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RHYME AND REASON:
A RHETORICAL, GENEALOGICAL EXAMINATION OF UNDERGRADUATE
MATHEMATICS
By

Sharon K. Strickland

A DISSERTATION

Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Curriculum, Teaching and Educational Policy

2008

ABSTRACT
RHYME AND REASON:
A RHETORICAL, GENEALOGICAL EXAMINATION OF UNDERGRADUATE
MATHEMATICS
By

Sharon K. Strickland

This is a rhetorical and genealogical examination of four undergraduate
mathematics courses: Real Analysis, Advanced Geometry, Complex Analysis, and
Discrete Mathematics. The framework of the study is rhetorical because I read course
artifacts from the perspective of literary analysis. The framework is also genealogical
because I was interested in the construction of subjectivity as performed in each of the
courses. To explore these issues I observed the four courses over a semester, looking at
the ways students and professors interacted, how mathematics operated in the courses,
and how supporting texts and documents such as textbooks, syllabi, course web pages,
and course handouts contributed to the visions of teaching and mathematics presented in
each course. This research presents an analysis of the construction of the subject (where
the subject is undergraduate mathematics students) in these four courses to highlight the
differences in pedagogical and mathematical approaches across them.

Epistemologically, the study takes a Foucaultian discourse position in which math
is what math does. The approach of this study rejects a more realist stance in which ‘real’
mathematics exists beyond material practices. Instead, the study allows for multiple
historical and temporal instantiations of mathematics, in which the subject of
mathematics (both student and discipline) is constructed differently in each of the

different settings.

This dissertation is a response to common stereotypes of mathematics courses as
teacher-centered lectures, sites of content knowledge delivery rather than pedagogical
experiences for students. This study takes the position that there is no pedagogy without
content, and there is no content without pedagogy. The study also responds critically to
assumptions that conventional lecture-based mathematics courses tend to be taught in
similar ways. Even though three of the four courses examined here were lecture based,
there was always more than lecture present, and even the lectures constructed different
possibilities for subjectivity.

To analyze the construction of the mathematical subject, I used a four-part
Foucaultian framework that asked: I) What aspect of the students needed to change
(substance), 2) How did the course invite students to take on these changes (mode), 3)
How did the activities (or regimen) of the courses act to initiate that change, and 4) What
a model or perfect outcome of the course might look like (telos).

This study concludes with a consideration of issues of inclusion and exclusion. It
suggests that mathematics is practiced differently across multiple courses and that these
differences allow for varied opportunities for students to engage with mathematics and to
be excluded from it. Mathematicians and teacher educators can benefit from awareness of
assumptions that shape subjectivity in mathematics courses, and such awareness can help

increase access to various types of mathematical thinking.

DEDICATION

For Brad,
Who egged me on,
And for Little Juice,

Who merely threw her eggs.

ACKNOWLEDGMENTS

I am beyond grateful to Lynn F endler who inspired me to take on this project and
helped me immensely as I fumbled my way through the theory and writing. She had faith
when [did not. She also had chocolate when I did not. Thank you Amy Parks, Steve
Tuckey, Brett Merritt, and the rest of the Critical Studies group for your friendship and
intellectual support all through this program. Your often irreverent and usually hysterical
observations sustained my interest in post-modern thought and kept me sane. Your shared
knowledge of science, mathematics, philosophy and life shaped this dissertation.

I also need to express my gratitude to the mentors, advisors, and professors who
have helped me along the way, many of whom I am glad to also call friends: Nathalie
Sinclair, Jon Star, Cleo Cherryholmes, and Sandra Crespo. My first mathematics
professor, Myrtle Lewin, deserves a medal for her dedication to her students. Without her
influence I doubt I would be here at all.

Finally, I want to recognize those who shaped my dissertation in less academic,
but in no less important ways. Nicole Johnson, thank you for your constant friendship and
frequent tech support. You kept my mind off my work when I needed a break and
prodded me forward when I needed to get back on track again. Without you and Mirya
Brion I could have never found the time to complete this research without the comforting
knowledge that my daughter was safe and happy.

I am fortunate to know and love you all.

TABLE OF CONTENTS

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

LIST OF FIGURES ....................................................................................................... ix

INTRODUCTION .......................................................................................................... 1

CHAPTER ONE

CONTENT AND PEDAGOGY IN UNDERGRADUATE MATHEMATICS .............. 10
Introduction ....................................................................................................... 10
Undergraduate mathematics as site of apprenticeship of observation ................. 13
Change in teaching ............................................................................................ 15
Lectures and mathematics education .................................................................. 19
Observing the courses ........................................................................................ 21
Analysis and theory ........................................................................................... 24

Why rhetoric .......................................................................................... 25
Rhetoric to Foucault ............................................................................... 31

Concluding remarks .......................................................................................... 39

CHAPTER TWO

REAL ANALYSIS: CONVERTS ................................................................................. 41
Introduction ....................................................................................................... 41
Prelude to the analysis ....................................................................................... 43
Substance: Epistemology ................................................................................... 45
Mode: Belonging ............................................................................................... 58
Regimen: Holding court .................................................................................... 66
Telos: Converts ................................................................................................. 72
Concluding remarks .......................................................................................... 76

CHAPTER THREE

ADVANCED GEOMETRY: FUTURE RESEARCHERS ............................................ 81
Introduction ....................................................................................................... 8 1
Prelude to the analysis ....................................................................................... 83
Substance: Self-concept ..................................................................................... 84
Mode: Aesthetics ............................................................................................... 93
Regimen: Play ................................................................................................. 101
Telos: Researchers ........................................................................................... 11 1
Concluding remarks ........................................................................................ 114

CHAPTER FOUR

Complex Analysis: PLATONIC DISCIPLES .............................................................. 117
Introduction ..................................................................................................... 1 17
Prelude to the analysis ..................................................................................... 1 18
Substance: Well-ordered rationality ................................................................. 119

vi

Mode: Pleasure in orderliness .......................................................................... 129

Regimen: Following rules and staying in line .................................................. 137
Telos: Platonic disciples .................................................................................. 145
Concluding remarks ........................................................................................ 148

CHAPTER FIVE

DISCRETE MATHEMATICS: TEACHERS .............................................................. 151
Introduction ..................................................................................................... 1 5 1
Prelude to the analysis ..................................................................................... 152
Substance: Self-concept ................................................................................... 154
Mode: Belonging ............................................................................................. 165
Regimen: Pedagogically effective prose .......................................................... 172
Telos: Teachers ............................................................................................... l 8 1
Concluding remarks ........................................................................................ 184

CHAPTER SIX

WHAT MATHEMATICS IS ABOUT ........................................................................ 188
Who does mathematics? .................................................................................. 198
Who was excluded from the course? ................................................................ 200
How was the course accessible? ...................................................................... 203
How did the courses speak to teacher education? ............................................. 206

REFERENCES ........................................................................................................... 212

vii

LIST OF TABLES

Table 6.1: Summary of analysis ...................................................................................... 197

viii

Figure 2.1:

Figure 2.2:
Figure 2.3:
Figure 2.4:
Figure 3.1:
Figure 3.2:
Figure 3.3:
Figure 4.1:

Figure 5.1:

LIST OF FIGURES

Dr. RA’s sketch Of x2 ..................................................................................... 49
Students’ sketch where f(a) and f(b) were maximums ................................... 69
Students’ sketch where f(a) = f(b) .................................................................. 69
Students’ sketch where f(a) and f(b) were minimums ................................... 70
Dr. AG’S emailed sketch ................................................................................ 93
Student’s hyperbolic square ........................................................................... 98
Dr. AG’s triangle .......................................................................................... 102
Dr. CA’s illustration of the points equidistant to a and b ............................. 122
Dr. DM’s pair and pairing illustrations ........................................................ 163

INTRODUCTION

My plans have changed often Since high school when I first declared to teachers,
family and friends that I wanted to be a literary critic and thus an English major. But to
their (and even my own) Shock I emerged from college with majors in mathematics and
classical languages. Despite my elaborate personal goals, I continued to find new
interests to thwart my previous plans at each next step in my education. Shortly after
college I began to teach and quickly became intrigued by the world of education. This led
to a master’s degree in secondary education and a minor in mathematics. Still not
satisfied and wanting to learn more I began a Ph.D. program anticipating a focus on ways
to make teaching mathematics better and so improve students’ learning. While taking a
seemingly innocent (in my opinion, at least) class during my second year I encountered
new ideas that would soon shift my academic interests yet again—Foucault and
postmodern thought. This in turn led to a desire to learn more philosophy in general.
Fortunately, amid my many academic meanderings and seeming digressions, analysis and
commentary (philosophical pursuits, both logical and rhetorical) continue to delight and
so anchor much of my work.

This dissertation takes my major academic interests into consideration—
mathematics as the discipline, Foucault (postmodern) and Aristotle (classical) as the
thinkers supporting the theoretical aspects, and education and teaching as the major site
of the research. These varied interests can combine to inform different aspects of the

project, as I will elaborate.

When I began this project my primary concern was to learn more about the
mathematics classes undergraduate students encounter. My overarching research
questions were: What kinds of mathematics teaching do preservice teachers witness in
upper level mathematics department courses? What could preservice teachers learn
about teaching in these courses? And just as my well crafted plans for my life—to
become a literary critic—changed, so did my thoughts about research and the sort of
study I wanted to write.

At first I was interested in what the students in the classes were learning about
teaching by taking discipline-focused, rather than education-focused, courses.
Specifically, I was interested in the practices that the students in upper-level
undergraduate mathematics courses might witness and therefore might (consciously or
unconsciously) adopt as future practices in their classrooms. To learn more about the
practices witnessed I observed four upper-level undergraduate courses that were either
required by the math department major program or that were often taken by preservice
teachers. Because the students in the teacher education program for preservice secondary
mathematics were required to earn math majors, I had plenty of reasons to suspect that at
least a few future teachers populated each of these upper-level courses. The preservice
teachers were my main focus—what could they be learning in their math courses about
teaching? But also, some of these students might go on to become graduate students in
mathematics or related fields and so might teach undergraduates one day themselves.
They too, then, might be learning from these experiences in their undergraduate courses
and might apply them to their future teaching. They might be building models of practice.

Finally, some other portion of the remaining students (those who were not preservice

teachers or who might not become future graduate assistants, instructors, or professors of
mathematics) might become parents, or community leaders, or businesswomen who come
to care about education even if education is not their primary field. Their expectations of
education for their future children and for students in general could be related to the
experiences they witness in these upper-level undergraduate courses. I had built a whole
net of causal relationships and possibilities on the backs of a few semesters Of
observation.

It is not that I no longer believe that these experiences have the potential for
informing future teaching decisions. I still believe that what anyone experiences informs
what he or she takes to be normal and abnormal, positive and negative, and desirable and
undesirable. Instead, I can no longer think of the relationships as linear or as causal.

My belief in the notion that what a professor does directly influences what a
student will do a year or more down the road of life is problematic for me. On the one
hand such a model of the relationship makes designing a classic research study very
difficult How could my observations of classrooms speak to what the students would go
onto do in their lives? With the limited time and funds of a dissertation I could not take
up the longitudinal study such a model would necessitate. I would not be able to identify
a cause-effect relationship. Instead I could describe the courses and hypothesize about
their future possibilities. This could be done and I do think this sort of work could still
inform researchers of teacher education. But more than the difficulties of research design,
methods, and the limitations of conclusions from that design, my problems are primarily

ones of power and knowledge.

As I mentioned before, I had taken a class in my second year of this graduate
program that began to unravel some of the ways I think about power. In that course I
came to read about Foucault’s notion of power. That it is not a fixed resource that some
wield and to which others bend. “power is neither given, nor exchanged, nor recovered,
but rather exercised, and that it only exists in action” (M. Foucault, 1989/1996, p. 89, p.
89). It is not that professors have power and students do not. Professors do not do things
that, voila, determine what students learn. Everyone and everything in a course sets up a
network of possibility and power relationships that change and move around. And so my
change in this dissertation and my analysis of my observations was to describe the ways
that the course and all of the parts and persons that make up the course come to construct
a subject through discourse. In particular my analysis describes the construction of a
model or model student. I have abandoned the idea that students are learning in the
course in the sense that they are consuming information and experiences directed by
professors. Rather, students, professors, course documents, rows, desks, chalkboards,
computers, and textbooks work together to construct a model student.

The guiding question of my revised research project is: What assumptions about
students and mathematics are embedded in the pedagogical milieu of undergraduate
mathematics courses? For the purposes of analysis I have reframed my guiding question
into one about the construction of the subject where the subject is the undergraduate
mathematics student. In what will follow I will describe the courses and the ways in
which these courses construct different model students. What sorts of things does the
course assume students need to change? That is, I take a course as an opportunity to

change some aspect of those in the course. What then does each course try to change?

How does the course invite students to take on these changes? What are students asked to
do in the course? From the answers to these questions I then describe what a model
student emerging from each course would look like, more specifically, what a model
student would look like that emerged from a perfect realization of each course.

I recognize that a model student is not real. He is not the same thing as a real
student that made an A in the course for example. The model student is a fiction of the
course. A way of expressing what perfection would look like if it were possible. That
model students are not real does not make them any less important. Take for example a
Barbie doll. There is no real Barbie woman, but many people would argue that Barbie
matters despite her plasticity. They would argue that Barbie’s existence in the discourse
still has effects on how girls and women think about perfection and themselves. Thinking
about a model can also help us better understand the different values and orientations of
the community that participates in that discourse. In the case of Barbie, the values and
orientations and priorities would be of American culture and those who participate in that
discourse. In my study the discourse is that of the undergraduate course.

This dissertation is an examination of what is said to students and about students. What is
said about them out loud and what is said about them in text. What is assumed about
them fiom course documents, patterns of interactions, and types of experiences initiated
in the courses. It is not a dissertation about the students themselves, or even about the
professors although each are described. This dissertation is about the stories of
mathematics, teaching, and students told by the courses.

My aim in this dissertation is to speak to several ideas about undergraduate

mathematics: that undergraduate mathematics courses look largely alike and share

common goals about proofs; that what evolves in these courses is natural, predetermined,
or expected; that because the courses are housed in the same departmental program that
they assume the same things about a model student; and that what we in teacher
education often think of as a dichotomy between content and pedagogy is far more
interrelated.

Before delving more deeply into the details and theory behind this project let me
quickly describe the work I did towards answering my original research questions. In a
span of two academic years (both in the fall) I observed the four courses described in this
dissertation. Each was at the 300 or 400 level, and due to the content, these were courses
mostly populated with majors in the field. Of course, although students of other fields
might enroll in these upper division courses, they are mostly the habitat of mathematics
majors. While in these classes I took field notes focusing on teachers and students and the
interplay between the two. In addition to the observational nature of the research 1 also
collected course handouts and textbooks as well as course websites when they existed.

The artifacts gathered for my initial research questions, What kinds of
mathematics teaching do preservice teachers witness in upper level mathematics
department courses? What could preservice teachers learn about teaching in these
courses? did not change, but rather my analysis and orientation to what story I would tell
from the artifacts. My analytical questions were about the substance, mode, regimen, and
telos of these undergraduate courses. The substance can be thought of through the
question, What aspect of students (or substance) does the course target? As I will
describe in more detail to come, I take a rhetorical approach to the analysis. As Aristotle

described so long ago, Rhetoric is the art of persuasion. Persuasion involves change.

What aspect of the student then is the course trying to persuade students to change? The
mode relates to the means by which the course enacts the persuasion. What techniques of
persuasion are used? How are students invited to take up the change proposed in their
substance? The regimen refers to the things students are asked or expected to do in the
course. In what acts do they engage? Finally, telos is about the constructed subject, which
in this study is a model student. What would a model, fictional, student who emerges
from this course look like? The model student is not the best student in the course. It is
not a real person at all, but an ideal, model, and perfect fabrication. In selecting these as
my analytical questions I am trying to highlight the ways the courses spoke to and about
students, not about the actual students themselves. Actual students were present, indeed,
often 20 or more in each course. Actual students contributed to these constructions of the
model, but the model student is no single one of them.

As I continue through this dissertation I address a relationship between teacher
education and undergraduate mathematics, why I chose to use a postmodern analytical
frame, and why these research questions can help educators think about teacher education
courses for future secondary teachers.

In the first chapter I describe my project in greater detail. I begin by describing
the different discourses to which this dissertation speaks: mathematics teacher education
and undergraduate mathematics education. I conclude by describing my theoretical and
analytical decisions.

Following this chapter there are four chapters analyzing each of the four courses,
respectively. I chose to order these chapters by the course number assigned to it in the

course catalog for this mathematics program for the purpose of imaging a possible

sequence through the program. Of course after the first required prerequisite courses
students could enroll in the later courses in a variety of orderings, but in as much as the
courses are numbered in an ordering that seemed to reflect a progression from
introductory to advanced, I have decided to order my chapters similarly.

Chapter two is about the Real Analysis course. Again this course was required for
all majors and was considered a prerequisite to the Complex Analysis course. As will be
elaborated further in chapter two, my work speaks only to the specific instance of the
course observed and does not intend to generalize to other courses with the same name
and number as each professor, environment, semester, and set of students constructs each
course experience new again. The Real Analysis course constructed the model student as
a convert to a particular vision of mathematics and the role of proof.

The third chapter describes the Advanced Geometry course, one quite popular
with preservice teachers according to my general knowledge of the program and the
students in it. The model student constructed in this course was a researcher who thought
about proofs as final outcomes in a messy path that started with conjecturing.

The fourth chapter focuses on the Complex Analysis course, one of two 400-level
courses observed. The model student constructed here was a Platonic disciple.

The fifth chapter describes the Discrete Mathematics course where the model
student was constructed as a teacher.

The final chapter pulls together the themes from the four courses to describe the
ways that some were similar to one another and some were very different from one
another. I will argue that each course in its own way excluded types of students not in

line with the model construct. I will also argue that some courses offered more

accessibility, or ways that students could feel at home than others. I will also comment on
the dilemma of mathematics departments that may want to either reduce or increase the
diversity of visions of mathematics, teaching, and students that math programs Offer.

Finally, to reiterate, this dissertation is concerned with the stories we tell about
students. It is concerned particularly the stories generated by the courses, which I see as
having multiple authors who exercise power differently at different times. This
dissertation is not concerned with the truth of the students; who they are. For example,
the advanced geometry chapter will speak of the substance in need of change as students’
self-concepts and the complex analysis chapter’s of students’ discipline. This dissertation
does not seek to judge whether these aspects of students’ substance are indeed the best or
most appropriate aspect in need of change. Readers may find that they think self-concept
is a poor substance to want to change. He might think that students’ skills in proofs are a
better substance. Or the reader might think it a very good thing to change but may think
that the mode or appeal to that change should be X rather than Y. This dissertation does
not seek to say that students’ self-concept is what is needed to change or that it is not
what needs to change. I do not argue whether the truth of the students is this or that, or
that the course got it right or wrong. This project is about the stories this particular set of
courses form about students and the model students these courses construct. Simply
stated this dissertation is about philosophy. In these four courses I describe I hope you
will notice four different articulations of the model student and will conclude with me
that answering, “what math is all about” is not quite as simple as proofs, and that

philosophy matters.

CHAPTER ONE

CONTENT AND PEDAGOGY IN UNDERGRADUATE MATHEMATICS

Introduction

Teacher education for prospective secondary teachers begins with mathematically
prepared students and aims to create mathematics teachers. Mathematics preparation
often occurs in the mathematics departments, while education departments assume
responsibility for the pedagogical aspects of teaching. However, the separation of these
duties is misleading as many educators and researchers acknowledge. Teacher education
courses for future mathematics teachers often incorporate mathematics preparation—
particularly mathematics related to the topics and subjects common in middle and high
school curriculum. Content and pedagogy merge in mathematics education courses but
they also intermingle in mathematics courses housed in mathematics departments.
Students of mathematics learn about pedagogy in these courses although these lessons
may never be explicitly stated as goals.

A famous merging of content and pedagogy occurs in the theory of pedagogical
content knowledge as articulated by Shulman (1986) where he first described
pedagogical content knowledge as knowledge that “embodies the aspects of content most
germane to its teachability” (p. 9) and continued to offer examples such as knowledge of
the most useful metaphors and representations, and familiarity with common
misconceptions about the content held by students. And while a merging of content and
pedagogy, pedagogical content knowledge, a valuable and powerful way of teacher

knowing, was also a separate thing from pedagogy and content. In Knowledge and

10

Teaching (Shulman, 1987 February), he describes seven minimum categories for teacher
knowledge that could form a knowledge base for the professionalization of teaching.
Three of these include pedagogical content knowledge, general pedagogical knowledge,
and content knowledge. According to Shulman, the articulation of pedagogical content
knowledge does not eliminate or absorb the distinction between content and pedagogy so
much as offer a possibility Of some overlap. Segall (2004), though, takes a different
stance towards the relationship between content and pedagogy. He points out the ways in
which pedagogical content knowledge as an idea has benefited teaching, but pushes on
the notion that content and pedagogy retain distinct features.
teachers’ pedagogies do not initiate the pedagogical act but add further
pedagogical layers to those already present in the text. In other words, the
instructional or pedagogical act does not begin with teachers in
classrooms, nor does the ‘content act’ end at the desk of the subject-area
scholar. Both produce pedagogical content knowledge, that is, content that
is always pedagogical and pedagogies that are always content-full. (p. 3)
Segall draws upon the work of McEwan & Bull (1991) to elaborate how all knowledge
(content) is pedagogical in that it is always trying to communicate.
there is no such thing as pure scholarship, devoid of pedagogy. The
scholar is no scholar who does not engage an audience for the purpose of
edifying its members...science, or any other form of scholarship for that
matter, is an inherently pedagogic affair. . .ideas are themselves
pedagogic...Explanations are not only 91‘ something; they are also always
fgr someone. (p. 331-332, quoted by Segall)
When acting as research mathematicians, university professors are still in the role of
educator with other research mathematicians acting as pupils. The mathematics research
is for the benefit of other scholars and so in doing the work, the mathematician is trying

to teach others about her ideas. The teaching takes place in journals, formal lectures,

conferences, letters, over coffee, and in a plethora of other locations and media. The

11

content is pedagogical. When this same mathematician enters the classroom she teaches
undergraduates mathematics that at some other point in time a person wrote for the
education of others (by or for Newton, Cauchy, Gauss, Euler, Pythagoras etc). Herein
begins some of the layering effect, which may have started much earlier in the
dissemination and dispersal of the mathematics through journals, alternate and/or
amended proof, examples, textbooks, previous coursework, etc. As preservice teachers
(and other for that matter) take an undergraduate mathematics course they cannot help
but witness pedagogy both in the teaching of the mathematics, but also in the layers of
pedagogy that worked to get the mathematics to that moment.

I want to reiterate that I am interested in studying the ways that undergraduate
mathematics classes “teach” pedagogy and “speak” about students, teachers, and
mathematics. Pedagogy exists in undergraduate mathematics because mathematics can be
seen as pedagogical. What we teach is how we teach it. How we teach is what we teach.
There is a bidirectional arrow between pedagogy and curriculum/content. I am belaboring

this point because students in the courses, including undergraduate mathematics, leam

both content and pedagogy]. To put it differently, although mathematics courses in
universities are housed in mathematics departments and not in education departments,
there remain things to say and research about possibilities for teacher education in these
courses. Future secondary teachers take mathematics courses in these departments. They

learn math. They also learn pedagogy.

 

I separate the terms content and pedagogy for emphasis rather than to distinguish them as two separate
and unrelated aspects of teaching.

12

Undergraduate mathematics as Site of apprenticeship of observation

At least since Lortie (1975), teacher educators have understood the power of the
“apprenticeship of observation.” As students, future teachers spend thirteen years in an
informal, but extensive “apprenticeship” witnessing teaching from the perspective of the
student. According to Lortie these experiences have a greater effect on their development
as teachers than any official teacher education or alternate certification program can due
in part to the disparity between the typically short duration of a certification program and
the longer time spent as students in classrooms. In this view then, the teaching that
students witness has the potential to inform their future teaching.

It is worth noting that Lortie’s concept of an apprenticeship of observation has been
found relevant to teacher education. Bird (1991) found that students in his elementary
methods course (in discussions of mathematics as well as other subject lessons) expressed
preferences for particular videotaped cases of teaching that were similar to the students’
previous experience as students with those types Of lessons. For example, after watching
a direct instruction lesson students talked positively of it, but after watching an episode of
Magdalene Lampert teaching mathematics, the students talked often about her needing to
have told the students more. That is, she should have offered more direction and
clarification of what they perceived to have been confused student ideas. Bird related
these preferences and critiques of teaching to the preservice teachers’ past experiences as
students. Similarly, Holt-Reynolds (Fall 1991) found that all of the preservice teachers in
her study consulted personal past experiences as students to make decisions about
instructional strategies. “Preservice teachers frequently report actual instances from their
pasts where teachers implemented activities similar to those advocated in their

professional course work. They recall the effects those activities had on themselves as

13

students. Corinne explained the process explicitly. “You just see what you liked, what
got you interested as a student, if I'd want to do [this activity] as a student. Then, you
correlate that to how you would teach.” ”(p. 5). Holt-Reynolds found that pre-service
teachers not only look to their past experiences as students to offer possibilities of what
they might do in classrooms, but also choose from these possibilities based on what they
enjoyed. Other researchers have also found that preservice teachers consult past
experiences as students in classrooms when planning to teach as well as when teaching
(Holt—Reynolds, 1992; John, 1996; Johnson, 1994; Richards & Pennington, 1998)

Not only is there research that suggests past experience is at least part of the web
of influences on teacher practice and theory, but also a host of teacher education
initiatives take the apprenticeship of observation and a theory of cyclic teaching as the
theoretical foundation of their work and recommendations for change. That is, if teachers
perpetuate a certain type of teaching, and if we in teacher education want to change that
teaching, we must interfere with the cycle. A widely influential publication, Everybody
Counts (NRC, 1989), places the undergraduate experience of preservice teachers as a link
in this model.

Undergraduate mathematics is the linchpin for revitalization of

mathematics education. . .Those who would teach mathematics need to

learn contemporary mathematics appropriate to the grades they will teach,

in a style consistent with the way in which they will be expected to

teach...Since teachers teach much as they were taught, university courses

for prospective teachers must exemplify the highest standards for

instruction. (pp. 39, 64, and p.65 respectively)

Alan Schoenfeld in A Sourcebook for College Mathematics (Alan H Schoenfeld, 1990)

also calls upon the notion of a cycle placing undergraduate mathematics at the pinnacle.

14

He states his opinion of the role of undergraduate mathematics in shaping K-12 teaching
practice very clearly.
The nation’s mathematics teachers learn their mathematics from us
[professors]. In our classrooms they learn much more than the subject
matter. They learn how to teach (we are, after all, their last models of
mathematics instruction before they go into the classroom), and they learn
what it is to learn and do mathematics. If prospective teachers experience
mathematics as a lively and engaging discipline, they may teach it that
way. If they experience it as dead and deadly, the odds are that their
students will too. Thus our instruction at the college level is a critical
factor in shaping future K-12 mathematics instruction. (p. ii)
Schoenfeld is not alone among mathematicians interested in changing teaching in K-12
settings by changing the teaching in undergraduate ones. Both Cuoco (2003) and Wu
(1999) encourage other mathematics professors to think carefully about their roles as
teachers and the lessons about teaching and mathematics it can impart on would-be
teachers. Others in mathematics education at the elementary, secondary and tertiary

levels also care about changing teaching and have found a variety of ways to contribute

towards such an effort.

Change in teaching

The notion that changing the teaching that preservice teachers witness will
revitalize teaching is very appealing. The allure is simple if the process is not. If we in
teacher education (and undergraduate mathematics!) can influence a generation or two of
teachers in such a way that we disrupt current and widespread teaching practices, we will
ring in a new era of better mathematics instruction. Or so the fantasy goes.

Herein lies my concern. I would like to claim that we would bring in an era of
more and possibly diflerent mathematics instruction. The idea that new means better is

problematic. Through this research I do not wish to critique professors with a goal to

15

change their teaching only because I feel that some other method is better. NO doubt, at
times I did think that there could be improvements in the classes I Observed, but for me to
dictate a type of teaching is just another way of being closed-minded. Rather than.
opening teaching to include a wider array of pedagogies such an exaggerated change in
teaching would merely be restricting it to but another monopoly, which would have its
own collection of problems. I find limited models of teaching to be an issue for concern
no matter what that single view may be. To deny a best system is not to denounce
organized education as an impossible undertaking. I do not want to replace one narrow
teaching approach with another equally narrow one. Rather, I want to consider the value
of having multiple different teaching approaches available to all mathematics teachers. In
turn, different approaches in teaching can allow for student access to mathematics in
more ways.

One thing that I learned quickly was that despite some prevalent biases towards
mathematicians in our culture (for examples of particularly nasty impressions see Picker
& Berry, 2000), these professors are human. The caricature of the mathematician is a
stereotype and so are the courses they teach. From a broad perspective three of the four
courses fell easily into the classification of direct teaching, but on closer inspection there
were striking differences. These differences tell stories about who the teacher is, what
roles students play, where these meet, and why study mathematics.

As I say above, depending on how closely you look at the classes they could be
seen as similar or easily distinguishable. One common notion is that mathematicians
teach undergraduate classes using a mostly lecture or direct instruction approach. In

reviewing several MAA publications (e. g. MAA, 1983; NAS, 1968; Rishel, 2000; Alan H

16

Schoenfeld, 1990; Lynn Arthur Steen, 1989), the authors also assume this to be the case
about their mathematician readership. By encouraging these professors to try new
methods they are tacitly acknowledging a presumed similarity in their teaching—lecture.
These assumptions might be accurate, but little research has been done on the nature of
teaching in these settings (Robert & Speer, 2001). A large segment of research on
undergraduate mathematics focuses on improving student understanding of
undergraduate concepts, mostly in calculus (e. g. Ferrini-Mundy & Graham, 1994;
Graham & Ferrini-Mundy, 1989; Lauten, Graham, & F errini-Mundy, 1994; Monk, 1994;
Tall, 1996), or describes big issues in the teaching, learning, and context of calculus (e. g.
Douglas, 1986; Alan H Schoenfeld, 1990; Lynn A. Steen, 1987). The Calculus Reform
movement, I feel, directly benefited me as a student. It was not mathematics the monolith
that captured my attention and formed my major, but the particular vision of mathematics
I saw in Dr. Myrtle Lewin’s Calculus 1 course my first semester in college. In no way can
I imagine that I would have continued past that one required course had it not been for
the work being done to reform calculus and the department. Though I do not know what
those professors read or what influences acted on their teaching and curriculum decisions,
much of what I read now about calculus reform resonates with my own experiences as an
undergraduate. My professors at my small liberal arts women’s college taught in a spirit
commensurable with reform. We used the Harvard Calculus texts through the three-
course sequence. We modeled functions on calculators, computers, and even carved
potatoes and fruit. We worked in groups, worked on application problems, and mostly
wrote pages of explanations per assignment if not technical proofs. The research by

educators about student thinking and about curriculum in general helped me then and

17

continues to do so now when I plan for teaching a course and seek out these sorts of
documents to prepare.

By saying that they do not address the teaching in upper level undergraduate
courses I do not mean that these calculus research projects did not influence it. The
authors were quite often advocates for change after all, caring deeply about their subject
and about students. What I am trying to suggest is that as a field, research in
undergraduate mathematics does not often go beyond introductory courses (i.e. calculus
and developmental courses such as college algebra) and nor into the teaching of upper-
level courses.

Much of the work in undergraduate mathematics (and the few I Offered as
examples are not presumed to be the limit of research in undergraduate mathematics)
takes the stance that changes can be made in courses either by better understanding of
student reasoning which will lead to better instruction towards aiding that reasoning, or
restructuring the environments of these classrooms to offer more interaction between
students, teachers, and mathematics. Behind these concerns are assumptions about needs
for some changes, which are implied directly or otherwise as changes from a direct
instruction approach to a more interactive one. I point this out not because I challenge the
interest in changing the teaching (I have already said a bit about that above) but to
highlight a widespread sense within both the mathematics and mathematics education
communities that undergraduate teaching is primarily a lecture-based environment and
the subtext is that this type of teaching arrangement is at best not ideal and at worst
damaging. These publications, which assume a mathematician audience, are far less

likely than those assuming a K-12 mathematics teacher audience to flat out discourage

18

direct instruction.

Lectures and mathematics education

In the K—12 realm, Chazan and Ball (July 1999) point out that admonishing
mathematics teachers not to “tell” (as in lecture, for example) is too simplistic and does
not consider the context of the class or learning. “As researcher-teachers, we claim that
what is needed is. . .less embracing or rejecting of particular lessons and more effort
aimed at developing understanding of and reasoning about practice.” Chazan and Ball
argue that describing a teacher’s action as telling students something is not enough.
Researchers need to understand what may have motivated that action, and what the
teacher may have hoped to have achieved by it. Disregard their message for a moment
and instead consider the presumed audience. Why write an article that seeks to illuminate
“telling” modes of instruction in a more nuanced light if there were not a segment of the
readership who believe that lecture is one-dimensional—and undesirable at that? The fact
of the article’s existence speaks to the status of lecture in popular education. While there
might be those who do not consider lecture simply “bad”, many in education are not so
kind to the method.

My personal experience as a mathematics educator ranges from being a student, a
high school teacher, 3 methods course instructor, and a researcher. In each of these
contexts I have encountered other educators who claim that lecturing is bad and must be
limited or done away with in education. When I was first observing mathematics classes
for this project a professor of education asked me what I was working on. As soon as I
told this person about my work in the undergraduate classrooms, this person made several

disparaging and even vitriolic comments about mathematicians and their terrible

l9

teaching. The comments went further than the teaching and bordered on character attack.
While this reaction was definitely extreme, I have encountered others who doubt that by
entering these classrooms I will learn anything interesting. They tend to assume wrongly
that all the professors are poor teachers—mostly due to the preferred style of lecturing in
that context, and that my goal therefore is to write a dissertation that vilifies this group of
teachers. My point is that although there may be favorable or more neutral positions on
lecturing, the bulk of writing in education at the K-12 level and a good segment of the
writing at the undergraduate level tend to prefer change away from this style of teaching.
When we move beyond what is recorded in publishable spaces, the comments and
opinions can become more drastic and biased against lecture and its practitioners.

In the methods courses where I teach and interact often with other methods
instructors from this and other universities, there are strong assumptions that because
these preservice teachers have likely seen lectures from their viewpoint as students (a la
Lortie) we do not need to discuss this style with these future teachers. Not only is it seen
as unnecessary to their education but also as potentially harmful. It is almost as if the
discussion of such a thing would act as an endorsement or as legitimizing by the methods
instructor. If we do not talk about it, they will not do it. While in their professional
academic lives educators might hold a balanced view of lecture, this is often not Shared
publicly with preservice teachers. It is much more likely to be ignored as a topic
altogether.

Too often the fact that many mathematics professors continue to use lecture-based
(telling) styles is also used to label their teaching as bad. There is a bias in the education

community that groups = reform = good whereas lecture = traditional = bad regardless of

20

what is communicated through these styles, what motivated them, or what goals the
teacher/professor had in mind. This myth hides the intricacies of what professors and
teachers who primarily use lecture actually do and say in these lectures. It is unclear
whether mathematicians, many of whom are educators in their own right, would agree
that lecture is bad or that groups are good. They might offer a different equation.
Whatever the teaching preferences of mathematicians may be, the fact that they influence
so many students, and preservice teachers in particular, their teaching warrants more
investigation.

I have tried to offer rationales for why undergraduate mathematics matters for
teacher education by arguing that preservice teachers learn pedagogy in both settings. In
researching what pedagogies preservice teachers may witness I am not trying to favor one
or more styles over others. This is relevant because I have also acknowledged that many
in education, a community I represent when I enter mathematics professors’ classrooms,
often disparage mathematicians for their teaching preferences. I have also tried to explain
why undergraduate mathematics is an interesting and worthwhile site for research into
teacher education, drawing upon notions of the apprenticeship of observation that has
been found relevant to teacher thinking and by pointing out that pedagogy and content are
not as separated as university department systems might suggest. Future teachers learn

about teaching by learning mathematics.

Observing the courses

Recall the research question: What assumptions about students, teachers, and
mathematics are embedded in the pedagogical milieu of undergraduate mathematics

courses?

21

Because I am interested in pedagogies embedded in undergraduate classrooms as
well as differences across classrooms I utilized a case-study approach where each course
acts as a case of constructions of the subject where subject is an undergraduate
mathematics student. There are two Sites of artifact generation:

° Observations with detailed field notes of four upper-level mathematics content
courses offered at a large Midwestern university

° Course documents including syllabi, schedules, textbooks, professors’ websites,
an article provided by one of the professors, and course assignments

The courses represent three important subfields in mathematics—algebra, analysis
(2 real and complex), and geometry. I Observed these classes approximately every other
week during the fall semesters of 2005 or 2006 and focused on the nature of the activities

during class time, the actions of the professor and students, who talked or asked questions

as well as what was said or asked, and the physical environment of the classz. Because I
was interested in more than just the professor’s lecture or class notes, I rotated the focus
of my observations with each visit. Some days the observations centered on the professor
(including content such as theorems, examples, and pictures presented verbally and/or
through written notes, comments, questions, and actions), some days on students (actions,
questions, and comments), and at times on the ways that students and professors
interacted to influence one another during class time. My field notes followed an
approximate chronological order of the class actions. If on some given day I focused on

the professor more than the students, I would record much of the work on the chalkboard

 

The observations focused on these aspects of the course because they are features of the classroom that
contribute to the discourse of the classroom. My use of the term discourse will be further discussed in the
analysis section.

22

as well as direct quotes. On those days when I focused on students more than professors,
I noted student actions, comments, and questions. At times during my Observations

students would approach me before and after class to share their thoughts about the

professor, an assignment, or some other aspect of the course3. On a few occasions they
even spoke to me during the official class time. Several students who sat near me would
turn to me during class to show me grades they received on quizzes, homework sets, or
exams. I consider these interactions as parts of my Observations. Documents collected
included course syllabi and schedules, assignments, textbooks and professor prepared
study sheets. Some professors also utilized a course or personal web page, which served
as an electronic document.

The two sites of artifact generation allowed me to explore a variety of features of
the classroom the students contribute to and witness. These observations are purely
phenomenological in that I focus on what I saw and heard. The observations picked up on
actions and speech of the people involved in each course and also the physical
environment of the classroom. Some of the documents were shared aspects of the course
I would not have been able to “see” or “hear” had I acted only as an observer.
Participants in the course had to seek out materials such as the textbook and the course
web sites.

Finally, by generating these artifacts I was not trying to use them to pinpoint and
therefore validate future claims. Instead they represented various aspects of the course to
explore and might even at times seem to be contradictory. For example, a classroom

scene I witness during an observation might contradict a statement made in the syllabus.

 

Only one course professor chose to announce or describe my presence in the class, but I did tell those
students from other classes who asked what I was doing there.

23

L;-fi..—-—.-W

Let us suppose the episode in question is that the syllabus states there will be a quiz every
other week, but the observations suggest that by mid-October only one quiz has been
given. In some projects this might be cause for concern. Which should be given more
weight: The intentions of the course design, or the actions? How should we reconcile
these differences? For the purposes of this research, coherence between what is intended,
what is said about what is intended, and what is acted is not required. The phenomena are
What the students experience. They see and hear and participate in these episodes during
class time and so becomes a part of the rhythm of a classroom (see Hawhee, 2002 for
rh3ftlam as pedagogy). How the teacher intends an episode may tell us something
in‘lI'.>()rtant about teaching, but it does not change what the students could see. By using a
Variety of artifact types I am representing a range of aspects of teaching: from the
phBr'Sically performed to the public textbook to the between student pre-class chit-chat.

They are not an attempt to pin down the truth of the teaching so much as to offer up more

Ways to analyze it.

Analysis and theory
The analytical tools I draw upon in this dissertation come largely from the
lritellectual traditions of rhetoric and continental philosophy. Some rhetoricians and
phi losophers that influence my work are Aristotle, Mikhail Bahktin, recently Kenneth
B I-‘~1‘l<e, and Michel Foucault—particularly his notions of discourse and the construction of
th '3 subject. I will address both why rhetoric, why construction of the subject, and how
th Qse relate to teaching. Before expounding on these questions, though, let me instead

I)

e - . . . . .
@1 n by exploring rhetonc as an Intellectual actrvrty that can complement current trends

’1)
I} I athematics education research.

L 24

 

Why rhetoric

To use a communication or rhetorical approach is not to deny other analyses as

.M ___ ._.——-__~

viable or useful. This is not in opposition to social science models of research or to socio-

cultural or socio-linguistic approaches. It is instead complementary. It is another way of

doing research—one that has long been established in academia, especially in the liberal
and communication arts. Because education can be thought of in terms of communication
the analyses are appropriate.

One way to investigate pedagogy is through the notion that teaching is about
persuading students to agree with and thus take up a new view of some issue. A student
encounters some fact, idea, concept etc. taught to them by a teacher. How can we in

education get this student to take this fact, idea, concept, etc. on as truth or worth their
ti111e? We persuade them of it and this persuasion is at the heart of rhetoric. But before
y 01.1 cry foul that learning and scholarship is more pure than strong-arming students to
some collective will, consider that there are many techniques of persuasion, not all of
which involve bullying or flattery. “Rhetoric may be defined as the faculty of observing
in any given case the available means of persuasion” (Aristotle, Book 1: Chapter 2).
L"Qgical reasoning may be persuasive to some students and many who find their way into
E‘c1\’amced science and mathematics find this form of persuasion very powerful indeed.
F or others a good allegory, personal anecdote, or retelling of an experience may be more
c: anincing. Different disciplines often favor different means of persuasion or accept
eértain ones as more trustworthy than others. History, for example, pulls from a variety of
SQ hrces such as personal narrative and legal documentation. Even within disciplines,
diz‘551‘7631ent sub-fields may prefer or rely on different techniques of persuasions. A

th
% D O G O O I I
()retrcal astrophysrcrst must use x-ray data to Infer Indirectly the exrstence of black

L 25

 

 

holes, for example, whereas an applied physicist can directly test the tension of a type of
chain.

Take the most “teacher-centered” form of talk as an example. A lecture is an act
of speaking. It has an audience of students whom the professor tries to convince of some
truth of the discipline. The mode of this persuasion in the classroom might be by appeal
to the authority of text/teacher, personal experience, worked examples, or logical proof.

According to Aristotle these modes of persuasion are chosen by the speaker in direct
response to the speaker’s sense of audience. “For of the three elements in speech-
nlalfing—speaker, subject, and person addressed—it is the last one, the bearer, that

determines the speech's end and object” (Book 1: Chapter 3). Each student ultimately
dec ides whether the lecture succeeds or fails. She will either take on the knowledge or
not- The teacher must work to persuade (or teach) each new student and class and the
pedagogies used are determined in part by what the teacher thinks will best suit the

material and student. Speaker, Subject, and Audience work together.

Furthermore, if we consider Bakhtin’s (1990) theory of the speech genre we can
t)etter understand lectures as responsive to student audiences. He says all speech, written

and verbalized, is composed and delivered with the expectation of an audience.

An essential (constitutive) marker of the utterance is its quality of being
directed to someone, its addressivity...Both the composition and,
particularly, the style of the utterance depend on those to whom the
utterance is addressed, how the speaker (or writer) senses and imagines his
addressees, and the force of their effect on the utterance. Each speech
genre in each area of speech communication has its own typical

conception of the addressee, and this defines it as a genre. (p.95)

AK teacher will choose words, phrases, postures, props, style, and tone in expectation of an

511.1 -
q 1 ence of students. Therefore a lecture—even if the teacher speaks the entire duration

26

 

of a classroom episode, filling the class with a single yet lengthy utterance—is crafted
and delivered in response to expectations of and assumptions about students. It can be
rare, though, that a teacher lecturing never opens the floor to other speakers. When
Students speak they also do so to an audience that includes their fellow students and
teacher. How they frame their speech likewise exhibits addressivity. This is but one way
in which rhetoric can help me analyze these undergraduate mathematics classrooms. The
rhetorical moves made by teachers and students in a classroom contribute towards the
persuasiveness of the lesson and/or interaction. Analyzing these moves offers a new way
t0 “read” teaching and thus understand the experiences of math majors who go onto
become teachers themselves.
Rhetorical studies are historically linked to issues of communication through
oratory and more recently (I) with the invention of printing and readily available printed
teXt, rhetorical studies can also work with and on text. A major concern is how
communication works and can thus be very pragmatic. That is, a rhetorical analysis of
tealChing can focus on what a lesson communicates, by what means, and for what
1) 1ll‘poses. In Aristotle’s use of rhetoric, the speaker, audience, and subject work together.
Although the speaker may be considered the leading character due to his obviously active
r0 1 e the listeners and subject are shaping what the speaker says and does. A speaker must
consider the audience and its relation to the subject as well as his own to both the
audience and subject. What position does the speaker take on the subject? Is the subject
known to be controversial with this audience? Generally accepted? Is the speaker credible
wi t1) this audience? If not (yet?), how to proceed? How does the subject suit the place,

81)
Q C e, and time? The three make up the scene and work together in ways that make

27

 

pulling them apart difficult if impossible. It may be possible to focus on one aspect for a

while and hold the other two at bay, but eventually their mutual dependence on one

another will emerge.
By contrast, a common model of teaching characterizes three separate but fixed

bubbles possibly connected by arrows to form a triangle—the teacher, students, and
curriculum/subject. In this representation their independence is held fixed and each
category is stable. We know who the teacher is, the student, and the mathematics. But

What if we took a position that allowed for fluidity? What could be said about teachers,

Students, and mathematics then?
A rhetorical analysis allows for multiple interpretations to be held simultaneously.

Mathematics and action merge. Teacher as student. Students evaluate. The notion of the
t5-"€l<.ther as independent variable and student as dependent falls away. Neither would we
pl ace mathematics as an independent variable and teachers as dependent. In fact, the
VOCabulary “independent” does not make sense. This is not an issue of including a
bi directional arrow as sides of the familiar teaching triangle. Instead, the vertices shift,
now and again collapsing upon one another to form a line or even a point before

e){lrnmding Angles are not maintained. Nor is length, or even form. Dynamic.

A rhetorical model of analysis, which traces its genealogy through the liberal arts
and continental philosophy, can offer much to educational research. Again, by taking this
ab131roach I am not denying other modes of research or analysis respect. Two major
S tITEiIIds of popular research in mathematics education are those that take the mind and
Hi i 111mg as primary — cognitive studies; and those that take the social aspect of learning

a& .
Dnmary—socio-cultural. The intellectual histories of research matters as attested to by

28

the recent discussion in Educational Researcher between Sfard and Juzwik concerning
different traditions’ uses of the terms sociocultural (as well as identity, discourse, and
narrative) (see Juzwik, December 2006; A. Sfard, December 2006). During this
exchange, Juzwik digs into differences in these words and traces intellectual genealogies
to Show the vocabulary we use in research pulls from different traditions and so imbues
the research with historical meaning. Sfard, though, explains how taking a macro view
can allow us to characterize large swaths of a field (e. g. participation vs. acquisition,
Where participation aligns broadly with sociocultural and acquisition with mind
independent research) to speak about similarity in seemingly disparate traditions. I will
try to take a more Sfardian approach (with bits of Juzwik for flavor) in my following
di Scussion of sociocultural and cognitive research in mathematics education. My focal
l'alflgth will be long and so might sacrifice a bit of magnification, ironically, to allow for
brevity.

Using chapters Of A Research Companion to Principles and Standards for School
Mathematics (Kilpatrick, Martin, & Schifter, 2003) as my macro guide, I find the
fol lowing chapters:

' Communication and Language (Lampert & Cobb, 2003)
' Implications for Cognitive Science Research for Mathematics Education (Siegler,

2003)

- Situative Research Relevant to Standards for School Mathematics (Greeno, 2003)
- A Sociocultural Approach to Mathematics Reform: Speaking, Inscribing, and

Doing Mathematics Within Communities of Practice (Forman, 2003)

29

 

Balancing the Unbalanceable: The NC T M Standards in Light of Theories of

Learning Mathematics (Anna Sfard, 2003)

In summarizing the language and communication studies, Lampert and Cobb (2003)
describe the purposes of such research as work towards making communication both a
goal as well as a method for learning and how this learning occurs (p. 246). Cognitive
research, according to Siegler (2003), focuses on “the acquisition of particular
mathematical procedures and concepts rather on broad philosophical issues...” but in so
doing allows for “relatively firm conclusions...” (p. 289). Greeno (2003) describes the
Situative perspective as concerned with combining psychology and social science
ethnography to research “how students become more successful in participating in
mathematical practices and how they develop identities as mathematical knowers and
Icarners (p. 315). Forman’s (2003) chapter describes current trends in sociocultural work

‘11 mathematics education as proposing “teachers need to understand the mathematical

l(r<l()wledge that children bring with them to school from the practices outside of school as

well as the motives, beliefs, values, norms, and goals developed as a result of those

lDI‘aCtices” (p.337).
As I read these chapters I began not only to appreciate the work of researchers

E19 ting within each tradition, reminiscing about my first readings of many of the originals
c :- ted, but also attempted a mini-Venn diagram in my mind. The overlap became too

llFilleth and the task quickly abandoned! At first I found much between the situative,

s -
Q Q i ocultural and language chapters that seemed commensurable. The authors and works

Ci
t'éci intermingled between these chapters often. I wondered whether the cognitive

30

chapter also related to the rest and soon found connections between it and the situative
through emphases on social science and psychology as research traditions. Sociocultural,
as described by F orman, also has ties to psychology through interests in concepts such as
belief and tracing motives. In the last chapter listed above, as well as the last of the
volume, Sfard (2003) declares, “[Educational theories] thrive only in the company of
other theories” (p. 355). She encourages readers to think of theories as “complementary”
rather than in competition with one another.
As long as the metaphor with which we think is not challenged by an
alternative metaphor, its entailments seem natural, self-evident, and
unquestionable. Only in the presence of another metaphor and another
theory can we become critical towards the seemingly obvious; only when
an alternative theory makes us aware of the metaphorical roots of our
present beliefs are we able to open ourselves to possibilities that we could
not consider before. (p. 355)
U tilizing rhetoric and the metaphor of teaching as persuasion allows me to investigate

di fierent aspects of teaching that other common research paradigms might not. Like in

teEtching, I advocate pluralism in research as well.

l‘hetoric to Foucault

When teachers take on the role of talking to students they become an audience
(. and vice versa). When thinking of teaching as rhetorical, a speaker (or teacher) must
ktlow something about or have expectations about the audience (students) that influence
what he chooses to say and how to say it (a la Bakhtin). The assumptions the teacher
h Olds about the students’ motivations, interests, needs, current knowledge, etc. influence
th e teacher’s decisions. Know thy audience. This is what Foucault calls the construction

Q 1‘ the subject. Here, the subject is the student/audience. For a listener (student) the

L 31

speaker is important. Why should I trust this person? By their authority, their words, their
actions? This is the construction of the teacher.

These constructions occur within classroom discourse, but undergraduate
mathematics courses and their classrooms are not places researchers expect to find rich
discussion or conversation. As will be further described in further detail in chapters of the
dissertation, five of the classes I observed did not dispel this common notion but these

classes did contain much more than just talk that was worthy of consideration. For
example, a professor’s action of looking students in the eye, the creation and distribution
Of a student petition for extra credit, and the voices of the textbook authors are each
examples I might have missed had I concentrated solely on in-class talk or written
Communication, and interviews. Much, though, of the research done in mathematics
e(Silication around notions of discourse does limit the definition of discourse to talk and
Words (Parks, 2007b). As Parks further points out many of these projects draw inspiration
and justification from NCTM’s Principles and Standards (2000) documents calling for
I1lore student engagement in classroom communication such as critiquing and creating
r1lathematical arguments. In these uses of discourse, things beyond talk are often ignored.
When other aspects of classrooms are investigated they are not usually called discourse
analysis, but are given other names such as studies Of teacher beliefs, textbook analyses,
and mathematical representation to name a few.
Sfard’s (2001) use of discourse does incorporate more than just talk. As she puts it
Slie “will try to show that there is more to discourse than meets the ears...” (p. 1). A
D1‘1" nary reason I do not start from her use of discourse is that she uses it as a theory of

1%
fining where learning mathematics is coming to terms with the mathematical discourse

L 32

 

as used by the established mathematical community and where researchers can measure
how close a student is to appropriating the discourse by reference to markers such as
extension of vocabulary. I do not want to align this particular project with a theory of
l earning and so avoid drawing too close to her work at this time, although I find it quite
interesting. Fortunately, in addition to Sfard’s use of discourse has more than words,
there does exist another notion of discourse that is much broader in scope than typically
used in mathematics education.

By framing my guiding question, What assumptions about students and
mathematics are embedded in the pedagogical milieu of undergraduate mathematics
courses? as an investigation into the construction of the subject (e. g. the student in
undergraduate mathematics) I can employ a notion of discourse formulated by the French
philosopher Michel Foucault (1978/ 1 990, 1983).

In this analytic, discourse refers to more than just talk or other verbal
Communication and also includes multiple actions and/or objects, which can be read and

interpreted as texts (see, e. g., Michel Foucault, 1978/1990; Mills, 1997/2004). Here I
Consider such texts to include people’s actions (students’ and professors’), the physical
Setting, classroom objects, verbal and symbolic communication, written documents, and
Websites. Through these multiple texts, constituting a discourse, a subject is produced or
C Onstructed. That is, who undergraduate students are (or should be) in upper-level
F1) athematics classes is neither a logical nor a natural phenomenon, but rather is an artifact
I) f discourse legitimated through repetition. This type of analysis does not intend to
cry. ti que current teaching practices with the goal of offering better pedagogies nor does it

Se ek to legitimize some practices as exemplary. Instead, this analysis attempts to rethink

33

i deas taken for granted. In this case, that undergraduate courses are not universal and
timeless. The educative goals, teaching, actions, and students can differ in important
ways. For example, in a 1968 publication of the National Academy of Sciences, the panel
on Undergraduate Education in Mathematics reports that elementary courses in
mathematics be taught in large lecture halls, not because this is best for students or most
logically commensurate with learning the subject, but for “utilizing staff more
efficiently” (p. 11). Contrast this with the more private tutorial systems of learning
mathematics in England. These differences can be attributed to the differing histories of
/ the subject as well as the histories of higher education in each location.
Before continuing, the idea of constructing the subject may need further
i llumination. A subject, here, does not refer to an individual, but to an abstracted category
made possible to talk about through discursive constructions. For example, Popkewitz
discusses as a subject the problem-solving child not as a specific child but as a type of
Child created through discourse (Popkewitz, 2004). “Neither ‘adolescence’ nor ‘problem
Solvers’ ...are objects that you can touch, but are ways of thinking, ‘seeing’ and feeling
about ‘the things’ of the world that were (and are) deemed important” (p.13).
Once the problem-solving child is invoked, researchers can examine the category
for behaviors, tendencies, preferences, and growth. They can also create more and more
fi ne—grained instruments and techniques for classifying “problem solvers’” thinking. In
C unent mathematics education the problem solver is a particularly desirable type of
S ta dent and is reinforced as a type and constructed through texts. Examples of texts might
in clude published articles, how-to books, textbooks, and collections of problem sets.

I\ u th ors can write books for teachers on how to best educate and stimulate problem

34

~_

solvers, how to write assessments that include good problems, as well as instructions on
how to assist other children into developing as problem solvers.

Recall though, discourse as understood in this analysis includes non-written
documents as well as actions, physical environments, and comments. I remember from
my time as a high school teacher how I contributed to the discourse constructing the
problem solver through my pedagogy. The school I worked for encouraged reform
mathematics teachers (another type of subject to explore as a construction) to emphasize
problem solving. In this school and district rote exercises were often considered in

opposition to problem solving exercises. Both were involved in the curriculum, but
problem-solving tasks were considered the better task and a priority. The school printed
posters for each mathematics class that outlined how-to steps for problem solving. As a
teacher I worked to get those students who were good at rote exercise to also improve
their problem solving skills; to become problem solvers. The students, who were
particularly skilled at problem solving tasks but not necessarily at exercises, were seen as
particularly creative. I encouraged them to do their homework and improve their grades
(because grades still relied on other types of tasks), but I upheld them as excellent
S tudents nonetheless and often in spite of their grades. When I attended professional
development workshops we discussed problems instead of exercises. We shared stories of
S tudents who had excelled at some particularly challenging problem or of other students
W11 0 had improved their problem solving skills and how we had managed this change.
We learned new techniques and lesson ideas to create problem-solving students. I point

lb ese experiences out not to declare that problem solving skills and tasks are bad, but that

35

through my participation in these professional development workshops, use of texts, and
teaching I reinforced the problem solver as a type of human.

Moreover I included questions on my exams that were considered problem-
solving questions as opposed to recall or application type questions. The textbook I used
included problem—solving style questions at the end of every set of assignments and
several sections of each chapter were devoted to problem solving tasks. I also assigned a
‘ problem of the week’. Many might think that this was good teaching and in many ways I
agree. I point these out not because I am ashamed to have taught what I did, but to
indicate that I was reinforcing the discourse of the problem solver and in particular I was
prioritizing this type of student. Again subjects are not individuals. In my example
offered above, many of my past students were individuals skilled at solving a variety of
problems. The difference between a subject and an individual is important. I had an
idealized, abstracted notion of the problem solver, which influenced the ways I acted on
and for particular individual students. For me, the problem solver was desirable because
they were nonconformists, used mathematics effectively in a variety of contexts, and

Would be able to confront life’s obstacles with success. Because I wanted these things for
my students, I wanted them to be problem solvers. By taking on this approach to my class
and my students I as participating in a discourse that privileged problem-solving and
C Onstructed problem solvers as a type of person.
Furthermore Popkewitz uses the term fabrication of human kinds (p. 13), to
91h phasize these human kinds (a term also used by Hacking, 1995) as types are fabricated

in the sense that they are both built and are also fictions. As a type of human, the problem

36

solver is a construction, an idea, and an ideal in mathematics education. When reified, a
subject can be talked about, researched, acted on, and further refined.

This extended example has focused on problem solvers as types of students
constructed through various texts including policy documents, textbooks, lessons, teacher
actions, and classrooms. The research for this dissertation explores the construction of the
student in the space of four classrooms. I interpret the actions of persons, activities,
documents, and dialogue in the classrooms for the purpose of describing the construction
of the student and teacher through these texts in each course. Although professors and
documents they create are explored as l write about students, this is not a study of only
professors, but also of how the subject of the student is constructed in these sites. At
times in the following chapters on the student it might seem that I am indeed focusing on
the professor due to the professors’ relative prominence in the classroom through their
choice of texts, creation of syllabi, and talk. In three of these classes (other classes not
observed may have been very different) the professors typically spoke much more often
than students and acted more visibly through their position at the chalkboard. The
following chapters will address construction of the student and in doing so will likely

refer back and forth between students and teachers.

Furthermore, the construction of the undergraduate student is specific to each

Course during this semester. That is, the construction of the student in the Real Analysis
(3 ourse, for example, shared some features with the construction of the undergraduate
S trudent in the Discrete Mathematics course, but they also departed in some ways. This is
i n fact, a primary reason why I chose multiple classes—to highlight difference. Just

b€=C=ause three of the four courses described here used a lecture—based approach does not

37

¥

guarantee that the construction of the student through the course is identical. It is even
possible if the same professor teaching the same course in another semester a new
construction of the subject might emerge due to possible differences in students enrolled,
texts, syllabus changes, and assigned classroom space.

In particular, I analyzed each course across the sites of artifact generation to infer

responses to the following questions4:

° Substance: What Aspect of the Student is Supposed to Change?

° Mode: How Are Students Invited to Learn?

° Regimen: What Are Students Asked to Do?

° T elos: What Does the Model Undergraduate Mathematics Student Look Like?
The answers depend on the respective class and in this way I am highlighting the
construction of the answer to these questions through the discourse particular to each
course. I am not attempting to answer these questions as generalities or recommendations
for students, parents, teachers, curriculum developers, or policy makers. Instead I
investigated the ways discourse in each professors’ classroom constructed the answers to
these questions.

Thus far I have discussed the notion of the construction of the subject (student) in

great detail, but there is a third aspect of this work—the mathematics. In using a
F oucaultian framework, issues of the mathematics will emerge as I describe the
C onstruction of the subject (students) but I will also devote some space to more specific

(1 i scussions of the mathematics in the classrooms in the conclusion chapter.

M

4.

F C>l_1cault (1983) proposes these questions as a framework for exploring how subjects get constructed.

38

Concluding remarks

In this dissertation I speak to discourses (mathematics teacher education,
mathematics education, and undergraduate mathematics) for the purpose of examining
commonly held notions. These include stereotypes about mathematics teaching at the
undergraduate level, namely that reform practices are better than traditional practices, and
that content and pedagogy are distinguishable. I also speak about discourses in the
Chapters for each course. I describe how the discourses contributed to the construction of
the undergraduate mathematics student. To do this I draw from postmodern and rhetorical
theories and analyses.

Each course presented different pictures of a model student that might emerge
from a perfect enactment of that course. We will see model students as converts,
researchers, Platonists, and teachers. In these four courses alone there were four different
articulations of the model student. This suggests that undergraduate courses are not as
alike as imagined. In these courses there were also multiple ways that students could do
mathematics partly because these courses presented different views of what it means to
do mathematics and who does it. Answering, “what is math all about” is not quite as
simple as “proofs”. The ways that the courses structured student access to the

mathematics suggested that different styles of teaching cab provide alternate and varied
entry points for students. Therefore a lecture method or an inquiry method (or others) can
both exclude certain types of participation and encourage others thereby excluding some
8 tudents and encouraging others. Those excluded and those encouraged were not the
Same across the classes. A best course, one that had the best model student, best vision of
m athematics, best accessibility for students, did not exist. Each had its own way. That the

‘30 llrses were not identical and that they were different implies that the notion of “best”

39

y

may not be very helpful for mathematics education and we might instead try to think of

courses as containing particular possibilities.

4O

 

CHAPTER TWO

REAL ANALYSIS: CONVERTS

I have had my results for a long time: but I do not
yet know how I am to arrive at them.
--Carl Gauss

Introduction
The Real Analysis course was the second of three courses required by all math
Inajors5 after the Calculus sequence and Linear Algebra. It was also the only Real
Analysis course available to students in the semester of this study. Again, like many
courses in this study, depending on a student’s entry into the program, a student could
have been in any year of their coursework, at or beyond the sophomore level. Given that
this was a fall term, most students then would have taken 3 Linear Algebra course the
previous spring or over the summer making them sophomores if they had tested out of
the first two Calculus courses. For students who only tested out of the first Calculus or
second course or neither, they would most likely be juniors when enrolled in this course.
To take this class in a senior year, while possible, would have meant that a student was
running close to not finishing on time. This course was a prerequisite for students
6 lecting to take Complex Analysis.
At the time that I observed this course I did not have access to records of the
S tmdents enrolled in this course to compare with those accepted into the certification

program. Most students intending to be student teachers would have gained acceptance to

 

 

A 1 1 students, that is, who were not taking the Honors program. Of the three courses—Linear Algebra,
Re 81 Analysis, and Abstract Algebra & Number Theory—only the latter two had Honors options.

41

g

 

the certification program in their junior year. Therefore any student who was a
sophomore would not have known his or her status in the application process. Because
the majority of students in the course were most likely sophomores or juniors it would
have been difficult to determine who was planning to be a teacher as they would not yet
be enrolled in either the mathematics methods course and/or not yet accepted into the
program. There were at least a few students in the course who were awaiting their
acceptance letters as evidenced by conversations I overheard. Because all preservice
mathematics teachers had to also be math majors, all preservice teachers took a Real
Analysis course at some time in their program.
The course met three days a week for 50 minutes each in the mathematics
building in one of the smaller rooms available. The rows of desks were faced so that they
DO inted towards a blackboard at the front of the room with a panel of narrow windows to
’th e left of the students. There was a computer station that linked to a projector mounted
Q n the ceiling and an overhead projector available, but these were never used. The room
W as next to the stairwell and a lot of traffic from the hallways generated noise so the
S i l'nge door on the right side of the room was often closed during class. The 27 enrolled
S hildents took up most of the available desks. The students were mostly talkative during
i h‘class group time and before class. While Dr. RA was lecturing they tended to only
g ‘3 eak when they had questions or were prompted by a question. The atmosphere was
% Qnerally still with bursts of talk when engaged by Dr. RA to do so. The students
a q(iressed Dr. RA by his title and last name and he addressed them by their first names.

B1

e seemed to know them well enough to comment on their interests. They seemed to

42

only know the names of the people who sat around them and often did not engage with

students not in their study group. The general atmosphere was fiiendly and academic.

Prelude to the analysis

Recall that throughout this chapter and the following three related to the four
courses observed, 1 will be structuring the analysis into four parts—the substance, mode,
regimen, and telos. The substance is the aspect of the student that the course design
attempts to change. What aspect of the student the course design attempts to remediate,
alter, or strengthen. The mode refers to the means by which the course design invited
S tudents to take on this change. The regimen relates to the things students were asked to

do that would presumably effect the change. The telos is a description of what a model or

perfect student might look like who emerged from the course.
Students in this Real Analysis course were positioned as residing within an

i ntuitive epistemic world. By that I mean that the students were addressed as if they

believed acquiring mathematical knowledge was a process of satisfying a personal
criteria of either intuition or trust in an authority. I am not saying that they were
D Qsitioned as intuitionists in the sense of the mathematical philosophy, although the

as sumed student position did not equate the truth of a statement with its provability. In

th is restricted way they could have been thought of as intuitionists. Yet they were not
S" Q en as being strictly intuitionists who also held that the law of the excluded middle, for

3‘: ample, was to be rejected.
The elements of the course suggested that their mathematical pasts have not

ained them to be careful about what they know and take to be true in mathematics. In

t h
to. absence of an authority such as a professor or textbook the students were supposed to

43

determine what counts as knowledge—accurate and true—by referring to their intuitions
or senses. The course pushed them from thinking about truth as either residing externally
in authorities such as professors and textbooks, or as an internal hunch or intuition.
Knowledge was not constructed as being accessible by intuitions. Rather, knowledge
was derived from a process of mathematical logic and argumentation. Students were
taught to develop a different epistemology: one where proof was the criterion rather than
authorities or intuition. It is not exactly clear where students were presumed to have
acquired this epistemology, but possibly from any prior course that was not a proof-based

course, such as some Calculus courses and a lot of K-12 coursework.

Students were invited to take this different view of how one arrives at truth by
appealing to their desire to participate in communities—both small group communities
and the larger mathematical community beyond the classroom. Students were asked to

j udge the truth and falsehood of statements, craft mathematical arguments that conformed
to the standards of the mathematical community, determine the relative persuasiveness of
di fferent arguments, and watch the professor do many of the above things. Overall, the
model student who would emerge from this course would be a convert to a rigorous

proof-based way of thinking about and participating in mathematical activities.

In that there was not a relationship between mathematics and some deeper
l'lleaning or realists or Platonic existence of ideas and that the doing of mathematics was
deeply human, the course design presented a social constructionist View. The formal

Work of analysis was presented as a human response to the perceived flaws of the

Newton and Leibniz Calculus, based on the needs of the changing community.

44

Substance: Epistemology

Many math majors find these courses to be among the most challenging

they take because the way of thinking may seem at first unfamiliar:

° Computation will not be the focus; most problems will not have a
number or formula as an answer.

' ' You can't master this material by memorizing rote procedures.
' Plausible reasons for expecting something is true, while to be desired,
will not usually suffice. Proof will ultimately be required.
The quotation above is from the syllabus of the Real Analysis course. It expresses a view
of mathematics that is utterly different from that expressed by Gauss in the epigraph of
this chapter. Students will need to begin to arrive at their conclusions and stop taking
them for granted. The students’ epistemology, way of thinking about and determining
what counts as knowledge, was the targeted substance of this Real Analysis course. As
the syllabus quoted above suggests, students enter the courses beyond Calculus with a
1 argely unquestioning orientation towards mathematics. They were assumed to think of
doing mathematics as calculations related to given theorems, given either directly from
the authority of a textbook or prior instructor, or from intuitions. Real Analysis both
Qlrlallenges the assumption of calculation and attempts to supplant it with a view of
1j‘lathematics that is based in proofs.

Issues of determining knowledge precisely may have been new to these students
bUt the formal proof approach was very much in line with the development of Real
Analysis as a field as well. Much of real analysis in the 18003 to the early 19005 took the
i Ssues of knowledge, particularly related to our knowledge of real numbers, real-valued
filnctions, and the underpinnings of calculus very seriously and hence the flourishing of
l3l‘ecise definitions and attempts to build or describe the foundations of analysis during

t1lis era. For example, the real analysis by Dedekind, Cantor, and Heine in 1872

45

challenged the axiom of completeness as an axiom (something to be taken as a
reasonable, intuitive starting point) and instead transformed it as a theorem of the
constructability of the real numbers from rational numbers. Furthermore, real numbers
were reimagined as sets of rational numbers and equivalence classes of Cauchy
sequences, removing them from intuitive associations with quantity and measurement.
This course was designed so that students, like the mathematicians during the
foundations era, would be provided with knowledge that they were lacking, namely a
rigorous way “to know” mathematics. The course began with a statement that the
students’ sense of what counted as mathematical knowledge was limited. As the syllabus
stated, the course was designed for students who knew a different kind of mathematics
than Real Analysis. The students were positioned as needing to adopt a new way to
know mathematics and so their epistemology was the target.

The textbook and the professor were both clear about this. On the professor’s
Website there was a statement in a section called “Comments on the Course”. It included
‘31) is description of some goals for the students:

To succeed in this course you will have to train yourself to think about the

logical structure of the subject matter and understand the definitions of

concepts and the statements and proofs of theorems. You will need to
understand a collection of key examples and be able to reason about their
properties. You will need to document much more of your thinking about
problems than you probably have done before. While a good intuition is
necessary to guide you to correct statements, just making correct
statements alone will not be enough. You must be able to prove them.

I 1:1 the above passage Dr. RA told students that they would need to “reason” and

‘document [their] thinking about problems” and acknowledged the positive role of

i htuition while also warning them that proof would become the standard tool for

determining knowledge, specifically mathematical knowledge.

46

The textbook directly stated that students coming into this course were seen as

lacking a rigorous epistemology. The textbook describes the challenge this way:

instructors today are faced with the task of teaching a difficult, abstract course to
a more diverse audience less familiar with the nature of axiomatic arguments.”

Later the author writes, “Having seen mainly graphical, numerical, or intuitive
arguments, students need to learn what constitutes a rigorous mathematical proof

and how to write one. (Abbott, 2001, p. v)

In both passages students are positioned as lacking a mathematically logical proof
orientation towards justification of their idea and arguments. The course is designed as if

students had probably encountered many graphical representations. The textbook, the
syllabus, and the course website address students as if they might have worked many
numerical examples of cases of an idea, and have intuitive arguments to support their

thinking. They are lacking proofs. This course worked to address this cavity in students’

mathematical knowledge.

The course challenged students to learn to think rigorously rather than

i ntuitively. To accomplish this the professor did not set out to rigorously prove
things the students already know or take for granted. To come to see the value of
Ij'lathematical proof as a determination of knowledge, the course did not attempt
to prove what seems obvious to them. Instead the focus is on using proof at times

When things are not already clear or familiar. One day in late October while the

. . . . 2 ,
Q lass was discussmg continuity, Dr. RA drew the graph of x on the board. “I m
I‘Ilot gomg to try to convmce you that x 18 not continuous. You ve known that it IS

Er years. It just isn’t uniformly continuous.”

“As the function gets steeper, the deltas must shrink and so they are not
uniform for each on value.” Dr. RA assumed that students were already familiar

47

. 2 . . . .
With the graph of x and that it was a continuous function. He did not choose to

prove that it was indeed continuous because as he stated, they likely encountered
this before, most likely in calculus. He chose instead to focus on the non-
uniformity of that continuity, something he was assuming that they had not
considered previously. After the picture argument (see Figure 2.1), Dr. RA went

on to provide a formal definition for uniform continuity and a formal proof of the

2 . .
x Situation.

48

 

 

4»—------------------

 

 

 

 

 

 

 

 

i»-----------—

o---—--

 

V

 

 

 

A

 

Figure 2.1: Dr. RA’s sketch ofx2

49

This episode reveals two general pedagogic moves found frequently in Dr.
RA’s lectures. First, he did not spend a lot of time trying to re-examine ideas that
the students already seemed to be familiar with or know. Proofs were not used to
verify ideas that the students already hold to be true. Instead, proofs were a tool
for dealing with unknown or unfamiliar mathematical terrain. Granted, for many

mathematicians having a proof of the continuity of a familiar and assumed

. . 2 . . .
continuous function such as x is valuable. But from a pedagogical perspective,

Dr. RA did not use class time to re-prove known things. He used proof for
contested ideas. To get students to value the need for a proof he used them when
they were needed in the eyes of the students. Second, Dr. RA tended to start with
examples (numerical sometimes and graphical in this case) and/or holistic
arguments to set up the situation before moving to the formal definitions and
proofs. Before providing students with the definition of a uniformly continuous
function, Dr. RA discussed an example of one, and with the graph he showed
What he meant by the non-uniformity of the delta neighborhood despite the
uniformity of the corresponding epsilons.

To get students to change their epistemology, the course needed to
negotiate a tricky position. Students must be convinced of the value of a more
rigorous definition-proof system for determining knowledge, and they must also
be taught how to work in this system. For students who have accepted certain

1“llathematics as already known and true from earlier experiences (take the case of

. . 2 . .
the continuity of x again as an example), to prov1de a proof of that same

50

knowledge might have had the effect of leading the students to perceive proof as
an annoying and complicated method for restating “obvious” ideas. Students had
probably encountered work in calculus courses where the continuity of certain
fiinctions was told to them. They were likely not skeptical of the source of this
knowledge (a textbook or past professor). In these cases, to rehash ‘known’
material would not necessarily be interpreted as enlightening but possibly as
redundant. Dr. RA’s tendency to prove things that were not obvious suggests that
he positioned students as needing to see examples of proof as necessary in their
eyes, not from the larger community’ eyes. This particular selection of problems
set up proof as a valuable epistemic tool rather than as a redundant technique or
empty exercise.

While the course was designed to get students to see the value in proofs as tools
for establishing knowledge, the course also needed to deal with the epistemological
problem that students take too much for granted. One common way the daily course
activities attempted to change students’ epistemological orientation was when the
professor provided a definition and then asked students to classify things according to
these definitions rather than according to their intuitions. For example, early in the
semester the professor provided a definition of a subsequence. He then provided several
examples of sequences and asked that the students decide which were subsequences and
which were not. The point of the task was for the students to become accustomed to
making their decisions based on the definition rather than on pictures or intuitions. The

definition became the authority to check statements against.

51

The book troubled this use of definitions as authority a bit as it described on
several occasions times in mathematics history where a definition was new, contested, or
completely absent. One such case the book referenced concerned the definition of a
function. The textbook provided a set-based definition of function and later in the
following paragraph declared that the one provided there is “more or less the one
proposed by Peter Lejune Dirichlet (1805-1859) in the 18305” (Abbott, 2001, p. 7). By
pointing out the date of the definition and its German mathematician author, “one of the
leaders in the development of the rigorous approach to functions that we are about to
undertake”, the text was signaling that definitions come from someone, somewhere and
somewhen. Furthermore the passage described why this definition is better than others.

Specifically it “allows for a much broader range of possibilities” than did function

interpretations based on the idea of a formula such as f(x) = x2 + 1. These prior

understandings were not so much definitions of functions (as we now think of
mathematical definitions) but generally accepted understandings. In this way, the
textbook noted that the concept of a function existed prior to the definition but it was in
defining the definition in this particular way that allowed mathematicians to formalize
concepts such as continuous and differentiable functions.

By describing historical accounts of mathematicians struggling with the adequacy
and accuracy of definitions, the textbook warned students of the problem of assumptions.
It told the history of mathematics as an attempt to avoid these mathematical pitfalls. It
also acknowledged the pragmatic aspects of mathematics history in that some definitions
are based less on absolute knowledge and more on the benefits of certain definitions over

Others.

52

Before continuing through the description of the course it seems fruitful to point
out how the professor’s statements and activities were generally well matched with the
book. On most occasions his definitions, theorems, and examples were taken directly
from the textbook. Many students often followed along during the lectures by reading the
book rather than taking notes. On one occasion he asked the class whether they would
like to see him prove Rolle’s Theorem. Some said yes. Some said no. One student
responded to her neighbor, “It’s already in the book isn’t it?” The students were
consulting their books frequently. From the students reactions in class, they seemed to
have read the chapters well enough to know which theorems the book proved and which
it did not. They also were asked to consult the text during class by Dr. RA. The students
often referenced the textbook in class and so they very likely also read the notes on the
history of the development of real analysis presented there, though I do not know what
weight they gave these commentaries.

Aside from the use of definitions and their histories as warrants for claims, there
were other reasons to suggest that the course worked on altering students’ epistemologies
about mathematical knowledge. Students were positioned as not knowing much about
formal notation and proof writing. When Dr. RA wrote proofs on the board he tended to
do so in full sentence and. paragraph forms. To use shorthand notation would have
implied that the students were comfortable reading and understanding this form of
writing. Instead, he wrote nearly every word that he also spoke. This is in contrast to the
other courses I observed where most spoken words in a proof were not also publicly
recorded. In fact, in these other courses—with the possible exception of Advanced

Geometry where pictures were also an important aspect of the work—words were rarely

53

written at all. Take the following proof that an open set’s complement is closed (which he

followed with a proof that a closed set’s complement is open) as an example:

1) Assume O is open. Let x be a limit point of OC. By definition of limit

point, every neighborhood of x intersects DC. In other words, no
neighborhood of x is contained entirely in 0. Thus, because 0 is open, x

cannot lie in 0. So x is a member of Oc and so Oc contains its limit points.

Therefore 0c is closed.

Writing out the words was not a reduction in rigor. Rather, students were treated
as if they lacked knowledge of notation or ability to follow outlined proofs. They were
treated as if they needed the fully detailed version, being positioned as unable to follow
an outlined proof or a highly symbolic one. If students lack an understanding of the need
for each and every logical step, why show them a proof that omits ‘obvious’ steps or
rationales? Everything was made explicit. By including all of the words and steps in a
proof, Dr. RA’s pedagogy implied that students were not prepared to understand the
distinction between outlined proofs and fully expounded ones. If he had only recorded
outlines of proofs the students might have been able to follow along or they might not
have noticed the unstated “obvious” assumptions. I do not know what the students might
have been able to understand had Dr. RA only used short hand proofs. Because he did not
use the abbreviated form in class, it is likely the course constructed students as needing to
be shown everything explicitly. The implicit curriculum was that students who did not
appreciate the logical progression of a proof might not have been able to discern which
steps in an outlined proof were “obvious and therefore not included” versus what was left
out because it was unrelated or truly unnecessary.

Dr RA’s proofs had a more prose-like feel than did other professor’s proofs I

observed. He used a lot of chalk in class. If he said it, he usually also wrote it. In

54

comparison, he did a lot of work to make the proofs he wrote readable and match spoken
mathematics. But, the textbook did more. The textbook also gave a proof that an open

set’s complement is closed.

Given an open set 0 Q R, let’s first prove that 0c is a closed set. To prove
0c is closed, we need to show that it contains all of its limit points. If x is
a limit point of OC, then every [emphasis in original] neighborhood of x

contains some point of 0C. But that is not enough to conclude that x
cannot be in the open set 0 because x E 0 would imply that there exists a

neighborhood v, (x) g 0. Thus, x 6 oc as desired. (Abbott, 2001, p.82)
Here again, is Dr. RA’s proof on the same result.

1) Assume O is open. Let x be a limit point of 0c. By definition of limit

point, every neighborhood of x intersects Oc. In other words, no
neighborhood of x is contained entirely in 0. Thus, because 0 is open, x

. . . C C . . . . .
cannot be in 0. So x is a member of O and so 0 contains its limit pomts.

Therefore 0c is closed.
The textbook’s proof begins similarly to Dr. RA’s. Both set up the given statement or
starting assumption on which the rest of the proof will rely. The textbook’s version also

specifies the set to which 0 belongs, the real numbers and it also states that the proof to

follow only tackles the proof that 00, the complement of O, is closed. Dr. RA then

moves to invoke the definition of a Iimit point. The book by contrast foreshadows what

the proof will do to prove that 0c is closed. “To prove 0c is closed, we need to show that

it contains all of its limit points.” Whereas Dr. RA jumped right into the proof, the
textbook set up a sentence to establish the general direction for the upcoming proof.
These sorts of preparatory moves suggest that the textbook author, like the professor in
many of his other proofs in the course, assume that students need this sort of guidance to

both understand a given proof and to begin writing their own. By providing well-formed

55

models for students, the course is designed to construct both the students’ lack of
understanding of proof-writing and a change in their epistemological grasp of
mathematics. Also, recall that this course was, for most students, the second course (after
Linear Algebra in this program of study at this university) past the traditional three-
semester Calculus sequence and so it is likely that students had had limited experience
writing proofs before.

A rather cheeky MAA review of this textbook acknowledges similarly that this
book offers a lot of support for new students.

This is a dangerous book. Understanding Analysis is so well-written and

the development of the theory so well-motivated that exposing students to

it could well lead them to expect such excellence in all their textbooks. It

might not be a good idea to create such expectations. You might not want

to adopt this text unless you're comfortable teaching from a book in which

the exposition will nearly always be clearer than your lectures. (Kennedy)
This review explicitly acknowledges that this degree of exposition is rare in
textbooks and in lectures alike, going so far as to warn instructors that elect to use
this text of the expectations for lectures students will come to posses. As has been
mentioned, other courses observed did not provide written detailed proofs for
theorems although the instructors may have verbally provided these. Other
courses tended to omit the English words from proofs written on the board and
only record the symbols. They would also not include phrases such as “First we
need to. . ..” or “To show X we first need to show Y.” Although these sorts of

anticipatory phrases were spoken in other courses, they did not get recorded

publicly. In Real Analysis both the instructor, Dr. RA, and the textbook included

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very detailed proofs that not only did not skip “obvious” steps but that also gave
commentary on the direction of the proof.
While the course was designed to change the students’ criteria for the
determination of knowledge, it also provided opportunities for students to learn
about issues in mathematics that remain contentious and where logic and rigor can
be pedagogically unhelpful. Take a definition of sets from the textbook as an
example. “Intuitively speaking, a set is any collection of objects” (Abbott, 2001,
p. 5). It goes on to provide examples of sets that are written as numbers, as words,
and as symbolic algorithms. After also describing the empty set, unions,
intersections, and complements, the textbook returns to the first sentence of the
section on sets.
Admittedly, there is something imprecise about the definition of a set
presented at the beginning of this discussion. The defining section begins
with the phrase “lntuitively speaking,” which might seem an odd way to
embark on a course of study that purportedly intends to supply a rigorous
foundation for the theory of functions of a real variable. In some sense,
however, this is unavoidable. Each repair of one level of the foundation
reveals something below it in need of attention. (p. 7)
Repeatedly throughout the textbook, the author comments on the pitfalls, benefits, and
implications of different definitions and issues concerning axiomatic systems in general.
The conclusion of the first chapter is an epilogue that describes problems mathematicians
have faced with set theory and the question of a complete axiomatic system. By striving
to change students’ epistemology, the book does not assert that all the troubles of the

world are fixed once a rigorous definition-proof epistemology is accepted. Rather, it

strives to continuously point out the problems of logic that mathematics continues to face.

57

What is exactly is a real number?...the set R of real numbers is an

extension of the rational numbers Q in which there are no holes or

gaps. . .We are going to improve this definition, but as we do so, it is

important to keep in mind our earlier acknowledgement that whatever

precise statements we formulate will necessarily rest on other unproven

assumptions or undefined terms. At some point we must draw a line and

confess that this is what we have decided to accept as a reasonable place to

start. (p.13)
Rather than falsely convey a sense that the epistemic system of mathematical proof is
foolproof, the course included discussions of times where mathematicians have struggled
to pinpoint ideas and definitions; places in history where fuzzy ideas were good enough,
and times where mathematicians toiled to improve these definitions. There was nuance to
this approach. The course suggested to students that rigor can be very beneficial and that
much of modern mathematics depends on the foundations movement of the 18th and 19th
centuries. At the same time, the course never went so far as to claim that this system is
error-free or complete. For a course aimed at students who were assumed to be novices to
proof and axiomatic systems, the course also treated them as if they were sophisticated
enough to deal with the conflicts and contradictions of the mathematical community’s
preferred epistemological tools. As I will later describe, this substance is different from

the Complex Analysis course’s take on substance, which stressed a rational faculty in all

things mathematical and personal.

Mode: Belonging

To invite students to take on a shift in their epistemology, the course appealed to a
sense of belonging and membership community both in the broader field and history of
mathematics and in the classroom. Despite a common stereotype that people who enjoy

mathematics are not terribly social, the students in this course, as they did in the

58

Advanced Geometry course, worked together often. In this way, mathematics was
constructed as a collective enterprise, not a solitary pursuit.

Although I was unable to be there on, the first day of class, I learned several things
about that first meeting. On the second day students who had not been present or enrolled
for the first class were instructed to meet with Dr. RA to get placed into a study group.

’ He explained that everyone else had already been assigned groups. These groups were to
meet each Monday night and he had arranged a study room in a nearby building to use as
the central location. He planned to be there and wanted everyone to attend. There would
be no lectures, but he would be available to talk with groups and assist them with ideas
needed to complete homework assignments or prepare for exams. These meetings fell
outside the scope of my class observations but from comments made by students it
seemed that they did attend these and were making use of this opportunity to get together
with their professor and one another to do mathematics.

One comment that particularly suggested their interest in this community
concerned what would happen on Labor Day, a Monday that should have been a study
group night, but a day where no classes would be held. A student asked Dr. RA if he
would still be there, suggesting an interest in attending despite the holiday. He replied
that they would still meet from 7pm -9pm.

Throughout the semester students continued to talk in class about their study
groups. On another day early in the semester as the class was packing up to leave, a
student sitting near me turned to other students, presumably those in her study group, and
asked them if they wanted to get together that night. They said that they did. “Good. I’ll

email you and we can set up a time.” In late October I again overheard students planning

59

a study session. A student suggested to her group mates that they get together Sunday
night so that they can arrive Monday with better questions for Dr. RA. On another
occasion a student walked in and told her seat neighbor, “I honestly thought about not
coming because of how I feel.” “I know”, replied her friend, “I came for you.” The norms
that had begun to be established for students working together outside of class and with
the professor seemed to be keeping many of the students interested in the course.
Belonging to a community was a strong aspect of the class and one that provided an
incentive to keep working, but the community of the study groups outside of class time
was not the only means of inviting students to take on the challenges of the course.
Groups were also used nearly every day of the course when I was present.

On most days Dr. RA began with a brief introduction to the ideas to be discussed
that day. Sometimes he provided a new example to consider or a new definition that they
would use. After this initial introduction to the day’s topic he would pose a question or
questions and have the students arrange themselves into their groups. After the first few
days of this, students began to sit together in the class near their group members rather
than taking any seat. They were deliberately arranging themselves physically to take part
in the group aspect of the course. They would usually pull their desks together into a
grouping and begin on the task. Some groups would have debates over the task and others
seemed to tend towards working alone before checking their answers with one another.
When at times the groups fell into this mode of silently working near one another rather
than working together, Dr. RA would exclaim, “I want to hear you talking!” and soon
they would begin to collaborate. This group time lasted anywhere from about 5-10

minutes on most occasions and on at least one day the whole period was devoted to their

60

groups. Again, this practice emphasized that mathematics was a collective endeavor.
The group work constructed mathematics not as a solitary, rational path, but as a field.
Participation in the field was a necessary part of the conversion to the kind of
mathematician that Real Analysis represented.

One day they were doing a review for an upcoming midterm. Dr. RA passed out a
sheet of paper that was divided into three parts: 1) Definitions 2) Short answer and 3)
Proofs. The definitions section included a list of twelve words of which four would
appear on the test that they would need to define. Examples included “closure”, “compact
set”, and “f(x) is uniformly continuous” For the short answer part they were given few
statements that they would be asked to determine as true or false and to either provide a
short explanation of their decision if true and a counter-example if false. There were
twelve of these as well, and here is an example: “An open set cannot contain any isolated
points.” The final section contained four things to prove such as “Prove that the Cantor
set C is compact.” The final item on the sheet was a direction to “Construct an increasing
function h: R 9 R, whose set of discontinuity is {l/n : n E N} .”

After he passed out the sheet he said, “By all means work with your neighbors.
It’s not meant to be quiet.” The students grouped up and began to work on the proofs
mostly. “The definitions are easy. Just make up some note cards and memorize them,”
advised a student.

In the syllabus Dr. RA described the course as challenging because it would
require students to think in ways that were different than they would have likely thought
about math before. Specifically, rather than thinking of math as computation with given

algorithms, they would need to think of ways to prove ideas as true or false rather than

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accepting them as such because an authority such as a professor or textbook said so. To
invite students into this change, the course was structured to appeal to student’s sense of
belonging to a community and group membership. The groups in the course were quite
formal compared to other courses studied. Advanced Geometry included students
working together in class but they were not assigned to these groups. That course also
had no formal post-class group arrangement. In Complex Analysis students were
“allowed” to work together on assignments but were discouraged from doing it; in
Advanced Geometry students worked together often in class but there no explicit
instructions to continue doing so outside of class. Only in Real Analysis and Discrete
Mathematics were students encouraged explicitly to collaborate. Dr RA went so far as to
organize students to groups both in class and outside of class and to formalize the
meetings of these groups.

The idea of students working together was not seen as an example of student
laziness but as a mode of invitation to take on the learning of the course. This was the
means of accomplishing the goal of altering their epistemology. In late October students
were milling around in the hallway waiting for the previous class to exit the room. A
student approached another student and asked him if he had finished a particular
homework problem and if, in class, she could look it over. She quickly added, “I just
want to know what is going on and I promise not to copy it.” These students seemed to
know that copying was taboo, but that you can learn from others without breaking that
taboo. She took care to let him know her purpose for wanting to see his work. The
community seemed to be seen by students as a way to connect ideas rather than as a tool

to a better grade despite not being warned by Dr RA. As in-class accounts of students

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describing their plan to meet and the actual grouping of students in-class suggested,
students were accepting this invitation to use community as a means to change.

While the groups were a formal and easily identifiable nod to community, the
class banter also suggested that the course acted as a social group as well as a learning
group. While it might seem a small detail that Dr. RA called on the students by name, this
act was often not shared in other classes. In fact, aside from the Advanced Geometry
course professors rarely if ever called on students by name. The instructors seemed to not
know who was in the class at all although each course observed only had between 20 and
30 students enrolled. In Real Analysis, Dr. RA did know the students by name and used
their names often. He could give papers back without needing to call on the students to
identify themselves; he could call them by name when they asked a question; and he
could “bless” their sneezes in a similarly personal way.

Not all work in the class was group work of groups or was cast in highly personal
vocabulary referencing students name or interests. There were also more subtle instances
of appeal to community. There were many times in a class period where Dr. RA would be
proving some theorem at the board as students watched. At this time the students were
positioned as audience members. They were not called to obviously and outwardly
participate. They sat and watched. Dr. RA’s pronoun choices were distinctive and
constructed students as belonging to a community on these occasions. Dr RA used we, us
or you as his primary pronouns. “Let me try to convince you with a picture.” “I do want
to hear you talking.” “These are the conditions that make us able to do in calculus what
we do.” “How could we talk about this?” He did use pronouns on occasion that did not

reflect a participatory assumption. “1 ’m going to do an example now.” These times were

63

fairly rare as most of his speech implied an interaction with the students through his
pronoun use or eye contact and body language.

This personal relationship and subtle pronoun nods to a relationship may seem to
be separate from the discourse of mathematics, but it constitutes an integral component of
a mathematics course. Someone might argue that they are artifacts of pedagogy and not
mathematics, yet the course design is a representation of mathematics in the world. The
Real Analysis course constructed mathematics as a human, personal, and relational
discipline. Mathematics was presented as something that people do together, not
something that exists in the ethereal world to be pursued within an isolated mind. In this
way the course represented a social constructionist view of mathematics and not a realists
philosophy such as Platonism. Doing mathematics is doing what people, mathematicians
in particular, do. If students wanted to belong to this group they needed to adopt the ways
of thinking about mathematics that this community shares. The community aspect of the
mathematics course makes mathematics seem more or less inviting to different kinds of
people, and it is therefore a mathematical construct, and not just a pedagogical construct.

There were also significant amounts of time before class devoted to casual talk
between the students and the professor and among the students. In other courses students
might chatter before the professor arrived, but once he or she did arrive, they tended to
grow quiet immediately. In contrast, when Dr. RA entered they would continue their
personal conversations although usually in softer voices or they would engage him in
their discussion. Sports—the university’s football team and the regional baseball playoffs
were favorites. Dr. RA also commented on changes he saw in them throughout the

semester. For example, one day a student who had had really long hair arrived with it all

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cutoff. Dr. RA’s first comment on entering the room was to joke about the student’s new
look. Aside from Advanced Geometry, no other course included this sort of lighthearted
chatter. Nor did those courses use student names as a part of the discourse. Although
these sorts of lighthearted conversations and jokes might not seem very mathematical,
they were a part of the community of mathematics present in this course.

In addition to the appeal of the community of students in this course (both in class
and outside of it), an appeal to become members of a larger community of mathematics
was also present. “In a recent rereading of the completed text, I was struck by how
frequently I resort to historical context to motivate an idea,” relates the author of the
textbook (p. x). The uses of history and tales of the larger mathematical community act as
enticement to the student to take on the shift in epistemology. As has been mentioned
before and suggested by this passage, the textbook included many accounts of the
development of real analysis in the history of mathematics. “This was not a conscious
goal...” he continues, “Instead, I feel it is a reflection of a very encouraging trend in
mathematics pedagogy to humanize our subject with its history” (p. x). Calling on history
is calling on a shared community. Others have struggled with these ideas and so too
might the students. Yet, whatever they face they are a part of something. Belonging to a
community is a unifying theme of the invitation to take on a new epistemology. The
grouping of students is a way to get them to take on this goal as their own. By grouping
them outside of class there is both a practical resource for finding help and a shared
obligation to help one another. Likewise, the in-class grouping activities create a

community or a set of sub-communities that the students come to rely on. Finally, to

65

become a part of the larger community of mathematicians is a further appeal to
community.

Students were not instructed to study hard to pass or to get a good grade (and
neither were they discouraged from this either). They were not invited to revel in the
beauties of mathematics as they were in Advanced Geometry or to discover pleasing
connections as they were in Complex Analysis. Instead they were called to join a
community. Doing mathematics is to be a part of something larger than the individual.
This draw of a community and a peer group acted as a motivating force. The smaller
class groups and the larger mathematical context provide act as an invitation to take on
the definition-proof system. It is the carrot-stick to the assumed reason they are taking
this course—to shed the old ways of thinking of calculations based mathematics and
begin to take on a proof based orientation. Furthermore, this group membership appeal
positioned the students as capable and contributing members of the community. Students
had a right to participate and a right to comment on mathematics; it was not reserved for

a select few.

Regimen: Holding court

The daily activities and general atmosphere of the class has already been
described in previous section, but there is still more to be said about the things students
did for this course. Primarily, they were asked to determine the truth. They were asked to
sit as judges, juries, lawyers, and sometimes as the general public in a mathematical
court. The students were asked primarily to engage in truth activities—judge the truth of
statements, be convinced of the truth of statements, argue for the truth of statements, and

watch similar proceedings of truth activities.

66

A very common way that the in-class group work unfolded was for the students to
be given a set of statements in a true/false format and to judge which were true and which
were false. The second meeting of class provided an early example of this type of
activity. Dr. RA instructed the students to group together and answer 1.3.9 from the
textbook. It included five parts.

a) A finite, nonempty set always contains it supremum.

b) If a < L for every element a in the set A, then sup A < L.

c) If A and B are sets with the property that a < b for every a E A and

every b E B, then it follows that sup A < inf B.

d)lfsupA=sandsupB=t,thensup(A+B)=s+t.ThesetA+Bis

definedaA+B= {a+b:aEAandbEB}.

e) If sup A s sup B, then there exists an element b E B that is an upper
bound for A. (Abbott, 2001, p. 18)

Three students near me pulled their desks together and began to address each statement in
turn. These students were rather quick to judge and within a few minutes had decided the
order went T, F, F, T, T. Other groups around the room were having more heated debates.
Whereas this collection of students near me acted as judges, other students were acting
first as lawyers. They were arguing their positions against one another before passing
judgment. All the while Dr. RA roamed the room and watched students, pausing to ask
questions, and generally gauging the direction of the groups. After about ten minutes he
called on students to offer their decisions; to have their say. Whether by purposeful
selection of students or by coincidence, each person who offered a response was in turn
judged to be correct by Dr. RA. This type of true/false activity represented most of the in-
class group activity I observed. True/false judging was also a staple of the exams. A
slight variation of this true/false work that would also appear would be like the statements
below assigned in late September. Students were asked to determine if the below requests

were possible and if so provide an example or if they were not possible and if so provide

67

an explanation. They were not exactly true/false scenarios but could be created/ could not
be created scenarios.

a) A Cauchy sequence that is not monotone

b) A monotone sequence that is not Cauchy

c) A Cauchy sequence with a divergent subsequence

d) An unbounded sequence containing a subsequence that is Cauchy

In addition to judging the veracity of statements, students also acted as juries
within the class. When the in-class group time was not spent determining statements as
true or false it was spent crafting arguments to prove some mathematical point or
theorem. In mid-November, Dr RA wrote Rolle’s Theorem on the board.

Let f : [a,b] 9 R be continuous on [a,b] and differentiable on (a,b). If f(a)

= f(b) = 0, then there exists a point e in (a,b) where f’(c) = 0.
A student stopped him and asked whether f(a) and f(b) actually needed to be equal to 0.
Dr RA replied that no, that condition was not needed, but it was still true. Again, this is
an instance of students having rights to speak about mathematics, question it, and refine

it. Dr. RA then erased the “= 0” from the initial assumption on the board to get:

Let f : [a,b] 9 R be continuous on [a,b] and differentiable on (a,b). If f(a)
= f(b), then there exists a point e in (a,b) where f’(c) = 0.

As Dr. RA was making his adjustment to the board he was also telling the class that the
proof could be found in the book for this more general case. When he turned around
again two students in the front row had their books open to the proof. With a laugh, Dr.
RA told them that they were cheating and that in a moment he wanted the class to prove
it without using the book. The two students also laughed and closed their books. The
students grouped up and began to prove the theorem. The group closest to me began by

considering a pictorial case (see Figure 2.2):

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Figure 2.2: Students’ sketch where f(a) and f(b) were maximums

They argued that in this case the endpoints f(a) and f(b) were the maximum points and
the minimum was somewhere in the middle so there existed a f’ (c)=0 in there (at the
minimum). Their next picture was a flat line and they argued that f(a) and f(b) were both
minimum and maximum points and that everywhere along the interval there were points

whose derivatives were 0. See Figure 2.3

 

Figure 2.3: Students’ sketch where f(a) = f(b)

The final pictorial case this group reasoned through was a reflection of the first
and their argument was similarly reflected. If f(a) and f(b) were the minimum points then
somewhere along the curve was a maximum, c, and at that point the derivative would be

zero. See Figure 2.4

69

0”

._‘

Figure 2.4: Students’ sketch where f(a) and f(b) were minimums

Although this group failed to consider cases where f(a) and f(b) were neither the
minimum nor the maximum, they were engaging in mathematical argumentation. This
sort of in-class proving was the common practice. At these times students behaved more
as lawyers than as judges. That is, they were in charge of developing the arguments to
show a desired conclusion rather than to judge the validity of a statement or case. Just as
tests included a judging aspect in the true/false sections, the tests also included arguing
sections where students proved given theorems.

To continue the courtroom metaphor, students also acted as jury members who
decided the value of an argument. On many occasions Dr. RA acted as the lawyer,
proceeding carefully through an argument or proof. His proofs were very detailed as has
already been noted, but they were also varied. He did not just present the statements and
rationales to prove his case, but also invoked first-hand testimony and graphical evidence.
On the same day that the class was proving Rolle’s Theorem, Dr RA also proved the
Mean Value theorem. To do so he first drew a picture of a generic function that
“wobbled” between f(a) and f(b) and talked through the picture as evidence that

supported his claim that there existed a point c along the interval that had the derivative

70

f(b)-f(a)
b—a

 

. When he had concluded this argument he proceeded to provide a second

way for the students to be convinced. He told a story about driving along the New Jersey
turnpike between Philadelphia and Boston. The speed limit was 55 mph and if you got on
at mile ten at 9:00 am. and got off at mile 50 at 9:35 am. you must have been speeding.
There must have been a point at which you were exceeding 55 miles per hour. He
concluded his argument with a formal proof of the Mean Value Theorem. As he
proceeded through the layers of evidence the students sat as jury members to be
convinced of his argument. He explicitly acknowledged their role by asking them twice
while writing the formal proof whether they were “convinced” and one time “ does
everyone understand?”

That students were juries as well as lawyers and judges is important. Dr. RA did
not take for granted that the students would necessarily be convinced of a mathematical
argument just because he claimed it to be valid. To have assumed that sort of trust would
have been to keep working within the system of thought that the course was trying to
override, a system where students took ideas as truth because they were told to do so by
trusted sources such as professors, teachers, and textbooks. For this course, it was
important to learn that the rules of mathematical logic and argumentation are valid forms
for attaining knowledge. Students were treated as if they might not yet fully believe that
his arguments and logical moves are in fact “proving” the theorem and so Dr. RA needed
to gauge their acceptance of his proof. Students were expected to have a say in what was
convincing and what was not. They were expected to speak up when something was not
persuasive or when they did not understand the proof. Doing mathematics includes being

convinced.

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In summary, students were expected to hold court regarding the truth and validity
of mathematical knowledge. They acted as judges when they assessed the truth or
falsehood of claims while working in small group sessions in class and individually on
exams. They acted as lawyers when they debated one another in these small groups or
when they crafted proofs or arguments of theorems. At times they were juries deciding
whether Dr. RA’s proofs and work at the chalkboard were convincing. In the remaining
times they were the audience watching the proceedings as Dr RA performed the roles of
lawyer, judge, and jury. To describe the students as either active or passive depended on
the nature or genre of the class time. In one respect being a jury member is both passive
and active. They typically sat still and quiet, but they were passing judgment through
body language such as slumped in chairs with hands propping up heads or leaning
forward with pencils in hand. Silence on their part was both active and passive. For a
course that worked to alter students’ sense of mathematical knowledge and how we come
to know, the kinds of activities students were asked to perform constructed the learning
of mathematics as a multi-dimensional practice. Students were not positioned as
recipients of mathematical facts or knowledge. Students acted as lawyers, juries, and
judges with respect to mathematical proofs. They practiced argument, documentation,

and evaluation as daily class activities.

Telos: Converts

By describing a model student I am not describing an actual student who was
found in the course. This is not a characterization of a real person, who for example,

made good grades. Instead it is a description of the imaginary model—a perfect

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realization of the perfect course under perfect circumstances. What ideal of learning
mathematics was constructed by the Real Analysis course?

A model student this course would be a convert to a community of formal proof-
oriented mathematics. This model person would no longer hold that math was all about
numerals and fancy calculations. Rather, in the ideal outcome of this course, students
would know that math is about number systems and real functions. It would be the
construction and sharing of proofs that rigorously described things most people took for
granted as true. The convert would believe in these ways of thinking and doing
mathematics. She would adopt proof as the criteria for knowledge and that the form of
writing a proof conferred knowledge. But she would not also necessarily worry about the
origins of this system. She would think that ideas/theorems needed to be grounded in
proof and logic but would not question the origins of that logic. This convert would not
think much about axioms and their consequences so much as use those axioms and know
when others used them well.

The conversion was an epistemological one—a change in what evidence counts to
create knowledge, but not whether that evidence is itself true. That is, rather than
question the legitimacy of calculus, this model student only questions how we know
calculus. Do we know it because we were told in Calculus I that the Mean Value
Theorem is true or do we know it is true because we can prove it to be so. So long as the
proof holds up to the standards of the mathematical community, it is deemed by the
convert to be true. The model student is not skeptical of the system—this convert has
faith in the system, but the model student is skeptical of all mathematical assertions until

that knowledge has been proven, “beyond a reasonable doubt,” to be of the system and by

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the codified rules of proving, a special genre of writing. As the textbook of this course
states, “At some point we must draw a line and confess that this is what we have decided
to accept as a reasonable place to start.” The model student of this course does not
concern himself with redrawing that line. He works to determine that once the line is
drawn all further ideas and theorems conform to it.

Part of this shift in belief from thinking about math as facts and algorithms to a
belief in tight definitions and logical proofs would be based on a trust in and a desire to
be a part of a larger mathematical community. Proofs are to be shared and communicated
with others. The convert wants to hear others’ ideas and wants to share her own. As a
judge she is willing to decide if arguments fall into the logical mathematical framework
and is willing to have her own judged similarly by others in the same community of
belief. To stretch the beliefs metaphor a bit, the model student would enjoy attending
group meetings with others who shared his faith and interests. He would be happy to
debate and be debated. He would also be happy to listen and observe others engaging in
mathematical conversations.

Although this constructed student of Real Analysis is open to having his proofs
judged and to judging others’ proofs, he is not necessarily a creator of new ideas. No
evidence from the class suggested that an ideal outcome would be for students to create
or formulate new mathematical ideas, theorems, or concepts. Instead, this course aimed to
create a model student who could operate and make decisions based on a particular code
of logic and argument—not someone who could create those codes or experiment with
changing them. A model student would act as a judge of mathematics but not as an artist

of it. She could determine the truth or falsehood of ideas based on whether those ideas

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were within the bounds of mathematical logic. In the same way that a judge determines
whether certain acts fall within the law or outside of that law, so too does the model
student use her codes of belief to assess the worth of an argument. But this constructed
students does not create new laws or codes of being. She did not push the boundaries of
established mathematics but stayed within them, like staying within the Constitutional
limits or within case law precedents.

In summary, the model student constructed in this Real Analysis course believes
in mathematics. She believes in an organized and logical system of mathematics. She
does not mind speaking her mind to question others’ ideas and does not feel slighted to be
questioned herself. His belief is not a gullible uninformed one. The convert is part skeptic
after all and enjoys discussing the minutia of a proof, but this part skeptic is still a
believer in the greater truth of mathematics. There is a line after all beyond which the
believer takes as faith. Anything on this side of that line though is open for skepticism
and demanding of proof. The model student is also social, reveling in conversation about
challenging mathematics and observing others likewise. To be a convert is not to be naive
or cowed by others. The model student is sure-minded and ready to be part of a
community and group of mathematics.

This Real Analysis course positions students as being in need of an epistemic
change. A model Real Analysis student no longer believes that mathematics is' to be
accepted at face value. Students are seen to be in need of a new way to determine the
truth of mathematics beyond accepting the word of teachers and textbooks. To
accomplish this change the course makes use of a community. The students are invited to

participate in small in-class and outside of class communities so that they can take on the

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challenges of this new episteme. Community is also conceived on a larger scale—that of
the wider mathematics community of mathematician and other mathematics
professionals. With membership in this larger community of students as incentive,
students are asked to take on the change in their knowledge system. As part of the work
to create this change students are asked to act as judges, lawyers juries, and audiences of
a mathematical court that rules on mathematical issues. Statements are judged, and proofs
are drafted and found either persuasive or not. Procedures are observed. After all of these
experiences a model student that would emerge from this real Analysis course would be a
convert to a logical proof-based system of mathematics. This convert would go on to

become an active member of the mathematical community.

Concluding remarks

Speaking about a model student can be a helpful way of analyzing the course as a
whole. It can tell us about the assumptions the course makes about students as a category
of person. What do they lack? How are they inspired? What should they do? Who should
they become?

But just about anyone who has experienced classroom life and real students,
whether they be young children, middle schoolers, teenagers, or college undergraduates
recognizes that perfection is rarely a part of the experience. There is no perfect class,
teacher, or students. Students come to classes with varied pasts, personalities, and hopes.
In recognition of these realities I would like to conclude this chapter and each of the
chapters describing one of the observed courses by speculating on what some real

students might think about this course.

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The emphasis on formal proofs rather than calculations or rote knowledge might
have turned many students away just as it could have excited students with very
analytical approaches to mathematics. The variety of types of tasks, though, would have
provided different opportunities to do mathematics for students with varied preferences.
Students who liked evaluation more than creation would have also fit in well when the
class engaged in true/false type determination. More of the student in work in class
tended towards this, but those who preferred writing proofs would also have had a chance
to participate. There was also a place for students who liked rote learning as tests
included the memorization of definitions.

Social students who enjoyed working with peers, particularly a consistent group
would have enjoyed this experience. They might have liked the study sessions as well as
the in-class group work not only for the mathematics but also because they could form
fi‘iendships. Those who wanted more one on one time with Dr, RA had a chance for this
in the after class study sessions as well. Students with a more outgoing personality would
have enjoyed the before class banter with Dr. RA and felt comfortable joking with him,
as the student who cut his hair was mildly teased about the dramatic change in his looks.
These sorts of students would have been well served by the appeal of belonging to a
community that the course design provided. Many students in the class did seem to
appreciate this aspect and made comments suggesting that they enjoyed working with
their peers.

During the in-class group time many groups were very animated and talkative.
But there were also groups that did not function very much like a group, but rather a set

of individuals who moved their desks closer together. The social aspects of the class were

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not so appealing to these students. They preferred to work alone and then share and check
their answers as a group rather than actually proceeding through the tasks together. On
one day when the class was reviewing and preparing for a test by working on a set of
tasks, one student was irritated that Dr. RA was not leading the review. “This is stupid. I
should have stayed home. I expected he was going to go over something.” The
implication was that without Dr. RA’s commentary or lecture this was pointless and that
the group work was not at all attractive. For a while she and another student discussed
their applications to the teacher education program She ended up leaving class before the
period ended.

The social aspects of the class did not seem to be helpful for everyone. In another
group the students seemed lost frequently and unable to make progress on tasks. It was
not that they did not work together or try the problems. Rather the collective did not seem
to be able to support one another. This group frequently called Dr. RA over to help them.

Both of these two cases, the girl who left class and the stalled out group, might
have preferred a more direct instruction approach. It may have been that each held the
assumptions about mathematics the course was trying to change—that the professor, an
authority was needed to tell them how to proceed. Yet there was a different flavor to
these two cases. In the first example of the student irritated by a group review worksheet,
she did not express a need for Dr. RA’s assistance. She implied that she could have done
the work at home by herself. It was that the model of mathematics in the class did not
match hers—a vision of a math classroom where the teacher talked and students took
notes. That she needed him in the sense of his help was not expressed, but that it was a

habit and an expectation about a normal classroom. The other example of the group that

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called Dr. RA over frequently seemed to be of an opinion that they needed him and that
without him they could not make progress. Possibly direct instruction would have been
more pedagogically appropriate for them or possibly it was the close interaction with him
that was wanted. Either way, the group time was not a positive time for them. Students
who did prefer lecture styles still had that available as class time alternated between two
types of interactions—group and lecture.

Just because a student might have preferred either group or lecture (or both or
neither), does not mean that they also enjoyed the removal of calculations fi'om the work.
As my analysis described, students were positioned as thinking that knowledge is the
result of executing some calculation type activity. Expressing your knowledge for these
students is executing those calculations well. Their skills would not have been valued and
they would not have been asked for. For student that did have this image of mathematics
and knowledge, this course would have been very abstract and possibly very annoying.
They might not have seen it as mathematics at all but maybe as tedious or nonsensical.
For others, the emphasis on proof was very positive. Take for example an exchange
between two students, “I’m at a place where I actually feel like I know what I’m doing,
which is a weird idea. I’m really excited about it.” “That’s where I was last week.”

This course had space for students who enjoyed social and/or individual
mathematics. In terms of how they do math, there were at least two possibilities: lecture
and group work. Again, with tasks, there was multiple ways for students to express what
they could do and liked to do: memorization, evaluation of arguments and statements,
and the construction of proofs. But when it came to a vision of what mathematics the

course design modeled there was only one option. Students who did not see proofs as a

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part of doing mathematics would. have very likely had negative or at least not so positive
experiences. But, others, like the students quoted above, found a new way to think about

math and kind of liked it.

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CHAPTER THREE

ADVANCED GEOMETRY: FUTURE RESEARCHERS

In mathematics the art of proposing a question
must be held of higher value than solving it.
-—George Cantor

Introduction

The Advanced Geometry course I observed was one of three courses in a set
where math majors had to select one as required by their plan of study. The other two
Options in the set were an Axiomatic Geometry course and an Ordinary Differential
Equations course. They could have taken multiple courses from the set, but these would
count as electives in the major. Students planning to also earn a teacher certification in
secondary mathematics were restricted in their options by being required to take one of
the two geometry selections. Therefore, preservice teachers could be assumed to be
largely present in these two courses. Furthermore, students who were not in the teacher
certification program (or were not planning to be) were required to take the Ordinary
Differential Equations course. This means that the only way students not intending to be
teachers would be in this course would be if they were taking it as an elective. Everyone
taking it as a requirement would be in (or planning to be in) the certification program.

As a 300 level course students could have been taking it anytime after they took
the Calculus sequence and Linear Algebra, the prerequisite for nearly all courses at and
beyond the 300 level. Therefore these are the only guarantees for students’ prior
coursework. Depending on a students’ particular entry into the mathematics program a

student could have taken this course in their sophomore, junior, or senior years. This

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made determining who in the class was intending to be a future teacher difficult as
students typically entered the teacher certification program sometime in their junior year.
Inferring from other research related to this course and from comments made by students
in the course the semester observed, 1 would estimate that about 50% of the 24 enrolled
students were either already in the certification program or had intentions to be so in the
future. Across the math department, the course was often spoken about as appropriate for
fiiture teachers because of the geometry content—a major high school and middle school
curriculum strand.

The class met in a computer lab in the education building, the only course
observed that did not meet in the mathematics building. Another unique feature of the
course was that there were two instructors. One was the professor of record as listed on
the university’s schedule of courses and the other was a person with particular expertise
in the geometry software, Geometer’s Sketchpad (Jackiw, 1991), which the course used
each day including exams. The secondary instructor was not present at each class
meeting, but the primary instructor was for each day I observed.

The room was arranged so that the computers faced the perimeter walls on all four
sides and the projector was directed at a spot over the computers at the front. The
instructors operated a laptop from the interior of the room but nearer the back (seats were
available behind them although students rarely chose those) and had a good view of each
student’s or pair’s screen. When the class engaged in whole class activities students either
faced the screen at the front or towards the middle to look at the instructors. The general
atmosphere of the class was informal and a bit noisy as most students spoke often either

in whole-class discussion or in small groups or pairs. Considerable portions of class time

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were spent in these smaller group situations with many people talking at once all over the
room. As the semester progressed most people, instructors included, seemed to come to
know most others by first name and the students referred to their instructors by first name

as well. The general feeling was informal but still academic.

Prelude to the analysis

The Advanced Geometry course constructed students as future researchers by
creating a mini research experience that simulated some aspects of a research
mathematician’s habits such as investigation, pondering, conjecturing, selecting
conjectures or ideas to pursue or reject, and proving. Unlike other courses observed in
this study, Advanced Geometry was the only one where students constantly initiated the
questions under investigation. Although there was a lot of comparative freedom in these
explorations, the two course instructors, Dr. AG and Mr. SP, worked to provide a
structure for the investigations to keep the class focused towards similar topics of
geometry. Within these ever changing boundaries between freedom and limitation,
students were asked to think less of themselves as students and more of themselves as
novice research mathematicians. Students’ self-concept —from student to novice
mathematician — was the focus of the course. It was the substance, the aspect of the
student that the course design worked to alter.

To further the student-as-research-mathematician construct, the course mobilized
aesthetics as the invitation (the mode) to fuel a desire to understand already established
mathematics and create new mathematics. Appeals to wonder about interesting,

surprising, beautiful, and unexpected relationships and possibilities were frequent.

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Students were asked to play in this course. Play was the regimen, or the theme
relating the activities students were asked to do. In particular, they were asked to play
games of “I Spy” and “What If.” That is, they looked for things of interest to pursue.
These might have been patterns or anomalies in a geometric scenario. They were also
asked to imagine mathematics if certain ideas were removed or altered. In the parts of
class devoted to non-Euclidean geometries this “What If” types of activities were central.
“What if the parallel postulate were ignored?” “What if I tried to make a kite on the
Poincare disc?”

The course’s work to change students’ self concept, the use of aesthetics to invite
students to make this change, and the conflation of doing math (working) with play
suggest that a model student (telos) that would emerge from this course would be a
novice research mathematician. Particularly, someone who is interested in creating new
ideas with a focus on the aesthetics of doing so, who can articulate them to others in a
style consistent with the greater mathematics community, who can defend them, and who

takes pleasure in all of the above.

Substance: Self-concept

Students’ self-concept of their own identity was the primary site of intervention in
this course. The course worked to move students from seeing themselves as students to
conceptualizing themselves as novice research mathematicians. That is, rather than act in
a conventional “student” way where they sat, listened, took notes, and mostly remained
quiet, the students in this course acted (or progressively did so throughout the semester)
in ways more closely resembling a novice research mathematician. Obviously, they did

not posses Ph.D.s in mathematics, but they were performing as researchers within the

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space of this class. They explored ideas, debated conjectures, offered critiques of one
another and of the instructors, and proved results.

One important way that these students were moving away from a traditional
“student” concept is through the role of questioning in the course. Students authored
many of the questions under investigation in the class, they directed their questions (and
expectations of a competent response) to other students, and they questioned ideas
proposed by the instructors. By taking responsibility for posing the problems and
conjectures and expecting their classmates to respond to their questions, they were
moving towards a self-concept of a mathematician—someone who not only studies and
knows established mathematics but also someone who creates it.

One example that illustrates the ways that students were seen as authors of
mathematics occurred in early November. A student suggested that the class try to
construct a square in taxicab geometry. Dr. AG suggests that they first construct one in
Euclidean and then use that to help them create the square in taxicab. Afier the students
alternated turns giving directions to Dr. AG for the Euclidean construction, three students
began tag-teaming what to do for the taxicab one. After each student offered a “next-
step”, Dr. AG rephrased her suggestions. “Jamie wants us to. . .”; “Sasha says. . .”; “Have I
done it right, Terri?” Both Dr. AG and Mr. SP constantly insert students’ names into their
comments. By referring back to the students there is a continual reaffirmation that these
ideas are student ideas. Carla then spoke up, “I’m going to disagree with you,” referring
to a comment made by Mr. SP that a square in both Euclidean and taxicab geometry act
the same. Dr. AG tells the class, “That is an interesting idea.” After she explains that in

Euclidean the circle that defines the construction of the square is everywhere

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symmetrical, but in taxicab it is not and therefore limits the square’s construction. Several
students as well as Mr. SP get involved in a conversation about the symmetry of the
objects in both geometries. Finally Mr. SP asks another student, Aaron, can you answer
Carla’s question more effectively than I can?”

In this episode many students offered ideas, they spoke directly to one another as
well as to the instructors. One also disagreed with the instructor and her idea drives the
conversation for about ten minutes. When Mr. SP was unsuccessful explaining his point,
he deferred to another student hoping that Aaron might be more successful. By deferring
to other students, and by taking up student disagreement seriously, the instructor was
signaling that the students had responsibility for the direction of the discussion. They
were not to just sit and listen in the ways that a common image of a student might.

Although the differences between the ways that students interacted with other
students in this class, and particularly in regards to questioning, were striking there were
other ways that the course acted on students’ self-concept as novice research
mathematicians. By deemphasizing the instructor’s role and providing many
opportunities for students to interact with one another the course laid a foundation for
students to seek out resources in addition to the instructors for assistance and ideas. When
students could begin to act without always relying on Dr AG or Mr. SP’s immediate
guidance, they could begin to move away from acting as students who were reacting to
the instructors. To shed a self-concept of the traditional student was not enough, but an
important step. The course also worked to get them to take on a self-concept of novice

research mathematician.

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The primary way that this was enacted was by having students choose the
questions that drove the curriculum. Students in Advanced Geometry were asked to
create conjectures—to question the geometric scenario under investigation and to
describe it in terms of a possible relationship or relationships. The first Friday of the
semester provides a simple example. After the announcements were over Dr. AG directed
the class to work for ten minutes on Chapter 1: Activity 5 from the textbook’s CD
supplement. It states:

If one side of a triangle is extended at one vertex, the angle it creates with

the other side at that vertex is called an exterior angle. Construct an

example of this in Sketchpad. Measure the exterior angle. Compare this

measurement with each of the interior angles at the other two vertices.

What do you observe? Express your observation as a conjecture. Is your

conjecture still true if you vary the triangle? (Reynolds & F enton, 2006b)

The students began working in small groups, pairs, or a few as individuals. The
two instructors as well as myself circulated around the room watching students work and
pausing to answer technical questions about the soflware program and the task in general.
After the exploration time drew to a close, the second instructor asked the students to
share out their conjectures. One group said that the exterior angle is less than or equal to
the sum of the other two. The instructor typed this into a textbox in the software that
projects onto a large screen at the front of the classroom. A few more groups chimed in
that the first group’s conjecture is false—the exterior angle is exactly equal to the sum of
the other two. Another group conjectured that the exterior angle is larger than any other
interior angle. The lead instructor initiated a discussion of vocabulary such as “other

two”, remote, and non-adjacent angles. When all of the conjectures offered had been

written on the screen the students were challenged to prove or disprove the conjectures

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under consideration. One student tackled the conjecture that the exterior angle will have a
measure greater than either of the non-adjacent interior angles, but his proof was based
on the idea that a triangle’s interior angles sums to 180 degrees. Dr. AG recorded the
proof be outlined but challenged the class to prove that a triangle’s interior angles
actually do sum to 180 degrees. If his prove relies on this, they should be able to
demonstrate it. She next walked them through a proof, pausing to prompt students to
offer the “next step.” A student interrupted this exchange. He amends the proof about
exterior angles, strengthening it to say that the exterior angle is equal to the sum of the
two remote angles and proceeded to provide a proof, again relying on the 180 sum. Dr.
AG then told the class that they have the mathematical tools to prove the 180 degree sum
conjecture, which was abandoned to follow the amended exterior angle proof. The class
continues onto the next activity, but ended before completing it. They were told that they
would return to it next week after the Labor Day holiday.

This day described above resembled most other days of the semester. The
class typically ran on a schedule of brief announcements, Dr. AG posed a
situation, student pairs or small groups created it and explored it either on paper
or by software, students publicly offered conjectures, then proved one or more of
the conjectures.

By posing their own conjectures students had the opportunity to author
the mathematics under consideration, but it is important to notice that the
instructors set boundaries in which the class could explore. Rather than tell the
students to “come up with some conjectures” in a completely free environment,

the instructors give a specific context and direct the conjectures towards those

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relating exterior angles to remote interior ones. This allows space for the students
to move away from traditional roles of sitting and listening to a lecture, for
example, and instead create the topics to discuss. This is a move that opens the
possibility for their self-concept to change to a novice researcher. Because they
are constrained by the specific conjecture scenario the students are not exploring
anything and everything about exterior angles. The direction to look at the
relationship between interior and exterior angles provides scaffolding and
limitations, hence the use of the adjective ‘novice.’ They remain students in a
classroom context with instructors and course material. There is, though, a move
towards allowing more authorship by the students, which in turn allows for the
possibility of their self-concept to alter.

As the class came to be more habituated to the practice of creating their
conjectures, the instructors gave more open-ended scenarios. One example of a
very open-ended task was the major assignment, The Book, worth 30% of their
course grade. In this assignment students were asked to create 3 conjectures and
their proofs. They were guided to consider modifying or extending a task
explored in class or in homework. They could also depart from class-based tasks
for their Book entries. This assignment exemplifies many features of the course,
some to be discussed in later sections. For now it is enough to notice how the
students choose what to include in their books. The students get to choose how
close they want to stick to topics from the course or how far they want to reach
beyond the course topics. By providing this range of possibility the course make

space for the students to alter their self-concept from student to novice

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mathematician. Just how far the students choose to progress away from a
traditional student concept and towards a novice research mathematician one is up
to each person.

Because students offered their own conjectures rather than the professors
offering them (in class and in The Book assignment), it was sometimes very
unclear whether a conjecture was true or false. That is, false conjectures made
their way onto the projected screen by the typing of one of the two instructors.
Ideas were not censored at this stage. To move towards a self-concept of a novice
mathematician, students not only needed to create conjectures and mathematical
ideas, they also needed to argue against them or prove them. The students,
prompted at first by the instructors and progressively throughout the semester by
other students, demand proofs in part because they have no other way to know the
truth of the statement. That is, they create the conjectures rather than the professor
stating them as theorems to be proven and so need a proof in a way that a student
who is told to prove a theorem does not need the proof. In the former method, the
proofs are necessitated by the uncertain nature of the conjecture. Granted, many
students were satisfied with “seeing” the results on the computer program and
accepted these as proofs (especially early in the semester), but the instructors and
other students continually pushed on these empirical observations.

In this Advanced Geometry course, it was the responsibility of the class to
debate conjectures by refuting them, disproving them, or proving them. In this
way the Advanced Geometry course shares features with the Real Analysis

course. Students here also act as lawyers, juries, and judges. But to steal, the

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judicial metaphor, students here also wrote the theorems and might be thought of
as members of congress. That students directed the content set it apart from other
courses. In this course students directed the content, within the boundaries
provided by the instructors and textbook—as well boundaries imposed by student
ideas.

By making student curiosity central the instructors could not have complete
control over the content although they directed it through the scenarios they dictated as
starting points for the student investigations. The pedagogy of eliciting student-driven
content created mathematical uncertainty that later needed resolution. Student formulated
idea were often non-rigorous in part because they were initial ideas and brainstorming,
but as the class period continued (or the week depending on time issues) the instructor(s)
and other students demanded that the conjectures be restated with more technical
vocabulary and proven if possible.

The course syllabus speaks of mathematics in terms of topics initially and later in
terms of investigation and processes. The first paragraph declares:

This course will begin with the geometry of the ancient Greeks, and will

progress through to more modern topics in geometry inspired by those

ancient roots. In the spirit of a mathematics defined first by its generating

tools (the ancient compass and straightedge) we will use in this course The

Geometer’s Sketchpad, a powerful modern visualization tool for

expressing and analyzing mathematical relationships. The topics will

include:

° Classical Euclidean plane geometry,

° Modern Euclidean geometry,

0 Transformational geometry,

- Projective geometry and the complex plane

0 Hyperbolic geometry

This list of topics tells us that the construction of the subject here in this course

needs to learn some topics, but does not quite say what these will be. The last four

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bullets refer to non-Euclidean geometries. By including different geometries the
course design is exposing students to new ideas (new compared to the Euclidean
they likely know from high school). Furthermore, these geometries often
contradict one another. A theorem in one will not hold in another. Like the
Discrete Mathematics course, this one is interested in expanding students’ views
of what counts as mathematics. It is a contrast to the Complex Analysis course
where students are exposed to a unitary and coherent vision of mathematics.

The syllabus later describes other aspects of the course:

In this class, you are being invited to explore geometry through computer-

based investigations, to make observations and conjectures about what you

see, and then to develop proofs to support or refute your conjectures. You

will be “constructing” many of the theorems and ideas we encounter, both

in class and through your homework. While we will be learning about

many results in geometry, some discovered long ago and some discovered

more recently, you will also be expected to initiate your own explorations

as well.
The first section quoted refers to topics to be explored in the course and makes explicit
mention of the ways that modern mathematics have been inspired by Greek mathematics,
including both topics and tools or processes as modern inspirations of ancient techniques.
There are topics and techniques or processes at play in the course. The processes seem to
be given greater weight during the class observations. Although students were invited to
learn theorems and definitions, it is the processes of exploring, conjecturing and proving
these theorems that take up most of the class time and conversation. Because the
Advanced Geometry students were as active as they were in creating the conjectures,

refutations, and proofs, they were acting less like traditional students and more like

contributing members of the mathematical community. Students were treated as if they

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were capable of making decisions about which mathematics to pursue, capable of

creating their own ideas, and capable of verifying them.

Mode: Aesthetics

If the primary aspect of the student this course works on is self-concept from
traditional student or observer of mathematics to novice researcher or creator of
mathematics, the mode of inviting students to take up a new self-concept here is by
invoking and highlighting aesthetic elements of mathematics.

In the above example from the first week of class the students work to create
conjectures and proofs fiom a situation involving interior and exterior angles of
triangles—one that they likely had encountered before in high school geometry. As the
semester continued the students began to work on mathematics not typical of high school
curriculum. In mid-October Dr. AG emailed a Sketchpad sketch to everyone before class
with instructions to download it for class discussion during the next meeting. It looked

like the one below in Figure 3.1.

fi.

 

Figure 3.1: Dr. AG’s emailed sketch

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As class began and before either instructor gave any directions about the sketch
students downloaded it and started dragging different points around to see what would
happen when changes were made in the position of P, P’ and the size of circle A. About
halfway through the semester the routine had become established that when given a
sketch or situation their job was to start asking questions about what changed and what
stayed constant. On this day the professor instructed them to focus specifically on the
relationship between P and P’. Developing curiosity is a central part of the course but
curiosity is a way into aesthetics, the means of inviting students to participate. By
providing opportunities to wonder—to be interested in, to doubt, and to speculate—and
time to follow that curiosity, the course attempts to strengthen students’ aesthetic
relationship with geometry and mathematics.

When given the opportunity to explore the students could take up issues they
found interesting that satisfied a personal aesthetic. Some students seem to revel in
strange behaviors of seemingly normal objects—points P and P’ in this case. Others took
up the aesthetic by methodically analyzing the behavior of the points. For the first group
of students the fun seemed to be the unexpected itself, for others the fun was to explain
the unexpected. There was space for students to follow their own personal sense of
wonder and aesthetic. Again, as will be examined in the chapter related to Complex
Analysis students are not invited to take on the challenges of the course by presenting a
well-ordered and structured sense of beauty—an aesthetic that values structure—but
instead one that had room for considering structures as aesthetically pleasing and/or for

considering strangeness beautiful.

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As Dr. AG wandered the room she stopped to work with a student who had an
idea but was unsure how to use the program to investigate it. Dr AG walked her through
the technical details but did not confirm or deny her conjecture, leaving her to keep
exploring. By refraining from judging the accuracy or merit of the idea, Dr AG kept
wonder available. The student did not request validation of her idea, only asking for
technical help in exploring it. She needed support with the program and was given it, but
her own sense of wonder and possibility kept her investigating her mathematical idea.
Another student was also working alone at a computer but instead of being excited by the
seemingly erratic behavior of P and P’, she seemed annoyed by it. Her aesthetic was to
know what was happening in a linear and predictable way. She preferred to know why
they were behaving as they were. Her annoyance revealed an aesthetic response, but one
of frustration rather than joy. Some students wanted more order in their aesthetic and
seemed to reject wonder for the sake of wonder. They wanted answers. These students
would possibly have preferred the well-ordered rationality of the Complex Analysis
course over this more open environment.

After about ten minutes students shared these observations at the request of Dr
AG:

1) A, P and P’ are on the same line.

2) P and P’ are “sort of” a reflection through the circle but distance
is not maintained.

3) As P gets really far away P’ approaches A but never gets there.
4) When you bring P inside the circle, P’ goes outside of it.

5) If you draw a straight line with P, P’ will trace a circle.

As the students offer these ideas Dr AG represents them on the screen by tracing out what

the students say. As she performs the 5th idea, a student exclaims, “That’s Cool!” and Dr

AG asks her to say that louder so that the rest of the class can hear. She wants them to

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know that mathematics is a place for coolness, here an expression of an aesthetic
response. Mathematical ideas can tap into feelings. As the class continues Dr AG gives a
mini lecture on the definition of an inversion, finally describing to the relief of the
annoyed student what transformation was at work in this sketch, and what a locus is and
how to use Sketchpad to create one.

Afier the lecture they were instructed to create loci for different positions of a line
under inversion. One student I approached noticed that a line inverts to another line when
the original line is on the circle. She seemed both fascinated and confused by her result
considering that most of the positions of the line results in an inversion that created a
circle. She spent a lot of time repeating her work and kept getting the same result. I left
her as she began to try to understand why this happens. She was visibly frustrated but did
not stop because of her emotional response. Instead it seemed to sustain her engagement.
We can wonder about beautiful and amazing things, but we can also wonder about
upsetting and confusing things. Harmony can be motivating but so can discord be a
driving force. Both forms of aesthetic possibility coexisted in this Advanced Geometry
course. Tapping into feelings of pleasure, annoyance, pride, and confusion were
techniques for inviting students to take on the work of authoring conjectures and
explaining or (dis)proving them.

A day in mid-September serves as another example of students using the
vocabulary of aesthetics as personal interest, particularly interest arising from surprise
that the same things kept happening. One student pulled the second instructor over and
shows him that in his sketch, whenever he connects the midpoints of the triangles’ sides

he gets a new triangle that shares the same centroid with the previous and original

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triangle. If he repeats the process of creating new triangles based on the new midpoints,
he continues to find smaller and smaller triangles that continue to share the same
centroid. “This is interesting,” the student declares. The student was articulating that
unexpected patterns were a type of aesthetic that he found important and that was moving
him towards thinking of himself (his self-concept) as a person who could describe
patterns and investigate them.

The class meetings devoted to hyperbolic geometry provided many opportunities
to exploit the unexpected as an invitation to do mathematics. In late October Dr. AG and
Mr. SP were leading a discussion of triangles in hyperbolic geometry. Dr. AG asked if it
was possible to have two non-congruent similar triangles in hyperbolic geometry. A
student cautiously replied, “Maybe,” bellying her uncertainty in this geometry. She went
on to explain that she thought you would need to alter your understanding of similarity as
understood in Euclidean space and that the trouble she was having was in thinking about
the nature of comparing angles between triangles in hyperbolic space. Other students
seemed to share her uncertainty. No one was rushing to share ideas. Dr. AG posed a new
question. “Will the triangle congruence theorems work here?” Again, the students seem
both intrigued by the question but hesitant to respond. One finally suggested that they
would need to determine if the congruence theorems relied on the parallel postulate. If so,
then, no, those theorems would not hold in hyperbolic geometry. With these introductory
questions, Dr. AG and Mr. SP set up the students to think about the differences between
Euclidean space and hyperbolic and the underlying assumptions of related theorems in

the two geometries. Many of the students seemed to be rather perplexed but intensely

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interested. Furrowed brows and students leaning far out of their seats towards the
projection screen filled the room.

On that same day Mr. SP instructed the students to take five minutes and
construct a square on the Poincare disc. When called together again a student provides
this set of steps: Construct a segment; construct the circles with the respective segment
endpoints as center and point; construct the line between the intersections of the two
circles; construct the segments between the four points. Below in Figure 3.2 is an

illustration of the final square when using this construction technique.

 

Figure 3.2: Student’s hyperbolic square

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The class embarked on a discussion of how this strange figure was a square
because each side was the same length, but that it did not look like a square because the
angles are not 90 degrees each. Mr. SP introduced the distinction between a regular 4-gon
and a square. He commented that in Euclidean they are the same idea.

Before the students could stew on that idea for very long a second student offered
another construction to consider. In this construction the square did have 90-degree
angles, but the sides were not equilateral. Mr. SP posed the dilemma: Do the students
prefer their squares to have 90—degree angles or four equal sides? A student asks if
parallelism of a square is part of the definition or a consequence of the definition. The
class discusses this point about parallelism and the consequences it has on constructing
squares in hyperbolic geometry. Sensing the growing discomfort about the possible non-
existence of squares in hyperbolic geometry, Mr. SP continues, “Part of the disquiet may
be that in hyperbolic we still use the term equilateral triangle although no equilateral
triangle in hyperbolic has three 60-degree angles. But in the construction, we do not
exploit the 60-degree idea. In squares, we do exploit it.”

Disquiet, as Mr. SP called it provided an opportunity for students to consider
ideas often taken for granted, such as the role of parallels in the construction of a square
in Euclidean geometry and the problems with using the same words such as square and
equilateral triangle across two different geometries. Discomfort and even horror are also
aesthetic elements and were used in this Advanced Geometry course to provide
opportunities for students to take on the challenge of constructing new self-concepts—

from a passive, note-taking student to an active participant in creating mathematics.

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Complex Analysis, included elements of aesthetics but horror, disquiet, and discomfort
were not seen as positive. They were to be avoided.

Students were curious is this class and they also found one another’s ideas “cool,”
but they found their own ideas both interesting and possibly troubling. Some students
seemed very annoyed by the whole process at times. These students might have been
more comfortable with the use of aesthetics found in the Complex Analysis course where
order, structure, and rationality were constructed as beautiful and inspiring. Mathematics
in this class is seen as having the ability to create emotional responses. Very often these
responses (coolness, interest, confusion) motivate further work. Not every student,
though, found these more negative aesthetics pleasing. Both excitement and discomfort
can lead to further interest and mathematics. By feeling excited or troubled they can
channel emotions towards curiosity in understanding why. The course worked to get
students to move away from traditional student roles and towards novice researcher roles.
Perturbation and excitement were some of the aesthetic responses used as invitations to
take on the new self-concept.

The Advanced Geometry course also drew on social aspects of doing mathematics
as did the Discrete Mathematics and Real Analysis courses. In both Discrete Mathematics
and Real Analysis the students were invited to belong to a community. Part of that
invitation was to highlight explicitly the values of the larger mathematical community.
Advanced Geometry did this at times, but not as frequently. Group membership was not a
mode in this course, but there was a strong sense of communal responsibility for the

doing of mathematics. All three of these courses constructed a vision of mathematics as

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about people working together. They had strong ties to social constructionism and

humanist philosophies of mathematics.

Regimen: Play

The students in this Advanced Geometry course were asked to play with ideas. As
much in the above sections suggest, students did a lot in this course. They talked, asked
questions, conjectured, proved, wrote and thought. In doing much of this they were
playing. Students played with their ideas by testing them out, trying this and that,
changing their mind, and revising their conjectures. Sometimes they played alone and at
other times with classmates and the instructors. There were many opportunities to be
wrong and to try ideas out in multiple ways.

The use of the word play might seem out of place for the dual reasons that this is a
university course, not a kindergarten, and that it is mathematics, not recess. Yet play, as a
description of the activity of the course, is not far-fetched. In a section of the textbook
titled, “To the Student,” we see this description of the course:

Playing can be a gateway to new ideas. In this course, we ask you to play

with geometric figures, to explore their properties, and to observe

relationships and interactions among those figures. As you play you will

be asked to make conjectures about what you see happening. (Reynolds &

F enton, 2006a, p. xix)

Furthermore, the passage says, “Playing can be a gateway to new ideas.” The focus is on
the play—the explorations, visualizing, conjecturing, curiosity. By allowing play to be a
theme of the activities students had space to come to new ideas. Recall that most class
days included the instructor posing a set of geometric figures and asking students to

create conjectures based on the objects in the scenario. Take this episode as an example

that came at the end of a class period. Dr. AG had triangle ABC projecting from her

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laptop onto the large screen in the front of the computer lab. Inside of the triangle were
four points labeled M0, B0, Jo, and Lo. She asked the class, “How could you figure out
by dragging, what L0, B0, Mo, and Jo, are? What triangle centers are they?” A long
conversation ensued. Students instructed Dr. AG to drag point B and watched the points
move around the screen, then they asked her to drag A, and C likewise. Several rounds of
such dragging continued, but by the end of class the students could only discover that Jo

was the circumcenter. The next day they began playing again on this task.

 

 

Figure 3.3: Dr. AG’s triangle

Dr. AG asked, “Any ideas on how we can find these points?” but no students
answered. She probed again, “Can we decide what Lo can’t be?” A student suggested,
“Centroid” and Dr. AG asked why, but again no one responded so again she probed.

“What is the centroid?” Finally, someone spoke, but rather than answer her questions the

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student told Dr. AG to “Move B closer to C. I want to see what happens.” She did and

the student followed up by describing what was happening as Dr. AG moved B closer to
C. “As this happens Mo seems to be the center of mass.” Dr. AG replied, “What else can
we learn?” A second student replied, “Bo may be the incenter because the incenter is the

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center of the incircle so Bo has to stay inside. “Great argument,” Dr. AG praised. Soon a
rush of ideas were shared:

“Well, when Lois inside all the rest are as well.”

“It [L0] is inside when all the angles are acute.”

“The altitude one is always inside when the angles are acute.”

“It would be nice if all the centers met at a point for an equilateral

triangle.”

“What would happen if it were isosceles?”

“Three of them are always on a line.”

In this episode the students were playing with the four given ‘mystery’ points.
Unlike most of these sorts of exploration activities, Dr. AG controlled the triangle and the
students gave her instructions on how to move it. They were not playing with their own
computers but were—through her—playing with the ideas. Their work was very similar
to a young child playing with the world. They watched to see if things changed, what
changed, and when they changed. They watched to see what stayed the same, and when.
They instructed Dr. AG to move things around to test out their conjectures. The task was
a big game of “What if” and “I spy” combined into a math lesson.
What if angle BCA was roughly 90 degrees? I spy Jo coinciding with point C.
What if angle BAC was roughly 90 degrees? I spy Jo coinciding with point A.

What if angle ABC was roughly 90 degrees? I spy Jo coinciding with point B.

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By making the activity an opportunity for play, the students were able to observe
the actions of the points and begin to combine their observations into conjectures about
the properties of the triangle centers. Play was the gateway to new ideas.

In other courses observed, students were usually told what mathematics was
important and needed to be proven. In both Real Analysis (during lecture time) and
Complex Analysis, the instructor would give a definition and then follow up with an
example, and then a theorem related to that definition. Under that model, an instructor
would have likely provided a definition for the orthocenter as the intersection of the
altitudes and then would have posed a theorem such as “For right triangles, the
orthocenter lies on the vertex of the right angle.” The instructor would then have likely
led the students through a proof. Instead, Dr. AG posed the task as a mystery game. By
doing so, the students observed the behaviors of the points for various types of triangles,
right, acute, obtuse, isosceles, equilateral, and scalene and used these observations to
deduce which center was which point. Sketchpad became a toy in the game—a toy that
maintained the geometric relationships inherent in the points’ constructions. By playing
the game students could formulate and formalize their ideas.

As evidenced by many of the above examples from the class meetings and
textbook, students are asked to play both by posing and answering questions related to
“What if” and “I spy”. The aesthetics of curiosity, wonder, and meaning are exploited to
draw students into these tasks.

Below are some questions and ideas projected onto the screen as students enter
class one day late in the semester.

Which triangle centers work in hyperbolic geometry and why?
When will regular polygons tile the plane?

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Which Euclidean theorems hold in hyperbolic geometry?
What does inversion look like in Taxicab Geometry?
How to solve CC.

How to solve CCC6.

What does Napoleon’s Theorem look like in taxicab geometry?
Constructing a hyperbola using power of a point.

How do polygons behave under inversion?

How to translate or reflect on the Poincare’ disk.

Creating a new metric.

Finding the perimeter of an ellipse.

Construct hyperbolic tools from scratch.

The ‘carpet’ problem extended.

What properties of cyclic quadrilaterals hold in hyperbolic geometry?

These questions were selections taken directly from students’ second entries in the
project, The Book, an assignment inspired by Erdos’ famous idea that God, or in his
vernacular The Supreme Fascist, boarded the world’s most elegant proofs in this tome. In
this assignment for the Advanced Geometry course students were asked to pay attention
to ideas of personal interest and develop a conjecture/proof or mini project of the
person’s own design to include in his or her personal Book. There were three rounds of
submission with the final and third round to be worth 20% (slightly more than a single
test) and to include revised versions of the earlier rounds along with a new entry. The
syllabus describes it thus:

When the great mathematician Paul Erdos wanted to express particular

appreciation of a mathematical result (a proof, theorem, or idea), he would

exclaim, ‘This is one from the Book!’ Over the course of the semester,

you will submit at least three geometric results. The following kinds of

work would be suitable for inclusion:

- A variation on a theorem or result we cover in class that you discover

while constructing, exploring or playing.

- A discovery based on one of the homework assignments.
0 A new way to visualize or prove a result covered in class or in the

 

This is in the context of Apollonius’ Problem—when given some combination of three objects, point,
line, and circle, find the circle(s) that are tangent to each. CCC refers to three given circles and to solve it
means to create a method for determining the circle(s) tangent to the three arbitrary given circles.

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homework

- An illuminating collation or integration of different ideas, proofs or

results from class

Your aim should always be to attempt to explain what you know about a

certain phenomenon and why or where it occurs. Proof is one language for

making such statements, but you may find others. You are encouraged to

work with others in the class during your investigations but you should

each submit your own presentation of the Book entry. The following

criteria will be used to assess this work: (a) originality and interest of

findings (b) presentation of findings, including clarity and articulacy. Feel

free to ask me if you’re not sure whether something is suitable for

inclusion.
The book entries quoted above that were projected onto the screen suggest that to be a
student in the course, specifically a graded student, meant that you must attempt
personally interesting and meaningful mathematics. To begin to have a sense of personal
interest in mathematics reflects the themes of changing self-concept to an identity of a
research mathematician with aesthetic sensibilities. The entries were self selected and so
could be about nearly any topic related to geometry although they tended to be extensions
of course topics. They were also teasingly similar to math papers, meaning that if these
things had never been published before, it would not be hard to imagine a paper titled,
“On Creating a New Metric.” To be a student was to create new (at least for the student)
mathematics and to prove or otherwise justify it. This major assignment also reflected the
theme of play. The Book was a place for students to play with ideas related to class ideas
and topics but out side of or extensions of class topics.

Very often in class students came up with ideas or shared observations not quite
aligned with the investigations’ focus that day. When these were shared Dr. AG would

often direct the student to consider playing with these ideas further in the Book rather

than in class. One day Mr. SP decided to stop class to make an announcement to the

1. 06

everyone that reminded them that although lots of them are coming up with interesting
observations worthy of further study, these were departing from a focus on the 1:2
distance ratio involved in the task. He suggests that they put those off-topic observations
aside for now and consider them instead for the Book.

When students play with mathematics, even when asked to focus in a certain
direction, their ideas could jump out of the space of the current investigation. The Book
servds as a place to hold these explorations. While they might not be related right now,
there is a set place, an assignment, where these extra ideas and explorations found a home
that remained within the space of the course. Students did a lot of conjecturing. Some of
these fit the goals of the class meeting. Others did not, but instructors and students also
did not discard them. Students could choose to return to these ideas as part of their Book
assignment. Therefore play was both a part of the day to day activities but it was also a
major assignment—one that served the purpose of continuing student play outside of
class and that can also respect play that is unrelated to the core tasks of the given class
period.

In addition to using play to further conjecturing and proving, students were also
asked to explain their ideas both in small group and pair situations and in whole class
arrangements. In other courses observed the students were also asked to share ideas
during class time, but in Advanced Geometry, there seemed to be more participation by
more students—even most. During the second week of class, pretty early in the semester,
the class observation revealed a high level of student-to-student and student-to-instructor

interaction. What follows are some abbreviated notes on the interactions that day.

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Dr. AG called on two students to walk the class through an explanation of the
previous class meeting’s discussion. They sparred as a verbal tag team, alternating back
and forth. A third student then commented on their idea. Then a fourth. Dr. AG next
called on three more students, who each responded with an observation. Dr. AG moved
away from collecting observations and asked the class if they could justify any or all of
these observations. Two more students did so. Ten minutes into class and nine different
students had spoken, either to offer an observation, a verification, or a comment. Next the
class turned to small group work and everyone was talking to at least someone else in the
class. Students asked one another questions such as, “What do we know about. . . .” And
“How can we use that?” Some students got up and walked over to other groups to watch
them work or to ask another question. Dr AG and I were also moving around the room
interacting with students when asked to do so. Students were pointing to their screens,
gesturing with their hands, and taking notes both on their computers and in notebooks. At
one point the laptop that was projecting the task onto the large screen went blank—it had
timed out. A student looked up, seemed to notice this and that Dr. AG was busy with a
group. He walked up to the laptop to wiggle his finger on the touchpad, reactivated the
screen and returned back to his group to continue playing. When the class returned to a
whole class discussion Dr. AG asked a student who had not yet contributed to the public
conversation to walk them through her proof. This student had used different notation
than was currently projecting on the large screen so Dr. AG adjusted her sketch to match
the students’. This student continued her explanation, but made a mistake. The class
waited while she collected her thoughts. When ready she began again. By the end of the

class thirteen students had spoken in the whole class format—some short answers or

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comments and some long descriptions or proofs—and everyone had contributed
something at some point in class. Play took many forms and everyone played in some
way—alone, in groups, and in whole-class discussion.

Some aspects of the course were impossible to observe in class such as studying
for exams, doing homework and working on their Books. Some examples I’ve already
discussed point to the role of the Book assignment as an important part of the course in
which students participate. The Book was worth 20% of their grade and was worth more
than any other single assignment. There were three in-class exams each worth 15% and
homework as an assignment group was worth 35%. This information comes from the
syllabus and hardly any class talk revolved around grades although non-Book
assignments were discussed every few observations (the Book was mentioned much more
frequently as a place to house ideas not suited to the class’ focus on a given day).
Although the assignments were not discussed often in class, they did reflect a solitary
aspect to serve as a small counterpoint to all of the collective play. In the syllabus,
students were “encouraged’ to work together on the Book. But whereas in class there was
a strong bias towards group work (students did work as individuals from time to time),
the homework assignments and Book were places where students might have continued
to play together or they could have chosen to work as individuals. Exams, though,
provided an exclusively solitary aspect to the course.

Class meetings just before an exam were used as time to discuss ideas relevant to

preparation for the exam, but also as time to keep doing math. On the day just before the

d . . . . .
2n exam and after a quick lecture on creating tessellations 1n hyperbolic geometry by

Mr. SP, Dr. AG signals a switch in class activities to exam preparation. The first question

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a student asks is about tessellations in Euclidean—whether they are possible or not. Mr.
SP quickly reminds the student of previous lessons on them and provides an example of
the classroom cinder block walls. The next student question, “Can we go over the
construction of a square in Taxicab?” opens up a long discussion of math that is related to
previous class discussions and activities but that is also new. Dr. AG verifies that the
class has not yet done squares in Taxicab, “so we can definitely do that now.” These two
questions and the instructors’ replies suggest that student questions about previously
covered material is not suited to exam review time, but that questions that are slightly
new are worthy of discussion. The former would have been review. The latter is extended
play. That is, exam preparation is not so much a review of old material but a chance to
play with past ideas modified. The class pursues the construction of a square for most of
the rest of the period and many different students get involved offering suggestions and
disagreeing on whether squares in Euclidean and squares in Taxicab are “essentially the
same.”

After a few quick questions of the type, “Will X be on the exam?” Dr. AG
wrapped up the hour by telling them a few details about the availability of Sketchpad for
the exam (yes, they can use it; yes, the Poincare disk will be available; no, they’ll need to
make their own tools rather than use saved ones; and yes, she’ll send them the sketches
they need by this afternoon) and instructed them to skip around as they take the exam—
do not necessarily work in the written order of questions. “Hardness is subjective,” she
says and she does not arrange the questions assuming a progression of difficulty.
Furthermore, she suggests, they should try to respond to each question if only in part

because “everyone likely has something to say about each question.” In other words,

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students should not work through the exam in a linear fashion but should play around.
They should skip here and jump there, but try to play at least a bit on every question.
Students did many things in class and outside of class. They did things with
fellow students, alone, and with the instructors. Play tied these varied activities together.
Play was a way to have students doing mathematics with a focus on curiosity and
wonder. The aesthetics of the environment sustained student engagement in this play.
Through all of these experiences students were provided opportunities to reconsider their

selves as research mathematicians.

Telos: Researchers

To describe the model student that would emerge from this course is not to
describe a flesh and blood real student from the course who was considered a good
student. It is to describe the imaginary. The constructed student is not a real person. He
and she are figments, fictions. The person the course may dream about producing. What
does this imagined person look like in this course? If the course were to have its way,
who would emerge from this experience?

The constructed student emerging from this Advanced Geometry course would
always be curious and endlessly fascinated with geometry. Life after this class would be a
constant hunt for interesting ideas and coordinating explanations. She would make every
observation into an opportunity to think about geometry. He would consider the
assumptions undergirding each conjecture. This imagined student would ask what, when,
and why. The constructed student was an apprentice research mathematician. This course

works hard to train students in the pre-proving aspects of mathematics—the preliminary

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necessities of a researcher’s work—the conjecturing, selecting, and eliminating of
mathematical ideas.

An apprentice research mathematician would not have the worries of tenure and
promotion. She would not need to teach courses or attend department meetings. This
researcher would only ever need to think about mathematics, but mostly geometry. He
would have long conversations with colleagues and mentors. In these chats they would
debate ideas and offer counterarguments. And when presented a counterargument to her
own idea, she would gracefirlly and gratefully accept it. This Advanced Geometry course
would have prepared him well for these episodes of social mathematics. The idealized
student would play well with others. Her experiences with her classmates and instructors
would have offered her experience in articulating her own ideas and listening acutely to
others’. When he felt the sting of a rejected idea he would remember that mathematics
was not a competition, but would nevertheless swell with pride when his ideas were
appreciated by others.

This future researcher, though, would continue to appreciate guidance from peers
and mentors. As much as the social gatherings were fun, they were also important for
structuring her thinking. Rather than create ideas from scratch the constructed student of
Advanced Geometry depends on the social interaction with others to spur inquiry. He
prefers it when someone offers a starting place. Like in her preparation in Advanced
Geometry where the instructors set the task into motion, the model student likewise needs
a good scenario. Once the student has this beginning point, she is capable of exploring it

and even running off in new directions.

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The model student would work seamlessly between her computer and paper,
valuing the benefits of both mediums for recording and exploring mathematics. This
apprentice research mathematician would spend his evenings tweaking the proofs
bandied about during his usual mathematics social hour. One of her favorite hobbies
would be to test out new ideas in different geometries. To do mathematics would be to
pose questions about mathematical objects and how these questions might be answered in
different systems and under differing constraints and assumptions.

A favorite mode of inquiry for this constructed student would be to draw and
sketch. He would spend quite a bit of time, in fact, on creating orderly sketches and
would make liberal use of color and line thickness to highlight aspects of the rendering.
The drawing would sometimes become very cluttered and would require purging and a
fresh start. Nevertheless, the sketching itself was an act of visible and physical thinking.

All of these sketches would be important for weeding out the obvious
counterexamples quickly and locating the extreme cases to be further investigated and
sorted. All of this would not be dreary work though, no matter how consuming. It would
instead be playful. She would delight in both the mystery and in the solution; in the
camaraderie and in the solitude of her ideas.

The constructed student would have been prepared well for this future by this
Advanced Geometry course. The daily class meetings, tasks within class, and
assignments constructed a model student who took up ideas for the pleasure of doing so;
who enjoyed social mathematics more than solitary mathematics; who strove for new
ideas and their explanations; and who emerged fi'om this course no longer thinking of

herself as a student of mathematics but as a person ready to begin creating her own.

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The course worked to move students away from seeing themselves as students
who watched mathematics into a self-concept of a capable, though far from perfect,
mathematician. They were asked to create their own ideas and conjecture to explore,
albeit within a narrow window defined by the instructor’s and the department’s selection
and arrangement of content. To take up this new self-concept the course encouraged
students to embrace a personal aesthetic of mathematics. They were asked to find
interesting relationships, with ‘interesting’ being highly personal. Unexpected and
confusing elements were not ignored but were likewise embraced as spaces for
fascination to grow. They were given opportunities to think of mathematics as a place for
the oxymoronic serious-play. It was serious in that the mathematics studied aligned with
accepted modes of proof. The class did not make up rules for proving and reasoning. The
mathematics explored was not frivolous but consisted of ideas common to other courses
of similar title. Yet it was also playful. Rather than take notes and listen, the students
drew pictures and shared ideas. They conferred with classmates and their instructors.
They messed around making errors and they took wrong or far too circuitous paths. This
messy mathematics, though, was important to the course. To construct a model student
who as an apprentice researcher the course needed to allow space for students to create

their own ideas and face their own, mistakes.

Concluding remarks

Again, I want to conclude by describing some real students in the course and what
sorts of things I heard them say about the class. When I cannot offer quotes from students
or examples of observations 1 will speculate from the position of a teacher educator on

the sorts of things students might say about the course if prompted to do so.

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As I circulated around the room during the first week one student pulled me over
to offer some insight. He tells me that the students’ and instructors’ conjectures and
language are not very rigorous and that Sketchpad is a tool to “observe a collection of
instances and is not 100% general.” By his tone I interpret that this is not a satisfactory
way of doing mathematics in his opinion. He even seems to be warning me. The
atmosphere of the class encourages students to form their own opinions and ideas and this
student wants me to be aware of his. This student saw himself as knowledgeable about
mathematics to a degree that he could articulate what aspects of the subject were present
or absent in the discourse. To this student, the students’ and instructors’ vocabularies
were not formal enough. That a major tool of the program was not a tool of generality but
of empiricism seemed to disturb him. This course was not functioning as he might have
preferred. His vision of a mathematics as a process from messy and uncertain to maybe
clean and known (after all many of the students’ conjectures never panned out) did not
seem to match his own. It is likely he would have enjoyed the Complex Analysis or Real
Analysis courses better for their more formal approach in terms of definitions and
generality.

Other students also seemed to not care for the collective nature of the
mathematics. Several students would mostly sit by themselves and maybe scoot over to
confer with a neighbor every now and again. Some students would immediately enter
class and sit in a seat to share a computer station with someone else. Different means of
participating allowed students to find the one that best fit their needs. Although students
bad choice in whether they worked alone all of the time, some of the time, or never they

did not have a choice in whether they were a part of the larger class discussions. Dr. AG

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called on students to offer thoughts when they had not contributed in a while. Some
students spoke out less frequently than others but to some extent they were required to
participate in this activity.

Furthermore, that students had to be a part of the full class conversations also
meant that they were excluded from activities such as lecture. It was mostly absent in the
course whereas other classes used this style of delivery frequently. Dr. AG and Mr. SP
did provide instruction, definitions, summaries and other sorts of lecture-like activities,
but these were rare and usually brief. Students who might have wanted for explicit
instruction more frequently (or exclusively) did not have access to this type of
participation.

Other students seemed to really enjoy the class. Many students came into the
classroom and immediately began to work on their sketches. Most students tended to
share ideas during class discussions without being prompted to do so. They sat forward in
their seats and leaned into the discussions, a sign I took for interest and enthusiasm. No
one personally came up to me to share positive comments about the course but they
would in class say things about finding certain ideas interesting, fun, and cool.

Inferring from most class sessions the students found the course to be an
interesting one, but I do not want to leave the impression that they all did as the examples
previously offered provide cases of students who were explicit in their negative opinions
of the course. Many students did seem to find the spirit of the class inviting, but not all

did. This vision of mathematics was not to everyone’s taste.

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CHAPTER FOUR

COMPLEX ANALYSIS: PLATONIC DISCIPLES

...from the time of Kepler to that of Newton, and from Newton to Hartley, not
only all things in external nature, but the subtlest mysteries of life and
organization, and even of the intellect and moral being, were conjured within
the magic circle of mathematical formulae.

—-Samuel Taylor Coleridge

Introduction

This Complex Analysis course met three days a week for 50- minute sessions. It
was a 400 level course and tended to be populated by seniors although juniors could also
be enrolled. Sophomores would have been very unlikely given that the course had Real
Analysis as a prerequisite. Therefore students would also have had taken the Calculus
sequence and Linear Algebra as well. As other courses in this study, the Complex
Analysis course was one of a set of options students could choose to fulfill a requirement,
an analysis stand in this case. Other options in the set were Analysis 11, Ordinary
Differential Equations II, and Partial Differential Equations. Again, students could take
more from the set to count as electives.

Seven of the 23 enrolled students were also taking the methods course for
secondary teachers through the College of Education. Others could have been planning to
become secondary teachers but not yet taking the methods course although this would
only have been the junior or possibly sophomore students. Of these seven students four
were also simultaneously taking the Discrete Mathematics course.

The class met in one of the larger rooms in the mathematics building. Like other

courses the noise in the hallway could become quite loud, but no one ever chose to close

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one of the two classroom doors unless it had been closed at the start of class. Like the
other rooms, the windows were to students left and the doors to their right. The rows of
desks faced a large chalkboard where Dr. CA stood. The computer station and projectors
were not used.

During class the students were very quiet compared to other courses and only a
few students tended to speak either to ask questions or offer comments. These students
tended to sit near the front of the room and were the ones Dr. CA would call by name.
Students never spoke to one another during class and only seemed to know a few others
in the room. After class the atmosphere lightened and students spoke about their

assignments and approached Dr. CA to ask questions.

Prelude to the analysis

The Complex Analysis course generally presented a view of mathematics that was
unified and linear—a progression of steps from one idea to the next. Students were
positioned by the course as in need of developing a fully rationalized existence (the
substance). The course was designed to help students improve the way that they think and
structure ideas to view mathematics as rational and ordered; and they were expected to
reform the ways that they physically studied and behaved to be more orderly themselves.
To entice students to take up these changes, the course design mobilized an interest in
orderliness (the mode). If students wanted to be orderly with neat and tidy proofs and
theorems they should also rationalize their lives fully. A structured and mostly
predictable life was presented in the course to invite students to learn to value the
rationality of mathematics not just so that they can know mathematics more completely,

but so that their lives will be more rational as well. To help students work towards this

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goal the course design asked students to follow the rules and stay the course. The regimen
was for them to follow the class rules for things from homework labeling to not being
disruptive in class. They also followed the model of Dr. CA who led by example, walking
step by step through well-ordered and tidy proofs. A model student (telos) who was a
perfect model of the perfect realization of this course would become fully rationalized

individual in their academic work and as well as in their physical and behavioral lives.

Substance: Well-ordered rationality

The aspect of the student that this Complex Analysis course worked to change
was their irrational existence. Students were very often positioned as sloppy in both their
thoughts and study habits—neither did they think carefully about ideas and nor did they
apply discipline to their studying or class work. To combat their disorderly tendencies in
both mind and body the course design highlighted connections between mathematical
ideas, content domains and techniques of proving theorems. Once students could begin to
notice these connections presumably they would begin to hunt for them on their own.
They would seek out the unity, order, and rationality of mathematics across domains and
within them. To change students’ haphazard behaviors, the course design targeted their
study habits and stressed the rules that students needed to follow in their behaviors. As
for study habits, the course instituted strict policies and practices to regulate attendance,
behavior during class, and outside of class preparation such as homework and studying
for exams. Mathematics, then, is an instance of mental discipline and mathematical
ability is an indicator of the quality of a student’s mental discipline.

The course website and syllabus outlined a string of topics (mostly types of

functions and their integrals) to be discussed throughout the semester. Course CA’s

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syllabus offered one other piece of information about the course content, which is the
first direct statement to the student: “You’ll need a good working knowledge of Calculus,
especially material on (1) infinite series, and (2) functions of several variables—their
continuity, differentiation, and how to integrate them along curves. If your background in
these areas is weak, you must be prepared to review when the need arises.” Immediately
students are positioned as persons with certain content knowledge and skills while
simultaneously acknowledging that students do not always have these backgrounds at the
skill level the course demands. Students who are in need of remediation in any area are
directed to attend office hours or seek help outside class. Discipline in their study habits
and preparation are expected. Furthermore, that students were directed to draw upon
these backgrounds from other courses suggests that the course will use student
knowledge of calculus regularly. That past material would be an important prerequisite
for the work of the course was made very explicit in the Complex Analysis course
whereas in others it was briefly commented on, or hardly mentioned. This course differed ’
from others with its focus on a linear and ordered progression of mathematics. Calculus
_ was not just a prerequisite it was a part of the linear progression of mathematics that this
course highlighted.

Having a good background in Calculus initially seems like a commonplace
requirement in an upper-level mathematics course given that it is a prerequisite to nearly
all courses at and beyond the 200 level, but for Dr. CA, this is essential in other ways.
Inferring from my class observations, his course would extend Calculus concepts and
techniques to reveal connections within mathematics the students had not appreciated

previously. The course was designed as if students had not learned enough in Calculus

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and Real Analysis. Furthermore, the course design suggested that what they did learn in
previous courses was not well integrated into a larger framework of the field of
mathematics. Students were treated as if they had done just enough to get by in their
other courses and did not have the mental discipline to seek connections across courses or
even within them. The official course documents emphasized math topics, but the course
itself included additional implicit and explicit goals. The students were expected to
strengthen or form mental connections between the new topics of complex analysis and
the previously experienced topics fi'om real analysis and calculus. They should begin to
see the unity and coherence of mathematics. Seeking out connections between
mathematical ideas was practiced and highlighted throughout the course. Students were
positioned as needing help in disciplining their minds towards an orientation that seeks
connections and does not just take topics as discrete ideas. They were asked to work hard
to integrate what they experienced in complex analysis with what they had already
experienced. Other courses in this study also had the prerequisite of Calculus, but in
several of these it was not explicitly called upon. The Advanced Geometry and Discrete
Mathematics courses promoted diversity of experience in mathematics whereas the
Complex Analysis course highlighted a progression of necessary concepts, definitions,
and theorems that built off of one another to form a unified whole.

Dr. CA emphasized connections between prior course content, current course
topics, and general techniques and reasoning in mathematics. For example, in one of the
first classes of the semester he was solving a question a student raised concerning a
homework problem. He stated that lz-al = lz-bl (where z are in the set of complex

numbers) is the set of numbers equidistant from both a and b. To illuminate this, he drew

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a picture of a point a and a point b with a segment connecting them. Next he sketched the
perpendicular bisector (see Figure 4.1) to indicate that the set of numbers, or points along

this bisector, are equidistant from a and b.

Figure 4.1: Dr. CA’s illustration of the points equidistant to a and b

He stated, “We should always be thinking in terms of geometry,” but continued
by explaining that although this geometric representation is “best” it does contain “at
least one significant problem”: it is a picture and not a proof. Because of this problem he
then proceeded to provide an analytical approach to the solution. Intuitive geometric and
formal analytic representations are offered as complementary techniques. In this course
sketching leads to understanding and deducing leads to certainty. Body and mind

contribute to changing students’ connections among mathematical ideas, but despite his

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claim that the geometric one was “best,” he continued to provide a better approach--
analysis. It is the more disciplined, rigorous, and rational approach. The geometric
analysis might be helpful but it is sloppy and needs improvement. While explaining that
points equidistant from a and b lie along the perpendicular bisector, Dr CA was
connecting geometric representations with formal ones. There is a nuanced relationship
here. One form is easily understood and a good model to share with students, but it is not
enough. In this course, if students were to use only the geometric models, they would not
grow to become more mathematically disciplined. They need both to be fully realized.
Calling the geometric model best seemed to mean a best first step in the context of his
overall point. The geometric understandings are ordered to the left of analytic ones on a
line of improving mathematical precision.

In another classroom episode Dr. CA brought in colored chalk to help students,

“get a feeling for what this exponential [e2] does, what it does to points in the plane. Try

. . . 2 .
to understand eZ through pictures by first exammrng e in the real plane.” Through these

episodes several elements of the curriculum and pedagogy of this course can be
interpreted. The pedagogy conveyed concern that students connect geometric and

analytic methods as two viable means to first understand and then. prove a result. By

. . 2
asking them to first examine e , the problem was reduced to the real plane before
returning to the complex. Geometric approaches are complementary to formal ones.
Students were first asked to feel to learn and only afterwards attempt an approach

consistent with mathematical logic. This tension between the two continually resurfaces

in class. Formal approaches are preferred, and this was one of the ways students were

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treated as if they were not quite ready for them. The course design functioned to bridge or
connect what students were assumed to already understand and what they need to
develop mathematically.

Consider the following episode from much later in the semester. The topic was
Green’s Theorem, which he described as, “A topic done in Calculus III and quickly
forgotten.” After the quick aside referencing students’ tendencies to forget what they
have previously been told, he continued by describing how a positively oriented domain
means that any time you “walk” on the boundary you keep your left hand towards the
domain, “typically counter—clockwise”. “This isn’t really a rigorous definition, but
without advanced topology we can’t do much better for now.” As he drew a stick figure
walking the boundary with orange chalk he explained that to create a formal definition in
mathematics is to “manipulate symbols and hope everything is OK.” The pedagogy of
this course went back and forth between acknowledging the current preparedness of the
students and teaching with their needs in mind—that they forget theorems and don’t
know advanced topology—and moving them forward. There is also a strong nod towards
formalism in the comment above. Mathematics is a symbol game without meaning, but as
we will see he also expressed Platonic views.

In course CA students were expected to learn the topics outlined in the syllabus
and in the department’s course website, but it was also important for students to learn that
previous course content (such as Calculus I, II, III, and Real Analysis) was going to be
expanded in the complex plane. One of the learning goals was for students to see the
continuity among all mathematics courses. Moreover, topics learned in Complex

Analysis could be returned to with new and deeper understandings after taking other

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upper-level or graduate courses. Successive courses illuminate new connections.
Connections between topics and techniques (geometric and analytic) combine to provide
understanding and rigor. Geometric and holistic arguments have a place in the course, but
they are seen as stepping stones to a more disciplined and rigorous approach. The course
both acknowledged that students are not ready for a fully rigorous approach but also that
they need to begin working with more rigor. They needed to begin to see mathematics as
linear and that learning mathematics is also progresses linearly and step by step from
informal and geometric arguments to formal analysis and proof.

The textbook acted as another text (here a literal one) for interpretation. The
explanatory sections and examples in the text often utilized geometric representations in
conjunction with algebraic equations. These explanations and examples also pointed out
places where the complex plane could be understood through examining connections and
analogies with the real plane. The section titled, “A Note to the Student,” explains, “The
presentation has been molded by my belief that what you have already studied in calculus
can be successfully applied to learning complex variables, which at its basic level is just
the calculus of complex-valued functions. Where there are strong, and even obvious,
analogies between the new material and calculus, I have pointed them out and arranged
the presentation to emphasize these analogies” (Fisher, 1990, p. xi). Notice that students
even need assistance noticing the obvious analogies. The textbook positions mathematics
as well ordered and linear by highlighting the ways that what students have learned
before will be called upon to learn the new things of this course. One cannot proceed to
the next step without the necessary prior experiences.

The syllabus also pointed out behavioral expectations for students that most other

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courses did not include. For example the syllabus told the students that the professor will
“assign problems each day, and will expect you to work on them so that we can discuss
difficulties that may arise in class the next class.” Although Complex Analysis was
definitely not the only the course to assign problems each class, it was the only one where
an explicit expectation that students would work on them was noted. In the other classes
it was an unstated assumption that students would work on the problems. By being
explicit, Complex Analysis was signaling an assumption about students’ work habits. It
was not taken for granted that students would work on the problems. This directive in the
syllabus is a way of constructing students as lacking mental discipline. If left to their
own choices, they would not do the problems. I am not claiming here that students in the
other classes did their work just as well as they did in Complex Analysis nor am I
claiming that if had Dr. CA not included these expectations in the syllabus that the
students would have worked more or less hard. The major point is that in making it
explicit we can discern a certain set of assumptions about students in this course that are
different from the assumptions about students’ work habits in the other courses. To have
disciplined work habits was to be desired in the course and this was made explicit here.
Students are positioned as needing step- by- step guidance in how they prepare for class.
There are other behaviors that the syllabus is explicit about reforming. “Repeated
disruptive classroom behavior can lead to failure in the course or dismissal from the
class.” Again, Complex Analysis was the only course whose syllabus mentioned
disruptive behavior in any way. To make a point of the consequences of poor behavior is
to make visible an assumption that such behavior is likely enough to need mention in the

syllabus, and more interestingly it is in the sub-heading on grading policies. Mental

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behavior such as performance on homework and exams is an aspect of the grade and so is
physical behavior. In the same section on grading, the syllabus also alerts students to the
impact that “exceptional effort, positive contributions to the classroom experience,
improvement over time. . .[ellipsis in original]” can have on their grades. Students are told
that these positive behaviors might be rewarded in the form of “raising your preliminary
grade. . .”. Disciplined behaviors that are constructed as positive will be rewarded but
“negative factors such as lack of effort, declining performance, or disruptive behavior can
lower it.” Grades do not just depend on growth in tests and homework but also in
behaviors. Students’ minds and bodies must become more ordered and rational.

These behavioral expectations construct mathematics as linear and step by step,
and that the learning of mathematics is likewise linear and step by step. Unlike in Real
Analysis where mathematics is a social enterprise, in Complex Analysis mathematics is
like a mountain to be climbed: you cannot reach the top without proceeding step by step
from the bottom. Complex Analysis is not constructed as a separate or unique sub-field
of mathematics. Rather, it is one step along the way of an orderly and linear progression
of understanding. Along this linear path, every individual must travel alone.

The dual issue of mental and physical aspects of the discipline also appears in
comments from the syllabus on grading policies. “In order to get credit for a problem set,
you must be present—both physically and mentally—for the entire class at which it is
due.” That is, a student must be present on the day that the homework is due and cannot
turn it in in absentia. Students must be present physically, but to be mentally present is
also a requirement and suggests that a person could be physically in class but not

mentally in class. Possibilities that might fall into this category include daydreaming,

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reading the newspaper, or sleeping. Yet these too are mental activities so to not be
present mentally is presumably to not be thinking about the mathematics of the course.
The course wants to limit off-mathematical physical and mental behavior and makes to
do otherwise into gradable offences. Again, no other courses made mention of mental
presence in class. This aspect of discipline was unique to Complex Analysis. By not
taking students’ behaviors (physical and mental) for granted, the course is suggesting that
students tend to behave otherwise and need reforming in the form of management. They
need well-defined steps and definitions. Being present is defined as being mentally and
physically in class. The behavioral guidelines mimic the mathematics of the course. To
improve in either you need to proceed along an ordered and explicitly defined path.

Students are positioned as lacking organized and structured models of
mathematics; they do not see the “obvious” analogies and connections between the
courses they have already had and this one. They are also seen as needing reforming in
their physical behaviors that are also connected to study habits like paying attention in
class and being physically and mentally present. Students, if they want to be
mathematicians, are also seen as needing explicit instruction regarding issues of
disruptive behaviors. To progress through mathematics as a student, he must also
progress through a set of rationally defined behaviors.

The course’s approach to remedy the physical and mental sloppiness is to
highlight connections between content areas and between techniques (geometric and
analytical arguments). There are also explicit guidelines for behaviors in class and
outside of class. Professor CA, in many ways, resembles a reformist. He points out that

students have not learned important information from previous courses and seeks to

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provide ways to (re)connect them to that material as well as introducing them to other
connections between geometric and analytic approaches to mathematics. Students’ past
experiences have been flawed, and the course design seeks to remedy this as best it can.
Overall, a well-ordered rationality is at the heart of what needs to change for these

students in this complex analysis course.

Mode: Pleasure in orderliness

How does Professor CA invite students to take up disciplined work habits, both
mental and behavioral? Through this course, students are constructed as in need of more
structured habits in how they attempt (or did not attempt) to make connections between
ideas and how they behave in class and outside of class. By highlighting connections
when they appear, Professor CA’s pedagogy was directed toward a desire for orderliness
as a reason for making changes in student orientations towards mathematics. Having
experienced the pleasure of an ordered environment, students would hopefully then seek
out further connections within content and therefore more orderliness, a self-sustaining
cycle. In addition to the pleasure of mathematical ideas students construct a different way
to think about pleasure—the enjoyment of getting good grades on tests and homework
sets brought about by taking on an ordered and rational approach to mathematics and
their habits. The unity of mathematics was constructed as highly desirable and the
recognition of the orderliness of that unity was the highest goal of mathematical learning.

After proving a tidy result related to a particularly complicated integral Dr. CA
narrated, “You expect some spirit to be overlooking this whole process and that allows us
to do this. To really understand the miracle you need to take further courses.”

Dr. CA often explained formal and rigorous definitions and theorems with their

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accompanying proofs in terms of a mystery or as magical, but instead of finding these
qualities distasteful, it is while he talks about these issues that he is most animated. His
eyes sparkle and his voice and pace quicken. He turns to the class, abandoning the board
for a moment and lets the chalk fall to his hip. In these moments when he is describing
mysteries and miracles in mathematics and how fun it is to revel in them and to explain
them, he looks students in the eye. The pleasure of mathematics is to be in awe of
unexpected connections, not in divergent unrelated phenomena. These unexpected
connections are byproducts of the unity of mathematics. Recall that earlier I described an
episode where Dr. CA seemed to express forrnalist philosophy of mathematics. In this
episode a Platonic philosophy is described. Math is sublime and out there somewhere. It
is exciting to stumble upon a problem where all seemed chaotic (like the very messy
integral) only to find it was already structured and fit nicely into the existing
mathematical framework. Seeking out the elusive connections between mathematical
topics, techniques, and ideas is presented as exciting and as a reason to do the work of the
course. When something seems unorganized and irrational it is just that the connections
have not yet been revealed. Finding these is desirable as positioned by the course design.

On many occasions Dr. CA would finish some proof at the board and comment on
a pleasurable reaction to that result. On one occasion he even referred to a proof as

“really quite cute” with a big grin. Sometimes the pleasure was to be in the simplicity of

. . . . . in
the result despite the intricacres 1n the proof. 6 — —l was a notable example.
Sometimes the pleasure was that a complicated earlier proof and result could be re-used
reinforcing the linear progression of mathematics. For example, after Dr. CA worked for

quite some time to establish Green’s Theorem for a triangle, he pointed out the joy to be

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had in noting that once the case of the triangle was established, many other regions could
also fall under the theorem by dividing up the region into a collection of triangles. That
is, any region that could be shown to be a region composed of multiple triangles, could
also be shown to fall under Green’s Theorem. An example of pleasure to be had in direct
connections between functions of real variables and those of complex variables was that

several rules the students might take for granted for real-valued functions are still

applicable for complex-valued firnctions. eZ+w = ezeW still held, for example despite the

introduction of the imaginary parts of the functions 2 and w. At other times the pleasure
was to be found in the implications of a theorem. The Cauchy-Riemann Equations were
one example of this type. Dr. CA pointed out with several examples how these equations
could be helpful in finding u when v is known or finding v when u was known. Finding
unexpected simplicity, transferable ideas (from the real plane to the complex), and utility
were pleasurable.

There was a duality to Dr. CA’s talk about the mysterious and sublime aspects of
mathematics. On the one hand it could sustain his interest and fascination with his work,
but on the other hand he diminished the mystery of mathematics by explaining that
further course taking would explain away any sense of wonder. What is wondrous now
might not be so wondrous after more hard work. It is and always has been coherent, but
we as humans might not always be in a position yet to see the coherence. Recall that he
told students that without advanced study in topology some current ideas taken for
granted could become illuminated. The mysteries can be understood with more hard work
and study. Topics learned at higher levels explain assumptions taken for granted

previously. But still, there remains a sense of wonder. Calling on a spirit to explain the

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mathematics can also be read to imply that he considers mathematics to be connected, not
because mathematicians designed it to be, but because it will and always has been. The
sense of inevitability pervades much of this discourse. Students grow by seeking out
explanations to a priori knowledge and doing this is pleasurable. Both the hunt for
connections and capture of them are thrilling.

The textbook is often different from the professor’s lessons in respect to pleasure
and mystery. Whereas Dr. CA made mention of the “amazing” or “exciting” results often
in his classroom talk, only the slightest hints of this theme are found in the textbook,
which is mostly restricted to providing definitions, theorems, proofs, examples, and
exercises and has very little commentary. One comment on the aesthetics of mathematics
comes from the section, “Preface to the First Edition ”: “Complex variables. . .is a
mathematical structure of enormous beauty and elegance” (Fisher, 1990, p. x) Notice that
by invoking the word structure a particular orientation towards mathematics is noted. It is
organized and structured, not diverse and haphazard. This short line in the textbook also
makes a claim that to study complex analysis is to participate in the beauty of the

structure. Much later in the textbook, in an appendix “Locating the Zeros of a
. . . . th
Polynomral” there rs a commentary on finding the zeros of polynomrals of 5 degree or

higher.

The reader may well draw the conclusion that the zeros of any polynomial
may eventually be found by a succession of steps that work for the
quadratic, cubic, and quartic cases—that is, by a sequence of substitutions
and extraction of roots. Quite surprisingly, this is not the case if the degree
is five or more. In fact, one of the more profound mathematical results of
the early nineteenth century was that there is no formula of this type that
will solve every fifth-(or higher) degree polynomial. (Fisher, 1990, p. 386)

This paragraph makes explicit that there is surprise in mathematics. Patterns do

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not always follow through in the ways that might become expected. There is an
implicit expectation that the reader or student is seeking out these patterns and
connections and that surprise is an emotional response to mathematics. In this
case, surprise that a pattern does not hold. Furthermore, the passage points out
that the theorem that says that not all roots of fifth or higher degree polynomials
can be located using a formula is profound. There are aspects of mathematics that
are profound and non-profound, surprising and expected. The invitation of the
course to get students to take on disciplined ways of thinking about mathematics
wants them to not rely only on their expectations and assumed patterns, but to test
those and be emotionally invested when they come up short just as they are
pleased when patterns hold. On the other hand, just because mathematicians
discovered that the pattern did not hold they also sought a proof that it did not
hold, not just that they seemed to not hold. Even when there are apparent
anomalies, there are also ways to codify those and bring them into the rational
structure through proof. All mathematics is organized into the structure even if we
humans at times have to work to find out the limits of the structure.

Both Dr. CA and the textbook position students as if they could be
inspired by the pleasure of finding connections within seemingly unconnected
ideas, patterns in what might have been chaos, and/or surprise when ideas thought
to connect in fact do not, or patterns that should have been were by proof not. The
non-existence of a pattern was still structured. To use surprise and pleasure—both
aesthetic emotional reactions—as an invitation to take on disciplined activity here

looked quite different than the aesthetics deployed to invite students in the

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Advanced Geometry course. In Complex Analysis students were often directed or
explicitly told what was interesting, surprising, elegant, and/or beautiful.

What was striking in the comparison between Advanced Geometry and
Complex Analysis was the rarity of student comments about what is interesting,
elegant, or beautiful in the latter course. In Advanced Geometry students often
expressed an interest in following up certain ideas, commented on the “coolness”
of an observation, and came to surprising conclusions of their own as evidenced
by their frustrations when expected phenomena did not materialize or when an
unexpected pattern did. In Complex Analysis these comments about aesthetics
were pronounced by the professor and the textbook, but not by students. This is
not to suggest that students preferred Advanced Geometry to Complex Analysis.
In fact, the two courses did not have students in common that I could ascertain
and no one ever made a comment to compare this course to another. In fact,
students in Complex Analysis did make comments about enjoying the course as a
whole, but they did not make comments about finding the mathematics
particularly interesting or surprising or some other aesthetic element. This tells us
that in the Complex Analysis course the aesthetic qualities of mathematics were
not for students to determine but to be told. The beauty of mathematics resided in
mathematics itself, and was not determined by humans. To come to a well-
ordered view of mathematics was in part to take on a particular notion of
beauty—that is structure or orderliness.

Student questions or comments during the class time never touched on the

pleasure of finding or describing connections between complex analysis, real analysis

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and/or calculus. Instead, their talk about pleasure involved the happiness of getting good
grades. This was pleasure in following the behavioral steps for studying outlined in the
syllabus. A group of students who sat a few seats behind me often expressed excitement
about problems they had solved correctly. A common interaction might begin by one
telling the others how s/he had figured out the solution to some homework problem with
what appeared to be pride. The others would ask more about this solution or share a
modified version that he had used. The student from this group that sat nearest to me
would frequently share that he enjoyed the course very much and often wanted me to
read his solutions. These examples of excitement were not so much about the theorem or
the connection(s) to other ideas so much as joy of finding the correct answer to a test or
homework question. This differs from the professor’s pleasure in mathematics in that his
joy was often about the simplicity or unexpectedness of a result. Where Dr. CA’s
emphasis on pleasure was related to the doing of mathematics—the fruits of disciplined
inquiry, the students’ take on pleasure was related to the fruits of behavioral discipline
and study habits that resulted in high grades. Pleasure was present as a reason to take on
the course but the emphasis differed between that related to the larger field (Dr. CA’s and
the textbook’s) and that related to individual success (the students’).

Although the students’ primary expressions of pleasure were related to grades, Dr.
CA also commented on grades as important aspects of the course. Throughout the
semester on days when he returns a graded test, Professor CA also drew on the board a
frequency table detailing how many students made which scores, and how many students
were currently making what cumulative grade. For example, one day the table indicated

that nine students were making 40’s, 11 were making 3.0’s, two were making 2.0’s and

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another two were making 1.0’s. The table never indicated a particular student’s grades,
but if a student knew that she had made a 3.0, then she might infer that she was
performing similar to most of her fellow classmates.

Students also spoke about their grades often. The one student who talked to me
before class would also show me his homework and test grades. These were usually quite
high, some even being above 100%. When he did miss a problem he would complain to
his friends (and they to him) and they would pass around their graded papers for each to
critique or admire.

The pleasure of doing well on tests seemed to be a big concern for these students
and others in the course. They would ask the professor for review problems to practice
and for study sheets to examine before upcoming test days. The five tests counted for 500
of the total 600 points considered in the final grade with the final 100 points reserved for
homework scores. On some occasions when Professor CA described a theorem and its
proof he would point out for the students that he likes to use this theorem on tests or the
final and therefore students need to study it. What was done in class related in an ordered
and connected fashion to the tasks on the exams. Also, that the grades were based on a
cumulative points model provides a strong analogy to the construction of mathematics in
the course. As a student passed through homework and exams, he or she accumulated
points. Each new assignment provided another step-by-step way to reach the top of the
mountain where 600 points was seen as perfect. The course policies on homework were
also fully rationalized. Not only the mathematics, but also the learning of mathematics
conformed to a rationally unified coherent system.

In addition to homework sets created from class lecture material and the textbook,

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Professor CA also used some extra-credit problems, which relied on the Maple software
program. In the syllabus he pointed out that these extra credit problems could
significantly alter the students’ homework average and thus final average. One reading of
this practice invited students to use the program by offering a better grade as the reward.

In summary, the course worked to create students who were fully rationalized in
many aspects of their life. Like mathematics, they needed a larger organizing structure
that could coordinate their mental activities and behavioral activities. To invite them to
take on these challenges the pleasure of order-- exemplified by finding connections in
mathematics --was highlighted by Dr. CA and the textbook. Recall that he described
further course-taking as a means to better understanding of ideas encountered in previous
courses. That is, by taking complex analysis, students could appreciate calculus and real
analysis in new ways. These sorts of comments offered reasons to learn new topics,
theorems and techniques. Pleasure in structure was the aesthetic element explicitly called
on in the textbook and in Dr. CA’s commentaries in class, but students also often
expressed their pleasure in getting good grades.

To invite students to make changes in their habits—to become more ordered and
rational in their study habits, orientations towards content areas, and behaviors—the
course highlighted the pleasure of mathematics. To have a fully rationalized mind and
body is constructed as pleasurable and therefore a reason to strive for the change the

course hopes to initiate.

Regimen: Following rules and staying in line

The course set up an expectation that students needed to become more disciplined

in their work in mathematics—seeking out connections in ideas, being attentive and well-

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behaved students, and transitioning from geometric arguments to analytic arguments. To
appeal to students to make these changes the pleasures of hard work are made explicit.
These are pleasures in the grades earned for doing well and the pleasures of connections
both expected and unexpected within complex analysis and across complex analysis real
analysis, and calculus. What then are students asked to do? What did the course construct
as the regimen to achieve these changes?

The major thing students were asked to do for the course was to follow the rules
of the course and follow the modeling of the instructor. Very often comments in class
directed students to alter some behavior or to be on their watch for undesired behaviors.
That the class was rather rule—oriented is not to suggest that Dr. CA came across as mean
or uncaring. Many students expressed a fondness for him and the course. This Complex
Analysis course asked students to become better disciplined by expecting more
disciplined behaviors. They were also expected to follow along with Dr. CA’s lectures.
To become more disciplined themselves they were to follow the example of Dr. CA.

Through the observations of the class, it was apparent that homework discussion
was an important activity. Each day, if not a test, began with time for students to ask
questions about the homework sets. This could take anywhere from 5 minutes to 20
minutes of a 50 minute class period. The previous class meeting Dr. CA would have
assigned a set of questions for students to prepare for the following class and he would
later (every following Monday) collect this work to grade. In early September Dr. CA
described the homework rules for the next Monday’s tum-in date (because of Labor Day
this Monday was not a turn-in date and the schedule was not yet as organized as it would

later become). He explained that he will select two problems from each of the smaller

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sets to grade in detail and he will spot-check the rest. “Make an effort to write up all the
problems and make it clear which are which. I’ll only spend a minute looking for it
before it’s a zero.” Students were instructed directly to not only do the homework but to
also follow the guidelines from the syllabus that outlines: “Problem sets must be written
up neatly and logically, with appropriate explanation provided.” Students were very
directly asked by Dr. CA to follow the rules for turning in homework.

There were other rules about homework related to the class time spent addressing
it. At the start of almost every day of the class, Dr. CA asked for students to tell him
which questions they had found troubling or difficult from the previous selection. At first
students would tell him a problem number rather than a question but after repeated
queries on his part for a question students began to adhere to this unspoken rule. Rather
than tell him the problem number that was difficult, they should ask him an actual
question. When students did ask him questions rather than assume that this time was for
him to work their homework problems, he gave their questions considerable attention.
One day a student shared with him a problem she was having with a homework
assignment. In particular she was having trouble understanding how two problems from
different but consecutive sections were related. She described how they seemed similar to
her but she was confused by what each was asking. Rather than address the two exact
questions that were the target of her confusion, Dr. CA talked about the general case the
incorporated both questions. He spent about ten minutes describing the general case that
would not only answer her questions but also a whole class of related problem types. He
drew pictures and worked back and forth between geometric explanations and more

symbolic ones. When he was done he turned back to the class and asked her if he had

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answered her question. She responded that he had. Although the start of the homework
question involved a little bit of discussion between the student and Dr. CA once he began
on the problem he worked on his own. The discussion turned quickly into a mini-lecture
where students were expected to follow along with his explanation. Most examples from
homework time followed this routine. A student would initiate a question and there
would be a few exchanges between the student and Dr. CA before he began to lecture on
the problem. As he did this, students would be writing in their notebooks presumably
following along with his usually lengthy responses.

When students were not able to formulate their questions well—when they did not
follow the rules for asking about homework, Dr. CA would not provide a lecture on the
subject. In mid-October a student asked about an integral from the homework set. “What
part is the real part?” To which Dr. CA responded, “I don’t know what your question
means.” The student responded, “I guess I don’t either.” After this statement Dr. CA
moved on to another student. To follow the rules of the homework discussions were
important and the consequences were to not be assisted. A student was expected to be
able to produce a coherent expression of his confirsion. Even confusion needed to be
ordered.

Following along as Dr. CA did mathematics was another important aspect of the
class as a whole and not just related to his mini-lectures during homework time. The
majority of non-homework time was new lecture time. During this segment of class Dr.
CA provided definitions, theorems, proofs, examples, and/or applications. The rules that
students were to follow were simple. “Don’t zone out,” as he instructed one day. He often

told them that he wanted them to “really pay attention.” The rule to follow was to follow

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along carefully and stay in line; he made this expectation quite explicit on several
occasions.

Students seemed to comply with both. Aside from one occasion when a student
was reprimanded for tapping his pencil students hardly moved in class. They never left
their chairs for any reason. Even when there were loud noises in the hallway that would
have prompted students in many other classes observed to get up to shut the door, no
student moved to do so in this course. Similarly, in other courses students would throw
away tissues or excuse themselves into the hallway, but not in Complex Analysis. They
did not read their textbooks or other books. They did not nod off or slump in their chairs
as happened in a few of the courses from time to time. Students were performing to the
expectations of the syllabus. They were behaving well by following the rules by staying
in line at least in their physical responses.

During lecture time Dr. CA often seemed rushed to finish an idea before the
period ended. In other classes observed the professor usually announced that the class
would finish whatever idea on the next day. In this Complex Analysis course, though,
students were instructed to follow along but just more quickly. Dr. CA’s words and
writing were both very fast. Even on days when he did not make outward comments
about being rushed, he still seemed hurried.

One day Dr. CA asked the class how much time was left and quickly answered
himself that there were 10 minutes left. “Good because now I can prove [Casorti-
Weierstrass Theorem].” A student yawned and was quickly encouraged to “Hang in
there. This proof is worth it!” Fifteen minutes later and five minutes past class he

announces that he wants to show them another theorem but warns them that if they leave

14]

they won’t have access to this information for their homework. He also warned them that
if anyone missed the next class (the Wednesday before Thanksgiving) that they “owed
him” but he did not explain this further. Following along in this episode includes
following a very fast lecture of a complicated proof that took fifteen minutes even in its
rushed form. Following also includes following the rules for being physically and
mentally present both for an in-session university calendar date and for post-class
lectures. That is they were being expected to be there on a day that school was actually in
session even though many students were positioned as planning to be absent. They were
also asked to stay late as Dr. CA finished a proof and even later as he began a new idea.
Students were expected to stay in line with the rules of the class.

When the course asked students to follow along with lectures they were not
always asked to stay late. On Halloween Dr. CA looked at his watch and turned to the
class. “Quickly, before the hour closes. Let’s give a proof of Cauchy’s Formula,” as he
promptly turned towards the board. Fourteen minutes later Dr. CA again turned back to
face the class, who had been following along as indicated by outward physical signs, and
announced with what seemed to be glee, “Four minutes to spare!” Following along and
doing so while Dr. CA did mathematics quickly was a common experience. There was a
flurry of chalk and notebooks on most days of the course. Students were expected to
follow the path of the mathematics as Dr CA performed it at the chalkboard.

That Dr. CA often seemed rushed to complete his class suggests that he had
ordered the lectures to begin and end with certain ideas. To have not finished a proof
before class would have set the ordered course off track. A slip in time on a given day

would set the class behind for each next new day. Since the topics were well ordered in a

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step-by-step path, skipping one was not an option. Time became very important. Students
were expected to follow along because doing so meant that they could follow the
progression of the mathematics to a final conclusion. If they missed a day they missed a
step. If they missed a step they could not arrive logically at the conclusion. Students
therefore needed to be present and paying attention or following along with what Dr. CA
was doing at the chalkboard.

Possibly because of this rushed state, students were explicitly discouraged from
asking questions during lecture time that commenced once homework time had ended.
New rules applied for this segment of time. Whereas Professor CA provided detailed
explanations for homework questions, often solving the entire problem for the class and
providing a mini-lecture on the general case, when students asked questions about the
lecture he would provide terse responses or state that he did not have time to go over it.
He would instruct them to visit him in office hours or after class, which many students
did as evidenced by the clusters of students who approached him after class.

These clusters of students were fairly common. On most days students
congregated around a large desk at the front of the room where he usually stood
collecting his notes and colored bits of chalk. On at least a few days some of the students
who formed these post-class discussions followed him from the room. The “following”
was not reserved for in—class only. There were several days during the semester where I
wondered if students had followed him into class from somewhere else as they entered
the room quite close on his heels. Just as students tended to gather around him after class,
many also did so before class began. It was entirely likely that some students had indeed

followed him to class given that office hours coincided with the hour immediately before

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class began and given that student comments before and after class suggested that many
of them made use of the opportunity for help from Dr. CA.

That students followed Dr. CA in and out of class is a possible indication of their
following a suggestion from the syllabus that was not quite a rule. The syllabus states that
if students were having problems with their homework sets they were invited to discuss
them during office hours and later in a section on collaboration students were warned
about the penalties of cheating off of other students although collaboration was
technically permitted. “You may work with others on the homework problems, but you
must acknowledge their contributions, and your final write-up must be your own.”
Although there was no rule prohibiting working with other students the choice between a
welcome invitation to work with Dr. CA or a tolerated but possibly penalized
collaboration with a fellow student might have been very easy for some students to make
and might account for the clusters of students seeking Dr. CA’s council before and after
class. That students sought out the professor for help was an indirect way of following
two different expectations. The first was that students should come to a rationalized and
ordered way of thinking about mathematics. The second was that they should not rely too
much on the assistance of other students in the course. To accomplish both students
would often seek out the professor for assistance.

Overall in the lecturers students watched Dr. CA point out connections in
mathematics and how ideas fit together or built off of one another. They did not do the
finding. He did. On the homework assignments students largely used the insights gained
from the connections described by Dr. CA, but again they did not find them. Students

were constructed as not yet ready to be full mathematicians, people who found the

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connections, themselves for they still had much to learn.

To follow the rules of the course was often to follow the modeling of Dr. CA.
These rules were quite explicit in both the syllabus and in class comments made by Dr.
CA. Students’ behaviors also suggested that they understood the expectations and that
they were following them. To become fully rationalized students of mathematics, these
students were expected to perform within a linear step by step course. They should
follow the steps of the mathematics as they watched Dr. CA solve homework problems
and give lectures. They should ask their questions about homework within a well-
organized framework. Everything they did needed to align with a structured approach to

the course.

Telos: Platonic disciples

What model student would emerge from this complex analysis course? Again the model
student is not an actual person and model does not refer to a student who happened to
make good grades or some other evidence of an actual student who might be deemed a
good student. The model student is constructed. He is fabricated and he is a fabrication.
He would be a person who would change from a lazy, ill-mannered, uncouth way
of life into a hard-working, polished, and articulate member of society. The model
student would see the errors of her ways and seek out the council of someone who was a
dynamic and inspiring force. The model student would be inspired both by the pleasure
of leading a disciplined life and would be interested in the worldly rewards of such hard
work. The constructed model student of this complex analysis course would be a Platonic

disciple.

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As the course goal was to create more disciplined students in their ways of
thinking about mathematics — the sublime connections, the mysterious breaks in pattern
and the well-ordered structures between and across mathematical ideas—the model
student would take up pursuit of both the mysteries and the explanations. The model
student would also become reformed with respect to behaviors and codes of living such
as living up to obligations like time commitments, giving attention and respect to others,
behaving respectfully towards others, and generally being what most of our current
American society considers polite. When respectful of others, the model student would
also be respected and given attention and care.

The model student as Platonic disciple would have come forth fiom the
experience of this class appreciating the pleasures and rewards of mathematics. These
pleasures would include the happiness and surprise of both the expected and unexpected
as well as seeking out answers to math’s riddles. The duality of reveling in the sublime
and other worldly, and in the explanation would be a part of the model student’s attitudes
towards mathematics. It is out there to be discovered. In addition to the pleasures of the
mind the model student would also find pleasure in external rewards and others’
approval. Of particular interest to the disciple would be the recognition granted by a
person of high regard and esteem. Just as this Complex Analysis course contained aspects
of non-rewarded personal pleasures and individual rewards in the form of grades so too
would the model student seek both pleasures. Math would be as the textbook described,
“a structure of immense beauty” (Fisher, 1990, p. x) and a field where personal

contributions were also valued in addition to more aesthetic elements.

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The disciple would conduct his life by following the codes of conduct as
described by the mathematical community and persons of high status in that community.
They in turn were following the model of the rationality and structure discovered in
mathematics. This mathematics was of a realist and Platonic philosophy. The model
student would follow the rules as written down and explicitly stated. He would also
follow the model and guidance of those in higher standing in the mathematical
community who also led lives by the rules. These models for life would be dynamic and
encouraging but also strict and demanding. The model student would seek this sort of
leader out. He would attend this person’s public seminars and lectures but would also
attend smaller study sessions and counsel.

The model student is not an unthinking and blind follower, but questions ideas
and speaks his mind. At the same time, the model student does not seek discord by asking
these questions. He wants to better understand and so the questions are seen as
contributing to his overall betterment. The questions the model student asks are also not
about posing new knowledge but are about better understanding the given knowledge of
the community. In time and after much hard work the disciple might become a leader in
the mathematical community but currently is positioned in that community as something
of a novice interested in learning more about the codes of conduct for life and
mathematics.

The model student wants to know more and pays very careful attention when
others who do know more than he are speaking. The model student seeks a more

disciplined life with the metaphysical and physical rewards of it by following the codes

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of conduct and the examples of others. The model student is a Platonic disciple of

mathematics.

Concluding remarks

As with the preceding chapter I will conclude by considering what real rather than
ideal or model students might say about this course and what their opinions of it might
be. In some cases I have examples of real students who made comments relating to their
likes and dislikes. At other times I get to speculate.

Students who found pleasure in order and routine would have enjoyed this
Complex Analysis experience. This type of student would not have minded the rules and
would likely have appreciated the structure and explicit set of expectations. Students who
had had positive experiences in Calculus and Real Analysis would also have found this
course interesting. They would have been able to use their past knowledge well and find a
place for integrating it with the new ideas of complex numbers and functions. Students
who thought of success in terms of getting good grades and who appreciated public
acknowledgment would also find a home here. Some students appreciate fully
rationalized grading policies, and those students would have found satisfaction in this
course. Many students in the course seemed to like the course very much.

As has been mentioned a group of students who sat near me seemed excited to
share their graded assignments and compare scores as well as the solutions they wrote to
get these scores. Another student, one of those also enrolled in the secondary math
methods course, told me that her experience was so positive that she had decided to leave
the teaching program so that she could continue into graduate school in mathematics. She

explained that she was torn a bit. If she continued in the certification program she would

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have needed to spend the next year interning in a school, but she was eager to go to
graduate school instead. In the end she decided that delaying teaching rather than
delaying graduate school would be a better option for her. This student described Dr. CA
as encouraging and inspiring. She said that she’d never really thought about graduate
school before this class, but that she liked what the class did so much that she wanted to
continue to have more of these experiences. She and the other students around me
enjoyed this class and the vision of mathematics it represented.

Many students never talked to me at all or expressed opinions about the course
that I could hear at least in my time in this class. They, too, may have enjoyed it. Maybe
they did not. Because I cannot offer particular examples of these students I will try to
describe the sorts of things students who might not have liked the class may have given
as reasons for thinking this way.

A student who was not interested in connections between mathematical ideas may
not have found these things inspiring at all, as the course nevertheless constructed. If
finding links between courses and ideas does not match a person’s sense of mathematics,
this course would not have seemed very interesting. Or if a student preferred to think of
mathematics as disconnected and isolated units, he might not have valued the emphasis
on the unity of mathematics.

Students who preferred to work in teams with others may have found the
individual nature of the course limiting. Those who enjoyed solving more open problems
without well-defined paths toward solutions would have likely found the explicit
discussion of method and proof restrictive. Similarly students who appreciated real world

concepts and tangible ideas might have missed that in this class. Students who were not

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linear thinkers may have had trouble coordinating their preferences with the step by step
approach offered in this Complex Analysis class. The structured approach to classroom
behaviors might have had given some students trouble staying seated all class; those that
liked to do work while listening to music, or in silence (as the frequently open doors
made for loud disturbances in the hallways often) could have also not done well in this
environment.

That the course constructed a model student is not to suggest that every student in
the course left the semester as that model. Some may have moved towards it like the
student who wanted to continue her progression up the mountain of mathematics by
entering into graduate study in math. Others may have been simply happy that the
experience was behind them. Some may even have rejected the notion of this model
altogether. The Complex Analysis course constructed a particular model and students
who found that model appealing were served well by it, but the diversity of humans
suggests that not everyone could have.

By pointing out that some students might not have liked the experience I am not
attempting to devalue the positive experiences of others. Nor am I trying to argue that the
class should become more inclusive of students with different orientations and
preferences. I do not hold that this model student or this vision of mathematics found in
Complex Analysis is a bad or a good thing. It is one of several orientations to

mathematics and the model student that found expression in my study.

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CHAPTER FIVE

DISCRETE MATHEMATICS: TEACHERS

Everything that is written merely to please the author is worthless.
--Blaise Pascal

Introduction

The Discrete Mathematics class was a 400 level course that met three days a week
for 50-minute periods. It was an option in a set of five selective courses in the algebra
strand. Students could also have chosen Abstract Algebra 11, Linear Algebra 11, Topics in
Number Theory, or Honors Algebra I to fulfill this requirement. Students could also take
this course to fulfill an elective if they had chosen one of the other four to meet the
requirement. Students taking this course tended to be juniors or seniors although the
prerequisites only stipulated that they must have had Linear Algebra first and so would
have had the calculus sequence as well. A sophomore could have been enrolled but this
would have been unlikely considering the program requirements and the schedule of
courses.

At the time I observed this class I had access to the number of students enrolled in
this course and simultaneously in the methods course for preservice secondary teachers.
Eight of the 24 students were jointly enrolled in these courses. Therefore at least eight
were seniors and were intending to be teachers. It is possible that others were in the
certification program but not yet enrolled in the methods course.

Like most courses taught in the mathematics building, this one was arranged with

the desks facing a large chalkboard in the front of the room with the row of windows to

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the students’ left. There were no computer stations or projectors available. This was one
of the larger rooms and had two doors on the right side; one in the back and one in the
front. Students entered through both doors and would occasionally exit through the back
door if they needed to leave before class ended. This room was on the ground floor near
the restrooms and therefore like the other classes described, tended to get lots of noise
from the hallway. Dr. DM usually shut the front door once class began and a student
would do the same for the back door. Once the doors were closed the room was still and
generally quiet.

The general atmosphere was pleasant as the windows were usually open and,
being on the ground floor were larger than in other rooms, let in a nice breeze. Dr. DM
usually addressed the class from a table at the front of the room or at the chalkboard and
did not engage in conversations with the students before class as some of the instructors
did. Groups of students knew one another and would chat before class and would help
one another during class. Dr. DM addressed the students by their first names and they
addressed him by his title and last name. This class tended to be more reserved than the
Advanced Geometry and the Real Analysis courses although they often would offer ideas

and ask questions in class.

Prelude to the analysis

The Discrete Mathematics course constructed the undergraduate mathematics
student as needing a different self-concept (the substance). The course constructed
students as needing a shift from a conception of themselves as calculators to one of
problem solvers able to articulate pedagogically appropriate explanations. Here, the

notion that students do not recall or understand theorems, techniques or other content

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from previous courses is not the main concern, as it was in Complex Analysis. Nor was it
that their belief system about knowledge needed changing like in Real Analysis. Rather,
students have not been exposed to “real” mathematics before this course. Real, here,
looked like problem solving and explaining. They may have been bright students who
have managed to navigate the college experience with success so far—they are in fact in
a 400-level course in what is viewed by most Americans to be a very tough subject, but
their ways of thinking about what it means to do math have been flawed. Their success
has been false and they need to alter their definition of mathematics. Whatever they may
have learned previously, it has not helped them lean to problem solve and explain their
solutions, which they should learn if they want to actually be members of the greater
mathematics community rather than students of formulas. Belonging to the community of
professionals who work in mathematics as well as membership in a classroom
community is set up as an appeal (mode) to get students to themselves and their work
differently. They need to not think of themselves as students concerned with calculating
solutions but instead as students interested in explaining solutions. To effect this change,
the course deemphasizes formal definitions and formulas and focuses instead on problem
solving and proving where proof is a tool of explanation. To accomplish this, the students
are expected to engage in a regimen of explaining themselves—their confusions, their
ideas, and their solutions to homework assignments, quizzes, and tests. Explaining here
means that students are rendering mathematical ideas into pedagogical forms.
Mathematics is explaining mathematics to others. The model student that would emerge

from a perfect realization of this Discrete Mathematics course would be a teacher.

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Substance: Self-concept

As in the Advanced Geometry course, students in this Discrete Mathematics
course were constructed as in need of a new self-concept. But whereas the Advanced
Geometry course worked to change students’ concept from traditional student to novice
researcher, students in this course were treated as if they needed a change from students-
as-memorizers and human calculators to students-as-problem-solvers. They were not
positioned as needing to improve their capacity to memorize definitions, theorems, and
proofs although that was certainly not discouraged. The course did include instances of
definitions, theorems, and lots of proofs, but whereas in some classes it seemed that
students were trying to memorize proofs as given by the professors of the courses, in
Discrete Mathematics the memorization was not emphasized by either the professor or
the students. To “do a proof” was not to memorize a proof or replicate it later. Instead it
was to use proofs as a form of explanation and a problem-solving tool. They also were
not seen as needing to get better at their calculation skills, but again this was also not
discouraged. The primary new self-concept targeted was to mathematize problems into
theorems and then prove them. This work was for the purpose of changing students’ self
concept about what it meant to be a student of mathematics. They should shed their views
of mathematics as memorization and calculations and instead take on a new self-concept
of a student who views mathematics as about solving problems and proving solutions.

Again, a distinction between the Advanced Geometry and this course might be
helpful. In the geometry course the self-concept change was from student to researcher.
In Discrete Mathematics it was from one type of student to another type of student. They
were still positioned primarily as students who did very conventional student-like things

such as watch, listen, take notes, and offer up ideas in class. They did not author their

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own conjectures as students in the geometry course did. That distinction is important to
the way that the self-concept target differed in each class. In geometry students decided,
within a given framework, what issues and mathematical questions they would pursue. In
the discrete course, the professor provided the questions that would be investigated and
solved. Who authored the questions differed between the courses—students or professors.

Although the targeted substance of the Discrete Mathematics course was from
students’ self-concept from a one type of student to another type of student, the
differences between these two types was still important. The course treated the students
entering the course as if they valued calculations and memorization of formulas,
definitions, theorems, and possibly proofs. The self-concept the course was designed to
instill is one where the students see themselves as problem solvers who use techniques of
proving to tackle both real-world and abstract problems. Proofs were presented as a way
to solve a problem and as a pedagogical tool, and not as an extraneous add-on to a
theorem.

The syllabus described the course as one that could conceivably have been

students’ first course in undergraduate mathematics and so there are no prerequisites7 or
expectations for students to utilize some previous course’s material. Whatever they may
have learned prior to this course is not seen as particularly valuable. That is, whatever
sense of mathematics the students come to the course with is not seen as necessary. In
fact, it is viewed as rather limiting and an impediment to getting them to adopt a new
self-concept of what it means to study and be a student of mathematics. Compare this to

the Complex Analysis course where students’ past courses were positioned as very

 

The university has a collection of prerequisites for the course, but the syllabus denies that they are truly
required background.

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important and absolutely necessary. Here mathematics was not viewed as a continually
progressing and building upon itself. Rather, it was discrete. Other than the above
statement, the syllabus does not describe the course material or content focus in much
detail. There is no list of topics to be covered, only a suggestion that certain chapters will
be discussed although many sections within those chapters may be omitted and other
topics added by the professor during lectures. The syllabus does not emphasize content in
the way that others did. By not including a list of topics the syllabus sends a message that
mathematics as topics and well-defined content is not the way of this course. They will
not progress linearly through a pre-set group of definitions, formulas, theorems, and
proofs. The syllabus works against such a notion of mathematics.

The textbook instructed the students that the course would not attempt to cover
topics related to discrete mathematics in depth and that such a goal would be “ill-defined
and impossible” (Lovasz, Pelikan, & Vesztergombi, 2003, p. vi). Instead, the course
would discuss a few topics and their related theorems and proofs. The book stated that
calculus might not be the best introductory course for students of undergraduate
mathematics. The authors suggested that calculus has been an important development in
mathematics, is worthy of study, and is appropriate (eventually) for future
mathematicians, engineers, and computer scientists, but may not be the best way to
introduce students to what mathematics as a field is “all about” (p. v).

Both the syllabus and the textbook agreed that what students encounter in other
courses might not be helpful to establishing what they suggest are appropriate self-
concepts as students of mathematics. The textbook calls calculus “very technical”

(Lovasz et al., 2003, p. v) and went on to explain that to start a program of mathematical

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study with all of the technical ins and outs found in calculus and analysis misses the
“feeling” (p.v) for mathematics and sets up a too narrow view of what it means to be a
student of mathematics. This course was designed to widen that View.

What then is mathematics “all about” in this course? What will this course try to
change about their self-concepts? The textbook’s first chapter begins by positioning
students of mathematics as problem solvers. The characters in the first problem, Alice,
Bob, Carl, Diane, Eve, Frank, and George are considering how many total handshakes
will have occurred after everyone shakes hands at a party. The solution and proof emerge
through conversation amongst one another and take wrong turns along the way, as new
ideas are suggested, rejected, or accepted. For example, in the handshake problem Bob
observes that he shook six hands, inspiring Carl to reason that because they each shook 6
people’s hands and there are seven of them present then the total number is 6 times 7
equals 42. Diane informs Carl that his number seems far too high and has used poor logic
because he shaking her hand and she shaking his is not two shakes, but one. Eve steps in
and declares that they should then divide 42 by 2 to account for the double counting. No
one argues with her and so the matter is settled.

In this example, the introductory problem, there is no definition or theorem given.
There is no summary of the chapter, which is titled Let ’5 Count. The action begins
straightaway with Alice inviting her friends to her party. Furthermore, this introductory
problem takes up 13 lines of text, barely a quarter of a page, eight lines of which are
dialogue. Mathematics then is someone being curious, asking a question, posing a
problem. To answer the question another person starts by making a personal observation.

This, then, is followed by some other person generalizing from that observation. Next,

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this hypothesis is critiqued based on feeling, a sense of what is reasonable, and logical.
Finally, a fifth person resolves the conflict and amends the hypothesis in consideration of
the critique.

This approach is interesting because it seems to disagree with many other
accounts of mathematics. First, there are people and they are in a conversation. People
talk, and this talk generates questions, ideas, critiques, and resolution. Sometimes the
ideas are experiential observations. Other times they are plausible and/or logical
arguments. At other times the talk is based on feelings such as when Diane suggests that
Carl’s number seems too high. She does not know the answer and so cannot compare his
number to the correct one, but it just does not feel right. People and language are
important in mathematics. Thinking about mathematics requires observations, feelings,
and logic. The resolution of the problem is another statement (Eve’s declaration to divide
42 by 2 and thus there were 21 handshakes)—one that is not challenged by contradictory
observations or feelings, and which stands up to every guest’s sense of logic.

Although the textbook begins to wean the reader off Alice and her friends
quickly by inserting a new character, the third-person omniscient narrators (a.k.a.
textbook authors), the course continues to emphasize the contextual nature of
mathematics as people solving problems. That is, that problems are best understood in a
human context, even if at times a bit bizarre. Consider for example, this episode from the
class observations, which involves a wonderfully absurd problem scenario.

The professor was standing in front of the six rows of student desks, not all of
which were filled. Those rows closest to the two doors connecting the room to the hall

were taken up completely as were the front desks. He paced back and forth with chalk in

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his hand with the textbook on the table in the front center of the room. “We will discuss
the Pigeonhole Principle.” The fluorescent lights create bright white glares on the dark
green and dusty chalkboard. He drew two squares on the board followed by a gap before
drawing a third. He told the class to assume that there were N boxes drawn. He then said
that the Pigeonhole Principle states that if we have N+l objects to place in N boxes, then
at least one box will have at least two objects in it. “Let us count the number of hair-
strands for every person in New York city. Claim—At least two persons have the same
number of hair-strands.”

The lesson continued, but before we follow the lecture, instead consider the
characteristics of the question under investigation in an upper-level undergraduate
mathematics course. The textbook began by stating that one of the goals of the course is
to examine the sorts of problems a mathematician might consider. Granted, not every
problem is supposed to be of interest to working mathematicians, but this one would
allow us to infer some decidedly juicy things about their preoccupations. If this is the
view of mathematics and what a student of mathematics should be thinking about, it is
quite different from a memorization and calculation based approach. Here math is
presented as a bit quirky by focusing on the number of hairs on people’s heads.

Like many problems in mathematics courses where contexts are provided, the
context can still be a bit fantastic. That is, yes, New Yorkers are humans, hair is a real
thing and it is countable, but to use this problem to motivate a reason to study or learn a
new technique is interesting. It is grounded in reality in the sense that these objects are
tangible, but just what constitutes a realistic problem is less definable. Mathematics

educators talk about a real problem as one that people care about, find relevant, or

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-.i . 21.... . lilii..-.lll.‘

otherwise intriguing. What may be a problem for Alpha may be no problem for Beta; but
what is compelling about this task is the ease of understanding the claim being made—At
least two persons in New York have the same number of hairs on their respective heads.
Simple to understand. Either every person has a unique number of hairs, or at least two

heads are adorned by the same number of hairs.

The simplicity of understanding the problems8 in discrete mathematics plays a
role in the expectations for students’ self concept shifts. In this course, students need to
prove things, but the professor points out that he thinks proofs should be introduced into
the curriculum not as an “artificial trick” but as a technique, the only technique for
solving a problem. Proof as a way of thinking through a solution. This requires a problem
that can be easily understood and thought through logically. Recall that the textbook
described calculus and analysis as too technical for students to get a “feeling” for what
math is “all about.” Many of the problems investigated in this Discrete Mathematics

course do not require really technical vocabularies or definitions to initially comprehend

and ponder. One should not waste time in understanding the problem9 if that time could
better be used in formulating the solution. When the problem can be easily understood
without the student needing to be trained in a lot of technical matters then different
strategies for thinking through the problem can be the focus. To move away from a
technical view of math as formulas, definitions, procedures and more to memorize, this

course enveloped students in problems that are simple to articulate but often more tricky

 

Not necessarily simple to solve.

Many mathematics education researchers and mathematicians (e.g., Polya, 1957; A. H. Schoenfeld, I992)

agree that understanding the problem at hand is extremely important and sufficient time should be given to
understanding the problem before solving it. In Dr. DM’s class, the problems do tend to be fairly simple to
initially understand.

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to solve.

Students were constructed as needing to view themselves as people who study
problems and how to mathematize real-world scenarios. They need to be able to
mathematize before they can solve or prove (explain) the solutions using mathematics.
Let us return to New Yorkers and their tresses. Where we left off, the professor had just
stated the claim under investigation, that at least two New Yorkers share the same
number of hairs on their heads. He next asserted that this is actually not a math problem
at all because “neither hairs nor New Yorkers are mathematical objects.” It is only an
excuse to talk about some mathematics. The task has to be mathematized, or abstracted

away from a problem about actual living breathing New Yorkers. After the class had

discussed why at least two New Yorkers have the same number of hair strands“),
Professor DM says that this problem about New Yorkers is ridiculous. We do not know
who New Yorkers are exactly or how many there are, no one will actually count their
hairs, and absolutely no self-respecting New Yorker would let us put him or her in a box
to make a point. The set up is crazy and so is the solution method, but if we pull away
from the New Yorkers, hairs, and boxes and think instead about the Pigeonhole Principle,
this is the real point. If you have more objects (mathematical or not) than categories, then
at least one category must have two objects in it. Although Dr. DM made a distinction
between real objects and mathematical objects in this example, students were still
positioned as needing to think about math as including the step of mathematizing. When

they think of math as too abstract right from the start—when they jump to formulas and

 

After a slight aside on the vagueness of the idea of a New Yorker, he states that there are over 10 million
of them, but it is a scientific observation that no person has more than 500,000 strands of hair so there are
500,001 boxes if we include baldness. By the pigeonhole principle if we tried to place the 10 million New
Yorkers in boxes representing the number of hairs they have, then 10 million far exceeds 500,001 and so at
least two must have the same number of hair strands.

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definitions and uncontextualized theorems they are seen as excluding much of the work
of being a student of mathematics. Their self-concepts need this mathematizing aspect
(re)inserted after so much prior coursework that has ignored it to the detriment of
students’ sense of mathematics as a field.

There was evidence from the class that students really did enjoy calculations and
formal definitions and vocabulary. In early September a student is at the board showing
his work on a counting problem—“What is the number of pairings (in all of the senses as
above [referencing an above situation with Alice and her party friends]) in a party of 10?”
(Lovasz, Pelikan, & Vesztergombi, 2003, p. 4)—and got to a point where he had

10*9*8*7*6
25

= 5*9*7*3 and began to calculate this to get a single number solution. Dr.
DM stopped him and told the class, “There is no need to compute numerical answers.”
To continue would be an exercise in multiplication. The problem had already been solved
in Dr. DM’s estimation and did not require further work despite the students’ desire to
keep going to a final conclusion with no operation signs. Such a move by Dr. DM was a
direct intervention against students’ sense that mathematics is the doing of calculations
and that their sense of self as students of mathematics was tied to how well they could
calculate numerical results.

Soon after the above incident a student asked how the problem just worked was
different from the handshake problem. Dr. DM began to describe the differences in the
problems and their solution methods using casual vocabulary and the difference between
a pair and a pairing. She asks further, “Is there a formal definition?” referring to the

definition of pairs. Dr. DM seemed a bit taken aback and stated a bit incredulously, “You

want a formal definition?” But rather than resist this request, he offered one and wrote it

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on the board as he also said it aloud:

Let A be a finite set. A pair is a two element subset of A.
{a,b} a,b E A, a a: b.

A pairing of A (which exists only if |A| is even) is a set of pairs, |Al/2,

such that their union is A.
Ex: A = {a,b,c,d} gives us {{a,b}, {c,d}}

He then turned the conversation quickly from the formal definitions just given and
suggested that they instead think of the difference between pairs and pairings graphically

and drew a picture (see Figure 5.1).

l.

a pair a pairing

 

Figure 5.1: Dr. DM’s pair and pairing illustrations

Although Dr. DM did indeed provide students with the formal definition when
requested (and is able to conjure it very quickly suggesting that he knows this well) he
tried to downplay the formal definition by offering instead that they consider a graphical
representation to help illuminate the distinction between a pair and a pairing. Contrast
this with Real Analysis and Complex Analysis where providing definitions was one of
the first things the instructors did before transitioning into a new topic. Altogether this
work on the pairing problem indicates that Dr. DM did not encourage students to focus

on performing calculations or memorizing definitions. He conveyed that they should be

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able to articulate a solution and posses general understandings about ideas—not

necessarily technical details. To do otherwise would be to lose sight of the forest for the

sake of the mathematical trees. The trees may be wonderful but they are not everything.
In another example of the course’s stance on calculations, Dr. DM had been

answering a student’s question about difficulty she was having with a homework

problem. In. the end Dr. DM ended up with 10010g2 10. He turns back to the class and

asks if someone would get out their “gadget” so that the class can get a sense of the scale
on this number. He goes on to add, “I’d never expect you to calculate that alone. Well,
maybe I would have 400 years ago.” The use of the gadget or calculator was a tool to

help everyone get a feel for the size of the answer, not to determine the answer. That was

already done. It was 10010g2 10. Furthermore, his comment also suggested that anyone

who would busy themselves with tedious calculations, especially without the assistance
of a gadget, would be centuries behind the times.

Throughout the course problems were posed, mathematized, and then solved
through proof. Primarily the course treated students as if they had come into
undergraduate mathematics with prior experiences that emphasized formulas and
calculations over proofs (if proofs at all) enter mathematics majors with no idea what sort
of work, problems, and techniques mathematicians really use. Furthermore, the course
conveyed that by taking a Calculus sequence in their first two years as undergraduates,
these notions are only reinforced. They had been stuck for several semesters in a study of
only a few trees when they could have been admiring the forest. Students should
preferably learn to dismiss their notations of mathematics as formulas and calculations, or

at least expand them to include a much wider range of problem types. Students were

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expected to alter their self-concept to include seeing themselves as students of a much

bigger problem-solving field.

Mode: Belonging

The Discrete Mathematics course shared elements of mode with the Real Analysis
course despite Real Analysis’s relationship to calculus, which the syllabus and textbook
in Discrete Mathematics suggest provided students a view of mathematics considered too
narrow. Both courses, though, invoked a sense of belonging to a community as an
invitation to students to take up the respective changes each course desires for students.
To get students to view themselves as students of a much larger field that is far more than
memorizing and calculating, the Discrete Mathematics course invited students to belong
to the greater mathematical community.

One way that Dr. DM set up the issue of belonging to a community was by setting
boundaries by saying what it did not value. In that way, the values of the community
were communicated indirectly. In problem sessions, such as that of the New Yorkers
discussed previously, Dr. DM pointed out small details of what mathematics is compared
to what it is not. In this case, the New Yorkers were not mathematical, but they were an
excuse to do mathematics. That phrase foreshadowed other distinctions about what math
is and what it is not; and who does it and who does not. Particularly, doing mathematics
and belonging to the math community is set up in opposition to doing science and
belonging to that community. One class in late October, Dr. DM is asked by a student to

share the proof of a problem in the text. He says he wants to do it both graphically and by

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induction1 1. He begins with the graphical version and draws a sequence of numbers
forming a Pascal’s triangle. He shows, by a series of examples, a property of the triangle
before saying, “Now we’ll do it formally by induction. Well, that was by induction as
well, but informally.” This phrase also sets up a distinction to come about the difference
between mathematical (formal) induction and scientific (informal) induction. Once he
completes the mathematical induction proof he turns to the class and describes how the
Pascal’s triangle example was “something different.” It was scientific induction, which
he explains as seeing that a pattern holds for some finite number of cases before making a
conclusion that it must always hold true.

This is the real danger in using scientific induction in mathematics—even
O I C I 10 I
in scrence! [Laughs] An experiment might hold up 110 trmes but does

not hold on the next term. We need a proof.

Through these seeming asides, he described the nature of mathematics as
distinguished by proofs from other fields including the sciences. He invited the students
to study these proof techniques so that they too could see the differences between
mathematics and everything else. Until they could appreciate this distinction, they would
not understand the expanse and boundaries of mathematics, which is what the course set
as a goal, namely that they would identify as a part of who they were.

On the one hand the course wants to widen what is presumed to be a limited view
of the discipline. But to know the fullness of mathematics also requires knowledge of the
boundaries. To conceive of everything as mathematics would be also to conceive of
nothing as mathematics. The word as a label would become superfluous and unnecessary.

Placing boundaries is needed here. To get students to take on this wider view was also to

 

l l
The mathematical proof technique.

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get them to recognize some limits for the community. Science is an example given on the
other side of the line from mathematics. The difference offered is that science is satisfied
with a large body of cases whereas those who belong to the community of mathematics
do not trust even a really big set of cases and desire a proof.

The issue of proof as a defining feature of the community was key in this Discrete
Mathematics course. “It is important to realize that there is no mathematics without
proofs.” (Lovasz et al., 2003, p. vi) The textbook’s comment echoed the sentiment that to
avoid the case where mathematics is so broad that it does not exist, we need some feature
by which to know it when we see it. It is proofs. That the community of mathematicians
care about proof is important. To work on students’ sense of self as problem solvers
where proof is a technique of explanation, the course mobilizes the values of the larger
community to entice students to take on new sets of personal orientations towards math.
If the larger community values proofs over calculation and memorization work, then
maybe these students should as well.

Proofs played a big role in class time. Usually each day began with students
asking questions related to homework questions that were usually of the form “Prove
that...” During this time Dr. DM would answer questions usually by doing a proof on the
board. On a few occasions he would ask a student to come up to the board and show what
they were attempting or what they were finding a struggle. That students were asked to
come forward publicly with their proofs was a nod towards belonging to a community on
a smaller scale. Dr. DM was not solely responsible for public talk and ideas although he
did seem to bear the brunt of the load. When Dr. DM or some student was not proving

some theorem or case, Dr DM was usually setting up a problem scenario (such as the

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New Yorkers, or Alice and her friends, or rabbits, or castles) and then posing a question
related to the scenario. Next he would begin to work on a proof that would resolve the
issue. On some occasions students helped along as he proceeded through the proof. They
might call out next steps or stop him to ask questions or clarification. They participated in
this community by coming to the board, asking questions, advancing the proofs, and by
helping one another. Many times throughout the semester students would lean over and
work with other students even while Dr. DM was working through something at the
board. They would lean over to ask one another questions, point out things Dr. DM was
writing, and stay after class to continue their work. Unlike in the Complex Analysis
course where students would stay late to talk to Dr. CA, in this course they most likely
stayed late to talk with one another. As one student told me one day before class started,
“There just aren’t a lot of math classes at [this university] that let you work with your
friends,” and went on to add that he preferred working with his fiiend Amanda. In this
course to belong to a community of mathematics (macro and micro) was to work
together. Doing math was not seen as a solitary path. In this example we see students
taking up this invitation to be a part of the community. The student quoted above was
explicitly commenting on a feature of this class that was appealing and that he did not see
in many other courses; he could work with a friend and this collaboration was welcome.
One day in late October as several students were taking notes Amanda had her
notebook paper out in a tidy stack and at neat right angles to her desk but she wasn’t
writing on it with any one of her three mechanical pencils also lined up ready to work.
Instead she was looking back and forth from her textbook to the board. She glanced up,

looked through her book, and then looked back again at the board. Every now and again

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she flipped pages as she passed from staring at the board to her textbook and back.
Finally, she asked Dr. DM a question and seemed satisfied with the response. But soon
after she asked her fiiend, Derrick, what page Dr. DM was working from and he leaned
over and pointed it out for her. She again went through her book-to-board glance dance.
She then turned to Derrick again and they carried out a whispered conversation. When
that ended she began to take some notes, but stopped soon after and put her hand up as if
to ask a question. Then back down again. Then up again. She seemed conflicted. Dr. DM
asked her if she had a question. They then engaged in a few exchanges, but neither
seemed to be understanding the other. Finally, Derrick told her that they could talk after
class. They did. All of this confusion it seems was the result of Dr. DM and Amanda not
sharing a picture of the problem, but assuming that they did. He had drawn on the board a
bird’s eye view of a staircase whereas she had assumed that the staircase discussed in the
textbook would have a side-on view (it was not drawn in the textbook). Coordinating her
idea of the staircase with his impeded her ability to comprehend the accompanying
notations and theorem and a fellow student took the time to acknowledge her need for
help and to act on it. But with the assistance of a friend in the class who took on the work
of helping her she was able to make sense of her confusion. Again, students were taking
responsibility for one another and were taking up not only the invitation to belong, but
particularly they were helping one another by explaining ideas and not stating answers.
The way that the above student helped Amanda was by explaining the mathematical
features of her model of the staircase and Dr. DM’s. He was solving a problem of

mathematics and of pedagogy.

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That students worked together was not unique to this course, but what seemed
different in Discrete Mathematics was that working together was not seen as a separate
type of class time. For example of contrast, in Advanced Geometry the students worked
together almost all of each class, but there were two basic types of class time—students
working in small groups or students and instructors holding group conversations. As
another contrast, in Real Analysis students were expected to work together during the
portions of class when told to do so either in small grouping-class sessions or in the once
a week study sessions. In Discrete Mathematics, however, the students were much more
flexible between whole-class time, professor lecture time, and student work time. In fact,
those distinctions do not really make sense for that class. Any moment it seemed could
turn into a small pair activity, or into a whole class conversation, or into a professor
lecture. The ways of belonging in this community were their own among the courses
observed. Belonging to the community did not mean that a student always needed to
work with a partner or group as they mostly did on the Advanced Geometry course. Part
of the appeal of this type of belonging was that students could create the ways they
wanted to participate. In Real Analysis Dr. RA largely defined the times when students
would switch from observer to actor. In Discrete Mathematics there were no assumed
best times for particular ways of belonging. That is a student could observe one minute
and then choose to work quietly with a neighbor and possibly switch back again as best
fit their interests.

The ways that students worked together, with the professor, and alone reflected
the variety of preferences one might have about work in social settings. Someone might

want to be alone all the time, or only work with one other person in particular. She might

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want to always work with others and prefer doing it with just about anyone. The syllabus
did offer some guidance on student collaboration. “While the discussions of the material
and homework assignments with your fellow students are encouraged, you must prepare
your homework papers alone.” Discussing material and homework then could be done
alone or with others and collaboration was actually encouraged rather than being
permitted as it was in the Complex Analysis course. It was only the writing up of the
homework solutions that was dictated to be a solitary activity.

Furthermore, an interesting bit of evidence that suggests that the course values
collaborative activity but without exactly defining who should take on which roles is on
the first page of the syllabus. There is a large sketch of three cartoon turtles working
together on a math problem. One turtle stands at a chalkboard and holds the chalk. He or
she is looking back at the other two, seemingly for confirmation. This turtle leans on the
desk of the third turtle and has his or her left leg casually crossed over the right ankle. A
second stands near the first scratching his or her head. A third turtle sits at a nearby table
drinking a cup of steaming coffee as a large question mark hovers above his or her head.
The setting might be a classroom, but there is only one table in view and it is slightly
larger than an individual classroom desk and has an unattached chair. In this scene it is
unclear whether any turtle is taking on the role of “teacher.” The first holds the chalk, but
the second stands nearby as if monitoring the work closely, and the third could be
interpreted as evaluating the work of the first and is finding it lacking. This scene is just
below the course title and instructor contact information. The syllabus sends a pictorial
message to students that collaboration is encouraged but is free to interpretation regarding

who teaches, who learns, and who evaluates.

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Belonging to a community was a way the course invited students to take on new
self concepts about mathematics and students were fairly free to choose how they wanted
to be in this community regarding their participation in class and in their discussions
about course material and homework assignments. On the other hand, they were
presented with a particular view of the field as understood by others outside of their class
and according to the view the criterion for membership or belonging was to think of
mathematics as about the study of and construction of proofs. Belonging to a community
on the small classroom scale was fairly open to student preference, but if students wanted
to belong to the larger community they would need to take on new views about math that

better matched those already members of the larger mathematics community.

Regimen: Pedagogically effective prose

Students were asked to do many things in this course and this regimen falls under
a theme. The students were asked to explain. Undergraduate math courses are known for
their emphasis on proof and this course was no exception. In several ways this course
shared a lot with both the Advanced Geometry course and the Real Analysis one. That it
targeted the students’ self-concepts about math it has ties to the Advanced Geometry
course. Yet in that course the emphasis on proofs was tied more closely to conjecturing
statements that would later become proofs if possible. In that it worked on changing
conceptions about what it means to do math—particularly away from a calculations and
memorization orientation there are echoes of the Real Analysis course. The Real Analysis
course, though, worked on students’ epistemologies—how they decide what is known.
This was done through getting students to judge the truth of statements, the

persuasiveness of a proof, and by having students write proofs to given theorems. In the

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Discrete Mathematics course the emphasis on proofs was in getting students to see the
explanative power of proofs. It was not so much that students needed to learn that proofs
determine knowledge, but how proofs can explain solutions and provide insight.
Mimicking this orientation towards proving, students were asked to explain in many
facets of their course experience using the discursive features of proof.

Recall that the textbook declared that there is no mathematics without proofs.
Throughout the textbook proofs are given as explanations of results more than as

validation of them. In the first chapter the textbook gives a theorem: “A set with n

elements has 2n subsets” (Lovasz et al., 2003, p. 10). The text goes on to provide two

arguments for the theorem. The first is a tree diagram that shows how to select the
possible subsets of a three-element set and goes on to describe this diagram and how to

modify it for an n—element set. The second proof shows how to enumerate subsets or label

. d
subsets With a number so that a person could speak about the 531‘ subset, for example.

After showing two such ways to make lists of subsets the textbook declares that there is a

problem with this method. What if there were ten elements and you were asked to name

the 233rd subset. Creating a list of them all would be rather tedious and so a system of

counting using binary notation is introduced. From this explanation of the use of binary

labeling an argument is forwarded to declare that for n elements there must be 2n subsets.

After the two proofs are given the textbook makes a remark.

You may have wondered why we needed two proofs. Certainly not
because a single proof would not have given enough confidence in the
truth of the statementl...The answer is that every proof reveals much more
that just the bare fact stated in the theorem, and this revelation may be
more valuable than the theorem itself. For example, the first proof given
above introduced the idea of breaking down the selection of a subset into

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independent decisions and the representation of this idea by a ‘decision
tree’; we will use this idea repeatedly. The second proof introduced the
idea of enumerating these subsets...we also saw an important method of
counting: We established a correspondence between the objects we
wanted to count (the subsets) and some other kinds of objects that we can

count easily (the numbers 0, 1, ..., 2n — 1). (Lovasz et al., 2003, p. 10)

Proofs are explanations as much as they are validations and possibly more so. When
students did the work of the course they were not just proving in the sense of verifying
and making a claim about knowledge. They were explaining ideas to others and
themselves. When submitting these homework sets, the syllabus clearly states:

...you will be required to write your homework and exam papers in full

and coherent sentences. Even if the solution of a problem consists mostly

of computations, you will be required to provide explanations of these

computations, like you will explain them to a fiiend... Papers consisting

of only formulas without any explanations will get 0 credit. You are

advised to prepare a handwritten draft before typing, and to clarify your

ideas during the preparation of the final version.
The expectation for students as they did their homework and constructed their proofs was
to explain their own mathematical thoughts in ordinary English sentences. The word
“explain” or some form of it is invoked three times. Doing the work of the class meant to
explain. On the day Dr. DM returned students’ first quizzes, he again pointed out that
students needed to explain their work. “F orrnulas and just answers are not enough to get
full credit.” Furthermore, these explanations needed to be in “full and coherent
sentences”, or in other words, in prose.

In late October Dr. DM begins a unit on graph theory by telling the class about
graphs and how they can be helpful visual representations of things that are hard to put

into words or are too cumbersome for other representations. He draws a simple 4-node

graph. He also gives a lot of examples, referring again back to Alice and her handshake

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party. He tells the class that he could give a formal definition of a graph but he really
prefers it if they “approach it with pictures. Informal language and the language of graph
theory are the best explanation.”

Students also demanded pedagogical explanations from Dr. DM and the textbook
assumes that they will likewise expect full explanations from it as well. One day Dr.
announced his plans to present two proofs for a particular theorem related to graph
theory; the first will use induction and the second will use double counting towards the
solution. He began the first one by using induction on the number of edges in the graph.
Once he completed this, a student asked him if this proof based on induction of the
number of edges was the only induction proof. Was it possible to do induction on the
vertices? The instructor replied that it was possible but that it was tricky, and then seemed
to be about to move on past her question. He paused though and announced that he would
do it her way. The student wanted to know more about induction in graph theory and
whether induction by edges was the only way, and if so why and if not, what would it
look like with induction on the vertices.

The instructor’s decision to follow her question was an instance of a student
having expectations for him and his explanations. That he tended to explain ideas through
his proofs was not just him modeling what he wanted students to do. It was also an
example of how a course constructs ideas, not just the professor. Students contribute to
the constructions and expectations as well. In this case they demanded more fiom him.
After he finished her proof and the second one (now third) that he had originally planned
for he commented, “A theorem does not need three proofs. One is enough to be

convincing, but these are illustrative for us as they shed light on different methods.”

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The textbook also assumed that students would demand better proofs and
explanations. These explanations should not just be about the truth of the statement but
should be helpful to the reader. In the chapter on Fibonacci numbers the textbook

presents a theorem for a formula to give an nth Fibonacci number:

. _2.((1”§)"—<“f))
‘5 2 2 (Lovasz et al., 2003, p. 71)

Here is the accompanying proof:

 

 

“It is straightforward to check that this formula gives the right value for n = 0, 1, and then
one can prove its validity for all n by induction.” (p.72). This is followed by an exercise
to do so. Just after the declaration that the reader should carry forth this induction proof
the textbook continues.

Do you feel cheated by this proof? You should; while it is, of course,

logically correct what we did, one would like to see more: How can one

arrive at such a formula? What should we try to get a similar formula if we

face a similar, but different, recurrence? So let us forget Theorem 4.3.1 for

a while and let us try to find a formula for Fr1 from ‘scratch.’ (p. 72)

The induction proof would verify the formula but it would not be an explanation
of where the formula came from or be helpful if students faced similar tasks in the future.
An induction proof would have been cheating them. They should expect better
explanations from others just as they were required to deliver better explanations to Dr.
DM and the rest of the class.

The professor and the textbook wants to explain ideas to students, wants them to
expect more from them, and in turn wants them to explain ideas back to him on their

assignments. They also need to be able to explain their ideas to one another. As has been

mentioned, students were asked to come to the board to show what progress they had

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been making on a problem or to come show the class what struggles they were facing.
Explaining was not reserved for student to professor alone and nor was it only for use
when you were sure of the result. Explaining could happen even when a student was lost.

At times Professor DM’s responses to student questions were brief for yes/no type
questions, and at other times he engaged the student to better understand the question.
Consider the following exchange from the second week of the semester just after
Professor DM asked for any questions about the homework assignment from the students.

Student: “1.8.8 (referring to an assignment number from the textbook)”

Professor DM: “So what’s the question?”

Student: “What’s my question?”

Professor DM: “Yeah.”

Student: “I did it but I don’t know if it’s right.”

Professor DM: “Yes. (laughs) What is not clear about it?”

Student: “The overcounting is not clear”
This exchange reveals some interesting expectations for him from the students and for the
students from him. He expected them to ask questions, but there was more. They were
expected to formulate the question in terms of pedagogically legible English prose—to
explain what it was they needed help solving or thinking about and to provide their work
thus far. By contrast, the students expected him to begin solving the problem when they
offered a problem number. For the first few weeks this pattern continued. Students
offered problem numbers instead of questions and he kept asking for a question or had
them come forward to show their work. As the semester progressed students began to ask
questions differently. Instead of only offering the problem number they began to follow
the problem number with an explanation of the difficulty they were facing.

That part of explaining included getting lost or muddled was reiterated in the

textbook. The authors advised the readers to be aware that learning mathematics requires

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“dirtying your hands” (Lovasz et al., 2003, p. vi). The metaphor suggests an active role
and possibly also some unpleasantness. To be able to explain something you need to try it
out and possibly fail from time to time. The nod to the role of failure might be a reason
Dr. DM dropped the lowest quiz grade and on a few occasions threw out a whole set of
quizzes and homework telling the class that he was not happy with using them as
measures of what they could do. This example also provides insight into the substance
targeted for change by the course requirements. The students’ self-concepts about
mathematics could include errors. Students of mathematics can make mistakes; they are
not always going to arrive at correct answers. Dr. DM’s grading policies reflected this
aspect of the substance.

The issue of grading and assessment in his course included a very interesting idea
related to another way that explaining emerged. Students in this class (and in others)
talked to me every now and again about the course and shared opinions on it and so forth,
usually in the form of pre-class chit-chat. One day a student was telling me about the
grading policies. Before I share the students’ take on grading let me offer what the
syllabus proclaims. “Your grade will be based on your homework, quizzes, and tests,
with approximately 1/3 of the grade determined by each of them.” According to this
student and with several nearby nodding in agreement, the unstated method for
determining grades was that near the end of the course Dr. DM would tell each person
what grade they would receive and then it was up to the students to set up a private
meeting to put their thinking into pedagogically effective, persuasive, prose. They should
argue why (if they felt it worthwhile) their grade should be altered. With no confirmation

of this by the professor I can’t declare whether this was the case or if it was a rumor

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passed around the class. On the other hand I was able to determine by observation that
neither of the guidelines in the syllabus about grading quizzes or homeworks was
enforced. Regardless, that students held this idea speaks to the ways that a culture of
explaining had formed. Everything, it seemed, revolved around making your thinking
pedagogical through explanations. It also speaks to the role of community in the class
that grades were possibly (or at least thought to be) negotiable in a face-to-face manner.
Students saw themselves as full members with rights and responsibilities for determining
their final grades.

Students also found other ways to explain themselves. Amanda told me one day
of her plans to petition Dr. DM for extra credit opportrmities, as she walked around the
room before class asking students to sign a paper. He had previously told the class that
there would be no extra credit. It seemed he had recently told Amanda that if she could
get everyone in the class to agree to want extra credit then he would create something.
She must have persevered and collected everyone’s signature because the next week Dr.
DM announced two options for extra credit. Amanda, in this example, had presented her
concerns to Dr. DM and he had been moved by her response to his concerns that the
whole class needed to in agreement. Her use of a petition was pedagogically effective for
him.

The first option offered by Professor DM was to submit an additional homework
set to replace a previous lower score. The second is to present a 25-minute (half the
period) lesson. To make room for these lessons Professor DM reserved three class dates
for a total of six possible presenters. Students were instructed that if they wanted to

present they needed to meet him to jointly agree on a lesson date, topic and for help in

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preparing the lesson. Only four students decided to pursue this option”. During these
four student lessons, students presented and proved problems/theorems with the usual
commentary and questions by fellow students while Professor DM sat in a desk in the
front corner of the room, adding comments and questions as well.

This extra credit option was an option about explaining. Students were to take on
the role of the instructor and explain ideas to their fellow students for half of a class
period. It was not explaining on their homework with words and maybe some symbols
and pictures. It was not explaining a quick idea while seated at their desks. It was to
explain on a grander scale. They would teach. Furthermore, to make space for students to
teach the class Dr. DM had to forfeit three days of time he would have largely controlled.
He would have chosen the topics, the order, the examples. Instead, getting students to
explain and teach the class was important enough that not only did they get the days but
they also, in consultation with him, chose the topics and what the explanations and
teaching would look like.

The expected regimen for students was to craft explanations. They explained
themselves through their assignments and proofs. They explained their ideas to the class
by calling out suggestions, coming to the board, and working together during lectures.
Dr. DM and the textbook also engaged in explaining through the text as presented in the
book and on the chalkboard and through Dr. DM’s in-class commentaries. Students’
explanations also factored into their grades. They made up the bulk of their homework,

quizzes, and tests. They were also important enough to the students that they understood

 

12
It is unclear how many decided to accept Option 1.

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their final grades to in part be determined by their abilities to articulate their concerns
about their final grades.

As in Complex Analysis the vision of mathematics as linear and rational extended
into their ways of behaving in class and on assignments, this course’s view of
mathematics as pedagogical explanations also extended into their behaviors. Students
were expected to explain themselves when confused, explain themselves on graded
assignments, and possibly explain their arguments for a different final course grade. And
finally, students were given the opportunity to take their explanations to the whole class

by teaching.

Telos: Teachers

The model student is not satisfied with learning formulas and computing answers.
This person wants to be able to explain why things work and seeks these answers through
proof. These proofs need to be both convincing and illustrative. A proof that only
answers whether something is true or false is not complete because it should also explain
why in such as manner that it can be understood by someone else. The model student
works towards creating proofs and listening to others’ proofs. Explaining her ideas and
expecting the same from others. The model student would not be satisfied with either
being given only an answer or submitting only an answer. The constructed model would
demand more of others: Would want explanations, fully articulated. The model student
would be a teacher.

A model student emerging from this course would have a very broad view of the
field. This person would have had the experience of calculus courses and analysis courses

and so would be versed in the technical details of that type of mathematics but would

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have added to his repertoire knowledge of other fields from computer science, the math
of games, social math, and topics of debate (Is the Four-Color Conjecture proven?) and
those from the typical canon of mathematical history (Gauss’s counting strategies,
Fibonacci numbers). The model student would have previously exhibited mastery of
calculations and memorizations but can now explain ideas and proofs from a wide
selection of sub-fields—some considered firmly in the field and some on the fringes.
Having a broad view can be helpful for a teacher who needs to be able to draw from a
variety of experiences and ideas for the purpose of making a pedagogically effective
explanation. The explanations must be for someone and so must take into consideration
what that person will likely find helpful, but the teacher must also be able to draw
accurately from her knowledge of the field to craft her explanations.

The bulk of the regimen section highlights skills a teacher would be expected to
have. The course design emphasized proofs as explanations. These explanations were not
to uncover hidden meanings, and not only to verify. They were explanations for someone
and so they needed to move between the abstracted mathematics and the reality of the
reader or student. Proofs as explanations were pedagogical. A model student would be
able to render mathematics into understandable prose while still remaining faithful to the
mathematics. This sort of work is the work of a teacher. Teachers try to make ideas
understandable to others by recasting ideas into language that is accessible to students.
The Discrete Mathematics course provided many experiences that would have served as
teacher preparation. Students in the course were expected to explain themselves to the
professor and also to other students in the course. A model student would be able to alter

explanations for different audiences. As constructed in this course a model student would

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be able to translate their problem-solving for people who might not need a lot of support
(like when real students of the course wrote up their homework proofs) and for people
who needed more support (like when real students in the course taught one another
formally as teachers-for—the-day and in small pairs). Teachers likewise need to be able to
offer explanations that are appropriate for people with diverse needs and degrees of
support.

Because the course design also expected students to expect better explanations
from others, the model student/teacher emerging from the course would be also expect
future students to be able to explain themselves in a well-articulated and coherent
manner. The teacher would need to be able to understand a student who was struggling
with an idea and would at times need that student to articulate his confusion in such a
way as to assist the teacher. This model student-as-teacher would also expect effective
explanations from students in their homework and assignments. The real students in the
Discrete Mathematics course did this by expecting well-crafted explanations from Dr.
DM. This would serve the model student-as-teacher by helping him better understand
what the student could do/explain and what the student had not yet been able to explain.
The quality of the explanation would be the key to the students’ learning and the
teacher’s response.

All of these things could be argued to fall within the domain of a teacher. A
teacher needs to know the canon of typical math courses such as calculus and analysis,
but also needs to know more about math as it is experienced by non-mathematicians. The
teacher should have interesting problems and ideas on hand to entice unenthusiastic

students she may encounter. The course’s use of discrete topics from many areas of

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mathematics combined with the assumed prior experiences constitute a very full
repertoire of experiences and knowledge from which the teacher may need to draw. This
model student would particularly know how to cast mathematics into real-world or at
least near-real-world experiences and applications for the hopeful benefit of future
students. Having a big picture view could help the model student select tasks and topics
well suited to the interests of particular students.

Overall, the experiences, skills, and orientations to mathematics that the model
student would have gained through the course would reflect the skills of a knowledgeable
yet flexible teacher concerned with sharing well crafted explanations and hearing

students learn to do likewise.

Concluding remarks

As with the other chapters I will close by speculating, from the perspective of a
teacher educator, what real students in the course may have found promising in the
course as well as what they may have found frustrating. Speculating on the experiences
of real students offers another way to think about a course. A model student can help us
imagine what perfection could look like, but perfection is hardly a human characteristic.
By considering real students we can see how students sharing features with the model
construction may have been more at home in the course and how those with different
sorts of mathematical dispositions may have been excluded.

The targeted substance was students’ self-concepts. Those who had found visions
of mathematics as calculations based comforting would have felt uneasy in the course. I
provided an example where a student was cut off from calculating his answer to single

numerical expression. Students who enjoyed this sort of mathematics and saw themselves

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as good at it may have found the lack of this type of work as well as the explicit and
implicit instructions to cease these activities, as limiting their ability to participate. They
might not have wanted to change their self-concept and may have rejected a new one that
seemed too much of a mismatch. Others may have felt that the view of mathematics as
problem solving and explaining was liberating.

On the other hand, the course was not too abstract. Application-style problems
that mimicked or at least came close to real world scenarios were abundant. Dr. DM, the
textbook, and students abstracted away from these scenarios to solve them, but the
problems were not presented as abstractions. Students that enjoyed being able to work
with. contextual problems would have enjoyed these opportunities. And again, students
who preferred working with mathematics typically referred to as “pure” mathematics
might have considered these problems to be too applied and not mathematical enough.
They may have thought of the many problems of the partygoers (Alice and friends) as
juvenile and tiresome.

Others, like Amanda, who asked for more formal definitions may have described
the class as not rigorous enough and relying far too often on assumed shared
interpretations. Others might have found the intuitive and graphical definitions as helpful
and relatable.

Another feature of the course that students could have taken issue with was the
requirement to explain. Students that preferred giving an answer and getting one might
have been disappointed that they had to also provide explanations. Students with this
viewpoint would have had trouble on the homework assignments, quizzes, tests, in class

lecture and homework discussions. As for the possibly negotiated final course grade,

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several students seemed to find that appalling. When the student was telling me about this
rumored grading scheme he seemed upset by it and others nearby also seemed shocked
that a course grade could be argued rather than earned or given. These students did not
like this possibility too much although one student to chimed in that it was a good thing
in her opinion because you’d only ever argue your grade up and not down. That is, it
couldn’t hurt. Those that didn’t like the idea, though, seemed upset that the grading
policy was too ill defined and they preferred a more fully rationalized structure.

Derrick commented on the ways that this course encouraged group work and that
he liked this feature. Those who preferred working with others had chances to do this.
They could collaborate outside of class or hold mini lessons together in class. Students
that wanted to work alone also had this opportunity as group work was not required and
was not a formal aspect of the course. Across this dimension, students could elect to
participate as isolated or as gregarious as desired. Belonging to the community was very
open to choice and so most students should have been able to find some means of
participation that was satisfactory.

This study was not intended to answer the question, “Who liked the course and
why.” I am speculating here from the viewpoint of a math teacher educator. But just as a
model student would be a realization of the perfect enactment of this course, the image of
the model student constructed by the course also provides us with insight into the
perfectly alienated student. Real students probably fall somewhere in between these two
extremes and might hold positions that look like the model student in one aspect and like
the anti-student in others. Like in the other courses described, some students may find

this Discrete Mathematics course to be a perfect or near-perfect match to their interests

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whereas others find it disagreeable. Whatever possibilities exist for students to embrace

mathematics in a course also sets up boundaries for others.

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CHAPTER SIX

WHAT MATHEMATICS IS ABOUT

We also want them to know what mathematics is about

and how mathematics is done.
—-Hung-Hsi Wu

One of the most reliable finding from research on
teaching and learning is that students learn what they
are given opportunities to learn.
--James Hiebert
While working on an assignment for a course one year I had the opportunity to
interview a very distinguished mathematician who is also a well-known mathematics
educator. My project partner and I were excited by the chance to ask him about the
different philosophies of mathematics, particularly Platonism, intuitionism, humanism,
constructivism, and formalism. We had planned a lot of follow up questions to our first. I
cannot recall exactly how we phrased it but it was not much more elegant than, “Tell us
about your take on the philosophies of math and their relationship to mathematics
education.” Again I do not recall the exact response but it was not more complicated
than, “I don’t have a philosophy of mathematics. I just do it.” We were fairly stunned.
There went the interview! I even recall being concerned about what we’d put in our
report. At first glance it seemed we had nothing to work with. But in retrospect I
recognize that was an important moment that contributed to the first stirrings of this
dissertation.

That we got such a response is not entirely shocking, really. I’ve heard many

similar reports in my encounters with students, parents, teacher educators, mathematics

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educators, and mathematicians. Math is often seen as just something you do or teach.
You might even enjoy doing it or teaching it or both. You might not. Math, though, just
is. Math teaching also just is.

Oh, yes, there are people who do think a lot about mathematics and their teaching.
This interview included one such person. He has thought far more about mathematics and
the teaching of it than I surely have. Yet, still, there remains a common notion that
mathematics is a single thing and that singular and unified nature is outside of
philosophy. Teaching, too, is commonly viewed as independent of philosophy (despite
the many requests I obliged to send my teaching philosophy to search committee chairs).
It is a practice, a real world, atheoretical thing. As a person deeply interested in
philosophies this piques me to put it bluntly.

This dissertation is part of my attempt to investigate some of these commonly
accepted notions about mathematics, mathematics education and particularly
undergraduate courses. Many people think of undergraduate mathematics classrooms as
mostly similar with different topics. For example, a Differential Equations course would
look roughly similar to a Number Theory course except the topics, definitions, theorems
and proofs would differ. The stereotypical model is that a professor stands at a
chalkboard and delivers information which students record in their notebooks. There
would be homework, quizzes, and tests. Students sit in rows. So forth. The point of the
class would be for students to learn the important theorems, ideas, and techniques of that
topic. This stereotype is communicated often in the ways that math educators talk about
math courses. We refer to courses as about something, about the topics typical to that

course. When we ask one another to tell us what they teach we usually tell one another

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the course title or the topics of the course. But these topics are only part of the
curriculum.

Undergraduate mathematics, as Hung-Hsi Wu refers to in the quote above (1998),
also presents a vision of mathematics, an implicit curriculum, a construction of the
subject that addresses philosophical issues about the discipline. What is it all about? How
is it done? The answers to these questions are often assumed to be the same just as the
surface features are so assumed. Math is about proofs. Yet again a simple statement hides
a lot of diversity. The example courses I have presented here illuminate just some of the
possibilities. Proofs are things to judge and critique; they determine knowledge. Proofs
are codified ways of describing phenomena for a pedagogical purpose; proofs are the end
product of conjecturing; proofs are the building blocks of a coherent and unified field.
These are not just differences of emphasis. They are differences of philosophy. They are
differences of mathematics.

“Social constructivism describes mathematics as a set of socially situated
discursive practices” (Ernest, 1998, p. 263). In many ways Emest’s articulation of social
constructivism as a mathematical philosophy resembles the F oucaultian perspective
undergirding my analysis. lBoth seem to be saying that math is as math does. In my
analysis of these courses math can be social, individual, Platonic, concerned with truth,
and concerned with beauty among others. I did not try to argue that one course’s vision
was the correct one, for example, but described the ways that mathematics was done in
multiple contexts as multiple possibilities for students to potentially adopt. “Persons
always learn or use mathematics within a range of such contexts and are multiply

positioned in them. Mathematical knowledge itself is recontextualized, reproduced, and

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operationalized within multiple contexts.” (p. 263) Emest’s statement is very much
aligned with my position. In this project I examined four courses to highlight the ways
that mathematics operated across different classroom contexts at a particular time in
history. I take the position that there is no one unitary mathematics and that there are
multiple ways that mathematics is performed, specifically in classrooms for this project.
This is a F oucaltian analysis in that I am not trying to argue for a particular truth, but am
exploring how visions of certain truths of mathematics operate and exist as possibilities.

On the other hand, I, like Ernest, am not trying to develop or propose a particular
philosophy of mathematics. He lays out several features of a mathematical philosophy
and then sets about evaluating his take on social constructivism with respect to these
criteria. In many ways he is trying to argue for a particular philosophy. In particular he is
arguing against what he terms absolutist (Platonism, for example) rather than fallibilist
philosophies of mathematics (humanism, for example). Similarly, Hersh (l997)uses
historic examples to argue that Platonism is not well-matched to the daily lives of
mathematicians or to the historical accounts of it. Although I am sympathetic to both
author’s work and positions, this dissertation is not an argument against a particular
philosophy of mathematics. I do not present these accounts of the four courses for the
purpose of claiming that certain philosophies are false. In fact, this dissertation is not
about the truth or falsehood of ideas. To take Platonism as an example (because it is
frequently the philosophy most under attack by philosophers of mathematics), I recognize
that elements of Platonism exist in the discourse of mathematics classrooms, particularly
in the Complex Analysis course. Its presence in the discourse is not something I

necessarily am trying to eliminate. In fact, its presence gives rise to the ability to speak

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against it (as Ernest (1998), Hersh and Davis (1980), and Hersh (1997) do). Platonism
makes counter-Platonic arguments possible and thinkable. By describing the courses in
this dissertation I am not trying to argue for or against a philosophy. I am not trying to
establish the truth of the discipline as socially constructed, for example (e. g. Ernest) and
so am not mobilizing my varied examples for that purpose. For me, the multiple visions
of mathematics do not form an argument for rejecting a particular vision, but instead
provide a way to think of mathematical discourse as containing all of these contradictory
positions and visions. That they are contradictory is not problematic for me as they may
be for philosophers interested in a solitary criterion-based truth.

In The Mathematical Experience (1980), Davis and Hersh explain their basic
conclusion, “It is reasonable to propose a different task for mathematical philosophy, not
to seek indubitable truth, but to give an account of mathematical knowledge as it really
is—fallible, corrigible, tentative, evolving, as is every other kind of human knowledge”
(p. 406). Although they propose a “different task” “not to seek indubitable truth” they go
on to suggest that their account is what “mathematical knowledge really is.” This sort of
rejection of truth and then a reclaiming of it is what I have tried to avoid in this
dissertation. Although these mathematician philosophers share many of my interests and
positions, this dissertation assumes a multiplicity of truths to describe the practices of
four courses as instances of some of the possibilities in mathematics, rather than an
assumption of a truth and the corresponding hunt for supporting multiple practices.

In the chapters devoted to each course I attempted to describe the construction of
the subject that these courses developed. By looking at the construction of the subject

rather than the learning of students I have tried to recast the issues of undergraduate

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education into issues of philosophy rather than psychology or even pedagogy. I believe
that this approach to research offers distinct advantages for educators. It allows us to see
the courses as more than students’ test scores or instructors’ teaching practices. It does
not assume that there is a right way to teach or a right single thing to learn. When we can
see the possibility of more than one best way (whether that be best lecture, best
curriculum, best groupwork, best assessments) we can imagine constructions that we had
not yet been able to articulate.

For now, though, let us return to the constructions that I did describe in this
dissertation and allow me to summarize those descriptions here.

The Real Analysis course constructed a model student as a convert—someone
who thought of mathematics as a way of knowing, of securing beliefs to evidence. That
evidence was the proof. Students were positioned as needing to alter their epistemologies
away from trusting in authorities such as professors and authors and away from their
intuitions. Their belief and trust needed to shift towards proof as a technique for
ascertaining knowledge. To get students to place their trust elsewhere the course drew
upon community. It called the larger community of mathematicians throughout history to
provide testimony for the new belief system. It also called on the community of the class
as a tool. On both scales, the macro field and the micro classroom, belonging to these
communities was seen as a desirable enticement for the students to proceed through the
course and effect epistemological change. Students were asked to act as a mathematical
court. They judged, crafted, and watched proofs. Mathematics was something to believe
in as much as it was something to do. When they crafted their proofs they were

performing the skills of proof-writing. They were following the norms and codes of the

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genre, proof-writing. In this course students were not asked to examine the logic and the
assumptions of that logic and axiomatic system. They were not asked to question whether
or how the proof as a written collection of symbols and signs conferred meaning and
truth. They were asked to draw a line in the sand and believe everything proven from that
point forward. That this genre, this way of writing, conferred truth was an act of faith.
The Complex Analysis course presented a similarly unquestioning view of
mathematics as well as a model student that presents analogies with religious vocabulary.
In this course students were constructed as Platonic disciples. In Complex Analysis the
substance of the course was in students crafting well-ordered and rational mathematical
lives. Real Analysis only dealt with students minds and beliefs, whereas Complex
Analysis targeted their bodies as well. Rational habits of mind and body were important.
They needed to see the coherence and unity in mathematics and take up that coherence in
their physical habits as well. Every aspect of their selves should be ordered and orderly.
To motivate this change, this course appealed to a particular aesthetic of mathematics—
the structured and connected aspects. Finding connections was pleasurable. Students were
asked to follow the linear structure of mathematics, follow the rules, and follow the
model of the instructor. They were not asked to deviate from the path or find their own
way. They were provided with explicit step—by-step approaches to mathematics as well as
their behaviors. Furthermore they were expected to stay in these lines. That I have
described the model student as a disciple includes an aspect of belief. There was an
implicit sense that in following these rules for their lives and the linear structure of
mathematics, they would believe that these were good and pleasing ways to proceed. But

where the Real Analysis students acted as judges and lawyers, there was an element of

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decision-making. The Real Analysis students also participated in a much more
collaborative community. They were participants in the process. By contrast the Complex
Analysis students followed and did not participate in those ways. The model student
followed the teaching of mathematics and the mentor. That this view of mathematics
encompassed a Platonic ideal of mathematics—it is discovered—made them disciples of
this particular vision of mathematics. In Real Analysis the convert has learned to trust in
the formalistic nature of proof—proofs determine what we know. When we write proofs
we are making decisions about knowledge. The Platonic disciple uncovers knowledge
through proof, but does not craft it. Finally, that the Complex Analysis course included
aspects of living, the disciple also takes on a way of being, a worldview not just a way of
thinking about mathematics.

The Discrete Mathematics course constructed the model student as a teacher—
someone who could render proofs into a pedagogical artifact. Proofs are ways of
explaining for the education of others and the self. The substance of the student that
needed reformation was their self-concept as a student. Rather than think of their roles as
students of mathematics in terms of calculations and memorization of formulas and
techniques, as the course assumed they did, they were to think of themselves as problem
solvers who could also translate those solutions into pedagogical forms. They needed to
see themselves as mediators between the problem and some audience or reader. The
mode of the course, like the Real Analysis course, was belonging. The course design
invoked membership to a larger community as a reason to take up the new self-concept.
And again, with ties to Real Analysis, there was an aspect of belonging to the classroom

community. This course asked students to explain mathematics. By explain I mean they

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were asked to put their mathematical ideas and arguments into a language and structure
pedagogically appropriate for the audience. Mathematics was a communicative exercise.
They needed to use their explanations to help others and themselves learn something—
not only verify it. They were not asked to study a unified set of topics. In fact, the topics
were discrete and the students jumped around from topic to topic. Mathematics was this
and it was that—a grab bag of ideas. What held the course together was the emphasis on
communicating the mathematics of the problems. They were asked to teach by rendering
their thoughts publicly. Even when unsure they were asked to craft their confusion into a
form that could teach the professor and other classes what was on the student’s mind. The
model student was a teacher. Mathematics was something to translate into pedagogically
effective prose.

Like the Discrete Mathematics course, the substance targeted in Advanced
Geometry was students’ self-concepts. But instead of seeing themselves as problem
solvers who could translate their work into pedagogical terms, this course worked on
students’ identities as researchers. Aesthetics, like in Complex Analysis, was the mode.
In Advanced Geometry though, there was no single aesthetic. Students could determine
for themselves what was interesting and pleasing. It could have been structure, form,
surprise, patterns, connections, or deviations. Students were asked to play. In class they
were given limits on the direction of their play when the instructors presented a particular
diagram, for example. For their Book assignment, they had more freedom of choice.
Within the wide or narrow space they decided what ideas were of interest to play around
with and what their conjectures would be. That personal aesthetics drove what they chose

to investigate was an important contrast with the Complex Analysis course where the

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aesthetic of unity was the only option. Students in Advanced Geometry not only
conjectured things that turned out to be true but also things that turned out to be dead
ends. They played to create conjectures. In this way the model student was constructed as
a researcher. Researchers have limits and restrictions in their work, but they also have
choice about what to pursue and what to abandon. That the Advanced Geometry course
was interested in play in the form of “I Spy” and “What If” reflects a researcher’s work
of noticing patterns and discontinuities and pursuing conjectures and ultimately, and

hopefully, theorems.

Table 6.1: Summary of analysis

 

 

 

 

 

l Real Advanced Complex Discrete
1 Analysis Geometry Analysis Mathematics
Substance Epistemology Self-Concept Well-Ordered Self-Concept
Rationality
Mode Belonging Aesthetics Pleasure in Belonging
Orderliness
Regimen Holding Court Play Following Rules Crafting
Pedagogically
Effective Prose
Telos Converts Apprentice Platonic Teacher
Researcher Disciples

 

 

 

 

 

 

 

 

 

After summarizing these courses (and see Table 6.1 for a tabular summary) 1 now

want to address several themes across them:

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0 Who does mathematics?

° What sorts of students might be excluded from the course objectives,
based on their respective model constructions?

° What degrees of freedom were present in each course design? That is, how

much space was allowed for a diversity of students to participate?

Wu’s comment quoted above construes that there is one version of what
mathematics is all about and how is it done. I hope that my analysis of these four courses
has shown that several visions of mathematics are at least possible and in these what
mathematics is (or rather could be) all about and how it is (or can be) done look quite
distinctive. But to Wu’s two-part comment I would like to add another. Who does

mathematics?

Who does mathematics?

The Real Analysis and Discrete Mathematics courses presented visions of
mathematics as communities and used membership in this community as an invitation to
students to delve into the course. A sense belonging and inclusion constituted the modes.
But these two courses also had students engage in communities within the classrooms, as
did Advanced Geometry. Real Analysis, Discrete Mathematics, and Advanced Geometry
did not just tell students about the wonderful collections of mathematician professors and
researchers that thought in X ways or believed Y ideas or did Z things. Their invocations
were not “trust me, this is how it is out there” sorts of arguments—the ones that ask
students to suspend belief and delay gratification. Instead, they included elements of

modeling this X, Y, and Z in the classroom. In these courses students could do

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mathematics. What that mathematics was, though differed, and so the ways that students
did math looked dissimilar.

In Advanced Geometry students did mathematics by working with a partner or
group (and alone in their Books) to observe phenomena such as patterns or variation.
They did mathematics by telling other people about their ideas whether true or false.
They did mathematics by refuting one another’s ideas or assisting each other by refining
or proving the other’s conjecture. To engage in mathematics was to engage in social
contact. The instructors were responsible for setting up the context and scenario and
students were expected to pick up from there.

Real Analysis also expected students to be actors in the mathematical court—
judges, lawyers, and jury. Math was an activity of judging the validity of ideas and
constructing proofs performed in a social setting. To evaluating a proof’s ability to
convince the student was to evaluate another student’s or Dr. RA’s work. The proof may
have been constructed in isolation at home or far off in the textbook author’s office, but
in class and within group sessions students, the proof was judged by peers or Dr. RA.
Mathematics required a form of interaction and interrelatedness. Work happed on the
individual level, but at some point it must pass through a social phase and so students
must play some role in this cycle. The students were put in positions to judge and be
judged.

To do math in Discrete Mathematics was to interpret problems into mathematical
language for the purpose of solving them and then translating them back into a
pedagogical form. Students were asked to do both of these things in the course. In their

homework and quizzes and tests they mathematized to get a solution and then they were

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required to explain it back again to Dr. DM. To fail to do either of these parts impacted
their grades. In that math was conceived in this way students did mathematics. Some
students also taught half-class periods. They did these things in front of their peers. So to
when they would come to the board to show their ideas or articulate their confusions.

Complex Analysis offers a strong counterpoint to the other courses in relation to
“Who does math?” In CA, doing math was about uncovering connections and finding
order. For the most part students watched Dr. CA do this and describe it. They did do
homework sets and exams, but the homework tasks were often either calculations or
simplifying expressions. Students were practicing the knowledge Dr. CA or the textbook
shared, but they were not finding the connections. They were using them, but never
finding them. Exam problems did have proofs but these were usually exact reproductions
of homework questions or proofs offered in class by Dr. CA. In that students followed the
well-defined sets of behavioral expectations, they were doing a part of mathematics. They
were ordering their lives to match the structure of mathematics.

Describing the possible ways that students in these courses did mathematics—and
‘doing’ looked different in them—is one way to see the courses as open. We can also
look at the courses as closed by examining what sorts of students and activities were not
encouraged. Who would be excluded from the course? What follows are mostly
speculations about the sorts of students the class did not construct as models and not

necessarily examples of particular individuals.

Who was excluded from the course?

Advanced Geometry was not a very conducive place for students who preferred

structured lessons with a definition, examples, theorem, then proof sort of path. Students

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who preferred a direct path to an idea would not have likely described the class as their
favorite undergraduate experience. Mathematics was not presented as neat and orderly; as
a progression through ideas. It was messy and spun off at tangents. Students were not
expected to necessarily follow ideas in a linear fashion. Someone who wanted a set of
directions or held a “just tell me so I can get on with it” sort of approach would have been
stuck in a sea of ideas that were not obviously connected at times. Students who felt
comfortable in a traditional lecture and note-taking environment may have been very
unsettled by the open environment. Students who preferred to work on well-defined
problems with a structured approach towards known conclusions were not the sorts of
students who may have found the course exciting, but possibly may have found it too
diffuse and too chaotic.

The Real Analysis course also excluded students. Those who wanted to know
why proofs as a form of writing led to knowledge would not have their questions
answered. Those wanting to know more about the systems of logic and axioms that mark
a proof from another form of argument would have been disappointed. Someone who
hated group work would not have liked the formal requirements to work together. These
students would have wanted class to “get on with it” during the group aspects in class. A
student that was good at the tacit logical procedures but who was not good at
communicating his or her ideas would have had some difficulty with the expectations to
share ideas in a group setting and write them up in assignments. On the other hand
students who did enjoy these aspects may not have liked the other parts of class that did
follow a set path of definition, examples, theorem, and proof. They may not have liked

the traditional classroom feel of these times, wanting instead more of the time to follows

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the small group work model where they played with ideas rather than watching Dr. RA
do so. Overall students with a desire for more than the foundational theorems of analysis,
who wanted to learn more about the warrants for proof as a measure of knowledge would
have been excluded in this aspect. Others preferring one or the other of the two types of
class interaction (lectures, groups) might not have enjoyed the back and forth between the
two and would have wanted the class to stick with whatever their preference was.

Students who were in the Complex Analysis course that did not like the rules or
the ordered paths would have been excluded. They would have wanted more explorations
off the track presented by the course. Nonlinear thinkers would not have had many
opportunities to depart from the direction the course took. Students who did not like to
remain still and quiet, who enjoyed working with others and wanted it to be encouraged
rather than allowed might have not felt comfortable. Students who wanted their questions
about the lecture answered in class or ones who wanted a less hurried pace would have
been excluded. An excluded student might also not have an aesthetic of mathematics that
highlighted structure and order but possibly diversity and novelty.

In Discrete Mathematics an excluded student would not be good communicators.
They would not find the requirements to explain themselves in effective prose easy.
Students who liked grades calculated according to the syllabus might have been troubled
by the possibility that it might not be. Those who wanted a syllabus that outlined a set of
topics and possibly a schedule for the course might have felt lost at regarding the
direction and purpose of the course. Students with a vision of mathematics as interrelated
and unified might not have seen the diversity of problem types as interesting but possibly

as incoherent and disorganized. Those that preferred formal definitions and had an

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orientation towards proof as verification rather than explanation and articulation would
likely not have found the Discrete Mathematics course very open to their visions of
mathematics.

By listing the possible preferences of students that may have been excluded in
each course I am not saying that there was one real life, non-model, excluded student that
found a particular course distasteful on each of these aspects, but rather that in different
ways and at different times and to different degrees some students may not have been
included in the discourse. That is, a student may have not felt comfortable with the
conj ecturing aspect of the Advanced Geometry but might have felt more comfortable
with the proving. Likewise, a student in the Complex Analysis course may have
appreciated the linear and organized lectures, but not have liked the emphases on
behavioral rules. Or in each of these, vice versa. This sets me up to now explore the
degrees of accessibility in each course. How much room was available in each course for
students to still participate despite having opposing views on mathematics or preferences

of classroom environment?

How was the course accessible?

That courses constructed a model student has been analyzed and speculations
about students who may have been excluded from each course have also been offered.
Now I want to consider the degrees of accessibility in the courses. How narrow or open
was the course to students of different orientations and preferences?

In Real Analysis students of different preferences about classroom environment
could find somewhere where they felt comfortable. Students could feel at home in the

group work or in the lectures. That Dr. RA used full sentences in his proofs may have

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been very accessible to some students. They could feel welcomed by the use of first
names and casual pre-class talk. The might like the idea of becoming part of a
community. They could have appreciated the references to history of real analysis and
they could have simply liked analysis and thinking of mathematics and proofs as
providing justification for knowledge. That also could have felt that their contributions
were welcome when they determined the truth and falsehood of statements, crafted
proofs, and critiqued them.

A student who wanted to feel at home learning the Discrete Mathematics could
find comfort in the individual nature of the lectures or in the opportunities to quietly pair
up with another student during the lecture, as both were available options. A student
could also feel supported by the ways Dr. DM responded to student needs by altering his
lesson plans to support their needs, how he answered their questions whether they were
asked during the homework discussions or in lecture, and how he responded to requests
for extra credit. A student with strong communication skills could appreciate the
opportunity to show his abilities of explaining mathematics in his assignments and
possibly by teaching the whole class. Students could also feel supported in how he
dropped low scores and entire quiz sets. They could also feel good about being invited
into a larger community. They could have enjoyed the problems and the contextual and
application types of them offered in the textbook and in lectures and assignments.

Students in the Complex Analysis course could have found the rhythm of rules
and structure of the classroom comforting. A student could enjoy the straightforward
exposition of the lectures. That Dr. CA spent a lot of time doing homework problems

might have been very helpful for a student. The model of mathematics as linear and

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ordered could have been very accessible for students; that he highlighted connections
between what they had learned before with what they were learning now could have been
quite illuminating. Students could have felt inspired by the references to mathematics as
mysterious and pleasurable.

Students in the Advanced Geometry course could have found a home in the
opportunities to work alone, in a small group, and as a whole class. They could have
enjoyed the use of technology. Students also might have liked being able to create their
own paths to explore and/or the encouragement to prove them as well. That there were
two instructors might have been very supportive. A student could have felt pride at
having her ideas honored publicly in the whole class discussions. Being allowed the
space to determine their own criteria for beauty could have been novel. That a major
course grade was not a test in the sense of a sit down exam but rather a cumulative
collection of their ideas might have been refreshing. A student could find the
opportunities to do non-Euclidean work to be liberating.

That the courses provided many ways for students to find a home is exciting.
There was likely something accessible to every student in each course. Although no one
student was enrolled in each of the courses, I imagine that if a student were to have
experienced each that the student, no matter his preferences or personal philosophies or
attitudes towards mathematics could have found some aspect of each course accessible.
The diversity of approaches to mathematics, pedagogy, and philosophy in the courses
speaks to the troubles of philosophy of mathematics. What “math is about” is too narrow
a question and presumes a single answer, but these courses offer many possible

interpretations.

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fail-l

How did the courses speak to teacher education?

As I began this dissertation with questions about what models of teaching
preservice teachers are exposed to in their undergraduate courses, I am happy to return to
that question, albeit modified. Instead of asking what they learned or how they might go
onto teach future students I want to ask what we in mathematics education can learn
about the preservice teachers we encounter in light of this dissertation.

On the one hand I hope that teacher education is excited by these courses. If we
could imagine a preservice teacher enrolled in each of these exact course observed, as I
am not implying all complex analysis courses, for example, look alike, that student would
have a wealth of experiences that we could draw from in a teacher education course. Just
about any method suggested in a typical methods course was here. There was small
groups, technology, lecture, whole class conversation, student presentations, pair-share,
inquiry, problem solving, proving, connections, communication, representation,
communities of practice, and alternative assessments. I may even have missed some!
From a teacher education perspective we can be happy to see that this hypothetical
student has experienced first hand so many things we value for K-12 classrooms. As
Deborah Schifter (1986) noted teachers need to learn in the same ways that we hope they
will also teach. That is, they need experiences as students in mathematics classrooms (or
professional development) that mimics the pedagogies and views of mathematics that we
teacher educators want them to go onto teach. There are a lot of things in my list above to
suggest that my hypothetical preservice teacher is getting these experiences in at least
some of their courses. When we look across this hypothetical teacher’s course load we

can see that this preservice teacher has been taught in ways that the mathematics

206

education community typically desires or has at least been exposed to different parts in
different classrooms.

Yet we also know that many of the non-hypothetical, non-model, utterly real
preservice teachers we find in teacher education courses seem to be clueless about the
methods we try to teach. To our eyes it is as if this is brand new stuff and even a tad
suspect. Why then don’t they recognize the methods when we talk about them or use
them in our own classes?

My study was in no way designed to answer that question but I do have some
thoughts about it. We largely assume that they have not had these experiences and act
accordingly. Our discourse in our field and in our classrooms does not consider their
mathematical preparation as sites of pedagogical possibility. In all of my work I have
never seen or taught a methods class that asks students to seriously consider their
experiences as undergraduate mathematics students in terms of pedagogy. The
mathematics classes they take are assumed by us to be classes about content and so
contribute to their content knowledge, which we distinguish from pedagogical
knowledge.

I hope that through this dissertation you, the reader, has noticed how philosophy
of mathematics, actions of mathematics, formulations of students needs, invitations and
motivations for learning all are present in undergraduate classrooms and that to draw nice
walls between pedagogy and content is artificial. Did Dr. RA use groups because that is
an effective teaching tool or because it reflected his collaborative and community-based
vision of mathematics? Did Dr. CA highlight connections between mathematical strands

because that reflects his deep content knowledge of calculus and analysis or because

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connections are an effective pedagogical tool? Did Dr. AG use inquiry because it
promotes student engagement or because it reflects an important quality of a
mathematician?

There is of course, Shulman’s (2004) theory of pedagogical content knowledge
that views the two as merged knowledge bases, but Segall (2003, p. 10) presents a nice
argument that the two categories are conflated, not overlapping. Because we, and I
include myself here, in teacher education think of pedagogy and content as separate but
possibly related, we may be missing out on opportunities to get our preservice teacher to
analyze their undergraduate experiences in terms of pedagogical/mathematical moves in
such a way that can be fruitful to teacher education.

On the other hand, there are a lot of things in these courses that we may not want
our students to go on to mimic. Yet, as Hiebert (NCTM, 2000) notes, students learn what
they have the opportunity to learn. They have been and will be learning things all of the
time that might not be advocated by reforms to mathematics education. Yet, again, as
Hiebert notes, students learn what they have the opportunity to learn. If we do not
provide opportunities for them to learn from their mathematical experiences in
departments of mathematics they may not be able to coordinate experiences from their
undergraduate courses with the sorts of experiences we want them to produce for
students.

In the sections above on who is excluded and what aspects of the courses were
accessible we could see that any course could exclude and include. I hope that you will
not think of X course as good and Y course as bad. Each included ways for students to be

at home in the course and each included ways to be exclusive. Each presented

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mathematics and students roles in mathematics differently. The diversity of these courses
allowed for multiple entry points for different students.

I want to return again to the list generated a few paragraphs above when I
declared that practices that teacher education tends to value were present in these courses.
Here they are again: groups, technology, lecture, whole class conversation, student
presentations, pair-share, inquiry, problem solving, proving, connections,
communication, representation, communities of practice, and alternative assessments.
This list includes many of the sorts of activities mathematics teacher education holds as
best practices. Our discourse and the methods courses we design promote these visions of
not just pedagogy, but of mathematics as well by argument of their conflation. But like
the undergraduate mathematics courses observed in this study, our courses too can serve
to exclude some students and provide access for others. Our discourse excludes
preservice teachers who find inquiry activities chaotic and those who prefer solitary
thinking. We are more accessible to students who enjoy describing their thoughts, and
giving presentations. Our discourse limits student engagement in some ways and
promotes others.

Just as the discourses that constitute undergraduate mathematics is more diverse
than many in teacher education assumes, our discourse in mathematics teacher education
is more narrow than we like to believe. Through our preferred pedagogies we also offer a
vision of mathematics and this vision may at times compete with those of our preservice
teachers. By setting up a system of explicit and implicit best practices we are telling them
that our vision is not only better but best. Because our preservice teacher students are

deeply invested in mathematics this could be coming across as a personal attack.

209

 

 

So we are faced with a dilemma. On the one hand the things mathematics teacher
education cares about have been shown to be present in the union of the four courses I
studied. The possibility that our preservice teachers witness these practices and have
learned mathematics from them should excite us. We may be drawn to highlighting these
practices through work in our courses that draws students to share and reflect on these
experiences. But when we call on our students to do this sharing and reflecting we are
reinscribing a particular discourse that by omission excludes some students. In trying to
make use of the knowledge that preservice teachers have had the sorts of experiences we
want to promote, we may be setting up an opposition between the mathematics that math
teacher education values and some of the mathematics that mathematics the disciplinary
field values. We are asking our preservice teachers to choose.

Parks (2007a) suggests several ways that mathematics teacher education might
begin to open up the discourse around good teaching: “giving assignments that
recognized mathematical knowledge and teaching practice as well as self-analysis, doing
research about teacher education in ways that framed practices not as best, but as
possible, and explicitly working to expand - in teaching and in writing -— visions of good
teaching and acceptable beliefs” (p. 170). I hope that her suggestions are as powerful to
you as they are to me, and I hope that through this dissertation I have managed to do the
last two reasonably well.

On the one hand Hiebert (yes, one last time) might be a good summary of this
dissertation. One of the most reliable finding from research on teaching and learning is
that students learn what they are given opportunities to learn. The experiences across

these classrooms provided different opportunities for students. On the other, I have tried

210

to cast this dissertation as a construction of the subject rather than as a description of
what students learned. By framing this as a construction of the subject I can call discourse
into play. By doing so I hope I have illuminated the ways that instructors, textbooks,
documents, and philosophies of mathematics as well as students work together to create a
discourse and a classroom experience both in undergraduate mathematics and in
mathematics teacher education. These discourses do not always agree but there is more
overlap than we might have supposed.

When I very first started thinking about the ideas that inspired me, that ultimately
found their expression here—now in a very different form than first imagined—I
intended to document the philosophies of mathematics present in undergraduate
mathematics courses and those found in secondary methods courses. My big plan was to
show how those philosophies found in the mathematics courses infected the preservice
teachers in such a way that they were poisoning our preservice teachers away from the
better philosophy! How things change, huh? Like Gauss, I had had my conclusions and
only needed to arrive at them. Through this dissertation I have learned to take a more
generous view of undergraduate mathematics courses and inversely, a more critical view
of mathematics teacher education. This is not because I have learned that the teacher
education courses are the ones doing the poisoning, but because it is too easy find blame

in others and perfection in yourself.

211

 

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