each of the others. D c

 

Figure 50: Trapezoid Task Version B

It’s not clear to me whether the accompanying figure was a part of the question as
it was originally posed or if it is something I added to aide in my understanding as a
student. Either way, there is a distinct difference in the information that is given—in the
words that are used and the way in which the important information is of a generalized
nature.

Each of the following differences highlights how Version A provided students
with a miscue—a characteristic of the task which can lead students to an incorrect
response or at least one that is different from the instructor’s intent. A miscue can take
the form of a misleading graphic, sentence structure that does not accurately
communicate the instructor’s meaning, or even an assumption that is not common to both
the instructor and the students (Shannon, 1999). Looking at the structure of Version B, I
was able to better identify some of the miscues that were a part of Version A of the
trapezoid task.

The first noted difference in Version B is the word “hypotheses.” Compared to the
use of “interesting things” in Version A, the prompt to develop hypotheses indicates to
me now (and likely indicated to me as a student) a more rigorous mathematical approach
to the task. For while a hypothesis is not necessarily something verifiable, it does connote
that the discoveries that are made should be more mathematical in nature. When I asked

students to find “interesting things” about the shape, I was opening up the door for my

148

 

students to observe things like how two of these shapes could fit together to make a
regular hexagon. Both ways of wording can lead to some interesting observations and
discoveries, but they are likely to be quite different from each other.

When I constructed Version A of the task, I remembered more about the kinds of
responses I gave to the task than the task itself. I specifically remembered making a
number of hypotheses about the figure and those responses were a part of how I was
reconstructing the task in my head. In writing my version of the task, I was trying to
create something that might get students to make some of the same kinds of conjectures I
did, without recognizing the fact that the students were not necessarily at the same
mathematical level I was when I worked on the task. The word “hypothesis” in Version B
carried a specific meaning for me—a hypothesis was the part of a proof that needed to be
proven. I wanted MTH 202 students to create hypotheses as well, but I knew that the
word hadn’t been introduced in class. My instruction to find “interesting things” was my
attempt to get students to make hypotheses (that could stand up to a geometric proof)
without using the word.

The second major difference in the language that is used is the naming of the
figure—a “special isosceles trapezoid” in Version A. By giving it this name before
students had a chance to investigate its properties, some of the discovery was taken away.
The word “isosceles” already implies something about the shape. Students may not have
been familiar with this word in this context (“isosceles triangles” are much more
common), but some carry over was to be expected. Some students even looked up
“isosceles trapezoid” in a dictionary and were able to find out some facts about general

isosceles trapezoids.

149

The inclusion of the word “special” in this case may have been problematic.
Students may have interpreted it to imply that this one trapezoid was special, and not the
set of isosceles trapezoids which have additional properties. There are other indicators in
the question as it was originally worded that compelled me to address the general shape
as a student. In that version, the side lengths of the isosceles trapezoid were described in
relationship to each other. The shape was described as a trapezoid with “three congruent
sides and a fourth side twice as long as each of the others.” By seeing the question posed
this way, I was given some immediate indication about what made this shape unique and
worthy of being investigated in the problem. But by giving students one copy of the
shape and asking them to make some discoveries about it, these special qualities were not
readily apparent. In fact, some students measured the sides and did not come up with the
relationships that were intended—finding the longer base not exactly double the length of
any of the other sides due to some measurement error or a distortion of my original

drawing due to the effects of a copy machine.

Conclusion
Task design is a complicated process. When done well, it recognizes the prior
mathematical background of the students and capitalizes on that background to build
deeper mathematical understanding. Good tasks are also designed with a clear vision of
what direction the teacher has for the students—making connections between
mathematical topics and laying the foundation for future investigations. In Simon’s
description of planning for effective teaching, he emphasizes the importance of the

teacher clearly identifying a hypothetical learning trajectory for his or her students,

150

which includes an understanding of the teacher’s learning goal, the teacher’s plan for
learning activities, and the teacher’s hypothesis of the learning process (Simon, 1995).
All of the issues are also important aspects of good task development.

There is little evidence on ways to support preservice teachers’ learning about
designing good mathematical tasks. But involving them in the process through a teacher
educator’s shared reflection is one possible place to start. It would also be helpful for the
teacher educator to have a clear understanding of some of the elements of task
design—the role that specific words and language in general can play, differences
between drawings and figures and what each can contribute to student understanding,
and the ways in which written language and geometric representations can be used

together to best convey the intended ideas of the task.

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CHAPTER 5

IMPLICATIONS AND LIIVIITATIONS

Implications

There are three major frndings in this study in which I examined the growth of my
own pedagogical content knowledge. First, this study highlights and problematizes
Shulman’s notion of representation. In mathematics there are mathematical and empirical
representations—classifications which do not map easily onto Shulman’s use of
representation. This study exposes some of those inherent distinctions and seeks to make
Shulman’s work more applicable to the field of mathematics. Second, this study describes
the importance of task design—a process that is particularly essential in teaching
mathematics—and finds that Shulman’s notions of PCK and the pedagogical reasoning
and action cycle miss or obscure its significance. And third, this study introduces the
notion of shared reflection to Shulman’s model for pedagogical reasoning and action
when it is applied to teacher education. It also finds that the act of going public with
one’s ideas through shared reflection can be a useful tool for teacher educators in the

development of their pedagogical content knowledge.

Complex Ways of Thinking About Representation
In his definition of pedagogical content knowledge and his description of the
transformation phase of his pedagogical reasoning and action cycle, Shulman makes

liberal use of the word representation. As a part of PCK, he includes “the most useful

152

¢

forms of representation of [the most regularly taught topics in one’s subject area]”
(Shulman, 1986, p. 9). In describing a teacher’s transformation of subject matter
understanding, Shulman points out that representation “involves thinking through the key
ideas in the text or lesson and identifying the alternative ways of representing them to
students” (Shulman, 1987, p. 16). In this respect, representation is both a product and a
process. These two aspects of representation may be useful to educational researchers and
teachers in a number of content areas, including science (Van Driel at al., 1996),
geography (Ormrod & Cole, 1996), and physical education (Griffin et al., 1996). In
mathematics, however, the word representation has another set of meanings that can
cloud the issue of pedagogical content knowledge.

There are many forms of representations that have been agreed upon by the
mathematics community—such as diagrams, tables, graphs, and formulas. These
mathematical representations can be powerful embodiments of mathematical ideas that
can provide some insight to the mathematics that might not otherwise be possible. Tables
of values can highlights patterns which signify whether a relationship is linear, quadratic,
or exponential. Symbolic representations and their manipulations can be useful in
determining whether two expressions are equivalent and whether one version of a
formula may be used in place of another.

But as students struggle with mathematical concepts and try to make sense of new
ideas and contexts, they sometimes make use of representations that do not carry the high
level of mathematical exactitude as mathematical representations do. However, these
empirical representations of the mathematics can also be quite powerful tools in helping

them make sense of what they are struggling with or to empirically justify an idea that

153

they may have previously seen. For example, most of the preservice teachers in my study
entered the class having some knowledge of how the area of a circle could be calculated

(various students reported on the first day of class that a formula that could be used was
“A = 1: r2,” “A = pie * r2,” or “A = %1r (r) squared”). After students used the radius square

method of estimating the area of the circle, virtually none of the students made these
mistakes again. Although it was not a formal proof of why the formula A = 1: 1'2 makes
mathematical sense, this empirical representation (covering the circle with pi radius
squares) helped to make the formula more plausible.

When I began to analyze the data I collected, I had very little appreciation for the
various types of representations I would encounter. First, I didn’t realize that there were
important differences between the mathematical and the empirical representations that I
and my students used. These differences became apparent, however, when I found that
some of the representations that students found the most powerful (using the radius
squares to cover a circle and making and squishing a clay sphere to see how its volume
related to a corresponding cylinder) were the ones that were far from rigorous. The
empirical representations were never offered as mathematical proofs, but instead as ways
for students to develop some informal sense of the formulas and other ideas we were
studying.

Second, as a mathematics teacher, I was aware of the use of the word
representation in mathematics. However, this prior understanding became problematic
for me as I encountered the multiple ways in which Shulman and others wrote about
representation with respect to pedagogical content knowledge. And even though Shulman

describes representation as both a product and a process, he is not referring only to types

154

of mathematical representations—as I initially believed—but to empirical representations
as well. As Shulman elaborates, “Since there are no single most powerful forms of
representation, the teacher must have at hand a veritable armamentariurn of alternate
forms of representation, some of which derive from research whereas others originate in
the wisdom of practice” (Shulman, 1986, p. 9).

If the language of pedagogical content knowledge continues to be applied to
mathematics education, more care must be given when the word representation is used
and there needs to be an increased awareness of the specialness of mathematics in this
regard. In describing PCK generically across the content areas, Shulman introduced some
language that confounds the issue when applied specifically to mathematics. With respect
to its importance in the transformative process of PCK, representation is a critical
element, but the term also masks the complexity behind considering mathematical vs.

empirical representations.

Growing Appreciation of Task Design

While the transformational phase of Shulman’s model of pedagogical reasoning
and action may prove problematic in its overgeneralization of the concept of
representation when applied to mathematics, it is also shortsighted in its omission of one
of the more critical elements of good mathematics teaching—task design.

A task can be designed with one of two major objectives in mind—as an
opportunity to perform or as an opportunity to learn (Shannon, 1999). When it’s intended
as an opportunity to perform, the task allows students to demonstrate something that they

have already learned, often in a new context. When the task is designed as an opportunity

155

to learn, it is intended to build upon prior classroom experiences for the students’
acquisition of new knowledge. Whether a task is constructed as an opportunity to
perform, an opportunity to learn, or both, it requires that teachers have a firm
understanding of students’ content knowledge, the mathematics of the task, and the ways
in which the task is designed to elicit student responses. As such, it is helpful to look
more carefully at the transformation phase of Shulman’s pedagogical reasoning and
action cycle to find if and to what degree it can be applied to task development.

As described by Shulman, the transformation phase—in which teachers develop
their pedagogical content knowledge through transforming what they know about a
subject into ways that their students can understand it—consists‘of four essential
processes: preparation, representation, selection, and adaptation. The preparation phase
focuses on reviewing the texts and other materials of instruction and examining the
accompanying educational purposes—as determined by the curriculum, the school, or the
teacher. The attention is clearly on using the resources that are available and determining
whether they fit what the teacher deems as appropriate. In the representation phase, the
teacher begins to consider various ways to represent an idea in transforming what he or
she knows about the content. The selection phase refers to the teacher’s consideration of
various instructional methods for presenting an idea to students—including lecture, group
work, projects, and discovery learning. And the adaptation phase concerns the ways in
which the teacher fits an activity to his or her students’ needs, making decisions based on
ability, culture, prior knowledge, skills, and other important factors.

Throughout each of these stages, Shulman describes ways in which the teacher

makes decisions about what he or she teaches and how the subject matter is presented.

156

This process is common to good teachers in all subject areas and it is helpful in
describing some of the basic commonalities of pedagogy from a general viewpoint. But
in attempting to apply the general to a specific case (such as mathematics), some
shortcomings begin to appear. Since a number of mathematics educators have described
their classrooms and their curricula as problem-based or problem-centered, and since an
increased attention is being given to the role that assessment plays in student learning, it
is understandable to consider the nature of task development and task design with respect
to Shulman’s model.

Unfortunately, the model is never very helpful in explaining what the teacher does
in designing tasks. Some of the process is hinted at in the preparation and adaptation
phases, but never to the degree that is necessary in making detailed decisions about
language, diagrams, types of questions (e. g., divergent vs. convergent), and format. In
Chapter 4, I described Version A of the trapezoid task I presented to preservice teachers
and the difficulties I had in eliciting the kinds of responses I anticipated from this task.
After looking at student work, designing an in-class activity with more explicit questions
around the task, and looking back on Version B of the task as it was presented to me in
my undergraduate studies, I found a number of ways in which Version A was not useful
in reaching the learning and process goals I desired. Version A was an example of a task
that was replete with miscues—misleading and rniscommunicative elements that
undermined students’ opportunities to perform and to learn. Examples such as this can be
helpful for teachers and teacher educators in learning to design tasks which: (a) use

vocabulary that is both understandable and unambiguous, (b) recognize diagrams as

157

either drawings (for specific examples) or figures (to represent a larger class of shapes),
and (c) are generally helpful in achieving the teacher’s learning goals for the students.
This process of instruction, reflection, and new comprehension is clearly outlined
in Shulman’s model of pedagogical reasoning and action, but the model falls short in
capturing the practice of initial task design and the difficulties that are entailed in making
important decisions even at the level of word choice and sentence structure. Perhaps what
is needed is a mathematics-specific version of Shulman’s model of pedagogical reasoning
and action, one that addresses the common aspects of good teaching across various
subject matters while attending to specific needs—like task design—of the mathematics

community.

Shared Reflection: Going Public

The decisions I made about what to teach and why, the struggles I encountered
when lessons did not go as planned or students interpreted my questions differently than I
intended, the alterations in lessons and tasks based on my past experiences and current
student feedback—these were all a regular part of my personal reflection as a teacher.
But when I decided that I wanted to attend to issues of teaching along with those of
content in MTH 202, I found that I could address these multiple concerns by sharing
those reflections with students.

I originally planned to use shared reflection with preservice teachers as a way to
focus on the development of their pedagogical content knowledge. With this approach, I
would act as an instructor of pedagogical practice—giving examples of ways in which I

transformed my understanding of the content in making it leamable for students. But

158

what I didn’t consider was the effect that this process of “going public” with my
experiences would have on my own PCK development. More than a mode of instruction,
shared reflection became another important stage in my growth as a teacher educator.

Instead of just making decisions about what to teach, I was now having to make
decisions about what to tell preservice teachers about those pedagogical decisions. For
example, when students responded to my initial trapezoid task by making measurements
of the drawing that was provided and not realizing that the shape was intended as an
example of a larger family of shapes, I went public the next class meeting, explaining that
the task was not a good indicator of what I wanted students to learn from the task or what
they should use as future teachers. But instead of telling students what I thought was
wrong about the task and what I hoped they would do with it, I decided that they might
benefit from questions that were a little more direct and exemplary of the ways I wanted
them to interact with the problem. I used some of the statements made by the students in
the class, framed as questions, to get at some of the issues of similarity, angle and side
length relationships, and applications of other mathematical ideas that many students did
not use on their own. I thus had over one hundred possible statements that I could have
used, but I had to make decisions about which ones would address the underlying
mathematics that I wanted to focus on and which ones would best move the students
toward their developing notions of similarity.

This process was helpful to me in sorting through what I thought were the critical
issues that I wanted to cover. Instead of opening a class discussion on what interesting
findings students may have noticed, I worked to focus their attention on several key

issues that I felt were a better fit in the learning trajectory I envisioned for the class.

159

These were issues that I might have taken up through my own personal reflection, but the
knowledge that I would have to justify these reasons for a classroom of future teachers
forced me to think harder about the decisions I was making and the reasons behind those
decisions.

Shared reflection can be a useful tool for the teacher educator who is interested in
attending to issues of both content and pedagogy. In particular, it is a powerful device for
helping the teacher educator frame issues that are important in the subject matter of

pedagogical content knowledge.

Discussion

Helping preservice teachers begin to establish their pedagogical content
knowledge is a worthy goal. Practicing teachers often struggle with making mathematics
meaningful for their students and some attention to their own PCK can assist in this
endeavor. And while its development naturally builds on lessons learned through
experience working with kids, PCK can begin to be addressed when teachers are learning
about teaching—most notably when they themselves are struggling with content and
learning to make connections within the domain of mathematics.

The teacher educator is also a teacher. His or her experience teaching
mathematics to prospective teachers also constitutes a site for the study of PCK
development. In this study, I set out to examine the ways in which I, as the teacher
educator, could use what I learned about my own PCK development with an eye on
attending to preservice teachers’ PCK. I found that my own PCK was underdeveloped in

many ways—in thinking about how various representations might enable students to

160

learn and understand mathematics, in addressing the importance of reasoning and
justification for preservice teachers and their future students, in looking at what is
involved in designing and implementing a task, and in considering the importance of
students’ abilities to generalize from a specific case to a larger class. The development of
my own PCK was thus both a model for preservice teachers in thinking about their own
PCK development and an important factor in making educational decisions about the best
avenues to follow in teaching MTH 202. The shared reflection further focused this
development for me as I thought about how much to let students in on my decision-

making process and to what extent they should be aware of the complexities of teaching.

Pedagogically Useful Mathematical Understanding

In recent years, some members of the mathematics education community have
begun to consider the interplay between subject matter knowledge and pedagogical
knowledge fi'om a somewhat different perspective. PCK, as it is widely understood,
represents a specialized middle ground between the content and the pedagogy. PCK gives
attention to the ways in which a teacher makes his or her content knowledge accessible to
students, thus recognizing the importance of the mathematics from the perspective of
teaching it. In this new perspective, that relationship is somewhat reversed, understanding
that the teaching of the mathematics is an important element in the perspective of
understanding the content.

Ball and Bass describe this not as a new knowledge base, but an important
subdivision of subject matter knowledge, labeling it pedagogically usefitl mathematical

understanding (Ball & Bass, in press). In other cases, people have talked about knowing

161

mathematics for teaching or applied mathematics for teaching. Whichever terminology
becomes widely-used, the underlying concept is relatively the same—that the important
element is the understanding of the mathematics in ways that are significant for teaching
it.

For years, mathematicians have talked about what mathematics is important for
specific professions. There are calculus courses that are specifically designed for the
engineer, geometry and spatial reasoning courses for the architect, and accounting
courses for the business major. There are also mathematics courses for the elementary
teacher yet, as many have found, the mathematics that is taught in these courses is not
regarded as a unique knowledge base for a specialized profession. Some researchers in
mathematics education have begun to address this issue by describing the mathematics
that teachers (and most specifically, elementary school teachers) need to know in terms of
what is pedagogically usefirl.

Ball and Bass contend that PCK is an important field of study, but that it does not
attend to all of the complexities of the practice of teaching. They argue that since
teaching often involves addressing more than one mathematical idea at once and, in many
cases, within the context of a new problem or novel situation, no amount of PCK can
adequately prepare a teacher for all of the circumstances which they may encounter. For
this reason, they argue, attention should be focused more on the underlying mathematics
that is involved and helping teachers understand it more completely. Unlike Ma (1999)
who describes the importance of teachers developing a “profound understanding of

fundamental mathematics,” Ball and Bass advocate a profound understanding of

mathematics that is pedagogically useful.

162

My understanding of PCK, however, does not employ such a narrow view of the
domain. PCK does not downplay the importance of reasoning nor does it advocate that
teachers establish a finite repertoire of strategies and answers. Instead, PCK is an attempt
to describe the process that teachers employ in making their content knowledge both
teachable and leamable. It recognizes that no two teaching situations are alike, and thus it
encourages teachers to develop a flexible knowledge of what it takes to teach the
mathematics—for example, by understanding ways that multiple representations of the
content can be used and understood, and by recognizing the complexities involved in

designing a task to meet the needs of the students at a particular time.

Using PCK As the Lens

This, then, raises the question of what a focus on a teacher educator’s pedagogical
content knowledge development can distinctly contribute to the field of mathematics
education. While it has helped to raise a number of issues for me in my teaching, I must
admit that attending to my PCK development was probably not the only avenue I could
have taken to realize these issues. It seems as if these concerns could also be
addressed—and perhaps more effectively—by centering on the content and making
decisions about what understanding is pedagogically useful. Ball and Bass (in press) talk
about the importance of decompression—a teacher’s ability to deconstruct his or her own
mathematical knowledge to elements which can be flexibly reorganized and adapted to
different educational purposes. Since teachers must contend with students’ developing

knowledge, they must be prepared to help students unpack the pieces which make up that

163

knowledge. To do so, the teacher must be confident and capable of looking at his or her
own knowledge in the same way.

It is important to consider how mathematics pedagogy may be both illuminated
and confused by using the lens of PCK. First, by giving attention to transformation—the
phase of the pedagogical reasoning and action cycle that centers on PCK
development—the teacher educator can increase his or her understanding of mathematics.
For me, this occurred in a number of places—like learning to distinguish between
mathematical and empirical representations and how these forms of representations relate
to the ways in which Shulman and others talk about representations, becoming more
aware of the ways in which mathematical language can interact with symbolic and spatial
representations, and distinguishing between drawings and figures and learning to use
them in appropriate ways when developing tasks. These were issues that may have been
raised even without focusing on my PCK development, but because I was using PCK as
my framework, it gave me some useful language to talk about the process and to relate
my personal experiences with those that are a part of the larger educational research
community.

Second, it forced me to consider whether Shulman’s model of pedagogical
reasoning and action was a useful framework specifically for mathematics and the work
that teacher educators do in mathematics education. I found that Shulman’s categories
were helpful, but also problematic when applied to mathematics—in the ways that the
multiple uses of representations could easily be confused and in how task development (a
critical element in mathematics instruction) is not distinctly attended to. I also found that

the pedagogical reasoning and action cycle is a good model even for teacher educators,

164

although it could be augmented with shared reflection to help move preservice teachers to

think of themselves not only as students, but teachers as well.

Limitations

As I recognized fiom the beginning, the fact that this study was done from the
first-person perspective provided me with some special insight to the subject in ways that
would not have been possible had the study been done by an outside observer. By the
same token, there were some limitations raised by the design of the study.

As much as I tried, it was very difficult to completely remove all subjectivity
from the analysis of my data. For example, even though I used pseudonyms for my
students when writing about their work in Chapter 4, I knew exactly who they were and
what my interactions were with them throughout the semester. In most cases, I would
have been able to identify their work by looking at the handwriting alone. This might
have been avoided had I analyzed the data afier a longer time period had elapsed or if
another researcher had chosen student work to analyze, rewritten it in his or her own
handwriting, or done some of the analysis of the student work him- or herself. This,
however, would have taken away from some of the richness of the data afforded by the
design of the study. One of the benefits of doing a self-study is proximity between the
subject and the researcher. More details can be gleaned, attitudes and feelings may be
recalled, and connections between seemingly disparate ideas might be better realized
when the researcher has such an intimate relationship with the data. While this proximity
can be a detriment to the effectiveness of the research, this limitation can be lessened

with careful acknowledgement of the difficulties from the onset of the study.

165

As a corollary, this mode of qualitative research carries the inherent problem of
sample size. As far as sample sizes go, it can’t get much less than n = 1. This can be a
limitation of the study if implications are made that the subject is representative of a
population and that lessons learned from the study are meant to be generalizable to an
entire population. This was never the goal of my study. As with most case studies, my
intent was to look closely and deeply at a particular event-in this case, my teaching of a
mathematics content course for preservice teachers. The event was also a case of the
teacher educator thinking hard about what constitutes his own PCK development in the
service of furthering the development of his own students’ PCK—a particular event that
had not been investigated in the research literature prior to this study. It was meant to
raise questions concerning the feasibility and implementation of such a course design and
not to imply that any other teacher educators use—or should use—the same approach. It
would be extremely helpful, however, to learn of some of the ideas learned from this
study can have an impact on other teacher educators and their students.

Another limitation around generalizability concerns the mathematical content of
the MTH 202 course described in the study. Since the course centered around geometry,
certain issues were naturally raised that contributed to this study that might not have been
so critical an issue had the course focused on another area of mathematics. Two such
examples come to mind: issues of representation and proof. While multiple forms of
representation are common in some other areas of mathematics, geometry seems to be
one of the most obvious mathematical subjects in which this issue becomes relevant. One
of the representations that played the biggest role in my analysis was the spatial

representation—a form that, while not unique to geometry, is hard to separate from it.

166

Likewise, the symbolic representations were an inherent part of geometry, through the
use of formulas for area, perimeter, circumference, volume, and surface area. On the
other hand, had the study been done in a course on algebra, it is clear that multiple forms
of symbolic representations would have played an important role, but not all non-
geometric symbolic forms lead to easily-accessible corresponding spatial representations
(e. g., polynomials of order higher than 3 and exponentials).l The fact that this course was
focused on geometry seemed to enable the study of representations better than other
topics would have.

Proof falls even more clearly as a particular focus of geometry than does
representation. Formal and informal proofs have long been a part of the study of
geometry and were clearly candidates for further analysis when considering the
mathematics content of MTH 202. But when proofs are considered a part of the broader
study of justification or even mathematical reasoning, all mathematical topics are fair
game. However, it is clear that the word “proof” carries with it some negative
connotations for a number of preservice teachers within the context of geometry. Issues
of justification, then, might be easier to address in a course that is centered around a study

of probability, number theory, or some other area of mathematics.

 

' Unless you consider graphs of these functions as a form of spatial representation. I tend to think of
graphs as a different category of representation in and of itself.

167

APPENDICES

168

APPENDIX A

Class Meetings with Mathematical and Pedagogical Content

169

 
     

 

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1/12/99 Comparing and Understanding definition of
Scaling 1.3 perimeter and area
Comparing shapes with same
area/perimeter
Giving basic formulas
(ggralizationsl
1/14/99 Comparing and Investigating constant area, Understanding some of the
Scaling 3.1, 4.1 changing perimeter history behind the NCTM
Investigating constant perimeter, Standards and mathematics
changing area reform in the US
Seeing that most “square-like” Developing and maintaining
shape has smallest perimeter for good classroom discourse
given area, largest area for given
perimeter (max/min)
1/19/99 Comparing and Discovering various estimating Establishing good
Scaling 2.1 methods for area of odd shapes definitions (e.g., formula vs.
equation)
Managing classroom data
collection
1/21/99 Comparing and Finding areas of parallelograrns Becoming familiar with
Scaling 5.1, 6.1, and triangles some of the commonly used
trapezoid formula Understanding how these area (or mentioned) mathematics
derivations formulas make sense curricula (e.g., Saxon,
Subdividing shapes into other UCSMP)
shapes to find areas Understanding “geometry”
Looking at one specific as more than the content of
reasoning of the trapezoid area high school geometry
formula derivation course
Looking at multiple
strategies for solving a
problem
Using concept maps to
assess students’ baseline
understandimf a subject
1/26/99 Measuring in Measuring and reading Helping students to be more
inches and centimeters and inches on rulers focused in writing about
centimeters, (Quiz mathematics—differentiatin
l) g between a learning log
and ajournal
1/28/99 Comparing and Understanding formulas for Discussing and using
Scaling 7.2, 7.4 circumference and area of circles etymologies of
(area as pi radius squares) mathematical terms

 

170

 

 

2/2/99

lat \ l\l \)
Circle area formula

Mathematical ('nnlcnt
Cutting a circle into smaller and

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Helping students learn

 

 

 

 

 

 

 

 

 

 

 

derivation, smaller wedges and rearranging metric prefixes using
Stretching and them to validate area formula whimsical situations
Shrinking 1.1 Developing informal Helping students make
understanding of similarity connections between
(lengths, angles, and area) mathematical ideas and
Identifying corresponding parts representations
of similar shapes Using manipulatives to help
students make sense of
some mathematics
2/4/99 Test 1
2/9/99 Stretching and Drawing shapes on coordinate Making sense of new
Shrinking 2.1 grids student strategies for
Formalizing understanding of solving problems
similarity (lengths, angles, and Understanding the purpose
general shape) of assessment
Using basic algebraic rules to Constructing good
describe a transformation assessments of student
Understanding scale factor understanding
2/11/99 3 rubber bands - Using coordinate geometry to Developing alternate
show using dot understand how shapes change representations to help
paper, Stretching with two or three rubber bands students understand a
and Shrinking 2.2, Understanding similarity using concept
2.3 ratios of corresponding sides
Using algebraic rules to
understand more complex
transfonnations of size change
and translations
2/16/99 Isosceles trapezoid Investigating answers to an Providing extensions to
observations, (Quiz open-ended question questions
2) Looking at interesting
relationships between angles,
lengths, etc. of a given shape
Developing hypotheses for
general cases from one example
(ggd and bad)
2/18/99 Stretching and Understanding how area Dealing with parental
Shrinking 3.2, 3.3 increases as the square of the misunderstanding and
scale factor community backlash to
mathematics reform
Using mathematical
recreations to develop
mathematical understanding
Using various tools to
measure and to create
circles
2/23/99 Basic conversions Looking at basic conversions Using student work to
and symbols used, within systems of measurement summarize class discoveries
Stretching and Investigating some real-world
Shrinking 4.3, 4.4 applications of similarity and
scale factor
2/25/99 Test 2

 

171

 

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Mathematical ("outwit

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NCTM Standards,
Stretching and
Shrinking 5.1, 5.2,
5.3

Using similar triangles to solve
various real-world problems

Reading and applying the
NCTM Standards to
course’s content at different

grade levels

 

 

 

 

 

 

 

 

 

3/4/99 Analysis of typical Creating flat patterns for unit Analyzing the typical class
class lesson, cubes and other simple boxes lesson for this section of
Filling and MTH 202 and using it as a
Wrapping 1.1, 1.2 model for writing lesson
plans
3/16/99 Flat pattern review, Reviewing concept of flat Using “guided discovery” in
Filling and patterns teaching students
Wrapping 1.3 Moving between flat patterns
and the boxes that are made from
them
3/18/99 Drawing oblique Drawing oblique and isometric Utilizing various models for
and isometric perspective drawings of various creating lesson plans
perspectives, (Quiz boxes and other simple
3) rectangular shapes
3/23/99 Filling and Finding various dimensions for Understanding various ways
Wrapping 2.1, 3.1 boxes with a given volume to establish connections
Establishing an informal within mathematics,
understanding that a cube is a between mathematics and
box with the smallest surface other subject areas, and
area for a given volume between mathematics and
Connecting surface area/volume the real world
relationship to perimeter/area
relationship
3/25/99 Connections with Validating a volume formula for Finding and discussing
geometry, Filling prisms ways in which geometry
and Wrapping 3.3 Investigating volumes of various content of MTH 202
prisms with a given lateral connects with other content
surface area, looking for areas
generalizations Being a reflective
practitioner
3/30/99 Cylinders - Filling Finding volumes and surface Understanding accuracy of
and Wrapping 4.2, areas of cylinders measurement and validating
ACE 13, 9 Connecting the concept of student answers
volume of cylinder to the Looking at mathematics
maximizing of prism volumes problems from different
with a given lateral surface area geometric perspectives
4/1/99 Big ideas in Identifying big mathematical Reading geometry and
Investigations, ideas in Investigations measurement units from
(Quiz 4) Investigations curriculum in
grade level groups
4/6/99 Filling and Discovering relationships Altering tasks for multiple
Wrapping 5.1, 5.2 between volumes of cylinders, purposes
spheres, and cones with Maintaining professional
congruent radii and heights development through
Validating volume formulas for NCTM teaching journals

 

 

spheres and cones

 

 

172

 

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4/8/99

'l‘usl‘t \)
Revision of grade
level big ideas,
Filling and
Wrapping 6.1, 6.2

Applying idea of scale factor to
solids

Understanding that the surface
area grows as the square of the
scale factor and the volume
grows as the cube of the scale
factor

Pedagogical ('nlltcltl l’( |\

Analyzing teacher texts
from Investigations to
identify big ideas in
geometry and measurement

 

4/13/99

Using literature in
math class,
Looking for
Pythagoras 2.1

Developing informal ways to
determine areas of shapes drawn
on square dot paper

Establishing that the “surround
and subtract” method is an
efficient strategy that can be
used in all cases

Finding ways in which
mathematical ideas from
MTH 202 get played out in
children’s literature

 

4/15/99

Looking for
Pythagoras 2.2,
2.3, (Quiz 5)

Finding all of the different sized
squaresthatcanbedrawnon3x
3 dot paper grids

Establishing the length of the
side of a square as the square
root of the area

Understanding irrational
numbers

Finding the length of a line
segment by relating it to the
square that has the segment as a
side

Using group quizzes to
assess student learning

 

4/20/99

Looking for
Pythagoras 3.1

Using dot paper to draw squares
of various sizes on given line
segments

Discovering the Pythagorean
Theorem using squares on dot
paper

Teaching students
elementary notions of
reasoning and proof

 

4/22/99

Test3

 

4/27/99

Pythagorean
Triples

Finding various whole number
triples which satisfy the
Pythagorean Theorem
Looking for patterns in
Pythagorean Triples to try to
develop rules or formulas for
generating other Pythagorean
Triples

Using concept maps to
assess grth in student
understanding

 

 

4/29/99

 

Pick’s Theorem

 

Looking at ways in which
mathematical formulas are
developed and/or discovered
Finding a generalization for
shapes drawn on dot paper and
their areas—Pick’s Theorem

 

Using formulas with
elementary students

 

173

 

APPENDIX B

MTH 202 Grading Policy

174

Homework will be assigned, but not collected or graded. Five quizzes over the
homework and classwork will be given. Each quiz will count 30 points for a total of 150
points. In addition, there will be one take-home projects which will require writing and
some research and will be worth 100 points. Students will also be required to turn in their
learning logs on a regular basis (about 8 to 10 times during the semester). Learning logs
will be evaluated and the entire logbook will be worth 100 points at the end of the
semester. There will also be three 100 point tests and a 150 point final exam, for a total of
800 points. The final will cover the entire course. Some quizzes and tests may have a
group problem solving component.

Grades will be determined at the end of the course on a total point basis. That is,
your quiz and test scores, your project grade, your learning log grades, and your frnal
exam scores will be added together to determine your grade. The percentage of the total
number of points that you will have accumulated will be calculated. Grades will be

determined as follows:

Points Percentages Grade
716—800 90- 100 4.0
676-715 85 -89 3.5
636 — 675 80 - 84 3.0
596 — 635 75 - 79 2.5
556 — 595 70 - 74 2.0
516—555 65-69 1.5
476—515 60-64 1.0

0 - 475 0 - 59 0.0

175

APPENDIX C

Initial E-Mail to Students

176

You are presently enrolled in the section of MTH 202 that I am teaching during the spring semester. I
thought I would let you know a little about the course before the class starts.

MTH 202 is a geometry course for elementary education majors and this will be the fourth semester that I
will have taught this course. As for myself, I am a doctoral student in mathematics education. I am at the
stage in my work here at MSU where I am beginning on my dissertation and I have chosen this class as the
site for my research.

What this means for you is that the course you are taking is not only going to be a learning experience for
you, but it will also be a learning experience for me. However, the things that I will be doing in the course
will be no different than what I would be doing if I were not using it for my own research. As a student in
my class, you will have the opportunity to choose whether you want to be a subject in my datanand you
will not have to make that decision until the end of the semester. This way, no one will have to worry that I
would be treating students differently based on whether they wanted to be a part of my study.

The fact that this course is a part of my research does not mean that you will be given extra work to do. As
I mentioned before, the course will proceed as it would have if it were not the site for my dissertation.

Now, a little about the class. My section will differ from the other sections of MTH 202 in several ways.
First of all, it will focus primarily--but not exclusively—-on the mathematics. In addition to the geometry
that we cover, we will also be talking a lot about what it takes to TEACH geometry to elementary and
middle school students. I feel that this is a critical piece for learning the subject matter and I have
approached all of my sections of MTH 202 this way.

To help with this, we will be using a series of books different from the one posted in the book stores (by
O'Daffer). You will be asked to purchase a series of student and teacher books that are a part of the
Connected Mathematics Project--a reform-based curriculum for upper elementary and middle schools. We
will use some of the problems in these books to launch our class discussions and I will often assign other
more challenging problems to follow them up. I have ordered these books myself and they can be
purchased from me (along with a measuring tool called an angle ruler) in my office for a total of around
$35 (the price is yet to be solidified).

And that's about it for now. I wanted to take this opportunity to contact you as early as possible to let you
know about the basic structure of the course, the books we will be using, and the fact that I will be
collecting data on the class for my dissertation. If you have any questions, feel free to contact me either by
email or by calling my office at 432-0054.

I look forward to meeting all of you on Tuesday at (ugh) 8:00am.

Jeff Wanko

177

APPENDIX D

Documentation of MTH 202

178

An important part of my work as a teacher involves examining our class discussions
closely as well as looking carefully at what you write to me and to yourselves. Having
records of our discussions and copies of your work is invaluable to me as a teacher.
These help me to shape the instruction of the class to be responsive to you and also help
me to continue to develop my own practice as a teacher. For myself, I write journal
entries about each of our class sessions that give me a record of my own thoughts and
reflections. I often share what happens in our classes and talk with colleagues about
teaching. These kinds of activities are an important part of learning in and from my own
teaching.

I do these things because I find that they help me to teach well and to continue to develop
my teaching. Although I have been teaching for a number of years now, I find that there
is always more to learn. These activities that I describe above are also an important facet
of being in a professional community and supporting the learning and development of
other teachers. They are the same kinds of things that you will be encouraged to do as an
intem—audiotaping your lessons, collecting students' written work, writing journal
entries about your class sessions, discussing your work with colleagues—and, hopefully,
you will have opportunities, resources, and support to do these kinds of things as a
practicing teacher. In many ways, this is an opportunity for you to get a sense of how one
teacher whom you get to watch closely (me) draws on these kinds of activities and
resources to inform and develop his teaching practice.

The materials we make and collect around this course will only be used for the teaching
of this course and CANNOT be used for research or presentations without your consent.

I understand that the records that are being made of my MTH 202 class—the audiotapes
and written documents—are part of the teaching practice of my instructor and cannot be
used as data for research or as a part of a formal presentation without my consent. These
documents will only be used to support the instruction of this course and the professional
development of my instructor. I understand that other teacher educators may be viewing
these documents for these purposes during the duration of this course.

 

Signature Date

179

APPENDIX E

Informed Consent for Study of MTH 202, Section 1

180

The purpose of this study is to gain a better understanding of the potential of the materials and
approaches utilized in your MTH 202 course to support and facilitate the development of your
content knowledge as well as your pedagogical content knowledge. Through this research, I hope
to learn more about the challenges and benefits of incorporating more innovative materials,
pedagogy, and curriculum to mathematics education of firture teachers. Because of this, I would
like to analyze data collected in your section of MTH 202. The things I would like to use to
conduct the research are:

Audiotapes of classes involving the discussions of the mathematics in the course: I, as a
researcher, will study the kinds of activities and interactions that occurred during the class,
examining what the students and instructor are doing and talking about. Transcripts of some
discussions may be made for the purpose of discourse analysis in the study. I will not identify you
by name in these transcripts and pseudonyms will be used.

Copies of your writing and work for the course: Another important source of information
about what students and instructors do in the course is to look at students’ writing and reflections,
students’ mathematical work, and instructors’ comments. I would like to use students’ work to
study students’ experiences and reactions to different class activities and events, and to analyze
what different students are exploring and learning at different points in the semester. These
formal analyses are for research purposes only. In the study, everything will be labeled with
pseudonyms so that you are not identifiable and the analyses that I conduct will not be used in
any connection with grades, placements, or other decisions about your development and learning
in your teacher education program. You have the option to provide all or any portion of your
work for the purpose of this research.

Some self-studies on learning: In TB 150, students are asked to pick one other class in which
they are concurrently enrolled and are asked to do a self-study of their own learning in that class.
If you are currently taking TB 150 and have chosen your MTH 202 class as the site for your self-
study, I invite you to share your self-study with me for the purposes of my own research. This is
done entirely on a voluntary basis and I will not ask for nor look at your self-study until after
MTH 202 is completed and final grades have been determined.

If you desire further information about this study, you may call Dr. Sandra Wilcox at 355-1741 or
Jeffrey Wanko at 432-0054. Results of this research will be shared with you on request.

Permission to participate and permission to use data collected in MTH 202 for research:

I have read the previous statement and agree to participate in this research by giving my
permission for Jeffrey Wanko to analyze this information collected during my MTH 202 class:
audiotapes of class sessions and photocopies of my class work. I understand that I may provide
all or only a portion (as opposed to all) of my work for research purposes, if I so desire. I also
understand that if I am also taking TB 150 and have chosen MTH 202 for my self-study, I may
also give permission to use my self-study as additional data for research. I understand that my
real name will never be used in any written reports of the research and my responses will be
treated with confidentiality to anyone outside of the investigators of the project.

Signature: Date:

 

181

APPENDIX F

Learning Log Prompts

182

Learning Log Prompt #1 _ Due 1/14/99

1. Write down the first three geometry formulas you think of and tell what they are the
formulas for (you will not be graded down if there are mistakes, but try to be
accurate). Describe some of the pros and cons for using mathematical formulas.
What do you think about using formulas with elementary school students?

Learning Log Prompt #2 _ Due l/21/99
2. Describe in detail an event from your high school geometry course which is
particularly memorable to you. Why do you think this event sticks with you?

Learning Log Prompt #3 - Due 2/2/99

3. One measurement attribute is weight. Some units of measurement of weight are
pounds, grams, and ounces. Make a list of as many measurement attributes as you
can and beneath each attribute list as many different appropriate units as you can
(customary, metric, and any others that may apply). Which of these units do you
think a third grader should know and understand and why?

Learning Log Prompt #4 - Due 2/11/99
4. Here is a special isosceles trapezoid. Write down all of the interesting things you
can find about this isosceles trapezoid (Figure 51).

/ \

Figure 51: Learning Log Trapezoid

 

Learning Log Prompt #5 - Due 2/23/99

5. When the midpoints of the sides of a quadrilateral are connected in order, another
quadrilateral is formed on the inside. What kinds of quadrilaterals can be formed on
the inside? (That is, can the quadrilaterals be squares? rectangles? parallelograrns?
rhombi? trapezoids? kites? irregular quadrilaterals? etc.) Which original
quadrilaterals produce which types of quadrilaterals on the inside? Experiment by
drawing many different quadrilaterals and connecting the midpoints of their sides in
order. Use these examples to make some general conclusions.

183

Learning Log Prompt #6 - Due 3/4/99

6.

In the draft of the NCTM Standards 2000, Standard 3 (Geometry and Spatial Sense)
and Standard 4 (Measurement) relate to some of what we have been discussing in
MTH 202 this semester. Pick one grade level band (either PreK-2 or 3-5) and
describe the things that we have talked about in class that are also recommended to
be taught in both Standards 3 and 4 in the Standards 2000.

Learning Log Prompt #7 - Due 3/25/99

7.

In the NCTM Curriculum and Evaluation Standards (1989), the notion of
“mathematical connections” is highlighted. We can think about connections
between school mathematics and real world situations, connections between
different areas of mathematics, and connections between mathematics and other
school subjects. Consider this last approach to connections and give examples of
some of the ways in which geometry and measurement connect to other school
subjects (possibly including science, social studies, language arts, music, etc.).

Learning Log Prompt #8 - Due 4/13/99

8.

In the article “Measuring Up with The Principal’s New Clothes” (fiom Teaching
Children Mathematics, April 1999, pp. 476—479), a teacher describes how she has
taken a children’s literature book and developed a mathematics lesson about
measurement around it for her third and fourth grade students. Describe another
children’s literature book or story and the lesson you could develop around it to use
in your elementary classroom. Make sure that you identify the grade level and
mathematical objective for your lesson, as well as brief description of the lesson
you would use (it does not have to be about geometry or measurement, although
that connection would be preferable).

Learning Log Prompt #9 - Due 4/20/99

9.

Describe in detail one event from your MTH 202 course which is particularly
memorable to you. Why do you think this event sticks with you?

Learning Log Prompt # 10 - Due 4/29/99
10. Write down all of the geometry formulas you know and tell what they are the

formulas for (do not look these up, do as many as you can from memory and try to
be accurate). Describe what you have learned this semester about using formulas
and explain some of the power and some of the problems of relying on formulas.

184

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