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This is to certify that the
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DISCURSIVE POSSIBILITIES: RE-IMAGINING REFORM
AND EQUITY IN ELEMENTARY MATHEMATICS

presented by

AMY NOELLE PARKS

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Doctoral degree in Curriculum, Teaching and
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DISCURSIVE POSSIBILITIES:
RE-IMAGINING REFORM AND EQUITY IN ELEMENTARY MATHEMATICS

By

Amy Noelle Parks

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Curriculum, Teaching and Educational Policy

2007

ABSTRACT

DISCURSIVE POSSIBILITIES:
RE-IMAGINTNG REFORM AND EQUITY IN ELEMENTARY MATHEMATICS

By

AMY NOELLE PARKS

Much of the research on preservice education has focused on finding and
remediating the problematic beliefs and inadequate knowledge of individual beginning
teachers. Research on children engaged in elementary mathematics has generally been
seen as a separate body of work. This dissertation seeks to explore the work of preservice
education by drawing the lines around it differently. First, it uses discourse as a
theoretical frame, which locates practices like teaching, problem-solving and
understanding mathematics, outside of individual heads and disperses them into multiple,
but always interacting, fields. Second, it uses children’s interactions with elementary
mathematics as a catalyst for thinking in new ways about how we might educate those
who teach them.

To explore these issues, I spent a year in a third-grade urban classroom, primarily
during math class, where I observed the students, the teacher, and the student teacher. I
also observed the elementary mathematics methods course that the student teacher
attended during the first semester of the year. Throughout this dissertation, I draw on my
experiences in these two classrooms, as well as on relevant written documents, such as

curricula, state standards, and academic writing in mathematics education.

In particular, this dissertation is intended to respond to two prominent strands
within the conversation about preservice education in elementary mathematics: those
about equity and reform-oriented teaching. In addition, the dissertation is also designed to
respond to current calls for “evidence-based” or “scientific” education research by
drawing on ethnographic, genealogical, and rhetorical research traditions. By highlighting
different research traditions in different chapters, the dissertation makes it possible to see
the analytical afi‘ordances of each of these research strands. In particular, the
ethnographic chapter argues that students who do not share the linguistic, cultural or
ethnic background of their teacher may have more difficulty answering open-ended
questions in competent ways. The chapters that draw on genealogical and rhetorical
traditions examine the role that metaphors of hierarchy have played in defining the ways
that we think about student learning in mathematics, the persuasive powers of different
kinds of problems commonly used in mathematics classrooms, and the purposes that
multiple genres of teaching may play in the mathematics classroom. The overarching
argument of the dissertation is that the current narrowly-focused consensus on the
definitions of equity and reform-teaching limits, rather than expands, possibilities for

students.

DEDICATION

For Perry,
Who kept asking me to explain Foucault,
And for Sophie,

Who didn’t.

ACKNOWLEDGEMENTS

This dissertation would not have been possible without the intellectual and
emotional support of my advisers, Lynn Fendler and Sandra Crespo. Lynn introduced me
to new ways of seeing not just my work, but my life, and seemed to have an instinctive
knowledge of when to push and when to back off. Sandra made room in the world of
mathematics education for an elementary school teacher and encouraged me to pursue
work that mattered to me, even if it put me at odds with the majority mathematics
educators. Always, they had my back.

I would also like to thank the two intellectual communities who significantly
contributed to my drinking about this work. I am deeply grateful to my friends at Critical
Studies, particularly Cleo Cherryholmes, Steve Tuckey, Brett Merritt, and Sharon
Strickland, who took me seriously -- even though they were so many turtles farther along
than I -- and who challenged me and agreed with me, depending on what I needed.
Finally, I want to acknowledge the role that Anne Haas Dyson and the students in her
Advanced Qualitative Research seminars played in the work of conceiving of this study.
You held my hand through the early days.

I have been very lucky and am grateful for the opportunity to have learned from

and with all of these people.

TABLE OF CONTENTS

LIST OF TABLES .............................................................................. vii

CHAPTER ONE

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

CHAPTER TWO

HOW MIGHT EXPLICIT QUESTIONIN G SUPPORT THE

LEARNING OF ALL CHILDREN IN MATHEMATICS? ......................................... 14
Theoretical Framework: Talking About Race and Language ..................... 18
Methods .................................................................................. 21
Thinking About Kinds of Questions .................................................. 29
Concluding Thoughts .................................................................... 40

CHAPTER THREE

DOWN THE RABBIT HOLE: MY OWN POSTMODERN TURN ..................... 42

CHAPTER FOUR

DISCOURSE IN MATHEMATICS EDUCATION ........................................ 59

CHAPTER FIVE

BAD BELIEFS OR DENSE METAPHORS? ............................................................... 76
Bad Beliefs ............................................................................... 78
Dense Metaphors ......................................................................... 81
Talking About Children: Metaphors of Hierarchy ................................... 84
Coherence among Metaphors: Or What Else Can We Think of as Ordered?....94
Possibilities for Teacher Educators .................................................. 100

CHAPTER SD(

PROBLEM SOLVING AS PERSUASION ................................................. 103
Problems as Persuasion ................................................................ 108
The Alchemy of Problem-Solving ................................................... 129

CHAPTER SEVEN

GENRES OF TEACHING ..................................................................... 138
Genres of Teaching ..................................................................... 143
Multiple Genres, Multiple Subjects ................................................... 156

CHAPTER EIGHT

DISCURSIV E POSSIBILITIES OR: HOW I LEARNED TO STOP WORRYING

AND LOVE THE ACHIEVEMENT GAP ................................................... 162

REFERENCES .................................................................................... 174

vi

LIST OF TABLES

TABLE 1: FOCAL STUDENTS ................................................................ 24
TABLE 2: SUMMARY OF DATA ............................................................. 26
TABLE 3: MOST FREQUENT CODES ........................................................ 27
TABLE 4: CATEGORIES OF QUESTIONS ................................................... 3O

vii

CHAPTER ONE
INTRODUCTION

In E.B. White’s novel Charlotte’s Web, a gray barnyard spider manages to save
the life of a pig intended for the dinner table. She accomplishes this by writing a series of
words in her web extolling the pig’s virtues. She calls Wilbur “radiant,” “terrific,” and
“humble,” rejecting the rat’s suggestion of “crunchy” for obvious reasons. The farmer
and the fair-going pubic agree that Wilbur is “some pig” — not to be squandered as bacon
— and he lives out his days taking care of Charlotte’s descendants. Of all the characters in
the story, only the farmer’s wife suggests that it is not the subject of the web messages
who is remarkable, but the author.

In this dissertation, I am concerned with the spider. I am not interested in whether
Wilbur is an unusual pig, but in how he came to be seen as an unusual pig in a particular
time and place; or, laying the analogy aside, I am not so much interested in elementary
students themselves, as with texts produced about them. To muddle things up even more,
I am interested in these texts not for what they can tell us about students, but for what
they can help us understand about preservice education. This dissertation is designed to
explore how students are constructed -- through problems, through teaching, and through
conversations about them -- and to consider what these constructions tell us about the
way we currently do preservice education in elementary mathematics.

This is a different way of thinking about the problem of preservice education
because the literature in this area has tended to focus on the deficiencies of individual
teachers. That is, it has mostly identified the work of preservice education as finding and
remediating the problematic beliefs and inadequate knowledge of individual beginning

1

teachers, while research on children engaged in elementary mathematics has generally
been seen as a separate body of work (often produced by psychologists, rather than
teacher educators). This dissertation seeks to explore the work of preservice education by
drawing the lines around it differently. First, it uses discourse as a theoretical frame,
which locates practices like teaching, problem-solving and understanding mathematics,
outside of individual heads and disperses them into multiple, but always interacting,
fields. Second, it uses children’s interactions with elementary mathematics as a catalyst
for thinking in new ways about how we might educate those who teach them.

To explore these issues, I spent a year in a third-grade classroom, primarily during
math class. In addition to spending time with the students, teacher and student teacher in
this urban elementary school, I also observed the elementary mathematics methods
course that the student teacher attended on Thursdays during the fall semester of the
school year. The course was taught by an experienced mathematics educator with a
background in mathematics. Throughout this dissertation, I draw on my experiences in
these two classrooms, as well as on relevant written documents, such as curricula, state
standards, and academic writing in mathematics education.

In particular, this dissertation is intended to respond to two prominent strands
within the conversation about preservice education in elementary mathematics: those
about equity and those about reform-oriented teaching. Currently, prospective teachers
are expected to develop knowledge, skills and dispositions that will allow them to teach
mathematics in ways that create learning communities, encourage discussion, and
promote reasoning and problem-solving (Ball & Cohen, 1999; National Council of
Teachers of Mathematics, 1991, 2000; Putnam & Borko, 1997). In addition, beginning

2

teachers are expected to develop understandings of students that will allow them to teach
in ways that work against current disparities in mathematics achievement, which have
been described primarily in relation to race, ethnicity, class and gender (Cochran-Smith,
1999; Gutstein et a1, 2005; NCTM, 2005). Despite wide agreement about the goals of
reform-teaching and equity, we know little about how working toward either of these
goals impacts the other during the teaching of preservice teachers. The NCTM Standards
assume that work toward equity is “interwoven” (p. 12) with the goals of coherent
curricula, learning with understanding and collaborative environments (N CTM, 2000).
Teacher educators may make similar assumptions, but we don’t know if these
assumptions are warranted. Popkewitz (2004) argued that the language of equity in the
Standards documents actually works against the stated goal of helping all children to
learn mathematics because the word “all” does not signify all children, but instead calls
to mind a certain type of child --one who is “disadvantaged,” and therefore unlike the
child who has the “capacities to learn, problem solve, and achieve” (p.23). His argument
raises questions about the role reform teaching plays in defining what equity is in the
teacher education classroom. Similarly, conversations about race, gender or class, which
may be held in methods classrooms to address the goal of equity, may impact the
meaning of reform teaching in mathematics.

For example, in the Principles and Standards for School Mathematics (2000) the
phrase “all students” occurs repeatedly in the section on “The Equity Principle.” On just
one page (p. 13), the report reiterates the phrase several times: “mathematics can and
must be learned by all students,” “all students should have access to an excellent and

equitable mathematics program,” “high expectations for mathematics learning must be

3

 

communicated to all students,” and “schools have an obligation to ensure that all students
participate in a strong instructional program” (italics added). However, other language
around these phrases seems to suggest that the authors are not referring to all students,
but to students with certain characteristics. The report’s synonym for all students is not
“students,” but “students who live in poverty, students who are not native speakers of
English, students with disabilities, females, and many non-white students” (p. 13). The
linking of these categories of students with the concern about helping all students to
understand mathematics works to reinforce the idea that certain kinds of students can be
expected to have difficulty learning mathematics and therefore require extra attention.
The relative absence of the phrase “all students” from the sections of the report
that address various mathematical strands reinforces this interpretation of all students.
The qualifier “all” is dropped in almost all descriptions of students in the sections on
mathematics content. Instead, it says “students must become fluent in arithmetic
computations,” (p. 35), “students should learn to formulate convincing explanations,” (p.
42), and “By the end of the second grade, students should be able to organize and display
their data through both graphical displays and numerical summaries” (p. 109). In these
and similar cases, the single word “students” seems to signify all students in the report’s
discussion of reform teaching. The student constructed through the description of reform
teaching is one who is fluent in computations, can form convincing explanations and can
organize and display data. Students who do not meet these criteria become “all students”
and need some sort of remediation or special attention. Thus, despite the frequent
exhortations to teach all students, the report works against a notion of equity that includes
a recognition that students come to school with varied ways of understanding and acting

4

 

on mathematics. If “students should be able to organize and display their data” or
“formulate convincing explanations,” those who cannot are inadequate. By emphasizing
what students should be able to do, the report works to create two kinds of students --
those who meet reform expectations and those who do not. The way educators talk about
certain students’ failures to meet these expectations and the expectations set out for
students through problems and teaching situations are topics that will be explored in the
dissertation.

Looking closely at how the subject of the elementary student gets constructed in
both the elementary and the methods contexts will allow me to explore ways that notions
about reform teaching --- which can be similar to or different than those in Standards
documents --- have consequences for how one thinks about providing equitable
educations for students in mathematics. My decision to take a critical stance toward
commonly held notions of equity and reform mathematics is based in part on my
recognition that the audience for this dissertation will be composed primarily of
university teacher educators, many of whom seek to change the attitudes and practices of
novice and expert teachers in relation to ideas of equity and reform-mathematics, while
seeing their own understanding of these ideas as unproblematic. Like Charlotte, who
considered the impact of “crunchy” on her audience, I am choosing to look at equity and
reform critically because I believe for this audience, it is the most educative stance. Were
I talking to teachers, politicians or parents, I might discuss these ideas in a different way.
In particular, I see the critical stance I am adopting as a way of creating openings in the
dense discourses around reform mathematics and equity in teacher education. I see this

move as productive because widespread agreement about goals, methods, and projects

5

necessarily limits innovation and possibility, and I believe that unpredictable innovations
are important to creating opportunities for change. When I am working in K-12 settings, I
often aggressively promote reform mathematics as a way of creating openings in the
dense discourse around traditional, algorithmic conceptions of mathematics. For the same
reason, when talking to teacher educators, I want to be critical of reform mathematics to
create openings for unpredictable innovations in the dense positive discourse around
Standards-based teaching.

In addition to contributing to the conversation about reform mathematics and
equity, 1 would also like this dissertation to respond to current calls for “evidence-based”
or “scientific” education research (National Center for Education Evaluation and
Regional Assistance, 2003, p. 1). Exactly what these calls mean is still under negotiation.
For instance, while the policy statements of some agencies have included only large
studies that use randomized control groups as appropriate (e.g., National Center for
Education Evaluation and Regional Assistance, 2003), the National Research Council
(2002) classified ethnographic as well as experimental designs as evidence-based.
Despite these differences in interpretation, most commentators on these policies have
generally accepted the principle of evidence-based research, although they have argued
simultaneously for a definition that is both clear and inclusive. For instance, in their essay
which leads the Educational Researcher focus issue on the NRC report, F euer, Towne
and Shavelson (2002, p. 4) said “Almost everyone can appreciate intuitively the
advantages of evidence-based policy; it is another matter entirely to make this concept
clear, operational and valid.” Others have argued that evidenced-based policies must
include room for small ethnographic studies (Erickson & Gutierrez, 2002), for the messy

6

ways of researchers who work in chaotic schools rather than orderly labs (Berliner, 2002,
p. 18), and for closer links between researchers and teachers (Burkhardt & Schoenfeld,
2003). While critical of the NRC report in various ways, these authors do not question
whether it is truly valuable to make a research in education “scientific.”

In the same focus issue, St. Pierre (2002, p. 27) offered a somewhat strident --
“The NRC report should scare us all to death.” -- response to the calls for evidence-based
research. Drawing on postmodern philosophies, she wrote “This latest attempt to
marginalize certain epistemologies and methodologies in order to control science, to
reduce it, to center it, cannot go unanswered” (italics in original). Her argument was that
science ought to include educational research that is not evidence-based, generalizable or
methods-dependent. I want to make a slightly different move in this dissertation, although
one that also draws on postmodern thinkers, by ceding the right to claim my work as
social science. Unlike the researchers cited above, I am not trying to broaden the
government definition of science, but to see what it might be possible to learn about
children and teaching by playing another sort of game.

F endler (2006) offered one possible alternative to social science as the means of
understanding the work of teaching when she suggested that the field of rhetoric could
provide useful tools for analyzing interactions in classrooms. Following this tradition,
teaching can be seen as “a kind of persuasion, or perhaps a way of moving people” (p. 2)
and analysis can examine the techniques used in this act of persuasion, including
beautiful language, shared assumptions, and available genres (discussion, lecture,
groupwork, etc.). In an essay on rhetoric as a research perspective, Leff (1 987, p. 24)

wrote:

(Rhetoric) implies a pluralism in which methods of inquiry and argument are
adapted to the particular subject under investigation. It seeks to solve

situated problems rather than to formulate abstract theoretical principles.
Explanation replaces prediction as the standard for verifying arguments. And such
verification depends on agreement within the community of those concerned
about the subject, not upon a process of matching evidence against a disinterested

criterion of proof. All abstract categories are treated with suspicion.

On first reading, this stance may not seem so different from those taken by many
qualitative researchers. For instance, Erickson (1986) described interpretive research as
skeptical of predictions, focused on the local, and methodologically diverse. However
rhetorical analyses differ from qualitative ones in important ways. Most significant may
be differences in claims made about the kind of work being undertaken. Ethnographers
collect data as evidence for arguments they will make about how people operate in social
situations. Although most ethnographers now acknowledge that the way they are
positioned in the world as human beings and as researchers impacts the way that data is
collected and interpreted, many also argue that careful attention to methods can produce
“findings” that reveal (at the very least, situational,) truths about the world (Bogdan &
Biklen, 2003; Emerson, Fretz & Shaw, 1995; Erickson, 1986). By framing teaching as
persuasion, or as an argument, rhetoric makes different claims about the work being done
by researchers. Rather than collecting evidence about the real world, a rhetorical analysis
identifies arguments and seeks to make explicit in a variety of ways the strategies used to
make these arguments.

Aristotle, one of the original rhetoricians, argued that persuasive power depends
on the character of the speaker and the perspectives of the audience, as well as on the

8

 

quality of the argument. Throughout the Rhetoric, Aristotle used examples to show that
decision-making (and therefore persuasion) is intensely personal. He wrote that blinding
a one-eyed‘man is worse than blinding a man with two eyes because the first man “has
been robbed of what he dearly prized” (1365b, In 17-18). This is not an argument based
on logic; it is an emotional one. Because the eye was valued more, desired more, prized
more, destroying it was worse. In this dissertation, I will draw on a handful of rhetorical
tools — persuasion, genre, and metaphor — to make arguments about the elementary and
methods classrooms I observed. To be persuasive to readers, Aristotle argued that I will
have to be able to “(1) to reason logically, (2) to understand human character and
goodness in their various forms, and (3) to understand the emotions-that is, to name them
and describe them, to know their causes and the way in which they are excit ” (1356a,
In 16-19). As you read this dissertation, I hope you will frnd the logical leaps I make
sensible; the ways in which I describe the human characters to be considerate and my
mobilization of texts honest; and the text, occasionally worthy of inspiring emotion,
including both anger and laughter. These are the means by which I seek to persuade you.
I am not — except for the second chapter, which I will discuss in a moment — asking that
you believe me because I followed standard ethnographic methods. I do not expect that
you or anyone else would be able to replicate this study, follow my ethnographic path, or
reach identical conclusions. In other words, I am not mobilizing the conventions of social
science in order to persuade you.

Poster-child for the postmodern Michel Foucault said he was not interested in
efforts to make Marxism or psychoanalysis into science. Instead, he asked “What types of
knowledge do you want to disqualify in the very instant of your demand: ‘15 it a

9

science?” (Foucault, 1980, p. 85). It is not just supporters of the No Child Left Behind
policy who are seeking to disqualify certain kinds of knowledge, but also those who
claim that ethnography or critical race theory are scientific. The insistence that one form
of knowledge production is valid always carries with it the shadow claim that another
form is not. Foucault’s work intentionally focused on these forms of disqualified
knowledge. He called this work genealogy, which he described as entertaining “the
claims to attention of local, discontinuous, disqualified, illegitimate knowledges against
the claims of a unitary body of theory which would filter, hierarchies and order them in
the name of some true knowledge and some arbitrary idea of what constitutes a science
and its objects” (Foucault, 1980, p. 83). In other words, the emphasis in genealogy is not
on what is true, but on how what we now know as true has come to be produced.
Examining truth as production requires attention to local, discontinuous and disqualified
knowledges because taking seriously these shifiing, plausible, overlooked alternatives
provides a way of seeing the rules of our own time and place as mutable, rather than
fixed.

I began my writing of this dissertation from within commonly accepted
ethnographic traditions. I collected data, coded and analyzed it, and made assertions. The
second chapter, which examines questioning practices in an urban third-grade classroom,
is the result of this work. However, I felt deeply uncomfortable about the ways in which
this chapter positioned the children in the classroom and myself as a researcher,
particularly in regard to issues of race and equity. The conclusions of the chapter — that
some children had more trouble than others answering open-ended questions (and I bet

you can guess which children) — seemed preordained, and my own work, despite its talk

10

of equity, seemed to reinforce long-standing assumptions. I turned toward rhetorical and
genealogical analyses in the following chapters as a way of unsettling the predictable
outcomes that the methodological traditions of social science (fieldnotes, data, evidence,)
and the accompanying theoretical constructs (culture, beliefs, learning) seemed to
produce. Despite this turn toward genealogy, I included this ethnographic chapter in the
dissertation for a couple of reasons. First, it provides a record of where my thinking
began and offers a contrast with the genealogical/rhetorical chapters that follow. This
serves a pedagogical purpose because the ways in which equity and methods of inquiry
are approached in this chapter differ from those that follow. For instance, in the
ethnographic chapter equity is coupled with the notion of culturally relevant pedagogy,
and therefore, the expectation that attending to the race, gender and economic status of
children is central to teaching them. Also, the ethnographic chapter relies on data, coding,
and assertions, while the following chapters make arguments without mobilizing these
terms. This contrast allows the reader to think about what he or she finds persuasive and
why and opens the possibility for thinking about what non-scientific modes of inquiry
might offer. A second reason for including this chapter, despite my reservations, is that I
see it as essential to my entry into the research community. Ethnographic methods (unlike
rhetorical and genealogical modes of inquiry) are widely understood and accepted by
educational researchers. As a novice, I wanted to demonstrate my mastery of these
methods before adopting a critical stance toward ethnography. In this way, the chapter
serves an important rhetorical purpose.

The majority of this dissertation is located within rhetorical and genealogical
traditions, rather than ethnographic ones, because these traditions create possibilities for

11

unpredictable outcomes. E.B. White’s Wilbur was the runt of the litter. Charlotte’s
response to his imminent death could have been to study the situation as it was and then
to work on remediating a pig, whose test scores for weight and height were well below
average, so that he might eventually have been seen as worthy of an award at the fair.
Instead, she thought about how truths are produced in the human world and deployed this
knowledge to come up with a solution that could not have been imagined based on all the
data collected up to the moment of her writing. In similar ways, my hope is that this
dissertation will create unpredictable openings for myself and my readers to act in our
own lives by both analyzing the problems of teacher education in elementary
mathematics in new ways and by demonstrating the possibilities for understanding the
work of education offered by non-scientific ways of knowing. To do this, I organized the
dissertation in the following way:

The second chapter is an ethnographic look at teacher questioning in the
elementary classroom I observed. I include this chapter both to discuss the ways that
different kinds of questions include and exclude students, but also to make it possible for
readers to contrast ethnographic and rhetorical/genealogical approaches to studying
teaching and learning.

The third chapter is a short follow-up to the ethnographic chapter, which
examines the ways in which I have come to see standard ethnographic arguments as
problematic and discusses my own postmodern turn.

The fourth chapter discusses in detail Foucault’s notion of discourse and its
relationship to research in mathematics education. This chapter lays the groundwork for

the following three chapters, which use rhetorical tools to pursue genealogical questions.

12

unpredictable outcomes. E.B. White’s Wilbur was the runt of the litter. Charlotte’s
response to his imminent death could have been to study the situation as it was and then
to work on remediating a pig, whose test scores for weight and height were well below
average, so that he might eventually have been seen as worthy of an award at the fair.
Instead, she thought about how truths are produced in the human world and deployed this
knowledge to come up with a solution that could not have been imagined based on all the
data collected up to the moment of her writing. In similar ways, my hope is that this
dissertation will create unpredictable openings for myself and my readers to act in our
own lives by both analyzing the problems of teacher education in elementary
mathematics in new ways and by demonstrating the possibilities for understanding the
work of education offered by non-scientific ways of knowing. To do this, I organized the
dissertation in the following way:

The second chapter is an ethnographic look at teacher questioning in the
elementary classroom I observed. I include this chapter both to discuss the ways that
different kinds of questions include and exclude students, but also to make it possible for
readers to contrast ethnographic and rhetorical/ genealogical approaches to studying
teaching and learning.

The third chapter is a short follow-up to the ethnographic chapter, which
examines the ways in which I have come to see standard ethnographic arguments as
problematic and discusses my own postmodern turn.

The fourth chapter discusses in detail Foucault’s notion of discourse and its
relationship to research in mathematics education. This chapter lays the groundwork for

the following three chapters, which use rhetorical tools to pursue genealogical questions.

12

Each of these chapters seeks to examine a commonly accepted notion —— such as, the belief
that process problems are better than algorithmic problems — in a way that makes the
concept seem less inevitable, natural and right. The common goal of each of these
chapters is to create more possibilities for teachers, children, and researchers. Although
each chapter is quite different, certain rhetorical maneuvers can be seen across the
chapters, such as the tracing of ideas through discourse broadly in embodied and written
texts and the avoidance of charts and categories. These strategies can be read as coming
out of rhetorical and genealogical traditions.

In particular, the fifth chapter addresses the issue of equity by examining through
the lens of metaphor, rather than belief, comments made about elementary students.

The sixth chapter addresses reform mathematics through the examination of the
mathematics problems used in the elementary classroom, the teacher education
classroom, and two elementary curricula, and focuses on the ways in which different
kinds of students are constructed by different kinds of problems.

The seventh chapter simultaneously addresses reform mathematics and equity
through a description and analysis of the genres of teaching present in the elementary
classroom, focusing on the opportunities for student participation in each genre.

The final chapter returns to the problems of reform teaching, equity and evidence—
based education to re-examine these issues in light of the arguments presented in the

dissertation.

l3

CHAPTER TWO
HOW MIGHT EXPLICIT QUESTIONING SUPPORT THE LEARNING OF ALL

CHILDREN IN MATHEMATICS?

Over the last two decades, most mathematics educators have moved toward a
shared belief that teacher talk which is open-ended, probing and neutral toward right and
wrong answers offers more opportunities for students to learn significant mathematics
than talk which is narrow, directive, and authoritative (F orrnan & Ansell, 2001; Lampert,
2001; Moyer & Milewicz, 2002; National Council of Teachers of Mathematics, 1991;
2000; Simon & Shifter, 1991). More recently, educators concerned with equity issues
have begun to question whether this more open teaching style, which I will call reform
teaching, is effective for all children, particularly for children who belong to minority
groups, who speak English as a second language, and who are poor (Ball, Goffney &
Bass, 2005; Lubienski, 2000). The purpose of this chapter is to explore one intersection
of reform teaching and equity issues by examining teacher questioning practices in a
reform-oriented elementary classroom located in an urban school with a diverse student
body.

The NCTM Standards documents (199]; 2000), which have played a major role
in shaping the shared conception of reform teaching, promote indirect teaching practices,
such as encouraging student discussion, asking for justification, and providing time for
exploration of challenging problems, rather than direct teaching practices, such as
lecturing, modeling, and drilling of correct procedures. These indirect teaching practices

are seen as strategies that promote process skills, such as reasoning, communicating and

14

problem solving, in addition to content knowledge of mathematical strands such as
number, geometry and data. Throughout this chapter I will use the word reform to
describe teaching that includes process skills as well as content and the word traditional
to describe teaching that is more narrowly focused on content. Many have argued that
reform practices lead to deeper learning of mathematics by all children (e.g., Blanton,
2002; Cobb et. al., 1997; Heaton, 2000; Lampert, 1990; Van de Walle, 2004; Yackel &
Hanna, 2003), and others have claimed that these practices are particularly important for
children who belong to groups who traditionally have not been successful in mathematics
classrooms (Davidson & Hammerrnan, 1993; Ladson-Billings, 1997; Moses & Cobb,
2001; Trafion & Claus, 1994; White, 2000). For instance, Habennan (1991, p. 290)
contrasted practices like the offering of challenging problems and the asking of open-
ended questions with the “pedagogy of poverty,” which “emphasizes repetition; drill;
convergent right-answer thinking; and predictability.”

In contrast, others have suggested that these reform practices, which may be
PrOductive for majority, middle-class children, may not best serve children from non-
dominant cultures (Ball, Goffney & Bass, 2005; Lubienski, 2000). Both inside and
outside of mathematics education, many have argued that children excluded from the
dominant culture need explicit instruction in content and practices that may be automatic
for Others. In her book on literacy, Delpit (1995) argued that children who are not taught
“the codes of power” (p. xvi) at home need direct instruction about these codes, which,
among other social practices, include ways of speaking and listening, ways of behaving,
and attitudes toward adults. Although Delpit has sometimes been cited as an advocate of

Skills—based teaching over reform teaching, her argument is more subtle. Calling the

15

opposition between reform and traditional teaching a “false dichotomy,” (p. 46) she
advocated a middle ground where children engage in work that promotes creativity and
critical thinking as well as in work that makes the standards of the larger community
explicit. Researchers in mathematics education have just begun to explore what this
middle ground might look like. In particular, Boaler (2002) has suggested that the fact
that some reform practices may disadvantage some students does not necessarily mean
that traditional practices would provide more learning opportunities for these same
students. Instead, she suggested that “the differences between equitable and inequitable
teaching lie within the different methods commonly discussed by researchers” (p. 240). In
her work, she identified several practices -- such as making real world contexts accessible
and teaching children to explain and justify -- that made mathematical discussions
productive for children in an urban school. She recommended that researchers continue to
explore ways that teachers can enact practices such as mathematical discussions and
problem solving with an eye toward variations that promote equity. This chapter seeks to
build on this work by exploring the ways that various kinds of questions can support
students in learning significant mathematics.
Research on teachers’ questioning practices has tended to emphasize moves away
fi‘om direct, single-answer questions toward open, probing questions (Lampert, 2001;
MCCrone, 2005; NCTM, 1991; Vacc, 1993). For example, Lampert (2001) talked about
beginning lessons with questions like “Okay, who has something to say about A?” as a
Way of teaching students “that mathematical talk can have a broad range, and not just be
about right and wrong answers to teachers’ questions” (p. 145). The 1991 NCTM

StE’llidards document presents a list of similar questions to Lampert’s as examples of good

16

questioning, including questions like: “How did you think about the problem?” and

“Does that make sense?” (N CTM, 1991, pp. 3-4). In addition to promoting reasoning and
justification, these questions can also be considered implicit, rather than explicit. That is,
not only are a wide range of answers possible, but a wide range of kinds of answers are
possible. As Lampert said, students could respond to her question of “Who has something
to say about A?” by offering an answer, providing a description of their thought
processes, asking a question, or evaluating the problem in some way. The question itself
does not provide clues about the kind of answer desired by the teacher. Thus, it can be
considered an implicit question, as opposed to an explicit one which does provide clues
about the kind of answer the teacher expects.

Research on questioning has not paid a great deal of attention to the ways that
context or differences in students might impact what is considered competent questioning
(Carpenter et a1, 1999; Mewbom & Huberty, 1999; Moyer & Milewicz, 2002). In their

study of preservice teachers’ question-asking during diagnostic interviews, Moyer and

Milewicz (2002) describe three hierarchical categories of questioning -- checklisting,

instructing rather than assessing, and probing and follow-up — which they use to evaluate
the questioning skills of the preservice teachers they studied. Their focus remained tightly
on the preservice teachers during the interviews, rather than on the ways that different
Stl-ldents responded to questions. Similarly, in their discussions of questioning, Carpenter
311d his colleagues (1999) emphasized the skills teachers must have in choosing problems
and selecting open-ended questions, rather than the skills teachers must have in adapting
me ir instructional practices to meet diverse needs in their classrooms. The current study

Seeks to re-examine teacher questioning in reform classrooms in light of equity concerns

17

about the need for explicit language and direct instruction by keeping both the teacher
and the students in focus. To do this, I asked the following research questions:

0 What kinds of questions do teachers ask in a reform-oriented elementary

classroom located in a minority-maj ority urban school?

What learning opportunities do various kinds of questions make possible for

students?
Theoretical fi'amework: Talking about race and language

This chapter is framed by the lens of culturally relevant mathematics pedagogy
(Gutstein, Lipman, Hernandez & de los Reyes, 1997; Ladson-Billings, 1995; 1997),
which asks researchers to attend to ways that students’ racial and cultural identities
interact with their learning of mathematics. Ladson-Billings (1995) described culturally
relevant pedagogy as teaching that was committed to developing students’ academic
excellence, cultural competence, and critical consciousness. Most relevant to this chapter
is the notion of cultural competence. Students of different cultural backgrounds and
language practices are likely to respond in different ways to classroom language and
i nstructional practices (Cazden, 1988; Dyson, 1997; 2003; Heath, 1983), and, therefore,
are likely to feel different degrees of cultural competence in the classroom. Ladson-
Billings (1997) argued that European American, middle-class students are more likely to
feel culturally competent in mathematics because current teaching practices tend to draw
on familiar cultural values, such as “efficiency, consensus, abstraction and rationality” (p

699) while many African American students might feel more competent working in

(:1 assroom cultures that emphasize other values, such as “orality,” “communalism,” and

“movement” (p. 700).
l 8

In this chapter, I paid attention to race in the analysis of student-teacher
interactions in a reform-oriented classroom because I wanted to make it possible to think
about ways that questioning practices might advantage some students and disadvantage
others. I am aware that pursuing this inquiry is dangerous work. Other researchers with
equity-oriented intentions have ended up in problematic places as a result of their efforts
to examine the ways that students of various races respond to language in the classroom.
In her seminal study, Ways with Words, Heath (1983) described differences in the ways
that children positioned differently in terms of race and class responded to the kinds of
questions commonly asked at school. By studying how children interacted with their
parents at home, Heath was able to argue that cultural differences produced these varied
responses and to suggest ways in which teachers could interact more productively with
all of their children. This work was important in challenging cultural deficit models,
where poor and minority students were seen as inadequate in reference to middle-class,
majority students. However, it also inadvertently suggested that race was an essential
quality, which could produce certain kinds of behaviors. More troubling, Orr (1987), who
studied the failure of Afiican American students in mathematics and science at a private,
progressive school, concluded that Black Vernacular English prevented students fi'om
thinking in mathematical ways. She suggested that this dialect, used by many Afiican-
Ammcan students in the school, did not have adequate vocabulary or grammar structures
to Support high-level quantitative thinking. Although this work was harshly critiqued by
linguists, in particular Baugh (1988, 1994), for both the quality of the scholarship and the
deficit-oriented stance, researchers have cited it (and continue to cite it) as part of the

useful literature on language in the mathematics classroom (e.g. Ball, 1990; Ma &
19

Kishor, 1997). My challenge in this chapter was to write about the ways that the African
American and Asian American students in my study responded to various questions
without (as much as is possible) either essentializing race and ethnicity or suggesting that
the ways that students responded were in some way inadequate.

I do not want to treat race, even implicitly, as an independent variable. That is, I
do not assume that there is a causal relationship between a child’s racial identity and his
or her performance in the mathematics classroom. However, I do want to acknowledge
the relevance of race in classroom interactions. In socio-cultural ethnographic research,
human interactions are seen as causal, rather than traits such as race, gender or
mathematical ability. Omi and Winant (2004) wrote about this as “the performative
aspect of race” (p. 10), where race is seen as continually acted out in various ways by
various people. Thus, race is understood as performed rather than as a fixed biological

category; however, “the enormous number of effects race thinking (and race acting) have
produ ” (p. 9) are possible objects of study.

This is the perspective I adopt as I work to describe classroom interactions

between European American teachers and the (mostly) minority children in their

Classroom. In doing so, I acknowledge that race is tied up in other features of classroom
life that may be important to understanding these classroom interactions. Race is highly
Visible (to American eyes); however, children’s differential knowledge of the rules for
Participation in classrooms may be more important to understanding instructional
interactions, but harder to see. For instance, Cazden (1988) argued that children’s
familiarity with the ways of speaking expected by teachers impacted how successful
ellildren were in participating in classroom conversations. These ways of speaking, which

20

can be thought of as registers, include word choice, timing, and syntax. Children who
speak in appropriate registers are more likely to be heard and responded positively to by
teachers. This is not to say that children’s ways of speaking are unrelated to their race and
culture, but that these ways of speaking may be one way of performing race and that it
may be as important to pay attention to patterns of language as to other performances of
race and culture (such as experiences at home, notions of appropriate adult-child
relationships, notions of self and other, etc.). My major concern in this chapter is to
explore the ways that various kinds of questions provided and constrained learning
opportunities for the particular children I studied, rather than to detail a pedagogy for
African American or other minority students. For this reason, I attend to the ways that
language was used in the classroom; however, because of the effects of “race thinking”
and “race acting” in our schools, I do not want to make race invisible.
Methods
Interpretive, ethnographic traditions emphasize close attention to the
particularities of human interactions within specific, and complicated, contexts. Erickson
( l 986) said that one of the benefits of ethnographic work is that it acts against the
“invisibility of everyday life” by “making the familiar strange and interesting” (p. 121).
These methods allowed me to closely examine the ways teachers spoke to children during
l‘ef‘tzbrm—oriented practices and to consider unfamiliar interpretations of their words — for
instance, that an explicit question promoted more rather than less mathematical thinking.
AS a researcher, I attended to the speech of participants, their interactions with each
other, and their use (and non-use) of mathematical texts and resources.

89 fling and Participants

2]

Blythe Elementary is a small K-5 school serving about 300 children. Like many
urban schools, Blythe’s student population is minority-majority. During the year of the
study, about 60 percent of its students were African-American and about 20 percent were
European-American. Asian American and Hispanic students made up nearly all of the
remaining 20 percent. About two-thirds of Blythe’s students received fi'ee- or reduced-
lunch during the year data was collected. The demographics of the class I studied
mirrored the population of the school as a whole. Diana Emerson, the teacher in the third-
grade class where my study was situated, had a great deal of experience, both as a

classroom teacher and as a part of a university teacher education program. She had been
teaching at Blythe Elementary for nearly 30 years; and throughout that time, she
participated in university study groups about mathematics, and welcomed student
teachers, university undergraduates and faculty into her classroom. Sara, the intern
assigned to Diana during the year of my study, took a year off in between her graduation
arid her internship, which she spent working for an afier-school program in another city
in the same state. Sara was described by former professors as a strong student who was
Committed to teaching. Both Sara and Diana were European American, as were the five
education undergraduates who visited Diana’s classroom one morning a week to teach
problem-solving lessons to small groups as part of their coursework.

Diana’s classroom was large, bright and well-stocked, although the carpet, ceiling
and visible walls revealed the age of the building, which had just celebrated its 40th
armiversary. The seating arrangement changed throughout the year, but student desks
were always arranged in groups, angled so students could see the board and overhead
screen at the front of the room. Shelves stuffed with books, math manipulatives and

22

science supplies lined the walls. Large desks for Diana and Sara occupied the back of the
room. One bulletin board ran along a side wall and typically displayed art work by each
student in the class, including self portraits, drawings of the future, and attempts to
represent the styles of various artists.

The year of the study was the first in the district’s adoption of a new elementary
mathematics curriculum, Math Advantage, by Harcourt School (Andrews et al, 2004).
Diana received 300-plus-page math books for each student, which sat on a shelf in the
back of the room for the first six weeks of the year until Diana decided she needed to pass
them out to avoid attracting the attention of her principal or other school district visitors.

These books were used for the first time in January during Sara’s lead teaching. In
addition to student books and teachers’ guides, Diana received a student manipulatives
kit for each child, which contained materials such as place value blocks, dice and rulers.
She did not ever pass these out, preferring to use the class sets of similar supplies she
already owned. In addition to the Math Advantage curriculum, Diana owned a complete
third-grade set of the Investigations curriculum by TERC, which both she and Sara
referred to occasionally.

To guide my observations of the mathematics classroom for the dissertation, I
chose five focal students in Diana’s classroom as an “anchor points” (Dyson & Genishi,
2005 , p. 49) for my attention. I alternated observations among my focal students, which
allowed me to see the classroom from multiple perspectives. When observing students, I
tyI’icfillly sat next to them or behind them. I took notes in a notepad on my lap and placed

my tiilae recorder in a visible but discrete location, such as on an empty table nearby.

Bee ause I wanted to observe diverse experiences in the classroom, I purposefully chose

23

focal students who were situated differently in terms of ethnicity, gender, perceived

mathematical ability, frequency of participation in whole-class conversations, and

physical location in the classroom. The chart below describes my five focal students.

Table 1:

Focal Students

 

Name

Race

Gender

Additional
Information

 

Aliah

 

Bi-racial: African
American/European
American

Female

Aliah left the
classroom twice a
week to attend the
district’s gifted and
talented program.
She spoke rarely in
whole-group
settings.

 

Caitlin

African American

Female

Caitlin spoke
infrequently in
whole-group
conversations, but
was often an
enthusiastic
participant in small-
group settings.
Diana described her

 

 

European American

Male

as “trying hard.”
Ben spoke
occasionally in
whole-group
discussions and
tended to work by
himself in small-
group settings. He
was described as an
excellent math
student by Diana,
Sara, and many of
the children.

 

 

African American

Male

 

 

Jerome participated
occasionally in
whole-group
discussions and
preferred to work

 

24

 

 

alone in small-group
settings. He was
described as
“struggling” by
Diana.

Marcus African American Male Marcus participated
fiequently in whole-
class discussions as
well as in small-
group settings.
Marcus was seen as

a good, although not
exemplary, student.

 

 

 

 

 

 

The class as a whole had 19 students for most of the year, including nine boys and
ten girls. Of these children, ten were identified as African American on enrollment forms;
four were identified as European American; three were identified as bi-racial; and two as
Asian American. Both Asian students were Hmong. One of these students, Mia, was the

only student in the class currently learning English as a second language, although
Charlie, the other Hmong student, spoke fluent English as his second language.
I decided to use the above chart as a rhetorical move, recognizing that readers

may need a quick way of keeping track of the students I spent an academic year getting to

know. However, it is important to note that the clean lines and categories of this chart are
pro blematic. Although it is not my intention, labeling Jerome as African-American, male
and Struggling works to make each of these categories stable (and worse yet, works to
conflate them). One of the challenges of this chapter was the struggle to write about
ethllicity, gender and ability in ways that recognized the role these categories played in
the life of the classroom while simultaneously working toward a theoretical stance that

reco gnizes these categories as continually performed rather than fixed.

25

 

 

Data Collection

At the end of September, I started weekly observations of Diana’s classroom

during math time. I visited weekly until Sara left the classroom at the end of April,

alternating observations among focal children in the classroom. I audio taped

observations and wrote fieldnotes immediately afterward, following ethnographic

traditions (Emerson, Fretz & Shaw, 1995; Erickson, 1986). I made copies of assigned

student work, assigned pages of the textbook, and pages of teacher’s guides, standards

documents and assessments that were referenced. I also took notes on what was written

on the board or the overhead. In addition, I took notes about the physical setting of the

classroom each week. The chart below summarizes data I collected.

Table 2: Summary of Data

 

 

 

 

 

 

[ Type Quantity
Observations of Elementary 22 observations of math classes
Classroom Breakdown of observations of focal students: Aliah 3;
Ben, 4; Caitlin, 6; Jerome, 5; Marcus, 4.
1 observation of science
2 observations of literacy
1 observation of recess
[Lesson Plans Copies of 4 months of plans from Diana
Copies of 6 months of plans from Sara
Interviews 5 informal interviews of Sara
3 informal interviews of Diana
1 formal interview of Sara and Diana together
Mathematics Curricula Mathematics Advantage by Hartcourt School, Third

Grade student book and teachers’ guide
Investigations by TERC, teachers’ guide

 

State Standards for
Mntary Mathematics

Grade Level Content Expectations by the Michigan State
Department of Education

 

Student work

 

15 pages of journal entries by each elementary focal child
2 classroom assignments by every elementary student in
2 classroom assignments by each elementary focal child

1 methods assignment by each student

3 lesson plans for methods class by Sara

 

26

 

Data Analysis

When I completed data collection, I began analysis by using open coding

(Emerson, F retz & Shaw, 1995; Erickson, 1986) on my fieldnotes. I developed several

major categories based on this coding. The most relevant for this chapter was “genres of

teaching,” which included whole-class mathematical discussions, small group work,

practice sessions, whole-class question-answer sessions and teacher-child individual

work. For the purposes of this chapter, I focused on episodes of teacher-child individual

work and whole-class discussions because these two genres allowed me to look at

practices with opportunities for reform-oriented learning, such as mathematical

discussions and problem solving, and at teacher questions. I coded episodes within these

two genres from across the year for kinds of questions asked, types of student responses,

and evidence of learning, and used this analysis to develop the assertions in this chapter.

The chart below lists the most frequent codes I used for the analysis; the meanings of

many of these codes will be explored in the following section of the chapter.

Table 3: Most Frequent Codes

 

What kinds of questions do teachers use in their
work with individual children in a reform-oriented
elementary classroom located in a minority-majority
urban school?

  
   

Explicit question, Single-answer
question, Multi-purpose question,
Implicit question, Reasoning
question, Analytic question,
Teaching question

 

 
   

What learning opportunities do various kinds of
questions make possible for students?

 

Student confused, student solves
problem, student makes
meaningful mathematical
statement, student continues to
work, student frustrated, student
gives up

 

27

 

When coding the data, I sometimes assigned multiple codes to a single segment of data.
For instance, a question might have been both “implicit” and “analytic.” After coding the
data, I sorted chunks of data by code. For example, I took all of episodes where explicit
questions were asked and compared them across students, kinds of tasks, and types of
responses. I made notes on key similarities and differences among several chunks of
similarly coded data and used these notes to write this chapter.
Researcher Role
At one time or another, I have occupied the roles of each of my participants. I
have been a third-grade student, a third-grade teacher, and a preservice teacher. The result
of this is both that I have some insight into (and some blind spots about) the challenges
and perspectives of each of these roles and that my own experiences shaped what I saw
during my observations. As a child, I attended a large integrated school in Milwaukee.
The school’s population was almost evenly split between neighborhood children, who
were mostly working-class and African American, and children bussed in, who were
mostly middle-class and European American (myself included). In junior high, I moved
to a wealthy, nearly all-white suburb, an experience which made both social class and
race visible for me. As an elementary teacher, I worked only in minority-majority schools
in both suburban and urban areas. In each of these roles, I’ve struggled with mathematics,
finding it distasteful as a child, boring as an adolescent, and sometimes mystifying as an
ed Llcator. I chose to work in mathematics not because I have found it easy, but because I
have found it difficult. My own struggles bring me insight into the struggles of students

and remind me of what it feels like to be an outsider.

28

As I approached my participants, I found myself drawing on each of these roles,
sharing my experiences as a beginning teacher with Diana and Sara and laughing with
students about the difficulty of a certain problem. My history also shaped my
observations. I found myself drawn to students who struggle in mathematics and looked
for undercurrents related to both race and class. Additionally, I found that I had to hold in
check my reactions to teaching. I have strong tendencies to like teaching that resembles
my own and to find teaching that strays too far from my own practice distastefill. My
goal was come in with a real curiosity that would help me to learn about the people in the
room, rather than to hold them up against my own experiences.

Thinking about Kinds of Questions
Initially, I had expected to find two broad categories of questions in my data:
implicit, reform-oriented questions and explicit, traditional questions. That is, I expected
that questions aimed at encouraging students to reason, communicate and problem solve
(reform questions) would be open-ended and unspecific, such as “why?” or “what do you
notice?” I also expected to find explicit, traditional questions -- questions where only one
correct answer was expected, where the teacher clearly signaled what kind of answer she
was expecting, and which focused on mathematical content, rather than process skills.
However during analysis, I found these two categories to be unproductive because they
contained too many disparate ideas. For instance, Diana frequently asked individual
Chil dren: “Do you agree with this solution?” when another child presented an answer to a
prob lem. This was an explicit question. It identified one student who was expected to
reSIDOnd, narrowed the range of appropriate responses, and directed the student’s attention
to eValuating another student’s answer. However, it was also a question that required

29

children to analyze the work of their classmates and, following the norms of this
classroom, to offer a sentence or two about their own reasoning. It was a question that
seemed to be both explicit and reform-oriented. As a result of trying to classify questions
like this one, I decided that I needed to work along two axes: fiom reform to traditional
and from implicit to explicit. Questions that I considered explicit were firmly embedded
in a context from which children could draw possible answers; whereas, implicit
questions required students to decide on a context in which to locate their answers. The

chart below classifies a few exemplar questions that Diana and Sara used in whole-class

 

 

 

lessons.
Table 4: Categories of Questions
L Implicit Explicit
Reform Why? Caitlin, can you say why you

What do you notice about this? disagree with Sienna’s answer?

Why does this make sense? Tell me why you’re adding 32 and

What’s a prediction you could 32.

make? Why would 26 not make any sense

What can you tell me about this? as an answer?

What do you think?

Traditional What do you do to add two-digit What is four groups of two?

numbers with regrouping? What digit is in the one’s place,

If you haven’t memorized your everybody?

facts, what can you do to get the What do we call the name of this

answer? coin?
Okay, in Celsius, what temperature
does water freeze at?

 

 

 

 

The questions in the top row require students to communicate their reasoning
about a problem or to offer justifications for their own thinking, while the questions in the
honour row require students to compute or recite memorized information or to discuss

Strategies for doing so. In some ways, it is difficult to evaluate these questions outside of

30

the classroom context in which they occurred. For instance, the question “If you haven’t
memorized your facts, what can you do to get the answer?” might be considered a reform
question because it offers the possibility for students to reason or justify their thinking.
However, this question was asked as part of the introduction to a timed test with the
stated goal of offering students ways to find answers if they had not memorized their
facts. The purpose was to promote fast computation, not to explore student thinking. I
classified it as implicit because the question did not signal the kinds of responses that
were expected, so some students did offer commentaries on ways they broke apart and
put together numbers before being redirected by the teacher toward test-taking strategies
such as counting on fingers, looking for multiple problems with the same answer, and
starting with easy problems.

Questions in the first column forced students to figure out a way to answer the
question from multiple possibilities. For instance, when asked “What do you notice?”
students might focus on patterns, the way numbers are written, correctness or
incorrectness, or other features of the problem. Similarly, the answer to “What do you do
to add two-digit numbers with regrouping?” might begin with lining up the problem or
with adding the ones column or with an explanation of the regrouping process. In
contrast, questions in the second column, direct students attention much more narrowly.
Rather than saying what she noticed, Caitlin must say whether she agrees or disagrees
with a given answer.

The questions in the reform-implicit box resemble those most often recommended
to teachers as effective questions (e. g., Vacc, 1993; NCTM, 1991) because they are seen
as eliciting higher-level thinking and because teachers of many grade levels and content

31

areas can incorporate them easily into their repertoire. That is, their acontextuality is seen
as a strength because these questions can be “good questions” in many kinds of lessons.
Many of the explicit-reform questions were more specific versions of their implicit
counterparts. For instance, instead of simply asking “why,” Diana asked a student to tell
her why he was adding 32 and 32. However, this difference in phrasing was important --
not only because the second question asked children to attend to particular features of the
mathematics being discussed, but also because students in the classroom responded
differently depending on the type of question asked. The following two sections explore
these different responses to both implicit and explicit reform questions. Understanding
how these questions operated in the classroom requires looking at their use in the context
of classroom interactions, which takes space. To provide this space, the role that
traditional questions played in the classroom is not discussed here.
Difi‘iculties in responding to implicit reform questions

By far, implicit reform questions were the most likely to be met with silence when
asked during whole—class lessons. Early in the year, Diana pointed to the numbers 5, 10,
15, 20, 25, 30, 35, 40, 45, 50 on the board asked students to tell her what they noticed. No
raised his or her hand except Ben, a European American child frequently identified as
“good at ma ” by the teachers. After a few moments, Diana remarked “I know more
than one person knows this.” When she made the question more specific by asking
students to focus on the ones place, several students raised their hands. Diana called on
Marcus, an African American child, who said that the ones place would “always be a
zero or a five in that pattern.”

In another lesson, Diana wrote the problem:

32

25
i2

on the board and asked: “What if you had to do this problem without pencil and paper?”
Many students, who had been eagerly raising their hands to answer earlier questions, put
their hands down and stared blankly at Diana. Ben and Charlie, who was Asian
American, raised their hands to answer and Diana remarked: “Honestly, sometimes I
think there’s only two people in here.” With her question, Diana had wanted to direct
students toward practice with mental addition (and in fact when she asked students for
“mental ma ” ways of solving the problem, several children raised their hands).
However, most students did not seem to immediately connect the idea of not having
pencil and paper to mental arithmetic. The question itself left a wide range of responses
possible responses (use a pen, ask to borrow some, get a calculator, etc.) and most
students seemed unsure of Diana’s goal.

These fiequent interactions worked to place a spotlight on Ben who was often in
the position of rescuing the class when no one else would raise his or her hand to speak.
(Incidentally, this was not a position that Ben occupied unwillingly. Typically, Ben
would not raise his hand to answer most questions asked, but would wait until he saw that
no one was willing or able to answer the question.) After one of these interactions in
January, Diana asked the class to listen closely to Ben’s explanation. Then she asked the
class, “Why should you listen to Ben?” Charlie called out: “Because you think he’s
smart.” Jerome added: “You think he’s right.” Diana quickly disagreed, saying that the

others should listen so they could learn to answer these questions as well.

33

At the end of the year, most students were no more likely to answer these kinds of
implicit-reform questions than they were at the beginning. However, many of them had
deeply entrenched beliefs in Ben’s mathematical ability, which were revealed in
interactions such as the one described above and in small group work, where students
routinely deferred to Ben, even when they had correct answers and Ben did not. Just as
these exchanges worked to frame Ben as a good math student, they also worked to frame
other students as struggling, at least in the public discourse. Marcus, an African American
student, routinely solved all problems asked of him, participated frequently in small
groups and quickly mastered his multiplication facts. However, in whole-class
discussions, he often appeared to be struggling when responding to implicit-reform
questions.

At the beginning of a lesson on rounding, Diana wrote the following on the board:

730 700
73 100
703 700

She asked students to write in the journals about what they observed and then
raise their hands. After a few moments, Marcus raised his hand and Diana called
on him.
Marcus: It’s even.
Diana: Why?
Marcus: Odd?
Diana: Why?
Marcus: Even?
Diana: Now they’re back to even. When I say ‘why’ do I mean you’re wrong?
Marcus: No?
Diana: You need to tell me why. If you don’t know, you need to say ‘I just
guessed. I don’t know.’

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Marcus: I don’t know.

Diana drew a number line on the board from O to 800, with slashes at each
hundred. She asked the students to look at the numbers and think about
rounding. Marcus put 73, 730 and 703 in correct places on the number line. He
drew arrows from these numbers to the hundreds numbers they rounded to;
however, he did not raise his hand again when Diana asked students what they
had done.

In this episode, Marcus did not get to demonstrate his mathematical competence
publicly, although he did privately in his journal and Diana made a point of looking at his
work and praising him. Marcus interpreted Diana’s “why” as a cue that his answer was
wrong. Diana expressly denied that this was her intent; however, like many teachers she
was slightly more likely to ask “why” when students answered incorrectly. In this case, it
is not clear if Diana was asking why Marcus said the numbers were even as opposed to
odd; why he chose the category of even/odd as important; or why he chose to focus on
the numbers in the second column (which were all even) as opposed to the numbers in the
first column. Because her question did not narrow possible responses for Marcus, he had
a wide range of interpretations to choose from, including the one he chose: that “why”
was code for wrong. Given his prompt response to the number line, it seems likely that
Marcus could have made a statement about the relationship between the numbers on the
board and the nearest hundred; however, the open question “why” left him struggling
over Diana’s intent, rather than the mathematics.

A conversation like this in the public space has consequences beyond Marcus. For
other students trying to decide whether they too might raise their hands and participate,
Marcus’s eventual admission of “I don’t know” after the confusing even-odd exchange

35

could seem discouraging. Throughout the year, only five students responded to implicit-
reform questions in whole-class discussions, despite the fact that all but two students
participated in whole-class discussions nearly every day when other types of questions
were asked.

It is tempting in looking at this episode to second-guess Diana’s teaching
decisions. However, she was working hard to enact the values of reform mathematics as
she understood them. Marcus opened the episode by saying the numbers were “even.”
This is true of at least one column of the numbers presented and Diana could have agreed
and then led him toward where she was heading with rounding numbers. Instead, she said
“why” because it was her practice to ask this question frequently and because she wanted
to offer Marcus the opportunity to make the connection between rounding and his answer
himself. In this episode, Diana also attempted to teach the children about her use of the
question “why?” by saying that it did not mean she thought the answer was wrong. Diana
frequently made these sorts of commentaries on her teaching strategies; however, many
children did not seem to internalize them. In April, after nearly a year of similar
statements by Diana, Jerome responded to a why question by immediately erasing his
answer.

In analyzing this and other episodes where students had difficulty answering
implicit-reform questions, it seemed that some students — most often, those whose race,
ethnicity, or ways of talking were different from Diana’s -- had to divide their attention
between the mathematics at hand and the interpretation of the language being used.
Sometimes, this interpretive work resulted in amusing miscommunications. For example,
after presenting the following problem:

36

You’re going to the store to buy carpet for a room that is 6 feet by 9 feet. The

perimeter is 30 feet. How much carpet do you need?
Diana asked: “What is most important here?” Charlie, an Asian-American student who
had learned English as his second language, shouted out: “To be polite!” However, more
often, these episodes seemed to leave children confused and, occasionally, the teachers
frustrated.
Supporting mathematical thinking through explicit questions

In contrast to implicit reform questions, explicit reform questions seemed to offer
students opportunities to reason, but also to provide the support necessary for more
students to enter conversations productively. In whole-group discussions, many more
children chose to respond to these kinds of questions and their participation tended to be
seen as far more competent by the teachers. For example, Caitlin, an Afiican American
student who rarely participated in whole-class discussions and never raised her hand to
answer an implicit reform question, spoke with confidence when Diana asked her to say
whether she thought another student’s expansion of 730 as 700 + 30 + l was correct.

Caitlin: No. No, it’s wrong.

Diana: Can you say why you disagree with Sienna?

Caitlin: There’s an extra one. 700 plus 30 is 730. There should be a zero, not a
one.

Here Caitlin publicly evaluated another student’s answer and provided a reason for her
correction, which demonstrated knowledge of place value as well as an ability to put her
mathematical thinking into words. Diana’s question focused Caitlin’s attention on the
answer to a particular problem and gave her specific directions about the kind of answer
expected. This confident exchange stood in contrast to a small-group interaction that

37

occurred after Caitlin had solved a problem that asked how many outfits could be made
with four shirts and three pairs of pants. Caitlin drew four shirts and wrote “3 outfits” on
each one. Then she wrote “3 x 4 = 12.” The final prompt on the worksheet said: “Explain
your work.” Caitlin, confused, questioned the student teacher, who replied: “What do you
mean you don’t get it? You already did the whole thing.” Caitlin said: “I know how to get
the answer, but I don’t get how to solve this problem.” For Caitlin, the requirements of
“solve” and “explain your work” were mysterious, and she seemed to classify these
questions as being a kind of mathematical work that she could not do, unlike finding the
answer to particular problems.

Unlike implicit questions such as “Why?” “What do you notice?” and “Explain
your wor ,” explicit reform questions removed ambiguity about the teacher’s purpose in
asking questions, which is what seemed to provide children like Caitlin with the
confidence to join the conversation. A number of specific strategies seemed to be at play
in the asking of these kinds of questions. Often, Diana attached a student’s name to the
question, which worked to invite particular children to speak. Also, these explicit
questions tended to identify what the teacher saw as important in the problem under
discussion (mental math strategies, rounding, Sienna’s answer). These questions also
seemed to follow other questions. That is, rather than opening a discussion, they were
almost always used in the midst of a conversation, which at least some of the time helped
students to think about their answers in terms of what others had said previously,
although sometimes these questions were used to redirect conversations that were not
going where the teacher intended, as Diana did when she directed students to focus on the

one’s place in the pattern episode.

38

Finally, many of these questions allowed students to respond silently as a group
before they committed themselves to speech. Diana frequently asked students to give her
a thumbs up or a thumbs down in response to questions. It is impossible to answer a
question like “why” in this way; however, Diana found ways to encourage students to
think when asking yes and no questions. Examples include: “Is there a way to solve this
problem without regrouping?” and “Did he find all of the arrays?” All children responded
to questions like these (admittedly, because Diana demanded that each student take a
stand). Some children looked carefully around the room before making a decision;
however, Diana frequently called on these students to explain why they had answered yes
or no. These students, buoyed by the knowledge that many of their classmates had
interpreted and answered the question in similar ways, often were able to articulate their
thoughts more clearly than when asked a more open question or when asked a question
that was unexpected.

Jerome, an African American boy whom both Diana and Sara described as
“struggling,” rarely answered questions as expected in whole-group discussions.
However, he seemed to be better able to participate in conversations that began with a
public yes/no question that allowed him to focus in on a specific issue and to feel
relatively confident that he was right, as evidenced by the answers provided by the rest of
the class. Early on in a discussion about a problem that asked students to figure out how
many vegetable pieces had been used in a pot of soup, Jerome shrugged and refiised to
answer when Diana asked him to tell the class something he had done to solve the
problem. However, a few minutes later she asked the class to put their thumbs up or
down to show whether they should add 50 carrots to the total number of vegetable pieces

39

in the soup. After looking around the room and seeing that most students had their
thumbs pointed down, Jerome pointed his down as well.

Diana called on him and asked why they couldn’t just write down 50 in their list.
Jerome: You have to double it! Double it!

Diana: Why do you have to double the carrots?

Jerome: Because it said to cut them in half.

Jerome had solved this problem in his journal before the discussion began; however,
when asked to say “something” about his solution, he had nothing to say. However, when
asked a specific question about what he had done and when given the additional support
of seeing that his classmates agreed, he participated with both competence and
confidence.
Concluding Thoughts

This study contributes to current conversations about whether reform teaching is
best for all children (Ball, Goffney & Bass, 2005; Boaler, 2002; Lubienski, 2000) by
suggesting that explicit questions are not in opposition to reform practices, but a possible
support for them. Open, ambiguous questions such as ‘that do you notice?” may be
appropriate for students who do not have to work to interpret the teachers’ language or
cultural practices but may provide an obstacle for students who must think about
language in addition to mathematics. In addition, students who are not fluent in the
register of school speech, as described by Cazden (1988), may need more explicit clues
about the timing, syntax and content of their responses. Future studies are needed to
identify what it is that allows some students to respond to implicit questions in ways that
are perceived as competent by the teacher. It may be that shared cultural practices
between teachers and students provide these supports or it may be that students’

40

familiarity with the register of school speech, regardless of the race or culture of the
teacher, provides these supports.

This study demonstrated that implicit questions can construct some math students
as successful and others as needing support. In a diverse classroom, overlaps between
culture, dialect and race, may cause teachers and children to make problematic
assumptions about mathematical ability based on students’ responses to implicit
questions. This study does not suggest that students from non-dominant cultures or who
use non-dominant dialects and languages cannot engage in abstract thinking and problem
solving. Rather, it fosters Boaler’s (2002) claim that they way teachers enact instructional
practices is central to whether these practices are equitable or inequitable. Questions that
require students to interpret teachers’ intentions and instructional goals, as implicit
questions do, seem to advantage some students.

In addition, this study also suggests that it would be fruitful to explore the
possibilities of implicit traditional questions (such as “What do you do to add two-digit
numbers with regrouping?) Although it was not the focus of this chapter, questions that
dealt with computation and memorization in ways that asked students to talk about
processes for performing these procedures seemed to operate in productive ways by
causing students to verbalize their thinking. Studying these sorts of questions would
contribute to the problem raised by Star (2005), who argued that we know very little
about what high-quality procedural knowledge looks like, because procedural knowledge
often has been conflated with weak understanding of mathematics.

Teaching mathematics in ways that are culturally responsive requires that
educators think both about the ways that race and culture mi ght be impacting

41

mathe
and u
teach
Thus.
math?

itisp

 

 

mathematics teaching and the ways that mathematics teaching might be impacting race
and culture. In this study, students’ familiarity with the language and practices of the
teacher seemed to have an impact on their willingness and ability to answer questions.
Thus, question answering became a way to “perform” race, which makes it important that
mathematics educators find ways to engage students in mathematical discussions so that

it is possible for all students to have competent public performances.

42

CHAPTER THREE
Down the rabbit hole: My own postmodern turn
“‘Then you should say what you mean,’ the Mad Hatter went on.
‘I do,’ Alice hastily replied; ‘at least — at least I mean what I say — that’s the same thing,
you know.’
‘Not the same thing a bit!’ said the Hatter. ‘Why, you might just as well say that ‘I see

what I eat’ is the same thing as ‘I eat what I see’! ”
-- Lewis Carroll in Alice ’s Adventures in Wonderland

“Q: You were ahead of others.
MF: Ahead of others? Not at all. I was below them.”
-- Michel Foucault in Foucault Live
My early encounters with postmodern thinkers left me feeling quite a bit like
Alice at the tea party: surrounded by people talking nonsense, picking on my language,
and playing games with serious work. Still, I found myself drawn in by the elegant prose,
the humor, and the unfamiliar ideas. The more I read, the deeper into the rabbit hole I
found myself, until eventually, my sensibilities changed and writers up on the surface
began to seem just as mad. The purpose of this chapter is to explore a few ideas in the
warren of postmodern philosophy that I have found meaningful with the goal of
transitioning between the ethnographic stance taken in the previous chapter and the
genealogical approach that will be adopted in the following ones.
Although I write now as a researcher, it was through my teaching that I came to
understand the possibilities of Foucault’s work. I entered graduate school after six years
as an elementary teacher, and in the beginning, I interpreted all writers through this lens.

At first, I was not sure what this French philosopher had to say about my time in the

43

classroom. His discussions of power, knowledge and subjectivity seemed both distant and
obtuse. In one of the first Foucault texts I encountered, I read: “power is neither given,
nor exchanged, nor recovered, but rather exercised, and that it only exists in action”
(1989/1996, p. 89). My reaction to this was something along the lines of “Maybe. But so
what?”

It was Popkewitz (2002) who answered this question for me. In an essay on the
teaching of school subjects, Popkewitz discussed the attention teacher educators have
given to the “urban child,” saying: “Teaching is to rescue the child with low self-esteem,
a division from the unspoken characteristics of high self-esteem. All the rescuings are
paradoxical. The system of reason makes it so that the child can never be of the average”
(p. 266). For me, reading this essay was like walking into a brick wall. The story I had
constructed about myself as a teacher began to unravel, and as I began to tell a new one, I
learned to understand and to value the conception of power Foucault had articulated.

Throughout my teaching career, I taught classrooms of predominantly Afiican
American students, first in a rundown, city school and later in a suburban showpiece. In
both places, I brought passion and certainty to what I did and saw myself as an advocate
for my students, whom I believed were oppressed both by society at large and by the
school in particular. In the first setting, my advocacy manifested itself in a fierce
commitment to the belief that my students could pass the state’s standardized tests. In
pursuit of this goal, I attended workshops, went on home visits, devoted half the day to
literacy, and learned ways to engage in mathematics that went beyond the four most
common algorithms. And many of my children did pass those tests. In my suburban
school, I lost some of my urgency, but still fought the gate-keeping of the gifted program,

44

refused to track my children and chose to work with the remedial classes during
intersessions. I was angry a lot of the time, but the truth is, I enjoyed the fights. I got into
teaching because I was on a mission.

When I read Popkewitz’s description of rescuings as “paradoxical,” it occurred to
me for the first time that my efforts to save my children might not have been wholly
positive. I saw that identifying children as poor, lacking in self-esteem, or without
academic resources emphasized what was “other” about them. Children who need
rescuing begin their school careers seen as deficient, powerless and needy. Popkewitz’s
argument helped me to understand Foucault’s (1989/1996) idea of power as a force that is
not limited to people who have been privileged by race, class or gender but that can be
exercised by nearly all people in nearly all situations. When I looked back on the children
I had taught, I began to redefine them as people who were at all times exercising power in
various ways and were not the helpless victims of their oppressors. Rather than thinking
about how to save them, I had to ask myself why it had been so important to me to teach
from the belief that the children in my classroom needed me to rescue them.

Working with this question caused me to redefine myself as a teacher and a
researcher. Rather than a savior, I now see myself as a story teller, whose words invite
others to recognize the ways in which they and those they interact with exercise power,
particularly in the classroom. I no longer see power as a scarce resource that I must
redistribute with my teaching and research, and as a result, I no longer need to rescue
school children or (to say the same thing in other words) to see them as inadequate in
order to justify my own work. Foucault said “to reveal relations of power is, in my

opinion at any rate, to put them back in the hands of those who exercise them”

45

(1989/ 1996, p. 144). This idea has become central to my redefinition of my work as
something other than a rescue mission. Right now teacher educators act in certain ways —
for instance, they “do” problem solving, promote discussions, and advocate “culturally
relevant” practices. These ways of acting, which can be thought of as technologies, are
means by which teacher educators exercise power. Exploring how and in what
circumstances these technologies are employed is a way of calling attention to power
relations and making it possible for teacher educators, including myself, to think
consciously about how we do our work. The justification for my work is not saving, but
creating openings for change by decreasing the sense that our current ways of acting are
inevitable.

You can’t spend much time reading about Foucault’s conception of power before
you have to come to terms with “power/knowledge.” (Thinking “tens of thousands of
words in the language and he has to make up new ones” is emotionally satisfying but
ultimately unhelpful.) The slash between power and knowledge reminds us not to go
around thinking that these are two separate things. Karl Von Clausewitz famously said
that war is the continuation of politics by other means; Foucault turned this around to say
that politics is the continuation of war by other means.

[I]t implies that the relations of power that function in a society such as ours
essentially rest upon a definite relation of forces that is established at a
determinate, historically specifiable moment, in war and by war. . . . The role of
political power, on this hypothesis, is perpetually to reinscribe this relation
through a form of unspoken warfare; to re-inscribe it in social institutions, in
economic inequalities, in language, in the bodies themselves of each and every
one of us (Foucault, 1980, p. 90).

46

What does this have to do with knowledge? What we know is a way of exercising
power — that is, research-based best practices are ways of reinscribing certain relations of
power within the social institution of the schools. What we know is also a product of the
ways that power has been exercised in the past — that is, we know about the American
Revolution rather than the American Rebellion because of power relations established at
a specific moment “in war and by war.” We know George Washington is a hero and not a
terrorist because of these power relations and each time we tell the story, use a dollar or
view Mount Rushmore, we inscribe the story a bit deeper. All of this means that the ways
in which we speak/think/act create words, categories, and ideas that function as truth. (I
borrowed Foucault’s slashes to make the point here, but he used the word discourse to
cover speech/action/thought, and I will adopt this efficiency from now on.) Recognizing
the conflation of knowledge and power has implications for how we think as researchers
about the creation of knowledge. Following Foucault, truth is not an absolute we uncover,
but a way of exercising power. If theories, facts and interpretations become true through
constant reinscription in discourse, then researchers have ethical obligations both to
analyze the ideas being reinscribed in their own communities and to consider the stories
that their own work will make true. This is a different ethical obligation than that required
by positivist theories, which ask that researchers tell the truth, no matter how unpalatable;
and it is different from the obligations required by modernist critical theories, which
demand that researchers refiame current truths in light of social and political inequities.
Postmodern theories see research as a will-to-truth. This is not so much the production of
knowledge that socio-cultural theorists talk about, but the recognition that all knowledge

is the exercise of power.

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Unlike the relationship between rescue and power, this is an idea that I understood
intellectually before I got it emotionally. It wasn’t until I picked up a Forbes magazine
left on my living room table by my mother-in-law that I began to take seriously the idea
that much of my own writing and teaching was working to reinscribe the truths I believed
I was arguing against. The Forbes piece was a typical right-wing rant about the education
establishment, which ran under the subhead: “The latest preposterous idea in
educationland is ‘closing the achievement gap.’ Educators everywhere are enlisting in the
campaign and somehow not noticing that it can’t possibly succeed” (Seligman, 2005, p.
120). The gist of the argument was that the achievement gap can never be closed because
some kids - those fi'om the middle and upper classes - are simply smarter than other
kids -- those who are “disadvantaged.” The story ran (charmingly) with a drawing of two
children, one with a head full of brains and another with, literally, nothing upstairs. I
went through several stages of vocal rage during my initial reading of this article, which
included a typical left-wing rant about people who conflate biology, culture,
environment, and upbringing. Then, in the calm after the storm, I saw it. It is only
because of liberals like me, who have worked — tirelessly - to produce the achievement
gap that Seligrnan and his colleagues at Forbes have anything to write about.

Just a few months before the Forbes piece, NCTM ran their position on “Closing the

Achievement Gap” (N CTM, 2005, p. 4). In it, they defined the achievement gap as:
disparities among groups of students usually identified (accurately or not) by
racial, ethnic, linguistic, or socio-economic status with respect to a variety of
measures, including attrition and enrollment rates, alienation from school and
society, attitudes toward mathematics, and test scores.

The constant repetition of the phrase “the achievement gap” works to create a real

phenomenon. An ERIC search for documents with the phrase in the abstract turned up

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more than 500 texts; a Google search produced over a million hits. This is reinscription
on its most basic level. Measuring the achievement gap etches this particular relation of
power into our social institutions, our language, and our bodies more deeply. What we
measure (alienation, attitude and math test scores), how we measure (multiple choice,
surveys, short answer) and what we report (scores by race and gender) are technologies
that allow educators to exercise power relations that were established in war and by war.
They remind some individuals of their inferiority (identifiable by “racial, ethnic,
linguistic, or socio-economic status,” [NCTM, 2005, p. 4]) and others of their
accomplishments. In fact, it is because of these relations of power that NCTM can refer
vaguely to racial, ethnic, linguistic and socio-economic categories and we all know who
they’re talking about. (Hint: it’s not upper-class white males.)

These technologies -— the methods of measurement, the content being measured,
and the way scores are (or are not) disaggregated — are not innocent, neutral or natural;
they do not simply measure what is true; they produce it. Typically, scores are not
reported by income-level, educational attainment of parents, hair color or height. We
choose which categories to make important. In one study on stereotype threat,
researchers told European American males that Asian Americans typically outperform
whites in mathematics. Researchers then gave these men a challenging math test. These
participants performed significantly worse on the test than men who had taken the test
without having this stereotype invoked beforehand (Aronson et al, 1998). For minority
students, the act of taking a standardized test may work to invoke stereotypes about
performance even without an explicit reminder because phenomena like the achievement

gap are so widely accepted as real. Thus, the achievement gap works in two ways in these

49

situations. First, its acceptance as a real phenomenon impacts student performance on
tests, and second, the tests then go on to produce evidence, in the form of test scores, that
the phenomenon is, in fact, real. That’s power/knowledge.

This relationship is also an example of what Canadian philosopher Ian Hacking
called “the looping effects of human kinds,” (Hacking, 1999a; 1999b). As social
scientists, the categories we use to describe those we study create kinds of people, and
unlike species of plants or animals, human kinds (minorities, females, urban children) can
learn about their kind and can change their behavior as a result. Hacking wrote:

They can make tacit or even explicit choices, adapt or adopt ways of living so as
to fit or get away from the very classification that may be applied to them. These
very choices, adaptations or adoptions have consequences for the very group, for
the kind of people that is involved. What was known about people of a kind may
become false because people of that kind have changed in virtue of what they
believe about themselves (Hacking, 1999a, p. 34).

In the mathematics education literature today, many human kinds exist. Researchers
discuss African-American students (Strutchens, Johnson & Tate (eds), 2000; Ladson-
Billings, 1997; Lee, 1998), Latino/Latina students (Ortiz-Franco, Hernandez & De La
Cruz, 1999; Gutstein, 2003), children with disabilities, (Thornton & Bley (eds.), 1994),
girls (Jacobs, Becker & Gilmer eds., 2001; McGraw, Lubienski & Strutchens, 2006),
English language learners (Lee & Jung, 2004), and urban students (Fuson, Smith &
Cicero, 1997; Secada, 1996). As intuitive as these categories may appear, they were not
always available.

Early research in mathematics education tended to differentiate students in far

fewer ways than we do today. Edward Thorndike (1922) referred only to “the pupil.” For

50

instance, we should try to find problems “which not only stimulate the pupil to reason,
but also direct his reasoning to useful channels” (p. 20) or “it is undesirable that the pupil
should regard ‘the crutch’ response as essential” (p. 112). Very rarely, Thorndike
discussed students in terms of their mathematical ability, calling students “gifted” (p. 54)
and referring to “the dullest twentieth of pupils” (p. 57). Writers today would probably
not use this phrasing (preferring perhaps students with disabilities); but, however it was
phrased, mathematical ability was one of very few criteria for Thorndike to use in
differentiating among students. Similarly, other writers who discussed problem solving in
the early part of the 20m century tended to see students in more unitary ways than we do
today. William Brownell (1938; 1948; Brownell & Chazal, 193 5) did not discuss
differences among students, except in regard to ways that they had been taught — through
drill or through meaningful instruction. George Katona (1940), who wrote about adult
problem solving, took a similar stance toward problem solvers. He discussed the
differences in his subjects’ abilities to remember card tricks in terms of the methods by
which they had been taught (rote versus with an organized structure), but not in terms of
demographic characteristics of the subjects themselves. Theorist Lynn Fendler (1999)
pointed out that this increased interest in demographics was part of a broad change across
disciplines in education, which was linked to increased use of statistical measures
designed to regulate increasing numbers of immigrant students in the school system.

No doubt, the proliferation of kinds of students has helped many of those working
in mathematics and mathematics education today to see themselves in the students being
written about in ways that were not possible in the research of the early 20th century.

However, as individuals recognize and read about their kinds, they must also find ways of

51

living in response to the category. Stereotype threat isn’t possible without the stereotype,
and research explicitly focused on kinds of students who have trouble in mathematics
can, without intending to, force individuals to “fit or get away from the very classification
that may be applied to them” (Hacking, 1999a, p. 34). It is because of the proliferation of
student kinds that the phrase “all students” has become so common in equity literature.
Thorndike and Brownell could write “pupils” or “students” and assume that their readers
would understand them to be talking about all students. However, now we have many
different kinds of students in our heads — boys and girls; Blacks, Whites and Asians;
urban and suburban — that the word “students” is likely to cause many readers to ask
“which kind?” Thus, the word “all” is added in order to work against the lines that have
been drawn. However, the word “all” now also acts to call to mind all of these categories.

My time in the postmodern rabbit hole has led me to a new stance toward equity.
In my writing, I no longer want to engage in arguments about the size of the achievement
gap, the reasons for it, or the way to close it. I no longer wish to be a part of reinscribing
the relations of power embedded in the notion of the achievement gap. I do not want to
repeat the story of race determining achievement - as I did, to some extent, in the
previous chapter. When I conceived of this dissertation, I imagined an ethnographic work
informed by postmodern thinkers, with a collection of chapters very similar to the
previous one on questioning. However, from where I stand now, I am acutely aware of
both the stance of rescue I adopted in that chapter and of the ways in which I reiterated
the beliefs I was seeking to write in opposition to.

This tension is evident in my treatment of the dilemma of rejecting race as a

causal factor and arguing for a culturally relevant pedagogy (which names race as a key

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factor in thinking about teaching). I tried to get around this problem in the chapter by
doing some serious back pedaling. For instance, I used a chart to summarize children’s
demographic characteristics and then acknowledged it to be problematic; I talked about
Afiican American culture, but then said that I was not treating race as an independent
variable; I warned readers of the trouble that others have gotten into, and then proceeded
to get in that same trouble myself. This sort of back pedaling is common in this kind of
work. Ladson-Billings, whom I cited in the previous chapter for her description of
African American culture as valuing “rhythm, orality, communalism, and spirituality”
(1997, p. 700), included footnotes in her own work where she stepped away from her
description of culture. She wrote “I do not mean to imply there is one monolithic Afiican
American culture” and “It is important that this notation not be read as the stereotypical
‘all Black people got rhythm’” (ibid).

I do not know what motivated Ladson-Billings to include these disclaimers;
however, I do know why I followed up the chart describing the demographics of my focal
students by apologizing for it. Whether I wished it or not, a chart that highlighted just
four qualities about students (their race, gender, mathematical ability, and level of
participation) reinforced the belief that these are the qualities that matter about human
beings and are important in making educational decisions. What I recognize in this
chapter is that my writing: “Labeling lemme as African-American, male and struggling
works to make each of these categories stable (and worse yet, works to conflate them)”
did not undo the work of the chart. The (not very) hidden argument of the previous
chapter is that African American children are different from European American children

in important ways that have to do with the learning of mathematics. This is not a story I

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want to tell because telling it reinscribes power relations that draw on racial inequity.
That is, the story makes race an important factor in thinking about who children are in
school, encourages people to make judgments about others based on race, and suggests
that being White is linked to successful performance in math class. Despite these
problems, I include the chapter in the dissertation for pedagogical purposes - aimed at
myself and my readers — because it will allow me to talk about the ways that genealogical
approaches to thinking about education might offer other perspectives on equity than
those made possible by ethnographic texts.

This strategy of simultaneously (or at least in sequence) writing in argument for
and against an idea is what French philosopher Jacques Derrida called writing “under
erasure” (Derrida, 1976, p. 60) and is undertaken when the concept under consideration is
both necessary and rejected. For instance, when talking about the relationship between
the sign and the signified, Derrida wrote: “But we cannot do without the concept of the
sign, for we cannot give up this metaphysical complicity without also giving up the
critique we are directing against this complicity,” (Derrida, 1978, p. 281). Here he argued
that we cannot talk about the pointlessness of differentiating between the sign (e. g., the
written word “child”) and the signified (e.g., the small humans we see in schoolyards)
without necessarily calling into play the concept of signs (as separate from signified).
Writing under erasure is a way of coping with this difficulty by explicitly pointing at the
tension. Similarly, the previous chapter was necessary to my argument for a number of
reasons. First, in order to establish credibility as a beginning researcher, I needed to show
that I could work within accepted traditions for qualitative work. Second, to illustrate the

problems that I see caused by calling race into play, I needed to call race into play.

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Having accomplished these goals, I have now spent this chapter unwriting my previous
work. Derrida sometimes signaled erasure by leaving a word that had been crossed out
within the text — showing it to be both necessary and challenged. This is how I see the
previous chapter in the context of the larger dissertation.

Although I am crossing out my use of race discourse, I am not arguing that one
should never mobilize the achievement gap, gender, or race to fight for equal funding,
better teachers or fairer tests. I recognize that I am giving some things up when I adopt a
postmodern critical stance. First, I lose the ability to speak directly about racism, sexism
and class difference, which in turn might make it easier for some to deny that these are
common ways of exercising power in our country today. In addition, I need to give up
some comfortable and relatively efficient tools. In the previous chapter, I drew on
ethnographic methods. I asked research questions, collected data in the form of fieldnotes
and student work, coded my data, and wrote assertions, which I backed up with evidence.
Following these forms, which are common in academic texts in education, provided me
with certain advantages. Readers could use these forms to identify the genre of my work
and to determine whether they wanted to trust my conclusions. The organization of the
chapter was easy because I could adopt a commonly accepted skeleton; and, I could
distance myself from my conclusions because they were what I “found” through analysis
of the data, recognizing, of course, the commonly accepted ethnographic caveat of
reflexivity, where my role as a researcher in collecting, writing up and making sense of
the data is openly acknowledged.

In writing about ethnography, anthropologist Clifford Geertz (1973) asked readers

to think about the difference between a wink and a twitch as a way of understanding

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culture, the “think description” of which Geertz argued is the central work of
ethnographic anthropology. He wrote:

The thing to ask about a burlesqued wink or a mock sheep raid is not what their
ontological status is. It is the same as that of rocks on the one hand and dreams on
the other — they are things of this world. The thing to ask is what their import is:
what is this, ridicule or challenge, irony or anger, snobbery or pride, that, in their
occurrence and through their agency, is getting said (Geertz, 1973, p. 10).

With tools like fieldnotes, coding and analysis, ethnography is good at closely describing
human interactions and making connections between these interactions. These
descriptions make convincing arguments about, for example, whether an observed muscle
spasm is a wink that indicates insider status or an involuntary reflex. As a result,
ethnographies do a good job of exploring questions about the meaning of people’s
interactions. For instance, studies have examined how children co-opted characters from
popular culture for their own purposes (Dyson, 1997); how young men in a low-income
neighborhood experienced race and schooling (MacLeod, 1995); and how the funds of
knowledge present in minority communities can be brought into the public schools
(Gonzalez, Moll & Armanti, 2005). These and similar studies have done equity-oriented
work by challenging cultural deficit theories, by articulating ways that race has been
made meaningful in our schools and society, and by describing culture as continually
shifting and negotiated, rather than as stagnant and monolithic.

However, there are other kinds of equity work to be done. In discussing the
differences between ethnography and genealogy, theorist Erica McWilliam (2003) wrote
that one of the greatest challenges of adopting a genealogical perspective is letting go of

the role of advocate. As an example, she noted that deciding to ask the genealogical

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questions “Why bullying now?” and “How bullying now?” rather than the ethnographic
questions “What is bullying?” and “How do we stop it?” can be unsettling (McWilliam,
2003, p. 60). In similar ways, the genealogical questions of “How has it become possible
to think about home culture as different fi'om school culture?” or “Why has it become
common to explain students’ schooling experiences in terms of race?” might seem like
less compelling ways of working toward an equity agenda than the ethnographic
questions that framed the studies described above.

As with ethnographic studies, genealogical studies do their equity work through
thick description. However, the object of description is not only human interactions in a
particular time and place, but also discourse in many times and places. The object of this
work (unlike ethnographic studies) is not to document the present through the
mobilization of evidence, method and theory, but to bracket the present through argument
so that the object of study appears contingent, permeable and historically dependent. Like
Geertz, the genealogist does not seek to determine the ontological status of the wink;
however, she also does not want to describe how it functions in its cultural context.
Rather, her goal might be to trace the use of winking as communication to show how it
became possible in a certain moment for one person to express “ridicule or challenge,
anger or irony” through the twitch of the eye. By revealing the historical contingency of
winking, it becomes possible to imagine a world without winking. This is how genealogy
does its equity work. Rather than reinscribing the importance of current social categories,
it makes it possible, however briefly, to imagine the world otherwise. The laying aside of
social science for rhetorical or genealogical tools is central to this work. Scientific

constructs such as data, analysis, and coding all function to make the objects under study

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real, rather than contingent. For this reason, in the following chapters, I will be using
rhetorical tools such as argument, persuasion, metaphor and genre to get out from under

the weight of social science traditions — or perhaps, to go below them.

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CHAPTER FOUR
Discourse in Mathematics Education
“Words, words, words.” -— Hamlet (Shakespeare, II, ii)

Mathematics educators have paid a lot of attention to discourse. Researchers have
looked at the discourse of reform classrooms in elementary schools (Cobb, 2000; Cobb,
Boufi, McClain & Whitenack, 1997; Forman & Ansell, 2001; Sherin, 2002; Stevens,
2000; Zack & Graves, 2001), the discourse of preservice classrooms (Blanton, 2002;
Crespo & Nicol, 2006; Danielewicz, 1998) and the discourse of practicing teachers
engaged in professional development (Fernandez, Cannon, Chokshi, 2003; Remillard &
Geist, 2002). In almost every case, the focus has been on the words of participants — both
those uttered in conversation, as in most of the classroom studies, and those uttered in
interviews as presumed proxies for thought (e. g., Ben Yehuda, Lavy, Linchevski & Sfard,
2005; Sfard, 2001).

Although some of the studies cited above do not use the word discourse, the
mathematics education community has come to define any study that focuses on
communication, language or conversation as a discourse study. In their literature review
for the Research Companion to the Principles and Standards, Lampert and Cobb (2003)
used “communication,” “language,” and “discourse” as near synonyms. In another piece,
Cobb (2000) defined classroom discourse as “the ways in which the teacher and students
talked” (p. 55). His definition of discourse as speech is shared by most of the authors
cited above. In addition, discourse studies are often justified by reference to the National

Council of Teachers of Mathematics recommendations that students engage in

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discussions where mathematical arguments are made and critiqued, as Lampert and Cobb
(2003, p. 237) did in the introduction to their literature review:

If school lessons are to involve learners doing mathematical work, classrooms
will not be silent places where each learner is privately engaged with ideas. If
students are to engage in mathematical argumentation and produce mathematical
evidence, they will need to talk or write in ways that expose their reasoning to one

another and to their teacher. These activities are about communication and use of

language.

Here, Lampert and Cobb made the argument that discourse studies are important
because of the role language plays in the learning of mathematics. Reform mathematics
practices, such as classroom discussions, small group work and student presentations,
have both driven much of the research on discourse in the classroom and have helped to
define how discourse is thought of in the community. Discourse and language have
become synonymous partly because studies that look at discourse tend to be located in
conversation-rich classrooms.

Most of these studies draw on one of two theoretical traditions, both primarily
developed in studies of language and literacy. Soda-linguistic analyses (Hymes, 1972;
Lampert, 1990; Mehan, 1982) tend to examine patterns of questions and answers,
exchanges of ideas or vocabulary, or conversational roles of teachers and students. Socio-
cultural analyses (e. g., Ben-Yehuda, Lavy, Linchevski & Sfard, 2005; Cobb, 2000;
Forman & Ansell, 2001; Heath, 1982; Lave, 1988) tend to examine the communicative
purposes of teachers and children, meaning-making by individuals in social contexts and
tensions between home and school discourses. Often relationships are drawn between

these phenomena and the participation structures and content of conversations. Although

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socio-cultural analyses in mathematics pay more attention to cultural tools (i.e., the use of
calculators) and practices (i.e., sharing journals), discourse is usually seen primarily as
language.

Cobb and his colleagues have done many socio-cultural analyses of interactions in
primary classrooms. I describe one of these studies here to highlight some of the
intellectual consequences of adopting a discourse-as-language framework. In this study,
Cobb, Stephan, McClain and Gravemeijer (2001) described first graders’ participation in
a measuring unit where children were asked to measure various objects using unifix
cubes. The context for the unit involved stories about tiny characters — Smurfs — who
used food cans the size of unifix cubes to measure. Together with the teacher, the
children also developed a ten-unit stick as a measure, which they called a “Smurf stick.”
Cobb and his colleagues reported the following exchange to argue that one child, Nancy,
saw measuring as an accumulation of distance and that her explanation, along with other
interventions by the researchers, helped two other children, Mitch and Megan, to adopt
Nancy’s (correct) way of reasoning. The episode opens with Nancy and Megan
measuring the white board with a Smurf stick and the teacher making marks at 10 and 20,
as the girls say those numbers. (The “R” in the dialogue stands for researcher.)

R: Where’s the 20? What does 20 mean?

Megan: 20 means 20 food cans.

R: That means 20 food cans? How much space would that be? Can somebody
show me how much space 20 cans would take up there? Mitch?

Mitch: About that long. [Indicates the space between 10 and 20].

Nancy: No. [Indicates the space from the edge of the white board to 20.] This is
because he [Mitch] did 10, not 20. [Mitch indicates he has changed his mind]
(Ibid., p. 141)

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Using this episode, the authors argued that Nancy’s way of reasoning became
commonly accepted in the classroom through the social participation of the students, the
teacher, and the researchers. They wrote:

In the case at hand, Nancy’s explanation was constituted as legitimate, whereas
Mitch’s proposal was treated as illegitimate. In responding to Mitch, Nancy
therefore contributed to the third mathematical practice, measuring by iterating
the Smurf bar. Similarly, Mitch contributed by making his proposal and by
indicating that he changed his mind. (Ibid, p. 142).

By looking at discourse in this way, the authors were able to make arguments
about what kinds of conversational and pedagogical practices might be important for
helping children to understand measurement and make mathematical arguments. In
addition, they used the concept of social space to get around arguments about transfer.
Rather than suggesting that ideas leapt out of Nancy’s head and into Mitch’s, they argued
that ideas about measurement in the classroom were constructed by Nancy and Mitch
together in collaboration with the other human beings present. These ideas then become
available for the teachers, researchers and students in this community to draw on during
future learning.

This analysis (and similar ones by other authors) bring mathematical learning out
of individual heads into the social space of the classroom, allow researchers to consider
ways that social norms contribute to (or inhibit) mathematical learning, and make it
possible for researchers to think about how ideas are constructed in the classroom.
However, analyses like this one, especially when they read together as a body of work,
have other consequences as well. First, the narrow focus on language ignores many of the

complexities of interactions in the classroom. Students respond not only to each other’s

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words, but also to social and academic status, which can be marked by past interactions
with the teacher, clothing, playground activities, accent, etc. As readers, we have none of
this information to interpret Mitch’s abrupt change of mind, and this lack of information
serves a purpose: it implies that the mathematical idea carried the day; however,
measuring by iterating the Smurf stick may not have been the only idea with
mathematical consequences constructed in Nancy and Mitch’s interaction. In one study,
choosing to focus on the explicitly mathematical idea, rather than on other ideas, may be
a small limitation. After all, researchers cannot study everything all at once. (A map the
size of the area is of very little analytic use.) However, when the majority of studies in
mathematics education define discourse in this way, they work to remove status (as well
as race, class, and other “silent” constructs) from the literature.

Second, the emphasis on the construction of ideas in the space of the classroom
appears to emphasize the social; however, it also works to reinforce the idea of the
individual existing independently from the community in which he or she is situated. In
the Cobb and colleagues piece, the authors describe Megan as a student who “reorganized
her reasoning” (p. 142) in response to Nancy and Mitch’s interaction. In describing
Megan in this way, the authors locate her as an individual within a social space. This
theoretical movement — which Latour (1999, p. 7) described as the change from “mind-
in-a-vat” thinking to “many minds in many vats” — doesn’t get us very far. It implies,
although the authors wrote that this was not their intent, that Mitch, Nancy and Megan
are variously able to appropriate ideas from the discussion not only because their
individual understandings about mathematics and measurement, but also because of how

each of them is situated in broader conversations. That is, their gender, race, family

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background, kindergarten teachers, breakfast, and tolerance for Smurfs all play a role in
shaping their verbal participation in math class. A steady diet of studies that emphasize
the ways individuals participate in conversation works to create a theoretical world where
individual knowledge, skills and beliefs are seen as the appropriate sites for intervention
and other possible sites (the openness of the classroom to students’ home cultures,
definitions of mathematical thinking, ways of relating on the playground) become
invisible.

In his historical analysis of the mathematical practices of abacus masters, Radford
(2003) critiqued the current emphasis on language in mathematics education research for
similar reasons, arguing that researchers have ignored the role that social practices play in
defining what mathematics is and who is good at it. He did this by looking at the kinds of
problems abacists solved during the Renaissance. For instance, he argued that the notion
of fair trade (where some amount of wax could be understood as mathematically equal to
some amount of wool) needed to be in place in order for certain problems to be posed.
Thus, the mathematical practice of students learning to use the abacus was shaped not
just by the conversations that went on between masters and their apprentices, but by the
social world of the time. In his conclusion, Radford argued that discourse was not a
robust enough tool to theorize mathematical understanding. He said researchers must
“take other elements into consideration — elements found beyond discourse itself” (p. 140,
italics in the original).

It is true that discourse-as-language is not an adequate tool to make sense of the
production of mathematical (or pedagogical) knowledge. As Radford (p. 141) said:

“[K]nowledge is also objectified by other means, such as sculptural forms, graphics and

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plastics, as well as by the habitual and historically constituted manner in which one acts
toward things and interacts with individuals.” However, a theoretical tool does exist that
takes all of the forms Radford mentioned (and more) into account — a tool that not only
makes it unnecessary for us to move “beyond” discourse, but, in fact, makes it impossible
to do so. Foucault (1978/1990; 1983) defined discourse quite differently than most
mathematics education researchers writing today. In his work, which looked at the ways
that big ideas such as sexuality and madness got constructed at certain moments in
history, Foucault read a wide range of activities and artifacts as discourse, including
published books, spoken language, symbolic language, the organization of physical
objects and people’s actions. In the first volume of The History of Sexuality, Foucault
(1978/ 1990) argued that the commonly-told story about western sexuality — that we’ve
moved fiom a repressive era where sex could not be spoken about to a more open one --
is invalid. Along the way, he discussed secondary schools in the eighteenth century:

On the whole, one can have the impression that sex was hardly spoken of at all

in these institutions. But one only has to glance over the architectural layout, the
rules of discipline, and the whole internal organization: the question of sex was a
constant preoccupation. The space for classes, the shape of the tables, the
planning of the recreation lessons, the distributions of the dormitories (with or
without partitions, with or without curtains), the rules for monitoring bedtime and
sleep periods — all this referred, in the most prolix manner, to the sexuality of
children (Ibid, p. 27-28).

In context of this dissertation, there are two important ideas to notice here. First,
the physicality of Foucault’s notion of discourse demonstrates the limits of notions of
discourse that consider only language. Whether or not the word “sex,” is spoken, hanging

curtains between beds brings sexuality into the conversation. Second, this wide-ranging

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take on discourse makes it impossible to think about standing outside of discourse for
purposes of analysis (or for any other purpose). Everything is discourse: architectural
plans and recreation curriculum. It is impossible for any of us to step outside of our
cultural and geographic time and place to make sense of an object. This should not be
read as an argument that innate objects — like rocks — do not exist, but as an argument that
we can only understand rocks —- and children, and, yes, mathematics too -- through
discourse. The purpose of bringing Foucaultian discourse to bear on social institutions,
such as schools, is to call attention to ideas that have become a taken-for-granted part of
our intellectual landscape, and thus, to make it possible to see those ideas as mutable
constructions.

Valerie Walkerdine (1988) engaged in this sort of analysis when she described
the discourses of rationality, desire and control at play in the teaching of mathematics to
young children. She used this analysis to argue that current notions about child
development and mathematics are neither inevitable nor natural. In talking about her
work, she said:

My argument has been precisely that ‘language’ and ‘cognitive development’ are
not descriptions of a real which takes place outside practices: all language, all
signs, concepts, and so forth are produced as and by relations in specific practices.
These practices therefore produce and read children as ‘the child.’ I will use the
concept of positioning to examine further what happens when such readings are
produced and how children become normal and pathological, fast and slow, rote-
learning and displaying real understanding, and so forth. In other words the
practices provide systems of signs which are at once systems of classification,

regulation and normalization (Walkerdine, 1988, p. 204).

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Here Walkerdine described “the child” as a subject produced through discourse.
Her goal in doing so was to show that our notions about who children are and what they
can do are constructed through practices like teaching, writing and parenting. For
example, Walkerdine described how popular theories of children and mathematics
constructed the subject of the child as someone who must be “developmentally ready” to
engage in particular mathematical practices, such as learning place value. She showed
how one teacher’s practice both contributed to and was shaped by this construction. In
doing this analysis, Walkerdine made it possible for readers to ask themselves: “What if
there is no such thing as developmental readiness?” and to consider the possibilities that
this question raises for the teaching of children. For instance, if developmental readiness
does not exist, then waiting for children to become ready is no longer a viable
pedagogical practice. Teachers must attend to children whom they may previously have
considered unteachable because the children were not ready. By using subjectivity
(looking at the child as a subject rather than at the identity of individual children) to
emphasize how current “truths” were produced, Walkerdine made it possible to think
about teaching young children in ways that went against the norms of the historical
moment.

Following Foucault and Walkerdine, I talk about subjects and subjectivity in this
dissertation, rather than identity. This is a deliberate choice meant to signify a
postmodern stance. Structural analyses portray individuals as shaped by social
institutions and forces, most particularly by class position. Many more recent critical
analyses (e.g., Levinson & Holland, 1996) see the relationship between individuals and

society as more fluid, where individuals are both shaping and being shaped by the

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societal institutions with which they interact. Researchers operating from this paradigm
often use the term identity to describe the individuals they study. From my perspective,
the trouble with identity (and the theoretical traditions from which it is drawn) is that it
reinforces an opposition between individuals and society, implying that either one could
exist without the other. Subjectivity is meant to suggest a being defined discursively.
There is not an independent entity influencing and influenced by; rather, there is a subject
continually under construction.

A non-mathematical example might help to illustrate the analysis of the
construction of subjects through discourse. As I write this, my kitchen is a mess, and
because I am sitting here typing rather than cleaning it, I feel guilty. As an academic, I
could examine the source of this guilt in multiple ways. Perhaps, the most obvious choice
would be the psycho-analytic route, where I would ask about the experiences and traumas
in my past that have contributed to my current feelings about the kitchen. Alternatively, I
could do a content analysis of the commercials I watch, looking at the messages
conveyed to me about cleaning the kitchen and also observing my own reactions to these
messages. I could look at my participation (or lack there of) in the rituals of cleaning the
kitchen and at the social meaning of cleaning in my household or at the relationship of
house cleaning to my identity as a white, middle-class woman.

Or, I could engage in the sort of discourse analysis I am proposing here,
examining the idea of clean kitchens by constructing the subject of the housekeeper. If I
pursued this project, I might find that at this time and place, there is a relatively dense
discourse around the idea of clean kitchens as important for housekeepers. That is, the

idea is reiterated in multiple ways. For instance, I do see many commercials advertising

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products for washing dishes and mopping floors. Also, when I moved into my house,
there was a gadget in the basement designed to hold mops and brooms, and there was a
latch on the cabinet under the sink, designed to protect children from the cleaning
products to be stored there. Routinely, I receive fliers in the mail from people offering to
clean my kitchen for me. And even when friends admit to messy kitchens, this admitting
reinforces the idea that we need to have clean kitchens. (My friends do not “admit” that
they do not sew their own clothing or fold their dinner napkins into swans).

If I wanted to pursue this problem, I might delve into historical records looking
for when it became possible to talk about keeping the kitchen floor clean --- sometime
after pigs and chickens were banished to the barnyard, I presume. I might also look for
other texts to examine: other people’s cleaning practices, the placement of products at the
grocery store, the conversations of mothers on the playground. Through this work, I
would be able to show how the subject of the housekeeper has been constructed and to
challenge the notion that clean kitchen floors are right, good and natural. In looking at
myself as a subject, I would not privilege my own thoughts, beliefs and actions about
housekeeping (identity-thinking); instead, I would see these as part of a larger discourse
that constructs a housekeeper.

Choosing to use a Foucaultian notion of discourse, rather than a socio-cultural or
socio-linguistic one for this dissertation allowed me to address different sorts of questions
than those that have previously been explored in relation to teacher education in
mathematics. Socio-cultural and socio-linguistic studies both have the goal of
understanding practices as they are currently enacted. This work is valuable because it

can capture the complexity of something like reform teaching in ways that studies that

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seek to show relationships between variables cannot. For instance, the work of Paul Cobb
and his colleagues has helped to show that reform teaching does not simply arise from
using a problem-oriented curricula or from allowing periods for student discussion, but
requires certain moves on the part of the teacher, such as the use of mathematizing
language to support students’ moves from concrete to abstract representations (e.g.,
Cobb, Boufi, Mc Clain & Whitenack, 1997; Cobb, Gravemeijer, Yackel, McClain &
Whitenack, 1997). Socio-cultural or socio-linguistic uses of discourse can also show how
practices that have been assumed to be homogeneous can vary in important ways across
contexts, as Gutstein, Lipman, Hernandez and de los Reyes (1997) did when they
described instances of reform teaching in mathematics that were shaped by the social
activism concerns of a group of urban teachers.

However, I am proposing a different sort of project. Rather than trying to
understand practices around reform teaching and equity as they currently are or trying to
show how certain instances of practice might vary from norms in the literature, I am
trying to highlight the unintended consequences of our current understandings of reform
and equity in the context of teacher education, to make visible norms that have become
taken-for-granted, and to create a space where we can think about teacher education in
ways that are not currently possible. To do this, I need a notion of discourse that supports
the tracing of big ideas across texts and contexts. In the three following chapters of this
dissertation, I use a Foucaultian notion of discourse to:

0 Examine metaphors used to talk about children and to discuss the
consequences of these metaphors for how beginning teachers see the

students in their classrooms.

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o Analyze the mathematical problems posed in both the methods classroom
and the elementary classroom to see how the student is constructed in both
sites by various kinds of problems and to explore the consequences of
these constructions.

0 Identify the variety of teaching genres present in an elementary classroom
and examine the possibilities each of the genres opens up for competent
participation by students.

As I worked on these projects, the Foucaultian notion of discourse provided new
opportunities for me to consider the ethical obligations of being a researcher in the
classroom. Initially, I conceived of my role in both classrooms as one near the observer
end of the participant-observer continuum. I expected that my presence would certainly
be noted by the people in the room, but thought that because I did not plan to speak, my
impact would be insignificant. Working with Foucault’s broad notion of discourse caused
me to understand my role in a new way: as part of the discourse whether or not I chose to
speak. In fact, it was not only Foucault who taught me this, but the children themselves.
About two months into the school year, I followed Caitlin to a neighboring table to watch
her work with a partner on one of the problems assigned by the university seniors. After
about ten minutes of work, her partner, Krystal, looked up at me and asked: “Why are
you watching us? Are we had?” I assured her that they were not bad, but “interesting.”
Krystal seemed to accept this and went back to work; however, as I continued to reflect
on this exchange, I realized that while none of my focal children had ever asked why I
was watching them, this did not mean that they did not have their own explanations for

why they were chosen. (Similarly, other children in the classroom almost certainly had

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their own explanations for my choices about whom to watch.) Some of these explanations
were likely to have been related to the very phenomena 1 was there to study: who is good
at math, who is struggling, what mathematical practices are worthy of attention.

The realization that I had no choice but to be part of the discourse caused me to
abandon my stance as a silent observer at some times. In early February, I observed
Jerome using tiles to discover how many arrays could be made from 17 tiles. After
several minutes of experimentation, he made one long row and wrote “1 x 17 = 17” in his
journal. As other children continued to work, Jerome stacked his tiles into two vertical
towers. Sara, the student teacher, walked by at this point, looked at Jerome’s notebook
and said: “Jerome, you can’t just copy that. You need to figure it out with the tiles.”
Jerome looked up, scowling. “I did do it,” he said and then he looked at me.

I had a choice then. As a researcher, it might have been interesting to see how
Sara responded to Jerome’s claim without an adult to back it up, since this would be more
typical of student-teacher interactions in the classroom. But Jerome knew I had seen his
work. He had watched me sketch it in my own notebook. To evade Jerome’s silent plea at
this point would have taught him something about the trustworthiness of adults in the
classroom and about their perceptions of him; I would have added density to a discourse
to which I did not care to contribute. So I said: “He did do it. He just didn’t draw the
picture.” Sara briefly apologized to Jerome and moved on. If we imagine that discourse is
something that one can stand outside of and analyze, this interaction becomes
problematic. The classroom is a different place after my interaction than it was before it;
in my study, I have to acknowledge the ways in which I as a researcher interacted with

the people in the room, detailing those interactions and the possible consequences.

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However, if we acknowledge that we cannot get outside of discourse, then it becomes
apparent that keeping silent would have been no less of a contribution to the discourse of
the classroom than speaking out. Only the content of my contribution would have
changed. In the following chapters, I analyze discourse not as an independent observer,
but as someone always in the midst of it.

In doing these analyses, I examined language used in both the elementary and
methods classrooms in addition to physical setting and social practices. In order to dig
deeply into the words of my participants, I borrowed some theoretical tools from Russian
literary critic, Mikhail Bakhtin. His work appealed to me because his view of language
recognizes the social world in which words are situated. He wrote:

All words have the ‘taste’ of a profession, a genre, a tendency, a party, a
particular work, a particular person, a generation, an age group, the day and hour.
Each word tastes of the context and contexts in which it has lived its socially
charged life; all words and forms are populated by intentions. (Bakhtin, 1981, p.
293).

Considering the “taste” of words provided a useful way of thinking about which

words got said where and to whom. Some words taste of reform mathematics, others of
the public school, and some of both. In looking at multiple sites — two classrooms,
standards documents and curricula -- I wanted to examine the ways that words used in
particular sites carried the intentions of those not present. For instance, Diana often asked
students “Why?” when students offered an answer to a problem. For Diana, “Why?”
carried the taste of reform, of the many university-sponsored study groups she
participated in; it was a question asked by good teachers who want their students to think.
For students on the receiving end of the question, the word carried different intentions.

When asked “Why?” most students assumed they were wrong and changed their answers.

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In these exchanges, both Diana and the children responded to the taste that “Why?” had
picked up in earlier interactions.

Many of my own words in this dissertation carry the taste of other writers. Most
particularly, I am informed by postmodern thinkers, including Fendler (2004), Foucault
(1977) and Popkewitz (2004). Given this influence, the goal of my project is not to
critique the current state of affairs to propose alternatives, nor is it to save either students
or preservice teachers from poor teaching, society or each other. Rather, through my
description and analysis, I hope to disrupt “old habits of thought that have been limiting
what it is possible to think” (F endler, 2004, p. 451). This can mean bringing a critical
stance to practices commonly seen as beneficial in teacher education, such as reflection,
practice-based learning, and collaboration, or it can mean bringing a similar stance
toward attitudes and practices in teacher education commonly seen as promoting equity.
These critiques can then serve as points of departure for others to re-imagine teaching and
learning. However, solutions will not be prescribed --- both because the offering of
solutions forecloses possibilities for unpredictable innovations and because the ability of
intellectuals to create emancipatory futures for others is called into question. This
skepticism toward progressive leaders is based on a history of failed attempts to liberate
the less fortunate. As Blacker (1998) said: "They promise us 'Good,‘ but things turn out
bad. This happens again and again and again. Maybe we shouldn't believe their promises
anymore" (p. 351). Researchers working out of this tradition are not confident that they
possess the solutions to other people’s problems.

In taking this stance for the dissertation, I am not abdicating my responsibility as

a teacher, a parent, a researcher or a citizen to act against practices that I see as

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inequitable. In fact, I expect that my work in this dissertation will help me to figure out
appropriate ways to act on the problems I am closest to in the contexts I best understand.
Nor am I suggesting that previous work in mathematics education that draws on linguistic
or socio-cultural traditions of discourse is misguided; it is merely different. What taking
this stance does mean is that I will not argue that the solutions that make sense to me -- in
the classes I teach or in the schools where I send my daughter -- should be universally
employed. In addition, it means that I see diversity of research perspectives as productive
for the field of mathematics education. Drawing on postmodern notions of discourse can
broaden, rather than supplant, understandings developed in other research programs. The
goal of the dissertation is to provide academics (including myself) a way of thinking
about the problems of teacher education in ways that will allow us to approach the
problems we understand best in new ways. It is an activist stance, although not a

prescriptive one.

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'E

CHAPTER FIVE
BAD BELIEFS OR DENSE METAPHORS?

Sara, Diana’s student teacher, left Blythe each Thursday to attend methods classes
in mathematics and literacy at the university with a cohort of 20 other beginning teachers,
called “interns” during their year of student teaching. Four of these beginning teachers
shared Sara’s placement at Blythe, while the other 16 interns had been placed in suburban
schools near the university. Early in the year, David, the instructor of the math methods
course, told his students that he would like to take them on a field trip to one of the
placement schools so they could watch him teach young children with the kinds of
problem-solving lessons that he had been demonstrating in class. Because he had
relationships with some of the teachers at one of the suburban schools, David arranged
for the class to meet there one afternoon to observe him teach geometry lessons to fourth-
grade and first-grade classes.

The fourth-grade room, which was large enough to easily accommodate the
visiting interns as well as the 22 students, had been arranged with the desks in groups of
four to five angled toward the white board at the front of the room. David began the
lesson by telling the children that definitions were important in mathematics and asking
them to help him come up with a definition of a chair. Students called out characteristics
and David challenged them through the use of examples and counter-examples until the
class agreed on a list of minimally defining characteristics. David then led them through
similar exercises with squares, rectangles, parallelograms and rhombuses, and finally

asked them to state relationships between these figures (i.e., a rectangle is a special kind

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of parallelogram). After the lesson, the interns gathered in the hallway to wait until the
end of recess, when they would go to see David teach a lesson in a first-grade classroom.

While waiting, Sara approached Jack, an intern assigned to another third-grade
classroom at Blythe, and remarked, “Can you picture one of our kids saying the
square is a polygon? They don’t even know that that means.” Jack said “no,” and
Sara turned to Katie, another Blythe intern, and continued: “This just shows me
how far behind our kids are. I mean, they’re third grade, but they won’t be here
next year. Diana is always saying our kids are so good, but they’re not like this.
Even when things are going well, when they’re excited about something, they just
don’t have the attention span.”

Katie: “They don’t raise their hands. Or rather they blurt out first and then they
raise their hands.”

Sara agreed.

This episode struck me because I was surprised by Sara’s comments -- first,
because she had always seemed to show a great deal of pleasure and pride in the
accomplishments of the students in her classroom, and second, because her and Katie’s
perceptions of the lesson that David had taught were so different from mine. I had
observed several students who called out without being called on; in fact, David had
seemed to encourage this by asking questions of the whole class and waiting for a choral
answer. In addition, while the class had produced a great deal of technical vocabulary,
including “polygon,” “quadrilateral” and “symmetrical,” most of these words had come
from just few students. In the lesson I observed Sara teach at Blythe the following week,

9’ 6‘

her students used the words “square number, repeated addition” and “strategy” while
describing their work. These words may not have been as unusual as the geometric ones,

but they showed appropriate uses of mathematical vocabulary for a third-grade unit on

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multiplication.l Yet, Sara was clearly struck by differences between her class at Blythe
and the suburban class she observed David teach. And she did not see these differences as
the results of participating in a different kind of lesson with an experienced mathematics
educator, of being a year older, or of studying a different content strand within
mathematics. Instead, she described the differences as innate in the students, calling her
own class “behind,” “not like this,” and lacking in “attention span.”
Bad Beliefs

The literature in mathematics education and in teacher education more broadly
has often explained episodes such as the one I described above as instances of beginning
(or experienced) teachers revealing their problematic beliefs. For instance, the literature
has described teachers as holding beliefs that embody dysconscious racism (King, 1991),
that promote color-blindness (Bell, 2002), and that link low achievement to inadequacies
in the students’ culture (Cooper & Jordan, 2003). In an review of research on prospective
teachers’ beliefs about teaching children of different races, ethnicities, and socio-
economic backgrounds, Gomez (1993) described a host of problematic teacher beliefs
that have been documented in dozens of studies, including beliefs that economic rewards
are fairly distributed in the United States, that low-income families do not support their
children’s learning, and that some students cannot learn.

Most studies of teachers’ beliefs in mathematics education have focused on
teachers’ beliefs about how mathematics should be taught or about what mathematics is,

rather than on beliefs teachers hold about students (e.g., Barlow & Cates, 2006; Mingus

 

' My goal here is not to say that Sara did not see the lesson for what it “really” was, but to point out that she
constructed differences that were meaningful to her. I also saw differences between this lesson and those in
Diana’s classroom: the teacher talked far more, students did no independent work, and children ahnost ‘
never spoke to each other. My production of these differences could equally serve as a point of departure
for study. That, however, remains a challenge for another day.

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& Grassl, 1999; Stipek, Givven, Salmon & MacGyvers, 2001; Timmerman, 2004;
Warfield, Wood & Lehman, 2005; Wilkins & Brand, 2004). However, a few studies

(F ennema, Peterson, Carpenter & Lubinski, 1990; Sztajn, 2003; Tiedemarm, 2002) have
looked at interactions between these belief systems. For example, Sztajn (2003, pp. 53-
54) wrote that teachers “have to put together, according to their own beliefs and
interpretations of existing rhetoric what they consider to be the best mathematics
education for their children.” To do this, he argued, teachers draw on both their sense of
what mathematics is and on their beliefs about who the children in their classrooms are.
Based on a case study analysis of two experienced teachers, he argued that the teachers’
beliefs about mathematics and children caused them to use drill-oriented work with
classrooms of low-income students and to assign activities that promoted higher-order
thinking to classrooms with children from more affluent families.

Framing teaching decisions and comments like Sara’s as instances of problematic
beliefs coming to light has meant that the interventions proposed by teacher educators
have often focused on finding ways to successfully change individuals’ beliefs (e. g.,
Artiles & McClafferty, 1998; Bondy, Schmitz & Johnson, 1993; Brown, 2004; Sleeter,
2001). For example, in their study of mentor-novice pairs, Achinstein and Barret (2004)
discussed beginning teachers’ tendencies to see their diverse classrooms as collections of
management problems. The authors of the study described ways that mentors were able
to help beginners reframe their thinking about students to include political and social
considerations, which caused the novice teachers to change their beliefs about their
students’ academic capabilities. Alternatively, studies have focused on identifying

qualities in beginning teachers that lead to either problematic or productive beliefs, such

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as previous experiences with cultural diversity, empathy for others, or valuing of
diversity (Garmon, 2004; Pohan, 1996; Smith, Moallem & Sherrill, 1997).

My concern with framing comments like Sara’s and questionable teaching
practices as the products of problematic beliefs is that this theoretical stance places too
much emphasis on the individual. Although most studies acknowledge that individuals’
beliefs are situated in larger social contexts, methods, such as surveys and interviews,
work to highlight the responsibility of single individuals. Similarly, interventions that
seek to change the beliefs of individuals through classes on cultural diversity, field
experiences, or reflections on experience make what goes on inside individual heads the
primary locus of change. I see two problems with this. Practically, it is slow work.
Following this model, teacher educators must diagnose and treat each prospective teacher
they encounter from now until the end of time. Theoretically, this model is also
troublesome. Whether one follows Vygotsky (1978, p. 88) who wrote that learning
“presupposes a specific social nature,” Bakhtin (1981, p. 293) who said that the word
“exits in other people’s mouths, in other people’s contexts, serving other people’s
intentions,” or the host of other thinkers who have argued that researchers cannot
consider the individual without the social (e.g., Lave, 1988; Rogoff, 1995; Wenger,
1998), studies of beliefs seem to give far too much weight to the individual, both in terms
of responsibility for problematic ideas and in terms of proposed interventions.
Educational philosophers Blacke, Smeyers, Smith and Standish, (1998, p. 88) in one of
the most eloquent articulations of this idea, wrote that “In part what is given in teaching,
in the initiation into a culture, is a gift that cannot be refused. What we come to know in

this way precedes the possibility of our autonomy.” In this chapter, I argue that Sara’s

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words comparing the children at her urban school with those she saw in the suburban
classroom did not reveal her “own beliefs” because the way she came to see her children
as “behind” preceded the possibility of her autonomy.

Dense Metaphors

Rather than framing this chapter as an analysis of Sara’s beliefs, I want to

examine Sara’s comments as a re-articulation of a common metaphor in the current
discourse about children. My hope is that by approaching the analysis in this way, I will
be able to simultaneously consider the actions of individuals as well as the social soup2 in
which those actions are located. To do this, I draw on the notion of metaphor articulated
by George Lakoff and Mark Johnson (1980), who argued that metaphors are more than
ways of using language, but also ways of understanding concepts. They used the
metaphor of argument as war to demonstrate both the pervasiveness of metaphorical
language in our everyday communications and also the ways in which metaphors can
influence how we see the world. Lakoff and Johnson pointed out that in our culture
people talk about attacking, demolishing, shooting down and winning arguments. They
plan counterattacks and defend their points. In contrast, the authors asked readers to
imagine a culture where argument as dance was the dominant metaphor for speaking and
thinking about disagreements.

[T]he participants are seen as performers, and the goal is to perform in a balanced
and aesthetically pleasing way. In such a culture, people would view arguments

differently, experience them differently, carry them out differently, and talk about

 

2 Speaking of metaphors, our lives as “social soup” is one of my favorites. Rather than seeing individuals
and discourse/society as separate, the soup metaphor merges these two constructs. While it is possible to
call individuals or discourse to the foreground (as a chef might consider the amount of salt or potatoes) the
soup cannot be separated fiom its component parts. Nor can we, as ingredients, get a vantage point outside
of the soup to make observations. Like the force, soup is always in motion.

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them differently. But we would probably not view them as arguing at all: they
would simply be doing something different (Lakoff & Johnson, 1980, p. 5).
Using this example, the authors suggested that metaphors are not simply instances of

poetic language that writers use to make their work more aesthetically pleasing, but are
ways of seeing the world that become more and more dense as they are re-articulated by
multiple speakers. These dense metaphors then become part of everyday language and
are often not registered as metaphors by listeners. That is, we hear the metaphor literally
so that if we hear a colleague say “I fought with the chairman over the budget,” we
typically would not consider the word “fought” to be metaphorical language. We
interpret fighting as the use of words in an angry way, not as physical struggle. However,
Lakoff and Johnson argued that the notion of “fight” works in the same way as more
unusual metaphors by highlighting some aspects of the concept of argument (the
disagreement, the expectation of winners and losers) while diminishing other qualities
(the shared purpose, the tum-taking). The frequency with which metaphors of war are
used to talk about arguments makes it easier for people to see only the aspects of
argument that are consistent with that metaphor.

Building on this work, Santa Ana (1999), who examined the prevalence of the
metaphor immigrants as animals in the media, argued that the more commonplace
metaphors become and the more prosaic they seem, the more powerful they become in
shaping the way people interact with the world. He wrote:

When an original, truly novel metaphor is used, the reader of the turn of phrase is
prompted by its novelty to evaluate the metaphor for its appropriateness,
creativity and utility. The mindful reader can choose to reject the linkage. If,

however, the metaphor does not draw attention to itself, then the reader is most

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often unaware that a conceptual linkage has been reproduced and is being
reinforced (Santa Ana, 1999, p. 237).

Lakoff and Johnson argued that many of our most common metaphors —— those that do not
“draw attention” to themselves -- are both based on our physical experience of the world
and can influence the meaning of other metaphors by drawing on the same physical
experiences. Many concepts we talk about metaphorically are structured in terms of up
and down, and forward and behind. That is, these metaphors draw on our notion of a
physical path that can be traveled in two directions, either vertically or horizontally, in
order to explain some other concept. For instance, conscious is “up” (wake up, early-
riser, up and at ‘em) and unconscious is “down” (fell asleep, dropped off, sank into a
coma).

Coherence among metaphors can be created when multiple metaphors draw on
similar physical concepts. For example, many concepts (e. g., good, more, awake, happy)
map onto “up” while other concepts (e.g., bad, less, asleep and sad) map onto “down” in
many of our commonly used expressions (Lakoff and Johnson, 1980). Sfard and Lavie
(2005) offered one empirical example of this kind of coherence when they told the story
of two four-year-olds who insisted that the mystery box they chose was “more” than an
identical rejected box. When questioned, the children agreed that the two boxes were the
same size; however, they stood by their claim that the chosen box was “more huge.” The
adults present could not understand the children’s statements, but Sfard and Lavie argued
that the children associated the word more with good and used it because they believed
that the chosen item must be by definition “more” than the less desirable item. This is an
example of coherence between the concepts of good and more, both of which are often

discussed in ways that draw on the physical concepts of “up” and “ahead.” In the

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following section, I will describe the ways in which Sara’s comments that her children
are “behind” and are “not going to be here next year” draw on the physical world to
shape thinking about students and mathematics and examine the prevalence of other
metaphors that draw on the physical world in similar ways. In addition, I will discuss the
aspects of children and mathematics that these metaphors emphasize as well as the
aspects they tend to hide.
Talking about children: Metaphors of hierarchy

In order for Sara to make her relatively straightforward comment that watching
the children in the model lesson made her realize “how far behind our children are,” she
needed to make several decisions: first, that an appropriate action when observing two
sets of children is to compare them; second, that an appropriate next step is to rank them
according to some criteria; and third, that an effective way to express that ranking is to
use a metaphor that draws on front-to-back directionality. Neither Jack nor Katie, the
interns Sara spoke to, seemed perplexed by her use of the word “behind.” They did not
seem to stop and evaluate the metaphor, as Santa Ana suggested that people may do when
confronted with novel metaphorical language. Both seemed to understand that Sara was
ranking the children in some way and Katie, at least, seemed to agree with the ranking,
offering her own evidence to support it: that the children in her Blythe classroom called
out without raising their hands. In this section, I would like to examine the prevalence of
metaphors of hierarchy in discussions about children in mathematics education literature,
elementary mathematics textbooks, standards documents, conversations in David’s

methods classroom, and Sara’s elementary classroom at Blythe.

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In the methods classroom, both David and the other interns often talked about
students in ways that created hierarchies by drawing on metaphors of the physical world.
For example, at various times in the semester, the following comments were made:

0 “What do you do with a second grader who’s really awesome in math — who’s
really far ahead of everyone else?”

0 “She’s in special ed. She’s really low.”

0 “Obviously you need to set up groups of kids in problem solving that can help
each other and pull people along who are having trouble.”

0 “Some of your kids — they’re going to be way up here (holding hand above head).
You have to rise up to meet them.”

0 “I’m afraid he’ll have to be retained. He can’t keep up.”
Each of these comments draw on a notion of children ordered in physical space, whether
horizontally (“really far ahead”) or vertically (“low”). Some of the comments put
students in motion. The line of students is marching forward and children must either
keep up or be pulled along. This is drawing on a similar sense of space as Sara’s
comment that her children “won’t be here” next year. In both cases, the mathematics is
portrayed as a path that must be traveled. Sara is saying that her third-graders will not
have made enough progress along the path in order to be in the same place as the fourth-
graders in a year’s time. The remark about “pulling children along,” which was made by
David, implies that some children will not be able to travel the path under their own
power and must be dragged forward by their more able classmates. Initially, it may seem
strange to think of these ordinary bits of language as metaphors; however, like more

poetic figures of speech, each of these phrases asks the listener to think of children in

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non-literal ways. Children are not really ordered in physical space. There is no path. No
one (contrary to political rhetoric) is actually being left behind. As Santa Ana argued, one
of the reasons this sort of language seems unremarkable is because it is so common. Sara
encountered these kinds of physical metaphors for children in her elementary classroom,
in her textbook and in documents from the state Department of Education.

For example, in one lesson Diana said: “Ben, you did it in a really sophisticated
way, but that might bring it back to other people,” “Come on, Jerome! You need to keep
up with the rest of the class” and (to the whole class) “You’re letting Tyler and Ben carry
you.” This last variation echoes David’s comment about pulling kids along. The children
in Diana and Sara’s class also used these kinds of metaphors. During independent work,
Aliah remarked that “Mia is ahead of me. She knows her division and her
multiplication.” Once, Sara even physically acted out the metaphor of mathematical
achievement as physical ordering on a path by calling children to line up at the end of the
lesson in an order that ranked their participation in a discussion. Students who spoke
frequently were at the front of the line, while Mia, who at this point in the year spoke
very little English, brought up the rear.

Although the language in state documents and mathematics textbooks tends to be
more formal than the spoken language, examples of phrases that drew on metaphors of
crossing physical space to talk about children’s learning of mathematics did occur. In a
handout for parents about mathematics standards in third grade, the authors wrote, “The
expectations were designed to ensure that students receive seamless instruction from one
grade level to the next, leaving no gaps in any child’s education” (Michigan State

Department of Education, 2006). As in the phrase achievement gap, the language in the

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parent handout implicitly draws on the crossing of physical space as a metaphor for
children’s learning in mathematics, where gaps in the path can be seen as dangerous. The
metaphor of physical space was also implicitly drawn on in the methods textbook that
Sara often used when planning lessons. Van de Walle (2004, p. 95) wrote “It remains true
that students will rise or fall to the level of our expectations.” Here students are compared
to an independent criterion (expectations) rather than each other, but still moving up is
good and going back, or falling, is bad.

All of the metaphors discussed above hang together coherently because each uses
the idea of a line in physical space, where being closer to the front or the top (or moving
in that direction) is good. This metaphor structures thinking about children and their
learning of mathematics in particular ways. Although saying a student is “really far
ahead” of everyone else hardly seems poetic, like other metaphors, it connects one
concept (knowing math) to another dissimilar concept (being physically in front). And,
like all metaphors, this connection makes some features of knowing math more salient
than others. Repeatedly using language that describes children’s learning of mathematics
as a journey or a ranking in physical space emphasizes the idea that students’ thinking,
learning and progress can be compared to each other. Just as location or progress along a
path can be identified through the use of landmarks, children’s mathematical progress can
be identified by where they stand relative to each other or by where they stand relative to
the content. This is how children can be thought of as “behind” or “ahead.”

These metaphors also work to portray mathematics as narrow path, which can be
traveled in only one direction (up and ahead). Sara observed David teach a geometry

lesson in a fourth-grade classroom and said that her children were “behind” the ones she

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had just observed. Geometry had not yet been taught in her classroom at Blythe; yet Sara
assumed that the suburban children’s performance in this geometry lesson revealed their
understanding of other mathematical strands as well. She did not say “Boy, they know a
lot of geometric vocabulary.” The ease with which she could perform the ordering of the
children in these two classes and the ease with which her ordering was accepted by the
other interns in part relies on the metaphor of mathematical achievement as a journey
along a narrow path. One cannot simultaneously be “ahead” in geometry and “behind” in
number. Each person has one location on the path.

The metaphor of travel along a path makes it difficult to see a student who
struggles with number as mathematically competent because of an ability to visualize
three-dimensional objects. This metaphor obscures multiple-entry points into
mathematics and makes it difficult for teachers to see student work as varied without also
seeing it as ordered. On a path, one is always nearer or farther from the front. Similarly,
this metaphor can cause teachers to interpret chunks of mathematical content as
necessarily ordered. In the methods class, one intern said that she could not move on to
multiplication because many of her students could not subtract with regrouping. David
asked what one had to do with the other and the intern replied “Subtraction with
regrouping comes first.” Subtraction with regrouping did come first in the mathematics
books used in Sara’s classroom (as in most third-grade math books); however, there is no
mathematical reason to assume that competence in the former is a prerequisite for
competence in the later. Children’s performance in different mathematical strands — such

as, geometry, measurement, data, and operations - can be quite varied, but using a

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metaphor that portrays mathematics as a “seamless” path with no “gaps” can make if
difficult to recognize this.

The metaphor of the path also contributes to understanding children in ways that
promote the idea that some children are good at mathematics and some children are not.
When mathematics is seen as a linear list of skills that must be mastered one after the
other so as to progress toward a single goal (the end of the path), then some children must
always be closer to that goal and others must be further away. When Ben offered a new
solution to an addition problem, Diana did not merely identify the solution as different,
but labeled it as “sophisticated,” and told Ben his thinking would “bring it back” to the
rest of the class. Similarly, students who are in different places than their classmates in
terms of understanding mathematics are seen as problems for the teacher and other
students. They must be “pulled along” or work to “keep up.” The metaphor of traveling
along a path obscures ways of understanding student differences as interesting, valuable,
or natural.

These ways of thinking are reinforced in mathematics education by many
concepts that may not use words that explicitly refer to positioning in the physical world,
but that nonetheless draw on the root idea that learning mathematics has one path that
individuals are more or less ably suited to traveling. In mathematics education, the notion
of development, which underlies theories and practices that expect all children to
progress through identical stages in identical orders (like travel along a path) is present in
much of the writing about children. Piaget’s theory (1928; 1950; 1952) that children
develop the abilities to realize that objects do not disappear, to use symbols, and to

conserve mass (among other abilities) at particular ages and in a predictable order has

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been particularly influential. David mentioned Piagetian concepts several times during
his lectures, suggesting that the prospective teachers use simple activities, such as asking
children to identify the larger of two sets of identical but differently spaced counters, in
order to make decisions about what sort of instruction might be appropriate.

Similarly, the van Hiele levels of geometric thought, which David lectured about
in class and which the author of Sara’s methods textbook devoted nearly forty pages to,
are grounded in a notion of development where all children are seen as progressing
through the same levels in the same order. Calling the van Hiele theory “the most
influential factor in the American geometry curriculum,” Van de Walle (2004, p. 347-
348) described the theory as saying that “the levels are sequential. To arrive at any level
above level 0, students must move through all prior levels.” In elementary school,
children are expected to move from Level 0, where they can identify shapes only because
they look just like other shapes they have seen (“It’s a square because it looks like one”)
to Level 2, where they can reason about the properties of shapes (“Because it’s a square,
all the angles must be 90 degrees”). Both the van Hiele levels and Piagetian stages
support ways of thinking that allow some children to be seen as “behind” and other to be
seen as “ahead,” either relative to their classmates or to benchmarks delineated in the
theories. In fact, this happened in the methods class on the day of David’s van Hiele
lecture when one of the interns volunteered that her daughter’s kindergarten homework
had been to name the features that made a circle a circle. David replied, “I don’t think
kindergarteners or even most first graders could do that.” The intern responded that her
daughter had said that circles don’t have comers, that they go all the way around, and that

they don’t have straight lines. David shrugged, adding “She must be very advanced.”

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Here, a 5-year-old’s ideas about circles were labeled as advanced because the uptake of
the van Hiele levels has been that kindergarteners are focused on visualization. The
theory created the opportunity for this child to be seen as ahead and at the same time
created opportunities for other children to be seen as behind. My goal here is not to argue
that the van Hiele levels are inaccurate, or that David’s use of them was incorrect, only
that the prevalence of developmental thinking in the mathematics education discourse
reinscribes the metaphors of hierarchy that Sara drew on to talk about her children.

In similar ways, our system of organizing students in classrooms by age
reinforces thinking about mathematics as a narrow path. Michigan’s Grade Level Content
Expectations “provide a set of clear and rigorous expectations for all students and provide
teachers with clearly defined statements of what students should know and be able to do
as they progress through school” (Michigan State Department of Education, 2006). In
other words, these standards define the path; they lay out what a third-grader must do and
make it possible for teachers to identify children for remediation (those who are behind)
and enrichment (those who are ahead). As Fendler (1999) has pointed out, this is a
different enterprise than looking at actual third-graders to figure out what it is they know
and are doing.

The metaphor of schooling as a path with grade-levels as landmarks dominates
not only practices in the classroom — Diana and Sara used textbooks written for third-
graders, prepared for the third-grade standardized test and followed the third-grade
standards - but also flames how educators talk about children. Diana frequently asked her
students to “act like,” “write like,” and “sit like” third graders. On one particularly noisy

afternoon, she looked slowly around the classroom until students quieted. Then she

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remarked. “Now it’s quieting down. That’s good because I was a little worried. I was
thinking somebody else might have snuck in here.” Concerned, Marcus asked “Who?”
Diana said: “Someone who wasn’t quite ready to be in third grade.” Jerome, who was
flequently chided, asked: “Like me?” Jerome had gone through three and half years of
schooling in the same building with many of the same classmates. He was eight years old
and knew himself to be in third-grade. Yet, the frequency with which his behavior was
corrected and answers revealed to be incorrect caused him to identify himself as someone
the teacher thought was not ready for third grade. In this instance, Diana reassured him
that he was, but Jerome seemed to not only recognize the metaphor of schooling as a
path, but also to be able to identify his place along it.

The third grade math books, which Sara used occasionally, also drew on this
metaphor. Each page in the math book represented one lesson. Students were expected to
proceed along a path from the first page to the last. In addition, each page in the math
book came with three additional worksheets, labeled “re-teaching” “practice” and
“enrichment.” Teachers were expected to assign each student one of these pages based on
the students’ position relative to the intended lesson. In a professional development
session about the curriculum, teachers were told that a child should never be assigned
more than one of these worksheet pages. Each student could occupy only one of these
positions. In addition, each page of the math book also had a section called “quick
review,” which students in the classroom completed at varying rates. This was a constant
source of flustration to Sara, who once exclaimed: “It’s Quick Review! What’s it

supposed to be? Quick!” Third graders, who were all supposed to be at roughly the same

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point along the path, were not expected to take different amounts of time to complete the
same task.

In the methods classroom, the prospective teachers flequently identified children
in their stories by grade level and if they did not when asking a question, such as “How
do you talk about least common multiples?” David would ask for the grade level. This
was not one of many bits of information solicited in order to make an informed teaching
decision, but most often, the only one. A students’ grade level was assumed to reveal
what a student would know and not know (his or her place along the path), which
allowed comments like “This is way too hard for first-graders” and “Fourth graders can
see relationships among shapes.” Although these statements do not explicitly use
language that compares mathematical learning to travel on a path, they do suggest that all
first-graders or fourth-graders should be in the same place in relation to learning
mathematics and thus are coherent with physical space metaphors.

David sometimes drew on notions of ranking and progress even when discussing
children beyond mathematics. As an assignment, David asked his interns to find out
something that their students did well outside of the classroom. When asked to report
back the following week, Jack said that he had learned that one of his students had a
beautiful singing voice. David then asked if she sang in a choir or a competition. Jack
said no; he had heard her on the playground. Then Sara offered that one her students
(Jerome) spent his weekends on a farm milking goats. She said this was unusual because
she taught at an urban school and she thought that most of her kids had never seen a farm.
David replied, “That’s a cool experience. So you got that category of cool experiences

and that other category of someone does something really well. You can’t milk a goat

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really well. I don’t know how you would do that.” With both Jack and Sara, David
seemed to diminish the importance of their stories because they did not support ordering.
David responded with much greater enthusiasm to stories about child gymnasts, bicycle
racers, and competitors in Irish dance, which all make it easy to talk about children in
hierarchical ways.

Coherence among metaphors: Or what else can we think of as ordered?

The prevalence of language that explicitly draws on the image of traveling along a
path to describe children’s mathematical learning as well as the saturation of mathematics
education with practices that reinforce this metaphor create certain discursive possibilities
and close down others. In other words, Sara’s choice to compare and order the children in
her placement classroom and the children in the model lesson classroom emerged from a
discourse community in which hierarchical thinking about children was common. Sara’s
adoption of that language can be seen as evidence of her participation in that community.
However, the prevalence of hierarchical thinking does not explain the order in which Sara
ranked the classrooms. That is, why did she see the students in her own classroom as
behind, rather than ahead?

The grade level of the students may have played a role. As discussed in the
previous section, most educators would probably see fourth graders as “ahead of” third
graders. However, Sara explicitly rejected this as a reason saying “I mean, they’re third
grade, but they won’t be here next year.” In addition, because the notion of grade levels
as markers along the path is so common, third graders who did not know as much as
fourth graders would not be seen as behind by most educators. They would be seen as

making adequate progress, relative to when they began their journey. Sara offered the use

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of geometric vocabulary and long attention spans as reasons for her judgment, and Katie
suggested that the frequency of hand-raising before answers as a criterion for ranking.
These factors may certainly have played a role. However, I would like to entertain
another possibility based on Lakoff and Johnson’s notion of metaphorical coherence.

Sara’s class at Blythe Elementary, a small urban school, was comprised of 19
students. Four of these students had been identified as White/Caucasian on school
enrollment forms. The other 15 students had been identified as Aflican American, Asian
American, or Bi-racial. Many of these children had brown skin and brown eyes. In
addition, many of the children of all ethnicities spoke non-dominant forms of English. In
some cases, this was the result of English being a second language. In other cases, this
was the result of speaking a dialect. For some students, the variations in their speech were
quite subtle - such as slight southern accents and using words like “y’all.” For others,
variations included grammar often deemed unacceptable in school, such as “He don’t
know.” In the class of 22 students at Northside (which hosted David’s model lesson),
most students (about three-quarters of the class) had the pale skin and facial features that
most Americans would probably identify as European-American. All of the children who
spoke used variations of English that would probably be considered standard. That is,
they used contractions fi'equently, but used grammatical constructions that would be
considered correct in formal, written English. Most children’s accents and inflections
mirrored those of David (and Sara and me).

In the preceding paragraph, I sketched out a few of the differences between the
classes at Blythe and at Northside. Of course, there were many other differences. The

architecture of the buildings and arrangement of furniture in the rooms was different. As

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group, the children at Northside seemed slightly taller. The children at Blythe tended to
talk to each other while the children at Northside tended to address only the teacher.
However, I did not randomly choose the differences I highlighted in the previous
paragraph. I described them because I believe they were differences that Sara noticed and
drew on in making her assessment. I would now like to examine some examples of
discourse about children flom the mathematics education community with the goal of
demonstrating how the ethnic, racial and class differences I identified above may have
become salient for Sara and may have contributed to her ranking the two classes in the
order she did.

More than a decade ago, NCTM issued what it calls its “Every Child” Statement,
which is posted on the NCTM website and quoted in Van de Walle’s methods textbook
(2004, p. 95), in order to underscore the council’s commitment to the ability of all
children to learn mathematics. The second half of the statement goes like this:

We emphasize that "every child" includes--

0 learners of English as a second language and speakers of English as a first
language;

. members of underrepresented ethnic groups and members of well-represented
groups;

0 students who are physically challenged and those who are not;

0 females and males;

. students who live in poverty and those who do not;

0 students who have not been successful and those who have been successful in

school and in mathematics.

The way the subjects of each of the above bullets are ordered is not coincidental. In the

first position are the kinds of students who have trouble learning mathematics: speakers

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of English as a second language, minorities, those with disabilities, girls, the poor, and
the unsuccessful. These statements would not have what Lakoff and Johnson called
“coherence” if they were all jumbled about. The category of “males” does not go with the
category of “poor” and “minority.” Although the language of ordering was not used
explicitly, the pattern of the bullets implies a continuum of students who range from
“those who have not been successful” to “those who have.” In addition, ordering the
subjects of each bullet in this way works to attach the categories of minority, female,

poor and physically challenged to unsuccessful.

Adding It Up, a report on elementary mathematics in the United States which was
authored by some of the most prominent mathematics educators in the field, also invoked
metaphors of hierarchy and linked notions of race and class with the ordering. First of all,
the section of the report that addresses issues of equity is called “Equity and
Remediation.” Immediately, this phrasing invokes the metaphor of the narrow path.
Remediation is only possible when some students are behind. The challenge of equity
becomes catching these students up to where they are “supposed” to be, rather than on,

say, learning to value where these students are. This section of the report goes on to say:

A number of children, however, particularly those from low socioeconomic
groups enter school with specific gaps in their mathematical proficiency.
Overall, the research shows that poor and minority children entering school do
possess some informal mathematical abilities, but that many of these abilities
have developed at slower rates than middle-class children. The immaturity of
their mathematical development may account for the problems poor and

minority children have understanding the basis for simple arithmetic and solving

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simple word problems (pp. 172-173, Adding It Up, 2005, italics and holding

mine).
The authors of this paragraph explicitly invoked metaphors of travel along a path with
the use of the words “gaps” and “slower rates.” In addition. they drew on notions of
development by referring to the “immaturity” of some students. Not only does the
phrasing of this paragraph support ways of thinking that place some children ahead of
other children, but the words chosen also attach demographic characteristics to positions
on the path. Poor and minority children are slow and immature. Another rhetorical move
in this paragraph is to conflate poor with minority, not only by using them in connection
with each other, but also by occasionally using one to stand for the other. The authors
wrote that “Overall, the research shows that poor and minority children entering school

do possess some informal mathematical abilities, but that many of these abilities have

developed at slower rates than middle-class children.” In the second half of this
sentence, the authors qualify the noun “children” with only the adjective “middle-class,”
although they began the sentence by referring to “poor and minority children.” This is a
little sleight of hand (or text). Consider the alternative: Poor and minority children

develop at slower rates than white children.

Replacing “white” with “middle-class” makes the sentence less jarring, but by
using “poor and minority” at the beginning of the sentence and by repeating the same two

adjectives in the following sentence, the authors order race along with mathematical

9’ ‘6 ,9 ‘6'

ability.3 In addition words like “poor, minority, rmmature,” and “slow” become

 

3 Astute readers might detect slightly less compassion on my part for the authors of this report than I show
for Sara and might (justly) point out that the authors of Adding It Up are no more reporting their “own”

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attached to each other in ways that one can stand as a proxy for the others. This
substitution of various qualities for race occurred in conversations in the methods
classroom as well as in written texts. For instance, when telling her story about Jerome
milking goats, she said: “This is an urban student - that’s what our school mostly is. You
would never guess by looking at him that he does this.” Jerome, who typically wore
glasses, button-down shirts and jeans belted at the waist, did not look like a stereotypical
farmer; however, he did not look particularly “urban” either, with the exception of his
dark skin. Sara’s comment demonstrates the ways in which race can get noticed and

remarked upon without using racially explicit language.

When multiple texts attach (however circuitously) concepts that typically would
be considered categorical rather than ordered — such as race and gender -- onto metaphors
of hierarchy, they work to create discursive expectations. The result is that when the
authors of the NCTM Principles and Standards for School Mathematics (2000, p. 12)
wrote that “all students, regardless of their personal characteristics, backgrounds, or
physical challenges, must have opportunities to study — and support to learn —
mathematics,” readers can understand this to mean that some students — those who “live
in poverty, students who are not native speakers of English, students with disabilities,
females, and many nonwhite students” (p. 13) can be thought of as struggling, or perhaps

behind. Through similar reiterations, the metaphor of students distributed along a path

 

beliefs than Sara is. All of us are situated in the current discourse about children in mathematics. However,
I do hold the authors of Adding It Up slightly more accountable for their reiterations of this hierarchical
discourse than I do Sara. First, they are speaking formally in a considered and public report, as opposed to
having a casual hallway conversation. Second, because of their stature, their contributions to the discourse
carry more weight.

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begins to describe not just individual children, but kinds of children. I would argue that as
a result of these layerings of discourse, Sara, as well as most other educators, would have
been inclined to position the students from Blythe as “behind” the students from
Northside because this ordering had more coherence with other metaphors of hierarchy
within the discourse of mathematics education than the reverse order. That is, it would
have been conceptually surprising to think of a class of mostly minority students from

poorer homes as “ahead of” a class of mostly majority students from middle-class homes.

Possibilities for Teacher Educators

This is not an argument intended to absolve Sara, or any of us, of responsibility
for our words. In fact, it is just the opposite. Seeing Sara’s positioning of her student as
the result of dense metaphors in mathematics education rather than as the problematic
beliefs of individuals changes the responsibility of teacher educators. Instead of (or
perhaps, in addition to) searching for ways to change beginning teachers, we can think
about the ways that our own teaching, writing, and conversation work for or against
hierarchical ways of thinking about children. In addition, we can be conscious of the
ways that we use demographic labels. This is not easy. As Lakoff and Johnson wrote,
metaphors are not “just talk” (1980, p. 4); they influence how we understand the world,
and with dense metaphors, like learning mathematics as travel along a narrow path,
revisioning the world requires new words, new metaphors and new practices that may
seem troubling, precisely because they are at odds with the ways that we have always

understood mathematics and children. In closing, I would ask readers to consider the

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following three questions as a way of beginning a conversation about re-imagining our
language and practices as teacher educators.

1. How could we talk about learning in mathematics without using metaphors of
hierarchy?

School learning, and not just in mathematics, may seem to necessarily contain within it
the notion of progress and of more or less, which is what leads to the ordering of
mathematical concepts and the ranking of children. One way of creating new metaphors
to talk about learning may be to talk about difference rather than progress. For instance,
Gardner’s (1983; 1999) work on multiple intelligences offered one possible language for
talking about intelligence without ordering. By identifying specific ways of thinking,
such as visual, logical and linguistic, Gardner argued that intelligence is not one
continuum, but a diverse grab-bag of abilities. Drawing on this metaphor, learning might
be looked at as a practice that adds diversity to one’s grab-bag. In mathematics, where
visualizing, making sense of the world quantitatively, arguing, asking questions, creating
representations, following procedures, and much more, all can be seen as essential,
drawing on metaphors that value diverse ways of understanding number, data and space
ought to be possible. Another possibility may be to hold each other accountable for the
meanings of the words we use to describe children. We can ask each other to explain
what it is we mean when we call children “low” or “immature.” What are we saying he
doesn’t understand? What is it that he does know?

2. How can we talk about learning in mathematics over time without drawing on notions

of development?

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Stage theories seem particularly dangerous in terms of positioning children. When
theories, like the van Hiele levels, are repeated in classrooms and textbooks, they take on
the power of physical law. Students who do not achieve stages on time or in the correct
order are seen as requiring remediation. Fendler (1999) asked why we choose to see these
children as behind, rather than seeing them as empirical evidence that theory is wrong.
Rather than labeling poor and minority children as “immature” because they, as a group,
do not solve problems in similar ways to middle-class, majority children, we might
instead ask explore these differences as a path toward understanding the variety of ways
that children come to understand the world quantitatively. For some time, literacy
researchers (e.g., Cazden, 1988; Dyson, 1995; 1997; 2003; Heath, 1983) have moved
away from talking about differences in the ways that children come to the written and
spoken language as evidence of maturity (or not). Instead, they have documented
differences with the goal of showing the many ways that children move toward
understanding. Dyson (2003, p. 5) wrote

I hope to turn this [developmental] view inside out, as I look from inside a
particular child culture out toward school demands. In this way, I aim to provide
conceptual substance for a different theoretical view of written language
development, one that normalizes variations in (as well as broadens conceptions
of) children’s literacy resources and learning pathways.

Similar work in mathematics education offers rich possibilities for new research.

3. How can we talk about the interactions of race, class and gender with the learning of
mathematics without reincribing metaphors of hierarchy?

Schooling resources, including money, challenging curricula and skilled teachers, have
historically not been equally distributed across all demographic groups of children. Acts

of racism, sexism, and discrimination against the poor have occurred and will occur

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again. School cultures can be varyingly welcoming to children depending on their lives at
home, and as researchers we need a language to talk about these phenomena However,
by repeatedly singling out certain demographic groups when talking about difficulties in
learning mathematics, we create a shared understanding that some groups, but not others,
have trouble learning. One rhetorical strategy may be to consciously create “incoherence”
in our writing, such as in the following phrase: “boys as well as girls, minority students as
well as majority students.” Small moves like this may at least work to challenge some of
the unconscious connections reinforced in a lot of writing. Another strategy may be to
become more localized in the ways that we talk about students so that we write not about
low-income, minority students, but about particular students in particular places -- who
have certain language practices, schooling histories, family lives, fliends, and personal
preferences in addition to races and socio-economic statuses. The more complicated
students become in our discourse the more difficult they will become to rank based on
any one characteristic.

My responses to the questions I posed are not intended to he answers so much as
openings to conversations. Like everyone else, I am immersed in the discourse I am
trying to change. The primary intervention of this chapter is to make visible metaphors of
hierarchy that previously may have gone unnoticed. As Santa Ana pointed out, the more
common our metaphors are, the less likely we are to question their usefulness and
accuracy. As long as our language of ranking remains invisible, it works to reinscribe a
particular set of power relations — one where some students (identifiable by physical
characteristics) are seen as mathematically capable and others are not. Foucault

(1983/1996, p. 144) said that to identify power relations was “to put them back in the

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hands of those who exercise them” and that is my hope for this chapter — that
mathematics educators recognize the ways in which our words and practices exercise

power in the shaping of how w -- and our students -- think about young children.

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CHAPTER SIX

PROBLEM SOLVING AS PERSUASION

 

“ We problem solve because we’re problem solvers. ”
-- George Bush, in the aftermath of Hurricane Katrina
Many references to problem solving in education literature contain little more
information or complexity than this tautology offered by the president. In the Principles
and Standards (2000) written by the National Council of Teachers of Mathematics, the
authors told us that “one cannot solve problems without understanding and using

mathematical content,” (p. 7) and that “problem solving in grades 6-8 should promote

 

opportunities for mathematical learning” (p. 256). The authors of Adding It Up, a report
on the state of elementary mathematics, contribute the statement that “problem solving
ability is enhanced when students have opportunities to solve problems themselves and to
see problems being solved” (2005, p. 420). It is difficult to take exception to or to be
surprised by the arguments, flequently made by those advocating reform mathematics,
that in order to solve math problems one has to use math; that students should learn math
when doing problems; or that to learn to solve problems students should solve problems
and see people solve problems. Few have argued, recently or in the past, that students in
mathematics should not do problems or that they should not learn flom them. This begs
the question of why there has been such a great need to make arguments in favor of
problem solving and problem-based teaching when there is so little evidence of anyone
seeking to do away with the use of problems in school mathematics.

In fact, the argument in the literature is not actually about the need for problem

solving in mathematics, but about the kind of problem solving that students should be

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doing. This can be seen in the modifying adjectives many writers attach to problems and
problem solving in an attempt to differentiate the kind of problems they are talking about
flom those described by some unnamed other. For instance, there are “real problems”
(Ball, 1988, p. 41), “genuine problems” (Hiebert et al, 1997), “non-routine problems”
(Adding it up, 2005, p. 126), “contextualized problems” (Ibid, p. 327), “challenging, but
accessible problems” (Ibid, p. 412), “interesting and well-selected problems” (NCTM,
2000, p. 182), and “rich and appropriate problems” (Ibid, p. 185). These adjectives might
be taken together to describe desired problem solving -- with routine, unchallenging,
boring, and fake problems as the opposite kind, although arguments in favor of these
sorts of problems are hard to find. In addition, most of the reform literature argues that
the good kind of problem solving is that which has content embedded in it. That is,
problem solving should not be taught as separate from content, such as number, geometry
or data, but used in the teaching of these mathematical strands (NCTM, 1991, 2000;
NCR, 2001; Romberg, 1999). The purpose of this chapter is to examine flom a rhetorical
perspective the kinds of problems used in the elementary class that Diana and Sara taught
and the methods class Sara attended. My aim is to provide new ways for thinking about
the kinds of problems educators use. By looking at problems as means of persuasion, I
hope to break out of binaries, where regardless of the particular adjectives chosen
(rich/boring, challenging/routine, genuine/decontextualized) some problems are seen as
good and others as bad. Instead, I want to offer new language for talking about the work
that problems are doing in mathematics classrooms.

If we see teaching as an act of persuasion designed - to list a few possibilities --

to cause students to adopt a belief, accept an idea as true, practice a skill, or feel pleasure

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while working (Fendler, 2006), then we can see problems as one of the means of
persuasion available to teachers. They are not, by any stretch of the imagination, the only
means available. Teachers could lecture, demonstrate, model, ask students to read
expository text, or employ a host of other pedagogical practices. Aristotle called rhetoric
as “the faculty of observing in any given case the available means of persuasion” (Book I,
Chpt. 2, ln. 1). This is one way of seeing the work of teaching — surveying and choosing
available means to achieve desired ends. In the case of the third grade classroom, the
desired ends were spelled out pretty clearly for Diana and Sara by the Grade Level
Content Expectations offered by the state, the Pacing Guide provided by the district, and,
should there be need for further guidance, the standards offered by NCTM. These
documents asked that third-graders be persuaded to do a number of things, including
recognize benchmark temperatures, read thermometers, use multiplication to describe
grouping situations, know multiplication facts from memory, talk about mathematical
ideas, explain solution methods, and make and test conjectures (Lansing School District,
2005). The extent to which these documents were persuasive to Diana and Sara is another
question, and one that is laid aside in this chapter to focus on other issues. Certainly, both
Diana and Sara did refer to these documents in their lesson plans and in conversations
with me, and many, if not all, of the lessons taught in their classroom could be reasonably
described as working toward one of the goals stated in the standards documents. David,
the university methods instructor, laid out his desired outcomes in his syllabus. He said
he wanted his intem-year students to “make problem solving an important part of their
mathematics curriculum,” to help children realize “that mathematics is all around them,”

and to understand “important mathematical concepts.” In setting out these goals, he also

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referred to the state and NCTM standards, and, like Diana and Sara, routinely used
problems as a means of persuasion in his classroom.
Problems as Persuasion

This chapter examines three kinds of problems used in both the elementary and
methods classrooms, which differed in the arguments they made and in their modes of
persuasion: practice problems, process problems and concept problems. I want to point
out here that I am not examining how teachers persuaded their students to complete these
problems. That is, I am not interested in how the teachers got their students to do their
work in the classroom. Rather, I am naming the argument that each of these problems
makes in relation to mathematics and examining the means by which the problems work
to persuade students. To do this, I adopted some analytic tools flom Aristotle, who
described three modes of persuasion open to speakers: logos, or logical argument; pathos,
or appeals to emotions; and ethos, or authority of the speaker and the context. Examining
these modes of persuasion allows differences in problems other than those typically
described (genuine or false; challenging or routine) to be discussed. The following
section of this chapter looks closely at each of these kinds of problems, identifying key
features, the arguments being made, and the modes of persuasion.
Practice Problems

The problems used in the elementary and methods classrooms that I identified as
practice problems are those many adults today would recognize as having made up the
primary mathematical work of their childhoods and are probably what many mathematics
educators have in mind when they issue calls for different kinds of problem solving.

These problems tended to be short, mostly (although not always) presented as “naked

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numbers,” assigned in sets, evaluated for correctness as well as speed, and little
discussed. Diana and Sara both frequently assigned practice problems in the elementary
classroom. Sometimes they were a feature of the textbook, which began each lesson with

a “quick review,” where students were asked to solve four or five similar problems, such

asthese:
l. 7x__=42
2. _x5=45
3. _x8=32

4. 9 x _ = 63 (Andrews et al., Hartcourt School, p. 368).

Sara, who taught the lesson where these problems were assigned, responded to
students both in terms of the correctness of their answers and the speed with which
they offered them. In the following episode, Sara asked Mia to solve the first
problem.

Mia: (almost instantly): Six.

Sara: Six. Good for you. See how quick Mia was with that. Did she have to skip
count in her head or use her fingers?”

A few children said “no.” And Sara added: “Nozz.”

Sara: She knows that fact and you should all be knowing your facts. Good job,
Mia.

Sara called on Ashanti next, after pulling her name out of a jar.

Ashanti: (after about ten seconds) “Nine.”

Sara: “You’re right. Good for you. Did you skip count in your head there?”
Ashanti nodded slowly.

Sara: That’s an okay way to do it, but you should still know the facts because it’s

way faster.

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Unlike her responses to students’ solutions of other kinds of problems, here Sara did not
explicitly value varied solution methods. In fact, rather than asking Mia and Ashanti to
explain how they solved the problems, Sara assumed what their process had been and
evaluated their effectiveness in terms of speed as well as accuracy. The role of speed in
defining practice problems can be seen both in teachers’ attention to how quickly
students completed the tasks as well as in the number of tasks given. In the lesson above,
Sara expected students to solve four problems in about four minutes. Timed tests of
addition, subtraction and multiplication facts were another common site for practice
problems in the elementary classroom. During these activities, students were expected to
complete a hundred problems in five minutes. Students tried each week to complete a
higher number of correct problems than in their previous attempts.

Although most practice problems involved one of the four operations emphasized
in elementary mathematics, some did not. For instance, over the course of the year
students completed sets of fifteen to twenty problems that involved reading
thermometers, measuring lines, and identifying shapes. The lack of discussion
surrounding these problems, the length of time given, the number of problems assigned,
and their repetitive nature marked these as practice problems despite non-operational
content. David assigned practice problems less often in the methods classroom than Sara
and Diana did in the elementary school. However, he did on a few occasions engage his
students in mental arithmetic, where he flashed a problem on the overhead for ten to
twenty seconds and expected students to rapidly compute an answer. As in the
elementary classroom, these problems, such as “28 x 31 =” and “46 + 87 =” were

relatively simple for students to interpret and each problem in the set was quite similar to

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others. Like Diana and Sara, David explicitly valued speed when assigning these kinds of
problems. David called on Sara after flashing a two-digit addition problem on the
overhead. Sara reported her answer (which was correct), saying that she had just carried
the numbers in her head. David replied: “But that’s not why we’re doing it. It will take
longer. You just did it by the algorithm, mentally. I’d like you to get good at doing it the
other way.” It would be possible to assign problems for mental arithmetic and to not have
them be practice problems. For instance, conversation afterward might focus on ways to
add that reinforced important place value concepts. However, here, David rejected Sara’s
method, not because it lacked explanatory power, but because it took longer.

If practice problems are a means of persuasion, then the argument they are
making is that a certain kind of mathematical knowledge needs to be automatic,
inexplicit, and unthinkable. The reading of thermometers, single-digit addition and
subtraction for third graders and double-digit addition for beginning teachers are not
problems that should be figured out but things that, in Sara’s words, you need to “be
knowing,” a statement with which the Lansing School District Pacing Guide concurs. Van
de Walle, author of the most widely-used elementary mathematics methods textbook,
wrote about fact mastery in terms of students “recalling the fact without being conscious
of using a strategy” (Van de Walle, 2006, p. 183). Mathematician Wu, who probably
would not agree with much in Van de Walle’s reform-oriented methods textbook, took a
similar stand on basic facts, writing that “the automaticity in putting a skill to use flees up
mental energy to focus on the more rigorous demands of a complicated problem” (1999,

p. 2). Practice problems are a means to an end; in order to pursue higher mathematics

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some mathematical acts must become automatic and unconscious so that attention can be
turned to other things.

If this is so, we can then ask about how these problems go about persuading
students to make certain knowledge bits (multiplication facts or mental arithmetic
processes) unconscious and automatic. The logical argument behind these problems is
that if students think a thought over and over again, it will soon become automatic. This
is mathematics as athletics. Hawhee (2002), a professor of English, talked about the
learning of rhetoric as “a bodily art: an art learned, practiced, and performed by the body
as well as the min ” (p. 144). Similarly, the knowledge intended to be created through
practice problems is a bodily art, where the response to a presented problem flows
automatically from the pen, rather than the mind. Like learning to shoot a basket or hit a
baseball, one small skill is practiced over and over, leading to mastery. The argument is
not this these problems are real mathematics, any more than taking swings in a batting
cage is real baseball, but that repetition will lead to habits that will be useful in other
pursuits. A student who completes a sheet of ten multiplication facts each day may be
persuaded to automatically recite that two times five is ten.

The emotional aspects of persuasion related to practice problems do not need to
be vocalized by either teachers or students in order to play a role in convincing students.
When doing long sets of practice problems, students routinely checked their own work
against that of neighboring students. Generally, this checking focused on the quickness
with which problems were completed rather than the answers being produced. Once
while working on a set of division problems, Aliah looked over at Mia’s paper and said:

“Dang. Mia’s on 21. I’m on 17. I do not get this.” During another assignment, Jerome

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finished a worksheet filled with problems on reading thermometers and looked around
the room. He noticed that nearly everyone else was still working and commented. “I’m
smart at math today!” In both cases, Aliah and Jerome’s emotional responses to the
assignment centered around speed rather than correctness, even though neither Sara nor
Diana made explicit reference to finishing quickly on those particular assignments.
Because speed was an important criterion on practice problems, students could feel pride
(or shame) by monitoring their own progress in relation to their peers without public
acknowledgement (or criticism) from the teachers. The emotional payoff of pride for
finishing quickly was available to anyone who worked steadily through an assignment
and worked to persuade students to come to know certain information by repeatedly
doing similar problems.

Practice problems drew on a number of accepted norms to persuade students to
come to know certain mathematical facts. The ethos of the elementary school in
particular is central to the persuasive power of practice problems. For example, the
student math book was filled with practice problems. In addition to the quick review on
each page, nearly every page offered a set of ten to twenty similarly-constructed
problems. Although these problems could possibly have been assigned for purposes other
than practice, the large number of problems and the relatively straightforward directions
on each page clearly marked them for this purpose. In addition, students came in to the
third grade expecting to learn their multiplication facts. Many reported using flash cards
at home and told stories about their older brothers and sisters learning the multiplication
facts. That these sorts of problems were an entirely expected part of third grade

mathematics played an important role in the problems persuasiveness. Students did not

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question having to do these problems over and over again. The repetition confirmed their
expectations. This was also true in the methods class, where interns routinely challenged
the instructor when doing problems related to proofs about prime numbers or algebraic
explanations of number games, but never asked David why they needed to be able to add
two-digit numbers in their heads. The expectedness of the problems was in and of itself
persuasive.
Process Problems

Unlike practice problems, process problems were not concerned with getting
students to know a particular bit of mathematical information, but with causing them to
execute certain ways of thinking that are currently seen as mathematical, such as
organizing information, finding patterns, explaining one’s reasoning, and using multiple
solution methods. Although Diana and Sara did not name the other kinds of problems
used in their classroom, they did call these “problem-solving problems.” Similarly, David
would mark these problems as separate flom the other work that went on in the methods
classroom by announcing that they were about to do “problem solving.” These problems
could be recognized as a kind because they were planned, were not linked to any
particular mathematical strand, and were always followed by discussions that focused on
solution methods rather than answers. In addition, when introducing these problems,
teachers called on students to solve them in multiple ways, unlike when introducing other
kinds of problems. A couple of examples of process problems flom the elementary
classroom include:

a You have 50 carrots broken into 2 pieces, 15 celery stalks broken into 4 pieces, 9
potatoes cut into 10 pieces and 2 onions cut into 50 pieces. How many whole

vegetables did we have? How many vegetable pieces did we have?

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o If A = 1 cent, B = 2 cents, C = 3 cents and so on, what is the value of your first
name? Are there any words worth $1.00?

And from the methods classroom:

0 In a room with 20 people how many handshakes would it take for all the people to

meet with each other? What if there were n people?

0 During a census, a man told the census taker that he had three children. When
asked their ages he replied, “The product of their ages is seventy-two. The sum of
their ages is my house number.” The census taker turned ran outside to look at the
house number and said, “Using the information you have given me, I cannot tell
their ages.” The man said, “I should have told you the oldest likes angel food
cake.” Hearing this, the census taker promptly replied, “Now I know their ages!”
What was his solution?4

For each of these problems, students were encouraged to solve them in multiple
ways, and, in fact, much of the discussion afterward focused on identifying the variety of
strategies students used. In both the methods and the elementary classroom, students were
called to the front of the room to present their work, which never happened with practice
problems and rarely occurred with concept problems. During these kinds of discussions,
the most common teacher question was: “Who has another way to solve this problem?”
Unlike practice problems, the revelation of the answer was not considered the final word.

That these problems were planned was also an important identifying

characteristic. Unlike practice problems or concept problems, these problems were

 

’ For those of you scratching your heads over this one, David’s intended solution involved students
generating sets of three numbers where the product was 72. The sets “2, 6, 6” and “3, 3, 8” both result in a
sum of 14 (presumably the house number); thus, the census taker’s confusion. The angel food cake clue is
intended to signal that there is an oldest so the answer must be “3, 3, 8.”

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always written ahead of time -- in plan books and on overheads for Sara and Diana and
on handouts for David; these problems were never produced on the fly in either
classroom. For the most part, these problems were also not written by the teachers ahead
of time, but culled from other sources. In fact, many of these problems have a handed-
down feel to them, which David referred to when he introduced these sorts of problems,
saying they were for “special lessons. And you’ve got to have file of these problems for
these special lessons, like when someone’s coming in to observe you and you want to
show real problem solving.” The “handshake problem,” which David used in the methods
class, is a particularly clear example of a “special problem.” The authors of Adding It Up
(2005, p. 107) used the problem as an example, noting that it is one of the more popular
problems found in the mathematics education literature. It can also be found in the Van
de Walle text, in a lesson shown on the instructional video Power-fill Practices in
Mathematics and Science put out by Carpenter & Romberg (2004), and in many other
books, articles, and curricula about problem solving (e.g., Krantz, 1997; Slavit, 1999;
Yarema, Adams & Cagle, 2000). With the guidance of a teacher, young students may be
able to find an algebraic expression or a numeric pattern that predicts answers. However,
the problem is not generally recommended for use with middle or high school students,
who may be working on formalizing patterns algebraically, but with much younger
children. (The lesson in the video involves third graders; David suggested it was
appropriate for children in the primary and intermediate grades.) This problem helps to
illustrate the focus on processes rather than mathematical concepts in these kinds of
problems. (Of course, the problem could be used for content—oriented purposes in high

school or undergraduate mathematics classes.) If the purpose of the handshake problem

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was to teach a concept, we might reasonably ask what content this problem is supposed to
teach. Certainly to solve the problem children would have to understand addition and
potentially numerical patterns. It could be used, with guidance flom the teacher, to build
algebraic reasoning. However, this problem’s prevalence in the elementary mathematics
education literature seems out of proportion with its usefulness in getting at any of these
concepts. In fact, the authors of Adding It Up speculate that the problem is seen so often
“because it can be solved in so many ways” (2005, p. 107). This virtue has little to do
with the problem’s effectiveness at revealing important mathematical ideas within a
particular strand, such as algebra, number or geometry, and much to do with the learning
of problem solving as a domain in and of itself.

Similarly, the vegetable soup and adding up names problems used in the
elementary classroom were not designed to teach students particular mathematical
concepts. These problems were assigned in the first weeks of the year, when Diana
alternated between doing these kinds of problems and reviewing for the state’s
standardized test, which was scheduled to be given in October. Although each of these
problems could have been used to teach particular mathematical knowledge, this was not
the focus of the discussions that occurred after problems had been solved. In the
vegetable soup problem, Diana called four students to the board whom she had
determined had each solved the problems in different ways. The first three students had
grouped numbers in various ways to add. Ben had used multiplication. In commenting on
Ben’s solution, Diana said:

Ben, you did this in a really sophisticated way. But that might bring it back to the
other people. This would be a wonderful thing to take notes on because Ben did
something that most of the rest of you didn’t do. Because you’re in third grade

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and you haven’t done times a lot I’m thinking a lot of third graders in early
October, you don’t know any times yet and that’s okay. But I still thought this
was a problem you could do, and some of you did do it in different ways. That’s

what matters.

Here, Diana did not use Ben’s solution to build on the connection between multiplication
and addition, as she and Sara would do later when they started the unit on multiplication.
Instead, Diana reinforced that there were many ways to solve the problem. She valued
Ben’s solution for being “something that most of the rest of you didn’t do,” and
suggested that students take notes on it for this reason, even while noting that most of the
children probably would not have enough experience to understand the multiplication.
She then went on to discuss other children’s solutions, in each case highlighting the
strategies students used, such as skip counting, drawing pictures or grouping numbers,
which made them different flom the previous ones.

The argument these problems make is that the way one goes about solving a
problem in mathematics is as important as the answer produced. Process problems seek to
persuade students both to do certain things, such as to record their thinking, to explain
their reasoning in words, and to interpret complicated problems; and to believe certain
things, such as to value multiple solutions, originality, and showing one’s work. The
logical argument here is that if a problem is complicated enough students will not solve it
in identical ways. Thus, the complexity allows students to see multiple solutions, to
pursue originality, and to perfect particular mathematical strategies or behaviors. Unlike
practice problems, which draw on athletic metaphors, process problems present
mathematics as a simulation. Students cannot actually do the work of mathematicians,

because they are not working on the edge of what is known; however, process problems

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are designed to create an environment where students feel and act as if they are doing real
mathematics. These simulations of mathematical work allow the behaviors and beliefs
revealed to become objects of discussion, which contributes to the problem’s ability to
persuade students to think and act in certain ways.

As in practice problems, feelings of pride and shame are drawn on as ways of
persuading students to adopt desired beliefs and to engage in expected processes,
although the way these emotions are incited is different. In process problems, students are
encouraged to feel pride when their work is revealed in public, especially if it shows an
original (or “sophisticated”) solution. When Diana called for volunteers to demonstrate
their answers to the vegetable soup problem, many children who raised their hands were
turned down because their solution was “the same as” someone else’s. Denying them the
opportunity to show off their work for this reason served to motivate students to attempt
something original in the future. An additional emotional aspect of persuasion in these
problems involved the ability of the problems to invoke feelings of “smartness” or
“durnbness” in students. Much more so than the practice problems, process problems
seemed to appeal to feelings of competence. This may be because all students were able
to have some success with practice problems, so that the speed with which they were
completed marked only degrees of competence. Whereas with process problems, students
who could not figure out what to do often had extreme emotional reactions. This was true
in the both the elementary classroom and the methods classroom, where, after reading the
angel food cake problem, one intern remarked: “I haven’t the faintest idea what to do.
And so now I feel dumb in math again.” In contrast, this same student’s inability to

compute the mental arithmetic problems as fast as David expected produced genuine

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laughter on her part. The teachers also had different expectations for competence with
these kinds of problems. When doing practice problems as a class, Diana and Sara drew
names randomly out of a jar to call on students, which demonstrated that they expected
all students to be able to solve these problems. However, with process problems the sticks
were never used and often Diana and Sara checked a students’ paper before asking him or
her to contribute to the discussion, communicating that all students were not expected to
be able to solve these problems, or at the very least that they were not expected to do so
in a way that would be interesting to the rest of the class.

The ethos of process problems was probably not as persuasive to children as that
of the practice problems. Using Bahktin’s language, process problems have the “taste” of
the university about them. These problems often trickle in from universities, either fl'om
reform curricula written by mathematics educators or else flom more informal
interactions. In fact, of the four lessons I observed that were centered on process
problems, two were introduced by university seniors who had been encouraged to use the
problems by their methods instructor. Process problems do not look like the mathematics
expected in the elementary classroom. They are not straightforward algorithms; nor are
they the typical three-sentence word problems seen most commonly in traditional
textbooks. Neither the third graders nor the interns saw these problems as expected parts
of the elementary mathematics curricula, and as a result, teachers in both contexts had to
do some extra work to persuade students to engage in these problems. Diana and Sara did
this by linking these problems with appealing activities, such as making vegetable soup
or eating watermelon (after doing a problem that required estimating the weight and

length of the fruit). Diana also made explicit statements to children about the value of

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these problems saying things like “with these problems you have to really use your
brain!” David made similar appeals to the interns, describing the angel food cake problem
as “not a real-world problem. This is a really-get-you-thinking problem.” Because
process problems were not an expected part of mathematics class, teachers had to draw
more explicitly on their own authority as experts when assigning these kinds of problems.
Concept Problems

Concept problems were the most difficult to identify because they had much
greater diversity in their textual features than the other two types and because they
revealed themselves as concept problems only within the contexts of particular lessons.
They are also a kind of problems that seems to disappear flom binary discussions of
problems that compare only practice problems and process problems. Concept problems
are not designed to make certain knowledge automatic like practice problems; nor are
they aimed at promoting particular mathematical strategies or beliefs. Concept problems
were designed to helped students build understandings of particular bits of mathematical
content, often by building representations, vocabulary, or connections among ideas.
Concept problems were typically assigned during a particular unit of study in the
elementary classroom or as part of a lecture on a particular topic in the methods
classroom. Sometimes they looked like process problems, as in these examples:

0 A million seconds is about 11 days. How many days do you think a trillion
seconds would be? (methods class)

0 You have 3 different shirts and 4 different pants. How many outfits can you

make? (elementary class)

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0 Mrs. Emerson bought 36 cookies with green flosting. She wants everyone in our
class to get an equal amount. How can she do that? Show your work. (elementary

class)

Sometimes they looked like practice problems:

0 Which is greater: 3_4 or Q
55 89 (methods)
0 21 + 7 = (elementary)

And sometimes they didn’t look like problems at all:
0 How can you show children that you can’t divide by zero? (methods)
0 Why do people usually start adding fl'om the right? (elementary)

What all these problems have in common is that they were aimed at helping students to
build particular concepts. Rather than valuing diverse ways of solving the problems, the
teachers in each of these cases were looking for students to call on particular ways of
representing or talking about the mathematics and used the discussions afterward to work
toward this purpose.

Concept problems seemed to embody the calls in Standards documents to teach
mathematics through problem solving (N CTM, 1991, 2000); however, these problems
did not often provide opportunities for students to solve problems in original ways or to
practice particular strategies, like organizing data or finding patterns. Because teachers
working on concept problems used the discussion time to build ideas related to the
content under study, mathematical processes often took a backseat. Students were
encouraged to solve concept problems in similar ways so as to build understanding of the

content under study. In the following episode, which occurred during the first week of a

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unit on multiplication, Sara gave students the problem: There are three flowers, each with
six petals. How many petals were there?” She asked students to represent their answer in
three ways.

After being called on, Ben reported that he did six times three and got 24. Sara
expressed surprise and then Ben corrected himself, saying “No, I mean 18.” Sara wrote
the expression “6 x 3 = 18” on the board and asked students how else they had

represented this problem. Charlie suggested counting by sixes.

Sara: Skip count by sixes, if you’re good. It’s kind of hard. That’s kind of a high
number to skip count with. But you could go 6 12 18.

Sara: How else can you write it?

Chacoria, Charlie, Khassan, and Evan raised their hands, and Sara called on
Chacoria, who said “Three groups of six.”

Sara: Right. Cause we have three groups of six (Sara wrote “3 groups of 6” on the
board.) You can easily say three groups of six. I can think of two other ways to
write what you have up here.

Caitlin and Khassan raised their hands. Jennifer called on Khassan.

Khassan: Count by twelves.

Jerome (calling out): Count by twelves?

Sara: Are there an even number of twelves in there?

Khassan: We already know the six plus six equals twelve, so-

Sara: Whoa, whoa, whoa. We can only skip count if there are an even number of
those things. Do you want to think about that a little bit, Khassan? (turning)
Caitlin?

Caitlin: You know six and six is twelve. So you can do twelve plus six is
eighteen.

Sara: So Caitlin does know that six plus six is twelve (gesturing at picture) so
she’s going to make these two twelve. And she’s going to add the last six to
twelve. You could do that.

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Aliah, whispering to herself: Six plus six plus six.
Sara: What Aliah?

Aliah: Six plus six plus six.

Sara (writing it on the board): What do we call that?
Many children, including Aliah: Repeated addition.

Sara: Good. How about you guys do one on your own?

In the two lessons previous to this one, students had been asked to draw pictures of
multiplicative situations and to write an addition sentence, a multiplication sentence and a
word sentence to describe the picture. According to Sara’s lesson plans, the goal of these
lessons was to help students develop an understanding of multiplication as a process of
adding equal-sized groups. In the episode above, Sara responded in different ways to
answers that used the vocabulary of the previous days’ lessons and those that did not.
Charlie suggested skip counting, and Khassan and Caitlin pursued this idea. These
answers could be considered original in reference to this problem because most children
in their journals used one of the representations that the class had used the previous day.
However, unlike with process problems, originality was not publicly valued here. Sara
cut Khassan off as he attempted to articulate his thinking about the skip counting. She
acknowledged Caitlin’s explanation of the solution as correct, but did not record it on the
board. Nor did she record Charlie’s skip counting by sixes. Instead, she responded to
Aliah’s whispers in order to get repeated addition on the board because the connection
between multiplication and repeated addition was an important part of this lesson. After
solving the problem in this way, Sara assigned a similar problem to the students. As they
worked, she encouraged them to produce as many of the representations that had already

been produced as possible, asking questions like: “Can you show it with a picture? What

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would the addition sentence look like?” The focus of the instruction was on building links
between various representations of multiplication rather than on problem-solving
strategies. This was not because Sara was unable to focus on mathematical processes like
reasoning and justification. She had learned from Diana to ask questions like: “How did
you get that answer?” and “Who thought about this in a different way?” however, in this
lesson, she did not ask these kinds of questions.

Over time, as children became more familiar with multiplication and the kinds of
story problems that are typically used to represent these ideas, problems similar to the
flower petal problem came to be treated as practice problems. During early lessons in the
unit, Sara did not encourage students to solve problems quickly. She rarely assigned more
than two or three problems in a 45—minute lesson. As students worked, she asked them to
go back and count, to draw lines between numbers and pictures, and to think about how
to describe the problem in words. However, a few weeks later, problems like these did
become practice problems for most students. Toward the end of the multiplication unit,
Sara passed out a worksheet of ten story problems all dealing with multiplication.
Although students were asked to “show their work,” most completed this worksheet
quickly and none of the problems were discussed as a whole class.

The logical argument behind content problems is that important concepts in
mathematics are best learned through solving problems that related to those concepts. In
our time, this seems like an obvious argument; however, in the early days of mathematics
education, educators believed that students learned mathematics deductively, that is, by
following a step-by-step argument. Edward Thorndike, an early mathematics educator

whose name has become synonymous with skill and drill, provided examples of both

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deductive and inductive ways of teaching mathematics in his book, The Psychology of
Arithmetic. A deductive explanation of multiplication (for the problem 623 x 3) would
be:

For convenience we write the multiplier under the multiplicand and begin with
units to multiply. 3 times are units are 9 units. We write the nine in the tens’ place
in the product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8 hundreds.
The 1 thousand we write in the thousands’ place and the 8 hundreds in the
hundreds’ place in the product. Therefore, the product is 1 thousand 8 hundreds, 6
tens and 9 units, or 1869 (Thorndike, 1922, p. 61).

In contrast, Thorndike argued that inductive ways of learning mathematics would be
more effective. For example, he described an inductive method of teaching with
multiplication as beginning with a problem like “The children of the third grade are to
have a picnic. How many sandwiches will they need if each of the 32 children has four
sandwiches?” (Thorndike, 1922, p. 61). He suggested following problems like this up
with multiplication problems written in algorithmic form, in words, and as repeated
addition. The process he described is not so very different flom what went on in Diana
and Sara’s classroom; nor is it so very different from the argument that many advocates
of problem solving make today. In their book on Cognitively Guided Instruction,
Carpenter and his colleagues offer problems like: “Our class has 5 boxes of doughnuts.
There are 10 doughnuts in each box. We also have 3 extra doughnuts? How many do we
have all together?” (Carpenter, et al, 1999, p. 60) as a strategy for teaching multiplication
to children. The extra three doughnuts make this problem slightly more complicated than
the ones offered by Sara and Thorndike; however, the heart of the argument, that children

learn mathematical concepts through working problems that connect those concepts to

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concrete representations (flowers, sandwiches, doughnuts) remains the same. To be sure,
Carpenter and his colleagues advised that students have plenty of time to invent
algorithms before being introduced to the formal procedures, but the differences in
beliefs behind how children learn do not seem to be as significant as are sometimes
described by current advocates of problem solving (N CTM, 2000) or algorithmic practice
(Wu, 1999).

Concept problems based their emotional appeal by linking feelings of pride to
inclusion and shame to exclusion. Because the goal of these problems was often to
develop shared representations of particular ideas, students worked much more
collectively on these problems than on others. In the counting petals problem, Sara wrote
some solutions on the board, but not others, despite the fact that they were equally
correct. The desire to be included in the classroom community for these problems was a
mode of persuading students to offer expected and shared representations of the
mathematics. This was as true in David’s classroom as in the elementary one. For
example, when he asked his students to develop a child-fliendly argument for why
division by zero was impossible, he did not break students into groups or partners, as he
typically did when doing process problems. Instead, he asked students to work
independently for a few minutes and then to share ideas with the class. He guided
students toward two arguments, one based on repeated subtraction and one based on the
impossibility of imagining zero groups of six. He outlined these two arguments on the
board as people added to them, but did not make notes about an argument based on arrays
put forward by one of the students. Inclusion and exclusion were also promoted through

the use of “we-oriented” language. For instance, when Chacoria and Aliah responded to

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Sara with expected representations, Sara replied to them with “we” statements. However,
when Caitlin made her argument about skip counting, Sara located it as something that
Caitlin was doing, although it would work. Similarly, if you go back and look at the
conversation presented in the section on practice problems, you will see that solutions are
attached to individual students, rather than a “we.”

As a mode of persuasion, the authority of the teacher was drawn upon most in
concept problems. Much more than with the other two kinds, Sara, David, and Diana
were likely to write these kinds of problems in the midst of a lesson, rather than relying
on plan books or texts. Often these problems were written to address a misunderstanding
or perceived need on the part of students. For instance, once in the methods class during
the discussion of a process problem, a student suggested comparing two fiactions by
seeing how far apart the numerator and denominator were. David stopped the discussion
about the process problem and wrote on the overhead: “Prove or disprove Hannah’s
claim.” This problem was designed in the moment, in response to a particular need; it
was not a “special” problem that needed to be taken from a file. The responsiveness of
the teacher in assigning these problems became a way of persuading students to learn the
concepts because the problems could be produced just in time. Similarly, in debriefing
discussions, the teacher’s authority on what was mathematically useful was much more in
play during these kinds of problems. Practice problems tended to have single answers, so
there was little role for the teacher to differentiate between subtle interpretations. In
process problems, virtually any student action might have been picked up as
mathematically interesting as long as it was different. But in concept problems, Diana,

Sara and David generally made decisions ahead of time about which representations or

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ideas would be most pedagogically useful and they used their authority as teachers to
focus student attention on these predetermined ideas.
The Alchemy of Problem Solving
In discussing school subjects, Popkewitz (2002, p. 262) argued that disciplines,
such as mathematics, science and music, undergo a transformation in schools so that the
mathematics taught in classrooms is no longer the discipline of mathematics as practiced
by mathematicians, but an entirely new discipline, that of school mathematics. He wrote:

[A]n alchemy occurs as the knowledge of an academic field moves into the
school. School subjects are organized in relation to expectations related to the
school timetable, conceptions of childhood, and organizational theories of
teaching. The question of academic or disciplinary fields is transmogrified into
school psychologies of instruction and theories for changing the dispositions and
characteristics of the teacher and child. The magic of the transformation is to
reconfigure the academic fields in schools so that only the namesake appears, as a

ubiquitous doorplate to mark a house.

Just as entire disciplines underwent alchemy when they were brought into the
schoolhouse, I want to argue that the practice of problem solving underwent a similar
alchemy. The problem-solving that mathematics educators write about today is not a
practice that has always been present in schools or in mathematics, but one that emerged
at a particular time in history through the work of researchers in mathematics education.
The presence of this new kind of problem solving can be seen in the three kinds of
problems that showed up in the classes I observed. My identification of practice, process
and concept problems is not an odd artifact of the particular contexts I studied, rather it is
a reflection of the broader discourse in mathematics education. The existence of this new

kind of problem solving — where the processes that students use to solve problems are

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considered as much a part of the content to be learned as the mathematical ideas the
problems address — is deeply entrenched. Process problems are assigned in schools
because practice problems and concept problems are not seen as adequately addressing
the content entailed in this new conception of problem solving. Initially, this statement
may not seem credible because of quantity of statements made that argue that problem
solving cannot be separated flom mathematical content. For instance, Van de Walle
(2004, p. 38) wrote: “It is important to understand that mathematics is to be taught
through problem solving. That is, problem-based tasks or activities are the vehicle by
which the desired curriculum is developed.” Similarly, in the section on problem-solving,
the writers of the Principles and Standards said that problem solving “should not be an
isolated part of the mathematics program” (p. 52). However, if one looks more closely at
how these (and other) texts are set up, at the type of problems discussed, and at what is
expected to be learned from problem-solving, the claim of the inseparability of problem
solving from content becomes less credible. And, it is in this separation of problem
solving flom content that the emergence of this new kind of problem solving can be seen.
For instance, although the Principles and Standards describes problem solving as
intertwined with the learning of mathematical content, the document separates
discussions of problem solving flom discussions of content. This means that in the
discussion of mathematics teaching in the primary grades, number and operations is
discussed flom pages 79 to 88, geometry flom pages 96 to 101 and problem solving flom
pages 116 to 121. If problem solving was truly intertwined with the learning of content,
there would be no need for a separate section because there would be very little to say

without a content area to flame the discussion. In the section on problem solving, the

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writers recommend that children learn “to develop a broad range of problem-solving
strategies, to pose (formulate) challenging problems, and to learn to monitor and reflect
on their own ideas in problem solving” (N CTM, 2000, p. 116). Similarly, both the
Michigan Grade Level Content Expectations and the Lansing School District’s Pacing
Guide list “problem solving” as a separate content area flom number, data, and geometry.
All of these documents expect students to learn a particular content about problem
solving. In the discussion of problem solving in the middle school, NCTM describes this
content as building “important problem solving dispositions - an orientation toward
problem finding and problem posing; an interest in, and capacity for, explaining and
generalizing; and a propensity for reflecting on their work and monitoring their solutions”
(Ibid, p. 258). To achieve these goals teachers need ways of monitoring students’
dispositions as well as knowledge, and become likely to select problems for use in their
classroom that make this monitoring possible.

It would probably be impossible to pinpoint exactly when this new kind of
problem solving emerged, but there can be little doubt that Polya’s (1957; 1968) work on
problem-solving heuristics was an important turning point. In a review of research on
problem solving, Kilpatrick (1985) claimed that “everyone in mathematics education who
works on problem solving must come to terms with Polya’s view of problem solving” (p.
7). Polya (1968, p. x) called heuristics “the study of means and methods of problem
solving” and argued that in addition to information (or mathematical content knowledge)
students should be taught “know-how” (or problem-solving strategies). In his work,
Polya outlined a general series of steps that problem-solvers might go through to solve a

problem, no matter what the content, including understanding the problem, devising a

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plan, carrying out the plan and looking back. The articulation of these steps led to
research that examined the impact of explicit teaching of heuristics zmd other general
problem-solving strategies to children. Cyert (1980, p. 5), in setting a goal for this line of
research, wrote “it would be highly desirable to have a way of teaching the problem
solving process.” One of the results of this work in classrooms was the emergence of
problem-solving posters, which outlined the steps of successful problem solvers. Students
could be expected to memorize these steps and to produce evidence of them in their
solutions to mathematical problems. In addition, they could be assessed on their ability to
problem solve, separately from their abilities to articulate understandings about number,
geometry or algebra. Although Polya studied the ways that mathematicians solved
problems in order to articulate his heuristics, few mathematicians have posters of
problem solving steps on their walls which they move through in linear ways. In his
journalistic account of mathematician Andrew Wiles’ struggle to prove Fermat’s Last
Theorem, Singh (1997) described Wiles’ solution path as including long pauses in the
work, walks through the park, and pursuits of seemingly unrelated mathematical ideas.
Many mathematicians might relate to these strategies, but it unlikely any of them will
become codified in schools as part of the curriculum of problem solving. This anecdote
points to one of the ways that we can understand school problem solving as a different
element than problem solving in the discipline of mathematics, for all that researchers
might draw connections between the two.

Lester (1980, p. 302), who reviewed the body of work that explored the teaching
of the problem solving process, listed the key problem solving behaviors identified by

mathematics education researchers: “divergent thinking, blind guessing, identifying a

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pattern, ability to employ abstract analytical reasoning, making use of drawings and
looking carefully at details.” Researchers working on heuristics tried to figure out how
these processes might be taught to children and to what extent teaching them might
support children’s efforts to solve mathematical problems. This work, along with Polya’s
and his followers, helped to create a content area for the teaching of problem solving that
was not part of the available discourse when Edward Thorndike and his contemporaries
were writing about mathematics education in the early part of the 20th century. For them,
mathematical knowledge could only be discussed in terms of the problems students could
solve. The way they went about solving them was not an object of study. For Brownell
(193 8, 1945), a student’s inability to articulate how she found the answer to 8 + 3 was not
indicative of a lack of problem-solving abilities, but of mathematical content knowledge,
which teachers would need to address by building students’ understandings of arithmetic.
Only after Polya’s work did it became possible for teachers to instruct children on a
series of strategies, such as making a chart or drawing a picture, in isolation flom content
in particular problems. The separation between content and process, which was
reinforced through studies of problem solving and problem solving standards, created the
necessity for claims today that problem solving and content must be interwoven. They
would have been impossible to make before this new kind of problem solving, which
focused on process rather than content, came into being.

What does it mean that we have this new kind of problem solving? One
consequence is that it is now common in conversations about reform mathematics to talk
about real problems, non-routine problems, and challenging problems. I believe these

words are intended to signify the difference between practice problems on the one hand

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and process problems on the other. Concept problems are overlooked because they do not

represent extremes in the binary description. Problems that require children to draw on

 

algorithmic or teacher-supported knowledge are seen as bits of potentially essential
content, but not as examples of problem solving. Thus, students who can successfully
engage with mathematical problems in wide range of content areas can also be seen as
lacking true understanding if they do not also demonstrate problem-solving skills. Few
educators, politicians or parents would expect 8-year-olds to be fluent in the sort of
algebraic knowledge required to generalize a solution to the handshake problem; yet,
because the problem is seen as requiring problem-solving knowledge rather than content

knowledge, it is seen as appropriate for third-graders.

 

In addition, multiple kinds of problems provide educators with multiple ways of
comparing students. While practice problems may not differentiate a great deal among
students — only revealing some to be faster than others — process problems offer a wider
scale for ranking, with some students providing original, “sophisticated” solutions and
others offering those that are the same as everyone else’s. Success at process problems
shows who is really good at math. Because this new kind of problem solving, which is
valorized in discussions of reform mathematics, requires that students reveal their
thinking (and preferably thinking that is different flom their classmates), students may
have more difficulty learning how to be successful at this kind of problem solving. It is
far easier to teach someone how to represent a multiplication problem as a picture than to
teach her to be original.

Another consequence of the creation of this kind of problem solving — where

process is seen as separate flom content - is that what it means to know mathematics can

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be mapped in binary terms, with the mathematically correct on one side (practice) and the
mathematically sane (process) on the other. Depending on one’s perspective, algorithmic
practice problems are “simple but sound” (Wu, 1999, p. 2) or “following rules without
thinking,” (Van de Walle, 2007, p. 166) and process problems that encourage invented
algorithms are “potentially harmful” (Wu, 1999, p. 7) or pathways to “rich
understanding” (Van de Walle, 2007, p. 219). Because these kinds of problems have been
set up in opposition to each other, there is little discussion of the role both kinds of
problems might play in the classroom; nor has there been much effort to develop a
vocabulary to talk about problems, such as those I call concept problems, that do not fit
neatly into a dichotomy between procedural knowledge and problem-solving strategies.
In writing about language, Bakhtin (1981) described authoritative and internally
persuasive discourses. He called authoritative discourses “acknowledged truths,” and “the
official line,” while he defined internally persuasive discourses as those we believe to be
true even though they are “backed up by no authority at all” (Bakhtin, 1981, pp. 342-
344). Bakhtin hypothesized that loud and frequent repetitions of authoritative discourses
indicate that the speaker does not find these discourses to be internally persuasive. This
notion raises questions about the reasons for the repeated calls to teach mathematical
content through interesting, appropriate and genuine problems. This is authoritative
discourse in mathematics education; it is the party line, repeated loudly and often. The
discourse that is internally persuasive to mathematics educators is less apparent. The
practice, process, and content problems identified in this study can be found in many
mathematical texts and assessments. In this chapter, I argue that each kind of problem

persuades students to adopt a different stance toward mathematics — whether it is making

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knowledge automatic as in practice problems, adopting mathematical behaviors as in
process problems, or developing understanding of new mathematical content, as in

concept problems. The alchemy that created the new kind of problem solving emphasizes

 

process problems rather than the other kinds. As a result, good teaching is that which
emphasizes process problems and good students are those who successfully solve them.
However, the fact that mathematics educators repeatedly separate out content, such as
number, geometry and patterning, from problem solving suggests that the community is
internally persuaded that other kinds of problems might be equally important.

Acknowledging this in the community’s party line on problem solving might provide a

 

broader field for defining school mathematics, which might in turn make spaces for more
students to see themselves as successful.

Mathematician Wu has been a continual critic of the movement in reform
mathematics, particularly those aspects that emphasize this new kind of problem solving,
where student thinking and explanations are seen as equally or more valuable than the
learning of time-tested algorithms. Wu (1999) has argued that students do not need to
make meaning through original invented algorithms, because standard algorithms are
standard precisely because they embed mathematical meaning within them. For some
time, I have been puzzled by the vitriol that Wu has directed toward the reform
mathematics movement. However, I have come to see that while Wu’s argument (which
echoes Thomdike’s) that meaning emerges flom repeatedly solving practice problems is
counter-intuitive to the way that I came to understand mathematics, it may very well
represent how he came to make sense of the discipline. If this is the case, then his

hostility to reform mathematics and its valorization of “problem solving” (as opposed to

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the solving of problems) may lie in the fact that his ways of knowing have been denied a
place in the party line — however internally persuasive these ways of knowing may be.

I began this chapter by saying that I wanted to draw on rhetoric to find new ways
of talking about problem solving. My goal was to show the practice of problem solving as
one that emerged at a particular time, rather than as one that has been a part of
mathematics since Plato first pulled the discipline down from the ether. By bracketing
problem solving in this way and by demonstrating the ways in which other kinds of
mathematical work persuaded students to enter into the discipline, I hoped to broaden
conceptions of what effective mathematics teaching might be. The greater the consensus
is around the kinds of problems deemed appropriate for schools; the louder and more

narrow the authoritative discourse; and the fewer the possibilities open to students.

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CHAPTER SEVEN
GENRES OF TEACHING

For some time now, teacher educators have been calling on teachers to make
substantial changes in the way they work with students in mathematics. Teachers have
been asked to create mathematical communities, to promote reasoning and problem-
solving, and to allow students to explore their own conjectures (National Council of
Teachers of Mathematics, 1991, 2000; National Research Council, 2005). To support
teachers in making these changes, researchers have paid a great deal of attention to ways
that classroom teachers interact with students. Many of these classroom interaction
studies, as well as the discussions about them, have tended to talk about observed
classroom episodes as representative of teachers’ practices as a whole. In other words,
studies have used particular classroom episodes to locate teachers on a continuum
between old teaching styles, which are seen as focused on procedures and right answers,
and new teaching styles, which are seen as focused on conceptual understandings and
mathematical processes. Similarly, procedural classrooms have been described as
producing students with algorithmic understandings; whereas, conceptual classrooms
have been described as producing students who can problem solve, reason, and justify
(e.g., Cobb et al, 1991; Heaton, 2000; Hiebert, Morris & Glass, 2003; NCR, 2004; Stigler
& Hiebert, 1999).

Most of these interaction studies have been shaped by a classic anthropologic
notion of culture, where features of classroom cultures (such as questioning practices,
treatment of right and wrong answers, and use of curriculum) are analyzed to see if they

support or work against students’ development of meaningful understandings of

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mathematics (e.g., Boaler, 1998; 2000; 2002; Cobb, Gravemeijer, Yackel, McClain &
Whitenack, 1997; Forman & Ansell, 2001; Lampert, 1990; 2001; Lubienski, 2000;
McCrone, 2005; Spillane, 1999). In many of these studies, classrooms are presented as
having one culture at all times, which has certain attributes. For instance, Lamwrt (2001)
described her classroom as a place where she was working toward the norm that
“multiple ideas, not right answers, were the ‘coin of the realm’ for buying attention” (p.
62). In a large-scale study, Spillane (1999) categorized the cultures of 25 different
mathematics classrooms as falling into one of three groups: classrooms that dealt with
significant mathematics in ways that supported students’ conceptual understandings,
classrooms that dealt with significant mathematics, but in ways that often undermined
conceptual understandings, and classrooms that focused on procedural skills. Similarly,
Boaler (2002, p. 255) contrasted one school culture, which she described as unproductive
for children because it “offered a structured, procedural approach,” with another culture,
which she described in positive ways because it emphasized “open-ended wor ” These
sorts of studies, grounded on the classic notion of culture, make it easy for researchers to
see relationships between certain kinds of practices and beliefs and desired outcomes for
children. However, this static view of culture has been challenged by many
anthropologists, including Rosaldo, who wrote:

Although the classic vision of unique cultural patterns has proven merit, it also
has serious limitations. It emphasizes shared patterns at the expense of processes
of change and internal inconsistencies, conflicts and contradictions. By defining
culture as a set of shared meanings, classic norms of analysis make it difficult to
study zones of difference within and between cultures (Rosaldo, 1989, pp. 27-
28)

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Rosaldo’s critique pointed to some of the assumptions underlying the classic
notion of culture. With its emphasis on “shared patterns,” the anthropological notion of
culture implies that classrooms or societies have a set of unchanging characteristics.
Because it supports the description of classrooms in fixed ways, using culture to talk
about teaching practices can also encourage hierarchical ways of thinking about cultures
and classrooms. For instance, anthropologists might have referred to cultures as being
arrayed on a continuum flom primitive to sophisticated, while mathematics educators
have used the words traditional and reform as well as procedural and conceptual to
convey similar ideas. Rosaldo’s solution to the dilemma of cultural descriptions that
appear stagnant and unitary is to put culture “into motion” (p. 91) by studying the ways in
which individuals within a culture cross borders, act in opposition to norms, and vary
flom one and other.

In this chapter, I am calling on the rhetorical concept of genre to do similar work.
I want to examine the diversity that existed within one classroom, rather than talking
about the teaching practice there in holistic ways. Instead of characterizing the culture of
the classroom I studied, 1 want to look at different genres of teaching that existed within
this single classroom. In particular, I use Bakhtin’s (1940, p. 945) notion of speech
genres, which he defined as the “relatively stable types of utterances” that develop in
particular spheres where language is used. (An utterance is one speaker’s turn in a
conversation.) For example, lecturing, ordering in a restaurant, chatting with a fliend, and
testifying in court are all speech genres where certain types of behaviors and utterances
are expected, but not others. In each of these situations, we do not choose what to say

from the infinitely-possible ways of arranging our words, but transform our ideas into

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what we perceive as the appropriate genre, considering content, word choice, and tone (in
addition to other qualities). In Bakhtin’s words:

We learn to cast our speech in generic forms and, when hearing others’ speech,
we guess its genre flom the very first words; we predict a certain length (that is
the approximate length of the speech whole) and a certain compositional
structure. If speech genres did not exist and we had not mastered them, if we
had to originate them during the speech process and construct each utterance at
will for the first time, speech communication would be almost impossible (Ibid.,
p.956)

Like all other social spaces, the classroom has its own expected speech genres,

with students and teachers continually in the process of predicting the genre of others’
speech and determining appropriate responses. In this chapter, I analyze classroom
interactions at the level of genre (rather than at the level of utterance or of classroom
culture), which makes it possible to think about the kinds of genres present in elementary
mathematics classrooms, the opportunities for participation within various genres, and the
expectations that different genres have for the student. By taking the spotlight off the
words of individuals and placing it on the kinds of conversations that people have,
Bakhtin’s concept of speech genres makes it possible to closely consider language in the
classroom in ways that support, rather than worked against, Foucault’s notion of
discourse, where both words and context are considered meaningful and where
individuals and communities are understood only in relation to each other.

Unlike the classic notion of culture, which emphasizes similarities within a group
or context, looking at genres emphasizes differences. Using this flamework, one
classroom can be seen as having many kinds of interactions, which sometimes work at

cross purposes from each other, rather than having a single culture that could be

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described as either reform—oriented or traditional. It is probably worth noting here that I
am using speech genre differently than Forman (1996) did in her analysis of classroom
interactions. She described the mathematics register as a particular genre of speech that
students had more or fewer opportunities to participate in depending on the speech
practices in their classrooms. The analysis in this chapter is slightly closer up, looking at
the multiple genres that exist in an elementary classroom, all of which might be
considered part of the mathematics register.

In the following section, I describe a few of the genres present in the elementary
mathematics classroom and examine the subject of the student constructed by these
genres, drawing on Foucault’s genealogical flamework, which identifies particular points
of focus, such as invitations to participate and expected practices, as useful in describing
a subject (Foucault, 1983). The goal here is to reveal some of the multiple subject
positions available for children in an elementary mathematics classroom. The
conversations I analyzed for this chapter came flom more than twenty classroom visits
between September and April; these conversations were recorded during my observations
and transcribed shortly after each visit. I began my textual analysis of the transcripts by
drawing on Tannen’s (1984) work in conversational analysis. After identifying speech
episodes, which I defined by topic of conversation and continuity of participants, I
examined these episodes for key features. These included topics of conversation, kinds of
questions asked by the teacher, kinds of responses offered by students, ways of physically
attracting (or deflecting) attention, kinds of mathematical work getting done, and pacing.
I grouped episodes that were similar across many or all of these characteristics as genres.

This resulted in my identification of eleven different genres in the elementary classroom,

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although I will look closely only at three in this chapter. I chose these three because they
were the most frequently employed in Diana’s classroom. I did analyze the genres of the
methods classroom as well, and found them to be quite similar; however, I give examples
only flom the elementary classroom here to help illustrate the difference between genres
is not that the teachers are different, but that the kinds of interactions are.
Genres of Teaching

This section of the chapter explores three genres of teaching observed in a third-
grade, urban classroom, which had been described as reform-oriented by university
faculty who had visited over several years. The teacher, Diana, had been teaching for
more than thirty years and had been involved in numerous university-led professional
development activities in elementary mathematics. The three genres described in this
chapter were not the only ones observed over the course of the year, and the goal here is
not to provide a comprehensive summary of all of the interactions that went on in this
classroom, but to explore the intellectual possibilities offered by thinking about
classrooms in ways that highlight the differences among teaching practices in one site
rather than discussing teachers’ practices in holistic ways.

Discussions

One common genre when Diana was teaching was the mathematical discussion.
During these conversations, the object of discussion was generally a mathematical idea or
process, rather than an answer; Diana flequently challenged students’ answers by asking
them to explain or by presenting counter examples; and many students participated,

sometimes talking to each other. The following episode occurred as Diana was reviewing

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addition with regrouping before the state standardized test. Children had discussed their
solution strategies for the problem:

25
+ 27

Diana then wrote the following on the board:

25
1
+27 and asked if this would be an okay way to solve the problem. Many

students called out “No!” Many also put their hands up in the air.

Diana: I’m going to call on someone and it might not be someone with
their hand up. It might be someone I’m curious about how they’re
thinking. Okay, Brian, I’m going to call on you. You have your

hand up, but you have a puzzled look on your face. What do you

think about this problem?
Brian: Um . .. you should switch them?
Diana: Why would you switch them?
Charlie: (calling out to Brian) It doesn’t matter. It’s just the same.
Diana: Brian, can you explain to Charlie what you’re thinking?
Brian: You would add up that side first.
Diana Don’t say side; use the place value name.
Brian: You would add up the ones first.
Diana: Do you want to show how you would do it?

Brian nodded and came to the flont of the room. He rewrote the problem in the

661”

conventional way, writing a at the top of the tens place and writing “52” as

the answer.
Diana: Why did you do that?
Brian: Seven and five is twelve. The “1” is the ten of the twelve (pointing

to the “1” and “2” in the problem). You have to put it in the tens

place.

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Diana: Charlie, you said it didn’t matter, where the one is. What about
what Brian is saying?
Charlie: The one is still a ten -— even if it’s in the middle. It’s the same. It

doesn’t matter.
Many children shook their heads.

Diana: It looks like some of you disagree. What do you think about Brian

and Charlie’s ideas?

In mathematical discussions, the invitations to speak came in the form of unusual
problems or ideas, challenges from the teacher, and requests for students to reveal their
thinking. For example, in the episode above, Diana told Brian she was calling on him
because he had a “confused look” on his face. Instead of looking for certainty on the part
of students, Diana wanted to encourage students who had thoughts about the problem to
voice them to the class. In addition, Diana invited students to speak to each other, both
through direct requests and by attaching ideas to particular children. In this episode, when
Charlie called out without raising his hand, Diana did not silence him as she sometimes
did during other teaching genres, but instead asked Brian to explain his drinking to
Charlie. Later, she opened up the conversation to the rest of the class by asking them to
comment on Brian’s and Charlie’s ideas, rather than on the problem on the board. This
move communicated to students that participation required engagement with each other’s
ideas, not just the teacher and the problem.

During the genre of discussions, students were expected to engage in the practices
of reasoning, justifying and communicating, much more than in practices closely linked
to particular mathematical domains, such as adding with regrouping. The conversation
about how students actually solved the problem 25 + 27 fell into a different genre, one I

call presenting student work. In this genre, students took turns describing the steps they

145

executed to find the answer and the teacher rarely encouraged students to speak to each
other and instead focused their attention on useful strategies that they might want to
record in their journals. However, in the discussion episode above, Diana brought in
Charlie’s idea about the placement of the 1 not mattering after Brian correctly used the
standard algorithm to solve the problem. At her prompting, Charlie said: “It’s still a ten,
even if it’s in the middle. It’s still the same. It doesn’t matter.” Diana’s goal in eliciting
this comment was not to teach children a new solution method for addition problems. She
later ended the discussion by saying that it was just easier to write the “l ” on top, noting
that this is why most people solved the problem in the standard way. She introduced the
idea of writing the “l” in the middle of the problem to give students a chance to justify
their ideas about place value and to reason about whether their classmates’ ideas made
sense. This is a different practice from solving problems. The discussion was designed to
provide an opportunity for students to make mathematical arguments and to critique the
arguments of others using mathematical reasoning.

Related to this emphasis on process skills, Diana’s concern during the genre of
discussions was with how students made their arguments or responded to the statements
of others, rather than with whether students obtained correct answers. Diana corrected
Brian’s language when he said “that side,” asking him to use the place value name
instead. Diana did not ask him what that column was called; she assumed he knew the
answer. However, by correcting Brian, Diana demonstrated the need to use precise and
mathematical language when making an argument. Similarly, in mathematical
discussions, Diana and the children often used the words “agree” and “disagree” to take

positions. In other genres, Diana might occasionally ask the class if an individual’s

146

answer was “right.” However, she rarely used this language in discussions, instead using
words that emphasized students’ commitments to particular positions, which she would
then call on them to defend.

The ideal student constructed during discussions is a mathematical apprentice,
someone who is learning the norms and conventions of the discipline. The genre of
discussions constructs a student who engages in the process skills, such as offering proof,
reasoning and justifying, described in the standards documents (N CTM, 1991; 2000) and
other reform-oriented texts (NCR, 2005; Van de Walle, 2004). The student is expected
not just to learn content, but to learn how to speak about the content in ways that echo
those of the community of mathematicians. Ball and Bass (2003, p. 29) wrote that
reasoning “is more than individual sense making. . . . Reasoning, as we use it, comprises a
set of practices and norms that are collective, not merely individual or idiosyncratic, and
rooted in the discipline.” In the discussion above, Diana’s students were expected to
engage in a collective process, where individual students did not produce answers to be
evaluated by the teacher, but where the class as a whole contributed to a shared
understanding. Students were expected to listen to and to comment on each other’s points
and the focus of the discussion as well as the vocabulary used was expected to be rooted
in the discipline. The third graders in Diana’s class will probably never be conflonted
with a similar problem to the one she raised on a standardized test or in adding situations
in their outside-of-school lives. The problem was designed so that students could act like
mathematicians, or at least act like mathematicians as described by mathematics
educators, by evaluating a novel problem and making public sense of it with the goal of

increasing their own and others’ understandings of mathematics.

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Game Shows

The expectations during other genres within this same class were often quite
different fl'om those enacted during discussions, particularly in the genre I am calling
game shows. Like discussions, this genre involved the whole class; however, the content
involved and the norms for participation differed. During game shows, students were
expected to answer quickly and correctly; students rarely spoke to each other, and the
purpose of the discussion was often to allow students to demonstrate what they already
knew rather than to explore new ideas together. The following episode occurred during a

lesson on rounding.

Diana: What is three hundred-sixty-two rounded?

Many children raised their hands.

Diana: Everybody?

Children: Four hundred!

Diana: Who can tell us why it’s four hundred and not three hundred?

Eight children raised their hands, some leaning forward in their seats. Others,

including Caitlin and Jerome, looked down at the ground.

Diana: Marcus?

Marcus: It’s past three hundred fifty and three hundred fifty is in the middle
of 300 and 400.

Diana: Very good. Everybody, what does nine hundred and seventy-eight

round off to? (About five seconds go by.) This is hard. Nine.
Hundred. Seventy. Eight. (said slowly).

Two children raised their hands. Diana called on LaTonya.

LaTonya: Ten thousand — wait (she looked at the ceiling).

Diana: 1 know you have the right answer. You just got confused.

LaTonya: One thousand.

Diana: Good. You got it!

Ben (quietly): LaTonya’s smart.

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Aliah (to Ben): Yeah. LaTonya is smart.

Diana: What does five hundred ninety six round off to?

This conversation continued for several more minutes, with Diana asking similar
questions, praising students when they answered correctly, and asking students to try
again when they answered incorrectly. After the conversation, students went on to solve
several similar problems on their own in their notebooks.

In the game show genre, students who believed they knew the correct answers
were invited to speak, while students who believed they did not know the answers were
invited to remain silent. In the episode above, many students raised their hands to signal
to Diana that they wanted to speak because they knew the right answers. Unlike in the
discussion, when Diana expressly said that she might call on someone who did not raise
their hand or who had a “confused look,” during game shows, Diana only called on
students who indicated that they had a correct answer to share — just as Alex Trebek only
asks for answers from contestants who have buzzed in. At the same time, students who
believed that they did not know the correct answer not only reflained flom raising their
hands, but also often attempted to remain inconspicuous, as Jerome and Caitlin did by
looking at the ground.

The practices students were expected to engage in during the game show genre
involved mentally solving mathematical problems, sharing answers, and sometimes being
able to explain the steps taken to achieve these answers. For instance, after the class
collectively answered that three hundred sixty-two rounded to four hundred, Diana asked
someone to explain how he or she knew. After being called on, Marcus said that three
hundred sixty-two was above three hundred fifty so the correct answer would be four

hundred. This explanation, as well as Diana’s response, was quite different than those put

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out by Brian and Charlie in the discussion example. Marcus was not expected to justify
his reasoning about why it made sense that three hundred sixty-two rounded to four
hundred, instead he was supposed to explain the procedure he used to find the answer.
Other children were not supposed to evaluate this procedure or to suggest that they were
thinking about it differently. Diana closed off conversation by saying “very good” after
Marcus’s explanation; whereas, in the discussion example, she asked Brian to think about
Charlie’s idea when he argued that the “1” could legitimately be placed in between the
addends. The genre of game shows was not concerned with the practices of reasoning and
justification, but with solving mathematically problems quickly and correctly. Marcus’s
explanation was a tool that might help some students to engage in the valued practice,
rather than an example of a child thinking like a mathematician.

During game shows, a great deal of concern was devoted to evaluating children’s
mathematical knowledge. This can be seen in comments flom the teacher — like “very
good” and “I know you have the right answer.” — as well as in comments flom students —
like Ben’s remark: “LaTonya is smart.” When LaTonya started off by incorrectly
answering “ten thousand,” Diana could have asked her if this answer made sense, a
question she was fond of asking in other contexts. However, in this case she made a
different choice. The comment “I know you have the right answer” indicated that Diana
was assessing LaTonya’s ability to round numbers, rather than her ability to explain it or
even to say the correct answer aloud, although LaTonya did go on to do this. Ben’s
evaluative comment about LayTonya was based not just on her correct answer, but on her
ability to answer a question that had hung in the air for several seconds because other

students had not been able to offer a response. These kinds of comments from students,

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both positive and negative, were common during the game show genre, but were absent
from many other kinds of conversations, which suggests that the students as well as the
teacher saw this format as particularly relevant for evaluating students’ competence.

The student constructed during game shows is one who takes standardized tests.
In Michigan, as in many other states, state standards demand that students be able to
perform a long list of mathematical procedures. In third grade, this includes identifying
the place value of a digit, comparing and ordering numbers to ten thousand, and
estimating the sum and difference of two numbers (Michigan State Department of
Education, 2006) — all skills touched on in the rounding conversation. Each year, from
third grade through high school, Michigan students take a standardized test, where they
are expected to perform these skills on a series of multiple-choice questions. During these
tests, students answer one question after another; they do not have opportunities to
discuss their thinking with others; and the ability to generate a right answer is prized.
Although the test in Michigan does ask a few open-ended questions, for the most part the
ability to reason, justify or prove is not tested. The genre of game show in the classroom
is a natural partner to the format of the standardized test, mirroring the evaluative
purposes as well as the tone.

Groupwork

In Diana’s classroom, about half of each lesson was spent in whole-class
conversations and about half the time was spent with students working at their desks.
Sometimes students worked by themselves, although they were always allowed to speak
with their neighbors, and sometimes students were explicitly asked to work in groups.

Most often these groups were comprised of the four to five students sitting at a cluster of

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desks; however occasionally Diana would assign students to work with particular
partners or would mix up children in the classroom. During groupwork, students usually
attempted a single or a small number of problems and they were expected to speak only
to members of their group. The teacher intervened very rarely, sometimes answering
questions when called over and sometimes managing logistical problems; however, most
conversation during groupwork occurred among students. In the following episode,
Caitlin and Evan had been assigned to work together on the following problem: Use tiles
to make three figures, each with an area of 16 but with different perimeters. Record your
answers on graph paper.

Before sitting down, Caitlin complained quietly about having to work with Evan
primarily because he was a boy and her best fliend was getting to work with a
girl. She threw her books down on the table before sitting down and sweeping the
tiles on the middle of the table toward her. She and Evan both took sheets of
graph paper. Caitlin arranged the tiles into a three by five array with one extra
tile in the last column. She turned toward Evan.

Caitlin: Can it be like that?
Evan: It can be like that, can’t it? (leaning over the table to look at the
tiles).

They both started drawing the shape on their paper. When she finished, Caitlin

started to rearrange the blocks even though Evan was not done. Evan looked over

at Caitlin’s drawing, but she had drawn hers vertically and Evan had drawn his

horizontally. He continued to draw on his own as Caitlin worked on rearranging

the tiles. Caitlin then colored her shape in, while Evan wrote the date on his paper.

Then she looked at his drawing.

Caitlin: Don’t do that! We’re supposed to do it the same way!

Evan ignored her and continued to draw.

Caitlin: You did it wrong! (She pointed to his picture, where he had drawn
two rows of five instead of three rows of five.) You missed some.

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Evan: I don’t get it. My head hurts (putting his head in his hands).

Caitlin: Fine. We were supposed to work together, but I guess that’s not
going to happen. Fine!

Caitlin rearranged the blocks into a four-by—four square. She started to draw

the shape and then looked back at Evan’s paper.

Caitlin: You did that wrong! I h0pe you know that. There’s supposed to be
another one down there. (She pointed, but Evan ignored her.)
Okay, you just do what you do.

Caitlin began drawing on her own paper. Evan looked at her work for a moment

and then wrote “A = 16” by his first drawing, even though the drawing showed an

area of eleven. He looked at his drawing for a moment and then counted the tiles.

When he got eleven, he added another row to his drawing and then counted the

perimeter. He wrote “P=18.” Caitlin looked at his paper and then counted the

perimeter and area on her first drawing. She wrote “A =16” and “P = 18.” She

started to count the second one, writing “A=16” and “P=17” for the four-by-four

square. She looked at Evan’s paper.

Caitlin: What did you get for the second one?

Evan: I got sixteen. The perimeter is sixteen.

Caitlin: Okay. I’ll count again. (She did.) You do get sixteen (surprised).

Evan: So I was right.

Caitlin: Yeah (smiling). Let’s see if we can do three and then the ones for
extra credit.

Invitations to participate in groupwork came in more informal ways than in genres

that involved the whole class. This was true both of invitations issued by the teacher and

of those issued by other students. In many of the groupwork problems, Diana and Sara

provided students with physical materials to use, which in themselves were a kind of

invitation because many students wanted to touch the materials. In addition, the more

relaxed atmosphere was a sort of invitation to participate in the mathematics. Because

students could chat with their classmates, although always about the task, many students

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engaged more actively in groupwork than in whole-class interactions. Also, the small
number of students in a group exerted a certain amount of social pressure to participate
because students could not count on classmates to fill the conversational space. In the
small group, students who opted not to participate were likely to be taken to task by their
classmates, as Caitlin did with Evan in the episode above. During the whole-class
discussion, Jerome and Caitlin could both avoid participating and avoid attracting the
negative attention of their peers, but Caitlin got vocally angry with Evan when his body
language suggested that he was going to disengage. Once he began to work again, she
conversationally asked him to report his answer. Reluctant students were often pulled
into groups in a variety of ways, including cajoling, threatening and requests for help.
The emotional tone of the invitations varied depending on the compositions of the group,
the problem at hand, and children’s moods. In this episode, Caitlin took a challenging
tone with Evan when he made a mistake and when he declined to respond to her
criticisms. However, when working with her best friend, Caitlin often took a playful tone
and with Ben, whom she viewed as being good at math, she was quite deferential. Much
more than in the teacher-led genres, students did not know what to expect in terms of
invitations when they entered into groupwork with their classmates. A wider range of
speech, gesture and tone was accepted in this genre than in those in which the teacher
was a central participant.

During groupwork, students were expected to have conversations with each other,
to solve mathematical problems, to represent what their group had done in writing, and to
resolve any difficulties that arose. Unlike the garneshow and discussion genres,

groupwork almost always involved the production of a product, such as the graph paper

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drawing Caitlin and Evan worked on in the above episode, a poster to share with the
class, or paragraphs written in individual journals. The practices important in groupwork
generally revolved around the production of this product. For instance, when Caitlin
observed that Evan had made a mistake in drawing his first figure, she insisted that be
change it. This would be a very unusual move in the genre of individual seatwork, where
students produced their own products and showed little concern for the work of others.
However, because of the genre, Caitlin perceived Evan’s work as a reflection of her own
understandings and became emotionally involved in his competence.

In the genre of groupwork, both students’ social and mathematical skills became
objects of concern. Diana flequently worked with group members around the
mathematics of the task at hand, sometimes suggesting a strategy to try, sometimes
clarifying directions; however, she did not engage with students on regulating how much
individuals should participate or how they should speak. Thus, when Caitlin grew angry
with Evan for making a mistake and when Evan grew frustrated with Caitlin for yelling at
him, neither called the teacher for help. Both knew that they were expected to solve
social problems on their own. Caitlin’s strategy was to keep working, while Evan’s was
to complain briefly about a headache and then to continue working. Diana said that one
of her goals during groupwork was that students would learn to work together in
productive ways; thus, these sorts of interactions were not seen as being less important
than the work around the mathematical task, but also part of the intended instruction.

The genre of groupwork constructs a student who is a collaborative worker. The
conversations that occurred during groupwork often resembled those that go on in many

offices. As with most adult workers, students did not choose their colleagues nor did they

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have a great deal of control about the task they were supposed to accomplish, the tools
for doing it, or the time flame. The desire to get the job done shaped many groupwork
interactions. Although Caitlin had been initially irritated about working with Evan, she
began by immediately responding to the task posed by Diana and by seeking Evan’s
validation that her response was correct. It was only when Caitlin saw Evan as getting in
the way of completing the job that she raised her voice to him. Toward the end of the
episode, Caitlin smiled at Evan and softened her tone when she recognized the value he
was bringing to the task in the form of correct answers. Her final comment proposing that
they attempt the “extra credit” also reflected a worker-like attitude. Rather, than engaging
in conversations about the mathematical relationships they may have been uncovering or
about a topic unrelated to school, Caitlin directed Evan’s attention toward getting more
work done. The mathematics in groupwork often functioned as a job students needed to
complete rather than as a source of interesting puzzles and questions. It was during
discussions that Diana took the tasks completed by groups and transformed them into
objects of curiosity.
Multiple genres, Multiple subjects

The three genres described in detail here do not by any means comprise the total
number of genres observed. In addition to individual seat work and presentations of
student work, which were mentioned briefly above, genres included private work
between teachers and individual students as well as public work between teachers and
individual students, where the teacher worked to remediate the misconception of a
particular child before turning attention back to the class as a whole. The genres used in

the methods classroom were remarkably similar to those in the elementary class. Like the

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third graders, the student teachers engaged in discussions, game shows, groupwork and
individual seat work. Lecturing, which was rarely used in the third grade classroom, was
a frequently used genre at the university. An analysis of these genres that drew on the
classic version of culture would probably seek to locate Diana’s practice on a continuum
that ranged from traditional to reform and would suggest possible ways of moving her
along that continuum. The goal would be to reduce the amount of time spent on genres
like game shows and increase the amount of time spent on genres like group work and
problem solving. However, I want to do something different.

Rather than seeing the variety of genres used as a negative feature of Diana’s
teaching, I would like to suggest that the multiple genres she drew on allowed many
students to find ways to engage in mathematics in her classroom. For instance, Caitlin,
who rarely spoke during whole-class discussions, demonstrated in the groupwork episode
described here that she could be an enthusiastic and dominant mathematical learner. An
analysis that focused solely on whole-class interactions might describe Caitlin as passive;
however, her leadership role with Evan works against that classification. Other students
participated reluctantly in groupwork, but enjoyed the chance to show off their
mathematical knowledge during game shows. As discussed in the beginning of the
chapter, calls in the mathematics education literature for teaching based on reasoning,
problem-solving, and other process skills have contributed to the idea that there is one
kind of good mathematical practice and have suggested that the goal of mathematics
educators ought to be to move all teachers toward one kind of practice. I would like to
suggest that it might be productive to work toward the goal of teachers having diverse

practices. Just as the genres of discussions, game shows and group work construct

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particular kinds of students, so do all other genres used in the classroom. The more
subject positions available for students, the greater the likelihood of all students finding
places for themselves in the mathematics classroom.

The concept of genre supports further inquiry along these lines, by allowing
researchers to look for features of particular episodes of teaching practice and to consider
what possibilities these genres make open up for students. In addition to exploring genres
commonly used in mathematics classrooms, researchers might also look for ways to
incorporate genres rarely seen. For instance, in neither the elementary classroom nor the
methods classroom did I observe a genre of teaching that primarily constructed the
students as appreciators of the beauty of mathematics. This value is commonly
recognized by mathematicians; however, it is rarely communicated to students either
directly or through our ways of teaching. At one point in the methods classroom, David
led the students through some shortcuts for finding the sum of all the factors of a number.
In fi'ustration, a student shouted out “What is the point! . .. What is the real point of
finding the sum of the factors of a number? What is that going to do in your life for you?”
David turned on the student, revealing more emotion than he had all semester and nearly
shouted: “What is the real point of going to look at a Van Gogh painting? Of trying to
analyze the color? What’s that going to do for you? There is a part of the mathematics
that should be sort of aesthetic. We’ve got this amazing relationship between numbers.”
However, none of the genres used in the methods class had constructed students as
appreciators of the beauty of mathematics. In fact, this episode was the first time in either
classroom that the word “aesthetic” was used. “Beauty” and “beautiful” were never used.

Although David did occasionally mention a proof that he found “elegant” in off hand

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ways, students in neither the methods nor the elementary classroom spent any time
talking about parts of mathematics they found beautiful, mystical or pleasing. Nor did
they engage in assignments which required them to judge mathematical problems or
solutions on aesthetic grounds. In her book on mathematics and beauty, Sinclair (2006)
offered some suggestions of what genres that constructed students as appreciators of
mathematics might look like. For example, she suggested that students might engage in
critical comparisons of solutions in ways that made aesthetic mathematical values
explicit. This practice would be different than the presentations of student work that I saw
in the two classrooms I observed (as well as in many other classrooms) where the goal of
instruction seems to be to show that there are many equally correct ways of solving the
problem. Instead, students would evaluate two or more correct solutions for aesthetic
properties. Teachers might make this aesthetic visible to children by asking specific
questions, such as “Does it explain better? Is it more general or predictive? Is it more
transparent?” (Sinclair, 2006, p. 174). Teaching in ways that construct students as
appreciators of mathematics could work to broaden the possibilities of what it means to
be a student in mathematics. Students who enjoy interpretive work with literature, music
and art might find genres that focus on developing an aesthetic sense of mathematics to
be ways to find room for themselves in the discipline.

Another set of genres that might work to make mathematics classrooms more
inclusive are those that would construct students as users of mathematics in the everyday
world. This is different from using real-world contexts in mathematics problems. As
Gerofsky (1996) pointed out in her analysis of word problems as a genre, despite their

use of real world objects and situations, these problems often have very little to do with

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how people draw on mathematical ideas in their everyday lives. For instance, I never do
put six oranges in my fl'uit basket, discover than I now have fourteen and wonder how
many I started with. Recently, when I described my work to my hairdresser, she
responded that she had never been very good at math in school and had not believed her
teachers when they told her she would need it in her adult life. She said that as it turned
out, she did use quite a bit of math, but it wasn’t anything like what she had done in
school. As Lave’s work (1988) demonstrated, people’s use of mathematics in their daily
lives is often more fluid and less resolved than much of school mathematics. People
grocery shopping stop halfway through calculations because they realize they are close

enough, and they care not about the lowest unit price in the world, but about which of

 

their two favorite brands is the better buy. Teaching genres that construct students as l
users of math in the everyday world would have to allow students to work in
mathematical situations — such as cooking and carpentry - without requiring that the
situations be represented as school mathematics.

The Standards movement in mathematics education has worked to build the
definition of what school mathematics is on the work of mathematicians. These
documents (N CTM 1991; 2000), and those who draw on them, argue that mathematics
must include proof, justification and reasoning, and much of the literature in the field has
been aimed at getting teachers to adopt these values and the practices that go with them.
These are not bad values. However, the consensus about what teachers should be doing is
dangerous because it narrows the possibilities of what it means to be a student who does
mathematics. This is not an argument that mathematics should be more like game shows,

more like art, or more like grocery shopping. It is an argument that mathematics is all of

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these things and that teachers who draw on multiple genres create more spaces for their
students to find productive relationships with mathematics than those who foreclose

possibilities by teaching in genres that represent a narrow view of mathematics, even a

Standards-based one.

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CHAPTER EIGHT
Discursive Possibilities Or:
How I Learned to Stop Worrying and Love the Achievement Gap

“The whole political thrust of postmodern criticism is to decrease the weight of
managerial discourses. Policy and solutions usually add to that weight. The tricky part,
for people relatively new to postmodern thinking, is to regard critique as a positive,
productive, generative, and helpful contribution Conversely, to regard proposed

‘solutions’ as authoritarian, managerial, instrumental, and exclusionary.” — Lynn

F endler, personal communication.

“We were worried about a Doomsday gap. ” — Russian Ambassador, Dr. Strangelove

In Stanley Kubrick’s (1964) movie, Dr. Strangelove, the entire world is plunged
into nuclear winter when a wacky American general sends a squadron of bombers to
attack the Soviet Union, triggering a Doomsday device that cannot be shut down. The
annihilation of all life on Earth occurs because of, rather than despite, the multiple
policies aimed at preventing just such an occurrence. Driven by the fear of a weapons
gap, both countries produced more and more weapons and implemented policies designed
to deter their opponent from ever putting these weapons to use. However, each of the
proposed solutions to the weapons gap -- the Doomsday machine, with its automated total
destruction, the emergency back-up plan to launch planes that cannot be recalled, and the
locking down of communication to military bases in order to prevent sabotage -- all
contributed to the horrible (if comic) climax. In the end, it was not merely the atomic
bombs themselves that were dangerous, but the lack of options available for response.
This dissertation -- while not a love song to the achievement gap -- is an attempt

to reduce the weight of the managerial solutions that have been proposed in the wake of

concerns about both equity and reform mathematics. Following Kubrick’s movie, I want

to argue that the closing down of possibilities for teachers and students — even as part of

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an effort to reduce the achievement gap or to improve mathematics instruction — could
ultimately be dangerous in unpredictable ways. In the second chapter of the dissertation, I
argued that students who did not share linguistic or cultural background of the teacher
were likely to be disadvantaged by the open-ended reform questions that are described as
part of quality mathematics teaching by much of the research community. I proposed that
instead teachers ask more explicit questions by naming students who should respond, by
identifying the part of the problem on which to focus, and by offering a narrow range of
possible answers. These proposed solutions might indeed allow more students to
participate in mathematics, but they could also work to close down possibilities for

students in the classroom. Students might never have the opportunity to tackle open-

 

ended questions or to find their own ways toward sensible conclusions. In addition, l
teachers might assume that they can look at the skin color of a child or listen to his or her
speech and know which kinds of questions are most appropriate. Students would have
another — albeit new — set of expectations to which they would be expected to conform.
This is an example of proposed solutions adding to managerial weight - what teachers are
supposed to know and do in order to demonstrate best practices — and of closing down
positions that students might occupy in the classroom.

In contrast, the other three chapters which describe classroom practice, while
dealing with quite different topics, all seek to open up possibilities for teachers and
students. In the chapter on metaphors, discourses of hierarchy are critiqued with the goal
of creating a space where educators might begin to imagine talking about children’s
learning of mathematics in ways that do not rank and order students. This act of

imagination allows not only new ways of thinking about children as having multiple

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abilities in terms of the ways that they interact with the world quantitatively, but also
allows for new ways of conceiving of mathematics, where the learning of the discipline
can be imagined not as proceeding linearly, but as wandering over a terrain where paths
and speed of travel might depend on individual interests or goals. In the chapter on
problem solving, I argued that the three kinds of problems seen in the methods and
elementary classrooms serve different persuasive purposes in reference to mathematics
and that the privileging of one kind of problem — those that emphasize mathematical
processes — is a relatively recent historical move in mathematics and one that is likely to
exclude some students. In the final chapter on genre, I argued that the diversity of genres
present in Diana’s classroom opened up opportunities to participate competently in
mathematics that would not be present if her classroom had included only those genres
approved by the Standards community.

My goal in each of these chapters was to challenge common assumptions held by
the teacher education community: 1.) That beginning teachers are the source of
problematic beliefs about students and that they need the interventions of mathematics
educators to change. 2.) That process problems resemble the work of real mathematicians
and are the kinds of problems all teachers should be focusing on in their practices. 3.)
That good mathematics teaching always has certain key features and that the work of
teacher educators is to get teachers practices’ as close as possible to this ideal. These
assumptions have certain discursive consequences that go beyond efforts to close the
achievement gap or to reform mathematics instruction.

The current conversation around equity has focused on identifying groups of

children who have trouble in mathematics and describing ways that their difficulties

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might be remediated (NRC, 2001; NCTM, 2005; Struchens, 2000). The discursive
consequences of this line of reasoning include the expectation that looking at children
will reveal something meaningful about their ability to do mathematics, the belief that
children can be categorized and linked with appropriate (or culturally relevant)
pedagogies, and the presentation of mathematics as a discipline that proceeds in linear
ways with some content and some ways of thinking as being more valuable than others.
This current discourse around equity contributes to Sara saying that you would never
guess by looking at him that one of her students could milk a goat and to the authors of
Adding Up saying that poor and minority children are developmentally behind. In
different ways, all three of the genealogical chapters seek to unsettle this discourse,
whether it is by arguing that certain problems (and therefore the children who
successfully solve them) serve valuable mathematical purposes or by arguing that the
commonly used metaphor of the achievement gap contributes to, rather than works
against, the ordering of children, even if the discussion is about how to close the gap.

In similar ways, this dissertation also worked to unsettle discourse around reform
mathematics. In the mathematics education community there has been broad agreement
that teachers should teach mathematics in ways that create learning communities,
encourage discussion, and promote reasoning and problem-solving (Ball & Cohen, 1999;
NCTM, 1991, 2000; Putnam & Borko, 1997). By analyzing the genres used in Diana’s
classroom and the persuasive purposes of different kinds of problems, I wanted to
challenge this shared agreement and to create a space where it was possible to consider
the dictum that good mathematical practices are those that promote discussion,

reasoning and problem-solving as not being universally true. Because of the widespread

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agreement about what mathematics classrooms should look like (NCTM, 1991 , 2000),
Diana felt it necessary to apologize when she taught in ways that deviated from the
practice described in Standards documents, and opportunities for Jerome to feel “smart at
math,” as he did when completing a temperature handout, came fewer and farther
between (possibly even as a result of my observations, as I was situated as a university
mathematics educator). This is another example of proposed solutions (have discussions,
use problems that focus on reasoning, hold children publicly accountable) adding to the
managerial weight. Diana felt that she had fewer degrees of freedom in the classroom
because of the widespread agreement of the mathematics educators who want to tell her
and other teachers what to do.

I struggled to wrap up each of my genealogical chapters without offering
solutions in ways that seemed meaningful and that rhetorically resembled endings. I was
not entirely successful in this. For instance, I recommend that we find ways of talking
about mathematical learning that do not rely on hierarchies; that we, as a mathematics
education community, articulate the purposes that we see problems serving in our
classrooms; and that we look for diversity of teaching genre in teachers’ practices. In
part, these recommendations are a failrn'e of conviction and imagination on my part. As I
think about communicating my work to the larger education community, I know they will
expect to read both a paragraph that resembles an ending on the last page of an article and
also that they will expect to get some insight flom me, as the author, about what my
analysis means to them. However, even with my postmodem influences, I am
comfortable with these conclusions because I see them as working to open up

possibilities in already dense discourses rather than shutting them down. That is, my call

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for diversity in teachers’ practices does not ask that teachers engage in or cut down on
any particular kinds of behavior. In fact, in pursuit of diversity, some teachers may adopt
practices that I would rather not see in mathematics classrooms. However, I believe the
recommendation to mathematics educators to support teachers’ development of diverse
practices could serve to give teachers room to move, experiment, and choose in their own
classrooms, rather than work to direct teachers and teacher educators to act in particular
ways in order to solve identified problems.

In the introduction to the dissertation, I stated that I wanted to explore the
possibilities that drawing on genealogical and rhetorical research traditions offered in
contrast to those offered by traditions that draw on social science. I believe that one
important role that rhetorical and genealogical traditions played in my study was that of
providing ways to think about educational research that worked against, rather than for,
the creation of categories. Social science research historically has drawn on scientific
paradigms, such as collecting data, forming theories, and gathering evidence. More
recently the “science” part of social science research in education has been emphasized
by governmental agencies (National Center for Education Evaluation and Regional
Assistance, 2003; NRC, 2002). The expectation that research in this tradition should
produce clear findings can lead to studies that categorize children and teaching practices.
For instance, teachers have been labeled as reform-oriented or traditional (Boaler, 2002;
Spillane, 1999) and students have been attached to particular pedagogies based on their
learning styles (Wilson, 2006), races (Ladson-Billings, 1995), and genders (Ambrose,

2002). These categories reduce discursive possibilities for students. In other words, they

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create expectations that students will learn (or not learn) in particular ways and it
becomes more difficult for students to act in ways not provided for by these categories.

Genealogical and rhetorical traditions, because they do not call on scientific
traditions of methods and findings in order to add persuasive power to their arguments,
are less likely to produce firm categories backed up by data. For instance, in the genres
chapter, I do not make an effort to create a comprehensive list of classrooms genres and
their features. I recognize that each teacher will enact genres in particular ways. Nor, do I
make any attempt to link particular kinds of children to particular genres. I suggest genre
as a rhetorical tool because I believe that it can support mathematics education
researchers in looking at teaching practice in ways that value diversity. I do not argue that
the genres that I “found” in Diana’s classroom are objective categories that “really exist”
or that other researchers would find the same ones. In fact, I would resist these sorts of
interpretations of the argument I present in that chapter or future studies that would try to
draw these kinds of conclusions. I believe it would be particularly dangerous to attempt
to proscribe genres for children based on race, gender, learning style, or temperament
because these descriptions would reduce rather open up possibilities for children and
teachers in mathematics classrooms. By drawing on literary rather than scientific
traditions, rhetoric offers researchers the opportunity to talk about the world in ways that
cause others to think differently about social interactions without etching boundaries
between people as the result of “evidence.”

I want to clearly state that the dissertation is not an argument that genealogical
and rhetorical perspectives are better than ethnographic or quantitative ones. Nor is it an

argument that traditional ways of doing mathematics are better than reform-oriented ones.

168

It is an argument that dense, dominant ways of thinking close down possibilities for
human beings living in these ways of thinking and that efforts to make currently accepted
truths seem permeable and historically contingent promote equity precisely because they
allow for more ways of knowing and being. This is why it is productive for mathematics
educators to challenge traditional ways of learning mathematics in many elementary
schools. And it is why it is productive for me to challenge reform-oriented teaching and
ethnographic methods in teacher education. And, why, someday, it may be productive for
someone else to challenge the dominance of rhetorical and genealogical ways of drinking.

To close, I want to turn my attention to the enterprise of teacher education. In the
Opening of the dissertation, I said that I wanted to theorize preservice education by
drawing on discourse and by turning my attention toward elementary children in the
mathematics classroom. Descriptions of the methods classroom showed up very rarely
throughout the dissertation, so it may seem a stretch to argue that I have met this goal.
However, I would like to spend this last section discussing the ways in which
mathematics teacher education could be informed by this work.

Whether or not the phrase is used, much of the research about teacher education
in general and mathematics teacher education in particular has been informed by the idea
of best practices. Researchers have set out to describe particular competencies that
teachers must develop (Ball & Bass, 2000; Hiebert, Morris & Glass, 2003; Lampert,
1990; Ma, 1999), to identify practices that will help them develop these competencies
(Blanton, 2002; Empson & Junk, 2004; Feiman-Nemser, 1990; Fernandez, 2002), and to
identify practices that will promote the beliefs seen as necessary for these competencies

(Ambrose, 2004; Warfield, Wood & Lehman, 2005). For the most part, researchers have

169

drawn the particular competencies, explicitly or implicitly, from the NCTM standards
documents, which say that teachers should to create learning communities, encourage
discussion, and promote reasoning and problem-solving (Ball & Cohen, 1999; NCTM,
1991, 2000; Putnam & Borko, 1997; Stein, Grover & Henningsen,1996). Researchers
have recommended a handful of practices to help teachers develop these competencies,
including using multi-media tools (Van Es & Sherin, 2002), interviewing (Moyer &
Milewicz, 2002), participating in lesson study (Fernandez, 2003), and letter-writing
(Crespo, 2003). In general, these recommended practices involve teachers looking at
artifacts, such as video tapes, transcripts and student work, and having conversations
about these artifacts over time.

This dense, unitary discourse around the practices which researchers want
beginning teachers to adopt as well as the relative agreement about the best pedagogical
practices for attaining these goals have certain consequences. First, this Standards-based
agreement offers a relatively narrow vision of good teaching to prospective teachers. So
much so, that the image of the “good teacher” is often personified in the form a single
person (Deborah Ball) -- through transcripts, video tapes and research articles. Second,
beginning teachers often experience the same sorts of learning activities over and over
again in their preparation classess, which privilege some students (those who like
participating in public conversations and private reflections) and disadvantage others
(those who have a distaste for self-analysis, regardless of their strengths in mathematics
or in working with children). Finally, teacher educators and their students often begin to

view their field placements in elementary schools and their methods courses at

 

5 When I taught my first methods course, I asked my students to tell me what they wanted to learn about
and what they were tired of. Overwhelmingly, they told me that they didn’t want to talk in groups and write
things on chart paper.

170

universities in oppositional ways. For instance, in their argument for reforming teacher
education, Ball and Cohen (1999, p. 6) wrote that teacher education needed “to become
sufficiently powerful to irnmunize teachers against the conservative lessons that most
learn from practice.” Here, teaching in the schools is flamed as a disease that university
folk need to eradicate. This kind of metaphor is an expected outcome of a discourse about
teaching that valorizes only a narrow range of practices.

The discussions of children’s interactions in the elementary classroom in this
dissertation call into question the usefulness of viewing teacher education through the
lens of best practices. Just as Diana’s use of the game show genre and of practice
problems allowed some students to feel competent in ways not offered by other genres
and problems, a more diverse vision of good teaching might not only welcome in more
prospective teachers, but might also support them in welcoming more students. Also, a
broader vision of quality teaching might make it easier not to draw such clear boundaries
between the teaching practices studied at the university and the teaching practices
observed in elementary schools. This more open discourse might flee prospective
teachers flom the expectation that they have to choose between their cooperating teachers
and methods instructors as mentors. Methods courses that themselves reflect a diversity
of practice in terms of assignments might also contribute to the opening up of visions of
good teaching. This opening up of assignments might require recognizing that a
prospective teacher could work with children in mathematically meaningful ways while
at the same time might be unable to write a compelling reflection about the lesson (a
heretical notion, I know). Opening up the discourse around teacher education would

require operating on multiple levels, including giving assignments that recognized

17]

mathematical knowledge and teaching practice as well as self-analysis, doing research
about teacher education in ways that flamed practices not as best, but as possible, and
explicitly working to expand — in teaching and in writing — visions of good teaching and
acceptable beliefs.

When asked about thinking in terms of good and evil, Foucault (1989/1996, p.
137) said:

I drink it is important to shift the boundaries, to make them indefinite, shake
them up, make drem flagile, allow for crossovers and osmosis. It isn’t possible not
to think in terms of good and evil, true and false. But you have to say every time:

and if it were the opposite, what if the lines were elsewhere

Here, he acknowledged the hmnan desire to categorize events, people, feelings and
teaching as good and bad, while at the same time arguing for the intellectual payoff in
considering the permeability of these boundaries. This is a maneuver I drink of as “But
what if that’s not the question?” What if -- “How do we get beginning teachers to adopt
Standards-based practices?” -- isn’t the question. What if -- “How do we close the
achievement gap?” — isn’t it eidrer.

Many years ago, when one of my closest friends was engaged to be married, her
mother decided to sew the wedding dress herself. Two months before the wedding, after a
particularly exasperating fitting, her mother exclaimed: “This is it. If it doesn’t fit next
time, then it’s you who’s going to change — not the dress!” It seems to me that in teacher
education we have a beautiful wedding dress and drat much of the research we’ve done
has been focused on getting the bride to fit. Many of our research questions have centered

on how we can make others change — whether we want them to adopt new beliefs, score

172

better on tests, succeed at different kinds of mathematics problems, or promote
discussions, reasoning, and higher-level drinking. All of these questions are based on
“gap” thinking, with our current situation and our desired outcome arrayed along a
continuum with more or less distance between. What this dissertation did for me — and I
hope for others — is to bring some of the consequences of this kind of thinking into focus
and to suggest drat there may be more possibilities for all of us if we abandon the notion

of gaps, whether they be about weapons or achievement.

173

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