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MATHEMATIZING, IDENTIFYING, AND AUTONOMOUS
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MATHEMATIZING, IDENTIFYING, AND AUTONOMOUS LEARNING:
FOURTH GRADE STUDENTS ENGAGE MATHEMATICS

By

Marcy Britta Wood

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

2008

ABSTRACT

MATHEMATIZING, IDENTIFYING, AND AUTONOMOUS LEARNING:
FOURTH GRADE STUDENTS ENGAGE MATHEMATICS

By
Marcy Britta Wood

Over the past decade, mathematics education researchers have used the lens of
participation to explore how student activity relates to mathematical learning. While this
lens is useful in drawing connections between mathematical activity that occurs during
lessons and subsequent mathematical activity, it does not provide a specific conception of
central concerns of mathematics education: student learning and thinking. Narrowing
participation to communication and equating thinking with communication addresses this
problem. Using this frame, discourse becomes the object of study and learning is defined
as a change in discourse. However, what remains unexplained are questions about why
students in similar circumstances engage in different activity or why the same student
engages in different activity on different occasions. Researchers have turned to identity to
tie student activity to who students think they are. By examining the interplay of
discourse about who students think they are (identifying) and discourse about
mathematical objects (mathematizing), it is possible to explore, elaborate, and clarify
connections across mathematical activity, identifying activity, and learning.

One tool that supports the analysis of mathematizing, identifying, and learning is
a theory of autonomous learning. Autonomy is a central goal of reform mathematics
education, yet it has not been well defined or well studied in classroom settings. Defining
autonomous learning as a constellation of mathematizing and identifying activities

reflecting curiosity creates a tool for evaluating students’ learning activities and

suggesting how mathematics educators might support more mathematically desirable
learning outcomes.

This study used the lenses of mathematizing, identifying, and autonomous
learning to probe the discourse of three fourth grade students engaged in a mathematics
lesson. The analysis of discourse evaluated changes in each student’s mathematical
discourse across the lesson, examined mathematizing and identifying activities might be
grouped and described as a kind of learning, and made connections between those
changes in discourse and kinds of learning. Finally, the kinds of learning and changes in
mathematical discourse were examined using the theoretical lens of autonomy to
determine what advantages the activities of autonomous learning might have for the
development of a mathematically desirable discourse.

The findings show five kinds of learning: engaged, directed, covert, watchful, and
guided. Each was characterized by a distinctive group of intertwined identifying and
mathematizing activities that could be connected to changes in discourse. The kinds of
learning also varied in their resemblance to the activities of autonomous learning, with
the kind of learning which was the most autonomous exhibiting the development of a
more mathematically desirable discourse.

The findings carry implications for how mathematics educators might construct
learning situations to encourage identifying and mathematizing to best support student
learning. They also suggest that the lenses of mathematizing, identifying, and
autonomous learning are useful tools for examining complexities of students’ actions and

interactions as they learn mathematics.

COPYTight
Marcy Britta Wood
2008

ACKNOWLEDGEMENTS

My mother has always said that it takes a village to support me through my
projects. Thus, it’s no surprise that I would rely upon the support of many people to
complete an endeavor as large as a dissertation. I truly appreciate the support, love, and
advice from everyone in my village.

I am immensely thankful for my advisor, Helen F eatherstone. Her support and
guidance throughout this project was extraordinary. She has taught me many lessons
about myself and about my writing. And she helped me find the pleasure in this project.
Thank you, Helen.

I was also fortunate to have support and guidance from three amazing faculty
members: Jack Smith, Anna Sfard, and Lynn Paine. Jack, Anna, and Lynn were on my
dissertation committee, for which I am immensely grateful. However, they were much
more than members of my committee. I deeply appreciate the relationships we developed
and their investment in both my academic and personal growth.

Three more very important people are Patrick Halladay, Lisa J ilk, and Ann
Lawrence, the members of my writing group. I am a much better writer and a saner
person because of the time I spent with them.

I would also like to thank Ginney Stokes for all that she did to make this
dissertation possible. Thanks also to my mom for telling me to get my big girl panties on.
Sometimes you just need a reminder to get going!

And finally, but not least, I want to thank my partner in life, Kristin Gunckel who

helped me see just what was possible and believed in me as I worked to get there.

TABLE OF CONTENTS

List of Tables ..................................................................................................................... ix
List of Figures ..................................................................................................................... x
Chapter 1 ............................................................................................................................. 1
Introduction ......................................................................................................................... 1
Problem Statement .......................................................................................................... 2
Overview of the Dissertation .......................................................................................... 8
Chapter 2 ........................................................................................................................... 10
Theoretical Framework ..................................................................................................... 10
The Participationist Lens .............................................................................................. 10
Narrowing Participation to Communication ................................................................. 12
Mathematical Activity and Identity .............................................................................. 18
A Definition for Identity ............................................................................................... 23
Moving fi'om Identity to Identifying ............................................................................. 33
Theoretical Links Between Mathematizing and Identifying ........................................ 37
Autonomous Learning .................................................................................................. 40
Research Questions ....................................................................................................... 50
Chapter 3 ........................................................................................................................... 51
Methods ............................................................................................................................ 51
Overview ....................................................................................................................... 5 1
Study Design ................................................................................................................. 51
Context .......................................................................................................................... 53
Choosing a site .......................................................................................................... 53

The Classroom Setting .............................................................................................. 55
School, Students, and Teacher .................................................................................. 57
Focal Participants .......................................................................................................... 58
Researcher Role ............................................................................................................ 62
Data Collection ............................................................................................................. 62
Classroom Observations and Student Work ............................................................. 63
Interviews .................................................................................................................. 64
Informal Conversations with the Teacher ................................................................. 65
Data Analysis ................................................................................................................ 65
First Pass through the Data ....................................................................................... 65
Second Analytical Pass ............................................................................................. 66
Third Analytical Pass ................................................................................................ 68
Transcriptions. .......................................................................................................... 71
Limitations .................................................................................................................... 72
Chapter 4 ........................................................................................................................... 74
Introduction to the Findings .............................................................................................. 74
Relevant Research on Area ........................................................................................... 74

vi

Overview of the Unit on Area ....................................................................................... 76

Description of the Focal Lesson ................................................................................... 78
Desired Mathematical Discourse .................................................................................. 81
Chapter 5 ........................................................................................................................... 83
Engaged Learning and Directed Learning ........................................................................ 83
Overview ....................................................................................................................... 83
Summary of the Lesson ................................................................................................ 83
Outcome of Learning .................................................................................................... 87
Initial Mathematical Discourse ................................................................................. 87
Final Mathematical Discourse .................................................................................. 89
Learning Activity .......................................................................................................... 96
Engaged Learning ..................................................................................................... 97
An Example of Engaged Learning ........................................................................ 97
Identifying the Audience .................................................................................... 102
Mathematizing .................................................................................................... 106
Departure from Engaged Learning ......................................................................... 115
Directed Learning ................................................................................................... 117
Identifying the Audience .................................................................................... 118
Transition Back to Engaged Learning .................................................................... 122
Autonomous Learning ............................................................................................ 123
Discussion ................................................................................................................... 125
Intertwining of Identity and Mathematizing ........................................................... 126
Kinds of Learning ................................................................................................... 128
Fluid Identities ........................................................................................................ 129
Development of Mathematical Discourse ............................................................... 129
Autonomous Learning ............................................................................................ 131
Chapter 6 ......................................................................................................................... 136
Covert Learning .............................................................................................................. 136
Overview ..................................................................................................................... 136
Summary of the Lesson .............................................................................................. 136
Outcome of Learning .................................................................................................. 140
Initial Discourse ...................................................................................................... 141
Approaching the Desired Discourse ....................................................................... 142
Final Discourse ....................................................................................................... 145
Learning Activity: Covert Learning ........................................................................... 148
Two Illustrations of Covert Learning ..................................................................... 149
Identifying the Audience ........................................................................................ 156
Teacher as Gatekeeper to Being “Done” ............................................................ 156
Segue to Peers are Not Teachers ......................................................................... 159

Not an Audience for Herself ............................................................................... 163
Mathematizing ........................................................................................................ 164
Adoption of Discourse ........................................................................................ 165
Production of Discourse ...................................................................................... 166
Substantiation of Narratives ................................................................................ 170

vii

Autonomy ................................................................................................................... 1 71

Discussion ................................................................................................................... 1 73
Intertwining of Mathematizing and Identifying ...................................................... 173
Development of Mathematical Discourse ............................................................... 174
Autonomous Learning ............................................................................................ 176

Independence Verses Decentering ...................................................................... 176
Knowing and Not Knowing ................................................................................ 179
Chapter 7 ......................................................................................................................... 181
Watchful Learning and Guided Learning ....................................................................... 181

Overview ..................................................................................................................... 18 1

Summary of the Lesson .............................................................................................. 181

Outcome of Learning .................................................................................................. 183
Initial Discourse ...................................................................................................... 183
Final Discourse ....................................................................................................... 185

Learning Activity ........................................................................................................ 192
Watchful Learning .................................................................................................. 193

Identifying the Audience .................................................................................... 197
Mathematizing .................................................................................................... 200
Departure fiom Watchful Learning ........................................................................ 207
Identifying the Audience .................................................................................... 215
Mathematizin g .................................................................................................... 2 16
Return to Watchful Learning .................................................................................. 219
Autonomous Learning ............................................................................................ 219

Discussion ................................................................................................................... 221
Intertwining of Mathematizing and Identifying ...................................................... 221
Development of Mathematical Discourse ............................................................... 224
Autonomous Learning ............................................................................................ 226
Reconceptualizing Jessica as a Struggling Student ................................................ 231

Chapter 8 ......................................................................................................................... 233
Discussion & Conclusion ................................................................................................ 233

Mathematizing, Identifying, and Learning Outcomes ................................................ 234

Autonomous Learning ................................................................................................ 239
The Importance of the Problem and the Problem of the Teacher- Approved Answer
................................................................................................................................ 242
Discourse for Whom? ............................................................................................. 248
Linking Discursive Features ................................................................................... 251

Assumptions and Contributions to Research on Identity ............................................ 254

Different Kinds of Learning ....................................................................................... 257

Reform and Traditional Mathematics Pedagogies ...................................................... 258

Beyond Area and Beyond Math .................................................................................. 263

Limitations and Future Research ................................................................................ 264

Conclusion .................................................................................................................. 266
References ....................................................................................................................... 268

viii

LIST OF TABLES

Table 3.1 Transcription conventions ................................................................................ 72
Table 4.1 Sequence of lessons involving mathematics during area unit .......................... 78
Table 5.1 Transcript excerpts and main events in lesson ................................................. 85
Table 6.1 Transcript excerpts and main events in lesson ............................................... 138
Table 7.1 Transcript excerpts and main events in lesson ............................................... 182

Table 8.1 Mathematizing and identifying activities of autonomous learning for each kind
of learning ............................................................................................................... 240

ix

 

LIST OF FIGURES

Figure 4.1 Scanned image of Figures H and I. ................................................................. 80
Figure 4.2 Scanned image of Figures J and K. ................................................................. 80
Figure 5.1 Jakeel’s initial work on Figures H and I .......................................................... 86
Figure 5.2 Jakeel’s final work on Figures H, I, J, and K .................................................. 86
Figure 5.3 Portion of worksheet on area of triangular figures. ......................................... 94
Figure 5.4 Illustration of J akeel’s geoboard showing the triangular figure partitioned into
a square and two rectangles. ..................................................................................... 95
Figure 6.1 Minerva’s first three solutions to the task involving Figures H and I. .......... 139
Figure 6.2 Minerva final work ........................................................................................ 140
Figure 7.1 Drawing of Jessica’s work at Excerpt 4. ....................................................... 186
Figure 7.2 The large square constructed by Jessica from two copies of Figure I ........... 188
Figure 7.3 Jessica’s final arrangement of Figures H and I. ............................................ 189
Figure 7.4 Bonita’s work as approved by the teacher. ................................................... 203

 

CHAPTER 1
INTRODUCTION

1 stop by unan ’s desk. He is a fourth grade student working on a math

problem that asks him to decide whether a rectangle has been divided into

thirds. Part of the solution to this problem includes careful attention to a

triangular shape arising from the partitions inside the rectangle. unan

has written his solution to this problem. However, when I ask him what he

has done, he is unable to explain his work. He is even unable to read his

written explanation. I puzzle for a moment about how he could have so

much done and yet have so little to say about it. Then my eye strays to

unan ’s neighbor Corey whose work looks almost identical to unan 's,

except for one peculiar change of words: In the sentence where Corey has

written “triangle unan has written “Hiangle I look more carefully at

Carey ’3 paper and realize that from the side, the “tr " in Carey ’S

“triangle ” does look remarkably like unan ’s “H Clearly unan has

been copying what he sees on his neighbor’s paper. (02/14/06)

unan’s copying was not inherently problematic: He was a student in a reform-
oriented mathematics classroom in which the teacher encouraged students to use each
other as resources. The teacher felt that students learned better when they had
opportunities to work together, support each other, and talk through their mathematical
ideas. Unfortunately, unan did not seem to be using the opportunity to interact with his
neighbor in ways that supported the kind of mathematical learning his teacher imagined.
Rather than learn mathematics from or with his neighbor, unan seemed to be avoiding
mathematical learning and focusing instead upon getting something written on his paper.

This vignette is not atypical of elementary mathematics classrooms. It illustrates
problems of mathematical learning that teachers and education researchers have thought
about, worried about, and worked on for many years. Why couldn’t unan explain the
mathematics of the task he was working on? Why was he copying from Corey instead of

discussing the mathematics of the problem with him? Was unan learning mathematics

by copying the right answers? What would need to change so that unan would engage

 

in learning mathematics in meaningful ways? These are the types of questions that have
motivated my work as a researcher. As a classroom teacher and as a classroom
researcher, I have been interested in finding ways to make sense of and support the
learning of students who, like unan, are engaging in mathematics lessons in ways that
result in less mathematically desirable learning than their peers. This dissertation is my
articulation, exploration, and demonstration of research lenses that I have found
particularly helpful in beginning to answer my questions about mathematical learning.
The remainder of this chapter provides an overview of how I’m framing the problem of
mathematical learning. I follow that problem statement with a summary of each of the

subsequent chapters in this dissertation.

Problem Statement

Within the last decade, researchers have begun to answer questions like my
questions about unan by considering how learning is an inherently social and cultural
activity. Rather than frame the individual as context-independent, researchers have
worked to develop theories that view learning as becoming a participant in a communal
practice (Sfard, 1998; Wenger, 1998). When applied to unan’s situation, these
sociocultural or participationist theories would consider how unan’s participation was
supported or constrained by the social, cultural, and historical context of his classroom
and his group. For example, researchers using a participation lens might ask how
interactions with Corey have influenced unan’s activity or how unan’s experience of
mathematics as a cultural activity impacted his engagement in learning.

The participationist lens is broad. It does not define what counts as participation

or what student activities should be studied. In particular, this broad net provides no

specific conception of student thinking or student learning. In order to operationalize
these terms, I have used Anna Sfard’s commognitive framework (2008) which —as the
name suggests — defines cognition as communication. Thus, student thinking is
communication, which need not be verbal and need not be with others. When thinking is
communication, the study of thinking becomes the study of discourse, with discourse
encompassing verbal and nonverbal communication. Learning is also defined in terms of
discourse, as a change in discourse. Finally, Sfard has tailored this framework to the field
of mathematics education by using the term mathematizing to indicate participation in
discourse involving mathematical objects, whether that participation is mathematically
appropriate or not.

The commognitive framework raises more specific questions about unan’s
activity in the opening vignette. What mathematizing did unan demonstrate? Did he use
mathematical words or work with visual objects? What features of the task was he able to
talk about? Has his discourse about fractions changed from earlier in the lesson or the
unit? While these questions would help us understand more about unan’s thinking and
learning, they do not help us understand why he was copying from Corey. He could have
asked Corey to explain the problem to him, he could have asked the teacher for help, or
he could have listened in on the conversation of other students to determine their answer.
The commognitive framework does not help us make sense of why unan engaged in the
specific activity of copying.

To answer questions like these about student activity, researchers have turned to
identity. By examining how student activity is related to who students think they are,

researchers have begun to make connections between student mathematical activity,

 

identity, and learning (Boaler & Greeno, 2000; J ilk, 2007; Martin, 2000; Nasir, 2002;
Sfard & Prusak, 2005). Identity, like participation, has historically been defined in ways
that do not specifically indicate what it is or what should be studied (Sfard & Prusak,
2005). In order to operationalize identity, I have used the definition proposed by Sfard
and Prusak (2005). They define identity as a significant, endorsable, and reified narrative
about a person. They elaborate on this definition:

The reifying quality comes with the use of verbs such as be, have or can
rather than do, and with the adverbs always, never, usually, and so forth,
that stress repetitiveness of actions. A story about a person counts as
endorsable if the identity-builder, when asked, would say that it faithfully
reflects the state of affairs in the world. A narrative is regarded as
significant if any change in it is likely to affect the storyteller’s feelings
about the identified person. (Sfard & Prusak, 2005, p. 16-17, italics in
original)

The use of narrative as a definition for identity allows me to emphasize two ways
in which I am elaborating others’ use of identity. In order to illustrate these two features,
let me offer a second story about unan. The day before I observed unan copying from
Corey, Andy, unan, and I were sitting near each other at a table.

Andy had realized that he could create smaller fractions by doubling the
denominator and halving the area of a known fraction. He was attempting
to calculate the new dominator for a fraction that was half of 1/96. He
spoke as he worked, “And so ninety-six times two equals. Hmm. Nine plus
nine is eighteen. ” unan noticed Andy ’s work and spoke to me, “You

don ’t plus those. Don ’t you just time those? Nine times nine. Cause he
said nine plus nine. ” I explained that Andy was saying ninety-six times
two which meant ninety-six two times. I continued, explaining that Andy
was adding ninety-six plus ninety-six. I then asked unan if nine times two
and nine plus nine would have the same answer. unan replied, “I guess ”
and then started working on a piece of paper, multiplying nine times two
and adding nine plus nine. After he finished these calculations, he
exclaimed, “0h! They do equal the same thing. ” (02/13/06)

unan’s activity on this day was quite different from the copying described in the

opening vignette. While there might be many reasons for this, use of an identity lens

suggests that in this moment unan was identifying differently than he was in the
opening vignette. Rather than rely on someone else’s mathematical drinking, unan
identified as someone who could ask questions and investigate ideas about mathematics.

Other studies linking mathematical activity and identity have not used identity in
a way that would capture unan’s shift in identities between these two vignettes. For
example, Nasir (2002) emphasized how an individual’s identity changed as they learned
so that as the person became more proficient at an activity, they identified with others
who were also more proficient. This conception of change in identity does not capture the
ways in which a person’s identification might differ as situations change across lessons or
even within a lesson. Researchers have not analyzed student activity for changes in
identity from moment to moment as context shifts. This example of unan illustrates
why this more static conception of identity is problematic: unan could be identified
very differently depending upon whether his identity was based upon the first vignette or
the second. An identity based on the first vignette would fail to capture unan as
someone who investigates mathematical conjectures and might only imagine unan as
someone who copies. Instead of declaring one identity for unan as he learns
mathematics, it is important to capture unan’s identities in both moments and to study
how the identities arise in the moment. This might allow for the construction of learning
environments that encourage investigation rather than copying. Thus my analysis of
identity emphasizes the ways in which identity seems stable and yet is fluid and
susceptible to change from moment to moment.

My construction of identity also emphasizes how identity arises from interactions

among students. Other studies of identity have not examined the role of student

interactions. For example, Boaler and Greeno (2000) considered how students came to
identify as math people or not math people based upon the match between their
conceptions of themselves as learners and the activities required in their math classes.
That analysis does not contribute to making sense of the difference in unan’s activity
between the two vignettes. I draw upon positioning theory (Harré & van Langenhove,
1999; van Langenhove & Harré, 1999) to elaborate how interactions among individuals
convey identities. For example, when I explained to unan why Andy was adding nine
plus nine, I asked unan if he thought adding two nines would have the same outcome as
multiplying nine by two. I could have told unan that the outcome of multiplying by two
and adding two nines would be the same or I could have told unan that he didn’t need to
know what Andy was doing. However, my reply to unan suggested a story in which he
was identified as someone who would be interested in and able to determine the outcome
of mathematical operations.

This study draws upon this elaborated notion of identity to examine how students’
mathematizing and identifying are connected to their interactions and to their learning of
mathematics. As the vignettes with unan illustrate, close examination of student activity
has the potential to reveal surprising variation and complexity in student identity and
mathematizing that could have important implications for learning.

This mathematizing and identifying lens has implications for reform mathematics
education. The Principles and Standards for School Mathematics (National Council of
Teachers of Mathematics, 2000), a guiding document for reform in K-12 mathematics
education, suggests that student interaction is essential for learning mathematics with

understanding. To facilitate interaction among students, teachers frequently place

students in groups to complete instructional tasks. However, students don’t always learn
well in group settings. Researchers have examined the challenges of students learning
from students through a variety of lenses such as status (Cohen, 1994), helping behaviors
(Webb & Mastergeorge, 2003) and distribution of resources such as information (Buchs
et al., 2004). However, researchers have not yet explored the role of identity in
facilitating or constraining learning in group settings. Because this project focuses on
mathematizing and identifying as it occurs in small groups, this project may add
information that could help scholars and teachers to make sense of group learning and to
design more productive group work situations.

Finally, the mathematizing and identifying lens I propose also helps us to study
and elaborate a central goal of reform mathematics education: autonomy. Mathematics
educators have stated the importance of autonomy in learning mathematics (Ben-Zvi &
Sfard, 2007; Karnii, 1994; National Council of Teachers of Mathematics, 2000; Piaget,
1948/1973; Warfield et al., 2005; Yackel & Cobb, 1996). I became interested in
autonomy as I looked for a way to capture the differences in student activity such as those
demonstrated by unan above. In the second vignette, unan acted in ways that were
consistent with descriptions of autonomy. He “[took] control of his learning”, “explore[d]
mathematical ideas”, and “reflect[ed] on his thinking” (National Council of Teachers of
Mathematics, 2000, p. 21). He wondered about Andy’s mathematics and, when I
answered his question, he did not merely accept my explanation, but investigated the
mathematics for himself.

Elaboration of the activity of autonomous learning through the lens of

mathematizing and identifying offered the possibility of capturing details of student

 

activity that seemed mathematically desirable and analyzing and suggesting changes to
activity that seemed less desirable. Thus I expanded my study of student learning to
consider the ways in which student activity resembled the activity of autonomous

learning and how that activity connected to changes in mathematical discourse.

Overview of the Dissertation

In the subsequent chapters of this dissertation, I elaborate my theoretical
framework and present my analysis of data before describing the implications of this
framework for research and teaching.

In Chapter 2, I add details to the theoretical framework sketched above. I begin by
explaining how participationist theories contribute to examination of learning
mathematics and then note challenges of using those theories. I discuss how Sfard’s
commognitive framework (2008) narrows participation to communication and focuses
analysis of mathematical learning on discourse. I then describe why and how identity is
important to making sense of student activity and I explain my construction of identity.
Finally, I connect mathematizing and identifying to autonomous learning, constructing a
definition for autonomous learning and describing the mathematizing and identifying
activities that indicate autonomous learning.

In Chapter 3, I describe my research process. I discuss the theoretical basis for my
methods and then describe the setting for this study including the school, classroom,
teacher, and students. This description is followed by details of the data I collected, my
analysis procedures, and limitations of my methods.

Chapter 4 orients the reader to the findings chapters. In this chapter, I provide a

brief description of research on the teaching and learning of area, the mathematical focus

of the lesson I examined. I then outline the lessons in the unit and indicate the position of
the focal lesson in the unit. Finally, I describe the focal lesson in detail and lay out
features of the mathematical discourse we might ideally hope that students would be able
to use by the end of the lesson.

Chapters 5, 6, and 7 present my findings. Each chapter considers the learning of
an individual student during the mathematics lesson. First, I analyze the change in each
student’s mathematical discourse across the lesson. I then characterize the kind of
learning in which each student engages and explore how the mathematizing and
identifying of the student compares to autonomous learning. The conclusion of each
chapter describes connections between mathematizing, identifying, and changes in
discourse. I also evaluate the ways in which the student demonstrates (or not)
autonomous learning and discuss implications of their mathematizing, identifying, and
change in discourse for our understanding of autonomy and learning mathematics.

Finally, in Chapter 8, I look across the three findings chapters to answer my
research questions. I discuss my findings, elaborate implications for teachers and

researchers, and discuss limitations and future research directions.

CHAPTER 2
THEORETICAL FRAMEWORK
In this chapter, I will build the theoretical background for my model by examining
a series of theoretical lenses and empirical studies. I intend to show how my model
integrates and extends the work of several other researchers. As I move from lens to lens,
I also refer back to unan and the two vignettes from Chapter 1 as a way of showing how

this progression of lenses allows an increasingly complex analysis of classroom events.

The Participationist Lens

Over the past decades, some educational researchers have adopted a perspective
on learning that emphasizes the ways in which learning arises from and is connected to
participation. Theories using this participationist perspective (Sfard, 2006) include
situative/ situated (Boaler, 2002a; J. S. Brown et al., 1989; Lave & Wenger, 1991), social
(Schoenfeld, 1999; Wenger, 1998), and sociocultural (Cobb & Hodge, 2002). These
theories define learning in terms of participation, as the increased ability to participate in
the activities of a community (Greeno et al., 1996).

The participationist perspective is frequently contrasted with cognitive or
acquisition-oriented lenses which view learning as something individuals acquire (Sfard,
1998). For example, an acquisitionist might ask whether a student learned and what the
student learned with responses to these questions framed in terms of immaterial,
knowledge-oriented objects that the learner might obtain. In contrast, the participationist
perspective assumes that people are always learning (Wenger, 1998) which eliminates
questions about whether students learned. Instead, this perspective asks what the student

learned and seeks to answer that question in terms of activity and interaction that arise

10

from examining how the student participated. For example, Boaler (2002b) examined
differences in learning between traditional and reform mathematics classrooms. Using an
acquisitionist lens, she found that the students in the reform classrooms appeared to have
learned more than the students in the traditional classrooms. However, as she examined
student learning through a participationist lens (2002a), she concluded that both groups of
students — those in traditional classrooms and those in reform classrooms — learned a
great deal but that what they learned was different. The students in traditional classrooms
learned “to watch and faithfiilly reproduce procedures and to follow different textbook
cues that allowed them to be successful as they worked through their books” (2002a, p.
43). In contrast, the students from the reform classrooms learned to use mathematics in a
variety of different, open-ended contexts. The participationist lens also enabled Boaler to
see how this difference in participation resulted in differences in test performance
between the two groups of students. The test that Boaler used to compare the students
required the ability to use mathematical knowledge in diverse ways and as part of
authentic activity. The students from the reform classrooms had learned mathematics in
ways that resembled the kinds of engagement necessary to successfully answer the test
questions. In contrast, the repetitive learning of the traditional students did not give them
experiences that were useful on the test. As a consequence, the students fi‘om the reform
classrooms outperformed the students from the traditional classrooms on the test. Boaler
concluded from this analysis that the ways in which students participate in learning
mathematics has consequences for what they learn and the circumstances in which they

can utilize their learning.

11

While participationist theories have been helpful in making sense of how student
learning varies as students participate in different activities and different contexts, the
broad brush of participation does not provide a means to specifically explore an
important concern of mathematics educators: student thinking. In order to make
statements about the mathematics unan understood, the vignette must be examined with
a lens that explicitly defines thinking in such a way that makes it accessible to
researchers. Furthermore, in keeping with participationist tenets, thinking, which is
usually conceptualized as originating from within the individual, must be defined in a
way that acknowledges how individual activity arises from and connects back to the
activities of the community. One theory that appears to be particularly promising for
defining thinking in ways that meet these challenges is Anna Sfard’s commognitive

framework (2001, 2008).

Narrowing Participation to Communication
Sfard’s framework defines thinking as communicating with oneself. Equating

thinking with communicating acknowledges the recursive relationship between the
individual and the community: Thinking is the individualization of the communication of
the community. This individualized communication might then be introduced to the
community where it could alter the communication of the community. For example, as
unan’s class studied fractions, one student, Andy, realized that new fractions could be
created by doubling the denominator and halving the size of an already existing fraction.
Andy’s thinking was connected to the mathematical practices established in this
classroom: Students had been constructing familiar fractions by determining their area.

Andy individualized this community practice by exploring how new fractions could be

12

constructed from these familiar fractions. Andy shared his thinking with the class and
although his explanation was initially resisted, some of the students eventually showed
interest in exploring his ideas, thus establishing the possibility that Andy’s thinking might
affect the activity of his classroom community.

Defining thinking as communicating makes discourse the object of study.
Discourse is broadly defined as communication, so gestures and other nonverbal or
nonlinguistic means of communicating are considered in a commognitive analysis. By
focusing on communication, the commognitive framework solves one dilemma of
participationist frameworks: setting boundaries on what to observe and how to focus an
analysis. Rather than make everything a student does a potential object for analysis, the
commognitive framework concentrates on communication involving mathematical
objects. These communications, whether mathematically appropriate or not, are called
mathematizing.

Using Sfard’s framework to analyze the vignette with unan would lead to
careful examination of unan’s mathematizing: what he communicated about
mathematics through writing, speaking, or gesturing. For example, unan’s written word
“Hiangle” would be investigated for its relationship to mathematical discourses including
discourses specific to this classroom. A researcher might want to explore the possibility
that “Hiangle” was unan’s individualization of the classroom discourse around
triangles. unan’s discourse as he explained (or, in this case, was unable to explain) his
use of the word would provide clues to unan’s thinking.

If the researcher had access to other moments in which unan communicated

about fi'actions, she might be able to make claims about a central, if not the central,

13

phenomenon of interest to mathematics educators: learning. Defining thinking as
communicating suggests that learning, as an outcome, can be defined as a change in
discourse. As students engage in the process of learning, they are working (even if
tacitly) to change their communication with themselves and with others. The
effectiveness of the process of learning can be evaluated by comparing the resulting
change in discourse with the mathematically desirable discourse. In the specific case of
mathematics, a desirable outcome of learning means that the learner is more able to
engage in an appropriate mathematical conversation with him/herself and others (Ben-Zvi
& Sfard, 2007). unan could be said to have learned if his communication around the
mathematics task changed so that he used a more mathematical discourse to explain
whether a given rectangle was partitioned into thirds.

The commognitive framework lists four features of discourse that can be used in
determining what changed in unan’s discourse (or that of another learner) and whether
that change is more mathematically desirable. These features are mathematical word use,
visual mediators, discursive routines, and endorsed narratives. I will briefly describe how
Ben-Yehuda, Lavy, Linchevski, and Sfard (2005) define each feature.

Word use considers how speakers use mathematical words. Ben-Yehuda et al.
(2005) examined each mathematical word uttered by their research participants. They
specifically considered whether mathematical words referred to written or spoken
symbols or to independently existing entities. For example, using area, a student could
use unit to refer to a repeated shape partitioning a figure drawn on paper into equal pieces
or to a concept not tied to a physical object (i.e. a unit of measurement, which is not

necessarily pointing to something concrete). When the discourse referred to

14

independently existing entities, the discourse is objectified, meaning that the learner
refers to the mathematical entity as an object separate from any concrete representation.
Objectified discourses are also impersonal. Rather than talk about what they did (i.e. I
counted an area of eight), students using objectified discourses use the mathematical
words as the subjects of utterances (i.e. The area of the figure is eight). Ben-Yehuda et al.
(2005) note that discourse of formal mathematics and that used by mathematicians is
impersonal and objectified and that students who used objectified discourses have been
found to have more effective mathematical performances. Thus, it seems important to
evaluate a discourse for objectivity as that could mean that the students using the
discourse have made advances toward a more mathematical and academically productive
way of talking about mathematics.

A second feature of discourse, visual mediators, are symbols, icons, or concrete
things that are the objects of the discourse (Ben-Yehuda et al., 2005). They are what the
discourse refers to, the objects (concrete or imagined) about which individuals are
communicating. For example, as students compare the area of two figures, they will refer
to the figures in their discourse. In order for the sentence, “They both have two,” to be
meaningful to everyone involved in the conversation, all interlocutors must understand
that “they” refers to the two figures (and that two refers to the area in units). The figures
thus mediate communication in this conversation. Mediators can also be iconic (such as
graphs or other pictures) or symbolic (such as ‘/2 or any other mathematical symbol,
whether present in written form or spoken).

Discursive routines are “a set of meta-rules that specify both when and how

repetitive discursive action is employed” (Ben-Yehuda et al., 2005, p. 203, italics in

15

original). As interlocutors develop a discourse, patterns emerge in what to say and when.
A common example of a discursive routine occurs when two friends meet each other.
Seeing a friend for the first time during the day (when) initiates a routine of greetings and
counter greetings (how):

Friend 1: “Hello. How are you?”

Friend 2: “Fine. How are you?”

Friend 1: “Good.”

This routine is complete after this exchange and the friends are free to engage in a
different discourse, which will have its own set of meta-rules and routines. Mathematical
discourses also have routines. For example, when asked for the area of a figure
partitioned into square units, a student might count the number of square units and then
announce the number. Having stated this piece of information the student would then
stop. The student has initiated a routine in response to a question embedded in a context.
The student proceeded through the routine to an answer and then terminated the routine.
Thus discursive routines have rules that govern when to initiate them, what to do and say,
and when to stop.

The fourth feature of mathematical discourses is the endorsed narrative. The use
of narrative in the commognitive framework differs somewhat from the use of narrative
in literacy. Narratives in literature are usually accounts of events. In contrast, narratives
in the commognitive fiamework are

a series of utterances, spoken or written, that is framed as a description of

objects, of relations between objects, or processes with or by objects, and

is subject to endorsement or rejection, that is, to being labeled as “true” or
“false” (Sfard, 2008, p. 300)

16

Endorsed narratives are statements that participants in the discourse believe to be true.
Math facts and definitions are two examples. Other endorsed narratives may be unique to
the particular discourse. For example, students and their teacher might develop a
classroom discourse around a particular task. Features of that task might be incorporated
in endorsed narratives. For example, the statement, “Two triangles make a square” is a
narrative because it describes a process with mathematical objects. It is not endorsable
across any two triangles. However, in the context of task in which all triangles are
conguent, isosceles, right triangles, this statement is true and thus, when individuals act
as though they accept this narrative as true, it is an endorsed narrative.

These four discursive features are helpful in examining changes in student
discourse and how student discourse differs from mathematical discourse. Using these
tools to make these comparisons allows claims about outcomes of student learning. If
these tools were applied to unan’s discourse, we could make claims about what he
learned and in what ways his learning outcome was mathematically desirable. However,
these tools and this framework as elaborated thus far does not provide a means for
explaining why unan’s activity was so different from his neighbor’s. Looking closely at
unan’s mathematical discourse cannot answer questions about why unan chose to
copy from Corey instead of engaging in the same work or why unan did not ask Corey
to explain the problem. The commognitive framework also does not address how
unan’s activity might result from interactions with Corey or others. To answer
questions about how students act and why they might act in different ways, researchers

have turned to the notion of identity.

17

Mathematical Activity and Identity
By examining how student activity is related to who students they think they are,
researchers have begun to make connections between student mathematical activity,
identity, and learning (Boaler & Greeno, 2000; J ilk, 2007; Martin, 2000; Nasir, 2002;
Sfard & Prusak, 2005). For example, Boaler and Greeno (2000) demonstrated the
interplay between mathematical activity, learning, and identity in their study of Advanced
Placement (AP) Calculus students. They interviewed several students who were learning
in classrooms characterized by Boaler and Greeno as didactic. The rhythm of these rooms
was predictable fi'om day to day: The students reviewed homework, watched the teacher
demonstrate the next procedure, and then practiced the procedure using problems from
the textbook. Students were discouraged from discussing the mathematics or the
problems and were instead directed to focus on their own work. Boaler and Greeno
interviewed several successful students from these classes. They found that some of the
students liked the classes and were identified as “math people” while several of the
students expressed a dislike for the ways of learning required by the class. This second
group of students noted that their math class asked them to act in ways that were not
congruent with how they saw themselves. They felt they were creative, verbal, and
independent and believed that if they continued in mathematics they would need to give
up these ways of being. According to Boaler and Greeno, the identities of this second
group of students were not consistent with the mathematical activity required in their
mathematics class. As such, they did not identify as math people. Both groups of students
Were successful in their calculus classes. However, their individual experiences of the

Classes varied, as did their identification as math people. Boaler and Greeno’s study

18

suggests that unan’s identification of himself as a mathematical learner might be related
to the match between his identity as a learner and the mathematical activities required in
his class.

The work of Boaler and Greeno suggests that students may participate in similar
mathematical activities and come to different conclusions about their identities. The work
of Sfard and Prusak (2005) adds more complexity to the relationship between identity
and mathematical activity: Rather than focus on how activity influences identity, their
work suggests that individuals’ identities influence their mathematical activity. Sfard and
Prusak looked at differences in activity and identity between two groups of Israeli high
school students, the OldTimers and the NewComers. The two groups had very different
ideas about both who they were to become (what Sfard and Prusak refer to as designated
identities) and about how to engage in mathematical learning. The NewComers felt that
learning mathematics was essential part of becoming completely human. In order to be a
whole person, they needed to understand mathematics. In contrast, the OldTimers felt
that who they were to become was fluid and could change. Mathematics was important
because it could open doors, but it was not essential to who they were. When each group
of students engaged in learning mathematics, the ways in which they participated were
substantially different. When the NewComers studied, they worked until they felt they
understood the mathematical content. When the OldTimers studied, they focused on
producing written evidence that they had done the assigned work, but they did not test
themselves or double check their understanding. Sfard and Prusak suggest that
differences in these students’ ideas about their designated identities could account for

these differences in their mathematical activity.

19

Sfard and Prusak’s proposal that identity influences mathematical activity is
supported by the findings of other researchers exploring connections between
racial/ethnic and gender identity and activity. Some of these other researchers hail from
the field of psychology, which means that their research methods are different from those
of educational researchers. However, their work is helpful in thinking about connections
between identity and activity. One team of researchers, McGlone and Aronson (2006),
started with the assumption that individuals have multiple identities. For example, a
person may identify as a woman, as an Asian American, and as a student at an elite
private school. (Multiple identities are important to my model. I will have more to say
about them later.) McGlone and Aronson proposed that reminding students of particular
identities could impact students’ performance on tests. They worked with male and
female undergraduates and gave them one of three tasks that asked them to reflect upon
their experiences relative to their gender, their elite private college, or the region of the
country in which they currently live (the control group). This task was meant to evoke a
particular identity (gender or elite college student). The researchers followed this identity
task with a spatial reasoning test. When they compared the women’s results from the
different identity tasks, they found that the group of women who answered questions
about their college scored the highest. The control group was the middle score and the
gender experiences group had the lowest score.

McGlone and Aronson use “stereotype threat” (Steele & Aronson, 1995) to
interpret these differences in performance. Stereotype threat suggests that individuals
may underperforrn in intellectual tasks because they are worried about “confirming or

being judged by a negative social stereotype” (Steele & Aronson, 1995p. 797). McGlone

20

and Aronson suggest that asking some of the women to reflect upon their experiences as
a woman made gender identity salient for them. When these women worked on the
spatial reasoning test, they underperformed because of their apprehension about
confirming the stereotype that women do poorly on spatial reasoning tests. In contrast,
the women who were asked about their college experiences were not in a context where
their gender was salient. Instead, they were reminded that they were high status students
at a private institution. McGlone and Aronson argue that priming this student identity
seems to have thwarted effects of stereotype threat and even enhanced student
performance.

This work of McGlone and Aronson draws upon and adds to the work of other
researchers who have found that racial/ethnic identity, like gender identity, can also
positively or negatively affect performance. Shih, Pittinsky, and Ambady (1999)
conducted a similar study with Asian-American women. They gave the women a task that
asked about either their gender experiences, their family heritage including other
languages spoken at home, or their preferences for cable television (the control group).
The women were then given a math test. The analysis of the math test results showed that
the lowest performing group was the group asked about their gender experiences. The
highest performing group was the group asked about their family heritage. Like McGlone
and Aronson, Shih, Pittinsky, and Ambady concluded that reminding women of their
gender identity invoked a stereotype threat that undermined their performance. In
contrast, reminding women of their Asian-American identity invoked a positive

Stereotype that enhanced performance.

21

The studies of McGlone and Aronson and Shih, Pittinsky, and Ambady provide
evidence that complements Sfard and Prusak’s suggestion that identity affects
mathematical activity and can be directly manipulated. Their work also extends that of
Sfard and Prusak by providing evidence that context can evoke particular identities. (This
extension will be important later when I discuss the importance of interactions.) Taken
together with Boaler and Greeno’s study on calculus students, this research illustrates the
bidirectional relationship between identity and mathematical activity and implies that
questions about learning cannot just focus on mathematical activity but must also
consider student’s identity. As students learn and change their activity, their ability to do
new things can lead to a new or changed identity. Also, a student’s identity may lead
them to seek changes in their mathematical activity and thus changes in their learning.
These relationships between identity, mathematical activity and learning are supported by
Nasir’s study (2002) of mathematical learning among African-American students.

Nasir focused on students’ out of school experiences with mathematics, observing
students as they played basketball and dominoes. She found that as basketball players
became more engaged in basketball, they became more interested and better skilled at
calculating their performance statistics, which lead to more engagement in basketball and
a shift in identity that reflected this increased engagement. As the students began to see
themselves as serious basketball players, they set new goals for themselves and worked to
change their participation to reach those goals. Thus, as the students learned more about
playing basketball, including calculating their statistics, their identity as basketball
players shified and as they identified as serious basketball players, they learned more

about basketball and statistics. Drawing from her observations of these students, Nasir

22

argued that learning, identity, and mathematical activity are intertwined and
simultaneously affect each other.

The research studies linking identity and mathematical activity that I reference
above are primarily focused on elaborating identity. As such, they do not provide a
specific definition of mathematical activity, participation, or learning. None of them draw
upon the commognitive framework or limit their examination of mathematical activity to
mathematical communication (although two of them (N asir, 2002; Sfard & Prusak, 2005)
include students’ communication as they engage in mathematics in their analysis of
mathematical activity). Their findings are still useful in constructing this theoretical
framework. However, my use of a more narrow conception of mathematical activity
means that my study may add more complexity and/or different features to understanding
the relationship between mathematical activity, identity, and learning mathematics. In
order to emphasize my focus on mathematical activity as communication about
mathematical objects, I will begin to use the words mathematizing or mathematizing

activity instead of mathematical activity.

A Definition for Identity
The relationship between mathematizing activity, identity, and learning can be
further clarified through a definition of identity. Because the definition of identity
organizes the connections between the elements of my model, I will devote some space to
developing and elaborating my definition. Sfard and Prusak (2005) note that many
researchers (in particular Gee (2001) and Holland, Lachiotte, Skinner, and Cain (1998))
have suggested “who a person is” as a definition for identity. While this definition feels

like a reasonable match with an intuitive sense of what identity might be, it does not

23

require or necessarily evoke a connection with participation. In addition, “who a person
is” suggests that identity is a stable essence of a person while I have already proposed
that identity is dynamic and changes with mathematizing (and that mathematizing
changes with identity). Finally, my definition of identity must also account for
stereotypes and the ways in which one person has multiple identities.

In order to meet the requirements listed above, I am drawing upon the definition
of identity proposed by Sfard and Prusak (2005). They suggest that identity is a collection
of reifying, significant, and endorsable narratives about a person. They elaborate on this
definition:

The reifying quality comes with the use of verbs such as be, have or can

rather than do, and with the adverbs always, never, usually, and so forth,

that stress repetitiveness of actions. A story about a person counts as

endorsable if the identity-builder, when asked, would say that it faithfully

reflects the state of affairs in the world. A narrative is regarded as

significant if any change in it is likely to affect the storyteller’s feelings

about the identified person. (Sfard & Prusak, 2005, p. 16-17, italics in

original)

For example, statements like “I am a woman” or “I am a math person” are identities for
me: I believe them to be both true and important to how I feel about myself. Sfard and
Prusak’s definition captures the link between mathematizing and identity through the
concept of reified narratives. Because this concept is vital to the articulation of
mathematizing and identity, I will spend some time elaborating the terms narrative,
reification, and reified narrative.

As I noted in my discussion of the four discursive features above, narrative has a
Variety of definitions arising from its use across multiple research disciplines and

traditions. My use of narrative to define identity draws upon the definition of researchers

Whose research interest and genre of data are similar to my own. Ochs and Capps (2001)

24

studied narratives arising in everyday conversations. They were interested in how
impromptu, co-constructed narratives help people understand themselves and others with
whom they interact. Their work parallels my focus on everyday conversations between
students as they negotiate their participation and identities in math class.

Ochs and Capps define narrative as an account of life events. They note that
narratives may have description, chronology, evaluation, and explanation, but that that
these elements do not have to be explicitly evident in order for a person’s speech to
qualify as a narrative. Ochs and Capps’ flexible definition of narrative means that
minimal statements are also narratives as long as they are an account of an event. Other
researchers have also argued this point. Dino Felluga (2003) along with Michael
Riffaterre (1990) argues that short sentences like “The road is clear” are narratives
because they suggest some sequence of events. The reader is drawn to wonder why the
road is clear, whether we should cross the road, or whether the clear road is a moment of
calm. This notion of a minimal narrative means that statements like “I am a math person”
are also narratives because they conjure images of what such a person has done or might
do. This connection between these minimal narratives and potential actions will become
important in my discussion of identities below.

Ochs and Capps’ definition of narrative points to an important feature of
narrative: Narrative is not a straightforward list of everything that occurred but is instead
the narrator’s abridged and edited account. This reworking of events is a necessary part
of how humans make sense of what happened. Events are not meaningful in themselves.
Instead, people work to understand events by focusing on particular parts of the event and

weaving these parts with different framings until the narrative offers an explanation of

25

the event that matches other narratives the author already has (Ochs & Capps, 2001). For
example, in the opening vignette, I told a particular narrative about unan. However, I
could have told a different story. I could have said that unan was worried that he would
look stupid if he asked Corey to explain what he was doing, so he was copying as a way
of saving face. Another alternative was to say that unan didn’t want to extend the
mental effort necessary to figure out the math and so he was copying. Either of these
scenarios is a possibility. However, I settled on my narrative because it resonates with
other narratives I have including narratives about myself, unan, his neighbor, this math
classroom, fourth grade students, and math learners. In addition, my narrative reflects my
ideas about narratives written for academic audiences, which means that I included as
little explicit interpretation as possible. However, if I told a story about this event to
unan’s teacher, my narrative might incorporate more interpretation and labeling of
unan. I might say that unan doesn’t think he’s smart or I might draw upon other
narratives that the teacher has told me about unan. Thus, I abridge and edit my account
of events in order to make sense of the event in light of narratives I have about both the
people directly involved in the event and the people involved in listening to my account
of the event.

Sfard and Prusak’s concept of reified narrative unites narrative with another
mechanism people use to making sense of the world: reification. According to Wenger
(1998), reification is “the process of giving form to our experience by producing objects
that congeal this experience into ‘thingness’ (p. 58). My mathematizing-identifying-
learning model is an example of reification. As I construct this model, I am making an

Object that freezes the transient actions of mathematizing and identifying and allows

26

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an

those actions to be manipulated, examined, and connected in ways that are not possible
when they are conceptualized as fleeting experiences.

As people tell stories, they tend to summarize the story’s action into statements
that describe or label people. These statements reify activity into human conditions. For
example, at the end of unan’s vignette, I could have added, “unan was reluctant to
engage math.” This sentence condenses unan’s copying and his inability to explain the
mathematics of the problem into a tidy description and interpretation of my perception of
unan’s problem. Once I have generated this reified narrative, I no longer need to tell the
entire story about unan’s actions. Instead, I have a short sentence that captures the
activity and turns it into a thing (unan’s reluctance) that I can use to make sense of my
interactions with unan.

This short sentence along with other significant and endorsable reified narratives
about unan is an identity of unan. It is important to note that as narratives, identities
have an author, an identified person, and an audience (Sfard & Prusak, 2005). For any
identified person, there can be multiple identities as different authors tell different
identities to different audiences. For example, depending upon my audience, I might
identify unan as a reluctant participant, as a low student, or as an African-American
boy. unan might also alter his identity depending upon his audience, narrating himself
as a talented football player, a caring friend, or an incompetent math student.

These multiple identities point to one of the complexities of identity: It is situated
and dynamic yet it appears to be a stable essence of a person. I can identify unan in
multiple ways at different times, each time making an endorsable and significant

identification matched to the situation. unan is an African-American boy when I’m

27

talking about demographics in a case study. He is a reluctant student when I’m discussing
his activities during math class with his teacher. He is a low student when his teacher
suggests that he receive extra tutoring support. Each of these different and changing
identities seems to be a stable and essential part of unan. This sensation of a stable
essence arises from identity’s connection to reification. Wenger (1998) notes that the
process of reification projects understanding into the world in such a way that it appears
disconnected from its human origin. The reified object seems to be a real, independent
thing rather than a subjective human creation. Also, as the object loses its history of
creation, it becomes timeless. Thus as narratives become reified into identities, they lose
their human author and gain a sense of permanence. No longer are they one person’s
perspective on a transient occurrence. Instead they are narratives that that stretch through
time and become something the person has always done and will always do.

However, because identities maintain their tie to actions, they are still situated.
Thus, a particular context can evoke a particular identity and not others. For example,
unan’s identity as a football player does not enter my introductory vignette because my
interpretation of the context and situation in the vignette did not evoke the narrative of
unan as a football player. If I wrote about unan at recess, I might see his identity as a
football player as more relevant to my narrative. (This is not to say that unan’s identity
as a football player is irrelevant when he is doing math. He might see this identity as
central to how he talks and interacts in math class. After careful study, I might also be
able to construct a narrative in which the football player identity is significant in the

context of the math class. However, I want emphasize that identities are dynamic and

28

situated and that there might be times when unan is seen as a football player and other
times when he is seen as a reluctant student.)

Thus far, I have elaborated upon the ways that my definition of identity accounts
for multiple identities and can be dynamic yet stable. I noted above that my definition of
identity must also account for stereotypes. It must be able to describe how invocations of
stereotypes work through identity to affect mathematizing activity as demonstrated in the
work of McGlone and Aronson (2006) and Shih, Pittinsky, and Ambady (1999). A
stereotype is a widely-circulating reified narrative that oversimplifies or exaggerates the
actions of a population of people. For some people, when they interact with members of
this population, these reified narratives are significant and endorsable and thus constitute
identities. This construction of stereotype relies upon two important features of identity
that I have yet to elaborate: identities as the tip of the iceberg and identities as the union
of the individual and the collective. I will develop both features and then return to
stereotype.

Wenger (1998) described reified objects as “the tip of the iceberg” (p. 61). What
is visible (or in the case of speech, audible) is connected to a more detailed and hidden
story of actions. Identities as reified narratives can be thought of as the tip of a narrative
iceberg: An identification of a person comes with expectations for how the identified
person has acted and will act. For example, if I say that unan is a low student, to the
extent that others agree with me, they will use that identity to construct unan’s past and
future narratives as a student. Another important feature of this iceberg metaphor is that
because the actions are hidden, the nature of the underwater iceberg is may be different

for different people. If I identify unan as a boy, I might be reifying a narrative about

29

unan’s preferences for recess activities. However, others might use that identity to
imply that unan’s participation in class resembles that of research about boys.

These different variations on the “underwater” narrative are an important outcome
of identity’s union of the individual and the collective. Because the individual and the
collective commingle in narrative, I will begin by describing narrative’s role in this union
before moving to identity. As I noted in my discussion of narrative above, an account of
events is constrained by its resonance with other narratives. While multiple stories might
be told about an event, the narrator who is interested in making sense of events and
communicating with others will construct an account of events that matches other
narratives that person has about themselves and others. These other narratives come from
a variety of sources. They may arise in a local community such as a classroom or school
or the narrator may draw upon national or international narratives. As the narrator uses
these other narratives, they incorporate collective perspectives into their individual
account of events. Also, as individual narratives are told and retold, they become part of
the narrative possibilities other authors can use in crafting their own narratives. Thus the
individual creation becomes a collective resource. This “communalization of the
individual” (Sfard, 2006, p. 23) is important because it provides a mechanism for
variation in narratives. It means that as the individual creates a narrative, s/he is not
necessarily constrained by the most popular national narrative. Instead, there are a variety
of narratives circulating, allowing for many different individual narratives.

Identities as endorsable reified narratives capture this same union of the
individual and collective. When I state, “I am a woman” I draw upon circulating

identities of women to make sense of and communicate my understanding of my actions.

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As narratives about individual women circulate and are reified (which especially happens
when the story is about an extraordinarily positive or negative event), they become
identities that are collectively available. For example, as stories are told and reified about
US. Speaker of the House Nancy Pelosi or NASA astronaut and alleged kidnapper Lisa
Marie Nowak, those individual identities (being a Pelosi or being a Nowak) could
become widely available and used to identify other women.

While this merging of the individual and collective provides occasions for
individuals to alter the collective understanding, it also provides a mechanism for the
collective understanding to limit the possibilities for the individual. Stereotypes are an
example of this. I defined stereotypes as widely circulating, oversimplified, reified
narratives about a population of people. As identities, they are imposed upon people
based upon the perception that they are members of a population rather than based upon
individual actions. As reified narratives, they are the tip of the narrative iceberg and thus
imply a particular set of past and future actions. However, because identities, as the tip of
the iceberg, can have different narratives attached to them, a stereotyped person may
agree with the reified narrative (such as African American, woman, or firefighter), but
have a different set of actions tied to that narrative. For example, as I discussed earlier,
one stereotype about women is that they are not as capable as men at tasks involving
mathematics (Shih et al., 1999). Someone could identify me as woman and I would agree
with that identity, but not with the accompanying stereotype about mathematical ability.
Instead, for me, being a woman is tied to narratives arising from a wide variety of
experiences I have had in which being a woman was salient. However, as research in

stereotype threat has shown (McGlone & Aronson, 2006; Shih et al., 1999; Steele &

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Aronson, 1995), people are not immune to the narratives of stereotypes: If a person is
concerned that they might confirm or be evaluated by the stereotype, they may actually
enact the stereotype narrative. Thus I may underperforrn on a mathematical task if I feel
that the task might confirm the stereotype about women even though my identity as a
woman and my past actions include narratives about strong mathematical abilities.

I engaged in this discussion of stereotype because I used research on stereotypes
to claim that identity affects mathematizing and I wanted to demonstrate how my
narrative definition of identity accommodates the connections between stereotype,
identity, and mathematizing activity. I want to conclude this discussion of my identity
definition with a summary of the ways in which identity and mathematical activity are
intertwined. Both begin with a person’s actions. Activities involving communications
about mathematical objects are mathematizing activities. An account of these actions, can
become an identity. In addition, identities, because they imply future actions, constrain
and enable options for mathematizing activity. Because learning arises from
mathematizing, the connection between identity and mathematizing is critical to learning.
As identity encourages participation in activities that lead to changes in mathematical
discourse, identity is implicated in learning.

Using mathematizing and identity together raises new questions about unan’s
activity in the opening vignette. How has unan identified himself and how have others
identified him so that copying is the appropriate form of activity for his identities? What
identities are constructed by unan and others based upon narratives about his copying?
What identities do unan and Corey have for Corey and how do those identities lead to

unan’s decision to copy from Corey? Thus far, these questions assume my perspective

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on events. However, unan and Corey might have very different narratives to describe
what happened. What narratives about unan’s participation might be told by each of
them? What different identities would arise from those narratives?

Underlying most of these questions is the notion that identifying is not solely an
individual activity. unan might identify himself in a variety of ways, but he also has
identities that are written by Corey, his other classmates, his teacher, his family, and
many others including myself. My framework as articulated so far assumes that an
identity composed by someone else might affect an individual’s activity. For example,
unan’s copying may arise from the ways in which his peers or his teacher have
identified him even though unan’s peers or teacher may not have specifically articulated
their identifications of him. However, my framework does not so far offer an explanation
of how one person’s activity might be affected by other people’s identifications of him or
her nor does it articulate how individuals can communicate identifications of each other
through interactions that don’t necessarily include reified narratives. To do this work of

tying together mathematical activity and identification, I draw on positioning theory.

Moving from Identity to Identifying
Positioning theory is an appropriate tool for extending my mathematizing-identity
framework because it uses story in ways that are parallel to my use of narrative in the
construction of identity. Positioning theory assumes that an individual’s actions can be
interpreted in multiple ways. In order to make sense of what is happening, people use
stories and they position themselves and any others involved in the activity in the story
(van Langenhove & Harré, 1999). For example, if one student says, “Let me see your

work” to another student, the second student might understand those words in different

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ways depending upon how s/he positions her/himself and the other student in a story
arising fi'om those words. The second student might draw upon a story in which s/he and
the speaker are fiiendly peers in which case the second student could understand “Let me
see your wor ” as a plea for assistance. Or, the second student might use a story of
intimidation and position the speaker as a bully in which case the second student could
understand the words as a command that must be obeyed. The story is important to
understanding because it interprets actions and explains who people are.

This use of story and position is similar to my use of narrative and identity except
that positioning theory assumes that all people involved in an interaction are identified
through the narrative/story enacted in the interaction, even if they are not explicitly
identified using a reified narrative. Instead of relying only on reified narratives,
identification through positioning is based upon the cluster of utterances and actions that
are appropriate for that person in context of the story (Harré & van Langenhove, 1999).
This connection between actions and identity draws upon the “tip of the iceberg”
metaphor discussed earlier. However, in this instance, the actions that are the underwater
part of the iceberg are used to determine what identity might be at the tip. For example, if
a student wants to be identified as a bully, they might march up to another student and
gruffly declare, “Let me see your work.” The bully is enacting a story of what bullies do,
part of which involves positioning others as helpless victims. Neither the bully nor the
student victim has been identified using a reified narrative, but if the bully has been
successful in their posturing, gesturing, and speech, they have communicated both their
identity as a bully and their identification of the other student as a helpless victim. In

addition, the bully may also identify anyone else in the room as either allies of the bully

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or other potential victims by either laughing and winking at the bystanders or by scowling
menacingly at them.

In positioning theory, even narratives about a single person identify both the
author and the audience of the narrative in addition to the subject of the narrative. In my
opening vignette about unan, I was identifying myself as someone who is interested in
elementary students who aren’t engaging in mathematical learning and I was also
identifying my readers as concerned about struggling students and what I might have to
say about them.

While positioning theory provides a mechanism for identifying everyone involved
in a conversation, it also provides a means for individuals to negotiate their identities. In
the example of the bully, the person identified as the helpless victim could dispute this
identity by refirsing to act in the ways that helpless victims might act. Instead of meekly
handing over their work, the “helpless victim” could stand up and firmly say, “No.” This
move shifts everyone’s positions. The “bully” has been identified by the student as
another student and not someone with particular power and the “helpless victim” has
become an assertive person. This positioning and repositioning could continue as the
“bully” and the “helpless victim” negotiate this interaction.

If the “helpless victim” had accepted this identification instead of disputing it,
they would have indicated their acceptance through their actions and speech including
their interactions with the bully. In other words, the helpless victim’s activity would
demonstrate their identification as the helpless victim while it would also assert the
bully’s identification as a bully. Thus, positioning theory unites interaction, activity, and

identity by describing how individual’s communication when viewed by others indicates

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how the individual is identifying themselves and others. This identification then opens
pathways for other individuals to act and interact in ways that either affirm this identity
or declare another identity.

Sfard’s recent work on identity (2007) elaborates ways of identifying that are
consistent with the indirect story-telling assumptions of positioning theory. Sfard has
described three ways in which people identify: direct, indirect-verbal, and enacted. Direct
identifying occurs as a person tells a reifying story about the identified person. I primarily
referred to this way of identifying as I elaborated my definition of identity in the sections
above. Indirect-verbal identifying is when a story is told about a person that does not
include reifying statements. Finally, a person may identify him/herself or another through
other activities that do not include story telling. Sfard calls this type of identifying
enacted. As students interact in classrooms, they rarely directly identify each other.
Instead, most identifications are enacted or in-direct verbal. I have used positioning
theory to provide a framework for elaborating these last two ways of identifying.

Positioning theory also supports the specific vocabulary I am using to capture my
thinking about identities. Rather than describe individuals as having identities, I draw
upon Sfard’s recent work (2007) and purposefully use the words identifiring, enacting
identities, identification, and engaging in identifying activity. These ways of talking are
consistent with the notion of positioning, which emphasizes the ways in which identities
are constructed by individuals in response to situations.

Positioning also captures the ways in which identities may appear to be stable but
are also situated and dynamic. An individual, through activity, both offers and authors

positions for him/herself and others. The positions are offered because they are consistent

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with reified narratives the individual has about him/herself and others, but those
narratives have been authored by the individual so while the reification suggests
permanence, the authorship indicates the human activity which could author and offer
another identity in another situation.

In summary, I define identifying as participation in discourse that communicates a
significant, endorsable, and reifying narrative about an individual. This narrative need not
be directly communicated: It may be suggested by what an individual says or does. It
may be accepted or rejected by other individuals involved in the communication process.

Finally, it suggests a storyline that guides the activities of the identified individual.

Theoretical Links Between Mathematizing and Identifying

My framework of identifying has many parallels with my description of
mathematizing. Because identifying arises from communication of stories, it, like
mathematizing, arises in discourse. Identifying and mathematizing are both focused on
the intersection of the individual and the community. Both are also central to my
definition of learning as a change in discourse. Let me elaborate this point. As a student
engages in learning, he or she both communicates about mathematics and identifies
him/herself as a kind of learner of mathematics. His/her communication about
mathematics simultaneously tells an identity story and provides the opportunity of
changing the learner’s mathematical discourse. Identifying that does not address a
mathematical object (and so is not mathematizing) tells how the student and others in
his/her audience might engage in mathematizing. For example, the student might
proclaim to another student, “You’re so fast at math.” This statement is about a student

and not about a mathematical object, but it identifies both the speaker and the addressee

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relative to their mathematizing. It implies that the addressee is a fast mathematizer while
the speaker is slower. This difference in speed might also be tied to perceptions of a
difference in competence so the speaker might also be identifying the addressee as a more
competent mathematizer. This statement about speed could affect the subsequent
mathematizing of the speaker and addressee: The addressee might accept the
identification as fast and then turn to see what the speaker is doing so that the speaker is
feeling slower. The addressee might offer some mathematical advice that benefits the
speaker. The addressee might also agree that he/she is fast and then race ahead through
more problems, leaving the speaker to feel slower, less competent, and less enthusiastic
about engaging in mathematizing. This example shows how identifying, even when it is
not directly about a mathematical object, can have implications for students’ engagement
with mathematical objects. These connections between identifying and mathematizing
suggest that learning arises from the interplay of these two activities (Sfard, 2007).
Theoretically, the claim that learning is the interplay of mathematizing and
identifying seems to be quite productive in making sense of connections between
students’ thinking about themselves as learners of mathematics and their thinking about
mathematics. However, this theoretical work has not yet been well explored in classroom
data. It is one of the goals of this dissertation to elaborate how this interplay among
identifying, mathematizing, and learning. For example, in the second vignette, I identified
unan as interested in and capable of exploring a mathematical idea about the
relationship of multiplying and adding. unan, by investigating this mathematical idea,
both tacitly agreed with my identifying statement and engaged in mathematical thinking.

His subsequent words, “Oh! They do equal the same thing,” seemed to suggest that

38

unan had changed his thinking about multiplying and adding. He seemed to realize that
adding two nines was the same as multiplying nine times two. While more investigation
of unan’s previous and subsequent discourse would further substantiate this claim,
unan’s statement seems to indicate different thinking from when he asked me why
Andy was adding nine plus nine. This analysis of this vignette with unan shows how
unan has engaged in the process of learning by both identifying and mathematizing.
Furthermore, it suggests that his participation in learning was effective as it changed his
discourse so that it was more mathematically appropriate.

In the three cases I present as Chapters 5, 6, and 7 of this dissertation, I
investigate this claim further. Specifically, in each chapter, I will elaborate a different
kind of learning and show how that kind of learning consists of a particular constellation
of mathematizing and identifying activities. I will also relate the kind of learning to the
effectiveness of the learning.

My analysis of learning, mathematizing and identifying includes one final piece
that might best be described by returning to the second vignette with unan. In this
vignette, unan engaged in mathematics in ways that reflected descriptions of ideal
mathematical learning. He worked to make sense of the Andy’s mathematics. He started
a conversation about something that seemed contradictory to him and he investigated my
response, rather than accepting my words based on my authority as an adult. unan,
through his questions and exploration, displayed curiosity about Andy’s mathematical
ideas. unan’s activity in this second vignette is important because it resembles the
descriptions of autonomous learning that mathematics educators have proclaimed as the

goal of school mathematics (Ben-Zvi & Sfard, 2007; Kamii, 1994; National Council of

39

Teachers of Mathematics, 2000; Piaget, 1948/ 1973; Warfield et al., 2005; Yackel &
Cobb, 1996). In the next section of this chapter, I elaborate my definition for autonomous

learning.

Autonomous Learning

In 1948, Jean Piaget proposed that autonomy should be the goal of education and
he elaborated this idea in the context of learning mathematics (Kamii, 1994; Piaget,
1948/1973). Constance Kamii reintroduced his ideas to mathematics education in the
1980’s and early 1990’s through a series of books focusing on children learning
mathematics (i.e., Kamii, 1985, 1989; i.e., Kamii, 1994). More recent work of
mathematics educators cites the work of Kamii and Piaget, concurs that autonomy is a
major goal of mathematics education, and begins to describe autonomous learning (Ben-
Zvi & Sfard, 2007; Warfield et al., 2005; Yackel & Cobb, 1996). However, this recent
work has not elaborated the activities of autonomous learning or explored the concept in
classroom settings. For example, Yackel and Cobb (1996) and Warfield et a1. (2005)
offer brief definitions of autonomy and Yackel and Cobb describe how autonomy is
related to judging mathematical contributions, but they do not explicitly explore the
connection between student autonomy and learning mathematics. Given the stated
importance of autonomy to mathematics education and the possibility that autonomy
could be a useful lens for differentiating and explaining student learning such as unan’s
activity in the two vignettes from Chapter 1, I propose that it is critical to further
investigate autonomous learning.

Researchers seem generally to concur with Kamii’s explanation of Piaget’s

definition of autonomy as the ability to think for oneself and to decide between truth and

40

untruth (Kamii, 1994). My definition of autonomous learning acknowledges the
importance of the individual’s thinking. However, I define autonomous learning in terms
of the learner’s focus on and desire to understand experiences including both physical
experiences and interactions with others. I draw upon the theory of learning elaborated
above to define autonomous leaming in terms of the interplay of mathematizing and
identifying. Specifically, I explore autonomous learning as a constellation of
mathematizing and identifying activities reflecting curiosity about how things are — both
what others think/say and what seems to be true.

My use of curiosity is purposeful because it captures the ways in which
autonomous learners compare their thinking with their observations and experiences.
Curiosity requires that the learner verify whether his or her thinking adequately describes,
accounts for, or matches his or her experiences. The Oxford English Dictionary defines
curiosity as “the desire or inclination to know or learn about anything, [especially] what
is novel or strange; a feeling of interest leading one to inquire about anything”
("Curiosity", 2008). The emphasis on the novel or strange highlights the autonomous
learner’s passion to make sense of ideas that he/she does not yet understand.

John Dewey argued that curiosity was necessary for reflective thought as it
provided the awareness and the store of facts and experiences from which thoughts would
arise (Dewey, 1910). Dewey characterized intellectual curiosity as “interest in problems
provoked by the observation of things” (p. 33, italics in original, 1910). For Dewey,
observation was not enough: The curious learner formulated questions about those
observations and was actively exploring and seeking material for thought. My definition

of autonomous learning draws upon these notions of the learner as active, problem

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seeking, and interested in the novel or strange. Curiosity means that the learner is not
satisfied with memorization or answers for the sake of other’s approval: The learner’s
autonomy is in part due to the learner’s own passion for understanding.

The autonomous learner’s curiosity leads him/her to explore problems arising
from observation of discourse. He/she investigates mathematical discourses and verifies
that the learner’s communication about those discourses is compatible with the
communication of others who are proficient in the discourse. This verification ensures
that the learner’s interpretation of the discourse is consistent with other’s use of the
discourse and reflects the learner’s attempts to make sense of the other’s discourse. It
ensures that the learner is decentering (Piaget, 1932/1960) or is doing more than
reflecting their own drinking: They are making sense of other’s thinking, whether the
other is the teacher, the textbook, or other students. Autonomous learning also involves
the quest for mathematical truth. The learner uses logic to examine a discourse for
contradictions both within the discourse and between different discourses. The learner
then works to resolve any contradictions by altering their thinking or by proposing
changes or additions to the discourses.

This notion of autonomous learning is consistent with the picture of autonomous
learning portrayed by Ben-Zvi and Sfard (2007). They emphasize that autonomous
learners explore the discourse of experts working to make the discourse-for-others into a
discourse-for-oneself Ben-Zvi and Sfard define discourse-for-oneself as “a discourse to
which the learner would turn spontaneously whenever it may help her in solving her
own problems” (p. 77). For example, the learner’s peers might try to explain to the

learner that two triangles can be put together to make a square that can be used as a unit

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for determining area. In order to make this statement into a discourse-for-oneself, the
learner needs to investigate this discourse. Perhaps he/she needs to assemble triangles
into squares to see how they relate to area. Perhaps he/she needs to attempt to determine
the area of a figure that is partitioned into triangles and squares. As the learner examines
this discourse and incorporates the discourse in his/her thinking, the discourse becomes a
discourse for the learner.

The concept of discourse-for-oneself is similar to Wenger’s (1998) concept of
ownership of meaning. Wenger defines ownership of meaning as “the degree to which we
can make use of, affect, control, modify, or in general, assert as ours the meanings that
we negotiate” (p. 200). As the learner transforms a discourse into a discourse-for-
him/herself, he/she becomes able to use that discourse to solve problems and accomplish
other tasks. This use of the discourse for the learner’s own ends demonstrates Wenger’s
ownership of meaning. The discourse is not merely words mechanically recited to get the
approval of another person, such as a teacher. Instead, when a learner owns the meaning
of a discourse, the discourse becomes a tool for the learner. Ownership of meaning
acknowledges the social nature of discourse: Ownership means the learner can use the
discourse to accomplish social ends. If a learner’s use of a discourse is not understood or
is rejected by others, then the learner does not own the meaning in a way that is valued
within the community. Wenger’s concept of ownership of meaning clarifies that Ben-Zvi
and Sfard’s notion of discourse-for-oneself does not mean the discourse is only used to
communicate with oneself or solve the individual’s problems in isolation. Instead
discourse-for—oneself emphasizes that the learner can communicate with herself as well

as with others using the discourse. Discourse-for—oneself and ownership of meaning are

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consistent with my definition of autonomous learning. The process of autonomous
learning results in a discourse-for-oneself and ownership of meaning as the learner
mathematizes and identifies in ways that reflect curiosity about others’ discourses and
mathematical truth.

As the learner engages in autonomous learning, he/she performs several
identifying and mathematizing activities that support and reflect his/her curiosity about
mathematical discourses. I will elaborate these discursive activities and explain how they
are connected to autonomous learning.

An individual engaged in autonomous learning identifies the audience of his/her
discourse in particular ways. The learner interacts with his/her audience in ways that
suggest they should provide support, evaluation, or verification of the learner’s
investigation. For example, the autonomous learner might position his/her audience as
experts in the discourse who can provide feedback on the fidelity of the learner’s
interpretation of the discourse. As another example, the learner might position the
audience as co-learners who are interested in what he/she has to say and who can ask
questions to clarify the learner’s meaning. In the eyes of the autonomous learner, the
audience is not a gatekeeper whose approval must be won through the meaningless
performance of a ritual. In addition, the learner him/herself is an important member of his
or her audience. As the autonomous learner communicates with him/herself, he/she can
make sense of the discourse for him/herself. This understanding for oneself is an
important goal for the autonomous learner.

The next three discursive activities of autonomous learning are connected to the

activity of mathematizing. First, a person engaging in autonomous learning faithfully

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adopts the mathematical discourse of those whom she considers as expert interlocutors.
This does not mean that the learner is not critical of the discourse. However, before the
learner can critique or examine another’s discourse he/she must be sure that his/her
interpretation and use of the discourse is consistent with other’s use. As I described
earlier in this chapter, the commognitive fi'amework describes four features that make
discourses distinct: word use, visual mediators, discursive routines, and endorsed
narratives. Faithful adoption of the discourse means that the learner’s use of the discourse
is consistent with others’ use of the discourse across these four features.

In addition to adopting the discourse, individuals engaged in autonomous learning
also produce new discursive features that build from the adopted features of the
discourse. For example, a discourse might focus on assembling triangles into squares to
be counted. The learner might realize that two right triangles assembled into a square are
each half of the square and might start to talk about each triangle as half. Use of the word
“half” both builds from the original discourse and the way triangles were used in this
discourse and is also a new word use for this discourse. Production of new discursive
features arises from and contributes to investigation of the original discourse. The
autonomous learner’s curiosity about how people use a discourse and how the discourse
relates to truth leads to explorations about what could be communicated using a
discourse. Exploration of what is possible with a discourse could lead to new discursive
features. Thus one more question to ask while investigating autonomous learning is
whether the learner mathematizes in ways that build from and are novel to the discourse

as used by others.

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The final mathematizing move related to autonomous learning is substantiation of
discursive narratives. Sfard (2008) defines a narrative as “a series of utterances, spoken
or written, that is framed as a description of objects, of relations between objects, or
processes with or by objects, and is subject to being labeled as ‘true’ or ‘false’” (p.
300). Examples of narratives include statements such as “two triangles make a square” or
“this triangle is half a square.” Narratives can also include statements about processes
like “you count the squares to figure out the area.” For example, in the second vignette
above, unan substantiated the narrative, “nine times two is the same as nine plus nine.”
This statement is a narrative because it describes two processes and their relationship and,
as unan demonstrated, can be labeled as true or false. Sfard notes that discourses have
meta-rules that direct their substantiation. In mathematics for example, discursive
narratives can be substantiated by a mathematical proof.

An examination of autonomous learning considers two aspects of substantiation:
whether the learner relies upon her own judgment and how well the learner employs the
meta-rules of the discourse. In school settings, students may not use their own discursive
resources to substantiate a narrative. Instead, they may rely upon other individuals such
as the teacher, another student, or the answer key in the back of the textbook. According
to Piaget (1932/ 1960), this reliance on others for right answers is an indication of
heteronomy, regulation or governance by others, and thus is not a trait of the
autonomous, or self-regulating, learner (who might co-learn with others, but does not rely
on other’s authority to substantiate narratives). My construction of autonomy and
substantiation does not define heteronomy and autonomy as discrete categories in which

the learner’s reliance upon another person means they are not autonomous. Instead, I

46

construct heteronomy and autonomy as ends of a spectrum. The learner could rely
entirely upon another to verify a statement in which case they are not investigating a
discourse or exhibiting curiosity. In this case, the learner is heteronomous and not
autonomous. However, the learner could ask another for an explanation of a statement. If
the learner explores the substantiation supplied by another or uses this substantiation to
work through their own substantiation, then the learner is more autonomous. Autonomy
does not mean that the learner does not interact at all with other individuals, but it does
require that the individual engage in their own examination of the discourse, even if that
examination is initiated or supported by another.

Just as autonomous learners may access varying degrees of support from others in
substantiating a discourse, they also exhibit varying degrees of proficiency at using the
meta-rules of a discourse to substantiate its narratives. Meta-rules define the pattern of
activity of a discourse (Sfard, 2008). They are the rules that guide how someone
determines the truth of a statement. For example, the rules for determining whether
people evolved from animals are very different in science and in the evangelical
Protestant religions. Meta-rules reflect social conventions and are not necessarily
imposed by external reality. A person who was familiar with a few basic narratives in
mathematics could not reason from those narratives to an understanding of how
mathematical truth is established. Instead, meta-rules are learned through interaction with
individuals who are proficient at the discourse. As learners become more proficient in the
meta-rules, they are better able to substantiate narratives in ways that are consistent with

the discourse. As a consequence, they will better able to evaluate the narratives they

47

produce in terms of the discourse and will become more autonomous in their ability to
communicate using the discourse.

In school settings, the meta-rules for substantiation of a mathematical discourse
may be different from mathematicians’ meta-rules for substantiation. For example, in a
mathematics lesson, verification of the area of a figure might require cutting apart the
figure and arranging the pieces into squares to be counted. In contrast, a mathematician
would accept multiple means of verifying the area of a figure and might not consider
physically cutting apart the figure to be an acceptable means. This difference in
substantiation means that student discourse should be evaluated to determine not only
whom the learner involves in their substantiation and how that person is involved but also
what rules the learner is using to guide substantiation.

In the paragraphs above, I’ve described four discursive activities (one related to
identifying and three related to mathematizing) that characterize autonomous learning.
These discursive activities provide a lens for examining student participation. For
example, in the first vignette, unan cannot explain his work, which suggests that he was
not an audience for his work. He copied what he wrote, so he was not producing new
discourse. He did adopt some discourse, but only words. He could not recite narratives,
enact routines, or point to visual mediators. Finally, he did not substantiate what he
wrote, relying upon Corey to produce an answer that reflected the truth. In summary,
unan did not enact autonomous learning in the first vignette. In contrast, in the second
vignette, unan identified both himself and me as audience members. He asked me for
information and then worked through two mathematical operations to determine the truth

of my response. As he worked through the operations, he both substantiated my statement

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and produced his own discursive features (the numerical statements) that built on my
statement. unan exhibited several mathematizing and identifying activities that were
consistent with autonomous learning. In addition, the outcome of his investigation
seemed to be a change in his discourse, indicating that learning happened. unan’s
activity and learning were very different in the two vignettes.

The theory of autonomous learning I articulated above seems effective in
highlighting differences in mathematizing and identifying that could lead to more
mathematically desirable learning outcomes. While a description of differences in
mathematizing and identifying could help researchers and teachers promote certain kinds
of mathematizing and identifying, a framework that theorizes the differences provides
more analytic power for invoking change in teaching and learning. The final project of
this dissertation is to use the lens of learning as mathematizing and identifying to craft a
definition for and description of autonomous learning that is useful in analyzing student
activity to sort out activity that promotes more effective learning. In each findings
chapter, after I have described the kind of learning engaged in by each student, I analyze
the learning for the four discursive activities described above and then draw conclusions
about both autonomous learning and the effectiveness of the learning for each student.

In summary, this dissertation seeks to understand the consequences of students’
classroom activities for the development of their mathematics. To do this, I use a
discursive lens and define the process of learning as the interplay of communicating
about mathematical objects (mathematizing) and communicating about individuals
(identifying). The outcome of learning is defined as a change in discourse about

mathematical objects and about the learner. Using these definitions, I probe the discourse

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of two groups of students engaged in a mathematics lesson. I focus on the learning
(process and outcome) of three students and describe the kind of learning each student
engaged in, relating the interplay of mathematizing and identifying and connecting each
individual’s learning process with the outcome of that process. Finally, I compare each
student’s mathematizing and identifying activity with a theory of autonomous learning in
order to both elaborate the theory and explore how a lens of autonomous learning might

help highlight limitations and affordances of each student’s learning process.

Research Questions
The theoretical framework elaborated has guided my investigation of the learning
of mathematics in an elementary classroom. My main research questions are:
0 What effect do the activities of mathematizing and identifying have on one
another and on the development of mathematics discourse?
o What is the advantage of autonomous learning for the development of
mathematical discourse.
As a reminder of how I am using the terms in my questions, I offer the following short
definitions: Mathematizing is participation in discourse (verbal and nonverbal) involving
mathematical objects. Identifying is participation in discourse that communicates a
significant, endorsable, and reifying narrative about an individual. Finally, autonomous
learning is the constellation of mathematizing and identifying activities reflecting

curiosity about how things are — both what others think/say and what seems to be true.

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CHAPTER 3

METHODS

Overview
This chapter elaborates upon both the tools I employed and my sequence of work
as I explored my research questions. I begin by elaborating my study design. I then
describe details of my research site and my participants, my data collection procedures,

and my data analysis.

Study Design

This study seeks to deepen our understanding of the sense making of students: As
students talk about mathematics or make statements about themselves or others, they
ascribe meaning to and inscribe meaning in each of these actions and interactions. These
meanings then serve as the basis for choices about how to act, react, and ultimately learn
about mathematical ideas. In order to make claims about what meanings students are
making during mathematics lessons, my research is based upon interpretive
methodologies that assume that people decide how to act and react based not upon other
people’s actions but instead upon the meanings or interpretations they make of other’s
actions (Erickson, 1986). For example, a student might decide to raise her hand to join
the many other bands that are already raised. Her decision to raise her hand does not
come from the activity of other students raising their hands but instead from her
interpretation of what it means that other students have raised their hands. This
assumption of interpretive methods matches my construction of narrative from my
framework above: People create accounts of events that provide meaning for the event.

They then react based upon that narrative. Thus an interpretive method not only supports

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the questions and assumptions in my study, it also parallels important features of my
theoretical framework.

There are several different interpretive methods, each of which focuses on a
different locus of meaning making. I have used case study because it provided a means
for exploring an abstract social phenomenon by examining the phenomenon in a specific
circumstance (Dyson & Genishi, 2005). In particular, focusing my study on four students
in one classroom over a limited period of time allowed me to concentrate on the complex
relations between the activities involved in learning mathematics (mathematizing and
identifying) and the activity and outcome of learning. In addition, case study is founded
upon the careful examination of the meaning—making of individuals. While it is not
possible for an outside researcher to be entirely accurate in their interpretations of
individual’s meaning-making, the close examination of individuals provides the case
study researcher with more opportunities to explore individual thinking than other
research methods (Hodkinson & Hodkinson, 2001). Finally, case study supports the
development of theoretical ideas (Hodkinson & Hodkinson, 2001). Case study supported
my effort to elaborate a theory of autonomous learning by allowing me to work back and
forth between detailed data and theory.

Case study describes my research approach and my assumptions about the sources
of meaning. However, what I use to make my case about the activity of learning is the
discourse, both verbal and nonverbal, among students in the classroom and between
students and the teacher. My focus on discourse stems from my use of Anna Sfard’s
commognition framework (2008). Sfard defines thinking as communication with oneself

and discourse as a specific type of communication regulated by rules for acting and

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reacting. Within this framework, the study of thinking is only possible through the study
of discourse. Given this framework, my study of the process and outcome of
mathematical learning must focus on discourse as both the means of engaging in the
activity of learning mathematics and as the object and outcome of mathematics lessons.
My study examines discourse during a mathematics lesson in order to construct the case

of learning mathematics.
Context

Choosing a site

To learn more about interactions among students, I needed a classroom in which
the students were allowed to interact during mathematics lessons. During a previous
research project (Wood, 2007), I had collected data in Mrs. Smyer’s classroom and I felt
that her emphasis on small group work during mathematics lessons would provide an
environment rich with student conversations. I had worked with Mrs. Smyer in a variety
of capacities, both on research projects and as a cooperating teacher for my
undergraduate students during their field experience over the previous four years. When I
approached her about this project, she agreed to allow me to collect data in her room.

Just before I was to begin data collection, Mrs. Smyer was asked to move to a
different classroom: She was needed to provide stability for a classroom of fourth grade
students who had been taught by several different substitute teachers. Mrs. Smyer’s
previous class (a group of third grade students) would be taught by Ms. Cramer, who had
just finished a year of student teaching in this third grade classroom.

Mrs. Smyer, and Ms. Cramer, and the principal were aware that the shift in Mrs.

Smyer’s classroom assignment had implications for this study and they invited me to visit

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both classrooms to determine which setting would be most appropriate. After visiting
both classes, I decided to follow Mrs. Smyer into the fourth grade classroom. I
considered several factors in making this decision. First, while Ms. Cramer was very
competent beginning teacher, she was a novice teacher and I felt that if I stayed in the
third grade classroom, my study would have to address her status as a new teacher. I felt
this could be a distraction from my focus on student interactions. In addition, I felt that
the trusting relationship I had established with Mrs. Smyer would provide me with more
opportunities for and more flexibility in making observations of students.

Third, after visiting the fourth grade class, I felt Mrs. Smyer was going to be able
to focus on teaching mathematics. This was a concern because her new fourth grade
classroom had a reputation as a “bad” class. The class had their own recess and lunch
times and while there were different explanations for this arrangement, Mrs. Smyer and
other teachers felt that the isolation was a result of the misbehavior of these fourth grade
students. I wondered whether Mrs. Smyer would need to focus more on managing student
behavior and less on teaching mathematics or whether she would shift her teaching away
from small groups and toward more individual work in order to manage student behavior.
My first visit to this classroom alleviated my concerns: The students were in small groups
and working on math and it was not evident to me why the class had a “bad” reputation.
The students’ behavior during this first visit proved to be fairly typical. While there were
two occasions during my data collection in which Mrs. Smyer stopped the mathematics
lesson to address the “disappointing” behavior of the students as a class, my concerns

about the time spent on managing student behavior proved to be unnecessary.

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In the end, collecting sufficient data was not a challenge: During the six weeks I
collected data, I was able to observe 34 mathematics lessons and record over 70 hours of
student interaction on videotape. Later in this chapter I will describe the details of the

data I collected.

The Classroom Setting

I originally selected Mrs. Smyer’s classroom because she encouraged students to
interact as they worked on math tasks. However, the kind of small group interaction Mrs.
Smyer encouraged and I sought for my data collection was not the norm for Mrs. Smyer’s
new fourth grade students. I will describe the changes Mrs. Smyer implemented as a
means of conveying what the students had experienced and what changed for the students
after Mrs. Smyer arrived. My data on the student’s previous experience comes from an
interview with their original fourth grade teacher, Mrs. Reeves. Because of the timing of
the change in teachers, I was not able to observe the classroom before Mrs. Smyer
arrived.

Mrs. Smyer changed the physical arrangement of the desks in the room. The
desks had been arranged in rows of single desks. Mrs. Smyer grouped the desks into
clusters of four desks. This physical rearrangement was tied to a change in how lessons
were organized. Mrs. Reeves’ lessons started with a whole group warm-up followed by
individual work on problems. Mrs. Smyer started her lessons as a whole group but this
was typically followed by work with a partner or a small group. Mrs. Smyer encouraged
students to discuss their work with each other and support each other in making sense of

the task and the mathematics.

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The kind of mathematical work also changed afier Mrs. Smyer arrived in the
classroom. Mrs. Reeves focused on developing computational skills such as proficiency
with the long division algorithm. She would model the algorithm, emphasizing the steps,
where to place numbers, and what operation (i.e. multiply, subtract, and bring down) to
perform. In contrast, Mrs. Smyer focused less on computational skills and more on
concepts. While she occasionally gave students worksheets with similar problems, she
also frequently gave students a task with one or two problems. Mrs. Smyer did not model
algorithms, preferring instead to have students explain their own thinking about a
problem.

Mrs. Smyer organized her math lessons into five different types of activities:
math, problem solving, centers, bell work, and Big 4. A “math” activity was one in which
the students worked in pairs on a specific task. The nature of the tasks varied across the
six weeks including sorting and classifying cut-out shapes, rolling dice to determine
probability, and completing worksheet pages on area and perimeter. “Problem-solving”
activities usually involved the students working in groups of three to solve a single
problem or logic task. During “centers” time, groups of three or four, the students rotated
through five different centers, spending approximately 15 minutes at each one. Many
centers featured games focused on mathematical concepts. Occasionally a center would
involve a solving a mathematics problem with the teacher. Some centers were arts
activities or focused on finishing work from earlier in the day. The last two math
activities usually offered practice on computational skills. Bell work consisted of a

worksheet students completed individually as they arrived in the morning. Big 4 usually

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occurred afier lunch recess. Mrs. Smyer would have four math problems that reviewed
concepts and skills on the overhead. Students copied and completed them independently.

The students were grouped differently for each of these activities. For bell work
and Big 4, they sat at their regular classroom desks. They sat at different desks and
worked with different students for centers, problem-solving, and “math”.

In a typical week, students had centers once, problem solving twice, and “math”
three times. They had bell work and Big 4 each day (although the bell work might not
focus on math each day). This schedule meant that on some days, students might be
involved in math activities at three or four different times. For example, students might
complete bell work in the morning, followed by a math activity before lunch, and Big 4
and problem solving after lunch.

Over Mrs. Smyer’s 15 years of teaching, she accumulated texts, lessons, games,
and manipulatives from workshops, conferences, and research projects. She drew from
across this wide variety of resources to construct the tasks and problems in her
mathematics units. During the units I observed, she used the district-adopted curriculum,
Harcourt (Maletsky et al. , 2004); Mouse and Elephant from Middle Grades Mathematics
Project (Shroyer & Fitzgerald, 1991); and Dot Paper Geometry (Lund, 1980). She also
used materials from a research project in which she was involved and she designed her

own problems and tasks based upon her assessment of student needs.

School, Students, and Teacher
The fourth grade classroom in this study was part of a K-Sth grade public school
located in a residential area of a midsize midwestem city. The student population of the

class was diverse in terms of both race/ethnicity and socioeconomic status. The

57

classroom had 20 students. Eleven students identified as African American. Seven
students identified as European American. One student identified as Asian American and
one student identified as Hispanic. None of the students were labeled as English language
learners. Twelve students received free or reduced rate lunches.

Most of Mrs. Smyer’s 15 years of teaching experience was teaching fourth or fifth
grade or a split 4/5 class. She did have some experience at the middle school level,
teaching 7th and 8th grade math for a year and a half. Mrs. Smyer was actively involved in
improving her teaching, participating in research projects in math and literacy education
at the nearby university, attending workshops, and working towards a master’s degree in
education. Mrs. Smyer’s principal described her as a highly effective teacher and her
school district concurred, recognizing her as the outstanding elementary teacher of the

year.

Focal Participants

In order to make sense of students’ interpretations of activity and interactions, I
focused my data collection on one group of students who sat together during the
instructional time the teacher called “math”. This group did not exist before I started data
collection: The teacher allowed me to select four students for a focal group. She then
arranged the other students into other groups. I chose the four focal students based upon
interactions I observed during my first few visits to the classroom. I wanted some
students who, like unan fiom the opening vignette, relied upon other students to support
their mathematical work. I also wanted students who, like unan’s neighbor Corey,
served as teachers or supports for other students. By focusing my study on students who

engaged in different mathematizing and identifying during the mathematics lesson, I

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hoped to make sense of the relationship between mathematizing, identifying, and
learning.

I selected my focal students based primarily upon interactions that occurred
during nine hours of observation over the first six days. My rationale for selection of
three of the students is best demonstrated by one interaction that occurred during my fifth
visit to the classroom. I will present the interaction in detail and then explain how this
interaction guided my selection of students. I have included a transcript of this interaction
below. While I will describe my transcription conventions later in this chapter, I will
explain my use of parentheses now so that the transcript makes sense. In situations where
I was not sure about what was said, I placed parentheses around the words that seemed to
constitute the spoken discourse. In situations where I could not make out the spoken
discourse I placed a question mark in the parentheses.

This transcript focuses on three students — J akeel, Minerva, and Nerissa — who
were sitting at the same cluster of desks, working on separate geometry tasks. J akeel was

trying to determine the name for a geometric solid he had pulled from an envelope. He

asked Minerva for help.

Line Speaker Spoken Discourse Gestures and Other

Number Details

1 J akeel Is this a sphere? He holds up the
cylinder and looks at
Minerva. Minerva
shakes her head.

2 Nerissa (It’s a) sphere

3 J akeel How do you spell it? He looks at Nerissa

4 Nerissa S-P-E-R—E (?) sphere

5 Jakeel Spell it again

6 Nerissa S

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7 Jakeel S Jakeel repeats

Nerissa
8 Nerissa P
9 Jakeel P
10 Nerissa E
11 J akeel E
12 Nerissa R
13 Minerva That’s not a sphere She is looking at
Nerissa.
l4 Nerissa Yes it is
15 Minerva It’s a cylinder
16 Nerissa (?)
l7 Minerva I wouldn’t listen to her She looks at Jakeel
18 J akeel I don’t know who I can believe. You’re He points at Minerva
my best friend
and she’s smart He points at Nerissa

In this interaction, J akeel identified himself as a learner by asking for help (Line
1). Both Minerva and Nerissa acted as teachers although they differed in both their
answer to J akeel’s question and their enactment of teaching. Minerva’s support for Jakeel
consisted of shaking her head (Line 1) and telling him not to listen to Nerissa (Line 17).
In contrast, Nerissa verbalized her answer. She further supported J akeel by offering her
spelling of sphere. However, Nerissa’s answer was wrong (the shape was a cylinder) and
her spelling was incorrect. Thus while Nerissa was a more willing teacher, what she
taught was problematic. In Line 18, J akeel articulated his identities for Minerva and
Nerissa and how those identities complicated his choice of whose answer to use.

This interaction illustrates the reasons I elected to focus on J akeel, Minerva, and
Nerissa. Jakeel was willing to ask others for support in engaging mathematical tasks.

Nerissa was willing to provide this support and J akeel saw her as smart even though, in

60

this instance, her support was not mathematically correct (J akeel did not yet know
whether she was right). Minerva’s mathematizing was more mathematically correct than
Nerissa and while she was willing to provide some support for Jakeel, her support was
minimal and was very different from Nerissa’s. Continued examination of the
interactions among these three students seemed to offer possibilities for making sense of
the relationship between mathematizing, identifying, and learning.

Once I had settled on these three students, I decided that I needed a fourth student
whose mathematizing was more like that of unan and J akeel than that of Nerissa or
Minerva. My research interests have been driven by students’ struggles to learn
mathematics and it seemed that I should complete this group with a student who relied
upon others rather than a student who was more self-sufficient. After reflecting over my
observations, I selected Jessica. 1 had observed Jessica working with other students and
noted her attempts to copy from her neighbor. I had also noted that a student complained
that she was a difficult partner because she played too much. I felt that Jessica exhibited
the kind of mathematizing that I wanted to further examine and so I included her as my
fourth student.

Consent from families to include children as focal students in the study also
influenced my choices. Although I obtained consent fi'om all families and students except
for one, some families indicated that they did not want their children to be focal students.
The families of my four focal students and the students consented to be part of the study.

While Jakeel, Jessica, Minerva, and Nerissa were my focal students, three other
students — Rebecca, Daren, and Bonita — play important roles in this study because of

interactions that occurred between the focal students and these students as they worked in

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problem solving groups. As I will describe later in this chapter, I focused my analysis on
a problem-solving activity. Because the students sit in different groups during math time
and problem-solving time, my focal students were not grouped together for problem-
solving time. Instead, J akeel was with Rebecca and Daren. Minerva and Jessica were
with Bonita. On this particular day, Nerissa was absent. Thus while my focal students
(with the exception of Nerissa) are central in my analysis, these other students play

important roles.

Researcher Role

Because my study focuses on student interaction, I worked to minimize my
interactions with the students. I was not unfiiendly, but I also did not initiate interactions
or provide assistance resolving mathematical questions or other issues that arose. I
assumed the role of “an unhelpful but attentive adult fiiend of children” (Dyson &
Genishi, 2005, p. 52).

I met with the four focal students during the first day they were grouped together.
I explained that I would be taping them each day and that the tapes were for my project
and not for the teacher or their parents. I told them that while I would be watching and
taping them, I couldn’t answer any questions they had. J akeel suggested that I was like a
ghost and I agreed that was a useful way to think about my presence. I would be there,
but I couldn’t help them.

The students were usually appeared to ignore the camera and microphone, but
they did occasional comment on my presence and on the video camera, sometimes

speaking directly into the microphone or signaling to the camera.

Data Collection

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I collected four types of data: student work, video-recordings of interviews with
students and the teacher, audio-recordings and notes on informal conversations with the

teacher, and several types of data documenting student interactions in the classroom.

Classroom Observations and Student Work

I will address two features of data collection around classroom observations: the
time period (number of days) over which I collected data and the focus of data collection
on each day. I originally intended to bound my data collection by collecting data across
two instructional units, capturing one unit in its entirety and the beginning or end of a
second unit depending upon the timing of my entry into the field. I thought it was
important to collect data across two curricular units because students may participate in
different ways when working with different mathematical content.

However, my timing shifted when Mrs. Smyer switched classrooms. This switch
was made eight weeks before the end of the school year and I lost data collection time
because I needed to gain consent and select focal students in this new classroom. Thus
my data collection was bounded by Mrs. Smyer’s time in this new classroom: I began
data collection her second week in the classroom and continued through the end of
school. During this time, I was able to collect data across three units: the end of a unit on
geometric shapes, an entire unit on area, and a final unit on probability.

I taped and collected student work for problem-solving, math, and center time.
However, I usually did not tape bell work or Big 4 because student work was primarily
independent during these times and I was interested in student interactions. I focused on
my focal group when they were together for math time. During problem-solving and

centers, 1 used multiple cameras to capture each focal student in his/her separate group. In

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addition to videotaping, I also recorded “scratch notes” (Dyson & Genishi, 2005, p. 67)
during my observations. These notes allowed me to supplement the videotape by
recording details not captured on the tape such as what was happening at other groups,
who was absent, and what other events happened in school that day.

At the conclusion of my data collection, I had over 70 hours of classroom video
tape from 28 activities (centers, problem-solving, or math activities) collected during 20

days of observation.

Interviews

At the conclusion of my study, I interviewed each of my focal students as well as
other students who had been involved in interesting interactions with the focal students.
Because I did not use this interview data as part of my analysis I will summarize the
interview questions rather than providing details. The interviews were semi-structured
and asked about classroom events. Each interview was approximately an hour long. I
asked the students to tell me about their experiences in school and describe their favorite
part and least favorite part. I asked if they were good at math and who in their classroom
was most like them in math. I also asked them to describe what math was like in their
classroom. I had the students complete two tasks that resembled work they had during
earlier mathematics lessons. Finally, I showed the students video clips from my data and
asked them if they remembered the day and then asked them to narrate the moment.

I also interviewed Mrs. Smyer and Mrs. Reeves (the fourth graders’ first teacher)
at the end of the study. Both interviews were semi-structured and lasted less than an hour.
I asked Mrs. Reeves to describe her experiences with the class including how she taught

math, what content she covered, and the history of substitute teachers in the room. I also

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asked her about her background as a teacher and any background information she had on
each of the focal students. I asked Mrs. Smyer some of the same questions I asked Mrs.
Reeves. She described the situation of the fourth grade class before she became their
teacher, her background as a teacher, and background information on each of the focal

students.

Informal Conversations with the Teacher

The teacher and I had several informal conversations about students and math
lessons. These conversations were important to this study because they included the
teacher’s narrative of events that occurred when I was not present in the classroom. They
also included the teacher’s interpretation of events, which was helpful in making sense of
students’ activities. Because these conversations were spontaneous, I did not purposefully
tape them. On some occasions when the conversation seemed particularly interesting and
the situation was appropriate, I asked the teacher for permission to audiotape. For the

other conversations, I made notes either as we spoke or shortly after our conversation.
Data Analysis

First Pass through the Data
Most of my data analysis focused on transcripts of my focal students’ actions and
interactions. However, these transcribed events are part of a larger context. In order to
record this larger context and situate my transcriptions, I constructed written fieldnotes
for each classroom observation. These fieldnotes drew upon my scratch notes, the student
work, the videotape, and the informal conversations with the teacher. The fieldnotes were
primarily descriptive, but they also included my reflections and questions about the

events during the observation. These fieldnotes also included a first rough transcription of

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conversations involving any of the students in my focal group. For example, notes of
whole class discussions reflected what my focal students were doing during the
discussions, but do not include transcriptions of what was said unless my focal students
were participating in the whole group conversation or in a side conversation.

I constructed separate fieldnotes for each lesson. I defined a lesson based upon the
teacher’s differentiation of math activities. Thus if the students participated in problem-
solving and centers on the same day, I wrote separate fieldnotes for each. My fieldnotes
totaled 369 single-spaced pages.

After I writing fieldnotes for each lesson, I wrote a reflective memo in which I
recorded my thoughts and ideas on events in the fieldnotes. As I explored different ideas
in these memos, I constructed labels for each idea. I then used these labels to track

emerging themes across numerous memos and lessons.

Second Analytical Pass

This first pass allowed me to select moments that were particularly interesting for
a second analytical pass through some of the data. This second pass considered the
learning of one student during one lesson and was comprised of three parts: an
exploration of the student’s opportunity to learn as defined by a difference between the
student’s discourse at the beginning of the lesson and the presence of other, more
mathematical discourses during the lesson; a description of the process of learning which
told the sequence of events involving the focal student during the lesson; and, finally, an
analysis of learning which identified factors that seem to support or limit the student’s

learning. As I conducted this second pass, I reviewed the video of each lesson I analyzed

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and constructed a more detailed transcript. I will describe the transcription process and
my transcription conventions in the section that follows this description of my analysis.

I selected lessons for this second pass based upon my evaluation of the
opportunity to learn for focal students in the lesson. I considered the student’s initial
discourse, the discourse of other students in the group, and the focal student’s final
discourse, looking for lessons in which the focal student’s discourse changed during the
lesson. The first lesson I chose to analyze was a problem—solving lesson on area in which
Jakeel was the focal student. It was clear from the fieldnotes I generated during my first
pass through the data that J akeel’s discourse had changed. In addition, I was also
interested in how he involved himself in his learning. My first impression was that he was
not copying from the other students but was asking them questions and working to make
sense of their explanations. I felt that his change in discourse and his engagement in
learning warranted a closer examination of this lesson.

As I indicated in the section on focal students, this lesson was a problem-solving
lesson, which meant that my focal students were not grouped together. J akeel was in one
group. Minerva and Jessica were in another group and Nerissa was absent. After
completing the analysis of Jakeel’s learning, I looked at the interactions and learning in
Minerva and Jessica’s group for the same lesson. It became clear that Jessica’s discourse
changed during the lesson, but her learning activities were very different from Jakeel’s
activities. I decided that close examination of her activity in the context of the same
lesson would provide an interesting and informative contrast to J akeel’s activities. I also

analyzed a third lesson, again focusing on J akeel’s learning, but I found in that analysis

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that the data were insufficient to construct a satisfactory argument about J akeel’s learning

and so I did not pursue this lesson beyond this second analysis.

Third Analytical Pass

As I examined my analyses of Jakeel and Jessica’s learning, it became apparent
that the factors I explored in this data could be sorted into the mathematizing in the group
and the identifying in the group. This insight helped to structure a third pass, which also
focused on the learning of one student during one lesson and had three parts: an
exploration of the focal student’s learning outcome as defined by changes from the
student’s initial to his/her final discourse; a description of the identifying and
mathematizing that occurred in the group and was related to the student’s learning
process; and a comparison of the student’s learning process and the activities of
autonomous learning. In the next few paragraphs, I will describe the analysis of
identifying, mathematizing, and autonomous learning.

Identifying and mathematizing were analyzed using different protocols. However,
because I was interested in learning activities, my analyses of identifying and
mathematizing were isolated to portions of the transcript in which students were engaged
with mathematical objects. For example, I analyzed portions of the transcript in which
students were talking about the task or working on cutting and gluing the figures, but I
did not analyze students’ conversations about church, soccer, or lyrics to rap music. To
analyze identifying activities, I partitioned each turn into message units.

Bloome, Carter, Christian, Otto, and Shuart-Faris (2005) define a message unit as
the smallest discursive unit that communicates meaning. They emphasized the

perspective of the listener in making decisions about where to bound message units.

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However, because I am interested in both the intention of the speaker and the listener’s
interpretations, I considered the speaker’s and the listener’s perspectives when making
decisions about where message units started and ended.

Message units are different from turns. Turns are bounded by changes in speakers.
A single turn might contain just one message unit or it might contain many. For example,
in the transcript I used above to introduce my focal participants, each of the first 17
numbered lines is a turn. Each is also a message unit because it communicates meaning.
Line 18 contains 3 units: “1 don’t know who I can believe”, “you’re my best friend”, and
“and she’s smart”. By partitioning the turn into 3 units, I can consider what J akeel might
be trying to accomplish with each unit and with the units together. I can also consider
how Minerva and Nerissa might interpret each unit.

In my analysis, I examined message units and any accompanying gestures or
nonverbal communication. For each unit, I considered the range of possible meanings for
the speaker and the listeners, seeking to answer the question about what story the speaker
and listener(s) might construct from the unit and how each person might be identified or
might identify him/herself within that story. When I found units that identified the focal
student as a learner in a consistent way, I pulled those units together, labeled them as a
kind of learning, and wrote a description of that identification in my analysis.

I want to emphasize that my analysis of identifying only included units and turns
that contained talk that was entirely mathematical. I was interested in not only moments
where students explicitly identified themselves and each other (like Line 18 above, when

J akeel called Minerva his best friend and Nerissa smart) but also in more subtle

identifying activity such as Line 1 when Minerva responded to J akeel’s question by

69

shaking her head and Line 2 when Nerissa provided a verbal answer. These different
responses suggest that Minerva and Nerissa were communicating to J akeel different
notions of their enactment of and willingness to be his teachers. Thus my analysis of
identifying activity included units in which students’ talk was entire mathematical like
counting or talking about two halves making a whole.

I also examined these same lines through a mathematizing lens. I wanted to
consider what mathematical discourse was available, how that discourse resembled the
desired mathematical discourse, and how changes in student discourse resembled both the
available and the desired mathematical discourse. In order to compare these different
discourses, I used the four discursive features Sfard (2008) elaborated in her
commognition framework (See Chapter 2). In my analysis I note differences and
similarities in these four features between different units of mathematical talk and
between mathematical talk and the desired mathematical discourse. These differences and
similarities allow me to make claims about student learning.

The third part of the third analytical pass explored autonomous learning. This
analysis developed as I attempted to characterize the differences between students’
learning processes. J akeel seemed to enact learning activity that was more like the
activity encouraged by reform mathematics pedagogies. I developed a lens for describing
autonomous learning as I worked to differentiate J akeel’s activity from that of other
students. I then used that lens to analyze the activity of each student. The features of
autonomous learning that I used in my analysis are described in Chapter 2. They are:

identification of the audience and adoption, production, and substantiation of

70

mathematical discourse. The development of the autonomous learning lens was a
recursive process involving using data to build theory and theory to analyze data.

I conducted this third analytical pass on three lessons. I returned to the lesson I
used in my second analysis and reexamined J akeel’s learning and Jessica’s learning. As I
explored Jessica’s learning, I realized that Minerva’s learning activities (Minerva was in
the same group as Jessica) were also quite different from Jakeel’s and Jessica’s.
Minerva’s learning activities seemed important to explore not only because they were
different, but also because they added complexity to the analysis and the theory of
autonomous learning. The third analytical pass resulted in three detailed and lengthy
memos, one about each student’s learning, that I rearranged and shorted to construct the

three findings chapters in this document.

Transcriptions.

As I worked on my second analytical pass through my data, I constructed detailed
transcriptions of each group during the entire lesson. During the second and third
analytical pass, I reviewed the portions of video I was analyzing and added more detail to
the transcriptions. I separated the transcript into numbered turns. I used turns to separate
and number the data because I wanted a way to track the sequence of events and where
each event occurred in the lesson, including the portions of the lesson in which students
were not involved with mathematical objects.

As the transcript included earlier in this chapter demonstrates, I used a table
format to organize line (turn) number, the speaker, the spoken discourse, and gestures or

other details. The fourth column primarily contains nonverbal gestures but I also used this

71

space to describe verbalizations that were not words and to note indications mood or
other details.
My transcription conventions draw upon Jefferson’s transcription notation as

described by Atkinson and Heritage (1984). and are indicated in Table 3.1.

Table 3.1 Transcription conventions

 

 

 

 

 

 

 

Symbol Interpretation

() Uncertainty about what was said

(?) More uncertainty about what was said or the speech was unintelligible
(pause) Pause in speech, approximate length of pause is included

= Interrupted speech

L Overlapping speech

 

 

Limitations

Any research method has limitations and affordances. The goal of the researcher
is to design a study that uses the affordances of the methods to illuminate a research
problem while minimizing the effects of the limitations of the methods. As I described
earlier, I have used cases to organize this study because they allow close examination of a
complex phenomenon. However this close examination means that the claims I make are
not generalizable to larger populations based upon this study alone. I cannot claim that
the learning activities I describe in this study will occur in other similar circumstances. I
also cannot make causal claims based upon careful control of variables. My findings
describe the activities I see in the groups I studied and suggest the possibility of
influences and effects, but I cannot claim that these same factors are at play across other
situations.

Also, my close examination of two groups in one lesson means that I do not
consider the effects of outside influences on the activity in this group. While I incorporate

information gleaned from across the classroom data I collected, I do not speculate about

72

 

how the students’ home circumstances, previous life experiences, or even prior
experiences during the day might impact their activity during the lesson. Because I am an
outsider attempting to make sense of other’s meanings, it is possible that there are
activities that I incorrectly interpreted because I lacked background information. I
attempted to counter this possibility by proposing and evaluating multiple interpretations,
discussing my data with other researchers, and looking for patterns in activities in order
to provide multiple points of data for an interpretation. These efforts reduce, but do not
eliminate, the likelihood that some of my interpretations are incorrect.

An additional problem with my research methods is justification of the selection
of data. I purposefully selected lessons that offered varying perspectives on learning
including autonomous learning in order to elaborate the detailed relationships between
mathematizing, identifying, and learning. However, if I had focused on different lessons
or collected data in the third grade classroom instead of the fourth grade classroom, the
findings from this study would probably elaborate different aspects of learning
mathematics. While this study could and should be expanded by examining other cases of
learning and by testing the theories in other populations and settings, the methods of my
study were justified in spite of their limitations: It was necessary to conduct a detailed
examination of a limited number of contexts in order to develop the theories and findings
that might be examined in future studies. This examination allowed the development and
elaboration of theory and ideas that can then be further explored through other studies

and other methods.

73

CHAPTER 4
INTRODUCTION TO THE FINDINGS
This chapter serves as an introduction to my findings. Each findings chapter
(Chapters 5, 6, and 7), I examine the discourse of one child as he/she works with a group
to solve a mathematical task involving area. In this chapter, I provide background for the
analyses in the subsequent chapters. This chapter begins with a review of research on
area that is relevant to focal lesson of this study. I then provide an overview of the area

unit taught by the teacher. Finally, I describe in detail the task assigned in the lesson.

Relevant Research? on Area

I limit my discussion of research on area to four points that are important to this
study. Three points focus on the construction, iteration, and counting of units of area
measurement. The fourth point relates to conservation of area as a shape is rearranged.
Because the math lesson described in this study uses figures that have been partitioned to
square grids (For example, see Figures 4.1 and 4.2), much of my presentation of research
focuses on how elementary age students use grids to determine area.

In order to measure the area of a figure, students must recognize that area is an
attribute describing the quantity of two-dimensional space inside the figure and they must
designate and iterate a unit so that it covers the space. Designating and iterating a unit can
pose challenges for elementary age children (Lehrer, 2003). According to Battista (2003),
children may have difficulties structuring spaces, which means that they struggle to
construct a regular grid that partitions a space into square units. Instead of drawing

regularly spaced lines to construct rows and columns, some students draw each square

74

independently or are inconsistent in their placement of lines. The resulting spaces may be
inconsistent in size and shape and, when counted, may not result in an accurate
measurement for area.

When students are presented with figures that have already been partitioned into
square grids, they may still have problems measuring area. If the figure is nonrectangular
or does not otherwise match the squares on the grid (for example, see J in Figure 4.2),
then some of the grid spaces will be partial squares. In order to accurately determine the
area of such a figure, students need to count the partial units in terms of the whole units.
Some students may not recognize the need for identical units and may mix units as they
count (Lehrer, 2003; Lehrer et al., 1998). For example, students may count all units,
partial squares and full squares, as one unit each. If a shape (like I in Figure 4.2) contains
spaces that are squares and triangles, students may count both as equal units (Piaget et al.,
1960).

Even when a figure is entirely filled with square units, quantification of area may
be problematic. While the students may accurately count the square spaces, they may be
focused on enumerating squares as discrete objects, rather than on understanding and
quantifying the spaces as units of area (Clements, 2003; Lehrer, 2003). If students think
about squares as discrete objects, they may not understand area as a measure of space.
For example, they may instead think about area as how many squares a shape can hold.
When students fail to see area as a measure of space, they may fail to consider some of
the important aspects of measurement such as the need to account for all of the space in
the figure. For example, when determining the area of a nonrectangular figure, they may

only count the grid spaces that are complete squares. Rather than think about area as a

75

measure of space, they may understand area as a count of how many squares happen to
be in the figure.

Finally, elementary students may struggle to make sense of how area can remain
constant as a shape is cut into pieces and rearranged. For example, Piaget, Inhelder, and
Szeminska (1960) presented children with rectangles partitioned into squares. They
rearranged the squares and asked the children if the rectangle was the same size. Some of
the children replied that the number of squares was the same but that the area was not.
The children were able 'to keep track of the number of squares, but they were unable to
conserve the area of the figure as its pieces were rearranged.

Students’ struggles with using identical units, counting partial units, relating area
to space, and conserving area are visible throughout my data. Some of these struggles
served as the basis for the teacher’s decisions about structuring the focal lesson. Others
become apparent as the students worked on the focal lesson. It is important to my study
to recognize these struggles not because I attempt to add to the research corpus on
children’s understanding of area, but instead because awareness of the challenges of
learning area helps us to evaluate students’ mathematical discourses to determine their
progress toward understanding area.

The remaining sections of this chapter present information on the focal lesson

including an overview of the unit on area and details of the lesson.

Overview of the Unit on Area
The unit on area consisted of eight lessons. In addition, there was one day of
centers in which two of the centers involved review of area concepts. The unit was

preceded by a unit on geometric shapes and was followed by lessons on probability.

76

The area unit began with a whole group introduction/review of the concepts of
area and perimeter. Students used square tiles to determine the area and perimeter of a
variety of shapes. The teacher designed the next two lessons to emphasize the
relationship between area and perimeter. Students were to construct all possible _.-.
rectangles given a certain number of squares and determine the perimeter of each
rectangle. The next lesson focused on determining area. The teacher gave the students
several shapes (which were called rugs) and asked which covered more floor. The
students needed to draw grids within the shapes. Most of the shapes were rectangular
although one was triangular. Some students struggled to make regular grids and the
teacher felt that problems with drawing grids were preventing students from exploring
questions about identifying and counting units. The teacher designed the next lesson — my
focal lesson — with this concern in mind. She gave the students pairs of pre-grided figures
and asked them to compare the area of the figures. Because this is my focal lesson, it is
described in more detail in the section below.

The math activity for the following day involved two centers, both focused on
area. One was a review of the lesson prior to my focal lesson. In another center, students
were to make a figure using square units and then determine the area of the figure. The
following two lessons involved determining the area (and sometimes perimeter) of
figures on geoboards. The first lesson involved rectangular shapes. The second day
involved triangular shapes. (This second geoboard lesson is mentioned in Chapter 5.) The
last lesson in the area unit was a review of area and geometric shapes. This sequence of
lessons is listed in Table 4.1. The table has information on whether the lessons were

organized as Problem-Solving, “Math”, or Centers. The table also includes math

77

activities that occurred during the same time as the lesson but whose content did not

include area.

In constructing these lessons, the teacher drew upon four resources: the fourth

grade Harcourt math textbook (Maletsky et al. , 2004), Mouse and Elephant Middle

Grades Mathematics Project curriculum (Shroyer & Fitzgerald, 1991), Dot Paper

Geometry (Lund, 1980), and materials from a university research project.

Table 4.] Sequence of lessons involving mathematics during area unit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Day I Math Lessons
Week 1
Tuesday Math: Introduction to area and perimeter
Problem-Solving: Recognizing Rhombi
Wednesday Power outage during school, math canceled
Thursday Math: Making different rectangles with an area of 12, measuring
perimeter
Friday Problem-Solving: Making different rectangles with an area of 24,
measuring perimeter
Centers (none involving area)
Week 2
Monday Substitute teacher: Independent work, not on area
Tuesday Substitute teacher: Independent work, not on area
Wednesday Math: Comparing areas of different shapes
Thursday Problem-Solving: Comparimg areas of different shapes (focal lesson)
Friday Centers: One center is review of the problem from Wednesday, in
another center students use squares to make a shape and then
determine the area
Week 3
Monday Math: Area and perimeter of rectangles using geoboards
Tuesday Math: Area of triangles using geoboards
Problem-Solving: Logic problem with coins
Wednesday Review worksheets on geometry including geometric shapes and

area

 

 

Description of the Focal Lesson

The focal lesson occurred on the Thursday of the second week of unit. In the

previous lesson, students were to answer the following questions which were printed on a

78

 

worksheet: Which of these rugs covers more of the floor? Which covers the least? Do any
cover the same amount? The “rugs” were four shapes printed beneath the questions: three
rectangles and one triangle. Many students struggled to find the area of the triangular rug,
prompting the teacher to construct the task described below — the task for the focal
lesson.

The teacher organized this task as a Problem-Solving lesson. During Problem-
Solving, the students worked in groups of three. Instead of giving instructions about the
lesson to the whole class, the teacher gave each group a one page “Task Card” which
listed materials needed, the task, group expectations, and the final product. Students were
to read the task card to determine their work for the lesson.

The theme of measuring rugs continued into this lesson, as indicated by the

information on the task card:

 

Materials
shape papers, scissors, glue, individual worksheet

Task

As a group, find which of the rugs covers more area or if they cover the same amount.
First compare rugs H and I. Prove how they are the same or different. Once you are
done with this, do the same task with rugs J and K. When you are finished with that
you may come up and grab the last worksheet and read what you are supposed to do.

Group Expectations

1. Everyone must show their work on an individual worksheet.

2. No one can touch their worksheet.

3. You may help by teaching but not by doing.

4. Everyone in the group has to agree on an answer before going and getting the next

paper.

Final Product
1. Use your paper to individually record your solution to the problem.

Challenge
Make up your own problem as a group.

 

 

 

79

To complete the task, students received scissors, glue, individual worksheets, and
copies of shape papers. The individual worksheets had a blank for the student’s name and
the following words: “Directions: Write how you solved the problem and show the
answer that you chose below.” The shape papers were half sheets of paper with two
figures on them. Students initially received papers with Figures H and I on them (see
Figure 4.1). Papers containing Figures J and K (see Figure 4.2) were kept at the front of
the room. Students were allowed to retrieve those papers and begin work on them as a

group after the teacher approved each individual student’s work with Figures H and I.

 

 

 

 

 

 

Figure 4.1 Scanned image of Figures H and I.
Figure labels were handwritten by the teacher.

 

 

 

 

{gun J’ 53“" "

 

 

Figure 4.2 Scanned image of Figures J and K.
Figure labels were handwritten by the teacher.

80

Desired Mathematical Discourse

To complete this task, students needed to determine which rug or figure covered
more area. To do this for Figures H and I, they could cut out each rug, cut Figure I into
one square and two triangles, and then rearrange the pieces of Figure I so that they cover
Figure H. The student could then state that because the shapes are congruent or fit on top
of each other without any excess, the figures have the same area. However, the teacher
designed this lesson so that students would wrestle with an issue that arose during the
previous lesson involving determining the area of a non-rectangular figure. During the
previous lesson, the students drew a grid on the non-rectangular figure. Many of the grids
drawn by the students were irregular and all of the grids contained spaces that were not
square. Most of the students then proceeded to determine area by counting each space in
the grid as one, without accounting for differences in the amount of area represented by
each space. The teacher’s intention for this task was to help students recognize that they
could not count partial units as whole units. Specifically, she wanted students to assemble
the triangular partial units into whole square units and then count each square.

Thus, to solve this problem using the mathematically desirable discourse, students
first needed to identify squares as the unit of measurement. They then needed to account
for the triangular partial units present in Figures I and J. The teacher expected the
students to physically (by cutting) assemble the partial units into whole units and then
count the units in each shape. Finally, students would state both the area of each shape in
square units and which figure covers more area. Successful mathematizing does not
require that students be explicit about each step in order. For example, students might not

state that they are using squares as the measurement unit until the conclusion of their

81

work when they state the area of each shape. As we shall see in the three subsequent
findings chapters, students experienced varying degrees of success in using the desired

mathematical discourse.

82

CHAPTER 5

ENGAGED LEARNING AND DIRECTED LEARNING

Overview

This case presents the learning of one student, Jakeel, whose identifying and
mathematizing changed dramatically during the course of the lesson. He shifted from
engaged learning which supported his work to craft a discourse-for-himself to directed
learning in which his mathematical discourse was entirely a discourse-for-others. I first
present the outcome of J akeel’s learning by contrasting his initial discourse with his final
discourse. I then describe how engaged learning was enacted by J akeel and supported by
his peers and how this learning shifted to directed learning and back to engaged learning.
I conclude by discussing the implications of these different processes of learning for

Jakeel’s participation in autonomous learning.

Summary of the Lesson

My analysis of Jakeel’s learning uses several excerpts taken from the transcript of
J akeel’s group. The excerpts are not presented in chronological order: Instead, I used
excerpts from across the lesson as necessary to support my argument. In an effort to
orient my readers to the position of the excerpts in the flow of the lesson, I have
summarized the main events in the lesson and constructed a table that describes the
content of the lesson and the excerpts and shows where the excerpts occur in the lesson
(See Table 5.1).

The teacher began the lesson by explaining the lesson and her expectations for
behavior to the whole group. She then dismissed students to work in small groups.

Rebecca, Daren, and Jakeel gathered at a table in the back of the room and started to read

83

the directions for the task. The teacher came to the group and asked questions about the
task. Once she was satisfied that they understood what to do, she allowed them to get
their materials. Jakeel picked up the worksheets with Figures H and I on them and the
Daren and Rebecca began to work. Jakeel asked Daren and Rebecca what they were
doing and Daren explained how he was thinking about the area of Figures H and 1.
Rebecca finished writing her solution to the problem and tried to get Jakeel to copy her
answer. J akeel refused and constructed his own solution. When he was done, Rebecca
told him that wasn’t what he was supposed to do and told him he would have to redo his
work (See Figure 5.1). He began again on a separate piece of paper and, with Rebecca’s
assistance, constructed another response to the task involving Figures H and I (See Figure
5.2).

The teacher came to the group and approved their answers for Figures H and I and
allowed the students to get the worksheet with Figures J and K. When the teacher
returned to the group later, Jakeel told the teacher he was done with Figures J and K (See
Figure 5.2). However, he was unable to demonstrate how count the spaces in Figure J so
the area was the same as Figure K. Rebecca explained her thinking about how to count
the spaces and after several counting attempts and explanations from Rebecca and Daren,
Jakeel was able to successfirlly count the spaces in Figure J for the teacher.

Table 5.1 matches this summary with the excerpts I use in my analysis. I begin
my analysis by considering how Jakeel’s discourse changed over the course of this

lesson.

84

 

Table 5.1 Transcript excerpts and main events in lesson

Line
number

Excerpt
Number

Excerpt Summary

Main Events

 

0

 

 

The teacher explains
the lesson .

 

 

 

 

 

 

 

 

100

 

The students move to
their small groups and
start work. Jakeel,
Rebecca, and Daren
read the task card and
get their materials.

 

 

 

Excerpt 6

Jakeel involves himself in the conversation .at
his table

 

 

Excerpt 1

Jakeel’s initial discourse

 

 

 

200

Excerpt 7

Jakeel refuses Rebecca’s directions

 

Excerpt 9

Jakeel uses rectangle

 

 

Excerpt 10

Rebecca tells Jakeel he has to rewrite

The students begin
work on the first
problem with Figures
H and I.

 

 

 

 

 

 

300

 

Excerpt 1 l

Jakeel asks what he was supposed to write

Rebecca tells Jakeel
his answer for Figures
H and I is wrong and
tells him what to

write.

 

 

 

 

 

 

 

 

 

400

 

 

 

 

 

 

 

 

 

 

500

 

The students work on
Figures J and K.

 

Excerpt 2

Jakeel’s final discourse: the same

 

Excerpt 8

J akeel counts and Rebecca explains

 

 

 

 

 

 

600

 

 

 

J akeel counts the
spaces in Figure J,
eventually counting
them correctly

 

85

 

 

Table 5.1 con’t
Excerpt 3 Jakeel’s final discourse: pairing triangles

 

 

 

 

Excerpt 4 Jakeel counts eight

 

The students clean up.

 

 

 

 

 

 

 

 

.- "or!“ J

. M“
aRvawmvfi'" _

 

 

 

 

 

Figure 5.1 J akeel’s initial work on Figures Hand I
This is student work. The text in the top left reads, “If you put the two triangles togeter

[sic] it make [sic] a square and it covers it”.

 

is.“ mama “NJ tum
m

day‘s-KW”

       

 

 

   

_l l‘ l5".—

 

l
L
.l
r .

 

 

 

Figure 5.2 Jakeel’s final work on Figures H, I, J, and K
This is student work. The text in the top left reads, “they cover the same amount” and, on

the second line, “two half [sic] make [sic] a square”.

Outcome of Learning

Initial Mathematical Discourse

The first time J akeel spoke about the mathematics of this task occurred 13

minutes after the beginning of small group work time. All three students — Rebecca,

Daren, and J akeel — were working on comparing the Figures H and I. The teacher arrived

at the group and began asking questions about their work. Rebecca responded that H and

I cover the same amount and the teacher turned to J akeel to ask for an explanation.

Excerpt 1

l 76 Teacher
177 J akeel

l 78 Teacher
1 79 Jakeel

l 80 Teacher
1 8 1 Rebecca
1 82 Teacher
183 Jakeel

1 84 Rebecca
185 Teacher
1 86 J akeel

Jakeel. What do you think? How can you

prove this? What could you do? What

materials do you have that could prove this?

H and I. They both have squares.

Okay but are they the same?

Yes

Do they cover the same?

I know
I mean Jakeel. How

Cause it’s a square

Cut this out

But this isn’t a square

No, so that’s why it won’t cover the same.

She picks up a paper
with Figures H and 1.

He points at the
figures on the paper
the teacher is holding.

The teacher puts the
paper down.

Daren and Rebecca
raise their hands.

Jakeel points at the

full square in Figure
1.

She takes a paper with
Hand] on it and
picks up scissors.

She points at the two
triangles in Figure I.

With his “Yes” on Line 179, Jakeel initially responded to the teacher that Figures

H and I had the same area. While this answer was mathematically correct, Jakeel was

87

unable to offer a mathematically appropriate explanation for why the two figures covered
the same amount. His explanation, “Cause it’s a square” (Line 183) suggested that he was
thinking about squares. The response provided a label for a geometric space, but it did
not explain how squares were important to area or to comparing areas.

Jakeel also did not attempt to explain how the two triangles in Figure I were part
of the area. He does not mention or indicate the triangles, raising the question of whether
he thinks the triangles are important for determining area. In addition, when pressed by
the teacher (Line 185), he changed his answer, responding that the figures “won’t cover
the same” (line 186). While this response was less mathematically accurate than the
“Yes” (Line 179) Jakeel first provided, it did begin to elaborate more upon notions of
area. Jakeel’s use of the word “cover” implied that he wasn’t only thinking of the squares
as labels for geometric spaces (Line 183), but that he recognized that an important
attribute of squares was that they offered coverage. The notion of coverage is consistent
with the notion of area as a measure of two—dimensional space. In addition, his use of
“the same” (along with his use of “both” in Line 177) indicated a comparison of the two
figures. A comparison is a component of the desired discourse. However, J akeel’s
comparison in this initial discourse seems to focus on descriptive geometry (the squares)
and not on quantifying or evaluating area.

In summary, J akeel’s initial discourse offered some mathematically appropriate
notions of area, but it was not the desired discourse. He wavered in his response about
whether the figures “covered the same.” He did not specifically label squares as the unit

of measurement for determining area. He also did not quantify the squares or offer a way

88

to account for the two triangular spaces in Figure 1. He did indirectly indicate that area is

related to coverage.

Final Mathematical Discourse
By the end of the lesson, Jakeel’s discourse had changed. I will draw upon four
moments to illustrate this change. Three moments come near each other at the end of this
lesson. The fourth moment is from a subsequent lesson that occurred five days later. The
first example of Jakeel’s discourse occurred 40 minutes into the small group work time.
Rebecca, Daren, and J akeel had finished their work comparing Figures H and I and were

close to completing their work on Figures J and K. The teacher came to the group and

looked at Jakeel’s work.

Excerpt 2

534 Teacher Okay. So, what’s the area of J? (2 second The teacher points to

pause) Jakeel ’s paper.

535 Daren It’s um same as=

536 Teacher What’s the area of J? The teacher looks at
Jakeel.

537 Jakeel The same as um Jakeel points at
Figure K on his paper.

538 Teacher What is it? Tell me what it is.
539 J akeel Eight
This excerpt suggests that Jakeel knew that Figures J and K had the same area

(Line 537) and that area was eight (Line 539). It is possible that Jakeel’s statement in

Line 537 was an echo of Daren’s statement (Line 535). However, in the following

moment, it became apparent that Jakeel expected both figures to have an area of eight.
One final point about this excerpt is that Jakeel was not explicit about what eight

was a count of. In other words, he does not elaborate on his eight with any noun that

89

would indicate what he was counting. This lack of the object counted was a consistent
trend in J akeel’s final discourse.

Just after Excerpt 2 above, the teacher asked J akeel to demonstrate how the area
of the Figure J was eight. J akeel counted the spaces in the figure but counted to ten.
Rebecca asked if she could help him and the teacher left as Rebecca began her
explanation. I have not included this transcript in this section because J akeel’s discourse
was very similar to his discourse in Excerpt 3 below except that Excerpt 3 illustrates
some additional features of his final discourse. I felt that it would be redundant to include
both.

Excerpt 3 occurred almost 10 minutes after Excerpt 2. The teacher returned to the

group and asked Jakeel again about his work.

Excerpt 3
621 Teacher Okay, show me on here which two I put She points at Figure J.
together
622 J akeel Those two Jakeel points at two
triangles in Figure J.
623 Teacher Can you draw a line or something Jakeel draws a line
connecting two
triangles.
624 Jakeel Yeah
625 Teacher To show me? Okay and what’s the other J akeel draws another
two? line, connecting the
two remaining
triangles. '

J akeel’s actions and words indicate that he was putting together pairs of triangles. This
action was important because it indicated that Jakeel was working with the triangles. This
is a shift fi'om his original discourse in which he only discussed squares. However, his
verbal mathematical discourse in this excerpt is limited. His only mathematical word is

“two”. Similar to Excerpt 2, he does not label what object the two enumerates. Thus

90

although I have referred to the objects as triangles, Jakeel has provided no information
about how he is thinking about what he is counting.
This excerpt continued with more questions from the teacher.

Excerpt 3, continued
625 Teacher Now show me how it’s the same area as that The teacher points to

Figure K.
626 Jakeel Okay look. one, two, three, four, five, six, He points to each
seven, eight. square in Figure K.
627 Teacher Now count that one She points to Figure J.
628 Jakeel one, two, three, four, five, six, seven, eight. He starts at the top of

Figure I and counts
the spaces in each
row, counting each
space, including each
triangle as one. He
pauses when he
reaches eight. He
doesn’t count the last
square or triangle.
Rebecca leans in
toward J akeel. Daren
watches hirn.

(Hold on?) Oh yeah. one, two, three, four, J akeel counts Figure I

five, six, seven, eight. in a more random
pattern, but again
counts the triangles as
one. Again he doesn’t
count each space.

When Jakeel counted Figure K, he successfully enumerated the area as eight
(Line 626). Because he did not describe what he was counting, we cannot be sure what he
was enumerating, but he was nonetheless confident that the area of K was eight. When he
counted the area of Figure J (Line 628), he also stopped counting at eight. However,
stopping at eight was not a signal that he had counted the entire area. Instead, J akeel
stopped at eight even though he still had two uncounted spaces. His hesitation and re-

counting indicated that he expected to end on the number eight. Jakeel’s actions here

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were consistent with his words from Excerpt 2 when he stated that the figures had the
same area (Line 53 7) and that the area was eight (line 539).

In this last transcript, it was apparent that while Jakeel was physically putting the
triangles into squares, he had not yet integrated the pairing of the triangles into his routine
for determining area. Rather than count each triangle as half or count two triangles
together as one square, J akeel counted each space as one, as a result, he was unable to
accurately determine the area of Figure I. As above, his discourse contains only counting
numbers. Thus my reference to triangles and squares may not represent how Jakeel
identified the spaces.

The teacher left the group shortly after the moment above and Rebecca and Daren
worked with Jakeel on counting the spaces in Figure J. During the next four minutes,
Rebecca and Daren explained how to count the area and J akeel practiced. The teacher
returned two times and each time Jakeel counted each space as he did in the transcript

above. On Jakeel’s fourth attempt in front of the teacher, he counted the area:

Excerpt 4

668 Jakeel One This word is elongated
as J akeel
simultaneously puts

his index and middle
fingers on two

triangles.

Two, three, four, five, six, seven He counts each square,
pointing with his index
finger to each square.

Eight He puts index and

middle fingers on the
last two triangles.

By pointing to two triangles with two fingers, Jakeel was able to count the

triangles as one and as a result, he was able to accurately count the area of Figure J. This

92

counting was a change from Jakeel’s discourse at the beginning of the lesson. While
Jakeel spoke about squares at the beginning of the lesson, he did not talk about or in
anyway indicate the two triangles. He also did not indicate that he understood area as
something to be counted. Thus his final discourse was a change from his initial discourse
and more like the desired course because he enumerated the spaces in the figures and he
included in that enumeration the triangles. In addition, his counting of the triangles
demonstrated that he understood that two triangles were the same as one square. One last
way in which Jakeel’s final discourse was like the desired discourse was that Jakeel was
able to use counting to establish that the two figures were the same.

There is one more important point about J akeel’s final discourse. It is unclear
from his discourse what he understood about area. His only mathematical words are the
counting words one through eight. While he was responding to the teacher’s request to
show the area, he does not mention the word area. In addition, he does not label what he
is counting. It is possible that J akeel was counting squares, but it was also possible that he
was counting the number of times he pointed (with one finger or two). If Jakeel was
thinking of his counting as the number of squares, he still might not be thinking of the
squares as units of area measurement: Instead, he could be thinking about them as objects
to be counted. Thus while Jakeel demonstrated many of the features of the desired
discourse, it is unclear what he understood about area and units of area measurement.

One final piece of evidence of Jakeel’s change in discourse was his activity on a
subsequent lesson. The following week, the students participated in another lesson
involving finding the area of triangular figures. (See Table 4.1 for a sequence of lessons.

The case study lesson occurred on Thursday of Week 2. This subsequent lesson occurred

93

on Tuesday of Week 3.) The students had worksheets with triangles drawn on a grid (See
Figure 5.3 for an example). They used rubber bands to recreate the triangles on
geoboards. They then determined the area of the triangles. One of the triangular figures

on the worksheet (the middle problem from Figure 5.3) was identical to Figure I from this

 

 

 

 

 

 

lesson.
. u a a__J_,.r a o . v o —--—-+———---f a I
____1- " I. .2 I"
-.-~“'-P r)" 't.
u..—r..=—-.—————.——‘-+ ' ' ' -'— .' If“ . . l. . 'f ' . .
. __/“ 3"
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a o o t e o c u o I l t / o o o c
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Figure 5.3 Portion of worksheet on area of triangular figures.

Jakeel was working with Minerva and Jessica. He created the triangle from the
second problem on his geoboard and divided it into a square and two smaller triangles
with rubber bands (See Figure 5.4). Jessica was working on the same problem and was

unsure how to determine the area of the figure. The teacher asked J akeel to show Jessica

the squares in the figure.

94

 

 

 

 

 

O O /P O O
0 _, I 0 o
./ . .
O o g g .
. O O a .

 

 

 

 

 

Figure 5.4 Illustration of J akeel’s geoboard showing the triangular figure partitioned into

a square and two rectangles.
Excerpt 5

450A] Teacher Now show her I a Jakeel. Show her how it
makes a square.

451A J akeel See these two make a square
452A Minerva Duh duh duh

453A J akeel And so it’s two squares

J akeel turns the
geoboard so he can
see it. He puts his
index and middle
fingers on the two
smaller triangles. He
then turns it back to
Jessica.

These are rude noises
that seem to convey
that Jessica should
already understand.
Minerva glances at
the teacher after she
makes the noises.
Jakeel removes his
fingers from the
board and places his
middle finger in the
full square.

In this excerpt, Jakeel was able to use words to describe what had only been a

pointing action in the previous lesson. In the previous lesson, Jakeel had pointed to the

 

' I have added the letter A to each of these line numbers as a reminder that this excerpt and these line

numbers are not from the same lesson as the other excerpts in this chapter.

95

two small triangles in the figure and counted them as one, without stating what they were
one of. In this lesson, in addition to pointing in the same way, he was able to state that the
two triangles made a square (Line 451). In addition, rather than only count the area,

J akeel specifically stated that there were two squares (Line 453). He still was not
explicitly connecting the squares with area, so it is uncertain whether he was only
thinking about counting squares or whether he understood that counting squares was a
measurement of area. However, this excerpt shows similarities to J akeel’s activity in the
previous lesson suggesting that he was able to draw upon that activity from the previous
lesson and continue to change his mathematical discourse.

In summary, over the course of the lesson comparing the various figures, Jakeel
learned to count partial units in triangular figures as one and he was able to draw upon
and elaborate this participation in a subsequent lesson. While J akeel’s final discourse was
not as mathematically rich as what was indicated in the description of the desired
discourse, he did learn during this lesson in ways that were helpful in his later
mathematical work.

The next section will look at the process of learning and describe the kind of
learning activity enacted by Jakeel and supported by his group during the lesson on

Figures H, I, J and K.

Learning Activity
I have partitioned J akeel’s learning activities in this lesson into three phases.
Initially, he enacted what I describe as engaged learning. He then went through a period
of directed learning before returning again to engaged learning. Because I describe the

process of learning as the interplay of identifying and mathematizing, I will describe each

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kind of learning in terms of the identifying and mathematizing activity of Jakeel. I will
also describe how Rebecca and Daren mathematized and how they identified Jakeel. I use
the five characteristics of autonomous learning as categories to help organize my
analysis. Thus for each kind of learning, I illustrate the key activities that distinguish the
learning, explain how Jakeel identifies the audience for his discourse, elaborate the
available and adopted mathematical discourse, discuss Jakeel’s production of variations

on discursive features, and finally describe J akeel’s substantiation of the discourse.

Engaged Learning

According to the Oxford English Dictionary, one definition for engage is to
involve ("Engage", 2008). This sense of involvement captures Jakeel’s activity as he
enacts engaged learning. Jakeel was involved with learning, not only in terms of actively
mathematizing and making his own sense of mathematical ideas but also in terms of
contributing to and exploring the ideas of his peers. My label of engaged learning also
draws upon Etienne Wenger’s (1998) concept of engagement. In particular, I want to
emphasize Wenger’s notion of engagement as mutual. For Wenger, this means that all
individuals are moving towards fully participating in a community by actively making
meaning and contributing to practices. As I will illustrate below, for Jakeel, learning was
a mutual activity and not an independent or ritualistic activity: He interacted with others,
contributed ideas, worked to make his own sense of discourse, and relied upon others to

support his quest for understanding.

An Example of Engaged Learning
My presentation of Jakeel’s engaged learning begins with a detailed exploration

of one excerpt in which Jakeel and his peers illustrate several aspects of engaged

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learning. This excerpt occurred toward the beginning of the small group work, before any
of the exchanges included above. Rebecca and Daren started writing their answers to the
task involving Figures H and I while J akeel looked around for his pencil. When J akeel
realized that Rebecca and Daren were working, he was puzzled.

Excerpt 6

147 Jakeel How we gonna get the answer if we didn’t He looks at Daren.
even read the paper?

With this question, Jakeel identified himself as part of a group working together to find

the answer. His “we” made him an active partner in this process. He also articulated the

first step toward getting the answer, reading the paper. J akeel could have asked what

Rebecca and Daren were doing or he could have watched them to see what they did, but

instead, he asked a question that positioned him as a contributing member of the group.
Daren responded to J akeel.

148 Daren We know. Look. He picks up a paper

with Figures H and I
and points at the

figures.

This is half. If we put this together that’s He points to the
two squares. triangles in Figure I.

149 J akeel That’s a obtuse.2 Jakeel points at
Figure I.

Daren’s response acknowledged J akeel’s question and positioned J akeel as
interested in making sense of the task. Daren could have ignored or dismissed J akeel’s
question (as we see other students doing in the next two chapters). He also could have
told J akeel to copy his answer. Instead, Daren explained his thinking including

demonstrating how his words point to the picture of the figures. Daren, like J akeel, also

 

2 During the previous unit on geometry, Jakeel erroneously began to call right triangles “obtuse”. He was
consistent in this mislabeling, suggesting that his use of obtuse in this moment, while mathematically
incorrect, correctly designated Figure I as being in the same category as other triangles Jakeel called
obtuse.

98

used “we” (Line 148) which might suggest to J akeel that he was part of the group and
that he could also work with the figures in the same way as Daren. Daren’s actions in this
excerpt supported J akeel in being engaged in learning by positioning him as actively
interested in making sense of the task.

In Line 149, J akeel demonstrated that his engagement in the learning process by
involving himself in the conversation. Rather than accept Daren’s explanation as the only
important mathematical information, Jakeel suggested he also knew information and had
something to contribute. It is unclear how J akeel was positioning Daren with this
utterance. He could have been trying to teach Daren or compensate him for sharing his
ideas by sharing one of his own. It is also possible that Jakeel was attempting to indicate
to Daren that he knew some information related to the task so that Daren would not think
he had to explain everything about the task. One message J akeel’s utterance seemed to
convey was that he was attending to and interested in mathematical features of the task.

Again, Daren responded to J akeel.

150 Daren We don’t care about that! (?) Look, look. Jakeel has picked up

his own copy of
Figures H and I.

You see how that Daren points at the
paper Jakeel is
holding.

151 Jakeel This is I He points at his
paper.
Daren’s response to J akeel (Line 150) evaluated Jakeel’s contribution and labeled

it as irrelevant. While this labeling could be seen as suggesting that J akeel was incapable,
Daren continued his explanation. Thus Daren’s utterance supported Jakeel’s sense
making by redirecting him to relevant mathematics and continuing to position him as

someone who can understand Daren’s thinking.

99

Jakeel indicated his continued interest in the task by picking up his paper (Line
150) and contributing again to the conversation (Line 151). With his statement, “This is
I” Jakeel indicated that he was attending to the figures on the paper and that he was an
active partner in this conversation. He also demonstrated that he could make
contributions that were more relevant to the task. Daren responded to Jakeel by

continuing his explanation.

152 Daren You see the one square and that’s got two. Daren is pointing at
You put these together. That’s two squares. Jakeel 's paper. It is
So they cover the same area. not clear what he is

pointing at. After this
statement, Daren
turns to his paper.

153 Jakeel Can you say that again? Daren puts his hands
on his face. Jakeel
looks at his paper.

With this question, J akeel acknowledged that he did not understand what Daren
had said and he indicated he wanted to hear the explanation again. It is possible that
J akeel wanted Daren to repeat his words so that he could imitate them or write them
down rather than so that he could hear them again in an attempt to make sense of them.
However, this possibility seems unlikely given J akeel’s previous contributions to the
conversation and given the other possibilities for what Jakeel might have said. For
example, J akeel might have asked Daren to give him the answer or he might have
shrugged his shoulders and tried to copy what Daren had written. Instead, this question
seems to reflect J akeel’s desire to make sense of Daren’s explanation.

Jakeel’s engagement in learning was also illustrated by his refusal to participate or
be positioned as someone who only wants the answer. For example, Rebecca suggested

that J akeel could copy her answer.

100

Excerpt 7

193 Rebecca So let’s cut these out. Wait. We’ll write Rebecca picks up her
first and then we’ll cut. Okay pencil. Daren and
Jakeel also stop
cutting and start
writing.

And then H. H has two squares. Hold on let Jakeel is writing and
me finish writing this and then I’ll show erasing
you what I’m writing.

Like I Rebecca puts her
paper in front of
Jakeel
See J akeel. J akeel, look. She points at her
paper.
194 J akeel I don’t want to write that Jakeel pushes the

paper away

I know what:

195 Rebecca You just won’t write

Rebecca placed her paper in front of Jakeel and directed him to look at her answer (Line
193). However, J akeel was emphatic in his refusal, stating that he didn’t want to write
what she had written (Line 194). In addition, the beginning of his next sentence suggested
that he knew what to write. Indeed, he was writing at the time Rebecca placed her paper
in front of him and he continued to write at the end of this interaction.

This excerpt was one of three interactions between Rebecca and Jakeel in which
Rebecca tried and failed to get Jakeel to follow her directions. In one instance, she was
very insistent that he and Daren copy her work, placing her paper between Jakeel and
Daren and commanding, “Write that. Write that. People write this.” (Line 160). She
continued to tell Daren and Jakeel to “write this” (Lines 167 and 169). Jakeel at first
deflected Rebecca’s command by teasing her about how she said his name. The teacher

then arrived and redirected the conversation.

101

In the second interaction, Rebecca told J akeel to make a square with the small
triangles he had cut from Figure I and tried to move his pieces for him. He told her he
knew what to do and proved it by putting the two triangles together. Both of these
examples, along with Excerpt 7, demonstrate Jakeel’s refusal to be positioned by Rebecca
as a passive learner who did as he was told. Instead, he wanted to be actively involved
and responsible for making decisions and making sense of interactions. His refusal to
follow her directions, coupled with his eagerness to contribute to conversations as
described earlier show how Jakeel was participating in engaged learning.

In the next few sections, I will discuss Jakeel’s audience and the mathematizing in
the group. My elaboration of these other features will continue to add nuance and texture

to my description of Jakeel’s engaged learning.

Identifying the Audience

In Chapter 2, I discussed the ways in which the autonomous learner identifies his
or her audience as important to the mathematizing of the learner. This relationship
between audience and mathematizing holds for other kinds of learning as well. In this
section, I describe J akeel’s audience for his learning activity, emphasizing the members
of J akeel’s learning audience and how he identified them.

An important member of J akeel’s learning audience was himself. While much of
his discourse was directed to others, there were moments when he was clearly talking to
himself. This was most apparent as he tried to make sense of how to count the area of
Figure J. This excerpt occurred almost 40 minutes into the small group time. The teacher

has just asked J akeel what the area was of Figure I. This excerpt continues Excerpt 2

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from above. I have included a large portion of transcript here so that I can use it to make

several points later.

Excerpt 8

538 Teacher
539 J akeel
540 Teacher
541 J akeel
542 Rebecca
543 J akeel
544 Rebecca
545 J akeel
546 Rebecca
547 J akeel

What is it? Tell me what it is.
Eight
How is that eight? I can’t tell that’s eight.

Because one, two, three, four, five, six,
seven, eight, nine, t-.

Hold on.

Can I tell him?

It’s

Okay, Jakeel. You see those little triangles.

Yes
Those are half a squares. Look and if you
put these together, those equal a square

and so it’s eight, it’s eight and eight.

Oh. one=

103

Jakeel points to the
spaces in Figure J as
he counts. He points
once at each space
with the pinky of his
right hand.

Rebecca has her hand
up. Jakeel points at
each space in K with
his pinky. This motion
suggests that he is
silently counting

Rebecca talks to the
teacher. The teacher
then leaves.

She points to Figure J.

Rebecca points to
Figure J on Jakeel ’s

paper.

Rebecca points to
Figures J and K.

Jakeel is pointing to
spaces in Figure J.
Rebecca interrupts
him.

548 Rebecca So write that if you put the two half squares Jakeel points at the
together it makes a square. If you put two spaces in Figure J.

half squares together, it makes a square. His motions suggest
Write that on your paper. Write that. Then that he is silently
you’re done J akeel. counting. It looks like

he points at squares
with one finger and at
triangles with two

fingers.
549 J akeel Oh, yeah!

There are two moments in this excerpt when Jakeel’s audience for his discourse
was primarily himself. On Lines 541 and 548, Jakeel silently pointed to the spaces in
Figures K (Line 541) and J (Line 548). Although J akeel didn’t audibly utter any words,
his actions were consistent with counting to himself. His “Oh, yeah!” on Line 549
seemed to indicate that he was successful in his counting. This utterance could have been
in response to Rebecca’s words (Line 548). However, Jakeel was looking at his paper and
his gestures while Rebecca explained which suggested that his exclamation was a
proclamation of his success counting rather than a declaration of his understanding of
Rebecca. Jakeel’s exclamation provides further evidence that he was an audience for
himself.

There were other moments in which J akeel included himself as an audience for
his discourse. On three other occasions, he demonstrated his counting of the spaces in
Figure J for the teacher. On two of those occasions, Jakeel stopped his counting before he
had counted all of the spaces. He was not interrupted or distracted by anyone else.
Instead, he stopped each time he reached eight. His voluntary halt to his counting
indicates that he was monitoring his activity and not relying on others to evaluate whether
he had reached the right number. Thus as J akeel enacted engaged learning, he was an

important audience for his discourse.

104

This excerpt illustrates that the teacher was also a member of J akeel’s audience.
He initially counted the spaces in Figure J in response to the teacher’s question about the
area (Line 540), so this first counting was specifically enacted for the teacher. However,
it was not enacted so that the teacher could tell Jakeel whether he was right or wrong.
J akeel knew that he should end his counting on eight. Instead of positioning the
classroom teacher as the person who should tell him if he was right or wrong or the
person who could help him figure out what he should do differently, Jakeel positioned
her as someone he performed for. He needed or wanted to show her that he knew how to
count the spaces in Figure J and he persisted in his counting and in demonstrating his
. counting to her until he finally was correct in his counting (See Excerpts 3 and 4 for
examples of J akeel’s counting for the teacher).

Jakeel’s positioning of the teacher contrasts with his positioning of Rebecca and
Daren. These two peers were evaluators and explainers for Jakeel. They would listen to
his discourse and then provide him with feedback including their thinking about the
mathematics. Excerpt 8 provides an example of this. J akeel counted the spaces in Figure J
for the teacher, but Rebecca observed his counting and was eager to explain to Jakeel
how to think about the spaces in order to end on a count of eight. J akeel was willing to
allow her to explain: He responded to her question about seeing the triangles (Lines 544
and 545) and, after her explanation, was successful in altering his counting. This
interaction is in contrast to the next interaction in which Rebecca repeated her thinking,
but couched it as a command. In Line 548, Rebecca was no longer the helpful explainer
but was instead directing J akeel about what to write. As I will demonstrate later when I

describe directed learning, J akeel was not always successful in rejecting Rebecca’s

105

commands. However, at this moment, when Jakeel was enacting engaged learning, he
ignored Rebecca’s command and focused on figuring out the counting, refusing to allow
her to occupy a position as director.

Daren and Rebecca served as evaluators and explainers on several occasions.
Excerpt 6 above provides examples of Daren evaluating and explaining. On Line 150,
Daren rejected J akeel’s contribution that the figure was obtuse and then explained his
thinking. Also, after each time that J akeel incorrectly counted Figure J, either Rebecca or
Daren explained their thinking about counting the triangular spaces. J akeel paid attention
to these explanations, asked questions when he didn’t understand (For example, see
Excerpt 6, Line 153), and, when he did make sense of their explanations, used them in
formulating his next utterance.

In summary, the audience for J akeel’s discourse had three different types of
members. First, Jakeel was a member. He listened to himself and compared his discourse
to his expected outcome. The teacher was a second member. She watched Jakeel’s
performances but was not responsible for explicitly evaluating him. Instead, his peers
served as this third type of audience member. Rebecca and Daren were evaluators and
explainers for J akeel, supporting him in altering his discourse. A common theme across
these audience members was an emphasis on supporting Jakeel in developing his own

discourse.

Mathematizing
The previous section of this chapter focused characterized engaged learning and

described J akeel’s audience for engaged learning. This next section focuses more

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specifically on the mathematical discourse, examining Jakeel’s adoption of the discourse,
production of new discourse, and substantiation of the discourse.

Adoption of discourse. Adoption of discourse means using the same narratives,
routines, words, and visual mediators as others using the discourse. There is evidence that
J akeel adopted a portion of the mathematical discourse available to him as he enacted
engaged learning. I will first describe the available mathematical discourse and then
evaluate J akeel’s adoption of that discourse.

During the small group time in this lesson, two important sources of mathematical
discourse for J akeel were Rebecca and Daren. Rebecca’s mathematical discourse was the
most mathematically sophisticated of the three students. This sophistication was apparent
as Rebecca read her written answer to the task involving Figures H and I to Daren and
J akeel.

160 Rebecca Whatever. H and I cover the same amount. Rebecca reads fiom

Figure I has one square and two half her paper.
squares that equals two squares.

While this statement did not specifically state that Rebecca was comparing the area of the
two rugs or the relationship between area and squares, the statement used mathematical
words to communicate how the partial units should be counted toward the total number of
squares. Rebecca specifically labeled the partial units as half squares. By using the word
half, Rebecca acknowledged the relationship between the partial shapes and the whole. In
contrast, when the partial shapes are labeled as triangles, there is no indication of how
they might be related to the squares or how they might be counted in the area.
Additionally, Rebecca’s use of “half’ as part of a number sentence (“one square and two
half squares that equals two squares”) offers further evidence that her use of half was

mathematically purposeful. She did not mean that the half units were just some part of a

107

square. Instead, the two halves could be quantified, combined, and then added to one to
equal two squares.

Rebecca’s statement in Line 160 hints at a narrative that she used repeatedly in
her interactions with Jakeel. On five occasions, when J akeel was enacting engaged
learning, Rebecca articulated a variation of “If you put two half squares together, it
makes a square” (Line 548). This statement was frequently accompanied by pointing at
the small triangles in either Figure I or J. Thus J akeel had multiple opportunities to hear
these words and see how Rebecca linked them to an image.

Like Rebecca, Daren also used “half” to describe the partial units in Figure I. For
example, he when he explained his thinking about the task to Jakeel in Excerpt 6, Line
148, he stated, “This is half. If we put this together that’s two squares.” This statement
suggests that Daren like Rebecca, was thinking that two halves together would count as a
square. Daren’s discourse was also like Rebecca’s in his gestures. In each of his
interactions with Jakeel, he physically and verbally pointed toward the figures on the
page. For example, in Line 148 above, Daren pointed at Figure I as he spoke, “This is
half. If we put this together that’s two squares.” Daren explicitly connected his words
with an image.

Daren was the most explicit about connecting area with counting squares. Later in
the Excerpt 6, Daren explained to J akeel, “You see the one square and that’s got two.
You put these together. That’s two squares. So they cover the same area.” (Line 152). By
enumerating the squares and then stating, “they cover the same area” Daren began to

connect counting squares with area although he did not explicitly state the connection.

108

Daren’s mention of area was only the second time (and only one of seven times)
that area was mentioned during this entire small group time. The teacher used the word
area five times, each time as either part of a question or request directed to Jakeel (i.e.
Line 534: What’s the area of J? and Line 431: Show me how it’s the same area as that).
While the teacher required Jakeel to react to her use of area, she did not explicitly
connect it with any unit of measurement, nor did she require that Jakeel or the other
students specifically label what they were counting as area.

In summary, the mathematical discourse available to Jakeel had instances in
which the objects counted were labeled as squares and in which tenuous connections
were made between area and the unit of measurement. What was repeatedly emphasized
in this group, both through words (especially the use of half) and through pointing, was
that two of the triangular spaces should be counted together as one space.

J akeel only adopted a limited portion of the mathematical discourse available in
his group. As I noted in the section on his final mathematizing, there was no evidence
that he made explicit connections between counting spaces and area. He also did not label
what he was counting or use the word half, except on one occasion that I will discuss in
the section on directed learning below. Instead, the portion of the mathematical discourse
that J akeel visibly adopted was counting the two triangular spaces as one. In particular,
what he adopted from Rebecca and Daren was simultaneously pointing at the two
triangular spaces while uttering a single counting word. While I will have more to say
later about the specifics of this pointing and how Jakeel intertwined adoption and
production, I want to emphasize the evidence suggesting that Jakeel’s pointing and

uttering were an adoption of Rebecca and Daren’s discourse. As I described in the section

109

on his final mathematizing, Jakeel was not able to appropriately count the area of the
triangular spaces until his fourth attempt. After each of his attempts, either Rebecca or
Daren re-explained with words and gestures that the two half squares should be counted
as one square (For example, see Excerpt 8, Line 546). Jakeel’s attention to and repeated
need for Rebecca and Daren’s discourse provide evidence that J akeel’s pointing and
counting was at least in part an adoption of Rebecca and Daren’s discourse.

Before I discuss J akeel’s production of discourse, I want to make two comments
about his adoption of discourse. First, while there was no evidence that he adopted other
portions of the discourse available to him, it is possible that if Jakeel had been asked
different questions or had more time he might have demonstrated adoption of more of
Rebecca and Daren’s discourse. Second, the portion of their discourse that he visibly
adopted reflected the teacher’s primary learning objective for this lesson. She wanted
students to learn to appropriately account for partial units when determining area. While
it is uncertain that Jakeel’s thinking included partial units, he was able to discriminate
between different size spaces and count them in ways that appropriately considered their
difference in size. Although it seems that Jakeel’s adoption of Rebecca and Daren’s
discourse was limited in terms of number of words, narratives, or routines, what he

adopted was significant in terms of the teacher’s mathematical objective for the lesson.

Production of discourse. Production of discourse refers to the learner’s
construction and use of narratives, routines, mediators, or words that were not already
part of the discourse presented to the learner. As I hinted above, when Jakeel enacted
engaged learning, he intertwined adoption and production of discourse. The best example

of this was his counting of the spaces in Figure J. He eventually adopted the routine of

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counting the two triangle spaces as one. However, he produced the hand gestures that
enabled him to correctly count. Rebecca and Daren pointed to the two triangular spaces
by using an index finger from each hand, drawing a line to connect them, or pointing
back and forth between the spaces with one figure. When J akeel accurately counted the
spaces, he used his index and middle fingers to simultaneously point to the triangular
spaces, counting one number when he pointed with the two fingers. When be counted the
squares, he pointed with his index finger and counted one number (See Excerpt 4, Line
668). These gestures were his invention and were a change from his initial counting in
which he pointed at each space, triangles and squares, with one finger (his pinky) or with
a pencil and counted each space as one. J akeel’s gesture demonstrates the integration of
production (of the gesture) with adoption of routine of counting two squares as one.
Another aspect of J akeel’s production of discourse was his word use. Jakeel rarely
copied or imitated another person’s words. Instead, he translated Rebecca and Daren’s
words into his own words. One example of this is the word half Although Rebecca and
Daren frequently used “half” as a label for the partial units in Figure I and Figure J,
J akeel only once used that word. This one use was during the portion of the lesson when
he was allowing Rebecca to direct his learning. He did not speak the word half, but rather
wrote it on his paper as part of a sentence Rebecca dictated to him. Instead of the word
half, the only time Jakeel explicitly labeled the partial units, he called them “triangles.”
He wrote this on his paper shortly after the conversation captured in Excerpt 7. He was
working on the task with Figures H and 1. Rebecca encouraged him to copy her writing,
but he refused, writing his own words, “If you put the two triangles together it makes a

square and it covers it.” While J akeel only used triangles only once, it was the first time

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anyone in the group had labeled the partial units using their geometric name. It provides
evidence that J akeel, as he enacted engaged learning, was determined to do his own
learning and to make his own sense rather than blindly adopt the discourse of others.
Here is a second example of J akeel forging his own discourse. He had finished
writing on his paper and was cutting out Figure I. The teacher came to the group and

began asking questions.

Excerpt 9

205 Teacher So what are you doing, Jakeel?

206 Jakeel I’m gonna cut these out. He is holding the
paper with Figures H

and 1. He points at it
with his scissors.

207 Teacher Why are you gonna cut them out?

208 J akeel Because I have to make these two a He points at his paper
rectangle just like these three. with his scissors. His
motions are hidden
from the camera by

his paper.
209 Teacher Okay. Why are you making it a rectangle?
210 J akeel Because if you don’t it won’t cover the
whole thing.

In Line 208, Jakeel stated that he had to make a rectangle. While his motions were
blocked from the camera by his paper, it seems that the “two” he wanted to make into a
rectangle were the two triangles from Figure 1. Instead of using square as the other
students had, Jakeel called the square a rectangle. In this line, Jakeel indicated that if he
assembled the triangles into a rectangle (a square), they would be just like the other three
squares already on the paper (one square in Figure I and two in Figure H). Jakeel’s use of
rectangle, like his use of triangle above was the only time (except for the teacher’s echo

of the word in Line 146) the word was used throughout the lesson. This again reflected

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Jakeel’s propensity to use his own words. Also, with his mention of “these three,” Jakeel
was also the only student in this group to count all of the full squares together. While
counting the three squares together did not relate to calculating or comparing the area of
the figures, it does reflect J akeel’s sense of the situation and his stance that he should
communicate his ideas.

J akeel’s reluctance to use the words of others while simultaneously using their
same image suggests that J akeel focused more upon the visual communication from
Rebecca and Daren than upon their verbal communication. If J akeel attended primarily to
Rebecca and Daren’s gestures (rather than their verbalizations) and linked those gestures
to his own words, then his verbal communication is more likely to consist of words and
narratives he produced. This is in contrast to Jakeel’s attention to words when he enacts
directed learning as I will later describe.

Substantiation of narratives. A final feature of mathematical discourse that I will
explore is the substantiation of narratives. Substantiation is how a person decides whether
a statement reflects the state of affairs (Sfard, 2008). When J akeel enacts engaged
learning he relies upon others to support his substantiation of statements. For example,
Rebecca supported Jakeel’s substantiation of the counting of Figure J. In Excerpt 8
above, Jakeel counted ten spaces in Figure I. In Lines 544 and 546, Rebecca explained
how to count so that the outcome was eight: “Okay, Jakeel. You see those little
triangles. . .. Those are half a squares. Look and if you put these together, those equal a
square and so it’s eight, it’s eight and eight.” Rebecca motioned to the figures on the

paper as she spoke.

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There are two examples of substantiation linked to this excerpt. First, Rebecca’s
explanation was her substantiation of the narrative that Figure J was eight. She explained
how she thought about the triangles as half squares which meant that two triangles could
be put together to make a square. Rebecca did not include in her substantiation counting
the spaces. Instead, she seemed to imply that the important evidence that Figure I had
eight was that two triangles could be counted together as a square. With this explanation,
Rebecca implied that substantiation was an apprOpriate activity and that J akeel could
make sense of this substantiation. Furthermore, her substantiation relied upon the visual
image. She used “see” and “look” and pronouns like “it” which were only sensible in
reference to the visual. Thus another possible message in Rebecca’s substantiation was
that it was important to link visual and the verbal communication.

J akeel could have attempted to memorize Rebecca’s substantiation or to interpret
her words into his own words to offer as an explanation to the teacher. However, as
J akeel enacted engaged learning, he constructed his own substantiation of the narrative
that the count for Figure I was eight. He altered his gestures so that he differentiating the
triangular and square spaces and then he was able to arrive at eight as he counted (See
Excerpt 4). J akeel’s substantiation incorporated Rebecca’s explanation of how to count
the two triangles, but was his own routine for counting. Thus Jakeel relied upon Rebecca
to support his thinking but he did not limit his thinking to Rebecca’s ideas. Likewise,
Rebecca’s explanation positioned J akeel as a learner who could use her ideas as a tool for
altering his own thinking.

J akeel’s enactment of engaged learning meant that he was an active learner,

asking questions, listening to feedback, and positioning his peers as teachers. He also

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displayed his thinking, contributing his ideas to the conversation and demonstrating his
responses to questions. He insisted upon doing his own work and not copying from others
and he attended to his own discourse, monitoring his counting to see if it matched his
expectations. While he faithfully adopted some pieces of his peer’s discourse, especially
the visual discourse, he tended to insert his own words rather than use the same verbal
communication as Rebecca and Daren. Finally, although he used others to help
substantiate the discourse, but he did not use their authority as the substantiation. Instead,
he built from their explanations to construct his own substantiation. His peers supported
Jakeel’s engaged learning by positioning him as someone who could make sense of their
explanations. Their explanations modeled discourse and connected the words to visual
objects, but did not explicitly state the answer, which created an opportunity for J akeel to
use their thinking to construct his answer. As J akeel enacted engaged learning, he was
involved in actively mathematizing and identifying as a learner who wanted to make his
own sense of mathematical ideas.

For the majority of this lesson (in terms of time and number of turns), Jakeel
enacted engaged learning. However, during the middle of the lesson, his activity shifted
to directed learning. This shift involved changes in his identifying and mathematizing
activity as well as the identifying and mathematizing activity of one of his peers,
Rebecca. I will first describe the circumstances leading to the shift in participation and

then characterize directed learning as enacted by Jakeel and Rebecca.

Departure flom Engaged Learning
During the early part of the lesson, Rebecca often (6 times) told J akeel what to do

and write. Each time, J akeel either ignored her words or he insisted that he would do his

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own work. For an example, see Excerpt 7, Lines 193 and 194. In spite of Jakeel’s resolve

to do his own work, Rebecca was eventually able to position him as needing to copy from

her: After Jakeel had finished his work on Figure H and I (See Figure 4.1), Rebecca

looked at his work.

Excerpt 10
254 Rebecca
255 Jakeel
256 Rebecca
257 J akeel
258 Rebecca

You wasn’t supposed to do that Jakeel.

Jakeel.

Jakeel what you was supposed to dooo is so

this. What we did.

Oh. Oh.

So you have to take it and rewrite it

and I’ll cut these out for you.

Rewrite what you wrote here on there.

I don’t want to rewrite.

Well you have to because you messed up.

She takes his paper

She puts her paper so
Jakeel can see it and
she points to it.

He looks at Rebecca ’5
paper.

She tries to pull up
Jakeel ’s glued pieces.

She points at an extra
paper.

Jakeel picks up his
pencil and starts
writing. Rebecca
starts cutting.

Rebecca’s feedback on Line 254 did not offer J akeel an explanation or an indication of

how he might think about the content of his work. Instead, as a rationale for her negative

evaluation of his work, Rebecca told Jakeel that he wasn’t doing what he was supposed to

be doing. This evaluation did not invite Jakeel to rethink his work or suggest that he

could make sense of what Rebecca had done. Instead the authority for deciding what to

do was embedded in “supposed” without any indication of how J akeel might make sense

of what he was supposed to do.

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Jakeel offered some resistance to Rebecca’s directive, stating, “I don’t want to
rewrite” (Line 257). However, Rebecca’s rejoinder (Line 25 8) left him with little room
for negotiation. Jakeel accepted this positioning, picked up his pencil and started to write
(Line 25 8). At this moment, Jakeel’s participation shifted from engaged learning to
directed leaming. For most of the next 17 minutes, Rebecca directed J akeel’s work on the
task. He did what she said and he asked her to tell him what to do. In the next sections, I
elaborate the features of directed learning as enacted by J akeel and supported by

Rebecca.

Directed Learning
In contrast to engaged learning, when J akeel engaged in directed learning, he was
more passive and reactive, allowing Rebecca to tell him what to do. The excerpt below
occurred after the excerpt above in which Rebecca told J akeel he had to redo his work.
Rebecca told Jakeel what to write, but he was distracted by a conversation with Daren.
He asked Rebecca to tell him again what to write.

Excerpt 11
319 J akeel What am I supposed to write again? He talks to Rebecca.
320 Rebecca You’re playing

321 J akeel What am I supposed to write? (Well you

play too much)
322 Rebecca They H and I cover the same amount of
floon
323 Daren Go get the other I and K. Daren talks to Jakeel.
324 Rebecca I mean just cover the same amount Jakeel writes. The
teacher comes by.
325 Daren We need J and K Daren talks to the
teacher.
326 Teacher So go up to get I and K. Daren gets up and
leaves.

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327 Rebecca H and I cover the same amount of floor. No She watches Jakeel
just they cover the same amount. write.

Same. Amount. Do you try to write small? Rebecca reads as

All right. That’s all you gotta write. Jakeel writes. Daren
returns with the
papers. Rebecca
reaches for the papers
and hands them out.

In Line 319, Jakeel asked Rebecca what he should write and then wrote what she dictated
on Line 322 and 327 (“They cover the same amount”). He did not attempt to use different
words, interject his own ideas, or connect Rebecca’s words to a visual image. Instead, his
activity with mathematical objects was limited to following Rebecca’s directions.

Jakeel’s activity when he enacted directed learning was markedly different from
when he was enacting engaged learning. As an engaged learner, he refused to follow
Rebecca’s directions, telling her that he knew what to do and that he wouldn’t do what
she asked (Excerpt 7, Line 194). He asked questions about explanations and contributed
to the conversation (Excerpt 6). In contrast, as he and Rebecca enacted directed learning,
they both focused on J akeel’s physical activity, whether it was writing, cutting, or not
playing around, rather than on his engagement with the mathematical discourse.

In the time that Jakeel engaged directed learning, there were five instances in
which Rebecca told J akeel what to do or write and he agreed. Each instance was similar
to Excerpt 11 with Rebecca giving directions for J akeel to follow. Because of the
similarities across these instances, I primarily draw upon Excerpt 11 to describe the

features of directed learning and how it is different from engaged learning.

Identijying the Audience
J akeel’s audience for directed learning was more limited than his audience for

engaged learning. In engaged learning, he addressed his peers, his classroom teacher, and

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himself. In contrast, when J akeel enacted directed learning, his audience for his content
discourse was either the classroom teacher or his peer, Rebecca. There was no indication
that he was attempting to communicate with himself. Instead, his communication around
the mathematical content was to write on his paper what Rebecca dictated (See Excerpt 7,
Line 327). This writing was done at Rebecca’s request, thus Rebecca was a member of
Jakeel’s audience. Since the writing was done on a paper to turn into the teacher, the
classroom teacher was also Jakeel’s audience. Jakeel, however, did not intend this writing
to be a communication with himself: He did not refer back to it. He did not question it,
examine it, or repeat it to himself. It might be argued that he understood what he wrote
and so he did not need to examine it. However, he asked Rebecca what he was “supposed
to write” (Line 321), implying that Jakeel did not know what to write. Furthermore, the
word “supposed” carries implications of expectation or requirement, suggesting that

J akeel was more concerned with how an authority might evaluate his work than
understanding it himself.

Rebecca’s role as a member of Jakeel’s audience was to monitor his physical
activity, telling him what to do and ensuring compliance. By doing this, Rebecca served
as a director for Jakeel’s activity. One example of Rebecca acting as a director was her
feedback to J akeel. In Excerpt 11, Rebecca accused J akeel of playing (Line 320). She had
told him to write, but Jakeel had talked with Daren rather than follow her direction.

J akeel confirmed Rebecca’s role as director by asking what he was supposed to write.
(Although be indicated that he wasn’t entirely pleased with her identification of him as
playing by retorting that she plays, too.) This interaction focused J akeel back on the

physical activity he was supposed to be doing. Rebecca’s feedback was not an

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explanation or a correction of mathematical discourse. Instead, it was feedback on
Jakeel’s progress toward the final product of the task. By contrast, when J akeel enacted
engaged learning, the feedback he received was an explanation of the thinking of his
peers, directed toward problems with J akeel’s mathematical discourse. Thus as J akeel
enacted directed learning, he provided support for Rebecca’s identification of herself as a

director of Jakeel’s physical activity.

Mathematizing. As Jakeel enacted directed learning, his use of mathematical
discourse was limited to writing the words Rebecca dictated. Excerpt 11 is typical of
J akeel’s discourse during directed learning. He spoke twice and each time he asked
Rebecca what he should write (Lines 319 and 321). His mathematical discourse was
limited to writing “They cover the same” (Lines 324 through 327). At other times when
he enacted directed learning, J akeel either wrote what Rebecca dictated or cut figures as
she directed. As a result of Jakeel’s limited mathematical discourse, J akeel’s adoption,
production, and substantiation of discourse were also limited. I will examine each of
these aspects of mathematizing.

Evidence of Jakeel’s adoption of discourse would be Jakeel’s use of the same
narratives, routines, words, and visual mediators as another. However, because J akeel had
limited mathematical discourse, there is no evidence that he adopted mathematical
discourse as he enacted directed learning. His writing of Rebecca’s words might be
construed as adoption. However, as I noted earlier, Jakeel wrote words as Rebecca
dictated. He seemed to be primarily recording Rebecca’s words rather than using them.
Also, in both instances in which Jakeel wrote Rebecca’s words, his next activity after

finishing his writing was playing with his glue. J akeel’s quick shift from writing to

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nonmathematical activity suggests that J akeel was focused on getting his writing done
rather than on exploring or investigating mathematical discourse, providing further
evidence that J akeel’s writing does not reflect adoption of Rebecca’s discourse.

Production of discourse is predicated on adoption of discourse. In order to
generate new discourse, the learner must first use accurately use the mathematical
discourse and then elaborate on that discourse to generate new narratives, routines, visual
images, or word use. As J akeel enacted directed discourse, he did not demonstrate
adoption of discourse and thus it was not possible for him to produce discourse.

Finally, as J akeel enacted directed learning, he did not directly substantiate
mathematical statements. He did not attempt to verify the truth of what he wrote or what
others said. Instead, he relied upon Rebecca to provide him with the right answer. For
example, in Excerpt 11, he asked Rebecca what he was “supposed” to write (Lines 319
and 321). His use of supposed indicated that J akeel was focused on what was the
expected or required activity. If J akeel had been attempting to substantiate mathematical
discourse, he might have asked how Rebecca knew that H and I covered the same amount
of floor or he might have counted or otherwise compared H and I to verify Rebecca’s
statement. Instead, he wrote what she dictated and then began to play with his glue. In
each moment in which J akeel enacted directed learning, he did what Rebecca directed but
did not engage in any activity that verified any statements she made.

J akeel’s mathematizing was an important characteristic of directed learning. He
and Rebecca focused on his physical activity rather than on his thinking about

mathematics. As a consequence of this focus, as J akeel enacted directed learning, he did

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not adopt or produce discourse and his substantiation of discourse relied upon Rebecca’s

authority.

Transition Back to Engaged Learning

Jakeel’s enactment of directed learning did not last until the end of the lesson.
Seventeen minutes after Rebecca was first successful in directing him, he switched back
to engaged learning. This return to engaged learning was evident in Excerpt 8. Prior to
this moment, Rebecca first directed Jakeel to cut out Figures J and K and then told him to
stop while she determined, out loud, the areas of J and K. She announced that they were
each eight and were the same (Line 358, not in this document). She then gave Jakeel
permission to cut out the figures: “Okay you can cut them up now. Don’t cut out the little
squares” (Line 378). J akeel followed these directions, cut out the figures and glued them
on his paper. At this point, his obedience to Rebecca’s orders reflected his enactment of
directed learning.

However, after J akeel had finished cutting and gluing, he transitioned back to
engaged learning as the teacher asked him questions about his work. Excerpts 2 and 8
show this transition. The teacher asked Jakeel about the area of Figure I (Excerpt 2, Line
534). He pointed at Figure K and replied that it was the same (Excerpt 2, Line 53 7) and
that it was eight (Excerpt 2, Line 539). The teacher asked J akeel to show her how Figure
I was eight (Excerpt 8, Line 540). J akeel counted the spaces in the figure, stopping as he
started to count ten. He said, “Hold on” and then counted the spaces in Figure K (Excerpt
8, Line 541).

At this moment, J akeel was not following orders or directions. He was evaluating

his own discourse and puzzling over his counting. He stated that Figures J and K should

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have the same count — a count of 8 — yet his counting was yielding a larger number. This
discrepancy over what he counted for J, what he thought the count should be, and what he
counted for K (Excerpt 8, Line 541) seemed to raise questions for Jakeel. His pursuit of
those questions demonstrated his shift back to engaged learning. He continued to
demonstrate engaged learning through the end of the lesson, working on counting Figure
J, receiving input from Rebecca and Daren and then recounting until he was able to count
to eight in front of the teacher (Excerpt 4).

This transition from directed back to engaged learning is an important piece of the
next section of this chapter, an exploration of the autonomous learning features of

Jakeel’s activity.

Autonomous Learning

Autonomous learning is the constellation of identifying and mathematizing
activities that reflect curiosity about what others think/say and what seems to be true. As
elaborated in my theoretical framework, curiosity is demonstrated by an interest in what
is novel, strange, or problematic. Evidence of curiosity can be found in the way the
learners identify their audience and in their adoption, production, and substantiation of
mathematical discourse. In this section, I will refer back to my descriptions of engaged
learning and directed learning to make claims about what features of J akeel’s learning
activity are consistent with autonomous learning.

Jakeel’s enactment of engaged learning contained several features were similar to
the activities of autonomous learning. His audience supported his exploration of the
problem of how to count eight when counting Figure J. He included himself in his

audience, demonstrating that his discourse was a discourse for himself. His peers

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provided explanations and feedback and J akeel monitored his own discourse. Jakeel
adopted and produced discourse, drawing upon Rebecca and Daren’s illustration of how
to count the triangular spaces in Figure I to construct his own gestures for successfully
counting and thus resolve the problem of how to reach 8 and not 10 when counting.
Finally, J akeel drew upon Rebecca and Daren’s substantiation of the discourse to create
his own substantiation. He was not willing to accept Rebecca’s statement about how to
count the triangular spaces in Figure I was as adequate proof that Figure I was 8. Instead,
he wanted to establish for himself that he could count the spaces and arrive at 8.

While J akeel’s engaged learning activity paralleled the activity of autonomous
learning in multiple ways, one important way in which he did not enact autonomous
learning was his lack of attention to the mathematical words used by Rebecca and Daren.
For example, Rebecca and Daren used half several times in their interactions with J akeel.
This word labeled the triangular spaces in Figures I and J in a way that captured the
mathematical relationship between the triangular spaces and the square spaces. J akeel did
not use the word half when be enacted engaged learning. He did not flame Rebecca and
Daren’s use of half as novel or strange. He did not adopt the word or produce new
discourse that integrated the word. His substantiation did not involve the word or reflect
his attempts to understand how half was an accurate description of the triangular spaces.
His discourse does not offer moments in which he explores or shows curiosity about
Rebecca and Daren’s use of half. While we cannot be certain that J akeel’s discourse
would have changed if he had been curious about half, his lack of curiosity meant that
there was not the possibility that he would explore the word and it’s implications for his

own discourse.

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J akeel’s curiosity about Rebecca and Daren’s discourse seemed to focus primarily
upon their visual communication. He explored how to point and count the pointing
gestures, but he did not attempt to use mathematical words to describe his pointing or
how it resulted in a different count for Figure J, suggesting that he did not explore
Rebecca’s verbal discourse in counting the triangular spaces. As I describe above, J akeel
connected the visual with his own words, but his learning outcome might have been
different if he had been more curious about their words.

As J akeel enacted engaged learning, his activity paralleled autonomous learning
in many ways. In contrast, his enactment of directed learning did not demonstrate any of
the activities of autonomous learning. His audience directed his activity. He did not adopt
discourse as a tool for himself. He did not produce discourse and his substantiation relied

upon Rebecca’s authority

Discussion

Research that has examined identity in mathematics lessons has focused on a
more macro view of identity and on identity as reported by individuals. These studies,
such as Boaler and Greeno’s examination of identity in high school math (2000) and
Jilk’s exploration of the intersection of salient identities and mathematics pedagogy
(2007), described student’s self-reported identities relative to learning math. This study
demonstrates what a microanalysis of discourse can reveal about identity and learning
mathematics. The analysis of engaged learning and directed learning presented in this

chapter demonstrates four claims about identity and learning mathematics.

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Intertwining of Identity and Mathematizing

First, this analysis suggests that identity and mathematizing are inextricably
intertwined. As students mathematize, they construct identities for themselves and others.
These identities then offer possibilities for the mathematizing of both the speaker and any
others who are identified. For example, in Excerpt 8 Line 546, Rebecca explained to
J akeel how he could think about the triangular spaces in Figure I in order to get the count
of 8. This mathematizing identified J akeel as interested in and capable of making sense of
how to count Figure J. It also provided him with the information he needed to alter his
counting gestures in order to count 8. As Jakeel responded to Rebecca, he identified her
as a teacher whose mathematizing could help him solve his problem of how to count. In
this instance, the mathematizing and identifying co-occurred and worked to construct
compatible learning and teaching narratives for J akeel and Rebecca.

Mathematizing can also identify individuals in ways that constrain mathematical
discourse. In Excerpt 11, Rebecca told J akeel what to write on his paper. Rebecca’s
mathematizing was a statement of the answer for Jakeel to write. It identified J akeel as
someone who was to write words and not someone who was to explore the mathematical
discourse. It identified Rebecca as someone who stated answers, but not someone who
explained or engaged other people’s mathematical discourse. Rebecca’s mathematizing
and J akeel’s willingness to write her words limited the mathematical discourse for both
of them. This limitation included not only the quantity of discourse, but also the range of
discursive features and the way in which they used the discourse. As Rebecca dictated
words, she did not try a variety of explanations to help J akeel understand. She also did

not use gestures to indicate how her words referred to visual objects. She focused on

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supplying J akeel with the words to write. J akeel was not a learner. He was a recorder. As
a recorder, he did not need to engage the discourse so that he could later use it to solve
his own problems. He needed to produce the appropriate outcome on his paper in this
moment. In this example, the ways in which Rebecca and Jakeel mathematized identified
each of them in ways that promoted a particular, limited way of using mathematical
discourse.

Finally, individuals can engage in identifying activity that does not refer to
mathematical objects, but constructs narratives that constrain or enable mathematizing.
For example, in Excerpt 11, Line 320, Rebecca told J akeel that he was playing.
Rebecca’s statement was not about mathematics although it addressed her perception of
Jakeel’s lack of mathematical activity. By commenting on Jakeel’s playing, she implied
not only that he was not working but also that he needed someone to monitor his activity
and direct him to appropriate activity. By focusing on J akeel’s writing, Rebecca limited
his mathematizing to the physical activity of writing words. J akeel’s next statement, in
which he asks what he was supposed to write (Line 321), indicated that he accepted her
limitations on his mathematizing. Rebecca could have asked J akeel what he thought the
answer was or how he was thinking about the problem or she could have explained how
she was thinking about the problem. While these might not be typical fourth grade
reactions to this situation, I suggest them in order to emphasize how Rebecca’s
accusation that J akeel was playing framed J akeel in a certain way that suggested he
should mathematize in a certain way: By commenting on J akeel’s playing, Rebecca
identified herself as a director and J akeel as someone who should be directed and their

mathematizing became about the physical activity.

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Kinds of Learning

Many conceptions of identities relative to learning math describe the learner as
either a math person or not a math person (see for example, Boaler and Greeno, 2000).
Because students (and others) use these identities to explain their engagement (or
reluctance to engage) mathematics, they are important identities to study. However, the
identities of math person or not a math person do not capture the different ways in which
people mathematize and identify as they enact these distinctions. Just as the use of a
higher power microscope lens changes the image so that new features are visible, the
microanalysis of student activity performed in this study can capture important details not
visible at lower resolution. Therefore, a second important claim made by this analysis is
that student’s identifications relative to learning math can be described in more detail and
in ways that capture important differences in activities in which students might engage.
This study of the learning activities of J akeel found two qualitatively different ways in
which Jakeel mathematized and identified: engaged learning and directed learning. The
analysis of these two kinds of learning shows how interactions with peers are important
in proposing and enacting these ways of learning. For example, J akeel could not have
enacted directed learning without Rebecca’s directions. This increased detail in thinking
about student mathematical learning also provides a means to examine what specific
mathematizing and identifying activities might support or constrain effective learning of
mathematics. Finally, this more detailed description of learning and identities offers an
alternative to the binary distinction between math person/not a math person that may

force students into categories that don’t capture how they think about themselves as

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learners of mathematics. Given what we have seen here, it would be hard to fit J akeel into

one of these two.

Fluid Identities

This analysis also captures the ways in which student’s identities may shift during
the course of a lesson. J akeel was an engaged learner for most of this lesson. However,
toward the middle of the lesson, be enacted an identity of directed learner and then
returned to engaged learner. This shifi in identities indicates a change in mathematizing
and talk about oneself and others. While it is possible that this change is new discourse
and reflects learning, it is also possible the change reflects enactment of a different, but
not new narrative about the person and their mathematical activity. For example, in a
previous lesson, Jakeel had enacted directed learning. Thus his enactment of the learning
in this instance did not reflect a change to a new discourse but rather a shift to an
identifying discourse that he had previously used. Thus this analysis shows that students

may enact multiple identities as they learn mathematics.

Development of Mathematical Discourse

A final claim resulting from my analysis is that certain kinds of learning activity
results in the development of a more mathematical discourse. Over the course of this
lesson, J akeel’s discourse changed such that by the end of the less he could differentiate
the triangular spaces from the square spaces and count two triangular spaces as one
square space. This change of discourse seemed to occur as Jakeel enacted the
mathematizing and identifying of engaged learning. J akeel’s enactment of engaged
learning involved identifying and working on a problem whose solution required Jakeel

to change his mathematical discourse. He became interested in the problem of how to

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count the spaces in Figure J. He listened to Rebecca’s explanation and then modified his
gestures. He counted and recounted, adjusting his pointing, until his count was only eight.
Both Rebecca and J akeel identified J akeel as an involved and interested problem solver
who could solve his problem by changing his discourse. By the conclusion of the lesson,
after a time in which Jakeel enacted engaged learning and was supported by Rebecca, he
changed his discourse.

In contrast, when Jakeel enacted directed learning, his mathematical discourse
was limited and showed no evidence of developing. As a directed learner, the problem
J akeel worked on was doing what Rebecca asked him to do. He and Rebecca both
emphasized J akeel’s physical activity. As a result, Jakeel was not identified and did not
identify himself in a way that required him to work towards changing his mathematical
discourse. Thus, when J akeel enacted directed learning, his mathematizing and
identifying and that of his peers focused him on activity that was not about changing or
developing his mathematical discourse.

I have defined learning as a change in discourse and noted that effective learning
means that the discourse has changed to be more like the desired discourse. Jakeel’s
discourse changed as he enacted engaged learning so that his discourse was more like the
desired discourse: He accurately differentiated the triangular spaces from the square
spaces in his counting. This was in contrast to the lack of a change in discourse as J akeel
enacted directed learning. This analysis suggests that J akeel’s engaged learning activity
resulted in more effective learning than his directed learning activity. This conclusion
demonstrates an important benefit of this study. The close analysis of discourse allows

for the coordination of mathematizing, identifying, and learning so that outcomes of

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learning can be linked to mathematizing and identifying moves. This linkage provides a
powerful tool for studying mathematical learning in classroom situations.

An important component in guiding my decisions about how to examine
mathematizing, identifying, and learning was my theoretical frame for autonomous
learning. By elaborating autonomous learning, I was able to develop a lens to examine
what mathematizing and identifying moves might be most desirable. I was then able to
use each case to clarify my autonomous learning framework. In the next section of this
discussion I will describe how the framework of autonomous learning supported the
examination of J akeel’s learning and how Jakeel’s learning added detail and complexity

to the autonomous learning framework.

Autonomous Learning

My framework for examining autonomous learning emphasizes the ways in which
individuals adopt, produce, and substantiate discourse with the construction of discourse-
for—oneself as the end result. This focus on discourse helped me to describe and
differentiate engaged learning and directed learning: Jakeel’s engagement with discourse
was very different as he enacted engaged learning and directed learning. In particular,
J akeel’s work to adopt, produce, and substantiate discourse as an engaged learner
demonstrated that he was developing the mathematical discourse as a discourse-for-
himself. In contrast, when he acted as a directed learner, his mathematical discourse was
a discourse-for-others. Jakeel demonstrated that his discourse during engaged learning
was a discourse for himself by including himself in his audience, by adopting Rebecca’s
visual discourse and then using it as a tool to produce his own routine for counting Figure

J, and by working to substantiate the count of Figure I using his own discourse. These

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features of J akeel’s activity show the ways in which Jakeel was enacting autonomous
learning as be enacted engaged learning. In contrast, as Jakeel enacted directed learning,
he did not adopt or produce discourse and he relied upon Rebecca for substantiation.
Also, he was not an audience for his discourse. Thus his enactment of directed learning
did not include any features of autonomous learning.

Using the lens of autonomous learning to examine J akeel’s enactment of engaged
learning and directed learning helps highlight which of Jakeel’s activities may have been
most helpful in supporting the development of his mathematical discourse. The change in
J akeel’s mathematical discourse occurred during the time in which his activities most
closely resembled the mathematizing and identifying of autonomous learning. For
example, Jakeel’s was curious about counting Figure J. He worked to monitor and change
his discourse for himself. He attended to, incorporated, and modified the explanations of
others. The overlap between the activities of autonomous learning and J akeel’s
development of a more mathematical discourse suggest that the enactment of autonomous
learning may support students in learning mathematics.

Ironically, Jakeel’s written discourse as he enacted directed learning was more
like the desired mathematical discourse than his discourse as he enacted engaged
learning. For example, while directed by Rebecca, he wrote, “[T]wo halves make a
square”. This statement because it included half and square as labels for the spaces in
Figure J, was more mathematical than J akeel’s pointing and counting of the spaces in
Figure J. However, the discourse Jakeel wrote was not a discourse-for-himself and there
was no evidence that he could use this discourse to solve his own problems. Thus even

though the discourse was more desirable, J akeel was not engaged in a kind of learning

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that made the more desirable discourse a useful tool. This suggests that it is insufficient to
consider what students say or write as an indication of their ownership of those words.
Instead, the context in which students engage in discourse and their ability to wield the
discourse as a tool is an important factor in evaluating student learning.

The lens of autonomous learning also helps us to critique J akeel’s engaged
learning. While J akeel’s discourse became more like the desired mathematical discourse
as he enacted engaged learning, he did not adopt the available verbal discourse in the
same way that he adopted the visual discourse. For example, if Jakeel had been more
curious about Rebecca’s use of half or the ways in which Daren verbally linked squares
and area, he might have adopted those portions of Rebecca and Daren’s discourse in
ways that could have helped him add those features to his discourse-for-himself. If Jakeel
had adopted more of Rebecca and Daren’s verbal discourse, his mathematical discourse
might have become even more like the desired discourse. Using the framework of
autonomous learning to consider how J akeel’s activity was or was not autonomous offers
a tool for considering ways in which J akeel’s mathematical discourse might be further
deve10ped.

While the lens of autonomous learning provided insights into J akeel’s learning,
the case of J akeel’s learning adds complexity to the framework of autonomous learning.
As Ben-Zvi and Sfard (2007) note, autonomy is typically construed as relying upon
oneself. For example, the definition of autonomy used by Warfield, Wood, and Lehman
(2005) is “students are capable of thinking about mathematical ideas without having the
ideas ‘explained’ to them and of solving mathematical problems without being shown a

method by another person” (p. 440). According to this definition, Jakeel’s activity could

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not be described as autonomous because he relied upon Rebecca and Daren to provide
him with feedback and explanations. However, Piaget’s description of autonomy, as
describe in Chapter 2, suggests that cooperative learning is an essential piece of
autonomy because it helps the learner decenter fiom their perspective and realize that
there are other ways of seeing, doing, and understanding (Kamii, 1994). This description
of learning together focuses on autonomy as an intellectual activity rather than as a
physical activity. Rather than examine who the learner learns with, Piaget’s autonomy
asks how the learner is engaging with ideas. According to this conception of autonomy,
J akeel’s engaged learning could be classified as autonomous. He examines the ideas of
Rebecca and Jacob and works to incorporate their feedback and explanations into his
discourse.

However, Piaget emphasizes that learners should not be coerced by those with
whom they learn: Learners should be equals, learning together (Kamii, 1994). This
condition does not describe the learning situation with J akeel. Rebecca and Daren were
not learning with Jakeel, they were teaching J akeel. These students were not on the same
footing regarding discourse around area. Their unequal footing was the reason that Jakeel
could learn from them. Ben-Zvi and Sfard (2007) note that autonomy is a possible
outcome for learners who are taught by other students. They elaborate a leamer-teacher
agreement in which the teacher is able to use a discourse that the learner does not yet
know. The learner must learn from the teacher. This situation can result in autonomous
learning if the learner works to make the new discourse a discourse for him/herself
through critical examination of the discourse. J akeel and Rebecca and Daren offer an

example of Ben-Zvi and Sfard’s learning-teaching agreement and thus offer evidence that

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autonomous learning, contrary to the conceptions of it as requiring the learner to be
working alone, can, and in some situations must, arise when one student learns from

another.

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CHAPTER 6

COVERT LEARNING

Overview

This case offers a contrast to the case in the proceeding chapter. Jakeel was
explicit about his learning. As he enacted engaged learning, he identified as someone
who was interested in understanding and coming to know. Even as he enacted directed
learning, he identified as not knowing what to do. In contrast, the focal student in this
case, Minerva, enacted covert learning in which she identified herself as already
knowing. The words covert learning emphasize her need to learn even as she presents as
not needing to learn. Minerva’s enactment of covert learning raises important questions
about the roles of independence and knowing in autonomous learning.

As in the last case, I begin by summarizing the lesson and then describing the
outcome of Minerva’s learning. I then elaborate covert learning and describe the
identifying and mathematizing of covert learning before discussing the implications of

covert learning for autonomy and for the development of mathematical discourse.

Summary of the Lesson
As in the last chapter, this analysis of Minerva’s learning uses excerpts from
across the transcript of the lesson in her group. Table 6.1 summarizes the lesson and
indicates the position of the excerpts relative to the turns in the lesson. In the next few
paragraphs, 1 provide a summary of the main events in Minerva’s group.
The teacher dismissed the students to work in small groups. Minerva, Bonita, and
Jessica gathered at a cluster of four desks near the entrance to the classroom. After

reading the task card and gathering the materials for the task, they began work cutting out

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the figures. The teacher came to the group and asked questions about the area of the two
figures. During this conversation, Bonita explained that the area was two because two
triangles made a square. Minerva disagreed with this. Bonita and Jessica explained how
two triangles could make a square. Minerva constructed her first solution which consisted
of Figures H and I glued down intact and a sentence about area. Minerva’s work on this
first solution (and her next two solutions) is in Figure 6.1. The teacher came to the group
and told Minerva that her paper didn’t show the area. Minerva constructed a second
solution that had Figure H and the two triangles from Figure I arranged in a square. This
solution was missing the square from Figure I. In addition, Minerva had numbered Figure
H as one and the two triangles from Figure I as one, counting two altogether. The teacher
told Minerva that her solution didn’t prove anything and Minerva started over on her
third solution. This solution had all of the pieces from Figure I with the two triangles
arranged as a square. It also had Figure H intact. Minerva had numbered each square in
Figure H as one and indicated that there were two. As she numbered the pieces of Figure
1, she first counted each triangle as one, but she then erased these numbers and wrote “1”
across both triangles and indicated that the triangles and the square from Figure I were
two altogether.

The teacher told Minerva that this solution did not show the whole rug, so
Minerva started over for the fourth time. In this solution, she glued the pieces from
Figure 1 next to the pieces from Figure H constructing a large square. (See Figure 6.2)
She indicated that the area of the both figures together was four. She and the teacher
discussed this solution and Minerva was able to get the teacher’s permission to go to the

next task (involving Figures J and K). The teacher then realized that Jessica did not yet

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have the answer to the task with Figures H and 1. She instructed Bonita and Minerva to

help Jessica, which they reluctantly did. They all then started work on Figures J and K.

Minerva’s solved this task by cutting the small triangles and arranging them into squares,

placing Figures J and K next to each other, and counting each of the squares to determine

the area. This work is also displayed on Figure 6.2.

Table 6.1 Transcript excerpts and main events in lesson

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Line Excerpt Excerpt Summary Main Events
number Number
0 The teacher explains
the lesson .
The students move to
their small groups and
start work.
100
Excerpt 1 Bonita states two triangles make a square.
200 Minerva disagrees.
Excerpt 4 Bonita and Jessica explain about the
triangles. Minerva says “Nah huhn”. Minerva works on her
Excerpt 6 Solution 1. Minerva is done. The teacher first solution.
disagrees.
Minerva works on her
second solution.
300
Excerpt 7 Solution 2. The teacher evaluates Minerva’s
second solution.
. Minerva works on her
Excerpt 5 Bomta states, “They’re the same.” third solution.
Excerpt 2 Solution 3. Minerva states she’s gotta explain
it.
400

 

 

 

 

 

 

 

 

138

 

Table 6.1 con’t

 

 

 

Excerpt 8 The teacher asks how to show the whole rug.
Minerva says, “Put it together.”

 

Excerpt 3 Solution 4. Minerva talks about connecting it. Minerva works on her

 

500 fourth solution.

 

 

 

 

_ . The teacher approves
Excerpt 9 Minerva teaches J essrca. Bonita and Minerva’s

 

 

solution to Figures H
and I

 

 

 

 

 

600 Minerva, Bonita, and
Jessica work on
Figures J and K.

 

 

 

 

 

 

 

 

I" (flax: AI? 3 ‘l 5

rs b '

0" é new:
IE)

  
 
  
    

4rl‘8’lplC)
first

77 9 I46.)

‘2 xtuere

 

 

 

Figure 6.1 Minerva’s first three solutions to the task involving Figures H and I.

This is student work. The first solution is in the top center. The writing reads, “The Area
is 2 beacuse [sic] theres [sic] 1 hole [sic] square and 2 triangles that makes a square”.
Minerva’s second solution is on the bottom left. Her third solution is on the bottom right.

139

 

 

 

 

 

 

 

 

Figure 6.2 Minerva’s final work

This is student work. Minerva’s fourth solution is on the bottom right. Her solution to the
task with Figures J and K is on the top left. The writing toward the top of the page reads,

“14 Area”. The bottom right reads, “4 Area”. The writing in the triangle (bottom middle)
is “Area 2”.

Outcome of Learning
During this lesson, Minerva’s solution to the task changed several times,
becoming more and then less like the desired discourse. Across these different solutions,
one feature remained constant and was a change from her initial discourse: her statement
and/or demonstration (through rearranging the pieces of the figures) that two triangles

make a square. In the following sections, I present Minerva’s initial discourse, the

140

moment in which her discourse is most like the desired discourse, and then her final
discourse. Across these three samples of discourse, I will emphasize Minerva’s use of the

narrative of two triangles making a square.

Initial Discourse
At the beginning of the small group work, there was little evidence of Minerva’s
discourse around area. This first example occurred toward the beginning of the small

group work. Minerva has told the teacher that they need help because they don’t know

what to do.

Excerpt 1

191 Teacher So what does the question say
that we have to try to do?

192 Bonita Compare them

193 Teacher Let’s read that task again.

194 Bonita First compare rugs H and I. Bonita reads from the directions on
the task card.
195 Teacher How can I compare them do
you think?
196 Minerva They’re different shapes.
197 Teacher Yeah so how could I compare
them

198 Minerva The area

Minerva’s response of “area” as a way to compare Figure H and I was mathematically
appropriate. However, the response was not detailed or elaborate. As a result, it is
difficult to determine what Minerva was thinking about area. The conversation with the
teacher continued.

199 Teacher Okay you’re going to compare their area. How
could I compare it?

200 Bonita Both the areas, they both have two.

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201 Teacher How do you know?

202 Bonita Because two triangles make a square Bonita points at the
two triangles in
Figure I.

203 Minerva No it doesn’t

204 Bonita Uhhunh. This is an
aflirmative
utterance.

205 Jessica Yes it does! The teacher leaves
the group

206 Minerva Nah hunh This is a negative
utterance.

Minerva’s statement on Line 203 was a direct contradiction of Bonita’s statement that
“two triangles make a square” (Line 202). While Minerva did not elaborate upon the
reasons for her disagreement, it was possible that Minerva either disagreed that two
triangles made a square or disagreed that the information was helpful in determining the
area of the figures. In either event, at this moment it seemed that Minerva was not
thinking about how triangles and squares would be useful in determining the area of the

figures.

Approaching the Desired Discourse
Excerpt 2 occurred approximately 16 minutes after Excerpt 1. During that time,
Minerva produced two solutions to the task involving Figures H and 1. Her first solution
was primarily written. It included the statement “The area is 2 beacuse [sic] theres [sic] 1
hole square and 2 triangles that makes a square” along with Figures H and I glued as
whole figures with no alterations. In this solution, Minerva explicitly used the narrative
of two triangles making a square in her writing, but she did not demonstrate this narrative

with the figures. The teacher asked Minerva to “show her” how the area was two.

142

Minerva complied, generating her second solution. In this solution, she cut the

two small triangles from Figure I and arranged them as a square. She glued them next to

another copy of Figure H that she glued intact. She explained to the teacher, “One, two

equals two. One and one equals two” (Line 288). As she said this she pointed first to

Figure H and then to the two triangles joined as a square. She seemed to be suggesting

that the answer of two came from counting Figure H as one and the two triangles of

Figure I as another one, resulting in a total count of two. In this solution, Minerva

illustrated by cutting and arranging the triangles from Figure I how the two triangles

could be a square, but her numerical solution of two included all of Figure H and just part

of Figure 1, raising questions about what how she thought about area and what her

intention was in counting. When the teacher examined and critiqued this work, stating to

Minerva, “ There’s no proof of to me of how they’re [Figures H and I] the same or

different. How am I going to prove they are the same or different? Minerva. How am I

going to prove it?” (Line 337).

After this input from the teacher, Minerva worked on a third solution. She glued

Figure H on her paper and then spoke to Bonita as she wrote. (See the bottom right of

Figure 6.1 for Minerva’s final product.)

Excerpt 2

378 Minerva One two equals two. I gotta explain it for her. Minerva talks as she
writes on her paper.
She writes on Figure
H, “I ” in the top
square, and 1 =2 " in
the bottom square.

“Her " refers to the
teacher.

While Minerva did not describe her counting as determining the area of Figure H, her

activity suggested that she was counting the separate pieces of Figure H and determining

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a number for Figure H. After writing on Figure H, Minerva cut out and cut apart the
pieces of Figure 1. She glued the triangles on her paper and then the square. These shapes
were glued so that they touched, but they were not aligned into a rectangle. (See Figure
6.1.) Minerva wrote “1” on one triangle from Figure I and “1:2” on the other triangle.
Minerva did not describe or explain this work to anyone, so any interpretation relies
solely upon her actions. This initial writing seemed to be a count of each triangle from
Figure 1.

Minerva next moved her pencil to the square from Figure 1. She held the pencil as
if to write, but instead she hesitated, and then erased what she had written in the triangles.

‘61”

She wrote again on the triangles, drawing a across both triangles and writing “1:2” in
the square from Figure I. This erasing and rewriting suggests that when Minerva realized
she should count the triangles together as one and that the square and the joined triangles
counted as two. Like her second solution, this third solution provided an illustration of
how two triangles could make a square. Unlike her second solution, Minerva’s counting
in this solution demonstrated that she was counting these two triangles together in the
same way that she counted the single square. Thus in this third solution, Minerva used
both a picture and words to indicate that two triangles made a square.

Minerva’s writing “l=2” on the square from Figure I was similar to the writing
she just completed on Figure H (Excerpt 2), suggesting that she was counting the
“squares” in Figure I in the same way as the squares in Figure H. At this moment,
Minerva seemed to be indicating that Figures H and I were the same, that they had the

same count. While this comparison was part of the desired mathematical discourse,

Minerva was not explicit about what object she was counting, how this counting related

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to area, or whether the figures had the same area. Thus while this discourse had many of
the desired mathematical features, it still lacked some components of the desired

discourse.

Final Discourse

Minerva’s final solution to the task involving Figures H and I was constructed
almost 38 minutes into the small group time. The teacher critiqued her third solution,
telling Minerva that it didn’t “show what the area is of the whole rug. This is only part of
the rug” (Line 477). The teacher pointed to the bottom of Minerva"s paper (Figure 6.1) as
she said this. The teacher’s statement is hard to interpret because Minerva’s work
contains all of the pieces fiom both figures. It seems that the teacher was suggesting
Minerva did not have all of Figure I. The teacher may also have been trying to articulate
that Minerva should have the pieces of Figure 1 arranged together as a rectangle (like
Figure H) instead of adjacent to each other as Minerva had placed them. Minerva
suggested that she could show the whole rug by “put[ting] it together” (Line 478).

This next excerpt shows how Minerva understood her words “Put it together”
(Line 478). After the teacher left the group, Minerva constructed her final solution by
cutting out Figures H and I again. She put glue on the back of the square from Figure I
and placed the square so that it overlapped Figure H, sticking Figure H to the paper (See

Figure6.2). The teacher arrived at the group.

Excerpt 3

492 Teacher So this is H, right? The teacher points at
Minerva ’s paper.

493 Minerva No, that’s H. This is 1. Minerva points at the

shapes on her paper.

494 Teacher So I’m wondering why you have I.

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495
496
497
498
498
499
500
501

Minerva
Teacher
Minerva
Teacher
Minerva
Teacher
Minerva

Teacher

I’m connecting it

Why are you connecting it?
Because you told us to.
Did I say connect it?

Yes.

Did I say connect it?

Yes.

I said I want you to prove how this is. I didn’t
say connect it. If I said connect it, I was wrong.

Minerva groans and
pulls Figure H from
her paper. The
square from Figure 1
comes up with it.

Minerva interpreted the teacher’s request to show “the whole rug” to mean that she

should connect or glue Figure H and Figure I together. By telling Minerva that she was

wrong if she said to connect it, the teacher indicated that connecting was not what she

intended (Line 501). However, Minerva continued to arrange the two figures together and

then to determine the area as the count of both figures.

Minerva’s understanding that the two figures should be put together is apparent in

her final work this task and on the next task involving Figures J and K. The bottom right

portion of Figure 6.2 shows how she arranged the pieces of Figures H and 1. After the

conversation in Excerpt 3 above, Minerva arranged the pieces of figure I and wrote “Area

2” on one of the triangles. Minerva placed Figure H as shown in Figure 6.2. Minerva then

wrote “4 Area” to the right of the connected figures. Minerva’s positioning of the two

figures and writing “4 Area” indicated that she was thinking about the area as both

figures together.

Her solution to the next problem also reflects her thinking about area as both

figures. The work at the top left of Figure 6.2 shows how Minerva arranged the figures

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from the next problem. She cut the small triangles from Figure J and assembled them into
two squares. She then placed the pieces of both figures together and counted the spaces in
each, writing “14 Area.” The area of both figures together should be 16. It is not apparent
from the video how Minerva counts to arrive at 14, but it is apparent that she counted
both figures in determining her number.) In both of these solutions, Minerva
demonstrated that two triangles make a square through her arrangement of two triangles
into squares and her counting of the triangles together as one.

A last example of Minerva’s understanding of area as joining the two figures
together is Minerva’s directions to Jessica. Jessica had Figure H glued on her paper and
was trying to determine what to do with the pieces of Figure I. Jessica had the square
from Figure I in her hand. Minerva told her, “Connect it to the two squares” (Line 585).
Minerva meant for Jessica to put the square from I next to the two squares in Figure H
already glued on Jessica’s paper. While Minerva does not specifically tell Jessica to count
the squares in both figures together, her command to Jessica, coupled with her use of the
“4 Area” and “14 Area” in her work shows that Minerva was thinking that area needed to
be both figures together.

At the start of the lesson, Minerva did not seem to understand that putting two
triangles together into a square could be an important piece of solving the task of
comparing the areas of the figures. However, she quickly incorporated that narrative into
her talk about and her work on area: Each of her solutions demonstrated that information
either in writing or in her arrangement of the pieces of the figures. Her use of “two
triangles make a square” demonstrated her learning. In addition, it demonstrated learning

that was an important part of the desired discourse: This narrative and Minerva’s use of it

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gave her a means of translating all of the pieces of Figure I into the same unit, which
could then be counted. However, Minerva’s final discourse also included a routine that
was not part of the desired discourse. She seemed to learn that the arrangement of the
pieces of both figures was important: It was important to connect the figures and count
them together to determine area. From Minerva’s actions, it is difficult to determine how
joining the two figures was related to her understanding of area. It is possible that she
interpreted area as a mathematical operation like addition in which separate pieces (in
this case from the two figures) were combined and counted. Whether this was her
understanding, it is clear that her final discourse, while more mathematical in some ways,
was mathematically troubling in others.

I would like to suggest that this final discourse, both the desirable and undesirable
components, was an outcome of Minerva’s mathematizing and identifying activities
during the lesson. In the next section, I describe Minerva’s learning activities and begin

to explain how they might account for the change in her discourse.

Learning Activity: Covert Learning

The word covert suggests something that is concealed, hidden, or disguised.
Covert learning is meant to imply that learning is happening without the learner openly
identifying as a learner or explicitly identifying someone as a teacher. Someone who is
engaging in covert learning is attempting to conceal their need to learn and their activity
of learning. The engaged learning Jakeel enacted is in contrast to covert learning. In
engaged learning, the learner identifies and is identified by others as someone who is
working to make sense of discourse and activity. In directed learning, the learner

identifies/is identified as someone who needs to be told what to do. In contrast, covert

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learning means that the learner recognizes (even if only tacitly and not explicitly) the
situation as one in which he/she can/needs to learn but that the learner does not want to
explicitly identify as a learner. He/she does not want others to position him/her as a
learner. Thus the learner makes moves that simultaneously communicate that he/she is
not a learner while also watching and listening to others in order to develop his/her own

ideas.

T wo Illustrations of Covert Learning

Throughout this lesson, Minerva engaged in covert learning: she did not explicitly
identify herself as a learner or another as a teacher and yet her subsequent actions
indicated a change in her discourse. In some instances, the change in discourse might be
attributable to an interaction with or between others. In other instances, Minerva seemed
to be developing her own discourse. The most apparent example of covert learning as a
result of an interaction with others occurred at the beginning of the small group work and
was included above as Excerpt 1. The teacher asked Bonita how she knew that the areas
of both figures are two (Line 201). Bonita responded, “Because two triangles make a
square” (Line 202). Minerva, without hesitation, exclaimed, “No it doesn’t” (Line 203).
Minerva’s disagreement was emphatic and explicitly directed at Bonita’s statement. It is
unclear what Minerva finds objectionable about Bonita’s statement. However, it is clear
from Minerva’s tone that she does not agree with Bonita’s statement.

As the interaction continued, Minerva’s tone changed. In Line 206, Minerva
continued to articulate disagreement with her peers. However, her utterance (“Nah
huhn”) was not as pointed or articulate as in Line 203. Instead, her “Nah huhn” had a

more generalized, mocking tone. Minerva did not argue a specific point, but instead

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offered a more generic counter to Bonita and Jessica’s insistence that Bonita’s first

statement was correct. Minerva continued to respond to Jessica and Bonita with “Nah
huhn”:

Excerpt 4
207 Jessica Yes it does!

208 Minerva Nah hunh

209 Jessica Let me see, let me show you

210 Bonita You cut this right off Bonita is holding Figure I. She points
to one triangle in Figure I
and put it there and motions next to the other triangle
in the figure.

In Lines 209 and 210, Jessica and Bonita both explicitly positioned Minerva as a learner
and themselves as teachers. Jessica’s “Let me show you” and Bonita’s demonstration of
how to rearrange the figure communicate their understanding that Minerva did not
understand something they both understood.

211 Minerva Nah hunh

212 Jessica Let me see. See. Look it. Let me Jessica takes Bonita ’s Figure I and

show her that it can make a her Figure 1.
triangle.
213 Minerva Nah hunh Minerva watches Jessica.
214 Jessica See. Look it. She places them together to make a
large square.

215 Minerva Nah hunh, nah hunh, nah hunh
216 Jessica Stop it.

217 Minerva Nah hunh Minerva puts glue on the back of
Figure 1.

Again in Lines 212 and 214, Jessica positioned herself as a teacher and Minerva as the
learner. During this interaction, Minerva watched Jessica and Bonita as they each

explained how “two triangles make a square”. While Minerva’s “Nah huhn”

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communicated that she disagreed with their utterances, the timing of Minerva’s “Nah
huhn” indicates that she was attending to what Bonita and Minerva were saying. She was
not interrupting their words, but responding (albeit negatively) to each of their
statements. Her “Nah huhn” was indiscriminate: She responded in the same way to each
statement of Jessica and Bonita’s regardless of the information, tone, or type of
statement: Jessica’s demonstration of “two triangles make a square” (Line 214) received
the same response from Minerva as her request that Minerva stop her utterances (Line
216). Minerva continued to respond to Bonita and Minerva with “Nah huhn” for a full
minute after the interaction above, uttering this phrase 32 times in total. Jessica and
Bonita both asked her to stop, but Minerva continued in spite of their repeated requests.
The repeated and indiscriminate qualities of Minerva’s “Nah hunh” suggest that
she was not communicating disagreement with the specific details of Bonita and Jessica’s
discourse. Instead, the “Nah huhn” seemed to be an attempt to portray her initial
disagreement in Line 203 as a jest or a joke. It seemed to suggest mockery of the serious,
explanatory tone set used by Bonita and Jessica. By disagreeing or denying everything
Bonita and Jessica said, Minerva’s initial negative exclamation (“No it doesn’t” Line
203) became more trivial. In addition, Minerva’s “Nah huhn” seems to send the message
that she does not need or want Bonita and Jessica’s explanations. In spite of Minerva’s
disagreement with Bonita’s statement that “two triangles make a square”, Minerva
adopted this narrative in her subsequent discourse. She incorporated these words into her
first solution to the problem (See Figure 6.1 above) and she repeated these words on two
separate occasions (Excerpt 5, Line 359 below, and 457, not included in this document).

Also, with the exception of her first solution, each of Minerva’s physical solutions to the

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task included the arrangement of the two triangles into a square. Minerva’s use of
Bonita’s narrative and arrangement of the triangles suggests that Minerva learned from
Bonita even while she professed disagreement with Bonita and Jessica’s statements.
Before continuing to elaborate covert learning, I want to address the possibility
that Minerva’s initial disagreement (Line 203) might have been the result of
misunderstanding or misspoken words. Perhaps Minerva understood and agreed with
Bonita but misspoke in the moment of Line 203. This scenario is unlikely for several
reasons. First, Minerva was absent during the previous lesson when the class first worked
on determining the area of triangular figures. Thus it is unlikely that Minerva had
recently (or possibly even previously) attempted to resolve the question of how to count
triangular spaces. Second, Minerva’s discourse prior to Bonita’s statement did not
indicate that Minerva had a solution to the task or recognized that she needed to account
for the triangular spaces in Figure 1. She had asked the teacher for help (Line 184) and
stated that she didn’t know what to do (Line 186). In addition, Bonita statement about the
two triangles (Line 202) was the first time triangles had been mentioned in this group. It
seems unlikely that Minerva was already thinking about how to work with the triangles.
Finally, if Minerva misunderstood Bonita or misspoke, it seems that a more effective
recovery would have been to offer her own explanation of how two triangles make a
square, as an illustration that she really did understand. Instead, Minerva’s “Nah huhn”
were timed so that they punctuated, but did not interrupt the explanations of Bonita and
Jessica. In addition, Minerva watched and listened to Bonita and Jessica. If she had been

less interested in what they were saying and doing, she could have been more dismissive

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and turned away from them. Instead, she attended to their words and actions and only
began to work on the problem after their explanations.

It seems that Minerva wanted to communicate that she didn’t need Bonita and
Jessica’s explanations while at the same time making sure that she had an opportunity to
hear and watch those explanations. It is this combination of sending a message of denial
about learning while also watching to see what others do and say that suggests Minerva
was attempting to conceal her learning.

Another example of covert learning occurred after Minerva completed her second
solution. The teacher was sitting with Minerva’s group asking questions about the task.
She asked them first about the area of Figure H. Everyone in the group agreed that the
area was two. She then asked about Figure 1. Before presenting this excerpt, I want to
comment that I will have more to say later about Minerva’s affect during this excerpt.
Her depressed tone is related to her moves to conceal her learning, but it is more relevant
to an earlier interaction than to the interaction below.

Excerpt 5

354 Teacher Okay. What is the area of Figure
I?

355 Bonita Two
356 Teacher How do you know? She looks at Bonita.
357 Jessica One

358 Teacher Tell me why it’s one She touches Jessica ’s arm.

359 Minerva Two triangles make a square. Minerva uses very little inflection as
she says this. She is supporting her
head with her hand. She is looking at

the desk.
360 Bonita Two triangles make a square.
361 Teacher So then what does that mean?
362 Minerva There’s another square there. Minerva puts her head down on her

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arm.
363 Bonita They’re the same.

The contrast between Minerva’s statement in Line 362 and Bonita’s statement in Line
363 is important. In her statement, Minerva communicated that the narrative about
triangles and squares was important because putting the triangles together made another
square. Minerva had demonstrated her understanding of this in her second solution when
she glued the two triangles together into a square. However, Minerva had not yet
articulated or demonstrated that Figures H and I were the same. Neither of her solutions
at this point compared the two figures. In contrast, Bonita’s statement reflected a
comparison of the two figures. Because Minerva had not yet articulated a comparison of
the figures, she might learn from Bonita’s narrative.

364 Teacher So how can I show it on my
paper that they’re the same?

With this statement, the teacher took up Bonita’s discourse and indicated that talking
about the figures as the same was appropriate.

365 Bonita Write it down?

366 Teacher What materials do you have that
would show you, help you to
prove it?

367 Bonita A pencil
368 Teacher How do you think, Jessica?
369 Jessica Um

370 Teacher How could I show those two are The teacher pulls out Jessica ’s paper.

the same? Jessica looks at it.
They said they’re the same. The teacher talks to Jessica.
Maybe they’re not the same.

You said they’re not

Again the teacher reinforced the narrative of the figures as the same. She then suggested

to Jessica that they might not be.

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371 Bonita [They are Minerva has picked up her scissors
and is cutting.

372 Minerva [They are. God

This statement was the first time Minerva stated that the figures were the same, indicating
a change in Minerva’s discourse during this interaction. This change was subtle, so it
could be argued that Minerva’s utterance in Line 372 did not indicate learning, but was
instead merely the first time she spoke these words. However, the notion that the two
figures were the same was a significant change fiom her earlier solution in which she
counted Figure H as one and the triangles from Figure I as one and then indicated that
together they were two. As discussed earlier, this solution did not include the square from
Figure I and did not compare the two figures. Describing the figures as the same was a
shift in discourse from this solution that counted the two figures together.

In addition, these words were more than a mimicking of Bonita. Minerva
illustrated that the two figures were the same in her next solution. In this third solution,

‘61,,

Minerva marked in each of the squares of each figure and indicated that both figures
were two (See Figure 6.1). Rather than count the two figures together, Minerva showed
how they both had the same number, demonstrating that they were the same. This change
from counting together to counting the same provides further evidence that Minerva
changed her discourse or learned during this interaction.

In spite of this learning, Minerva did not position herself as a learner during this
interaction, nor did anyone else position her as a learner. Instead, she portrayed herself as
knowledgeable and confident, communicating the expert response (Two triangles make a

square) to the question of why the area of Figure I was two (Line 359) and taking up the

approved discourse that the figures were the same (Line 372). Just as in Excerpts 1 and 4

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(with the “Nah huhn”), in this excerpt, Minerva simultaneously communicated that she

did not need to learn while also learning, again demonstrating covert learning.

Identz'fizing the Audience

Minerva’s audience was a key feature of enacting covert learning. She was very
careful about her performance, working to convince her audience that she was
knowledgeable. Presenting as knowledgeable meant that she could successfully complete
the assigned work, a decision that was in the hands of the teacher. Thus teacher was the
primary member of Minerva’s covert learning audience. Minerva’s peers were important
for the ways in which they were explicitly not members of her mathematizing audience.
Minerva was also rarely an audience for herself. I begin my exploration of the audience
for Minerva’s covert learning by describing her framing of the teacher. I then describe

her peers and herself as audience members.

Teacher as Gatekeeper to Being “Done ”
Minerva positioned the teacher as the person to persuade that she had finished her
work. This was evident early in the small group work on the task. Minerva finished her

first solution and announced this to the teacher:

Excerpt 6

257 Minerva Done Minerva smiles at the teacher and
writes her name on the top of her
paper.

Minerva’s announcement and her smile indicate her confidence that she had completed
the task. Minerva did not attempt to explain her work or her thinking. Instead, she wanted
the teacher to know that she was finished. The teacher examined her paper.

258 Teacher How is this? I can’t tell anything She points at Minerva 's paper.
about the area by this, Minerva

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259 Minerva I said the area’s two. She writes “2 " under the copy of
Figure I on her page.

In spite of the Minerva’s pronouncement that she was done, the teacher asserted that
Minerva had not yet answered the question (Line 258). Minerva might have replied to the
teacher by explaining her thinking about area to the teacher. Instead, she emphasized
what she had done and then added to her work (Line 259). Minerva could have invited
the teacher to inquire about how Minerva knew the area was two. Instead, Minerva
continued to position the teacher as the person to persuade that she was done, rather than
as someone interested in her thinking.

The teacher disagreed:
260 Teacher But how can you show me by

this, how can you actually show
me?

261 Minerva Cut it out. The teacher leaves the group.
Minerva picks up a paper with H and
I and starts cutting.

The teacher’s words in Line 260 support Minerva’s enactment of covert learning: The
teacher persisted with her message that Minerva’s solution was inadequate. However, the
teacher did not position Minerva as someone who needed to learn. Instead she implied
that Minerva had more to show her. Minerva interpreted the teacher’s words as a request
for physical activity rather than for explanation of her thinking. She stated that she would
cut (Line 261) to show how the area was two. Minerva’s response also indicated that she
had stopped trying to convince the teacher that she was done (which was a change from
Line 259). Instead, she accepted the teacher’s message that she had more work to do (not
more or different explaining to do) and laid out what work she should do.

While this conversation was short, it demonstrates Minerva’s emphasis on getting

done and her framing of the teacher as someone to be persuaded that she was done. This

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emphasis is consistent with covert learning. Minerva presented herself as knowledgeable.
She did not indicate that she didn’t understand or ask for the teacher’s help in making
sense of area. Instead, her final response suggested that she knew both the answer and
what more she needed to do to satisfy the teacher. The teacher’s words can also be
interpreted as agreeing with Minerva’s positioning: Minerva did not need to learn from a
peer or offer a better explanation. She needed to do more to show her solution.

The next time Minerva completed her solution to the task, the teacher’s evaluation
was more critical. Minerva had finished her second solution (See Figure 6.1) when she
received the following feedback from the teacher.

Excerpt 7

335 Teacher So what are you trying to do There is no answer.
with these two rugs?

What are you trying to do with There is no answer.
these two rugs?

Okay I’m trying to compare the Minerva picks up her pencil and

area and see if they are the same writes “I” on the two triangles on the
or if one is bigger than the other. bottom left of her paper.

And you need to use these

shapes, but you need to prove it

on your paper.

336 Minerva Idid!

With this response, Minerva expressed both confidence that her answer was complete and
disagreement with the teacher’s evaluation. As in the example above, both Minerva and
the teacher identified the teacher as someone who evaluated whether Minerva had done
what she needed to do.

The teacher continued to evaluate Minerva’s work:

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337 Teacher This does not prove anything to She points at various parts of
me. This is H. This is H. This is Minerva ’s paper.
I. I don’t even know what this is.
There’s no proof of me on here
which one is,

Do you think this is the same as She points at the two triangles

this because I don’t. There’s no assembled as a square and then at
proof of to me of how they’re Figure H. Minerva writes “I” on one
the same or different. How am I of the two triangles on the bottom left
going to prove they are the same of her paper. Minerva drops her

or different? Minerva. How am I pencil and puts her hands on her
going to prove it? face.

338 Minerva I don’t know.
By stating “Do you think this is the same as this because I don’t” (Line 337), the

teacher indicated that Minerva’s discourse was not the teacher’s desired discourse and
positioned Minerva as someone who needed to change what she had done. Again, the
teacher did not explicitly label Minerva as a learner: She did not tell her that she wasn’t
making sense or that she needed to ask her peers for help in understanding the task.
Instead her questions “How am I going to prove it?” implied that Minerva could generate
an acceptable solution. Thus while the teacher did declare that Minerva had not
adequately completed the task, she did not explicitly position Minerva as needing to

learn.

Segue to Peers are Not Teachers

Minerva’s response, “I don’t know” (Line 338), was part of a pattern of
responding to the teacher’s feedback on her work. Minerva received negative feedback
from the teacher on four of her solutions to the problem. Her response to the teacher on
three of those four occasions was to state that she didn’t know what to do, just as she did
in Line 338 above. (The exception was her response after her first solution as elaborated

above with Excerpt 6.)

159

Minerva’s tone as she spoke her “I don’t know” was depressed and deflated. She
was slumped over her desk with her hands on her face. This response might be seen as
reasonable in light of what could be described as harsh criticism from the teacher. The
teacher did not cushion her feedback to Minerva. Instead she was direct in her
disagreement, stating that Minerva offered “no proof” of how H and I were the same or
different (Line 337). It is possible that the depressed tone of Minerva’s response was in
reaction to the teacher’s harsh tone.

However, Minerva’s depressed response may also be tied to her covert learning
moves. First, as part of Minerva’s portrayal of herself as someone who did not need to
learn, it was important to be done with the task. The teacher’s feedback indicated that not
only was Minerva not done with the task, but also that her solution was wrong. The
teacher was unequivocal in pointing out that she did not agree with Minerva that the two
triangles and Figure H were the same (Line 337). In contrast to the teacher’s feedback on
her first solution, Minerva needed to do more than elaborate her solution: She needed a
different proof. Not only was Minerva not done with the task, but the moves she had
made to covertly learn what to do to solve the task had been inadequate.

Another reason for Minerva’s depressed response may have been that she could
not see a way out of her situation that would allow her to continue to overtly deny her
need to learn. The teacher implied that Minerva could generate a solution (Lines 335 and
337), but Minerva’s “I don’t know” indicated that Minerva did not have another solution
in mind. She was unwilling to admit that she needed help or that she didn’t understand

which left her with very few options for generating a new strategy for solving the task.

160

This difficulty demonstrates Minerva’s reluctance to position her peers as part of
her audience. If she had been willing to acknowledge that she was a learner and that her
peers might have something to teach or might be able to learn with her, she could have
turned to them to talk about the mathematics of the task. However, Minerva’s enactment
of covert learning meant that such an overt display of not-knowing was not possible.
Instead, Minerva was eventually able to construct another solution for the task by
attending carefully to interactions between the teacher and Bonita (See the discussion
above accompanying Excerpt 5). This sequence of moves surrounding this second
solution was repeated for Minerva’s next solution. She generated the solution, received
feedback from the teacher, insisted she was correct, responded with despair, and then
covertly learned from interactions between other students and the teacher.

Minerva’s reluctance to position others as her teacher was also evident from her
lack of questions during the lesson. Minerva asked only two questions related to the task
throughout the whole lesson: Very early in the small group work, Bonita stated that they
had “to know how what rugs can cover up the floor” (Line 90). Minerva responded with
“Rugs?” indicating that she wanted clarification that they were talking about rugs. Then,
a short time later, Minerva asked the teacher, “What do we do? We need help.”
Throughout the lesson, Minerva did not ask for any other clarification of the task or of
mathematical content. She did not ask for explanations or definitions. Any of these types
of questions would have identified her as a learner and the person to whom the question
was directed as her teacher. Indeed, in this second question, the one time Minerva asked
for help, she used “we” rather than “1”, providing additional evidence that she did not

want to position herself alone as a learner in need of help. Minerva’s lack of questions

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was consistent with her enactment of covert learning and her identification of others as
not her teacher.

Minerva’s reluctance to engage in peer teaching also included her reluctance to
serve as the teacher of others. For example, when Minerva started on her second solution,
she began to cut out another copy of Figures H and 1. Jessica noticed this and asked,
“Gotta cut another one out? Why are you cutting another one out?” (Line 263). Rather
than respond to Jessica’s question with an explanation, Minerva dismissed her question,
saying, “Because I’m doing something” (Line 264). This response ended their
conversation. (This conversation is analyzed further in the next chapter.) It is possible
that Minerva did not yet have the words to explain what she was doing or why to Jessica
or it may be that Minerva was embarrassed or disappointed that she had not yet finished
the problem and did not want to be slowed down by explaining her work to Jessica. It is
also possible that Minerva resisted teaching because she worried that teaching would
expose her lack of understanding and show others that she actually needed to learn.

Regardless of her motivation, Minerva was unwilling to explicitly serve as a peer
teacher. Minerva’s reluctance to teach was displayed again later in the lesson. After
Minerva had constructed her final solution to the task with Figures H and I, the teacher
demanded that Minerva (and Bonita) help Jessica. The teacher had to discipline Minerva
to get her to begin to help Jessica. Minerva’s teaching was minimal. Instead of
encouraging Jessica to explain her thinking, asking questions that would help Jessica
understand the task, or inviting Jessica to ask her questions about the task, Minerva
directed Jessica’s actions with short ambiguous commands. Jessica was able to complete

the task to Minerva’s satisfaction, but she needed the support of Bonita and the teacher to

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do this work. I describe this interaction in more detail in the next chapter, but for now, I
want to make the point that as Minerva engaged in covert learning, she was reluctant to

identify herself as a teacher and others as learners.

Not an Audience for Herself

One final consideration for Minerva’s audience is whether she identifies herself as
a member of her audience. I noted in the previous chapter that Jakeel’s identification of
himself as an audience for his discourse seemed to support his learning. Thus it is
important to consider whether Minerva seemed to talk to herself as she mathematized.

On one occasion, Minerva’s gestures indicated that she seemed to have herself as
an audience. This occasion occurred as Minerva was working on her third solution, the
solution in which she marks a “l” in each square of Figures H and I (See Figure 6.1).

‘61”

Minerva first glued down Figure H and then wrote in the top square and “1:2” in the
bottom square. Minerva then glued the two triangles from Figure I and finally the square
from Figure I. Here is my description of Minerva’s actions after she glued the pieces
from Figure 1:

She picks up her pencil and writes a “1” on one triangle from Figure I and

“1:2” on the other triangle. She moves her pencil to the square from

Figure I and hesitates. She then erases what she wrote in the triangles and

draws a “1” across both triangles and writes “1=2” in the square from

Figure I. She puts her pencil down. (Line 379).
Minerva’s initial writing on the triangle (“1” and “1=2”) suggests that she was thinking
about counting each triangle as one. However, when she went to write on the square, she
changed her mind and marked both triangles together as 1. Writing “1:2” on the square

suggests that she was counting the two triangles in the same way that she was counting

the square, with the result that there were two. The hesitation and then change in

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Minerva’s writing suggests that she was an audience for her discourse. While she did not

utter or write words, the change she made in mathematical symbols indicates that she was
asking herself whether her communication made sense and then changing what she wrote
when she found it lacking.

The solution Minerva constructed as she communicated with herself was the most
mathematically correct solution she generated. She indicated that both figures could be
counted in the same way to arrive at the same number, 2. While I cannot conclusively
connect Minerva’s thinking to herself with her increase in mathematical discourse, it is
interesting that the two coincide.

Minerva does not provide other clear examples of communicating with herself.
While this does not exclude the possibility that Minerva talked through her ideas in less
visible ways, it does suggest that exploring her thoughts was not a frequent enough or
central enough activity for her to engage in it in explicit ways. This is consistent with her
enactment of covert learning: Minerva did not explicitly identify as a learner. She focused
on getting done, convincing the teacher that she was done, and, when the teacher was
unconvinced that she was done, using her peers as sources of ideas that she could use to

generate new ways for getting done.

Mathematizing
In order to get done with the task, Minerva needed to demonstrate mathematizing
that satisfied the teacher. To do this, she adopted discourse, produced discourse, and

relied upon others to substantiate discourse.

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Adoption of Discourse

Engagement as a covert learner required that Minerva adopt the discourse of
others. She was not open about this adoption: Unlike J akeel, she did not ask questions or
appear to listen to explanations. However, the changes in her discourse provide evidence
that she attended to and then used the discourse of others. When I first introduced
Minerva’s covert learning, I discussed in detail two examples of Minerva’s adoption: her
use of two triangles make a square (See the discussion of Excerpt 4) and they ’re [both
figures] the same (See the discussion of Excerpt 5). In both of those examples, Bonita
used discourse that Minerva later exhibited.

Another example of adoption of discourse is Minerva’s use of “the area is 2”.
Bonita was the first person to announce the area of Figures H and I. In Excerpt 1, Line
200, Bonita stated, “Both the areas, they both have two.” Minerva seemed to adopt this
statement, using a variation of it in each of her solutions to the task involving Figures H
and I. For example, in her first solution, she wrote, “[t]he area is 2”. In her subsequent
solutions, she emphasized counting to two. This statement seems to be something
Minerva adopted rather than something she produced because the statement seemed to be
a starting point for Minerva’s solutions rather than the result. In her first solution, she
started her written statement with “[t]he area is 2” and then proceeded to elaborate how
the area could be two, but without explicitly connecting these words to the figures: “The
area is 2 beacuse [sic] theres [sic] 1 hole square and 2 triangles that makes a square”. Her
next solution involved counting Figure H as one and the two triangles from Figure I as
two and then declaring that the perimeter was two (Line 288, not included in this

document). While I believe that Minerva meant area when she said perimeter, what she

165

counted was not the area of either figure. Instead, she seemed to know that she needed to
reach an answer of two and she was attempting to count different things to demonstrate
that answer. Because Minerva could not initially demonstrate “the area is two,” I believe

that she adopted this statement from Bonita.

Production of Discourse

In addition to adopting discourse, Minerva also produced discourse. However, in
many instances her production of discourse did not reflect faithful adoption of another’s
discourse. Instead of building from discourse she adopted from others, Minerva’s
production of discourse as she enacted covert learning was, at times, not consistent with
others’ discourses or the desired mathematical discourse.

For example, as I described in the section on Minerva’s final discourse, she
demonstrated a routine for determining area that involved cutting apart the pieces of the
figures, joining triangles in pairs to make squares, connecting all of the pieces of both
figures, and counting all of the squares of both figures to determine area (See Excerpt 3
and Figure 6.2). This routine had not been demonstrated by Bonita, Jessica, or the
teacher. Instead, Minerva seemed to generate it in response to the teacher’s evaluation of
her third solution.

Minerva’s third solution had the two triangles of Figure I joined as a square. Next
to this square, Minerva glued the whole square from Figure I. She also glued Figure H
intact nearby (See Figure 6.1 and the discussion accompanying Excerpt 2). The teacher
looked at Minerva’s work and told her, “But this doesn’t show me anything” (Line 431).
The teacher then asked Jessica several questions before turning back to Minerva’s work.

The teacher spoke to Minerva:

166

Excerpt 8
477 Teacher What did you do on your paper that Minerva points to the
showed that two triangles equaled one? bottom of her paper,
perhaps to the two
triangles assembled
into a square.

So how could I show, what the area is of The teacher points to
the whole rug. This is only part of the rug. the bottom of
Minerva ’s paper.

478 Minerva Put it
[together

After this interaction, Minerva cutout Figures H and 1, arranged the pieces so that they
were connected, and counted all of the squares.

When the teacher asked her why she was doing this, she told the teacher that she
was “connecting it” (Excerpt 3, Line 495) “[b]ecause you [the teacher] told us to”
(Excerpt 3, Line 497). However, the teacher had not told Minerva to connect the pieces.
Instead, the teacher had asked Minerva how she could show the area of “the whole rug”
(Line 477). Minerva had translated the request to show the area of the whole rug into a
request that she put all of the pieces of both figures together.

The teacher’s confusion over why Minerva was connecting pieces of the two
figures suggests that Minerva’s production of discourse was not based upon a shared
understanding of the teacher’s comments about the whole rug. Minerva’s reluctance to
ask questions, talk about her work with others, or seek explanations prevented her from
asking the teacher for clarification about what she wanted or what she meant by the
whole rug. She also did not explain her thinking about her solution and so the teacher
could not build fi'om Minerva’s thinking to convey her concerns about Minerva’s

solution. Instead, Minerva seemed to assume that she understood both the teacher’s

167

concern with her work and the meaning of the whole rug and she produced a routine that
addressed that concern.

Minerva’s production of discourse was similar to J akeel’s production in that she
did some translation into her own words. For example, Minerva used “connecting”
(Excerpt 3, Line 495) which was not a word used by anyone else in her group, including
the teacher. Minerva, like J akeel, also counted without verbally indicating what object
she was enumerating (See for example Excerpt 2 and the accompanying description of
Minerva’s writing). Significantly, however, Minerva’s production of discourse differs
from J akeel’s in that Jakeel’s discourse was clearly tied to a visible object while in most
instances, Minerva’s discourse lacked explicit links to images or gestures. For example,
in Excerpt 8, Minerva states, “Put it together” (Line 478). This statement was not an echo
of words uttered by anyone else during this lesson: They seemed to be Minerva’s own
production. However, Minerva’s understanding of “it” is not clearly conveyed by any
gesture and, as becomes evident in her final solution (discussed above), Minerva’s
understanding of it was quite different fi'om the teacher’s.

As another example, while directing Jessica on what to put on her paper, Minerva
also used pronouns and did not clarify her words with gestures. The teacher had just

asked Jessica how she knew the area of Figure I was two.

Excerpt 9.
555 Jessica This is two because if you put two triangles As Jessica says
together that’s a square. triangles, she claps
her hands together
over her head.
556 Teacher But where on your paper does it show me

that?
557 Minerva Gosh, Jessica, put it together.

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558 Teacher Where on your paper does it show me that? With both hands,
Jessica pats
Figures H and I
that she has glued
on her paper.

But that doesn’t show me that.
559 Minerva Put it on there

Minerva’s statements in Lines 557 and 559 were not connected any visible object.
Minerva did not she motion with her hands to indicate what object Jessica might put
together or put on there and Minerva’s words were too vague to point to an object by
themselves. “Put it together” in Line 557 might have been a reference to the two triangles
mentioned by Jessica in Line 555. If this is what Minerva intended, then “it” should have
been “them” in order to refer to the two triangles. Rather than support Jessica in
understanding her intentions, Minerva’s directives in Lines 557 and 559 required Jessica
to make connections without providing enough information to be sure that Jessica would
think about the same object as Minerva. This example was one of fifteen utterances by
Minerva where her words either directed someone to do something or described what she
had but without any accompanying gesture that would help more precisely communicate
Minerva’s message. In contrast, there were only three occasions in which Minerva used
gestures to specifically indicate the object of her talk.

While Minerva produced her own discourse, this discourse was usually not
explicitly connected to an image, leaving the object of her discourse unclear (for both the
researcher and her listeners). This ambiguity could support Minerva’s enactment of
covert learning as it allowed her to produce discourse that could be interpreted in a
variety of ways. Her listeners, including the teacher, could make assumptions that

Minerva was talking about the same object they referred to or were thinking about. Thus

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Minerva could appear to be knowledgeable without needing to be explicit about what she

knew.

Substantiation of Narratives

The final aspect of Minerva’s mathematizing I will probe is her substantiation of
narratives. Substantiation is how the learner decides whether a narrative reflects the state
of affairs (Sfard, 2008). For Minerva, whose focus was on appearing knowledgeable and
finishing the task, it was important to use narratives that accomplished these two goals.
Thus, for Minerva, narratives reflected the state of affairs if they seemed useful in
completing the task. Thus, her substantiation depended upon her interactions with the
teacher.

Her reliance upon the teacher for substantiation is evident in the interactions
around Minerva’s work. Minerva relied upon the teacher to determine whether her
solution was adequate (See Excerpts 6 and 7). She would craft a solution, the teacher
would critique it, and Minerva would start over with another solution. If Minerva’s
substantiation relied upon her own explanation or investigation of the narratives in her
solution, she might have explained her solutions to the teacher, arguing for her solution
and how it met the teacher’s expectations, rather than only insisting that she was done.
Furthermore, Minerva started over with each solution. Rather than build on what she had
written or the figures she had glued, she cut out and glued additional copies of Figures H
and I and wrote new text for each of her four solutions. It is possible that Minerva felt
each solution was constructed from a logically flawed foundation and needed to be
restarted in order to correct this flaw, but there was no moment in which she appeared to

evaluate her solutions and she did not state what might be wrong. Finally, Minerva’s first

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and third solutions had mathematically appropriate foundations. If she had been logically
evaluating her solutions, looking to substantiate them based upon mathematical
explanations, she might have realized that she had appropriate beginnings that needed to
be elaborated rather than abandoned. Instead, she seemed to be relying upon the teacher
to indicate whether her work, including any associated narratives, reflected the
appropriate mathematics.

Relying upon the teacher to substantiate narratives and approve her work is
consistent with Minerva’s enactment of covert learning. The need to appear as knowing
precludes activities that would demonstrate uncertainty or learning. Thus as Minerva
enacts covert learning, she cannot engage in activities such as exploring or verifying a
narrative. Instead, her decisions about whether narratives reflect the state of affairs relies

upon her observation of how others, and in particular the teacher, respond to narratives.

Autonomy

At first blush, it seemed that Minerva enacted autonomous learning. She was
focused on the mathematical task. She worked persistently and independently toward a
solution and she was creative in generating multiple solutions. However, examination of
her audience and her mathematizing suggest that very few of her activities could be
described as autonomous learning.

The moment in which her mathematizing and identifying mostly closely
resembled autonomous learning was as she worked on her third solution. I describe this
in detail in the earlier section on Minerva as an audience for herself. As Minerva worked
on this solution, there was a moment in which she hesitated and then changed her writing.

In this moment, it seemed that she observed a problem with what she had written and

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then worked out how to change her discourse to address this problem. Her activities
reflected autonomous learning in several ways. She identified herself as the audience for
her discourse. It seemed that she attempted to substantiate her first narrative. When she
found it lacking, she constructed a new solution that was logically consistent and drew
upon narratives of triangles, squares, and area used by others. Unfortunately, the teacher
did not recognize the ways in which the solution was mathematically appropriate and
Minerva did not argue for or explain her work. Instead of continuing to act in ways that
reflected autonomous learning, Minerva enacted covert learning and relied upon the
teacher to approve her work.

As Minerva enacted covert learning, she did not demonstrate activities of
autonomous learning. She did not position her audience as interested in explanations,
investigations, or co-leaming. She did not appear to be an audience for herself. As she
enacted covert learning, she adopted some discourse and produced discourse, but she did
not investigate either what she adopted or what she produced. As a result, she did not
always use narratives and routines in ways that were consistent with other’s use or that
were mathematically appropriate. Her substantiation of discourse relied upon the
teacher’s approval rather than explanations. In summary, Minerva’s enactment of covert
learning, while it exhibited independence, was not an independence that enabled or
reflected autonomous learning. Thus while Minerva could enact autonomous learning (as
demonstrated by her work on the third solution), for much of this lesson, her learning

activities were not autonomous.

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Discussion

Intertwining of Mathematizing and Identifying

This case of covert learning adds evidence supporting my claim that
mathematizing and identifying are intertwined. In the previous chapter, when I discussed
mathematizing and identifying in reference engaged learning and directed learning, I
emphasized the ways in which the mathematizing and identifying of one student, J akeel,
was related to other students’ mathematizing with and identifying of him. In contrast, this
case of covert learning demonstrates how one student (Minerva) identifies herself
through her reactions to the mathematizing of other students.

Minerva identified herself as knowledgeable by resisting mathematizing that
identified her as a learner or teacher and by covertly attending to mathematizing that
might be useful later. Excerpt 4 illustrates both of these points. Bonita and Jessica
attempted to explain to Minerva how two triangles make a square. Minerva, through her
“Nah huhn,” indicated that she was not willing to be identified as a learner and yet her
subsequent work showed her use of the narrative two triangles make a square. The
mathematizing of Bonita and Jessica positioned Minerva and provided discourse for her
to adopt. She adopted this discourse, which allowed her to appear knowledgeable, while
at the same time rejecting the ways in which the mathematizing positioned her as
unknowledgeable.

This example also shows how Minerva’s mathematizing communicated a certain
identity. She used Bonita’s narratives (See for example Excerpt 5, Lines 359 and 372) to
convey that she understood the mathematics. Also, her ability to work independently on

the problems demonstrated that she was mathematically competent. She did not need to

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ask questions or seek explanations. Instead, she was able to complete solutions to the task
without seeming to need input from others. However, Minerva’s struggles to construct a
solution that satisfied the teacher demonstrated how this same identifying constrained her
mathematizing. Minerva’s appearance of competence, knowledge, and independence was
not a mere facade. These identities were significant and endorsable for her. Evidence of
this lies in her firm reaction each time the teacher evaluated her work. Minerva insisted
that she was done. The teacher had to insist just as forcefully that she was not done before
Minerva was willing to back down. (See for example Excerpts 6 and 7). Minerva’s
confidence in her mathematizing meant that she did not investigate her discourse or that
of others. She did not ask questions or seek feedback on her work. As a consequence,
Minerva’s mathematizing was limited. Minerva’s identity and mathematizing were

intertwined so that each kind of activity had implications for the other.

Development of Mathematical Discourse

As I suggested above, Minerva’s mathematizing and identifying activities were
related to the development of her mathematical discourse. Final discourse shows that her
discourse changed to include the narrative two triangles make a square and a routine for
using it to assemble smaller triangles from the figures (I and J) into squares. Her
discourse also changed to incorporate a less mathematically desirable routine, counting
the squares of both figures together as the area. Each of these changes in her
mathematical discourse can be traced back to her enactment of covert learning.

The first example in the section introducing covert learning described when
Minerva first encountered the notion of how joining triangles might be important to

solving this task. Bonita stated and then explained this narrative. (See Excerpts 1 and 4).

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Minerva at first disagreed with Bonita (Line 203) and then resisted her explanation with a
series of “Nah huhn”s. However, in her first solution, Minerva incorporated this narrative
about triangles. Minerva’s identifying as someone knowledgeable meant that she
attended to and used this narrative when it seemed helpful, but it also meant that she did
not explore more of Bonita’s explanation. Thus Minerva’s enactment of the identifying of
covert learning meant that her discourse changed, but that the change was limited.

The less desirable routine that Minerva developed can also be tied to her activity
of covert learning. She did not learn this routine by listening to another: She developed it
in response to the teacher’s request to show the area of the whole rug. This request came
from the teacher’s evaluation of Minerva’s third solution. This solution was the most
mathematically appropriate. It displayed both rugs and indicated how the area of each
was two. However, the pieces of Figure I were not arranged together in a rectangle. (See
Figure 6.1). The teacher incorrectly assumed that Minerva did not have all of Figure I in
her solution. If Minerva had not been enacting covert learning, she might have asked the
teacher what she meant or she might have attempted to argue that her solution did show
the whole rug. However, Minerva’s enactment of covert learning meant that she did not
explain and she did not explicitly connect words with visual objects. Instead of probing
the teacher’s reaction to her work, Minerva assumed that she needed to construct a new
solution in which the whole rug was Figures H and I together. It seems that Minerva’s
reluctance to investigate discourses as part of her enactment of covert learning meant that
she abandoned her more mathematically correct solution and, without investigation or

verification, developed a less mathematically desirable solution.

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As Minerva engaged in covert learning, she developed both desirable and
undesirable mathematical discourses. Her evaluation of discourses as a covert learner
seemed to depend upon their role in getting the task completed to the teacher’s
satisfaction rather than upon their logical or mathematical consistency. Thus the
development of her mathematical discourse was not guided by mathematical
appropriateness but instead by her desire to identify as already knowing which was
accompanied by limited mathematizing. This case of Minerva’s covert learning
demonstrates how identity and mathematizing intertwine with consequences for the

development of the student’s mathematical discourse.

Autonomous Learning

As 1 indicated earlier, Minerva rarely engaged in learning activities that could be
described as autonomous learning. In addition, her final discourse was a shift away fiom
a more mathematically desirable discourse and included a routine that was not
mathematically desirable. Based upon these findings, it would be easy to portray Minerva
as an unsuccessful student. However, her confidence, independence, and creativity in
generating solutions are attributes of a successful student and might also be attributes one
would ascribe to an autonomous learner. Exploration of this tension between the
autonomous and non-autonomous aspects of Minerva’s learning activities helps to clarify
the activities of autonomous learning. I will focus on two particular tensions:

independence versus decentering and knowing versus not knowing.

Independence Verses Decentering
Some researchers define autonomy in terms of independent action. For example,

Warfield, Wood, and Lehman (2005) define autonomy to mean that “students are capable

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of thinking about mathematical ideas without having the ideas ‘explained’ to them and of
solving mathematical problems without being shown a method by another person” (p.
440). This definition seems to be a very good match for Minerva’s enactment of covert
learning: She did not want to have ideas explained to her and she solved the mathematical
task without relying on others. However, it would be troubling to label Minerva as
autonomous given her solutions to the task and especially the ways in which her final
mathematical discourse was not mathematically desirable.

Minverva’s covert learning argues for a construction of autonomy and of
autonomous learning that emphasizes a different aspect of independence. We want
learners to be independent thinkers, but at the same time they need to learn canonical
mathematical discourse. A construction of autonomy needs to negotiate this tension
between individual learning and collective ideas. Piaget’s theoretical construction of
autonomy offers a means of negotiating this tension. For Piaget, a key feature of
autonomy is decentering or learning to think about the perspectives of others (Kamii,
1994; Piaget, 1932/ 1960). Students become aware of and able to adopt other perspectives
as they examine others’ ideas and explore others’ perspectives. Their thinking becomes
less egocentric and subjective and more objective. They become capable of thinking in
other ways and imagining other possibilities. Autonomy does not mean the learner is free
to think anything. It means that the learner is capable of understanding and evaluating
ideas including the ideas of others.

For Piaget, the only way to gain understanding of other’s perspectives is to learn
with and from others. Decentering is not possible in a coercive environment: When

learners are obligated to act, they learn to evaluate ideas through the lens of reward and

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punishment rather than through rational examination of the logic of ideas. Instead,
learners need cooperative learning environments in which “the obligation is to consider
all viewpoints, to be coherent and rational, and to justify one’s conclusions” (Kamii,
l994,p.53)

When Minerva’s learning activities are viewed through Piaget and Kamii’s lens, it
becomes clear that her activities were not autonomous. Minerva’s focus on the teacher’s
approval meant that she constructed learning as coercive and not cooperative. She was
focused on what she needed to do and evaluated ideas in terms of reaching the reward of
being done rather than for their logical consistency. Although she adopted the ideas of
others, she did not subject these ideas to rational examination. Thus while she was
competent, independent, and creative as she enacted covert learning, these attributes
came from her confidence in her ideas and not from her work to coordinate her ideas with
the thinking of others. Minerva’s covert learning helps to clarify that autonomous
learning is only about independence in so far as it means that the learner has chosen for
themselves to explore and logically evaluate the perspective of others.

This definition of autonomous learning also clarifies the meaning of “discourse-
for—oneself”. Sfard (2008) defines discourse-for-oneself as “a discourse in which one
engages of one’s own accord while trying to solve her own problems” (p. 297). Minerva
might be said to have a discourse—for-herself: She produced her own narratives and
routines and used them to solve her own problems. This discourse was meaningful to her.
However, discourse-for-oneself does not mean one’s own discourse. It means that the
individual’s discourse is decentered. It arises from critical examination of others’

discourses (Ben-Zvi & Sfard, 2007). It is the result of adoption, production, and

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substantiation arising from investigation of the logical consistencies of other discourses.
It means one can use the discourses of others as tool for oneself. As Minerva engaged
covert learning, she did not logically examine other discourses. Instead, her discourse
arose from her apparently unexamined interpretation of the situation. For example, she
did not ask the teacher what she meant by “the whole rug.” Minerva constructed her own
understanding. Minerva’s final discourse was also not a “discourse-for-others” (Sfard,
2008): The discourse was not sensible to others, including the teacher.

Minerva’s enactment of covert learning clarifies that autonomy and autonomous
learning does not mean that the individual learns in isolation or without assistance.
Instead, autonomous learning is predicated upon the learner’s interest in making sense of
the discourse of others as well as of the situation or problem she investigates. As
Minerva’s covert learning demonstrates, an individual might develop a mathematical
discourse without engaging in autonomous learning, but the development of a
mathematically appropriate discourse-for—oneself requires exploration of other

discourses.

Knowing and Not Knowing

Minerva’s covert learning clarifies one more aspect of autonomous learning: the
tension between knowing and not knowing. Minerva worked to appear knowledgeable:
She appeared to avoid being seen as needing the help of her peers. Minerva’s desire to be
seen as knowing is consistent with the emphasis in schools. Students are seen as smart
and are rewarded for having answers, not for needing or asking for help (Duckworth,
1996). This need for knowledge can become entangled in definitions of autonomy. For

example, Warfield, Wood, and Lehman’s (2005) definition of autonomy (See above)

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might be construed as already knowing or at least, not struggling to know: A student who
can think about mathematical ideas without explanations from others and solve problems
without being shown how by others must not engage in significant struggles to know.
However, Duckworth (1996) notes that an important part of learning is what a person
does when they don’t know. Schools emphasize knowing and right answers, but students
only know things because of what they have done when they didn’t know. The definition
of autonomous learning I have constructed also emphasizes not knowing. Curiosity is
founded on recognition of what the learner does not know. The autonomous learner,
because of his/her curiosity, embraces what is novel and strange, exploring problems of
his/her own creation. This exploration leads to knowing, but not as end in itself. Instead
knowing leads to more questions about what is not known.

Covert learning offers an important contrast to autonomous learning. In enacting
covert learning, Minerva emphasized knowing and independence. While these might
seem like positive, autonomous traits, this case shows how enacting knowing and
independence and hiding learning resulted in a final discourse that was mathematically
problematic. If Minerva had recognized and admitted to the teacher had her peers that she
didn’t know, had asked questions, and had explored her discourse and the discourse of
others, she might have proceeded differently in her work on this area problem and

learned a more mathematically appropriate discourse.

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CHAPTER 7

WATCHFUL LEARNING AND GUIDED LEARNING

Overview

Minerva’s group contained one other focal student: Jessica. This chapter explores
the same lesson in the same group from the perspective of Jessica’s learning. Like Jakeel,
Jessica exhibited two different kinds of learning during the lesson. At the beginning of
the lesson, she enacted watchful learning: During this part of the lesson she made
decisions about what to do by observing Minerva and Bonita. Later in the lesson, the
teacher facilitated a transition to guided learning in which Bonita, Minerva, and the
teacher assisted Jessica in arranging the pieces of Figure I on her paper. Jessica’s case
shows the ways in which student interactions can constrain mathematizing and how
students, through their mathematical activity, might identify in ways that are rejected by

others.

Summary of the Lesson

This case examines the same lesson and same group as the previous case. I have
summarized the activity and main events below. While much of this summary is the same
as in the previous chapter, I have added details that are relevant to Jessica’s work during
the lesson.

After explaining her expectations for the lesson to the whole class, the teacher
dismissed the students to work in small groups. Minerva, Bonita, and Jessica gathered at
a cluster of four desks near the entrance to the classroom. After reading the task card and
gathering the materials for the task, they began cutting out Figures H and I. The teacher

came to the group and asked questions about the area of the two figures. During this

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conversation, Bonita explained that the area of both figures was two because two
triangles made a square. Minerva disagreed with this. Bonita and Jessica explained to
Minerva how two triangles could make a square.

After this explanation, Minerva attempted a variety of solutions to the task.
Jessica watched her and mimicked her cutting and gluing activities. The teacher visited
the group several times during the lesson and asked each student to explain her thinking.
Minerva disengaged alter the teacher criticized her work on her fourth solution and
Jessica offered Minerva encouragement.

Bonita and Minerva finally received the teacher’s permission to go to the next
task (involving Figures J and K). However, the teacher then realized that Jessica did not
yet have the answer to the task with Figures H and 1. She instructed Bonita and Minerva
to help Jessica, which they reluctantly did. After Jessica constructed the appropriate
arrangement of the pieces of Figure I on her paper, they all started work on Figures J and
K. During their work on Figures J and K, the teacher returned to the group and asked
Jessica to talk about her solution for Figure I.

Table 7.1 Transcript excerpts and main events in lesson

 

 

 

Line Excerpt Excerpt Summary Main Events

number Number

0 The teacher explains
the lesson .

 

 

The students move to
their small groups and

Excerpt 7 Jessica watches Minerva and Bonita. She start work.
talks to Bonita about cutting.

 

 

 

 

 

 

 

100

 

 

 

 

 

 

 

 

 

 

 

 

182

 

Table 7.1 con’t

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Excerpt ll Bonita first uses area is two and two triangles
200 make a square.
Minerva works on her
first solution.
Excerpt 8 Minerva cuts out the figures again. Jessica
asks why Minerva works on her
Excerpt 9 Minerva gets the teacher’s approval. Jessica second solution.
copies her activity.
300
Excerpts 1 Jessica first talks about the area of Figure 1. Minerva works on her
& 2 third solution.
400 Excerpt 3 Jessica again talks about the area of Figure I.
Excerpt 10 Jessica becomes a cheerleader for Minerva. Minerva works on her
500 . , . fourth solution.
Excerpts 12 The teacher approves Bomta 3 solution.
& 13 Jessica celebrates.
_ . . The teacher approves
Excerpts 4 Jessrca’s final discourse on F rgure I. Bonita and Minerva’s
8‘14 _ solution to Figures H
Excerpts Bonita, Minerva, and the teacher guide and I.
15-20 Jessica.
600 _ . _ Minerva, Bonita, and
Excerpt 5 The teacher asks Jessrca about Figure I again. Jessica work on
Figures J and K.

 

 

 

 

 

Outcome of Learning

Initial Discourse

Jessica first talked about the mathematics of this problem 16 minutes after

students began work on the task. The teacher came to the group and inquired about their

work.

183

 

Excerpt 1
348 Teacher How am I going to prove, Jessica, She points to a paper with
either one of these rugs is bigger Figures H and I.
than the other one or if they are
the same? How am I going to
prove it?

349 Jessica Find out the area

350 Teacher Okay, so what’s the area of Figure Jessica points back and forth to
H? the two squares in H.

What is it?
351 Jessica Two

Jessica’s motion (line 350) and her statement of “Two” indicated that she could
determine the correct area of Figure H. It seemed by her motion that she was counting
squares. However, she did not explicitly state that she was counting squares.

The teacher then asked her group about the area of Figure I.

Excerpt 2.
354 Teacher Okay. What is the area of Figure
I?
355 Bonita Two
356 Teacher How do you know? She looks at Bonita.
357 Jessica One

At this moment, Jessica stated an incorrect area for Figure I. If she was counting
squares, as she seemed to be doing when she determined the area of Figure H in Line
351, the response, “one” while still not the correct area, was a reasonable answer for the
number of whole squares in the figure.

A short while later, the teacher again asked Jessica’s group about the area of
Figures H and I.

Excerpt 3
396 Teacher How could I find out the area?

397 Bonita We already found the area.

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398 Jessica If you count the squares.

This is two. She points at H with two fingers,
putting one in each square.

That looks like three. She points at 1, drawing a circle
through the three pieces.

In this excerpt, Jessica was more explicit about her routine for determining area.
In line 398, she mentioned counting squares. She then demonstrated how counting
squares would result in an area of two for Figure H and three for Figure 1. Her
determination of Figure H’s area was consistent with her mathematizing in the previous
excerpt (Lines 350 and 351). However, her statement of the area for Figure I changed
from one in Line 357 to three in Line 398, indicating that Jessica was uncertain what to
count as squares in Figure I. It appeared that in the first excerpt, she only counted whole
squares (resulting in one),while in the second excerpt, she counted each of the spaces as a
square (resulting in three).

Jessica’s initial mathematizing had some of the features of the mathematically
desirable discourse for this task: She explicitly identified squares as the unit of
measurement and articulated a number for the area of each figure. However, in order to
arrive at a more mathematical determination of area, Jessica needed a means of
accounting for the two triangles in Figure I in a way that didn’t either ignore them or

count each one as a whole.

Final Discourse
By the end of the lesson, Jessica’s mathematical talk included both a
mathematically correct number for the area of Figure I as well as an explanation that
began to account for the contribution of the two triangles to the area of the figure.

However, careful examination of her words and gestures raises questions about Jessica’s

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understanding of her own mathematical talk. In order to demonstrate Jessica’s final
discourse, I use excerpts of transcript from two events at the end of the lesson.

This first event occurred toward the end of the group’s work on Figures H and I,
43 minutes into the small group work time. The teacher had just approved Bonita and
Minerva’s solutions. She turned to Jessica and asked her to explain her answer. Figure

7.1 is a drawing of Jessica’s paper at this moment.

 

 

 

 

 

 

 

Figure 7.1 Drawing of Jessica’s work at Excerpt 4.

This is student work.

Excerpt 4

550 Teacher What’s, what’s not, what’s the She points to Figure I glued on
area of this Jessica? the bottom of Jessica ’s paper.

551 Jessica It’s two.
552 Teacher How is it two?

553 Jessica Because (?) She points to all of the pieces of
I at the top of her paper.
554 Teacher I don’t know how that’s two.
555 Jessica This is two because if you put As Jessica says triangles, she
two triangles together that’s a claps her hands together over
square. her head.

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556 Teacher But where on your paper does it
show me that?

557 Minerva Gosh, Jessica, put it together.

558 Teacher Where on your paper does it With both hands, Jessica pats
show me that? Figures H and I glued on the
bottom of her paper.

But that doesn’t show me that.

Jessica demonstrated in Lines 551 and 555 that she could state the correct area of Figure
I. In Line 555, she also stated a narrative (“if you put two triangles together that’s a
square”) that began to explain why the area of Figure I was two. However, this
explanation was incomplete because it did not show or describe how putting two triangles
together resulted in an area of two for Figure I. In addition, as the teacher noted in Lines
556 and 558, Jessica could not point to any of her work to show how Figure I was two or
how two triangles made a square. Instead of pointing to her paper or to Figure I, when
Jessica spoke about triangles she clapped her hands over her head (Line 555). Thus while
Jessica had words to talk about the area of Figure 1, her use of these words did not seem
to refer to any particular object.

Jessica had a series of other exchanges with the teacher and Minerva shortly after
the interaction above. These exchanges provided further evidence that Jessica was not
connecting her words about triangles and squares (Line 555) with the parts of Figure I.
For example, the teacher and Minerva talked about how Jessica could “make it two”
(Line 564, not included). They were referring to Figure 1. However, Jessica picked up
Figure H and began to cut it out. This work on Figure H suggests that not only was
Jessica not connecting her words to the parts of Figure 1, she was also unsure whether her

words referred to Figure H or Figure I.

187

When Jessica did work on Figure 1, she placed two copies of the figure together
(See Figure 7.2). This move illustrated Jessica’s understanding of “if you put two
triangles together that’s a square” (Line 555). Rather than focus on the parts of Figure I
and how the two small triangles could be used to make a square, Jessica understood
triangles to mean all of Figure I. Jessica’s actions in this moment offer additional
evidence that she had not yet made sense of how triangles and squares could be useful in
explaining the area of Figure 1. While Jessica’s words in Line 555 offered a
mathematically appropriate explanation for how the area of Figure I was two, Jessica’s
actions did not indicate that she connected these words with the pieces of Figure I or with

the area of the Figure.

 

 

Figure 7.2 The large square constructed by Jessica from two copies of Figure I

With support and direction from the teacher, Bonita, and Minerva, Jessica was
eventually able to place the pieces of Figure I so that the two triangles made a square
(See Figure 7.3. I discuss this in detail in the section below on guided learning).
However, she did not talk about this arrangement until 1 1minutes later when the teacher

inquired about the work.

188

 

 

 

 

 

 

Figure 7.3 Jessica’s final arrangement of Figures H and I.
This is student work. The additional copies of Figure 1 shown in Figure 7.1 have been
removed by Jessica and Minerva.

Excerpt 5.
612 Teacher Okay, Jessica. So how is it two? She points to Figure I on
Is this how you made it? Jessica ’s paper. Jessica barely
nods her head.
So tell me how it’s two.
613 Jessica Because Jessica smiles.

if you put two triangles together it Jessica waves her scissors in the

makes two. air and then points at the two
triangles joined into a square on
her paper.

Unlike Jessica’s earlier explanation for the area of Figure I (Line 555), it seems that her
words here refer to the objects on her paper. She talks about two triangles and then points
to the two triangles on her paper (Line 613). However, it is difficult to tell from this

excerpt what Jessica meant when she stated, “it makes two”. It is possible that the process

189

of rearranging Figure I allowed Jessica to make sense of how the area of Figure I could
be two squares. However, it is also possible that Jessica could be referring to the two
triangles when she says “it makes two” or that, like in Line 555, she was uttering the
words and making motions that would satisfy the teacher without really understanding
how Figure I has an area of two.

One last interaction that supports the latter interpretation occurred during a
subsequent lesson during the next week. (See Table 4.1 for a sequence of lessons. The
case study lesson occurred on Thursday of Week 2. This subsequent lesson occurred on
Tuesday of Week 3.) This is the same lesson I described during my description of
J akeel’s final discourse. It involved determining the area of triangles depicted on
geoboards (See Figure 5.3 and 5.4). Jessica was in a group with Jakeel and Minerva.
They were working to determine the area of a figure identical to Figure I when the
teacher came to the group.

Excerpt 6
426A Teacher Okay, so you have how many
squares in there Jessica?
427A Minerva Two
428A Jessica One
429A Minerva Two
430A Teacher [Okay, why do you say two? She points a pencil at Minerva
431A Jakeel [Two His speech overlaps with the
teacher ’s.

Jessica’s answer in Line 428A was mathematically accurate: There was only one square
in the figure. However, as demonstrated by Minerva and Jakeel’s answers, the teacher
intended for students to count the two triangles as a square, for a total of two squares.
Jessica’s answer provides further evidence that, unlike J akeel and Minerva, she had not

yet changed her discourse to include a routine for addressing triangular spaces in figures.

 

3 I have added the letter A to each of these line numbers as a reminder that this excerpt and these line
numbers are not from the same lesson as the other excerpts in this chapter.

190

The interaction continued with Minerva and the teacher discussing how to teach
Jessica. Jakeel volunteered to show his thinking.

451A Jakeel See these two make a square Jakeel turns the geoboard so he
can see it. He puts his index and
middle fingers on the two smaller
triangles. He then turns it back to
Jessica.

452A Minerva Duh duh duh These are rude noises that seem to
convey that Jessica should already
understand. Minerva glances at
the teacher after she makes the
noises. Jakeel removes his fingers
from the board and places his
middle finger in the full square.

453A Jakeel And so it’s two squares

454A Teacher Do you agree, Jessica? Do Jakeel continues to hold up his
you understand really or not board
you’re not sure?
455A Jessica A little bit
Jessica’s final response captures my assessment of her final discourse. While her
discourse changed “a little bit” over the course of the lesson, the changes did not include
links between the narrative about triangles and the area of the figure. They also did not
include a mathematically appropriate routine for determining area. Instead, as
demonstrated by this second lesson, it seemed that Jessica had learned the responses that
met the teacher’s expectations for the first lesson, but she was unable to use this discourse
in subsequent lessons.
In summary, Jessica’s final discourse had some features that were more
mathematically desirable than her initial discourse. She was able state the appropriate
area for Figure I (Excerpt 4, Line 551) and she was able state how the partial units in the

Figure might be arranged so they can be counted as a whole unit (Excerpt 4, Line 555 and

Excerpt 5, Line 613). However, her final discourse did not fully explain how she arrived

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at her area measurement nor did it explicitly state what she was measuring. In addition,
Jessica seemed to be unable to demonstrate how her words were related to the picture on
her paper and she was unable to use this discourse during the subsequent lesson. In
contrast, in her initial discourse, Jessica’s verbal discourse was tied to images on her
paper. Also, she stated what object she was counting (squares, Excerpt 3,Line 398). Thus
Jessica’s final discourse had some changes that were more mathematically desirable and
other changes that were less so.

This section has explored the change in Jessica’s mathematical discourse through
close attention to her mathematical talk. Before making claims about connections
between Jessica’s learning and the interactions of the group, I will first describe Jessica’s

mathematizing and identifying activities.

Learning Activity

Jessica’s learning activity changed during this lesson. There was a constellation of
identifying and mathematizing activities that I have labeled watchful learning. These
activities transitioned to other learning activities in which Jessica’s peers and the teacher
were explicitly focused on Jessica’s actions. By the conclusion of the lesson, Jessica
again enacted watchful learning. As I did in the chapter on Jakeel, I first offer examples
of watchfirl learning. I then describe the audience and mathematizing for watchful
learning before describing the transition to guided learning. My analysis of guided
learning includes an explanation of the audience and mathematizing for guided learning

before describing the transition back to watchful learning.

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Watchful Learning

Throughout this lesson, Jessica frequently identified as a learner. However, the
activities that identified her as a learner were not explicit requests for help. Instead
Jessica’s learning moves were subtle and not overt. Rather than ask questions or engage
her group members, Jessica watched the other students in order to determine what she
should do.

The following excerpt offers an example of Jessica’s watching. The excerpt
comes from the beginning of the small group work. Minerva has handed out copies of the
papers with Figure H and 1. Jessica examined the paper and then set it in front of her and
watched Minerva hand out other materials. Jessica continued to watch Minerva for 2.5

minutes. During this time, Bonita rejoined the group after a trip to the restroom.

Excerpt 7

171 Minerva Here. (?) She passes out individual
worksheets. Jessica writes on
her sheet. Minerva starts cutting
out Figure H. Bonita picks up a
paper with Figures H and I.

172 Bonita I don’t get this. We cut these Bonita starts cutting.

things out?
173 Minerva Sure Jessica writes her name on her

individual worksheet. Jessica
looks at Minerva and Bonita and
picks up her paper and scissors.

174 Jessica Why you cutting (?) huh, Bonita, This is said like a chant and not
huh, Bonita as a question. Jessica looks at
Minerva who already has one
shape cut out. Jessica starts to
cut. Bonita finishes cutting.

Jessica did not start to cut out the shapes until after both Bonita and Minerva had started.
In addition, before she started to work, Jessica looked at both Bonita and Minerva (Line

173). Of the three students in this group, Jessica was the student who could have started

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work the earliest. She had time to examine the figures while Minerva sharpened her
pencil and passed out materials. She was not away in the bathroom like Bonita. She had
the paper and the scissors and yet she waited until both Bonita and Minerva had started
cutting before she started (Line 174). Jessica’s hesitation to begin coupled with her
observations of Minerva and Bonita seem to indicate that Jessica was watching Bonita
and Minerva to see what to do.

This second example of watchful learning demonstrates not only how Jessica
watched her peers to see what to do, but she also watched to see what activity, including
what answer, was approved by the teacher. Jessica did this by carefully attending to
interactions between the teacher, Bonita, and Minerva that indicated the teacher’s
approval of activity or answers. The following interaction occurred almost half way
through the lesson. Jessica had glued Figures H and I to her paper. She was watching
Minerva and the teacher. Minerva just completed her first solution and told the teacher
that she was done. (Lines 258 through 261 were included in Chapter 6 as Excerpt 6).
Excerpt 8

258 Teacher How is this? I can’t tell anything She points at Minerva 's paper.
about the area by this, Minerva

259 Minerva I said the area’s two. She writes ”2 " under the copy of
Figure I on her page. Bonita
cuts on scrap paper.

260 Teacher But how can you show me by
this, how can you actually show
me?

261 Minerva Cut it out. The teacher leaves the group.
Jessica glues down her Figure
H. Minerva picks up a paper
with H and I and starts cutting.
Jessica looks at Minerva. She
sniflfs the glue stick and twists
the bottom. Bonita has small

194

262
263

264
265

Bonita

Jessica

Minerva

Bonita

Like my nail?

Gotta cut another one out? Why
are you cutting another one out?

Because I’m doing something.

She’s playing tic tac toe with it.

pieces of paper. She puts glue on
one and attaches it to her finger.

Jessica smiles. There is a pause
during which Jessica watches
Bonita and Minerva.

Jessica’s questions (Line 263) indicated that she had noticed Minerva cutting out

another copy of Figures H and I but wasn’t sure why Minerva was doing this. Jessica did

not yet start to imitate Minerva’s actions. Instead, she engaged in a conversation with

Bonita about her church. This conversation continued for several minutes until the

teacher walked by the group. During this next excerpt, which immediately followed the

conversation about Jessica’s church, Jessica watched the teacher, Bonita, and Minerva.

Excerpt 9

283 Teacher
284 Bonita
285 Teacher
286 Minerva
287 Teacher
288 Minerva

Okay. Okay, so I’m not sure what
you’re doing right now Bonita.

I’m trying to find out what to do.

So what you need to ask, what are
you guys doing right now.
Minerva what are you doing? Tell
me what you’re doing.

I cut the triangle apart so I could I
could show how you could make
it

Make it?

Show it’s two. Two perimeters.
One, two equals two.

195

Jessica looks at the teacher.

Jessica looks at Bonita.

Jessica looks at Minerva.
Minerva puts glue on the back of
Figure H and glues the figure on
the left middle of her paper.

As Minerva counts one, two, she
points first at the I by Figure H
and then at the I = 2 by Figure
I.

One and one equals two. So I just Minerva puts glue on the small

(did this?) triangles from Figure I and
glues them on the left bottom of
her paper. Jessica continues to
look at Minerva.

289 Teacher Okay. Do you understand what She looks at Bonita. Bonita
she just said? shakes her head.

Okay she’s not sure what you’re She talks to Minerva.
doing so can you say it again?

290 Minerva I’m gluing this down so (they?) Jessica starts to cut out another
can understand how it’s two copy of Figure H and 1.

During these two exchanges, Jessica paid careful attention to Minerva’s activity.
However, she hesitated to imitate Minerva until after the teacher had indicated her
approval of Minerva’s activity. This approval came in Line 289, when the teacher asked
Minerva to explain what she was doing to Bonita. The approval was quickly followed by
Jessica’s decision to cut (Line 290). It is possible that other factors were involved in the
timing of Jessica’s cutting. For example, she might have been planning to cut out another
copy of the figures but was distracted by the interaction with the teacher. However, the
timing of her activity suggests that the decision to cut was based at least in part upon her
observation of the interactions among the teacher, Minerva, and Bonita and the teacher’s
approval of Minerva’s actions.

Watchful learning has some activity in common with covert learning: Both kinds
of learning rely on observation of others for examples of new discourse. Both emphasize
the teacher as the arbiter of right and wrong answers. A key difference between the two is
that as Jessica enacted watchful learning, she did not actively work to hide or deny that
she was learning. She was willing to ask questions that indicated that she didn’t know

(See Excerpt 8, Line 263). I argue (see Identifying the Audience below) that her watchful

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learning was, in part, an outcome of Bonita and Minerva’s refusal to respond to her
questions and explain their activity (See for example, Excerpt 8, Line 264). In contrast,
Minerva enacted covert learning through her refusal to ask questions, listen to
explanations, or indicate that she didn’t know. While both watchful learning and covert
learning depended upon the mathematizing of the learner’s peers, watchfiil learning was
Jessica’s reaction to interactions with peers while covert learning was a more proactive
position in which Minerva worked to identify herself in particular ways.

The next sections will analyze the specific identifying and mathematizing moves

of Jessica’s watchful learner, starting with her identification of her audience.

Identifying the Audience

Toward the beginning of this lesson, Jessica made three attempts to engage Bonita
and Minerva in conversation about the task. In Excerpt 7, Line 174, she spoke with
Bonita about cutting the figures. In Excerpt 8, Line 263, she asked Minerva why she was
cutting more figures. Finally, in Chapter 6, Excerpt 4, she attempted to explain to
Minerva how two triangles make a square. Each of these examples resembled the
mathematizing activity of J akeel as he enacted engaged learning. Jessica, like J akeel,
demonstrated that she was interested in the activity of her peers and that she had
mathematical ideas to contribute to the conversation. However, in each of these instances,
her overtures were ignored or rebuffed. Bonita ignored Jessica’s comment about cutting
(Excerpt 7, Line 174). Minerva dismissed her question about why she was cutting
(Excerpt 8, Line 264) and Bonita, by replying that Minerva was playing tic-tac-toe made
Jessica’s question seem silly and trivial (Line 265). Finally, Minerva, with her series of

“Nah huhn” communicated that Jessica’s attempt to explain her thinking was not

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important (See Chapter 6, Excerpt 4 and the accompanying discussion). Jessica’s
attempts to identify her peers as interested and helpful members of her audience were
unsuccessful.

Instead, as the lesson continued, Jessica was more passive. Rather than position
Bonita and Minerva as members of her discursive audience, Jessica positioned herself as
a watchfiil member of their audience. Bonita and Minerva became models for her activity
rather than teachers or co-learners.

The exception to this positioning was when the teacher arrived at the group. With
the teacher’s arrival, Jessica’s interactions changed: She became a cheerleader for
Minerva, building from what the teacher said to encourage her in her work. For example,
the teacher was talking to Minerva about her fourth solution. The excerpt below
continues Excerpt 3 from the previous chapter.

Excerpt 10
501 Teacher I said I want you to prove how Minerva groans and pulls Figure

this is. I didn’t say connect it. If H from her paper. The square
I said connect it, I was wrong. from Figure I comes up with it.

Just show me H on your paper.
Just show me H.

502 Minerva lcan’t

503 Teacher Yes you can. Jessica is rearranging parts of
Figure I.
Just take this off. Show me The teacher points at Minerva ’s
Figure H. paper. Jessica is rearranging

parts of Figure I on her desk.
504 Jessica Yes you can.

On Line 504, Jessica echoed the teacher’s words from Line 503, encouraging Minerva in
her work. On three other occasions, Jessica supported Minerva’s efforts to solve the

problem (Lines 468, 472, and 515, not included in this document). Each of these four

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moments came while Minerva was in despair after the teacher’s critique of her work (See
for example, “I can’t” in Line 502). Jessica’s cheerleading identified Minerva as capable,
but needing moral support. This cheerleading may have been a social move, enabling
Jessica to contribute to the work of the group, or it may demonstrate Jessica’s sympathy
for Minerva. In either case, the cheerleading was consistent with Jessica’s watchful
learning: Minerva was a model for Jessica’s mathematical work. Jessica needed Minerva
to demonstrate the answer and obtain the teacher’s approval so that she would know what
to do.

Jessica’s cheerleading also demonstrated how the teacher was an important
member of Jessica’s audience. Jessica was interested in supporting Minerva in getting the
teacher-approved answer on her page. Jessica’s attention to teacher approval indicated
that Jessica identified the teacher as someone who knew the right answer and who would
determine whether Jessica had the right answer. Other evidence of Jessica’s interest in
teacher-approved activities was evident in Excerpt 9 above, when Jessica waited until the
teacher had endorsed Minerva’s activity before she began to mimic Minerva. Also,
Jessica only began to use Bonita and Minerva’s narratives about Figures H and I after the
teacher indicated her approval of their answers (I describe this moment in more detail in
the section below on adoption of discourse). Through these actions, Jessica indicated that
the work she produced was primarily intended for and ultimately needed the endorsement
of the teacher.

One final point I want to make about Jessica’s audience members for her watchful
learning discourse is that Jessica did not appear to include herself in her audience. As she

enacted watchful learning, she cut and glued figures, but she did not write on her paper.

199

Aside from her explanation of how two triangles make squares, all of her mathematical
discourse was in response to the teacher’s questions. Neither her explanation to Minerva
nor her responses to the teacher were directed at herself or primarily intended for herself.
Jessica may have communicated with herself about mathematical ideas at times or in
ways that were not visible. However, if such moments occurred, they were not a
prominent part of Jessica’s mathematizing. Unlike the moment in Minerva’s third
solution or J akeel’s engaged learning, Jessica did not seem to communicate her

mathematical ideas with herself.

Mathematizing

Adoption of discourse. Jessica’s final discourse had two features that were
adopted from the discourse of Minerva and Bonita: the area of Figure I and the use of two
triangles make a square as justification for that area (See Excerpt 4, Lines 551 and 555).
Bonita was the first person to articulate these two narratives and the routine of using them
to account for the area of Figure I. Toward the beginning of the work in the small group,
the teacher came to the group and asked questions about the task. This excerpt was
presented and discussed in the previous chapter as Excerpt 1.
Excerpt 1]
199 Teacher Okay you’re going to compare

their area. How could I compare
it?

200 Bonita Both the areas, they both have
two.

201 Teacher How do you know?

202 Bonita Because two triangles make a Bonita points at the two
square triangles in Figure I.

Bonita’s statements on Lines 200 and 202 became Bonita and Minerva’s routine for

talking about Figure I. On two other occasions, when the teacher inquired about the area

200

of Figure I, Bonita and Minerva responded that the area was two and then they offered an
explanation that was a variation of Bonita’s narrative “Because two triangles make a
square” (Line 202).

Jessica adopted Bonita and Minerva’s discourse after the teacher indicated her
approval. This approval came more than 40 minutes into the small group time. I will first
present the transcript that shows the teacher’s approval of Bonita’s solution. It also
demonstrates Jessica’s attention to this interaction.

The teacher came to the group and looked at Minerva’s fourth solution, which
was on a separate paper from her earlier work (See Figures 6.1 and 6.2). Minerva claimed
that she had done what the teacher asked on her first paper. The teacher and the students

examined Minerva’s first paper (Figure 6.1).

Excerpt 12
516 Teacher How is this two? The teacher talks to Minerva.
517 Bonita Let me see something. All you Bonita pulls Minerva ’s paper
got to do is make these two. towards herself She points to
the two triangles in I.
518 Teacher So do that on your paper! Jessica looks at Bonita.

This remark seems to indicate the teacher’s approval of Bonita’s solution. The
conversation continued.

519 Bonita I already did.

520 Teacher Where, where is it? Jessica stands up to look. Bonita
points at her paper where she
has a line connecting the two
triangles in I. See Figure 7. 4.

521 Jessica Bah, butta, spa, bah These seem to be celebratory
sounds. There are no
accompanying gestures. Jessica
is continuing to look at Bonita ’5

paper.

201

When Bonita pointed to her answer on her paper (Line 520), Jessica stood up to look. In
addition, she made a series of funny, celebratory noises (Line 521) that might indicate her
enthusiasm that Bonita was close to having a teacher-sanctioned right answer.

522 Teacher So you just drew a line Bonita nods her head. Jessica
looks down at her own work.

So you didn’t actually out it out Bonita shakes her head. Jessica
looks at Bonita.

Is that okay to do? Bonita shrugs her shoulders.

Do you think it is? Bonita nods her head. Jessica
looks at the teacher.

Okay. So you showed me that. Jessica picks up her pieces of

Right? Everyone in your group Figure I and places them on her

has to paper.

With the words, “Okay. So you showed me that” (Line 522), the teacher approved
Bonita’s solution. Immediately after that approval, Jessica began to copy Bonita’s
solution, picking up the pieces to Figure 1. She looked back and forth between her paper
and Bonita’s paper and arranged Figure I on her paper so that it matched Bonita’s. This
copying suggests that Jessica was beginning to adopt Bonita’s visual solution. However,
Jessica only copied the position of the figure and not any of the accompanying writing. I
will say more about this as l summarize Jessica’s adoption of discourse.

Shortly after this moment, Minerva finished her work and the teacher told her to
go get the next task (Figures J and K). Jessica celebrated the conclusion of their work on
Figures H and I.

Excerpt 13

539 Jessica We did it. This is said with enthusiasm.
There is a pause.

Y’all got it right?
Jessica’s statement, “We did it” (Line 539) indicated her recognition that they

finished the problem. Her question, “Y’all got it right?” (Line 539) sought confirmation

202

that Bonita and Minerva had a correct answer. It is after this moment that Jessica first
stated that the area of Figure I was two (Excerpt 4, Line 551) and gave the reason, “This
is two because if you put two triangles together that’s a square” (Excerpt 4, Line 555).
Jessica continued to use these narratives at one other time during this lesson that she was

asked about the area of Figure I (See Excerpt 5).

 

 

 

 

 

T‘- ‘ M

\‘2 {({tju /

Figure 7.4 Bonita’s work as approved by the teacher.
This is student work. The top two figures are labeled “Figure H” (on the left) and “Figure

I” (on the right). The text in the middle reads, “They cover the same amount!”. At the
bottom of the page, Bonita has written, “equals 2”.

 

As Jessica enacted watchful learner, she carefully observed the actions and interactions of

Minerva and Bonita and imitated their activities, no longer using her previous discourse

203

to respond to the area of Figure I. This imitation occurred only after the teacher had
explicitly approved Minerva and Bonita’s activities and discourses.

Jessica’s adoption of these narratives and the routine of when to use them did not
include connections to the images on the paper. Jessica could use the words in
appropriate moments, but when asked to illustrate the words with the pictures on her
paper, she couldn’t. In Excerpt 4, when the teacher asked Jessica to show how two
triangles make a square, Jessica patted her paper, but was unable to show how the figures
she had glued on her paper could illustrate this narrative. Her copying of Bonita’s
solution (Excerpt 12) also illustrates this disconnect between words and images. Jessica
adopted part of Bonita’s solution, carefully arranging her Figure I, but she did not copy
any of the lines or words from Bonita’s paper. Jessica’s adoption of discourse as she
enacted watchful learning careful attention to what to adopt, but not exploration or
connection from words to pictures.

It is important to note that earlier in this lesson, Jessica showed Minerva how two
triangles could make a square (See Chapter 6, Excerpt 4). She had not integrated this
narrative into a strategy for determining area, but she stated this narrative, seemed to
understand it, could demonstrate it using all of Figure 1. Therefore, part of what Jessica
needed to learn about this narrative was that the small triangles in Figure I could be
combined in the same way as the large triangles. At this later moment in the lesson, her
adoption of the narrative about triangles and squares did not seem to be connected to
either the small or the large triangles.

Production of discourse. Toward the beginning of the lesson, Jessica used

discourse that was not an echo of what others had said during the lesson. However,

204

toward the end of the lesson, afier she had begun to use the teacher-sanctioned discourse,
she only used words, narratives, and routines that had been enacted by others and
explicitly approved by the teacher. As I described above, her final discourse involving the
area of Figure I was an adoption of what Bonita and Minerva had said.

As I noted in the section on audience, there were moments early in the lesson in
which Jessica attempted to initiate conversations. For example, she explained how two
triangles made a square (Chapter 6, Excerpt 4) and she asked Minerva why she was
cutting (Excerpt 8, Line 263) but Minerva and Bonita rebuffed Jessica’s attempts at
mathematizing. These dismissals of Jessica’s mathematizing not only positioned Jessica
as someone who should not talk or ask about mathematical objects, they also indicated
that Bonita and Minerva were not interested in mathematizing with Jessica or with each
other. Exploration and discussion of mathematical discourse was not encouraged for
anyone in this group. Unlike J akeel’s peers, the members of Jessica’s group showed no
interest in constructing an environment conducive to producing discourse. Jessica’s
interactions with her peers around mathematics were limited to watching.

Substantiation of narratives. As Jessica enacted watchful learning, her
substantiation of narratives relied upon the teacher’s indication of approval. My
discussion above of her adoption of discourse demonstrated how Jessica began to use the
narratives of Bonita and Minerva only after the teacher signaled that their solutions were
correct. In addition, Jessica’s activity as a watchfiil learner included no exploration or
explanation of discourse (beyond her demonstration of how two triangles make a square

for Minerva). In order to use logic to substantiate a narrative, Jessica would need to

205

explore or explain the narrative. However, Jessica’s activity does not provide evidence of
verifying narratives except by looking for approval from the teacher.

Jessica was not alone in relying upon the teacher to substantiate narratives. In the
previous chapter, I argued that Minerva relied on the teacher as she enacted covert
learning. Bonita also waited for the teacher’s approval before committing to a narrative.
Bonita crafted two solutions to the task with Figures H and 1. Her first solution was much
like Minerva’s first solution: She glued H and I intact on the paper and wrote, “They
cover the same amount!” Much later in the lesson (while Minerva was working on her
fourth solution), she glued another copy of Figure I on her paper, drew a line connecting
the triangles, and wrote “equals 2”. Bonita did not present this work to the teacher until
after the teacher had approved Minerva’s solution (See Excerpt 12). Bonita didn’t explain
how her work showed the right answer. Like Minerva, she stated that she had done what
the teacher requested (Excerpt 12, Line 519) and then nodded and shrugged her way
through the rest of the teacher’s questions (Line 522), letting the teacher work her way to
endorsing her work. Thus Bonita, Minerva, and Jessica depended upon the teacher to
substantiate narratives.

This point is important for two reasons: First, it hints that Minerva and Bonita’s
refusal to mathematize with Jessica was connected, at least in part, to their reluctance to
mathematize at all. While there is evidence that they were frustrated with Jessica (I
discuss this below), this frustration does not appear to be the primary reason not to
mathematize with her. Second, Jessica’s reliance upon the teacher may have come in part
from the reluctance of Bonita and Minerva to engage in mathematical conversation with

her. Jessica, as I have described earlier, initiated conversations about mathematics and the

206

tasks on several occasions. Neither Bonita nor Minerva continued those conversations.
Jessica could have attempted to substantiate narratives on her own. However, it seems
that it would have been challenging for Jessica to make sense of what Bonita and
Minerva were doing and their narratives if they were unwilling to share their thinking
with her (or with anyone else). Thus Jessica’s dependence upon the teacher for
substantiation is like other watchful learning activities: It was not one option of many laid

at Jessica’s feet. Instead, it seemed to arise through interactions with Jessica’s peers.

Departure from Watchful Learning

In Chapter 5, I described how J akeel’s learning shifted during the lesson. He was
initially engaging in learning. However, part way through the lesson, his activity shifted
so that Rebecca was directing his learning. Jessica also experienced a shift in learning
during this lesson. For most of the lesson, she participated in watchfiil learning. However,
after the teacher approved Bonita and Minerva’s solution, she realized that Jessica did not
have the solution on her paper. She required Bonita and Minerva to help Jessica complete
her work. As the focus of the group became Jessica and her work, the interactions in the
group and Jessica’s learning changed.

Jessica’s shift in learning began after Bonita and Minerva had solved the task
involving Figures H and I. They were preparing to start the next task when the teacher
asked Jessica for her solution to the task. Jessica was able to explain that the area was two
because, “if you put the two triangles together that’s a square” (Excerpt 4, Line 555).
However, Jessica was unable to show this on her paper (Line 558). The teacher
responded:

Excerpt 14

207

560 Teacher

561 Jessica

Some how I need you to show me
that on your paper.

Well there ain’t nobody helping
me.

With her statement on Line 561, Jessica identified herself as in need of help and

identified her peers and perhaps the teacher as not being helpful. Earlier, as Jessica

engaged in watchful learning, Bonita and Minerva had served as models for her learning

so they were not explicitly or directly involved teaching her. Jessica’s statement here

seems to indicate Jessica’s readiness for interactions in which others were more

supportive of her learning.

The teacher accepted this invitation and required Bonita and Minerva to literally

and figuratively position themselves as helpfiil to Jessica.

562 Teacher

Okay. So what do you need help
with?

So Bonita, come over here please.

You know what, you need to stop
it right now. I don’t know what
happened (last night?) but it
needs to stop. (2 second pause)
You are not listening to me and it
needs to stop.

Come over here Minerva. Come
over here Minerva. Get up and
move and come over here
because I want you to actually
come over here. Come over here
please Bonita.

Okay, how is this two?

The teacher sits down next to
Jessica.

Bonita stands next to Jessica 's

desk

The teacher talks to Bonita.

The teacher motions to Jessica 's
desk. Minerva gets out of her
seat and stands next to her own

desk

Minerva and Bonita’s physical positioning was important to the ways in which they could

identify and communicate with Jessica about her work. Bonita’s position next to Jessica’s

desk provided her with access to Jessica’s work. She could both see what Jessica was

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doing and could point at Jessica’s work as she spoke. In contrast, Minerva stood by her
own desk, diagonally across from Jessica’s desk (See Figure 7.5 for their seating
arrangements). From her standing position, Minerva was as far from Jessica’s desk and
her work as she could physically be. The difference between Minerva and Bonita’s
physical locations coincided with differences in the ways they identified Jessica. I will
first elaborate on Minerva’s identifying activity and then contrast it with Bonita’s

activity.

 

 

 

Minerva

 

 

 

 

 

 

Jessica Bonita

 

 

 

 

 

 

 

 

 

Figure 7.5 Seating arrangements during this lesson
The interaction from the episode continued as Minerva responded to the teacher’s
question about how the area was two.

563 Minerva It’s not.
564 Teacher How can she make it two?

565 Minerva My gosh! Look! Minerva sounds exasperated.
She grabs her paper and holds it
up.

566 Teacher Is that the way we teach The teacher talks to Minerva.

somebody? Is that the way:

567 Minerva We’ve been talking about it for (2
second pause) hours.

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In this episode, Minerva continued the identification of Jessica as a watchful learner and
herself as a model for Jessica. She held up her paper (Line 565) and told Jessica to
“Look!” sending the message that Minerva had the answer on her paper and if Jessica just
looked at Minerva’s paper, she could also get the answer. Minerva’s next statement,
“We’ve been talking about it for hours” (Line 567) also reinforced the message that
Jessica had access to the necessary information. Minerva was communicating that she
had been available as a model for Jessica. If Jessica had attended to Minerva’s modeling,
she would already have the answer. With this statement, Minerva not only identified
Jessica as a watchfiil learner, she also identified Jessica as unsuccessful in her watchful
learning.

Minerva then shifted her identification of herself from model to director and of
Jessica from watchful learner to directed learner. Her next several utterances were a
series of commands: “Not like that” (Line 572), “Put them together!” (Line 578), “Put
them together. Cut them put them together. Glue them together. Gosh.” (Line 582) and
“Connect it to the two squares” (Line 585). Each of these commands was directed toward
Jessica, telling her what she should or should not do.

One further piece of evidence of Minerva’s identifying was a statement she made
to the teacher. This interaction occurred just a few moments after the interaction in the
previous transcript. Minerva had just told Jessica to connect a square she was holding to
the two squares:

Excerpt 15
585 Minerva Connect it to the two squares. Minerva is exasperated.

586 Teacher You can’t say it like that. That is She talks to Minerva.
so mean.

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587 Minerva I’m just sitting here telling her She responds to the teacher.
what to do.

On Line 587, Minerva summarized her activity in this episode: She was sitting and telling
Jessica what to do. Jessica could be successful by merely following the orders issued by
Minerva. Jessica did not need to reflect, explain, hypothesize, or describe. She just
needed to follow Minerva’s directions.

Jessica might have been willing to enact the directed learning Minerva proposed.
However, Minerva did not provide Jessica with enough information to follow her
directions. For example, Minerva, Bonita, and the teacher were trying to help Jessica
understand that she needed to take the two small triangles in Figure I and put them
together to make a square. Jessica had just demonstrated that she could take two copies of
Figure I and put them together to make a large square (See Figure 7.2). Jessica did not yet
understand that she needed to work with the smaller triangles in Figure I. Minerva
directed Jessica:

Excerpt 16

578 Minerva Put them together! Minerva emphasizes each
syllable in each word.

579 Jessica I just did. You said it was wrong.

Although Minerva attempted to tell Jessica what to do, her use of “them” (Line 578) was
not specific enough to direct Jessica to work with the small triangles. Instead, Jessica
protested that she had followed Minerva’s directions and that Minerva was contradicting
herself (Line 579).

Minerva and Jessica had the same type of misunderstanding just a few turns later.
Excerpt 17
582 Minerva Put them together. Cut them put Jessica cuts the triangles from 1.

them together. Glue them
together. Gosh.

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Thank you

583 Jessica How am I going to put them Jessica is holding up the square
together when I only got one of from 1.
these.

Minerva issued a series of commands and it seemed that Jessica understood what
Minerva wanted because Jessica started cutting the small triangles from Figure I.
However, Jessica’s question in Line 583 demonstrated that Jessica was still not
interpreting “them” as the two small triangles from the figure. Instead, Jessica was
focused on the square she had cut from Figure I.

These two interactions demonstrate how Minerva’s attempts to position Jessica as
a directed learner were unsuccessful because Minerva’s mathematical discourse was not
specific enough. Jessica was attending to Minerva’s directions. However, because she
was unable to make sense of “them”, Jessica was unable to do as Minerva asked.

Instead of enacting directed learning as commanded by Minerva, Jessica enacted
guided learning as supported by Bonita and the teacher. The teacher and Bonita did not
give Jessica orders but instead reacted to her moves with questions and guidance that was
tailored to helping Jessica bridge the gap between what she was doing and what Bonita
and the teacher wanted her to do. For example, when Jessica placed the two copies of
Figure I together so they made a large square (Figure 7.2), both Bonita and the teacher
responded in ways that guided Jessica toward what they wanted her to do.

Excerpt 18
573 Teacher We are comparing just this piece She points at I and then at H.

[Figure I] with just that piece
[Figure H].

574 Bonita How could you make a square Bonita points at Jessica ’s paper.
right here?

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The teacher pointed out that Jessica needed to compare Figure H with Figure l (Line 573)
and Bonita asked how Jessica could make a square (Line 574). These responses provided
Jessica with information about what to compare and what she should be attempting to do
without providing her with specific directions about what to do. In these responses, the
teacher and Bonita identified as helpfiil teachers who, like Minerva, knew what Jessica
should do, but unlike Minerva, expected Jessica to use their questions and feedback to
make her own decisions about how to proceed toward the end they had in mind.

Throughout this episode, Jessica enacted guided learning. She did not have
enough details from any one person to be directed. Instead, she needed to make sense of
what each person was trying to get her to do. She enacted her interpretation of the
directions from Minerva and the guiding questions and comments fi'om Bonita and the
teacher. In moments where she could not make sense of the directions or questions, she
either protested (“I just did. You said it was wrong.” Excerpt 16, Line 579) or asked for
clarification (“How am I going to put them together when I only got one of these?”
Excerpt 17, Line 5 83). These responses indicate that Jessica was trying to follow the path
indicated by Minerva, Bonita, and the teacher.

The kind of learning Jessica enacted here was different from the directed learning
enacted by J akeel. When J akeel was directed, he focused on following directions and not
on the mathematical discourse. His mathematizing was limited to writing what Rebecca
uttered without investigation or verification of the discourse. He was content to keep the
mathematical discourse as a discourse-for-others rather than work to turn it into a
discourse-for-himself. In contrast, in this episode, Jessica could not merely copy or repeat

the mathematical discourse provided by others. Either the discourse was too vague (in the

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case of Minerva’s commands) or it was a question or piece of information that she
needed to work from. The mathematizing of others required Jessica to respond by
drawing upon the mathematical discourse she owned. This mathematizing is a significant
reason why Jessica was not engaging in directed learning during this episode.

Jessica was also not participating in engaged learning. When J akeel enacted
engaged learning, he performed several specific mathematizing and identifying activities.
In particular, he produced his own new discourse, positioned his peer teachers as co-
leamers, and refused to be directed. Jessica’s activity during this episode did not reflect
these same features. She did not produce new mathematical discourse. Jessica had four
turns during this episode. Two of these turns, Lines 579 and 583 above, reflected
Jessica’s confusion over what she should do. The other two turns, as demonstrated below,
were short responses to questions about geometric shapes:

Excerpt 19
576 Teacher What two shapes make a square?

577 Jessica A triangle

Later, as Jessica continued to move shapes on her paper, she had this conversation with
the teacher:

Excerpt 20
588 Teacher Take the two triangles.

Now what is that Jessica? Jessica moves the
two triangles so they
form a square.

589 Jessica A square

None of Jessica’s responses show Jessica producing new mathematical discourse. Jessica
also did not position her peers as co-learners. Instead of sharing ideas with Bonita and
Minerva and expecting them to learn with her, Jessica turned to her peers and her teacher

to help her get their answer on her paper. Jessica’s activity throughout this episode

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demonstrated that she was working to physically construct the solution known by others.
Her focus on generating what others wanted and not on using the interactions to develop
her own discursive tools is a key difference between Jessica’s enactment of guided
learning in this episode and J akeel’s enactment of engaged learning in his group.
Jessica’s enactment of guided learning had some similarities to and some
differences from watchful learning in terms of Jessica’s identification of her audience and

her mathematizing. I discuss in the next sections.

Identifying the Audience

As Jessica enacted guided learning, she identified Minerva, Bonita, and the
teacher as helpful guides. She directed her discourse toward Minerva, Bonita, and the
teacher. She was attempting to construct the image they had in mind. She needed their
feedback on whether her actions were leading her toward the end they had in mind. She
attempted to follow their directions and hints, arranging and cutting Figure I in response
to their feedback. She was willing to respond to their questions (See Excerpt l9 and 20)
and she held them accountable for their feedback. When they provided her with
seemingly contradictory requests, she called them on it, protesting that she had done what
they said (Excerpt 16, Line 579) or asking how what they wanted was possible (Excerpt
17, Line 583). She was working hard to follow their guidance: They needed to be sure it
made sense.

While Jessica’s identification of Minerva, Bonita, and the teacher were different
as she enacted watchful learning and guided learning, her identification of herself was
consistent: As in watchful learning, when Jessica enacted guided learning, she did not

seem to explicitly include herself as a member of the audience. There was no moment in

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which she visibly (even if silently) spoke to herself or monitored her communication with
others (as J akeel did when he stopped himself before he counted all of the squares in
Figure J). While it is possible that Jessica monitored her words or communicated with
herself in ways that weren’t visible, her lack of explicit examples indicates that she was

not a primary audience for her discourse.

Mathematizing

Jessica’s mathematizing as a guided learner was different from her mathematizing
as a watchful learner. During the time in which she enacted guided learning, her focus,
and that of her peers and the teacher, was on generating a particular arrangement of the
pieces of Figure I on her paper. This focus and the shift in interactions resulted in
different adoption, production, and substantiation of discourse for Jessica.

As I noted earlier, Jessica’s verbal discourse as she enacted guided leaming was
limited to four turns. In these turns, she did not demonstrate that she adopted discourse.
Instead, she named shapes (Excerpt 19, Line 577 and Excerpt 20, Line 589) or reacted to
problems with directions (Excerpt 16, Line 579 and Excerpt 17, Line 583). However, the
emphasis of this guided learning time was the production of a particular arrangement of
Figure I on Jessica’s paper and not on her verbal discourse. If Jessica had copied the
arrangement from Bonita and Minerva’s paper and then demonstrated that she was using
the image, I could argue that she had adopted the picture as a part of her discourse.
However, the interactions of guided learning were focused on Jessica’s production and
not adoption of the arrangement. (As I noted earlier, Minerva’s contribution to Jessica’s

guided learning had more to do with Minerva’s vague discourse than upon a desire to

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have Jessica work to figure out the preferred arrangement.) Jessica was to take the
feedback from the group and use that to generate the desired image.

She was eventually able to do this. However, her creation of the image was highly
structured and guided by Minerva, Bonita, and the teacher. As a result, Jessica’s
generation of the image did not seem to arise from earlier adoption of discourse,
indicating that the image was not a production of discourse for Jessica. Instead, it seemed
largely a result of guidance and input from the teacher, Minerva, and Bonita. While
Jessica was the one who arranged the pieces, others told her when her moves were right
or wrong: When Minerva and Bonita disagreed with her placement of the figure, they
simultaneously exclaimed “Not like that!” (Lines 571 and 572). As Jessica got closer to
constructing the desired picture, Minerva watched her and, when Jessica had the triangles
together as a square, Minerva told her “You got it!” (Line 591). This feedback meant that
Jessica did not have to decide whether her picture was complete or not: She could rely
upon Minerva and Bonita and did not have to reason from her adopted discourse about
the area is two because two triangles make a square. Further evidence that she did not
produce the image by reasoning from her adopted discourse was her limited ability to
connect the image and the verbal discourse when asked by the teacher (See Excerpt 5,
Line 613).

The connection between the image and the verbal discourse was an important
piece of Jessica’s substantiation of narratives as she enacted guided learning. Jessica
needed to create the picture that Bonita, Minerva, and the teacher were describing with
their words. Ultimately, whether a statement reflected the state of affairs depended upon

Bonita, Minerva, and the teacher’s opinions. However, as Jessica worked to translate

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their words into a picture, she also evaluated whether their words made sense. On two
occasions, Minerva told Jessica to do something that seemed problematic to Jessica:
Minerva’s commands contradicted Jessica’s understanding of how things were. For
example, Jessica had just put two copies of Figure I together and was told this was wrong
when Minerva told Jessica to “put them together” (Excerpt 16, Line 578). To Jessica, this
command contradicted the information she had just received about what she had done
and she protested, “I just did. You said it was wrong” (Line 579). At this point, Bonita
intervened and showed Jessica how she was to put two triangles together (Line 580, not
included in this document). Minerva’s command to put them together had many possible
interpretations, one of which did not match Jessica’s understanding of the state of affairs.
Jessica needed Bonita to show her what was the correct interpretation. A similar sequence
of events happened on the second occasion when Minerva’s commands were problematic
for Jessica. In Excerpt 17, Line 582, Minerva uttered a long sequence of commands,
which involved doing various things to “them”. Jessica was confused because she was
focused on what to do with a single square and she couldn’t make sense of how to put
them together when she only had one (Line 583). The teacher intervened to help Jessica
understand that she was supposed to be working with two triangles (Excerpt 20, Line
588). In both cases, Jessica substantiated narratives and found that they did not reflect
what she thought was possible with the picture. She was then shown she was wrong in
her substantiation. While she was more actively involved in substantiation when she
enacted guided learning than when she enacted watchful learning, her substantiation still

relied upon the evaluations of others.

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Return to Watchful Learning

Jessica’s enactment of watchful or guided learning largely depended upon her
interactions with others. When her peers refused to mathematize with her, she enacted
watchful learning. She transitioned from watchful learning to guided learning when the
teacher intervened and required Bonita and Minerva to support her work with
mathematical objects. She transitioned back to watchful learning when her arrangement
of Figures H and I met the standards of the teacher, Bonita, and Minerva. Once Jessica
had the teacher-approved arrangement on her paper, Minerva handed out the papers for
the next task with Figures J and K. Jessica watched as Minerva began cutting out the
figures and then began cutting her own figures. Minerva and Bonita were no longer
guiding her by issuing commands or asking questions. Instead, Jessica, if she didn’t

already know what to do, was to learn by watching.

Autonomous Learning

As Jessica enacted both watchful learning and guided learning, there were
moments in which her learning activities suggested the possibility of autonomous
learning. For example, early in the lesson, Jessica explained how two triangles make a
square to Minerva (Chapter 6, Excerpt 4). Jessica was sharing her understanding with
another student and indicating her interest in explanations. If Minerva or Bonita had
encouraged her explanation, asked her for more detail, or allowed her to wonder how her
explanation was different fi'om Bonita’s (who focused on the small triangles in Figure I
while Jessica focused on Figure I as a triangle), the group might have embarked on

activities resembling autonomous learning. Jessica’s willingness to explain and put forth

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her ideas was a first move toward autonomous learning. However, neither Minerva’s nor
Bonita’s subsequent actions supported the autonomy in Jessica’s first move.

Jessica provided another opportunity for autonomous learning when she asked
Minerva why she was cutting out another copy of Figures H and I (Excerpt 8, Line 263).
Jessica noticed that Minerva was doing something that didn’t make sense to her and so
she asked for an explanation. Minerva could have responded in a way that would have
initiated a mathematical conversation or would have validated Jessica’s curiosity. Instead,
Minerva’s dismissive remark (“Because I’m doing something”, Line 264) sent the
message that exploration of her activities was not welcome.

Instead of continuing to ask questions or offer explanations, Jessica watched her
peers and adopted their words when sanctioned by the teacher. This watching, adoption
without production, and substantiation by teacher sanction were not autonomous learning
activities. While Jessica’s initial activities during this lesson showed the potential for
autonomous learning, the reaction of her peers did not support the growth of this
potential.

Jessica’s enactment of guided learning also seemed to have potential, but not
realization, of autonomous learning activity. As in watchfiil learning, there were two
occasions in which Jessica noticed, and commented on, a problem between the discourse
of Minerva and what Jessica thought was possible with the picture (Excerpts 16 and 17).
Jessica comments could have opened the possibility for conversation around what Jessica
thought was problematic, what Minerva intended, and the implications for each
perspective for the area of Figure I and for comparing the area of the two figures.

However, in both moments when Jessica questioned Minerva’s directions, she was shown

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(by Bonita and then the teacher) what Minerva meant in a way that indicated that
Minerva’s thinking is correct. While these responses were helpful and supportive of
Jessica, neither response encouraged Jessica to continue her exploration of the
discrepancy between her ideas and Minerva’s. Instead, Bonita and the teacher indicated
that Minerva was right, Jessica was wrong, and Jessica needed to work with the two
smaller triangles in order to get the right solution. Jessica’s enactment of guided learning,
like her enactment of watchful learning, contained moments in which Jessica raised
questions about problems she noticed with discourse. Like watchful learning, the
response to Jessica’s problems did not encourage autonomous learning and Jessica did
not continue to ask questions. Thus Jessica’s opportunities to engage in autonomous

leaming were, in part, constrained by the reactions of her peers and the teacher.
Discussion

Intertwining of Mathematizing and Identifiing

Jessica’s enactment of watchful learning and guided learning contain several
instances in which mathematizing and identifying were intertwined. In Chapter 5, I
emphasized how mathematizing identified J akeel in different ways as he enacted engaged
and directed learning. In Chapter 6, I described how Minerva resisted the identification of
learner as constructed by the mathematical explanations offered by Jessica and Bonita.
This case of Jessica’s learning continues the theme from Chapter 6: There are multiple
moments in which mathematizing fails to construct identities because of the reaction to
the mathematizing.

As I described earlier, Jessica attempted to initiate mathematical conversations on

three different occasions and on each of these occasions, her mathematizing met with

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resistance (See Excerpt 7, Excerpt 8, and Chapter 6, Excerpt 4). Jessica seemed to be
identifying as someone who wanted to be engaged in learning with others. She wanted to
share her ideas and talk with others about what they were doing. With her attempts at
mathematizing, Jessica also identified Bonita and Minerva as interested in talking about
mathematics and the task. However, both Minerva’s and Bonita’s replies rejected
mathematizing and this identification: Minerva’s “Nah huhn” dismissed the explanation.
Bonita first ignored (Excerpt 7) and then downplayed (Excerpt 8) Jessica’s attempts to
start conversations around the task. Minerva and Bonita rejected Jessica’s positioning of
them and proposed instead that they not talk about mathematics (or in Minerva’s case,
that they not talk at all). With these rejections, Minerva and Bonita did not explicitly
identify Jessica. They did not suggest that she act in a particular way. Instead, they
communicated that she should not act in ways that required them to explicitly act as her
co-learners or as her teachers. Because Minerva and Bonita were unwilling to
mathematize with Jessica, Jessica was left to enact a kind of learning that did not require
interactions. She could have worked independently. However, as suggested by her earlier
mathematizing, Jessica seemed to prefer to learn with and from others. As a result, she
enacted learning activities that allowed her to learn from her peers without requiring them
to explicitly teach her. Minerva and Bonita’s unwillingness to mathematize with others
did not identify Jessica in particular ways, but it did restrict the ways in which she could
identify.

Minerva also mathematized in ways that offered an identification of Jessica that
was not taken up by Jessica. During the time in which Jessica enacted guided learning,

Minerva attempted to direct her by issuing commands. These commands identified

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Jessica as “under orders”: Minerva was telling her what to do. Jessica did not need to ask
questions or investigate. She just needed to do what Minerva commanded. However,
Minerva’s commands were too vague. Jessica could not make sense of how to “put them
together” (Excerpt 16, Line 578 and Excerpt 17, Line 582). Jessica could not act as a
directed learner. In contrast, the mathematizing of Bonita and the teacher identified
Jessica as someone who could and should reason from information to figure out the
answer (See Excerpt 18). They were willing to provide her with some guidance (“We are
comparing just this piece with just that piece”, Line 573) and ask questions (“How could
you make a square right here?” Line 574). These words left enough ambiguity that
Jessica was required to reason from this information and what she knew if she was going
to figure out what she was supposed to do. Minerva’s commands, because they were
vague, also required Jessica to reason. Thus Minerva seemed to want identify Jessica as
directed but her commands unintentionally supported Jessica’s identification as guided.
Finally, Jessica identified herself and others such that the mathematizing in the
group changed. Toward the end of the work on Figures H and I, the teacher realized that
Jessica did not have a solution to the task. Jessica was able to state the area for Figure I
and the narrative that two triangles made a square, but she could not show this on her
paper. The teacher told her that she needed the answer on her paper and Jessica replied,
“Well there ain’t nobody helping me” (Excerpt 14, Line 561). With this statement,
Jessica identified Bonita and Minerva as unhelme peers and identified herself as
someone who wanted help. The teacher reacted immediately and endorsed both of these
identifications: She required Bonita and Minerva to get out of their seats and help Jessica

and she sat down next to Jessica and began to guide her learning (Line 562). The

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resulting interactions involved more mathematizing than occurred in this group when the
teacher was not present. While Jessica said very little, Bonita, Minerva, and the teacher
watched her work and provided feedback on her actions. Thus Jessica’s identification as

needing help resulted in a change in mathematizing.

Development of Mathematical Discourse

Jessica’s mathematical discourse changed during the course of this lesson. By the
end of the lesson, she could state the area for Figure I and state a narrative that addressed
the triangular spaces. She also had an arrangement of the pieces of Figure that that
assembled the triangles into a square and placed the squares together as a rectangle.
These changes have features of the desired mathematical discourse. However, as I
discussed in the section on Jessica’s final discourse, there are some aspects of these
changes and Jessica’s final discourse that are troubling. She was not clear how
assembling the triangles into a square helped her determine the area, she did not connect
her words with her picture, and she was not able to use this discourse during a subsequent
lesson involving a problem with the same shape. These outcomes suggest that the
activities of watchful learning and guided learning, while useful in supporting Jessica in
attaining narratives and a picture, were not sufficient to help her learn the discourse so
that it could be a tool in other situations. There are two points I want to make about how
the mathematizing and identifying in this lesson seemed to have consequences for
Jessica’s final discourse.

My first point is specific to Jessica’s enactment of watchful learning. As a
watchful learner, Jessica had opportunities to observe the mathematical discourse of

others, but she did not have opportunities to explore or practice the discourse. Bonita and

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Minerva’s refusal to mathematize with Jessica meant that Jessica did not ask questions,
respond to others’ questions, try the discourse, or receive feedback on the discourse.
Jessica’s work was to learn the right discourse through observation only: Others would
mathematize and Jessica could use that to determine the right answer. While Jessica
succeeded in mastering the right answer, her limited mathematical interactions and
Bonita and Minerva’s refusal to explicitly identify as teachers meant that Jessica did not
have opportunities to develop the mathematical discourse as a tool for herself.

Second, as a watchfirl learner and as a guided learner, Jessica’s goal was to attain
the right answer, either the right words (watchful learning) or the right picture (guided
learning). The focus on attaining the right answer meant that Jessica (and her peers) did
not work toward what was mathematically logical and did not build from what she knew.
Instead, she watched to see or listened to figure out the right answer. Furthermore, when
Jessica enacted guided learner, she only worked (and was only supported in working) on
getting the right picture: There was no attempt by Jessica, her peers, or the teacher to
connect the picture to the original task or the narratives about area. This focus on getting
the right answer suggests that Jessica, Bonita, Minerva, and the teacher understood
learning in terms of a metaphor of acquisition (Sfard, 1998). Rather than emphasize what
Jessica was doing during the lesson, everyone (including Jessica) emphasized whether
Jessica could recite the right answer and had the right picture. This emphasis on what
Jessica had at the conclusion of the task rather than upon what she did as she worked on
the task helped Jessica develop her mathematical discourse as a set of answers to tasks,

but not as a tool for problem solving.

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Autonomous Learning

The lens of autonomous learning offers additional insights into connections
between Jessica’s learning activities and the changes in her discourse. I defined
autonomous learning as the constellation of mathematizing and identifying activities that
demonstrate curiosity about what others say or what was true. In order to demonstrate
curiosity, Jessica needed to investigate what others said and pose problems based upon
her observations. As I have emphasized earlier, there were moments in which she seemed
curious, but her curiosity was not encouraged by her peers. While Jessica was interested
in what others said, for most of the lesson, she did not visibly investigate the discourse of
others or pose problems about her observations. Her interest in others’ discourse was, by
the end, an interest in whether their discourse was the right discourse. While enacting
guided learning, she protested and asked questions when she did not understand, but
again, her curiosity and problem posing was dismissed as Bonita and the teacher guided
her to the right answer.

Jessica’s activities as a watchful learner, aside from her initial explanation and
attempts to start conversations, did not resemble the activities of autonomous learning. As
a watchful learner, Jessica’s audience was ultimately the teacher who could approve her
work: She was not a member of her audience. She adopted discourse but did not produce
discourse and she relied upon the teacher to substantiate her discourse. Jessica’s watchful
learning activities did not resemble that of autonomous learning and the outcomes of her
learning were also not the outcomes of autonomous learning. In particular, Jessica’s final
discourse was a discourse-for-others and not a discourse-for-herself. According to Sfard

(2008), discourse-for-others is “discourse in which one engages only with those for

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whom this discourse makes sense and for the sake of communication with these other
people (as opposed to practicing this discourse in self-communication)” (p. 297). Jessica
only mathematized when asked by the teacher and she repeated the teacher-approved
discourse used by Bonita and Minerva. When asked by the teacher, Jessica could not
explain the narratives she uttered or link them to the work on her paper. Jessica’s final
discourse was not one that she used to communicate with herself. Nor was it one that she
probed and evaluated. Jessica’s final discourse was used to communicate to the teacher
that she had the right answer and not that she understood the mathematics of the task: It
was a discourse-for-others.

Jessica’s final discourse might have been different if she had been encouraged to
continue the curiosity she demonstrated early in the lesson. Dewey defined curiosity as
“interest in problems provoked by the observation of things” (p. 33, italics in original,
1910). Jessica displayed such an interest when she asked Minerva why she was cutting
(Excerpt 8, Line 263). If Bonita and Minerva had responded to this overture and Jessica’s
other attempts at mathematical conversation with interest in her explanations and
mathematical responses to her questions, Jessica might have continued to be curious
about the task. Through exchanges with Bonita and Minerva and continuing observation,
she might have explored their discourse and used it to communicate with herself. Instead,
Jessica’s preference for learning with others coupled with Bonita and Minerva’s
preference not to overtly teach or learn meant that Jessica primarily watched and adopted
discourse and did not pose or explore problems on her own.

Jessica’s enactment of guided learning also lacked many features of autonomous

learning. However, guided learning initially seemed to be very similar to autonomous

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learning. Bonita, Minerva, and the teacher did not tell Jessica what to do, how to
construct the final arrangement, or what the final arrangement was (Minerva attempted to
more direct but her discourse was too vague to make clear what Jessica should do).
Jessica was not expected to copy someone’s work. Instead, Bonita, Minerva, and the
teacher provided Jessica with information and clues and Jessica was to reason from those
clues. This reasoning could be seen as an autonomous learning activity because Jessica
was expected to do intellectual work. This expectation matches Warfield, Wood, and
Lehman’s definition of autonomy as the ability to solve mathematical problems without
being shown the solution by another person (Warfield et al., 2005). This definition
suggests that the key feature of autonomy is the lack of input fi'om others. It implies that a
teacher interested in promoting autonomy should help his/her students become more
independent in their work. However, Jessica’s learning suggests that working
independently is not a necessary or sufficient criterion for autonomous learning. Even if
Jessica had been able to construct the desired arrangement of Figure I without input from
her peers or the teacher, this activity could fail to be autonomous if her purpose was
limited to constructing the right arrangement.

Autonomous learning is not merely about requiring students to reason on their
own and it does not have the right answer as its end goal. Autonomous learning involves
investigation of problems the student has posed based upon observations. This does not
mean that the student won’t construct the right answer, but it does mean that the goal of
the student’s work is solving problems related to discourse. Arriving at the right answer
is just one of the outcomes of this problem-solving work. Jessica’s work to construct the

right arrangement of pieces did not involve her in a problem arising from her

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observations. Instead, the problem she needed to solve was figuring out what Bonita,
Minerva, and the teacher had in mind as they guided her toward the desired arrangement
of Figure I. Jessica was not identified (and did not identify herself) as someone who was
critically evaluating discourse. Instead, she was someone who had failed to arrange
Figure I on her page and who needed to craft that arrangement. While the teacher and
Bonita (and, inadvertently, Minerva) had the best intentions as they guided (rather than
directed!) Jessica toward the desired arrangement, this guiding learning, because it had
the right answer as an end goal, could not provide the problem-posing environment
necessary for autonomous learning.

The activities of guided learning also failed to resemble autonomous learning in
other ways. Jessica identified her audience as helpful guides who were monitoring her
work. They were not co-learners, fellow explorers, or teachers interested in explanations
which might be the case in autonomous learning. Also, an autonomous learner includes
him/herself in their audience: Jessica did not seem to have herself as a member of her
audience. Also, the mathematizing of guided learning differed from that of autonomous
learning. Both kinds of learning emphasize production of discourse, but Jessica was
expected to produce a particular image by following hints rather than producing discourse
by investing and exploring narratives and images. Jessica was also not expected to adopt
the discourse used to guide her. She needed to follow the guidance, but there were no
opportunities for her to demonstrate its use. Finally, Jessica’s substantiation of discourse
during guided learning relied upon others. She asked questions or commented when the

clues she was given did not match her picture. However, she did not explore and was not

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encouraged to explore her questions or conflicts. Instead, she was guided toward seeing
what she should be thinking.

As in watchful learning, the activities that contributed to getting the right image
did not support Jessica in communicating with herself about the image. She crafted the
image that others wanted. As a result, the arrangement of pieces was discourse-for-others
and not discourse-for-herself. This became evident as she was later asked to show how
the arrangement showed Figure I as two (Excerpt 5). Jessica’s gestures indicated that her
verbal explanation of “two triangles together” makes two (Line 613) might not have been
connected to the arrangement on her paper and to the area of Figure I. Also, her inability
to use the discourse in the subsequent lesson provides further evidence that the discourse
was not a discourse-for-herself.

As Jessica enacted both watchful learning and guided learning, there were
moments in which she showed curiosity and posed problems based upon her
observations. However, her curiosity was not supported by her peers or by the end goal of
reaching the right answer. Jessica could have attempted to explore the discourse on her
own or she could have persisted in asking mathematical questions of her peers, but she
seemed to want to learn with or from her peers and she was responsive to their refusal to
answer her questions or discuss mathematics with her. In the end, very few of Jessica’s
learning activities were autonomous learning activities and her final discourse was a
discourse-for-others. Perhaps if she had persisted and had been allowed to persist in her
curiosity her final mathematical discourse would have been a more mathematically

appropriate discourse-for-herself.

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Reconceptualizing Jessica as a Struggling Student

The careful study of Jessica’s activities enabled by this data suggests the need to
take a closer look at the notion of “struggling student”. I initially selected Jessica as a
focal student because I was interested in what I had framed as her struggles to learn
mathematics: I had noticed that she copied from her neighbors during math lessons.
Furthermore, her peers complained that she played too much. I had assumed that copying
and playing were strategies to avoid engaging in intellectual work involving mathematics.
However, copying and playing are not truly struggles to learn. Instead, struggling to learn
would mean the learner was confronting challenges in investigating and exploring
discourse. Thus the label of “struggling student” is problematic because as a euphemism
for “failing student” it hides the ways in which students may not be struggling to learn at
all because they have given up and are copying from their neighbors or because they do
not have opportunities or encouragement to investigate discourses. In addition, the use of
struggle to label failing students implies that learning should not be a struggle. It
reinforces the notion of the need to already know examined in the previous chapter:
Minerva, as she enacted covert learner, did not want to be seen as struggling to know and
yet the struggle to know should be a central focus of activity in schools.

Having just argued that I should not have originally labeled Jessica as a struggling
student, I will now argue that struggling student is an appropriate label for her at times
because it captures her efforts to engage with discourse. During this case lesson, Jessica
demonstrated that she was interested in discussing mathematics and she had questions
about her work and the work of others. At moments, her mathematizing and identifying

indicated that she was curious about mathematical discourse. However, her peers were

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unwilling to accept her mathematical overtures and Jessica seemed unwilling to risk the
social consequences of insisting upon a math—talk agenda. Thus Jessica did struggle to
learn. However, her struggles to learn mathematics were not about what we would want
her to struggle with (the discourse) but were more about her peers’ unwillingness to
mathematize or to identify as co-leamers or teachers.

This conclusion represented a shift from my original thinking. It was only
possible through examination of the whole lesson: If I had only sampled parts of the
lesson, I might not have seen Jessica’s attempts to engage her peers and might have
assumed that her watchful learning reflected her reluctance to mathematize. Instead,
detailed examination of the whole case demonstrates that Jessica did struggle to learn
mathematics, but that her struggles focused on the engagement of her peers and not on
the kind of struggle we might want to encourage in schools: struggles to investigate

discourses.

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CHAPTER 8
DISCUSSION & CONCLUSION

In Chapter 1, I present a vignette that described how unan acquired the answer
to a mathematics problem by copying from his neighbor. unan’s copying was not a
unique or isolated event: I have seen students borrow answers from others on numerous
occasions, including in my own third and fourth grade classrooms. My research goal was
to make sense of students’ activities as they engaged in learning activities, including the
activity of copying from others. In Chapter 2, I elaborated a theoretical framework that
described how learning mathematics could be seen as the interplay of mathematizing and
identifying activities. Sfard’s commognitive framework (2008) provided the foundation
for describing how mathematizing, identifying, and learning could be defined and
examined through discourse, with the outcome of learning defined as a change in
discourse. I also constructed a theory of autonomous learning that considered what
mathematizing and identifying activities might lead to mathematically desirable changes
in discourse. These frameworks refined my questions about student activity and learning:
I looked at the effects of the activities of mathematizing and identifying on each other
and on the development of mathematics discourse. I also considered what advantage
autonomous learning offered for the development of mathematical discourse.

In Chapter 3, I explained my research methods, outlining my data selection,
collection, and analysis process. I described the classroom, teacher, and students involved
in the study; how I chose which lessons to analyze, and how I examined discourse to
make claims about mathematizing, identifying, autonomy, and leaming outcomes. The

fourth chapter provides background on the lessons and introduces the three findings

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chapters. Throughout the findings chapters, I illustrated how my theoretical framework
and analytic tools provided insights into learning activities, suggesting how student
activity constituted and was connected to interactions with peers and with the teacher and
how those activities related to changes in mathematical discourse.

Each findings chapter offered a discussion of one student’s learning activities
including a comparison of those activities to my theoretical construction of the activities
of autonomous learning. In this chapter, I summarize the claims in the three findings
chapters and look across all three to draw other conclusions about students’ activities and
changes in their discourse. I will first specifically address my two research questions and

then I will discuss other claims that arise from my examination of the data.

Mathematizing, Identifying, and Learning Outcomes

My first research question asked what effect the activities of mathematizing and
identifying have on one another and on the development of mathematics discourse. As I
elaborated in each findings chapter, mathematizing and identifying are intertwined such
that sometimes they are the same activity while at other times the activity of one is tied to
activity for the other. Several studies that have explored the connection between student’s
mathematical activity and their identities haven taken a larger scale view of identity,
looking at multiple individuals across classrooms or at individuals across an extended
period of time. They have looked at how students come to see themselves as math people
(Boaler & Greeno, 2000), how the intersection between pedagogy and identity supported
students (J ilk, 2007), how identities and stories about the future are connected to
engagement with mathematics (Sfard & Prusak, 2005). A smaller scale view offers the

possibility of adding complexity to the constructions of identity already elaborated by

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these researchers. This smaller scale view can capture the ways in which identities might
be more fleeting or fluid (for example, Jessica’s few moments as an explainer of
mathematics toward the beginning of her lesson). It also provides a glimpse at
mechanisms of identity construction (for example, how Minerva learns while conveying
that she already knows) and how interactions between students construct identifications.
These cases of micro-analysis also show connections between learning activities and
learning outcomes, suggesting what kinds of mathematizing and identifying activities
might be more conducive to mathematical learning.

In each findings chapter, I discussed how mathematizing was also identifying. For
example, as Rebecca explained to J akeel how she counted the half squares in Figure J
(Chapter 5, Excerpt 8), she identified herself as knowledgeable and as interested in
helping J akeel. She also identified J akeel as capable of understanding and acting on her
explanation. In addition to examining in detail how mathematizing identified others,
close analysis of the interactions also showed that all three cases contain examples of
how students rejected the identifications constructed by others through mathematizing.
For example, Rebecca repeatedly attempted to direct J akeel by telling him what to write.
While he did eventually allow her to tell him what to do, he initially rejected her
directions (3 times), telling her that he knew what to do (Chapter 5, Excerpt 7). In the
case of Minerva’s learning, Minerva undermined the explanations of Jessica and Bonita,
disagreeing with their identification of her as not understanding how two triangles make a
square (Chapter 6, Excerpt 4). Finally, in the case of Jessica’s learning, she attempted to
mathematize with Minerva and Bonita, identifying them as helpful, but they refused her

overtures (Chapter 7, Excerpts 7 and 8). These moments of disagreeing with

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identifications provide glimpses into the identities students were trying to communicate
and provide insights about student activity that might not be available if I had constructed
identity as a more static feature or as only visible across a larger span of time.

This close examination also shows the social work accomplished through
mathematizing. Historically, mathematics has been viewed, even celebrated, as acultural:
l + 1 = 2 around the world. More recently, this view of mathematics has changed and
some mathematics educators recognize the ways in which mathematics is socially
constructed, cultural knowledge (Bishop, 1988). However, mathematical discourse
maintains some of the sense of aculturalism through its objectified, impersonal sentence
structure. Sfard (2008) notes that alienation is one outcome of the objectification that
occurs in mathematical discourses. The sense of human agency is removed through the
use of the passive voice and use of mathematical nouns as subjects in sentences. These
cultural conventions create the impression that mathematical utterances are entirely about
mathematical content and free of messages about the speaker and his/her audience even
as they simultaneously identify the speaker and the audience. For example, when
Rebecca states, “H and I cover the same amount. Figure I has one square and two half
squares that equals two squares.” (Chapter 5), she does not mention herself, Daren, or
J akeel. However, with this statement, she identifies herself as knowledgeable and begins
to position Daren and J akeel as needing to copy from her. As I began this study, I
partitioned student activity into mathematizing activities or identifying activities.
However I soon realized that students were sending messages and accomplishing social

work with talk that seemed to be only about mathematical objects. The close examination

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of student’s mathematical utterances and the reactions of other students made it clear that
mathematizing activities were also identifying activities.

Identifying activities can also have implications for mathematizing, even when
they do not specifically address mathematical objects. Each of the cases illustrates this. In
Chapter 5, Rebecca identified J akeel as playing (Excerpt 11) and implied that he needed
to write the words she dictated, suggesting that J akeel’s mathematizing should be limited
to recording her words. Minerva’s sequence of “Nah-huhn”s (Chapter 6, Excerpt 4)
identified her as not interested in mathematical explanations (even as she timed her
utterances so that she could hear the explanation). Finally, in Chapter 7, Jessica identified
Bonita and Minerva as not helping her (Excerpt l4), prompting the sequence of
exchanges in which they guided her learning. In addition to demonstrating a link between
identifying and mathematizing, this study also demonstrates the ways in which
identifying activities affect multiple individuals. For example, when Jessica identified as
needing help (Chapter 7, Excerpt 14), her request changed the mathematizing for
everyone in this group. Immediately following the request, Bonita, Minerva, and the
teacher began to ask Jessica about and provide guidance for the mathematical task.

Finally, this study examines how mathematizing and identifying are involved in
the development of mathematics discourse. As students identified themselves and others,
they suggested and enacted different mathematizing activities. The mathematizing
activities provided or prevented opportunities to explore and practice discourse, which
seemed to be linked to changes in discourse. For example, as J akeel enacted engaged
learning, he identified as interested in exploring discourse. He asked questions and

worked to change his discourse. In the end, the change in his mathematical discourse was

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the most desirable of the three students. In contrast, Minerva’s enactment of covert
learning prevented her from asking questions or investigating other’s mathematical
discourses. She seemed to want to appear as knowledgeable and did not mathematize
with Bonita or Jessica unless required to do so by the teacher. In the end, the change in
her mathematical discourse contained some mathematically undesirable features.
Across the five kinds of learning, the mathematizing and identifying that seemed
to be linked to the most desirable change in discourse were ones in which the learner
identified and was identified as capable of exploring other’s ideas and of working from
those ideas to his/her own ideas. In contrast, the learning outcomes that were the least
mathematically desirable were those in which the mathematizing focused on right
answers and the learner relied upon the teacher or peers to approve their work. This
conclusion supports Ben-Zvi and Sfard’s learning-teaching agreement (2007). Ben-Zvi
and Sfard argue that changing the learner’s discourse arises from an unwritten agreement
about how the learning is to happen. They specify three requirements for positive
learning outcomes: The learner and his/her teacher (including peer teachers) must agree
on whose discourse is to be learned, on their respective roles of learner and teacher, and
on how the change will proceed. Ben-Zvi and Sfard emphasize that the learner cannot
passively adopt the discourse of the teacher but must rationalize and critically examine
the discourse. J akeel’s enactment of engaged learner demonstrates this. He did not merely
adopt the discourse of Rebecca and Daren, but asked questions and produced his own
discourse based upon their discourse. In contrast, as Jessica enacted watchful learning,
she adopted the narratives of the available discourse but did not examine, rationalize, or

work to connect them to the pictures of the figures.

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This study also shows the challenges learners face in taking on the examination of
others’ discourses. First, this examination requires that learners acknowledge that they
don’t already know the discourse. As I discussed in Chapter 6, this identification as not-
yet- knowing runs contrary to what schools teach implicitly about what it means to be a
good student: When work is assigned, good students are supposed to already know how
to do it. Minerva’s enactment of covert learning displays this tension. Her presentation as
knowledgeable means that she cannot ask questions or examine others’ discourses. If
learners are going to engage in thoughtfirl exploration of others’ discourses, then they
must be willing to identify in ways that may not be valued in many classrooms. A second
challenge of examining others’ discourses is obtaining the support of peers. Jessica’s
enactment of watchful learning and J akeel’s enactment of directed learning demonstrate
that peers may be reluctant to engage with the learner in examining their discourse.
Minerva and Bonita refused to mathematize with Jessica and Rebecca, in many instances,
pushed J akeel to copy from her rather than to ask her questions or explore her discourse.
In order for learners to enact the learning-teaching agreement, they need more than the
willingness to take on critical examination of others’ discourse; they need teachers
(especially peer teachers) who are willing to support them in examining the teachers’

discourses.

Autonomous Learning
My second research question asks about the advantage of autonomous learning
for the development of mathematical discourse. As a reminder, autonomous learning is
the constellation of identifying and mathematizing activities that reflect curiosity about

what others think/say and what seems to be true. My study shows that autonomous

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learning can be a useful lens for evaluating learning activities and outcomes. Examining
the similarities between autonomous learning and each of the five kinds of learning
shows that the most autonomous learning was also the learning with the most
mathematically desirable change in discourse. The lens of autonomous learning offers
reasons for this outcome. I elaborate each of these points after I introduce a table that
summarizes this comparison.

Table 8.1 shows the mathematizing and identifying activities of autonomous
leaming across each of the five kinds of learning I describe in this study. I list the kinds
of learning horizontally. The vertical axis lists each of the mathematizing and identifying
activities of autonomous learning: the learner’s identification of his/her audience,
whether the learner includes him/herself in his/her audience, adoption of discourse,
production of discourse, and substantiation of discourse. The yes or no in each cell
indicates whether the activities of the kind of learning match that of autonomous learning.
The table also includes the discursive outcome of the learning: whether the discourse is
for the learner, others or neither.

Table 8.1 Mathematizing and identifying activities of autonomous learning for each kind
of learning

 

 

 

 

 

 

 

 

Kinds of Learning
Engaged Directed Covert Watchful Guided
Identifying the Yes No No No No
on Audience
,5 Self as Audience Yes, .
5 Yes No limited No No
.3 3 Adoption 0f Yes No Yes Yes No
a .. Discourse
g 5 Production of
E :1!) . Yes No Yes No Yes
2 Discourse
3 Substantiation of
é, Narratives Yes No No No No
”scours" for Himself Others Own Others Others
whom

 

 

 

 

 

 

 

 

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In my discussion of mathematizing and identifying above, I claimed that J akeel’s
enactment of engaged learning produced the most mathematically desirable discourse. As
the table demonstrates, his engaged learning also had mathematizing and identifying
activities that most closely resembled autonomous learning. He identified his audience as
evaluators and explainers, supporting him in his exploration and elaboration of their
discourse. He included himself as a member of his audience, which meant that he was
working to make sense of the discourse for himself. He adopted the discourse of his peers
and built from that discourse to produce his own discourse (the counting and pointing
routine). He substantiated his new discourse himself. By the end of the lesson, his final
discourse was one that he could use to communicate with himself and with others
(discourse-for-himself).

These autonomous learning activities suggest reasons why J akeel’s enactment of
engaged learning may have resulted in the most mathematically desirable discourse. He
worked to make sense of and then use the mathematically appropriate discourse of his
peers. He then verified that their discourse and his new discourse reflected the state of
affairs. He engaged with discourse in order to change his discourse so that it was an
accurate tool for counting figures. J akeel could have asked Rebecca to show him how to
count to get the right count of 8. Then he could have mimicked what she did. Instead, his
activities reflected curiosity about the discrepancy between what he counted and what he
knew to be the correct count and an interest in crafting a change to his discourse such that
it counted 8.

In contrast to engaged learning, the other kinds of learning did not demonstrate as

many activities in common with autonomous learning. Directed, covert, watchful, and

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guided learners did not identify their audience as explainers or co-learners. For the most
part, the learner did not communicate with him/herself (with covert learning as a minor
exception). When the discourse of others was adopted, the learner did not use it to
produce new discourses, so he/she did not explore the implications of the adopted
discourse (see watchful learning). When the learner produced discourse, he/she did not
build from or in reaction to adopted discourse (see covert learning). Instead, the new
discourse was not probed or investigated using the logic of other discourses. Finally, each
of the four other kinds of learning relied upon the teacher or another student to
substantiate the discourse: The learners did not use their logic to determine whether the
discourse described the state of affairs. In summary, as learners enacted directed, covert,
watchful, or guided learning, they did not attempt to engage mathematical discourse in
ways that probed what others had said or whether the discourse seemed to be true. The

discourse was not the object of activity.

The Importance of the Problem and the Problem of the T eacher- Approved Answer
The advantages of autonomous learning for the development of mathematical
discourse might be linked to the problem the learner worked to solve. As I noted above,
as J akeel enacted engaged learning, he was working to solve a problem of his own
construction: how to alter his counting and pointing so that he could count to what he
knew was the true answer. In contrast, for each of the other kinds of learning, the
problem faced by the learner was determining the teacher-approved answer. As Minerva
enacted covert learning, she was working to generate an answer to the task that was
acceptable to the teacher. As J akeel enacted directed learner, he was working to write

what Rebecca dictated in order to get what he was supposed to have on his paper.

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Jessica’s enactment of watchful learning focused on finding the teacher-approved answer
as articulated by Minerva and Bonita. Finally, Jessica’s problem as she enacted guided
learning was attaining the right arrangement of pieces on her paper. For each of these
kinds of learning, the problem of getting the teacher-approved answer did not focus the
learner on exploring or evaluating discourse for truthfiilness.

Focusing on the problem of getting the teacher-approved answer provides limited
opportunities to enact the mathematizing and identifying of autonomous learning. I will
elaborate two reasons for this. First, autonomous learning emphasizes exploration of
discourses. As J akeel enacted engaged learning, he worked to change his discourse. He
listened to and incorporated Rebecca’s explanation and he experimented with his
discourse until he was satisfied. In contrast, as the students in this study enacted directed,
covert, watchful, and guided learning, they did not investigate discourses. The students
watched or listened to others to determine the correct answer, but they did not ask for or
offer explanations. They did not question whether something was true, but instead relied
upon the teacher or another student to substantiate or correct their work. As students
focused on the teacher-approved answer, it limited their mathematizing so that they were
not engaging in the autonomous activity of exploring discourses.

A second contradiction between autonomous learning and seeking teacher-
approved answers is the reliance upon others to evaluate the truth of discourses and to
pose problems. Piaget (1932/1960) described perils of reliance upon others, what he
called heteronomy. When a child obeys others, he/ she does not seek truth or question
what others say. Instead, he/ she interprets the directions and follows the directions using

his/her own perspective. In order to decenter or become less egocentric (and more

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autonomous), the child must not blindly obey, but instead must determine truth for
him/herself. This requires examining and trying on others’ perspectives in order to realize
the multiple ways in which a situation might be viewed. The students in my study are in
similar situations to Piaget’s child. As they focused on teacher-approved answers, they
followed the directions and cues of others and did not seek truth for themselves. They did
not use the task as an opportunity to explore other perspectives or decenter. While in
some instances they were able to adopt the narratives of others (for example, Jessica’s
watchful learning), they were unable to use these narratives as a tool for themselves.
Thus, while the discourse of the learners changed, their reliance upon others to indicate
the teacher-approved answer seemed to result in changes that were not as mathematically
desirable or as useful to the learner as the changes in the discourse of the most
autonomous learner (J akeel as he enacted engaged learning).

The quest for the teacher-approved answer also meant that students did not pose
problems for themselves. They accepted the problem presented by the teacher. Brown
and Walter (1990) suggest that problem posing is central to learning mathematics. Not
only does the act of posing a problem allow the learner to decenter, it also contributes to
the sense of that the learner is in charge. The learner is not a passive recipient of orders
and directions: He/she chooses what to learn. This identifying matches the identifying of
autonomous learning. As the learner determines what questions to ask, he/she positions
him/herself as curious, as someone who investigates discourses, and as a member of
his/her own audience. As learners pose their own problems based upon their observations

of mathematical discourse, they may engage in more autonomous investigation of

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discourse, which could contribute to the development of the learner’s mathematical
discourse.

My data offer illustrations of this claim. As J akeel enacted engaged learning, he
pursued the problem of changing his discourse. The teacher initiated the problem by
asking J akeel to count the area of Figures K and J. However, the teacher did not point out
that J akeel’s count of Figure J was problematic: J akeel realized this as he counted and he
began to work on changing his count without prompting from the teacher. Rebecca and
Daren supported J akeel in his identifying and mathematizing. They offered explanations
but not solutions, allowing J akeel to invent his solution. As J akeel pursued this problem
he posed to himself, he investigated his mathematical discourse. His peers were supports,
but did not tell him what do what or whether his investigation was done. In the end, he
changed his discourse so that it was more mathematically desirable. Furthermore this
change in discourse was not temporary: J akeel used this same discourse in a subsequent
lesson on area.

In contrast, when J akeel and the other students enacted learning activities that
focused on the right answer, they identified as more reliant upon others and less in charge
of their learning. They needed the teacher and their peers to approve their work. Their
teacher and peers concurred with this identification and served as judges of whether
students were done. The learners also relied upon their peers to supply them with the
teacher-approved answer, which limited their mathematizing. For example, Jessica
initially identified as a problem poser: She asked Minerva why she was cutting another
copy of the figures (Chapter 7, Excerpt 8). However, Minerva and Bonita rejected

Jessica’s attempt to position herself as responsible for investigating what didn’t make

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sense and to position Minerva as someone who would be helpful. Rather than insist on
pursuing her question, she became more passive and accepted the problem of getting the
right solution to the teacher’s task. As she watched Bonita and Minerva, she looked for
the teacher-approved answer and did not ask questions or explore ideas. She did not
indicate that she was in charge of her learning and she did not choose what to change
about her discourse. As a watchful learner, Jessica’s mathematizing was limited: She did
not have the opportunity to explore Bonita and Minerva’s discourse and they did not
explain their thinking. In the end, the change in Jessica’s mathematical discourse was not
as desirable or as useful to her as the change in J akeel’s discourse. As Jessica’s watchful
learning illustrates, as students enacted kinds of learning that focused on the teacher-
approved answer instead of on posing their own problems, they relied more on others,
which resulted in different mathematizing and different interactions with peers and the
teacher. In the end, the changes in their mathematical discourse were not as
mathematically desirable and, as I will discuss below, were not tools they could use to
generate or solve future mathematical tasks.

This study is not the first to note that student learning is less desirable when
students focus on getting the teacher-approved answer. However, this study elaborates
two points about students and teacher-approved answers. First, students are not always
focused on the teacher-approved answer. J akeel, Jessica, and Bonita illustrate this point.
J akeel switched between engaged learning, which was not focused on teacher-approved
answers, to directed learning, which was entirely focused on the answer. At the beginning
of the lesson, Jessica and Bonita both enthusiastically explained how two triangles made

a square. Later in the lesson they were both focused on the teacher-approved answer.

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These shifts between mathematizing for the teacher—approved answer and mathematizing
to explore and explain suggest that shifting students from a focus on teacher-approved
answers to a focus on exploring discourse may require as little as providing situations
which support students in the mathematizing and identifying activities they already know.
A second point is that providing students with the right answer might create an
opportunity for mathematical exploration and thereby promote rather than discourage
autonomous learning. J akeel knew that the squares in Figure J should total 8: Rebecca
had already counted both J and K and arrived at 8 for both figures. J akeel counted 8 when
he counted Figure K and he expected Figure J to have the same count. J akeel was not
working to figure out the teacher-approved answer, he was working to figure out what he
needed to do differently to arrive at the count he knew to be correct. Confirmation of the
right answer might have helped Jessica as well. Creating the final arrangement of the
pieces for Figure I did not help Jessica understand the area of the figure. If she had been
shown this arrangement and then asked to explain how it related to the original figure and
the area of the original figure, she might have connected the picture to the discourse and
explored the discourse in ways that could have made it a tool for herself. Teachers and
mathematics educators have sometimes dealt with students’ focus on the teacher-
approved answer by refusing to disclose their answer, believing that if they don’t
emphasize the answer, students will need to trust their own logic and problem solving
skills rather than the authority of the teacher or the textbook. However, learning
mathematics involves developing a mathematically desirable discourse. Having the right

answer might provide students a means of evaluating and learning to trust their logic and

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discourse (as it did for J akeel) and might raise questions and curiosity about
mathematizing rather than foreclosing opportunities to think.

This discussion suggests that problem posing is entangled with identifying and
mathematizing activities and that it is a critical activity for autonomous learning and
perhaps for the development of mathematically desirable discourses. In particular, it
seems that focusing on the teacher-approved answer prevents the identifying and
mathematizing activities of autonomous learning. Many mathematics educators have
lamented the negative consequences of student’s attention to right answers (6. g. Bishop,
1991; Brown & Walter, 1990). What this study contributes is analysis of the connection
between seeking teacher-approved answers, identifying, mathematizing, and changes in

discourse.

Discourse for Whom?

The lens of autonomous learning highlights variations in the discursive outcomes
of learning activities. The five kinds of learning activities resulted in three different kinds
of discourse: discourse-for-others, discourse-for-the-learner, and the learner’s own
discourse. In my discussion above of J akeel’s enactment of engaged learner, I described
how engaged learning resulted in a discourse-for-the-leamer (a discourse-for—himself).
Ben-Zvi and Sfard (2007) describe a discourse-for-the-learner as the result of the
learner’s efforts to critically examine others’ discourse. As a result of this examination,
the learner can use the discourse to communicate with him/herself as well as with others
using the discourse. They can also use the discourse to solve their own problems (Sfard,
2008). These are the outcomes of autonomous learning. J akeel’s final discourse, resulting

from his enactment of engaged learning, met these criteria.

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In contrast, a discourse-for-others is not a tool for the learner, but instead is used
in ritualized communication with others to whom the discourse makes sense (Sfard,
2008). Jessica’s enactment of watchful learning and guided learning produced such a
discourse, as did J akeel’s enactment of directed learning. Finally, as I described in
Chapter 6, Minerva’s covert learning resulted in her own discourse. It had peculiar
routines and narratives that did not match the discourse of anyone else.

These students also had differences in interactions with peers as they enacted
these different kinds of learning. As J akeel enacted engaged learning, he was the most
interactive. Rebecca and Daren offered evaluations and explanations. J akeel responded
and contributed his ideas. Jessica’s enactment of watchful learning was somewhat less
interactive. While she carefully watched Bonita and Minerva, she had limited
opportunities to talk about mathematics with them. At the opposite end of the continuum
from J akeel’s engaged learning was Minerva’s covert learning. Minerva actively worked
to limit interactions with her peers. One well-discussed example is her dismissal of
Jessica’s question about what she was doing.

This continuum of interactions correlates with the spectrum of discursive
outcomes, with the discourses connected to more interactions being more mathematically
desirable and more useful to the learner. Piaget offers an explanation for this. He
(1932/1960) suggested that learning with others was essential to autonomy: By co-
operating with others, children learn to decenter, or take on others’ perspectives, making
their own thinking less egocentric (Kamii, 1994). As students decenter and become less
egocentric, they are able to critically examine their own thinking and to use the ideas of

others. Thus as J akeel enacted engaged learning, he considered the ideas of Rebecca and

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Daren and received their input on his ideas. His final discourse was one that he could use
to successfully communicate with himself and others. It was a discourse-for-himself.

Jessica’s one-sided interactions as a watchfirl learner allowed her to take on the
discourse of others but not to use or critically evaluate it. She developed a discourse-for-
others. Finally, as Minerva enacted covert learning, she did not explain her ideas and she
limited the explanations of other students. She had few opportunities to decenter and
critically examine others’ discourses. Her final discourse was meaningful to her, but not
to others. She developed her own discourse. Piaget’s theory suggests that if Minerva had
been able to identify as wanting to learn from/with others, she might have engaged in
mathematizing with others that explored discourses and provided her with the
opportunities to develop her discourse as a discourse-for-herself.

This examination of Minerva’s independent work and development of discourse
bears some similarities to Stanley Erlwanger’s (1973) well-known study of Benny’s
mathematical learning. Benny’s sixth grade mathematics curriculum was Individually
Prescribed Instruction (IPI) Mathematics. Students worked individually through the
curriculum, solving practice exercises. When a student felt he/she had mastered the
material, he/she could take the accompanying test. If 80% of his/her answers were
correct, the students could proceed to the next material. By the standards of this
curriculum, Benny was a successful student. He had completed more units than many
other students in his class. He worked independently and when he experienced difficulty
with material, he was able to determine the pattern for correct answers and change his
answers. However, Erlwanger found that Benny’s explanations of mathematical concepts

were not mathematically appropriate. Benny had generated some interesting and highly

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unconventional routines for a number of mathematical processes. Erlwanger suggested
that Benny’s troubling success in this program was a result of the program’s focus on
right answers rather than on the individual’s process. An examination of Benny’s
situation through an autonomy lens would concur that the focus on right answers is
problematic (as I discussed in the section above), but an autonomy lens also suggests that
the individualized nature of the curriculum was also problematic. The curriculum was not
structured to allow students to mathematize with each other and they only mathematized
with the teacher if they couldn’t get right answers. Benny did not discuss mathematics
with others. He had no opportunities to evaluate other’s discourse or examine other
perspectives. He did not decenter and, like Minerva, he developed his own mathematical
discourse. Erlwanger’s case of Benny contributes to the evidence from Minerva’s covert
learning that autonomous learning is not synonymous with independent work and that
student’s learning outcomes can be more problematic when they don’t engage the

discourses of others.

Linking Discursive Features

One final feature of autonomy that played a role in each case was connections
between the verbal discourse and images drawn on paper. Students’ adoption, production,
and substantiation of discourse is enabled by their use of visual mediators to support their
use of words. Sfard (2008) notes how one student was able to use the visual image of a
table to support himself in solving a function problem even as he confused the words
slope and intercept. Sfard also notes that the visual image can serve as a powerful tool for
supporting students in generating new narratives. J akeel’s engaged learning illustrates

this. As J akeel enacted engaged learning, he seemed to focus on Rebecca and Daren’s

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gestures and pictures, linking that visual to his own words, narratives, and routines. In
contrast, as be enacted directed leaming, he only focused on words, writing what Rebecca
dictated without creating connections to images. Minerva and Jessica also demonstrated
limited connections between words and images. As Minerva enacted covert learning, she
worked hard on generating the picture desired by the teacher. However, she rarely
gestured as she spoke and she frequently used pronouns that made the visual mediator for
her words unclear. Finally, as Jessica enacted watchfirl learning, she mastered the
narratives of the teacher-approved answer, but did not construct the matching image. Her
guided learning resulted in a teacher-approved picture, but did not provide her with an
opportunity to connect the picture to her words. With the exception of J akeel’s engaged
learning, there were few strong links between what students said and what they indicated
visually on paper. Even J akeel’s engaged learning, which did link his words with his
pointing, might have resulted in more mathematically desirable discourse if J akeel had
attended more to Rebecca and Daren’s use of words (such as half). These findings lead to
a conjecture that is worth future exploration. They suggest that students’ engagement in
the autonomous learning activities of adopting, producing and substantiating discourse is
facilitated by their work to link words with visual mediators.

The possible need for this connection suggests that teachers should attend
carefully to both students’ use of words (such as when they might use pronouns instead
of nouns or fail to use nouns at all and only use counting words) and to how they link
those words with visual images. Teachers might also present students with tasks that
explicitly require students to connect words and images. One activity that has been used

successfirlly in classrooms is a task in which students work in groups to demonstrate four

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different ways to solve a multi-digit multiplication problem. Students must then make
connections across the different solutions. As students link solutions, they discover how a
solution that uses an algorithm can be represented as an array, groupings of objects, or
repeated additions. As students verbalize the connections they see, they necessarily use
words (although they could still use vague or inappropriate words if the teacher is
inattentive) to describe mathematical processes and visual representations. As they
produce narratives and substantiate them as they make these links. While it is possible for
students to engage in this task in unproductive ways, the task illustrates a problem that
specifically promotes autonomous activities.

In my discussions at the end of each chapter, I described how my examination of
the mathematizing and identifying activities enacted by the students added detail to a
description of autonomous learning. J akeel’s engaged learning illustrated how students
could engage in autonomous learning while learning from others. Minerva’s covert
learning demonstrated that enactment of identities of successful students (independence,
persistence, creativity) does not necessarily result in autonomous learning. The case of
covert learning also suggests the value in identifying as not knowing. Jessica’s watchful
learning shows how interactions among students can limit student’s attempts at
autonomy. Finally, the work of her peers and her teacher during guided learning
illustrates how an emphasis on the teacher-approved answer can prevent students from
engaging in autonomous learning even when they are given the responsibility for
reasoning to that answer. Using the lens of autonomy to examine the mathematizing and
identifying in each of these cases has resulted in an elaboration of autonomous learning

that seems useful in evaluating and encouraging student learning.

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Assumptions and Contributions to Research on Identity

In my theoretical framework, I constructed a theory of identity that described how
identity could seem to be static but could also be fluid. I wanted to be able to account for
how the same individual might respond in different ways to very similar circumstances.
For example, J akeel shifted from engaged learning to directed learning and back to
engaged learning. Why was it that Rebecca was eventually successful in getting him to
copy from her and then why did he return to engaged learning? My notion of identity as
fluid allows this transition. J akeel initially identified as capable and knowledgeable.
However, when he had the wrong arrangement of pieces on his paper, Rebecca firmly
asserted, based upon the incorrect solution J akeel had created, that he was not
knowledgeable. Rebecca identified J akeel in a different way and he acquiesced. Later,
when the teacher asked him to show her how to count Figure J, he realized that his
counting routine was problematic. He also worked to try to change his counting and
Rebecca supported him in this, identifying him as capable of changing his counting.

J akeel’s transitions in identifying activity illustrate the ways in which individuals
can foreground and background different identifications. Acting as knowledgeable or as
in need of direction were not new identifying activities for J akeel. My data provide
evidence that he had enacted engaged learning and directed learning in the past. He
mobilized those identifications and stories as situations changed. Jessica’s shift between
watchful learning and guided learning reflected this same mobilization of identifications.
When Jessica’s attempts at questioning failed, she became a learner who watched.
However, she later claimed that nobody was helping her (Chapter 7, Excerpt 14, Line

561) and identified as a learner who needed more explicit help. Jessica and J akeel

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identified and acted differently at different times sometimes placing one identification
forward and sometime another depending upon circumstances.

This notion of how identifying activity foregrounds and backgrounds different
identifying stories has two important implications for mathematics educators and
researchers. First, teachers and researchers (and students) should not assume that
individual students are only one particular kind of learner. As teachers identify students
in particular ways, especially if those ways are negative, they might not notice or provide
opportunities for students to enact other more positive identities. They may also interpret
student activities through those negative identifications such that they might respond
differently to same question from different students. This more fluid notion of identity
also provides opportunities for researchers to examine identity at higher resolution.
Rather than account for an individual’s activity over an extended period of time,
researchers can examine how identity unfolds in a moment. This also opens the
possibility for examining how interactions foreground certain identities. For example,
Rebecca was able to identify J akeel as needing to copy by telling him that he didn’t do
what he was supposed to do. She could have told him that he needed to rethink his
answer or she could have asked him to explain his thinking. Instead, she insisted he
didn’t know what to do.

A second implication of this notion of foregrounding and backgrounding
identities involves identities that such as race, ethnicity, gender, and class. These social
identities are frequently intertwined with academic identities. For example, for some
women, at some moments, their gender is salient in their mathematical activity as they

worry about whether others think they are as capable as the man sitting next to them in

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math class. This worry may prevent them from asking questions or engaging in
conversations about mathematics. However, at other times, gender may be backgrounded.
For example, a woman who is identified as good at explaining a problem may foreground
this academic identification and background her gender. Thus it is important to recognize
how students might foreground or background these social identities, sometimes allowing
them to constrain or enable their mathematical activity.

In the three cases I present, I did not provide examples of student’s identification
as black, white, boy, or girl because my data for this lesson included only one instance in
which students were engaged in mathematics and specifically invoked a racial or gender
identity: Toward the beginning of the small group time, J akeel told the teacher that he
had not yet “cutted” out Figure H (Line 212). Daren teased him about his use of “cutted”
and Rebecca smiled. J akeel turned to Rebecca, “(I see you Rebecca you smile) what’s so
funny? A black brother can’t talk?” (Line 216). As this example illustrates, students’
identifying activities may specifically acknowledge race. In this moment, J akeel
foregrounded his racial identity. However, he did not tie it to his mathematizing, but
instead seemed to use it to engage Rebecca (and not Daren, who was the one who was
most active in teasing J akeel). I do not mean to imply that J akeel was not indicating his
race through his use of “cutted”, but that until Daren teased him, he seemed to be
backgrounding his racial identity.

I could have examined my data through a lens that specifically examined
discourse for features (such as the word “cutted”) that would indicate race, gender,
ethnicity or class. However, in addition to the lack of data indicating foreground of these

social identities, I also lacked data that determine what it might mean to an individual

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student to identify using one of these social categories. Different students might draw
upon different narratives of race, ethnicity, gender, or class to guide their identifying
activities. Thus, what one student might see as acting black might be understood in a very
different way by another student. J ilk’s (2007) research on Latina immigrants supports
this claim: She described both the diverse identities and the diversity in narratives
associated with one identity label used by her participants. In order to analyze my
classroom data for these features I would need to collect that that would allow me to
specifically notice how each student narrates these identities.

It is important to recognize how students and others foreground race, gender,
ethnicity and class in order to avoid situations in which these identities are foregrounded
in ways that limit mathematizing. For example, McGlone and Aronson’s (2006) studies
of stereotype threat suggest that asking students to identify their race and gender at the
beginning of a standardized test may foreground these identities in ways that affect the
student’s performance on the test. I suggest that we need to carefully examine learning
situations to see whether and how students are foregrounding race, whether there are
other factors that are bring race to the front for students, and how that foregrounding

affects their mathematizing and learning outcomes.

Different Kinds of Learning
One significant contribution of this study is the description of different kinds of
learning. These kinds of learning are useful because they offer vocabulary for describing
learners beyond the broader categories of smart! struggling or math person/not a math
person. These categories run the risk of identifying students in ways that make the labels

seem like inherent traits of students. Smart becomes who the student is, rather than a

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word describing the student’s activities (Sfard & Prusak, 2005). By describing learning in
terms of student activity, we avoid labels like smart, slow, struggling and instead focuses
on how student activity (including interactions with others) connects to changes in
mathematical discourse. Identifications become transient outcomes of identifying activity
rather than predictions of necessary future activity. By focusing closely on and
categorizing learning activity, this study can propose changes in activity that might lead
to changes in learning outcomes.

One limitation of my construction of kinds of learning is that it came from close
observation of a few students. I cannot make claims that other students will enact the
same cluster of identifying and mathematizing activities. Indeed, the complexity of
individuals and interactions strongly suggests that a careful analysis of many students in
many learning situations would identify many different kinds of learning.

More investigation across more individuals and more activities may reveal
broader categories of kinds of learning. For example, my analysis could be used to sort
student activity into autonomous and nonautonomous learning. These categories could be
helpful in focusing on how student activity supports the development of desirable

mathematical discourses.

Reform and Traditional Mathematics Pedagogies
This discussion would be incomplete if I failed to address the implications of this
study for reform mathematics pedagogy and for the debate between reformers and
advocates of more traditional pedagogies. I recognize that the terms reform and

traditional may be used in different contexts to emphasize different mathematical tasks.

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As I use these terms, I draw upon Suzanne Wilson’s (2003) distinction between the
pedagogies. She describes reform math pedagogies as

based on a Dewey-ian conception of the child and curriculum as two sides

of the same coin. This position places emphasis on process as well as

content, on the child as well as the curriculum. It also places emphasis on

students and teachers working together, and on teaching “hi gher-order

thinking” or “conceptual understanding” as well as the basics (pp. 17-18).
Traditional math pedagogies use “more teacher-dominated instruction, more focus on
skills and mastery of the basics, and more emphasis on a “canon” of legitimate
knowledge” (p. 17).

This study addresses two concerns of reform pedagogies: student interactions and
autonomy. Reform pedagogies make extensive use of student talk. Ostensibly,
interactions among students promote understanding of mathematics as students propose
and argue about mathematical ideas, reason through conjectures and alternative solutions,
and evaluate the thinking of others (National Council of Teachers of Mathematics, 1991,
2000). However, students don’t always learn mathematics when they work with and talk
with others: The theoretical promise of interaction does not consistently result in the
desired outcomes. Determining what factors might be implicated in the success or failure
learning through interactions requires a framework that illuminates the complex
cognitive, affective, social, historical, and cultural elements that might come into play in
an interaction. Using discourse to consider mathematizing, identifying, and learning
outcomes provides a lens for coordinating the study of these complex factors. For

example, the lens allows examination of how a student in an interaction might

simultaneously communicate social information, mathematical content, and emotion.

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The findings in this study confirm that interactions are implicated in learning.
Students who interacted more had more desirable changes in their mathematical
discourses. However, some interactions were also problematic: Students sometimes
foreclosed the mathematical conversations of others by refusing to participate or by
insisting on directing the learner’s activity. These findings are not new to the field: Many
researchers have examined interactions among students and teacher’s struggles to
promote productive interactions (e. g. Cobb et al., 1997; Cohen, 1994; Hufferd-Ackles et
al., 2004; Lampert, 2001). Some of this research has adopted the stance that students
need to be taught how to interact in productive ways. For example, Yackel and Cobb
(1996) suggest that students can (and should) be taught how to evaluate each other’s
mathematical contributions. While teaching students specific ways of mathematizing and
identifying surely leads to desirable changes in mathematical discourse, my study
suggests that we might first consider encouraging the mathematizing and identifying
activities that students already bring to mathematics lessons. For example, each student in
these two groups demonstrated productive mathematizing and identifying activities:
Jessica explained her drinking and asked questions, Rebecca and Daren explained their
ideas in ways that promoted J akeel’s thinking, J akeel asked questions and explored
discourse, Minerva examined her third solution and changed her writing, and Bonita
explained her thinking. Perhaps a first step in promoting a reform classroom is helping
teachers and students recognize ways in which they already act and interact that support
learning mathematics.

That last sentence was easy to write and given my time spent in analyzing this

classroom data, I might feel comfortable noting productive actions and interactions.

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However, as many very competent teachers have demonstrated, making sense of what
kinds of activities to promote in classrooms can be a difficult task. What can be
especially confusing for teachers enacting reform pedagogies is their role in encouraging
the development of autonomy in their students. Many mathematics educators have noted
that autonomy is a central goal of mathematics education (Ben-Zvi & Sfard, 2007; Kamii,
1994; National Council of Teachers of Mathematics, 2000; Warfield et al., 2005; Yackel
& Cobb, 1996). However, as I have noted earlier, what constitutes autonomous learning
has been poorly defined. It is frequently discussed in ways that connect it to student
independence. Some teachers interpret this construction of autonomy to mean that they
should design lessons in which students work to discover math without teacher guidance.
Learning without the teacher becomes possible when students learn from each other in
small groups. Thus the reform notions of student interactions and small group work lead
to an enactment of autonomy as students learning without the teacher. However, as Ben-
Zvi and Sfard (2007) note, because mathematics is socially constructed, there are some
aspects that can only be learned from someone who is already fluent in the desired
mathematical discourse. Teachers must be involved in modeling the expert discourse
students should adopt.

This study proposes modifications to the conception of autonomy as
independence and elaborates what student interactions (including interactions with the
teacher) are essential to autonomous learning. As I noted in my analysis of Minerva’s
covert learning, autonomous learners are not learning by themselves. Nor are they
focused exclusively on finding the teacher-approved answer. Instead, students enact

autonomous learning when they choose to work on a problem arising from their

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observations. J akeel demonstrated this as he worked to change his counting. He did not
do this work without support: However, the support was not a demonstration of what he
needed to do nor did Rebecca coach J akeel toward one specific way in which he should
change his gestures. Instead, Rebecca provided discourse J akeel could adopt and she gave
him feedback on his work. In addition to providing examples of autonomous activity, this
study has elaborated a definition of autonomy and a list of mathematizing and identifying
activities that might help teachers conceptualize, promote, and analyze autonomous
learning activities.

This study’s emphasis on student interactions (along with the researcher’s own
predilections) indicates a bias toward reform pedagogies over traditional pedagogies. It
addresses some of the concerns proponents of traditional pedagogies voice about reform
pedagogies. For example, some traditionalists are concerned that reform math requires
students to discover mathematical concepts without guidance from the teacher
(Mathematically Correct, 2005). I have shown that guidance from the teacher is essential
to learning mathematical discourse. However, it is also essential that students pose their
own problems about mathematical discourse rather than only seeking teacher-approved
answers on teacher-assigrred tasks.

This study also suggests that reform pedagogy can be hard to enact: not all student
interactions are productive and not all enactments of autonomy have supported student
learning. Furthermore, it is possible for students in traditional classrooms to act
autonomously. They might be curious about the discourse the teacher is using. They
might wonder about and investigate patterns across multiple problems they’ve worked.

However, traditional pedagogies often fail to encourage such curiosity and to promote

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interactions that encourage students to decenter. If learning mathematics requires students
to develop a more mathematically desirable discourse, students must be allowed and
encouraged to engage in mathematizing and identifying activities that feature
examination of mathematical discourse. This means that the traditional focus on the

mathematics canon is appropriate as is the reform focus on interaction and autonomy.

Beyond Area and Beyond Math

Because this is a case study of student learning across a lesson, the scope of the

 

lesson content is necessarily narrow. My interest in mathematics education has led me to
focus on mathematics. However, my focus in this study on the mathematical concept of
area resulted from the possibilities presented by the data I collected, rather than from a
essential connection between my theoretical framework and area. Focusing on area was
useful because it necessarily contains a visual component, which allowed me to see how
students were (or were not) making connections between their words and the possible
images. However, my theoretical framework is not tied to any specific math content.
Presumably, I could have conducted my analysis on a lesson from one of the other math
units (geometric shapes or probability) fi'om which I collected data and potentially find
connections among mathematizing, identifying, and learning outcomes.

There is also nothing in my theoretical framework that limits it to exploration of
learning mathematics. While the framework is founded on research describing
mathematical contexts, the assumptions underlying the framework could apply to other
content areas. For example, the notion that the outcome of learning is a change in
discourse and that the activity of learning arises from discourse about content and

learners could apply in science or history classrooms. Indeed, connections between

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discourse about the content and discourse about the learners might be more explicit in
content that focuses on human thought and culture literacy or history. For example,
Wortham’s study of social identification and academic learning (2006) examined a class
that was a joint English and history class. Wortham examined how the social categories
available in the curriculum were used to identify students. Wortham used a larger time
scale than I used in my study: He considered learning and identifying across a year and
considered how students learned content through their use of the content to identify other
students. He also had a different construction of learning: the use of “new combinations
of resources to represent or react productively to subject matter across events” (p. 105). It
would be possible to examine Wortham’s data through my theoretical framework to see
what connections might be found across discourse about humanities, identifying, and

outcomes of learning on a smaller time scale.

Limitations and Future Research

I designed this project as a case study in order to explore what seemed to be
complex connections between mathematizing, identifying, and changes in discourse. As I
described in my methods chapter, this study design allowed detailed probing of student
activity, but it only allowed detailed probing of the activities of three students during one
lesson. Certainly there is more that could be learned through additional case studies,
studies that consider a range of mathematical content, and studies that connect these
theories to larger populations. In the paragraphs below, I describe three specific
limitations of this study and how they could be addressed by future research.

First, this study lacked examination of highly successful learning activities: None

of the three cases presented learning outcomes that entirely matched the desired learning

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outcomes. While these three cases were interesting and helpful in examining connections
across mathematizing, identifying, and change in discourse, their stories could be
enhanced by comparison to a situation in which the student’s learning activities resulted
in a more mathematically desirable change in discourse. For example, during the lesson
with J akeel, Rebecca and Daren, I noted that Daren had figured out that the triangular
spaces were half of the square spaces. This seemed to be a significant (and notable to the
students) change in discourse. It would be interesting to see what activities surrounded
this change. However, Rebecca and Daren were not focal students and I did not collect
data on them during the previous lesson. Examining a few lessons with more successful
outcomes could enrich my findings.

Another limitation of this study was the narrow time. Studying one lesson allowed
me to notice and investigate detail. However, it leaves unexamined a question I have
about my framework. Both Nasir (2002) and Wenger (1998) suggest that learning
involves a change in identity. This seems quite reasonable given my framework: If
learning is a change in discourse and identity is discursively constructed, then one
outcome of learning could be a change in discourse. I hoped to examine this in my data,
but I was persuaded that one lesson was too short a time to expect students to construct
new stories about themselves and others. While I did see changes in identifying activity, I
was persuaded that these changes reflected foregrounding and backgrounding of
identities and not construction of new identities. It would add complexity to the
framework of mathematizing, identifying, and change in discourse to examine changes in

discourse that reflect new stories about individuals. This study would require data

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collection over a more extended time frame in order to capture the perhaps more gradual
process of students writing new or modifying existing stories about themselves.

Finally, in this chapter and in the findings chapters, I make several suggestions to
teachers about what they might do differently to encourage autonomous learning. These
suggestions are based upon my three detailed but limited cases, which means both that
they may not work in other situations and that much could be learned by examining the
implementation of these suggestions. I propose that a design-based research project
(Design-Based Research Collective, 2003) would provide an opportunity to enact these
suggestions, study the outcomes, and modify both the theory and the teaching
suggestions. Engaging in a research cycle that allows implementation, investigation, and
modification would help strengthen the connection between my theoretical framework

and practice.

Conclusion

This study raises questions about how and whether the kinds of learning I
elaborated might be enacted by different students in a different setting and about how my
suggestions for teachers might affect student interactions and activity. However, the
study has also answered questions I’ve had about making sense of student activities and
interactions. I especially wanted a tool that would help me account for differences in
student activity such as unan’s copying and his later investigation of mathematical
operations. Examination of discourse for mathematizing, identifying, and learning
outcomes (changes in discourse), I was able to suggest ways in which the intertwining of
mathematizing and identifying link to student activity and changes in discourse. I was

also interested in examining the differences in moments when students might choose to

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copy or when they might choose to examine discourse in order to evaluate what student
activities and interactions might encourage mathematically desirable changes in
discourse. The development of autonomous learning and its constellation of identifying
and mathematizing activities provided a tool for highlighting more productive student
activity and suggesting what activity and interaction to encourage during mathematics
lessons.

This study could be seen through a pessimistic light: The students demonstrate
changes in discourse, but none of the students demonstrates all of the features of the
mathematically desirable discourse. Each of the students enacts one or more kinds of
learning that is far removed from autonomous learning. However, I see cause for
optimism: The lens of mathematizing and identifying and the special case of that lens,
autonomous learning, allows teachers and researchers to see how undesirable learning
outcomes are not inherent traits of students but instead arise from a predominance of
nonautonomous learning activities and interactions. Each student also demonstrated
autonomous learning activities, which suggests that success in learning mathematics does
not require overcoming cognitive deficits or teaching students entirely new learning
skills. Instead, we might find that by seeking, encouraging, and designing for moments in
which students display curiosity, we can support students in enacting more activities that

connect to mathematically desirable changes in their discourse.

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