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STORIES OF CHANGE: NARRATIVE PERSPECTIVES ON
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STORIES OF CHANGE: NARRATIVE PERSPECTIVES ON ELEMENTARY TEACHERS
IDENTIFYING AND IMPLEMENTATION OF NEW MATHEMATICS TEACHING
PRACTICES
By

Joy Ann OSlund

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Teacher Education

2009

ABSTRACT

STORIES OF CHANGE: A NARRATIVE PERSPECTIVE ON ELEMENTARY
TEACHERS' IDENTIFYING AND IMPLEMENTATION OF NEW MATHEMATICS
PRACTICES

By
Joy Ann Oslund

Recent mathematics education reforms (N CTM, 2000) have resulted in increased
opportunities for teachers to learn new teaching practices. However, the relationship
between teacher professional development and the actual implementation of new
practices is unclear. I posit that a teachers’ decision to implement newly learned practices
is strongly connected to her identity. However, the relationship between experienced
teachers' implementing of new practices and their ways of identifying has been largely
ignored.

How teachers learn to implement mathematics pedagogies that promote equitable
student outcomes is a particular concern. Complex Instruction (CI), one such pedagogy,
is a set of principles and practices for organizing cooperative group work in
heterogeneous classrooms (Cohen, 1994; Cohen & Lotan, 1997). In this dissertation, I
report on four elementary teachers who chose to implement CI. Building on theories that
link learning, practice, and identifying (e. g. Wenger, 1989), I analyze the identifying
stories of the teachers using tools from narrative analysis (e. g. Gee, 1991; Labov, 1972)
in order to understand how identifying and change in teaching practice impact each other.
I elicited narratives in three contexts: individual interviews about CI lessons, group

discussions, and presentations about CI to other teachers.

This study explores how teachers come to identify as new kinds of teachers while
implementing new practices and how those ways of identifying become reasons for the
teachers to continue those practices. Teachers identified as CI mathematics teachers by
reifying narratives of their CI mathematics teaching experiences and integrating them
into existing identifying nan'atives or negotiating between new narratives and existing
narratives.

This study provides a language for the difficulties inherent in teacher learning
(i.e., implementation of new practices) that is respectful of teacher knowledge and
experience. The explication of the impact of mathematics education reforms on teachers'

lives and practice has implications for policy and professional development.

Copyright by
JOY ANN OSLUND
2009

To
Hannah and Lena, who motivated me;
Dave, who supported me;
and I AM, who inspired me.

Thank you.

ACKNOWLEDGMENTS

“No one of us alone is as smart as all of us together.” This is a guiding principle
in Complex Instruction. Truly, this dissertation would not be what it is without the
smarmesses of many people who provided support, read drafts, suggested ideas, and
inspired my writing.

I especially want to thank my advisor and dissertation director, Sandra Crespo,
who read countless drafts and encouraged my creativity. Thanks also to the rest of my
committee, Helen F eatherstone, Ralph Putnam, Anna Sfard, and Dorinda Carter. Each of
you pushed and encouraged me in important ways.

My weekly meetings with Ann Lawrence and Deb VanDuinen were indispensable
to my completion of this project. I am deeply grateful for the constancy of wisdom and
fi’iendship.

This dissertation would not have been possible without my CI colleagues. I am
deeply thankful that Lisa Jilk brought CI to Michigan State University and that Marcy
Wood and Helen F eatherstone worked with her to bring it to local teachers and interns.
The CI week in the summer of 2006 was a turning point in the story of my development
as a teacher and scholar. My deepest gratitude also goes to the CI teachers who let me
into their practice and their lives. You have inspired me with your ingenuity, dedication,
and energy.

To my second- and sixth- grade students and colleagues in the Negaunee Public

Schools: the joys and sorrows of our daily work inspired and motivated me throughout

vi

my Masters and Doctoral studies. I am so thankfiIl to have Spent eight years teaching in
the best place in the world.

To Jeremy Dowsett, especially for your time and energy with edits. But you and
the rest of Blacksoil have also been essential to the integration of my spiritual and
scholarly identities. Thanks also to my neighborhood group: Julie, Dave, Irene, Bruce,
Brigette, and Kim. My weekly dinners with you have sustained my emotional, spiritual,
and intellectual energy in a way that nothing else possibly could have.

To my parents, Duane and Rita Palomaki: You both demonstrated a passion for
equity and inclusion in education that has impacted my life story. Your constant prayers
were effective. Thank you is too small a phrase to next to the immensity of your impact.

Hannah, Lena, and Dave, I have not forgotten you, but have chosen to save the
most important people in my life for last. You have given so much; this dissertation is
yours as much as it is mine. Hannah and Lena, watching you grow and learn has been
more inspiring to me than any class I have taken or any book I have read. You are
geniuses! Dave, you have partnered with me in this endeavor in every way. Your faith
and love have kept me going.

Above all, thanks to God, the first I AM statement and author of the narrative that

changed my life.

vii

TABLE OF CONTENTS

List of Tables ................................................................................................................ xii
List of Figures ............................................................................................................. xiii
Chapter 1: Introduction .................................................................................................... 1
Dragging Teachers Up the Ascent .............................................................................. 8
John, Rita, the Cave Dwellers, and Me ..................................................................... 12
The Story of Teachers‘ Learning .............................................................................. 14
Chapter 2: Literature Review ......................................................................................... 18
Mathematics Teacher Professional Development, Equity, and Reform ..................... 18
Complex Instruction in Mathematics: The Professional Development Context... 19
Teacher Learning Situated in Teaching Practice ....................................................... 24

A Participationist Lens ....................................................................................... 24
Learning as Situated in a Particular Context ....................................................... 26
Identity .................................................................................................................... 27
Defining and Operationalizing The Work of Identifying ............................... 29
Identifying and Implementing New Mathematics Teaching Practices ....................... 33
Life Stories and Changes in Practice .................................................................. 34

The Impact of Implementation on Identifying .................................................... 36
Research Questions .................................................................................................. 37
Chapter 3: Method ......................................................................................................... 38
Phase 1 .................................................................................................................... 39
Phase 1 Participants ........................................................................................... 39

Phase 1 Data Collection ..................................................................................... 40

Phase 1 Data Analysis ........................................................................................ 40

Ways of Identifying ..................................................................................... 40

Figured Worlds ............................................................................................ 42

Phase II .................................................................................................................... 43
Phase II Participants ........................................................................................... 44
Jonathan ....................................................................................................... 44

Glynnis ........................................................................................................ 45

Joanna .......................................................................................................... 46

Patricia ......................................................................................................... 47

Phase 11 Data Collection ..................................................................................... 48
Individual Interviews ................................................................................... 48
Groupwork Interviews ................................................................................. 49
Presentations About C1 ................................................................................. 49

Summary of Data Collection ........................................................................ 50

Phase II Data Analysis ....................................................................................... 51

viii

Direct Identifying Statements ....................................................................... 51

Narratives .................................................................................................... 5 l
Narrative Structures ..................................................................................... 54
Ethnopoetics and Metaphors ........................................................................ 55
Chapter 4: To Implement or Not to Implement: Life Stories .......................................... 57
Overview ................................................................................................................. 57
Cases of the Four Teachers ...................................................................................... 60
Gina ................................................................................................................... 60
Ways of Identifying ..................................................................................... 60
Mathematics Life Stories, Figured Worlds, and Choosing Not to Implement
CI ................................................................................................................. 64
Kelly .................................................................................................................. 65
Ways of Identifying ..................................................................................... 65
Mathematics Life Stories, Figured Worlds, and Choosing to Increase Students'
Opportunities to Talk to Each Other ............................................................. 68
Shirley ............................................................................................................... 69
Ways of Identifying ..................................................................................... 69
Mathematics Life Stories, Figured Worlds, and Choosing to Implement CI..72
Jonathan ............................................................................................................. 74
Ways of Identifying ..................................................................................... 74
Mathematics Life Stories, Figured Worlds, and Choosing to Implement CI..77
Discussion ............................................................................................................... 78
Parallels Between Identifying Stories and Stories of Implementation: When
Implementation Allows the Performance of an Identity ...................................... 79
Contrasts Between Identifying Stories and Stories of Implementation: When
Implementation Allows the Revision of a Story ................................................. 81
The Failing (Foreclosure) Dilemma ................................................................... 82
Implications for Further Study ........................................................................... 84
Chapter 5: Narratives of Implementation: Challenges and Persistence ........................... 86
Overview ................................................................................................................. 86
The Challenge: Integration and Denial of Identity .................................................... 88
Joanna: A Giver Steps Back ............................................................................... 88
Jonathan: An Insider Experiences the Periphery ................................................. 94
Glynnis: A Non-Conformist "Tweaks" CI .......................................................... 96
Patricia: An Expert Sticks Her Ground ............................................................. 101
Persistence: The Expression of Identity .................................................................. 104
Joanna as Project Person .................................................................................. 104
Jonathan as Rigorous and Inclusive .................................................................. 106
Glynnis as Subversive ...................................................................................... 108
Patricia as Organizer of Materials and Behavior ............................................... 109
Discussion ............................................................................................................. l 12
Difficulties of Identity Integration .................................................................... 113
Performing Identities ....................................................................................... 1 15

ix

 

Chapter 6: Generalized Narratives: How—I- Teach Stories as Integrated Identities ......... 117

Overview ............................................................................................................... 117
Generalized Narratives .......................................................................................... l 18
Patricia ............................................................................................................ 118
Jonathan ........................................................................................................... 120
Joanna .............................................................................................................. 121
Glynnis ............................................................................................................ 123
Discussion ............................................................................................................. 124
Integrated (Expert) Identities: What it Means to Identify as a CI Teacher ......... 125
The Reification of Experience: Present-Tense Verbs .................................. 125
Integrated Identity ...................................................................................... 126
Expert Identities ......................................................................................... 127
Chapter 7: Zooming In: Identifying and Mathematics-For-Teaching in Number Line
Narratives .................................................................................................................... 128
Overview ............................................................................................................... 128
The Two Stories .................................................................................................... 131
Identifying ............................................................................................................. 133
Glynnis ............................................................................................................ 133
Participant and Spectator Roles .................................................................. 133
Jonathan ........................................................................................................... 140
Participant and Spectator Roles .................................................................. 140
Identifying: Discussion .................................................................................... 143
Mathematics-For-Teaching Competencies ............................................................. 144
Ethnopoetics and Mathematical Discourse ................................................. 145
Connecting Mathematically Important Ideas and Moving Between Metaphors
.................................................................................................................. 147
Mathematics-For-Teaching Competencies: Discussion and Conclusions .......... 169
Discussion: Identifying and Mathematics—For-Teaching ........................................ 170
How Identifying and Mathematics-For-Teaching are Interlinked ...................... 171
Glynnis: Uncertainty in Her Mathematical Knowledge and the Tentative
Nature of Knowledge ................................................................................. 171
Jonathan: Being Verbs and Certainty .......................................................... 172
Identifying and Perceptions of Competence ................................................ 173
Knowledge of Mathematics-For-Teaching as a Culturally-Laden Term ............ 174
Implications For Inclusive Approaches to Mathematics Teacher Education
Practice and Research ...................................................................................... 175
Chapter 8: Discussion and Conclusions ....................................................................... 177
Contributions of the Dissertation ........................................................................... 179
The Relationship Between Identifying and Implementation .............................. 180
Identifying Narratives and Implementing or Not Implementing CI ............. 181
Identifying Narratives and Teachers' Understandings and Implementation of
Teaching Practices ..................................................................................... 183
Identifying and Mathematics-For—Teaching Impact Each Other and Are Often
the Same Thing .......................................................................................... 185

 

Teaching as Identifying .............................................................................. 186

Obstacles and Persistence ................................................................................. 186
Assigning a Location to the Problem .......................................................... 187
Overcoming the Obstacle and Sustaining Implementation .......................... 190

The Dynamic Nature of Identifying and Implementation .................................. 191

Limitations of the Dissertation ............................................................................... 193
Implications ........................................................................................................... 195
Directions for Future Research ......................................................................... 198
Conclusion ............................................................................................................ 200
Appendices .......................................................................................... 202

Appendix A: Interview Protocol for Phase I Life Story Interviews ................... 202

Appendix B: Protocol for Phase 11 Individual Interviews .................................. 205

Appendix C: Round Robin Protocol For Teacher Discussions .......................... 209

References ................................................................................................................... 210
xi

LIST OF TABLES

Table I. Data Collection ................................................................................................ 51
Table 2. Types of Data and Dates Collected .................................................................. 54
Table 3. Gina's Direct Identifying Statements and Figured Worlds ................................ 63
Table 4. Kelly's Direct Identifying Statements and Figured Worlds ............................... 67
Table 5. Shirley's Direct Identifying Statements and Figured Worlds ............................. 71
Table 6. Jonathan's Direct Identifying Statements and Figured Worlds .......................... 75
Table 7. Constructed Dialogue in Glynnis' Story ......................................................... 138
Table 8. "Someone Doing Something To Something" Template .................................. 151
Table 9. "Someone Doing Something to a Mathematical Object" Template ................. 152
Table 10. "Someone/Something Is/W as Something" Template .................................... 152
Table I I . Glynnis' Metaphorical Shifts ........................................................................ 156
Table 12. Constructed Dialogue in Glynnis' Story ........................................................ 158
Table I 3 . Jonathan's "Someone Doing Something to Something" Template ................. 162
Table I4. Jonathan's "Someone Doing Something to a Mathematical Object"

Template .............................................................................................................. 163
Table 15. Jonathan's "Someone/Something Is/W as Something" Template .................... 163
Table 16. Metaphors in Jonathan's Story ...................................................................... 166
Table I 7. Comparing Glynnis and Jonathan's Stories ................................................... 168

xii

LIST OF FIGURES

Figure 1. The Story Plot Map ..................................................................... 13
Figure 2. Common Plot of Chapter 1 Stories ................................................... 14
Figure 3. Plot of Dissertation Story ............................................................... 15
Figure 4. Components of Complex Instruction .................................................. 20
Figure 5. Chapter 4—Act I and Plot Point 1 .................................................... 60
Figure 6. Chapter 5—Act II and Plot Point 2 ................................................... 88
Figure 7 Chapter 6—Act III ..................................................................... 119
Figure 8. The Students' Number Line Work ................................................... 134
Figure 9. Original Concept of Identifying and Implementation ............................. 192
Figure 10. The Relationship Between Identifying and Implementation of CI ............ 189
xiii

CHAPTER 1: INTRODUCTION

In 1991, I graduated from high school and left my childhood home in the rural
Upper Peninsula (U.P.) of Michigan to study at a university in a wealthy suburban area.
For the first years, I struggled to feel like I belonged with my classmates, many of whom
had grown up in the area around the university. I spoke differently (folks from the UP,
or "Yoopers" as we are colloquially called, often have an accent that others describe as
almost Canadian, but not quite), dressed differently (fashions take longer to reach the
UP, where places to buy clothes are fewer), and struggled to understand my peers'
assumptions about their suburban lifestyles.

As hard as it was to leave the UP, it was even harder to return, because I had
changed while I was away. After graduating fi'om the University and teaching for a
semester in a suburban elementary school, I was offered and accepted a position at an
elementary school near where I had grown up. I was relieved to be going home. I
expected to be received as an insider. And yet, people pointed out that I spoke differently
and dressed differently. Several of my students' parents distrusted me from the start. They
perceived that I was trying to impose my big-city (progressive) ideas about education on
them.

This impacted my perception of my role as a teacher. A particularly memorable
moment in my early teaching stands out as one that shaped my identity. A second-grade
student had been talking about how he did not plan to continue school any longer than
legally required; after all, his parents were doing well without degrees. As I launched into
a speech about the importance of education both for economics and for life, it hit me. By

inviting my students to become "school people," I was possibly asking them to

experience the same heartache I was experiencing. I could be inviting them to participate
in an activity that could potentially change how they participated with their families and
community. I was asking them to change themselves.

This narrative is my own; however, there are older narratives with similar plots
told by other storytellers. W.E.B. DuBois, in the story, "Of the Coming of John"
(1994/1903), wove the tale of a young African-American man who, in the early 1900's,
left his village in the south for a schooling in the north. Early in his time at school, John
had trouble adapting to the norms and expectations of the academy. W.E.B. DuBois
writes about John's troubles when he first arrived at school:

Up in Johnstown, at the Institute, we were long puzzled at the case of John
Jones. For a long time the clay seemed unfit for any sort of moulding. He
was loud and boisterous, always laughing and singing, and never able to
work consecutively at anything. He did not know how to study; he had no
idea of thoroughness; and with his tardiness, carelessness, and appalling
good-humor, we were sore perplexed. One night we sat in faculty-
meeting, won'ied and serious; for Jones was in trouble again. This last
escapade was too much, and so we solemnly voted "that Jones, on account
of repeated disorder and inattention to work, be suspended for the rest of
the term" (pp. 143).

After a time, John Jones was allowed to return to school. DuBois describes the changes
that ensued:

Thus he grew in body and soul, and with him his clothes seemed to grow
and arrange themselves; coat sleeves got longer, cuffs appeared, and
collars got less soiled. Now and then his boots shone, and a new dignity
crept into his walk And we who saw daily a new thoughtfiilness growing
in his eyes began to expect something of this plodding boy. Thus he
passed out of the preparatory school into college, and we who watched
him felt four more years of change, which almost transformed the tall,
grave man who bowed to us commencement morning. He had left his
queer thought-world and come back to a world of motion and of men (pp.
144).

While the changes in John allowed him to succeed in school and allowed his instructors
to talk about him as a student from whom they could "expect something," they became a
hindrance when John returned to his home community in the south. On the day John
arrived home on the train:

He stepped to the platform, and paused: a little dingy station, a black
crowd gaudy and dirty, a half-mile of dilapidated shanties along a
straggling ditch of mud. An overwhelming sense of the sordidness and
narrowness of it all seized him; he looked in vain for his mother, kissed
coldly the tall, strange girl who called him brother, spoke a short, dry word
here and there; then, lingering neither for hand-shaking nor gossip, started
silently up the street, raising his hat merely to the last eager old aunty, to
her open-mouthed astonishment. The people were distinctly bewildered.
This silent, cold man, --was this John? Where was his smile and hearty
handgrasp? "'Peared kind 0' down in the mouf," said the Methodist
preacher thoughtfully. "Seemed monstus stuck up," complained a Baptist
sister. But the white postmaster from the edge of the crowd expressed the
opinion of his folks plainly. "That damn Nigger," said he, as he shouldered
the mail and arranged his tobacco, "has gone North and got plum full 0'
fool notions; but they won't work in Altamaha." And the crowd melted
away.

In this story, the changes that John had to make to become a good student were changes
that caused him to be distrusted by his community when he returned home. John had
changed how he identified, and the community-members perceptions of him changed as
well, making for a tragic narrative.

Another story about the impact of schooling on how one identifies and is

perceived by others is the 1983 film, Educating Rita (Cartlidge & Gilbert, 1983). The

film1 tells the story of a 26-year-old housewife from a working-class family who decided

to go to college for the first time. As she studied, she became distant from her husband,

 

l
The film was based on the stage comedy of the same name by British playwright Willy
Russell, which premiered in London in 1980.

who did not understand her growing passion for literature and learning. At the narrative's
climax, they ultimately divorced.

Why would a person like Rita continue to passionately pursue her schooling, even
though it meant losing her husband? In the movie, Rita is depicted as saying to her
professor:

I'm beginning to find me. It's great. It is, you know, Frank? It might sound
selfish but all I want for the time being is what I'm findin' inside me. . .I see
[my husband] lookin' at me sometimes. I know what he's thinkin'. I do,
you know. He's wonderin' where the girl he married is gone to. He even
brings me presents some times, hoping that the presents'll make her come
back. It can't. She's gone. And I've taken her place.

Rita is articulating something about how her study of literature is both helping her to
discover who she is and also changing who she is. About half way into the movie, there is
a scene in which Rita brought an essay to her professor, Frank, to discuss:

Frank: I don't know what to say.

Rita: Yeah well try and think of something Frank. I don't mind if you tell
me it was rubbish. I don't want pity. Was it rubbish?

Frank: No, no, it wasn't rubbish. It was a totally honest, passionate account

of your reaction to the play.

Rita: Sentimental you mean.

Frank: No it was too honest for that. It was almost moving. But in terms
of what you're asking me to teach you, in terms of passing
examinations, oh, God you see, ah,

Rita: Say it! Go on. Say it.

Frank: In those terms, it's worthless. It shouldn't be, but it is. But in it's
own terms it's, it's wonderful.

Rita: It's worthless, you said. And if it's worthless you have got to tell me
because if I want to write essays like those on there. I want to learn
and pass exams like they do.

Frank: Yes but, if you're going to write that sort of stuff, you're going to

have to change.

Rita: All right. What, just tell me how to do it.

Frank: Yes, but I don't know if I want to tell you. I don't know that I want
to teach you. What you have already is too valuable.

Rita: Valuable? What's valuable? The only thing I value is here! Coming

here once a week.

F rank: But don't you see. If you're going to write that sort of stuff, pass
examinations, you're going to have to suppress, perhaps even
abandon altogether, your uniqueness. I'm going to have to change
you.

Rita: But don't you realize I want to change. Is this your way of telling me

_ ' that I can't do it? That I'm not good enough?

Frank: No of course you're good enough. It's not that at all.

Rita: Because if that's what you're trying to tell me, I'll go now.

Frank: No. Rita, I promise you. You are good enough.

In this scene, Frank was articulating what I had intuitively realized when I was teaching
my students—that learning results in loss. At times, it may result in the loss of previous
paradigms, notions that become outdated, or scientific "facts" that get revised based on a
new study. It may sometimes result in the loss of one's previous self-perceptions as one
takes on new ways of identifying and changes one's life story.

The changes in Rita were noticeable to others as well. Later in the movie, when
Rita encountered her ex-husband and his new wife on the sidewalk, he said of her:

I believe you're doin' really well at the college now, eh? I hardly

recognized you. You really look the part. Doesn't she, eh? Look the real

student. You'll be on drugs and demonstration next, ah hah.

Indeed, by the end of the movie Rita was dressing differently, talking differently,
responding to her teachers differently, and even identifying by a different name.

Long before DuBois and Educating Rita, long before the assumption that
education was to be for all people in a democracy, Plato articulated similar ideas in The
Republic (Reed & Johnson, 2000). He employed the metaphor of emerging from a cave
into the sunlight to conceptualize education. He told a story of uneducated people in this
dialogue:

Next, said I, compare our nature in respect of education and its lack to

such an experience as this. Picture men dwelling in a sort of subterranean

cavern with a long entrance open to the light on its entire width. Conceive
them as having their legs and necks fettered from childhood, so that they

 

remain in the same spot, able to look forward only, and prevented by the
fetters from turning their heads. Picture further the light from a fire
burning higher up and at a distance behind them, and between the fire and
the prisoners and above them a road along which a low wall has been
built, as the exhibitors of puppet shows have partitions before the men
themselves, above which they show the puppets.

All that I see, he said.

See also, then, men carrying past the wall implements of all kinds
that rise above the wall, and human images and shapes of animals as well,
wrought in stone and wood and every material, some of these bearers
presumably speaking and other silent.

A strange image you speak of, he said, and strange prisoners.

Like to us, I said. For, to begin with, tell me do you think that these
men would have seen anything of themselves or of one another except the
shadows cast from the fire on the wall of the cave that fronted them?...And
again, would not the same be true of the objects carried past them?...If
then they were able to talk to one another, do you not think that they
would suppose that in naming the things that they saw they were naming
the passing objects? Then in every way such prisoners would deem reality
to be nothing else than the shadows of the artificial objects. (p. 11)

Plato's description of uneducated people as naive and unable to perceive reality is one
that is prevalent today. It is not unlike the perceptions of the faculty in "Of the Coming of
John," who spoke of John as "unfit for any moulding" because of his exuberance—a very
different perception than Frank in Educating Rita, who saw the beauty that Rita brought
to the world before her schooling. Plato then continued to describe education as emerging
from a cave into the light, a painful experience:

When one was freed from his fetters and compelled to stand up suddenly
and turn his head around and walk and to lift up his eyes to the light, and
in doing all this felt pain and, because of the dazzle and glitter of the light,
was unable to discern the objects whose shadows he formerly saw, what
do you suppose would be his answer if someone told him that what he had
seen before was all a cheat and an illusion, but that now, being nearer to
reality and turned toward more real things, he saw more truly?...And if he
were compelled to look at the light itself, would not that pain his

eyes. . .And if, said I, someone should drag him thence by force up the
ascent which is rough and steep, and not let him go before he had drawn
him out into the light of the sun, do you not think that he would find it
painful to be so haled along, and would chafe at it. . .(p. 12).

In Plato's analogy, students do not willingly learn, but must be dragged by force. Seeing
the world as it really is as opposed to the shadows of reality causes pain for the learner.
Plato described the experience that would allow this person to see outside of the cave. He
then went on to describe the inner changes in the newly educated person that would cause
him to pity those who remained in the cave, and eventually return to them to tell them the
narrative of what he had learned. However, he would be spoken of as foolish and
incompetent by the cave dwellers:

And consider this also, said I. If such a one should go down again and take

his old place would he not get his eyes full of darkness, thus suddenly

coming out of the sunlight?

He would indeed.
Now if he should be required to contend with these perpetual

prisoners in 'evaluating' these shadows while his vision was still dim and

before his eyes were accustomed to the dark—and this time required for

habituation would not be very short—would he not provoke laughter, and

would it not be said of him that he had returned from his journey aloft

with his eyes ruined and that it was not worth while even to attempt the

ascent? And if it were possible to lay hands on and to kill the man who

tried to release them and lead them up, would they not kill him?

They certainly would, he said (p. 13).

Although I had not yet read Plato's work when I was teaching second grade, I
realized that there could be tragic consequences to perceiving my students as "cave-
dwellers" in darkness and trying to "drag [them] thence by force up the ascent which is
rough and steep." I began to have mixed feelings about my role as a teacher. On the one
hand, I saw that choosing not to engage in educational pursuits would limit the choices
my students had as adults. On the other hand, I saw that my students and their families,

unlike Plato's cave-dwellers, brought something valuable and beautiful to the world, and

like Frank in Educating Rita, I feared that schooling could destroy it.

Dragging Teachers Up the Ascent

For a long time, mathematics teacher education has been concerned about
teachers' knowledge of mathematical content and their beliefs about mathematics,
teaching and learning, and students. Of course, there are good reasons to be interested in
what teachers need to know and how to help teachers learn these things. Like many
others who write about teacher education, my ultimate desire is that students would learn
mathematics. Therefore, I am interested in helping teachers know how to teach
mathematics to children rigorously and equitably.

However, in speaking and writing about any learners, including teachers, teachers
and researchers are faced with a number of choices for metaphors to describe learners. In
my teacher education courses, I ask preservice teachers to reflect on whether they view
children as empty buckets to be filled or as children who come to class with a complex
web of knowledge, experience, and beliefs. In the same way, professional developers
have the choice to think about and treat teachers as lacking something important or
bringing something important. One way to attend to what teachers bring to these learning
environments is for teacher educators to elicit and pay attention to teachers' stories,
including their life stories and stories of practice. As seen in the popular accounts
summarized in this chapter, narrative is a powerful tool for communicating the impact of
life experiences on identity. Unfortunately, both in the mainstream media and teacher
education literature contain a plethora of messages that would have people believe that a
majority of teachers lack the knowledge and beliefs necessary for effective teaching. By

collecting and analyzing teachers' narratives about their teaching experiences, I illuminate

some of the some of the complex work that teachers do as they begin to implement new
practices.

Of course, there are teachers in schools who should not be there. However, we
need to be careful about how we talk about the goals of teacher professional
development, so that we do not, even inadvertently, portray teachers as cave dwellers. It
is difficult to change teaching practice. Much has been written about US. teachers' lack
of content knowledge (6. g. Ma, 1999), and their incapacity to change their teaching
practice to align with current recommendations for reform (Cohen, 1990) or their
resistance to innovations. And much of the literature assumes that learning is exclusively
for creating certain kinds of teachers in order to impact student outcomes. However,
Featherstone (2005) argues that this utilitarian View is a "pitfall in the work of teacher

education" (p. 135) and that seeing teachers as only a means to an end as opposed to the

ends themselves is immoralz. One way I view and treat teachers as ends instead of as
means is by adopting narrative methodology, which allows me to listen to the richness of
knowledge and experience teachers bring to their practice.

As well, much of the literature assumes that learning is always and only a
positive—or at least a neutral—experience for learners. When I became a mathematics
teacher educator, I brought with me, based on stories of my elementary teaching
experience, a sense of the enormous implications of inviting my students to learn (in this
case, learn to teach). These implications are corroborated by the stories of Rita, John, and

the cave dwellers. Those who study preservice teacher education are aware of the identity

 

In making this argument, F eatherstone used Immanuel Kant's argument that it is never
moral to regard a human being as a means to an end.

work (the construction of a new identity) involved in the transition from student to
teacher (e. g., Britzrnan, 2003). However, this line of study has not been adequately
carried forward into the study professional development of experienced teachers. The
narratives of experienced teachers who are attempting to become new kinds of teachers
have not been widely analyzed. I hypothesized that some of the same principles of
identity work involved in preservice teachers’ transitions from student to teacher would
be at play in experienced teachers' transitions from one kind of practitioner to another
kind of practitioner. When we ask experienced teachers to continue to learn about
mathematics teaching, we are asking them to change their identities in ways similar to
those I asked of my rural second-graders.

I hypothesized, however, that teachers’ construction of new narrativized identities
in professional development (PD) courses would often be subtler and less easy to uncover
than it is in preservice teacher education. Our preservice teachers are identifying as
teachers for the first time, but experienced teachers in PD already identify as teachers and
have many narratives of teaching to tell. However, they identify as certain kinds of
teachers in those narratives. The performance of new narratives necessary to construct an
identity as a new kind of teacher is what I was interested in studying. In this dissertation,
I do this by analyzing teachers' narratives during a time that they are implementing new
principles and practices for mathematics teaching. By analyzing teachers' narratives of
teaching to see how they portray themselves in their stories, I gained insight into the
kinds of teachers they perceived themselves to be.

At the same time, it is important for all students in K—12 mathematics classrooms

to have equitable opportunities to learn and achieve at rigorous mathematics content—

10

 

and there are a lot of reasons to believe that this is not currently happening in many
classrooms in the US. There are a lot of reasons for inequities concerned, and only so
much that individual teachers can do to solve these issues. However, firmly believing in
the power of an individual teacher to provide rigorous and equitable learning experiences
for her students, I wanted to be involved in effective teacher professional development
that helped teachers learn to teach in ways that are equitable and rigorous. These are the
goals of the professional development course I studied. This is not a simple way of
thinking about the purposes of teacher education. On the one hand, I am interested in
teachers' learning in and of itself, for the development of teachers as human beings,
without simply seeing teachers as the means to an end (in this case, equitable outcomes
for students). It is morally imperative to see teachers holistically and be concerned about
teachers' experiences and the quality of their lives. At the same time, it is morally
imperative to work for equity in K—12 schooling, and a part of this involves teaching new
pedagogies to teachers.

Recently, some researchers interested in studying equity in mathematics education
have focused on the relationship between identity and learning in mathematics classes.
Much of this literature has focused on how students' mathematics identities impact their
choices to participate (or not participate) in learning mathematics (6. g. Martin, 2006;
Nasir, 2002). The stories I have presented so far in this chapter, however, draw an arrow
in the other direction; they theorize that choosing to participate in learning impacts
individuals' identities. Therefore, identity may be theorized as a recursive process;

learning impacts identity, and identity impacts learning.

11

In this dissertation, I tell the story of a group experienced elementary teachers
who had attended a Professional Development (PD) course in which they had the
opportunity to learn principles and practices for teaching mathematics that were new to
them. These teachers had come to identify as certain kinds of teachers in their narratives
of teaching, and those ways of identifying impacted their perceptions of the
recommendations presented in the course and their decisions to implement them (or not).
First, I conducted life story interviews of four teachers, two who had decided to
implement CI and two who had not, to look for the relationship between their life stories
and decisions about CI. Then, over the course of several months, I followed a group of
four teachers who told me narratives about their experiences with implementing the
principles and practices taught in the PD. I did this in order to study how their identifying
narratives about mathematics teaching were impacting their day-to-day learning in and
from implementing the new teaching practices and how their learning was impacting their

mathematics teaching identities.

John, Rita, the Cave Dwellers, and Me
In this dissertation, I tell the story of a group of teachers who were implementing
mathematics teaching practices that were new to them. Their story is parallel to the
stories above. The stories of John, Rita, and the Cave are useful in helping to shape and
make sense of these teachers' stories. In fact, many teachers use the following structure
(Bieda, 2009, based on Hill, 2009):
0 Act I: The set up. (Characters and setting are established.)
I Plot Point 1: A tension or problem

0 Act II: The confrontation. (The tension escalates.)

l2

I Plot Point 2: A climax
0 Act III: The resolution.

A way to graphically represent this plot structure is shown in Figure 1.

 

 

 

 

 

 

Act I: The Set Act III: The
Up Act II: The Confrontation Resolution
Plot Point 1 Plot Point 2

Figure 1: Story Plot Map

The narratives that I summarized in the beginning of this chapter were similar to each
other and told parallel stories that might be generally summarized in the same way:

0 Act I: The set up. An individual is identified as being a certain type of
person.

I Plot Point 1: The tension or problem. The person experiences some
type of dissatisfaction or longing that motivates the person to go to
school. (Or, in the case of "The Allegory of the Cave", is forced to
become educated.)

0 Act II: The confrontation. Schooling. The learner a number of fulfilling
learning experiences along with a number of painful experiences resulting
from changing identity and relationships.

I Plot Point 2: The climax. The learner experiences an identity crisis

and must decide whether to become a "schooled" person or

13

continue to relate to family and friends in the ways they did
previously to schooling.
0 Act III: The resolution. The learner is a changed person. The learner has

new ways of being, and has had to come to terms with what has been lost.

A graphic representation of this plot is show in Figure 2.

 

 

 

 

 

 

Act I: An
individual is Act II: Schooling ACt 1113 A changed
identified person
Dissatisfaction or Identity crisis: With
longing whom/what do I

participate/ identify?

Figure 2. Common Plot of Chapter 1 Stories

The Story of Teachers' Learning

In the remainder of this dissertation, I will tell the stories of a group of teachers
who took a Professional Development (PD) course on using Complex Instruction to teach
mathematics. Their story is structurally and thematically parallel to the stories above. In
terms of the plot structure, the story of these teachers is parallel to the stories in Chapter
1, especially to the stories of John and Rita.

In Chapter 2 I elaborate my theoretical framework. I begin by discussing the
history of the problem of teacher learning for equity in mathematics education and for the
current mathematics education reform movement. I describe the Professional
Development (PD) event that is the context for this study and its connections to those
problems. Next, 1 frame this study within a sociocultural view of learning, describing

theories of participationism and situated learning that inform my study design and

14

analysis. Next, I describe the theories of identity as narrative on which I draw throughout
the study. Finally in Chapter 2, I list my research questions.

In Chapter 3 I describe my methods for narrative analysis. I describe the
participants, how they were chosen, and the contexts in which they teach. I provide
details of the data I collected and the procedures I used to analyze the data.

In Chapters 4, 5, 6, and 7 I present the analysis of the teachers' narratives. The

story I tell in these chapters may be graphically represented as in Figure 3.

 

Act I: How the _ . .
teachers identify Act 11: Stories of implementation Act 111: CI

 

 

 

 

 

(life stories) of CI in math . Teachers
Teachers decide to Decision to persist with
implement CI (or implementing CI (or not)

not)

Figure 3. Plot of the Dissertation Story

In Chapter 4 I tell about Act I and Plot Point 1. I report on my analysis of the life
stories of four teachers who took the CI course. Two of those teachers told narratives
about implementing Complex Instruction after the course and two did not. I analyzed the
teachers' life stories, narratives of the CI course, and their discussion of why they did or
did not choose to implement CI after the course. I compared what teachers said about
who they were at the time of the interview with who they said they were previously. This
included both comparing how teachers identified themselves in narratives about their K-

12 schooling with how they identified themselves in narratives of their teaching, and how

15

they identified themselves as teachers before and after the CI course. In this chapter, the
teachers' narratives function to propose reasons for deciding to implement C1 (or not).

In Chapters 5, 6, and 7 I report on the narratives of four teachers who decided to
implement CI. This is a different group of teachers than in Chapter 4, with the exception
of one teacher who is in both groups. In Chapter 5 I tell Act II and Plot Point 2 of the
story of implementation of CI. I report on their narratives of implementation of Complex
Instruction, including a description of the challenges that ensued when implementing CI
(Act II) and their reasons for deciding to persist through these challenges (Plot Point 2).
In this chapter, I report on my analysis of teachers' narratives, comparing what teachers
say about who they are—and more specifically, what kind of teacher they are--with what
they say about implementing new practices. The narratives in this chapter function to
explicate the identity construction that has to happen for the new practices to become a
part of their teaching repertoires.

In Chapter 6 I tell about Act III of the story of teachers implementing CI. After
implementing CI for some time, some teachers began to tell narratives identifying
themselves as CI teachers, which is the topic of Chapter 6. In this chapter, I theorize
about the reification of experience into integrated, expert identities. In this chapter, the
narratives function to claim and integrate some new ways of being teachers.

In Chapter 7 I move back to Act I of the story. Specifically, I argue for adopting a
view of teachers coming into PD events as bringing valuable competencies with them,

instead of adopting a deficit model (or cave-dweller model, as per "The Allegory of the
Cave") in teacher PD. In this chapter I move into a micro-discourse analysis of small

parts of two teachers' narratives about the same lesson and compare what these two

16

teachers say about who they are with how they say it. In this chapter, the narratives

function to perform knowledge that is (or is not) explicitly recognized by the teacher.
Finally, in Chapter 8 I look across the four findings chapters to answer my

research questions. I discuss the implications of my findings, limitations of my study, and

directions for future research.

17

CHAPTER 2: LITERATURE REVIEW

Mathematics Teacher Professional Development, Equity, and Reform

If learning involves loss, then why encourage teachers to continue learning,
attending teacher professional development opportunities, and implementing new
teaching practices? As the knowledge base of mathematics grows, and as the US. moves
into a post-industrial economy, there is a great deal of change recommended for both the
content and the methodology of mathematics pedagogy. To serve students well, teachers
must continue to learn as both doers of math and practitioners of mathematics teaching.
The current wave of mathematics education reforms (N CTM, 2000) has brought an
increased call for teacher Professional Development (PD) (Ball, 1996).

A particular concern exists for increasing the mathematics achievement of students ’
who have not been well served by traditional mathematics teaching methods. Teachers
are confronted with what Ladson—Billings (2006) calls the "education debt," that is, the
accumulated effects of inequality in education over the course of our nation's history.
Teaching a high level of mathematics to all students is more than an economic issue, it is
a moral one. Mathematics is a gatekeeper for many economic opportunities, including
higher education. It is also a necessary component of successful participation in
democracy (Ball, Goffney, and Bass, 2005). Teaching high levels of mathematics to all
children, regardless of race, language, gender, social class or ability, is therefore an issue
of economic and social justice.

The link between PD and teacher learning has been studied by assessing changes in
knowledge of mathematics for teaching (Hill, Rowan, & Ball, 2005), in students' learning

(Fennema, Carpenter, Franke, Levi, Jacobs, Empson, et al., 1996), in teacher beliefs

l8

(Ambrose, Clement, Philipp, & Chauvot, 2004), and in teaching practice (Drake, 2006).
Studying these changes separately has been informative; however, it is far from clear
what kinds of teacher PD are effective at impacting teacher practice and improved
student achievement.

As teacher educators and researchers our challenge is to work toward teacher PD
programs that ultimately effect student equity in math education—in ways that honor
teachers as ends in themselves and not merely ends to improving student achievement. It
would be inconsistent to treat teachers otherwise while simultaneously claiming to be
working for equity and justice on behalf of students. Our methods of researching and
implementing teacher PD must treat teachers with the same dignity we hope they afford
their students.

So in approaching my research questions of what makes for effective PD and
effective teacher learning, I wanted to choose a PD program that has been shown to
effectively impact the learning of students who have historically been underserved (ex:
Boaler, 2006; Gutierrez, 1999) and which also respected teachers as learners with various
identities. Complex Instruction (CI) is a PD program that met these criteria. CI is a
particular type of cooperative groupwork, used in heterogeneous mathematics classes to

promote equitable and rigorous outcomes (Cohen & Lotan, 1997).

Complex Instruction in Mathematics: The Professional Development Context

Mathematics teachers at "Railside" school in California used CI to de-track their
high school mathematics courses. The mathematics program at Railside school, which
has a high concentration of Latino/a immigrant students and students in poverty, had a

positive impact on outcomes such as students' choosing to take upper-level mathematics

l9

 

classes and going on to colleges and universities (as compared with other schools with
similar demographics) (Boaler & Staples, 2005).

C1 was developed by Elizabeth Cohen and Rachel Lotan, and had primarily been
used in elementary and middle school contexts (Cohen, 1994; Cohen & Lotan, 1997).
The teachers at Railside worked with Cohen and Lotan to adapt the principles and
practices of CI to the context of high school mathematics. The PD course, entitled
Designing Groupwork in Mathematics, engaged teachers in learning the specific ways CI
had been enacted at Railside school to promote rigor and equity in mathematics. The
framework for using CI to teach mathematics that was presented in this PD was
organized into three overlapping components: Curriculum, instructional strategies, and

status. These are demonstrated in Figure 3 (Teachers Development Group, 2005).

    
  
   
   
     

Instructional
Strategies
Norms, roles,
interventions

   

  
  
 

Curriculum
Groupworthy
tasks

   
 

 

Status
Perceptions of
competence

Figure 4: Components of Complex Instruction

20

Curriculum in CI consists of what are called groupworthy tasks, which are
mathematics tasks that require a number of different abilities to solve, such that it is very
unlikely that any one individual in a group will be able to complete the task without the
others. For example, Jonathan developed a task which asked students to solve the
problem "15 X 49" in four different ways, and create a poster that demonstrated those
four solution methods. Other CI teachers have extended that task so that students also
highlight connections between the four solution methods using color, lines, the
arrangement of the solutions on the paper, and other visual rhetoric tools. Such a task
requires students to construct and use alternative algorithms, find similarities and
differences, create and understand a number of representations, organize information, and
more (depending on how they choose to solve the problem). Teachers generally give
students a one-page task card that explains the task and expectations for the product
without being too explicit about the specific steps students should take to solve the task
(which can be solved in multiple ways).

Instructional strategies include roles, norms, and interventions. Students are
assigned roles, which have been specifically designed to engage them in the mathematics
at stake. An example is the resource monitor, who makes sure all group members have
access to the materials, calls the teacher over for group questions, and is responsible for
engaging the group members in clean up. Another role is facilitator, who is responsible

for making sure that all group members contribute equally to the solution and. understand

21

the work of other group members3.

Groupwork norms are established that communicate expectations for what it means
to do mathematics in groups. For example, some teachers have used the norms, "no one is
done until everyone is done," "helping does not mean giving answers," and "group
questions only." In addition, a number of interventions are used. Teachers in the PD
course spend time learning about how and when to intervene in a group, how to step back
and watch a group carefully in order to diagnose status and mathematical issues in the
group before intervening, and where to position oneself when choosing to intervene in a
group. The purpose of all of these instructional strategies (roles, norms, and
interventions) is to promote interdependence among members of a group and autonomy
from the teacher.

The third component is status. A status issue happens when people in groups rank
themselves and each other according to a set of unspoken criteria, and when such a
ranking interferes with equal participation in the group (Cohen, 1994). In classrooms, two
different types of status are at play: academic status and social status. Status interferes
with students' participation in group mathematics tasks because often the students who
are perceived as more mathematically competent (have higher academic status) are likely
to do much of the work, while the students who are perceived as less competent (have
lower status) are likely to allow students with high status to dominate the space,
materials, and ideas. This third component is not separate from curriculum and

instructional strategies, as both of those components are geared toward disrupting status

 

Other forms of cooperative learning have used roles; however, CI differs in that every
role is specifically engineered to require the individual to engage in the mathematics of
the task (not to simply clean up, for example).

22

 

in groups and therefore equalizing participation and learning.

In addition to using groupworthy tasks and instructional strategies to disrupt status,
teachers attend to status by thinking about the abilities that students bring to math class
that may not be traditionally mathematical but may be leveraged for solving
mathematical tasks. By building on these abilities (for example, in the multiplication task
above, students must use tools from visual rhetoric to communicate mathematical ideas),
and by making them explicit to students, teachers help students to see each other as
mathematically competent. Over time, such experiences help to disrupt status in the
mathematics classroom.

The CI math PD differs from other math teacher PD opportunities in its focus on
status. CI takes a comprehensive approach, engaging teachers in all aspects of
mathematics teaching, including defining mathematics broadly, rethinking what it means
to be "smart" in math, learning to implement instructional interventions, establishing
norms and roles for problem solving, and creating rigorous curricular tasks. In other
words, they are asked to reframe and rewrite their narratives about mathematics, their
students, and themselves as doers of mathematics. Cl and creates space for students with
many different kinds of identities to participate fully in mathematics classrooms (Jilk,
2007) and has been shown to have profound effects on student learning (Boaler &
Staples, 2005; Cohen & Lotan, 1997; Horn, 2005).

The subjects of my study are elementary teachers who chose to attend a 30-hour PD
course on using CI given at MSU. I hypothesized that the reason some teachers have
committed to changes in practice after this workshop is that it allows many different ways

of identifying to happen in the workshop. 1 studied teachers ways of identifying using

23

 

narrative methodology to learn more about this. I note below that their narratives of
successful participation in the CI math PD context are resources some of these teachers
used to implement CI and sustain these changes in their mathematics teaching practices.
But what does sustained implementation of CI in mathematics teaching do to teachers?
How does it change their narratives (or not)? And what do the teachers do to change C1
(or not) in response? Understanding the implications of learning new practices on

teachers is important to respecting teachers and simultaneously promoting equitable

outcomes in school .

Teacher Learning Situated in Teaching Practice

A Participationist Lens

Learning may be described as becoming able to participate in or take part in
discursive norms and routines of communities of practice (Wenger, 1989). The
participationist perspective (Sfard, 1998) posits that learning and participation are thusly
related. This theory is often contrasted with what may be called the acquisitionist
perspective—an idea that individuals who are lacking some knowledge acquire it from
other individuals who impart knowledge to them. While an acquisitionist sees a learner as
lacking important knowledge, a participationist acknowledges that, because all
participation is linked to learning, all learners bring knowledge, identities, and practices
with them. Each learner has a background through which he or she understands and
interprets new experiences. Therefore, teachers' narratives of their experiences should be
of interest to teacher educator's and researchers interested in teacher learning.

Studies have looked at teachers' learning in and from PD courses, either by

measuring teachers' knowledge and beliefs before and after a PD course, or by studying

24

the fidelity of teachers' implementations to the reformer's intended implementation. One
concern that such studies have not fully addressed, however, is that even teachers who
want to implement recommendations for reform may not make essential changes and may
simply weave some new practices in with their old ways of teaching (Cohen, 1990).

From an acquisitionist standpoint, teachers who attend a PD course should have
been given what is needed to fully implement the new practices. In that view, if the new
practices are not implemented in the ways the professional developers intended, two
reasons are possible: Either, 1) the PD did not adequately disseminate the knowledge,
skills, and beliefs that are necessary for implementation, or 2) the teachers did not
adequately acquire them.

However, a participationist lens provides more opportunities to look into the
dynamics of implementation. A participationist lens focuses attention on the backgrounds
that learners bring to a situation—practices, knowledge, and identities—through which
learners may interpret new experiences. From this perspective, it is important to listen to
teachers' narratives of experience and understand how they interpret those experiences,
which I do in this study by adopting analytic methods from narrative research.

An important idea is how learners reijjz participation—that is to translate concrete
experiences into abstract objects that have thing-ness and are spoken of as existing in
ways that physical objects are spoken of as existing. For example, a person who has
many experiences in mathematics classrooms may reify those experiences into a
statement about her "mathematics ability," assigning the noun "ability" to speak of the
idea as an object. Or a teacher who tries to implement a new teaching practice, such as

Complex Instruction (CI) may tell a number of narratives about teaching CI lessons.

25

 

Eventually she may directly identify herself as "a CI teacher," which is a reification of
her experiences with implementing CI. In this study, I analyzed teachers' narratives for

the reifications of their experiences.

Learning as Situated in a Particular Context

An idea important to studying learning as participation is situated learning (Lave
& Wenger, 1991). That learning is situated means that particular meanings are situated in
particular contexts. Not only do teachers come to PD events with different backgrounds
and experiences, but when they leave the PD course and return to their classrooms to
implement the recommendations made in the PD course, they return to different contexts
with different students. They have different experiences as they participate in
implementing the recommendations. While they are implementing the recommendations
and telling narratives of these implementations, they are negotiating different meanings in
practice. Simply looking at teachers' implementation, knowledge, or beliefs before and
after a professional development intervention therefore gives only a thin slice of the
picture of teachers' learning about PD recommendations over time. By analyzing
teachers' narratives of practice over a period of several instances of implementation, 1
give a fuller, more nuanced picture of what it takes to implement new practices and
sustain that implementation over time.

In this dissertation, the object of study is the learning of teachers' in their
implementation of new practices in their classrooms. There has been a call for increased
attention to teachers' learning in and from teaching practice (which is different from
teachers' mere experience) (Ball & Cohen, 1999). Teachers' knowledge and practices are

linked with the particulars of the contexts in which they occur (Putnam & Borko, 2000).

26

This dissertation also builds on research that reports on teachers' learning in and from
day-to-day interactions in their classrooms (Horn, 2005). Therefore, I studied teachers'
narratives of day-to-day classroom interactions, analyzing their narratives for the contexts

in which the teachers set them.

Identity
Above, I mentioned that learning may be defined as becoming able to participate “a
in a community of practice. Similarly, identity may be defined as participation in a

community of practice (Wenger, 1998). People identify by choosing to participate in a

 

"7’ '.l n'WV' x11 . a

particular community of practice (such as a club, church, school, or other organization).
They tell narratives about their participation, recounting their actions and experiences.
And they may reify these experiences into objectified ideas about themselves (such as "I
am a kindergarten teacher.") People also identify by non-participation in groups.
Additionally, people may identify by affiliating with groups whether or not they
physically interact with a particular community of practice (for example, affiliating as a
Democrat is participation, although it does not necessarily mean that an individual is
participating in activities of the Democratic party).
Participationist views of identity and learning (Lave & Wenger, 1991; Wenger,
1989) imply boundaries between communities of practice, and different positions within
communities (such as legitimate peripheral participation and full participation).
Individuals may identify as being a part of a community of practice or not, but also on the
periphery or fully participating. This idea of position in a community of practice relates
with a notion of identity as positioning (or aligning oneself with some communities,

people, and ideas and distancing oneself from others) (vanLangenhove & Harre, 1999).

27

In analyzing the teachers' narratives, I attended to how their portrayals of themselves
positioned them with or against others, such as the students, colleagues, administrators, or
the mathematics.

Identifying and Narrative in Figured Worlds

Anthropologists have illuminated the contextualized and recursive nature of
identifying stories and actions (Holland, Lachicotte, Skinner, & Cain, 1998). They have
defined identities as stories of self-understanding, which have "strong emotional
resonance" (Holland, Lachicotte, Skinner, & Cain, 1998, p.3) and are created as "people
tell themselves who they are and then try to act as though they are who they say they are"
(Holland, Lachicotte, Skinner, & Cain, 1998, p. 3). People improvise identities using the
cultural resources available to them. Conflicts between stories, histories, cultures, and
situations complicate the work of identifying. Conflicts between different theoretical
contexts, or figured worlds, are one such complication.

A figured world is "a socially and culturally constructed realm of interpretation in
which particular characters and actors are recognized, significance is assigned to certain
acts, and particular outcomes are valued over others" (p. 52). Figured worlds may exist
within spaces where groups of people actually meet and live, or may exist figuratively in
dramatized or narrativized worlds such as books and T.V. Examples of figured worlds are
Alcoholics Anonymous or the world of idealized romance. Materials, words, and actions
take on different meanings in different figured worlds, and different agents are expected
to take part in certain actions, the meaning of which is specific to the figured world. For

example, eye-blinking may be interpreted to be flirtatious eyelash-batting in a romance

28

movie. In a Windstorm or in an optometrist's office, the same action might be interpreted
very differently.

Figured worlds in which teachers may participate (and index in their narratives of
practice) include a certain discourse of mathematics education reform (such as that
portrayed in NCTM, 2000), the school community, and the particular PD context in
which this study is situated. Teachers may identify themselves differently in these
different figured worlds, because of the different cultural resources available to them in
each of those places, the different things that they perceive to be valued in those places,
or the different outcomes that are seen as meaningful in those places.

While Holland, Lachicotte, Skinner, and Cain (1998) highlight the conflicts of
individuals in different figured worlds, they do not consider the ways in which an
individual's practices across these contexts impact their identifying narratives. Seeking a
framework for completing a fme-grained analysis of the narratives teachers tell in
practice, I turn to work done on student learning in mathematics education (Ben-Yehuda,

Lavy, Linchevski, & Sfard, 2005; Sfard, 2006a & 2007; Sfard & Prusak, 2005).

Defining and Operationalizing The Work of Identz'fizing

People may identify themselves directly or indirectly by constructing narratives
(which I will explain in the following two sections), or by acting (Sfard, 2007). In this
study I analyze teachers' direct and indirect identifying narratives. People identify
directly through statements about themselves, such as "I am good at math", and indirectly
by telling stories of their experience. I will first discuss the theoretical basis for indirect
identifying, which people do by telling narratives. Then I will discuss the theoretical

basis for direct identifying.

29

Indirectly identifying by telling stories. People are identifying whenever they
speak because identity-building is one of the tasks of discourse (Gee, 1999/2005). People
act out particular selves in the telling of narratives, using them to construct and transform
themselves (Wortham, 2006). Narrative style is associated with one's cultural identity and
presentation of self and making changes in narrative style is a tool for changing one's
social identity (Gee, 1989). Attention to the structure of narratives and how the structures
set up interpretive questions and limits for answering those questions reveals the ways
people are making sense of themselves, the figured worlds in which they participate, and
their interactional positioning (Gee, 1991). As a teacher tells a story of an event that
occurred in her classroom and how she responded, she does so according to how she
wants her listener(s) to identify her as a teacher.

Directly identifiing using identifying narrative statements. Identifying narrative
statements are the result turning stories into labels, such as I am a capable mathematics
student or I am a kindergarten teacher (Ben-Yehuda, Lavy, Linchevski, & Sfard, 2005;
Sfard, 2006a & 2007; Sfard & Prusak, 2005). This definition of identifying narrative
statements should not be confused with narratives as defined by sociolinguists such as
Labov (1972) (Juzwik, 2006), which require a number of temporally or causally related
statements to be strung together. Rather, direct identifying narrative statements are
statements made on the basis of the reification (objectification and detemporalization) of
implied narratives (Sfard, 2006b).

Identifying narrative statements are statements about a person that are reified,
endorsed, and significant. Reified narratives are those that a person considers to be an

object that exists regardless of time. ("I am a kindergarten teacher" is a statement

30

objectifying experiences of teaching at a certain grade level). Endorsed narratives are
those that the person would accept as true. (The person making the statement would have
to agree that "kindergarten teacher" is an appropriate way to describe herself). Significant
narratives are those that the person would have an emotional reaction to or would find
salient and important to their self-understandings. (Being a "Kindergarten teacher" as
opposed to simply "a teacher" or "an early-grades teacher" would have to be salient to the
individual's self-perceptions). One's collection of identifying narrative statements can be
classified into two types of identity: designated and current identity. Designated identity
refers to narratives about what the person expects or desires to be in the future. (The
kindergarten teacher may desire to be "a student-centered teacher") and current identity
refers to narratives about the person in the present. Learning may be defined as "closing
the gap" (p. 19) between designated and current identities (Sfard and Prusak, 2005) (e. g.,
the kindergarten teacher may come to see herself as "the student-centered teacher" she
desired to become).

This theory has been used to examine students' mathematics learning. Sfard and
Prusak (2005) studied the identifying narratives of students in high school mathematics to
gain insights about achievement differences between two groups of students: students
whose families had recently immigrated to Israel from Eastern Europe, and students
whose families were native to Israel. They noticed that the newcomers to Israel generally
attained higher levels of mathematics achievement than those from non-immigrant
families. They highlighted a difference between the identifying narratives of immigrant
and non-immigrant students. Students from non—immigrant families did not construct

designated identities involving mathematics learning, but students from immigrant

31

families included mathematics as a central part of their designated identities. Sfard and
Prusak posit that this difference in the construction of narrative identities is linked to the
observed difference in mathematics achievement between the two groups of students.

In another study, Wood (2008) studied children's identifying and mathematizing
(participating in mathematics instruction) in an elementary. classroom where CI was
being used to teach mathematics. She showed that the children's mathematizing was also
a process of identifying.

This work has not been widely applied to the study of teachers. It specifically
speaks to the dynamic nature of identity but has not yet been used to examine how
people's ways of identifying change or remain the same from context to context. Because
Holland, Lachicotte, Skinner, and Cain (1998) posit that people identify differently in
different figured worlds, one might assume, when Operationalizing identity as identifying
narratives, as Sfard and Prusak (2005) do, that people construct a variety of identifying
narratives when those narratives are set in different figured worlds. Using the concepts of
identifying narratives and figured worlds together will potentially afford a more
contextualized description of identifying.

Identifying and learning are interdependent, dynamic, and continually negotiated in
practice. Complicating this process are conflicts between past histories with which people
are comfortable, present discourses that tell conflicting stories, and different rules for the
different figured worlds in which they participate. For example, a teacher may be caught
between the figured worlds of reform mathematics and views of the school staff, and may
be receiving messages from administrators and parents that conflict with both of these

perspectives. These conflicts may affect this teacher as she improvises practice in the

32

flow of a mathematics lesson (e. g., as she intervenes in student groups) and evaluates this
performance during and after the lesson (e. g., as she identifies as a teacher who
intervenes quickly when she sees students struggling). If she objectifies or reifies her
evaluation of this performance (e. g., as a "good" performance), and she is emotionally
committed to that reification, it will become an identifying narrative that may affect her
firture learning, identifying, and practice (e. g., she may identify as a successful teacher
who is quick to intervene). Understanding how teachers identify and how this is related to
learning (i.e., changes in discourse or practice), helps us to understand how teachers'
narrative identities, constructed in PD events or in practice, provide teachers with both
resources and constraints for changing their teaching practice in order to provide
equitable, rigorous instruction for students in math. A logical next move in this line of
scholarship is to use the notion of identifying as narrative to study issues of knowledge,

culture, and context in teacher learning in practice.

Identifying and Implementing New Mathematics Teaching Practices

As mentioned earlier, identifying may be defined as participating in a community
of practice (Wenger, 1989). The teachers who participated in this study all chose to
participate in the community of practice that was the CI math PD course. Therefore, in
one sense, they were identifying as CI course participants. However, after the completion
of the course, some teachers decided to continue participating in C1 by implementing it to
teach mathematics in their classroom, and some did not. Because implementing CI is
essentially participation in a practice, and because participation and identifying are
closely interlinked, teachers' implementation of CI and their other ways of identifying are

also likely to impact each other.

33

Additionally, people make sense of new ideas through their salient identities. Lisa
Jilk (2007) studied Latina women who had graduated from Railside High School (see
also Boaler & Staples, 2005). She found that the students had interpreted their high
school mathematics experience through the lenses of their salient identities. Likewise, it
is reasonable to expect that teachers would interpret their experiences in the CI
mathematics course through the lenses of their salient identities. Because of this, it is P
plausible to hypothesize that teachers in this study will have different stories of CI

implementation based on their different ways of identifying. Indeed, research theorizing

 

links between teachers commitment and identity has discovered that when teachers see
the relationship between their values and the directions that are recommended by
reformers, they are more likely to become intellectually and emotionally engaged in those
directions (Day, Elliot, & Kingston, 2006). However, the connection between those

commitments to reform and implementations of reform are largely unstudied.

Life Stories and Changes in Practice

Researchers have found connections between teachers' mathematics life-stories and
their implementation of a reform-based curriculum (Drake, Spillane, & Hufferd-Ackles,
2001; Drake, 2006). Defining identity as a life story (McAdams, 1993), they studied
elementary teachers' mathematics and literacy life stories (Drake, Spillane, & Hufferd-
Ackles, 2001). They found three different plot structures for teachers' mathematics life
stories: the turning point, failing (foreclosed) and roller coaster stories. These researchers
discovered a relationship between teachers' life-story plot structures with their
implementation of reform. Teachers who had failing (foreclosed) stories taught

traditional lessons and focused on superficial aspects of reform; those with turning point

34

stories focused on affective aspects of reform; and those with roller-coaster stories
focused on both affective and content-based aspects of reform.

Turning-point stories were found to be particularly influential on teaching
practice. Tuming-point stories are those in which a teacher recounts early experiences of
failure and current positive perceptions of themselves as learners and teachers of
mathematics. By more closely examining teachers with turning point stories, Drake
(2006) found that the content of teachers' turning point stories impacted their
implementations of reform. Teachers who had turning point stories based on mathematics
content (as opposed to mathematics tools) felt that they should learn along with their
students and be transparent with students, were trying to incorporate new understandings
of mathematics into their teaching practice, and were looking for new pedagogies and
curriculum that fit their new understandings of mathematics. They looked at reform
curriculum materials and their students as a way to learn mathematics. Their observed
teaching practices were coded as more strongly reform-oriented. Teachers who had tool-
based turning point stories believed that manipulatives were crucial for making
mathematics accessible and enjoyable. They often changed the conceptual ideas in
reform-based lessons in order to include more manipulatives, actually making the lessons
more traditional in nature.

Much about the relationship between teachers' stories and teaching practice is still
unknown. How implementing recommendations for reform impacts experienced teachers'
identifying narratives has not yet been examined, nor has the impact of the culture of

schooling on this process.

35

The Impact of Implementation on Identz'fi/ing

The research on life stories and implementation draws a one-way arrow from
teachers' life stories to their implementation of a reform-based curriculum. However, in
this study, I am also studying how implementing CI will impact teachers' ways of
identifying.

The idea that the practice of teaching effects teachers is not a new one (Waller,
1932). This idea has been studied in preservice teacher education. For example, Britzman
(2003) found that student teaching had an effect on intending teachers' identities.
However, the effect of implementing a new teaching practice on an experienced teacher's
identifying narratives is relatively unstudied.

One reason to hypothesize that implementing a new mathematics teaching practice
will impact teachers' ways of identifying is because of the intertwined nature of
participation and identifying. As stated earlier, implementing CI may be spoken of as a
way of participating with the CI community of practice. As such, while implementing C1,
the teachers will have new experiences with participation and therefore new ways of
identifying (new narratives).

Identifying may also be conceptualized as story-telling (Holland, Lachicotte,
Skinner, & Cain, 1998; Sfard & Prusak, 2005), or making sense of experience through
the telling of narratives. The implementation of CI will result in new stories of
mathematics teaching practice for teachers to tell. The telling of the stories is one way
that teachers are likely to make sense of the experience (Ochs & Capps, 2001). Therefore,
the telling of narratives in interviews for this study is also likely to lead to new narratives

in the future.

36

 

Another reason to hypothesize that implementing a new mathematics teaching
practice will impact teachers' identifying narratives is because learning and identifying
are intertwined (Sfard & Prusak, 2005; Wenger, 1989; Wortham, 2006). That is, how one
identifies impacts ones participation in learning activities, and learning impacts how one
identifies. It is likely that teachers will learn something from the new experience of

teaching mathematics with Cl, and that this will impact their ways of identifying.

Research Questions
A goal of this dissertation is to discover more about the relationship between
implementing a new mathematics pedagogy and teachers ways of identifying. Therefore,

The research questions that guide this dissertation study are as follows:

1. How do teachers' ways of identifying and their stories of implementation of a new
mathematics pedagogy interrelate?

2. How do teachers' ways of identifying become obstacles (or not) to teachers' attempts
to change their teaching practice? How do they make sense of the obstacles and
overcome them?

3. How do teachers' ways of identifying change (or remain the same) as teachers

implement CI to teach mathematics?

37

CHAPTER 3: METHOD

In order to learn about the links between teachers' implementation of CI and their
ways of identifying, I designed a two-phase study of teachers who were implementing CI.
I generated data through interviews about teachers' life stories and stories of mathematics
teaching practice in order to study their ways of identifying over time and in varied
figured worlds.

The data collection and analysis for this dissertation was done in two phases.
Phase I was a pilot phase designed to investigate the links between teachers' life stories
and their decisions to implement (or not implement) CI. In Phase 1, four teachers from the
first cohort of the CI math PD course were chosen. I chose two teachers who had
reportedly been implementing CI to teach mathematics, and two had had not. I conducted
two mathematics life story interviews with each teacher to look for parallels between
their mathematics life stories and their stories of implementing CI to teach mathematics.
The results of Phase I are reported in Chapter 4 of this dissertation.

In order to answer the questions that I generated based on the results of Phase I, I
designed Phase II of the study, which makes up the bulk of this dissertation (Chapters 5,
6, and 7). I designed Phase II to study the relationship over time between teachers' stories
of CI implementation and changes in identifying. For Phase II I invited teachers from
both cohorts of the CI math PD course who decided to implement CI to teach
mathematics after the PD course. The data for Phase II was generated in three contexts, in
order to attend to the contextualized nature of identifying and the situated nature of
learning. I interviewed teachers individually about CI mathematics lessons they planned

and taught in their classrooms. In addition, the teachers gathered together to share stories

38

 

of mathematics teaching with Cl, and the teachers also presented their work on using CI

to other teachers who had not yet taken the CI course.
Phase 1

Phase 1 Participants

Participants were chosen based on their reported and observed implementation of
CI during ongoing collaboration in the months following the CI workshop. I chose
teachers who expressed an interest in continuing to implement CI to teach mathematics

over several months, in order to study the impact of this implementation on the teachers'

ways of identifying over time. Two teachers, Shirley4 and Jonathan, had reported making
substantive changes to their classroom. Two others, Kelly and Gina, had reported using
little or no CI. Kelly said she intended to do more groupwork in the future; Gina did not. I
chose participants who had responded differently to the workshop so a range of workshop
and classroom experiences could be studied.

Gina and Kelly taught in the same school in an urban district, with 215 K—5
students, of which 47% qualified for free or reduced lunch. The largest racial groups were
Caucasian (61%), Latino/a (19%), and African American (15%) (Standard and Poors,
2006). Gina was near the end of her second year teaching kindergarten, and her 28th year
of teaching. Kelly was transitioning from kindergarten to first grade and had been

teaching for seventeen years.

Shirley had taught for twelve years in a suburban district. She was

teaching sixth grade mathematics and science in a school of 255 fifth and sixth

‘

All names of teachers and schools have been replaced by pseudonyms.

39

 

graders, of which 25% qualified for free or reduced lunch. The main racial groups
were Caucasian (65%), Afiican American (20%), Asian/Pacific Islander (7%) and

Latino/a (6%) (Standard and Poors, 2006).

Jonathan had taught for fifteen years in a suburban district and was
teaching fifth grade mathematics and social studies in a school of 400 fifth and
sixth graders, of which 29% qualified for free and reduced lunch. The main racial
group in the school was Caucasian (83%), with the remainder of the student body
fairly evenly distributed between multi-racial, African American, Latino/Latina

and Asian/Pacific Islander students (Standard and Poors, 2006).

Phase 1 Data Collection

Phase One began as a pilot to the dissertation. I developed a semi-structured, two-
part life story interview protocol based on Drake, Spillane, and Hufferd—Ackles (2001).
The protocol is included as Appendix A. During the first interview I elicited stories of
learning and teaching mathematics and stories of participating in the CI workshop, in
order to look for links between teachers' identifying narratives and stories of CI
implementation. Participants were also asked to tell stories about their practice after the
workshop and their implementation of CI. In the second interview, I followed up with

questions that arose when transcribing and reading the first interview.
Phase I Data Analysis
Ways of Identl'fi/ing

Plots of Mathematics Life Stories. In order to understand teachers' ways of

identifying at the beginning of the study, I compared teachers' stories about learning and

40

 

teaching mathematics against the plots described by Drake, Spillane, and Hufferd—Ackles
(2001) and categorized them as failure (foreclosed), turning-point, or roller coaster
stories. The turning-point stories I then compared with the further subcategories in
Drake's (2006) study. I compared teachers' reported implementation of reform-oriented
mathematics teaching in general, and implementation of CI specifically, with those of the
teachers in that literature.

Direct Identifizing Statements. I noted direct identifying statements within the
stories. A direct identifying statement is one that is reified, endorsable, and significant, or
to which the speaker was emotionally attached. Such statements often were phrased as "I
am" or "I was" followed by an adjective or a noun, or "I can" or "I can't" followed by
something general or overarching, such as doing math in general. Statements in which
participants stated that they were unable to carry out a particular mathematical task were
not counted in this category. Other identifying narratives included those in which the
participants used the words "always" or "never" in describing themselves. Consider the
following two statements; both said by Jonathan near the beginning of his interview when
he was asked to tell his mathematics life story:

"I took only as much mathematics as I had to take."

"And I wasn't very successful in elementary school or middle school."

Both of these are strong statements with which Jonathan is identifying; however,
one fits the definition of direct identifizing statement (Sfard & Prusak, 2005) and the other
does not. When Jonathan said, "I only took as much mathematics as I had to take," he was
describing something he did (or in this case, did not do). The statement has not been

reified—that is, he has not objectified it into a quality of himself. The second statement,

41

"I wasn't, " is not about something Jonathan did, but something he was. Jonathan had
reified many things he did (and did not do) so as to construct himself as being a certain
type of person. Therefore, I did not code the first of those two statements as a direct
identifying statement, but I did code the second one as such. Again, this is not to say that
Jonathan is not identifying himself in the first statement. Of course, whenever we talk, we
are deciding how to portray ourselves to others (Gee 1999/2005), and so the first
statement contains messages about identity. Additionally, Wenger (1989) equates identity
with participation, and the first statement is about participation, so in that sense, it may be
recognized as being identifying. However, what I was concerned with were direct
identifying statements as defined by Sfard and Prusak (2005) specifically, in order to
further their line of research about a particular way of identifying that includes statements

that are reified, endorsable, and significant.

Figured Worlds

In order to attend to the contextualized nature of identifying, I analyzed the
identifying narratives to see if they invoked particular figured worlds. Figured worlds
included the CI community, Teacher Education, mathematics reform (for which there
seemed to be at least two different figured worlds, including different iterations of reform
or teachers interpretations of those), schooling in the past, the community of school
colleagues, the political/administrative realm, and social situations outside of school.

An important note is that I did not intend to draw firm lines between categories of
identifying narratives or figured worlds. 1 analyzed them as such because I noticed that
teachers often told contradictory stories of themselves, and I hypothesized that they were

identifying differently when telling stories set in different figured worlds. However,

42

sorting and categorizing the identifying narratives was not at all a definitive or clean
process. I placed several statements into multiple categories of figured worlds. At times,
it was difficult or impossible to identify with certainty whether a particular statement
referred to one particular figured world or more. The exercise did help me to see patterns
in identifying narratives, but should not be considered the only or even the best way to

classify identifying narratives.

Phase II

In Phase II I deliberately collected teachers' narratives of teaching with Cl that they
told in different contexts and analyzed these to illuminate the relationships between
identifying and changes in teaching practice. I studied a group of teachers who had
chosen to make CI a part of their mathematics teaching practice. I examined issues of
context as I interviewed these teachers in a variety of contexts and around artifacts of
practice from February-October, 2008. The teachers participated interviews in three
contexts: in individual interviews, group interviews, and presentations of their work to
those who had not yet taken the CI course. The purpose of the different interview types

and contexts was to gain more insight about the impact of context on identifying.

43

Phase II Participants

Four teachers participated in Phase II from the beginning to the ends. In
the sections that follow, I report on the cohort of the CI math PD course in which
each teacher participated, how they had been using CI to teach mathematics, and

the demographic characteristics of the teachers and the contexts in which they

work.
Jonathan

Jonathan was the only teacher who participated in both phases of the
study. Jonathan had participated in the first cohort of the CI math PD course
(summer, 2006), and had been using CI to teach mathematics for a year and a
semester. During his first year of CI math implementation, he reportedly taught 12

CI math lessons, all of which he designed himself by adapting other mathematics

tasks.

At the beginning of Phase II, Jonathan had taught for sixteen years in a
suburban district that I call Cornerstone. He is a white male in his late 405 who
had entered teaching as a second career (previously he was a manager at a retail
store). During Phase I and the beginning of Phase II (F ebruary-June, 2008) he was

teaching fifth grade mathematics and social studies. However, at the end of Phase

 

5 A fifth teacher (Kelly, from Phase I) began Phase II with the other four teachers.
Between Phases I and 11 she reported using several CI mathematics tasks in her
classroom, and expressed interest in participating the study, especially in order to
participate in the group discussions with other CI teachers. However, after participating
in one set of individual interviews and one discussion group session, Kelly decided that
study participation did not fit into her schedule, and decided to stop participating.
Although she consented to allow me to analyze the data, I decided to report in this
dissertation only on the four teachers who participated in Phase II from beginning to end.

44

 

II (August-October, 2008) he was teaching only fifth grade mathematics.

At the beginning of Phase I Jonathan's school had 400 fifth and sixth
graders, 29% of whom qualified for free and reduced lunch. The main racial
group in the school was Caucasian (83%), with the remainder of the student body
fairly evenly distributed between multi-racial, African American, Latino/Latina
and Asian/Pacific Islander students (Standard and Poors, 2006). By the end of
Phase II, the demographics had changed; 39% of students qualified for free and
reduced lunch. The breakdown of racial groups was Caucasian (78%), African
American (11%), Latino/a (7%), Asian/Pacific Islander (3%), Multi-Racial and

Native American (1% combined) (Standard and Poors, 2009).
Glynnis

Glynnis participated in the second cohort (Summer, 2007) of the CI math
PD course. She had reportedly used one of the CI structures—roles—for a couple
of lessons at the time Phase II began, and her classroom norms were similar to

those taught in the CI course. Glynnis said that she wanted to participate in this

study because she wanted to do more with C1.

At the beginning of Phase 11, she had been teaching fourth grade for eight
years in an urban district. She is an African American female in her 503, who
entered teaching as a second career (before that she worked in the district's

administration office).

Glynnis was teaching in a school I call Riverside, in which 275
kindergarten through fifth grade students were enrolled, 65% of whom qualified

for free and reduced lunch. Racially, the students were African American (66%),

45

Caucasian (15%), Latino/a (12%), and Asian/Pacific Islander (7%). Glynnis
reported that the demographics of the school had changed at Riverside during her

tenure there, having gone from a student body that was primarily Caucasian to a

student body that was primarily African American6. The school is located near
the edge of the city. When she began teaching at Riverside, she was one of two

African American teachers in the school. At the time of this study, she was one of

three.

Joanna

Joanna, like Glynnis, had participated in the second cohort (Summer, 2007) of the
CI math PD course. She had been using CI mathematics tasks in her teaching for about
one semester when Phase II began. Joanna reported using CI tasks occasionally, usually
using tasks from the Group Solutions, Too (Goodman & Kopp, 2007) book that was
distributed at the CI course.

At the beginning of Phase 11, she was in her tenth year as a teacher. She was in her
eighth year of teaching third grade in her current school, which is a rural school I call
Edgefield Elementary. Prior to that school, Joanna had taught for one year at another
rural school, and had had two semester-long positions elsewhere. She is a white female
who entered teaching after her children entered school.

Edgefield Elementary enrolled about 360 students in grades kindergarten through

fifth, 12% of whom qualified for free or reduced lunch. The main racial group in the

 

I am reporting here only on the demographics of the school during the duration of the
study. Therefore, I have not included statistical data demographics of the school during

Glynnis' entire teaching career, other than to report what she told me about changes she
had noticed.

46

school was Caucasian (90%). Five percent of the students identified as African American,
2.5% as Hispanic, and the remainder Asian/Pacific Islander or Native American
(Standard and Poors, 2009). The Edgefield school building was built as an open school.
Therefore, there are no walls between classrooms. Instead, teachers have arranged their
fumiture (such as cabinets, bulletin boards, and the like) in order to designate spaces for

each teacher's classroom".
Patricia

Patricia was another teacher who had participated in the first cohort

 

(Summer, 2006) of the CI math PD course. She had been using CI mathematics
tasks in her teaching for one year and one semester. Patricia reported teaching one
CI mathematics task each week, which she took directly from Group Solutions

(Goodman, 2000), a resource distributed to teachers in the PD course.

At the beginning of Phase 11, she had been teaching for 32 years in a
suburban district. She is a white female who was in her 505 and had begun her
teaching career in a rural district as a special education teacher specializing in
emotional impairments. She had also worked as a teacher consultant before
moving to her present district, where she first taught in a special education
classroom. At the beginning of Phase 11 she had been teaching first grade (not

special education) for seven years.

Patricia was teaching in a school I call Circle Pines, in which 330
kindergarten through fifth grade students were enrolled, 12.4% of whom qualified
for free or reduced lunch. The main racial group in the school was Caucasian

(64%), with the second largest racial group being Asian/Pacific Islander (24%).

47

African American students comprised 7% of the student body, and Latino/a
students 5%. (Standard and Poors, 2009). Circle Pines is located in a district that

borders a university town.

Phase 11 Data Collection

Individual Interviews

In order to collect narratives that were situated as closely as possible to the
teachers' practice, I conducted approximately three sets of interviews with each teacher
from February to October, 2008. Each set of interviews consisted of a pre-interview
before teaching a CI mathematics lesson and a post-interview after it. I focused the
conversation as closely as possible to particular episodes of classroom practice.

I asked teachers to bring one or more artifacts from teaching the lesson (such as
student work, a lesson plan, a video- or audio- recording, journal notes, etc.) to facilitate
the telling of narratives.

I had intended to do four interviews, thinking that I wanted to limit the number of
interviews because teachers find that it takes significant time and effort to plan for each
CI lesson, especially when first beginning to use CI. One teacher who has said that he is
very committed to including CI in his practice reported that he was able to teach about
twelve CI mathematics lessons in his first year following the CI workshop. Therefore, I
realized that teaching and discussing four lessons using CI over the course of one
semester would be a significant undertaking for teachers. My assumption was correct,
and in fact I was only able to conduct three sets of interviews with each teacher. (See

Appendix B for the protocol for an interview around a CI lesson).

48

Groupwork Interviews

Teachers also met together three times to share stories of using CI in math, solve
problems of practice together, and participate in focus-group interviews. This provided
another context and audience for teachers' storytelling, and allowed me to gather a
broader range of stories of teaching practice. I suggested that the stories could be about
the same lessons they focused on in their individual interviews, or different lessons,
depending on the teachers' wishes.

The discussion was loosely structured using a "Round Robin Protocol" (Teachers
Development Group, 2005), which teachers used during the CI workshop. The purpose of
the protocol was to provide the teachers' with some structure for continuing the
conversation but allowing me to give fewer (or no) oral promptings during the group
discussions. (See Appendix C). At the first meeting, I gave the teachers a copy of the
protocol, explained that I wanted to hear stories from each teacher and that the purpose of
the protocol was not to rigidly enforce a structure but to communicate to them what I was
hoping to hear. The teachers agreed to the structure of the protocol. At the beginning of
each of the two subsequent meetings, I reminded teachers of the protocol, asked them to
take a few minutes of quiet time to think about their stories, and then simply asked who

had something to share. After starting the discussion I rarely spoke.

Presentations About CI

In order to provide a third context and audience for teachers' storytelling, I invited
teachers participating in this study to present their work on C1 to teachers who had not yet

taken the CI course. Jonathan co-presented at a regional Fall Educator's Conference with

49

me in October, 2008. (I presented in person, while Jonathan was present via Skype).
Glynnis and Joanna were interested in presenting at that same conference, but could not
get time away from their classrooms to do so. Therefore, they videotaped presentations to
be played at that conference. Also in October of 2008, all four teachers presented their
work on using CI to teach mathematics to new mathematics education instructors, who
were interested in learning more about CI. These presentations were audio and video
recorded. The purpose of asking teachers to present in these venues was to generate

stories told in a different context to a different audience because identifying is contextual.

Summary of Data Collection

In order to broaden the contexts in which the stories were told and thus attend to
the contextual and dynamic nature of identifying, different participants participated in
individual interviews at different times and in different places. This was also a result of
my desire to avoid interrupting the teachers' schedules when possible. I arranged to
interview the teachers around their CI lessons whenever they had planned to teach them.
Additionally, not every teacher was able to be present at every teacher discussion group
meeting, due to conflicting schedules. The data collection events, dates, and teachers'

present are reported in Table 1.

50

Table 1. Data Collection

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Patricia Joanna Glynnis Jonathan
Life story 03-11—08 & 9-14-07 9-12—07 6-11-07 &
Interview 7-9-07
Individual 03-11-08, 4-18-09 & 3-5-08 & 4-01-08 &
Interview 1 03-18-08, & 4-25-08 3-25-08 5—05-08
(pre- and post-) 03-21-08
Individual 4-22-08 & 5-2—08 & 4-23-08 & 5—05-08 &
Interview 2 4-29-08 5-15—08 4—30-08 5—23-08
re- and post-)
Individual 5-22-08 9-26—08 & 10-06-08 & 9—25-08 &
Interview 3 10-20-08 10-26-08 10-07-08
(pre— and post-)
Group Participated Participated Participated Participated
Discussion 1
(4-15-08)
Group Participated Participated Participated Absent
Discussion 2
(5-15-08)
Group Participated Participated Participated Absent
Discussion 3
(8-18-08)
Round-Table Participated Participated Participated Participated
Presentation to
Instructors (10—
20-08)
Fall Educator's Videotaped Videotaped Presented live
Conference presentation presentation via video
10-10-08) conference
Phase [I Data Analysis

Direct Identifiiing Statements

As in Phase I, I noted the direct identifying statements uttered in the interviews,

discussions, and presentations. See Phase I for a description of the method.

Narratives

I set apart the stories told by teachers from the other types of discourse (I use

"stories" and "narratives" interchangeably.) Some researchers have posited that defining

51

 

narrative according to a set of structures is too constraining to be helpful; however,
leaving narrative undefined, and therefore considering everything to be narrative, is
equally unhelpful (Georgakopoulou, 2007). The Oxford English Dictionary (2003)
defines narrative as "An account of a series of events, facts, etc., given in order and with
the establishing of connections between them; a narration, a story, an account."
Recounting events in order is indeed a part of some definitions of narrative (Labov,
1972). However, narrative analysts have disputed whether the order of events needs to be
chronological. In many cases, narrators use techniques such as foreshadowing and
backshadowing to build coherence or suspense in a story, or to explain events within a
story. Or events may be linked causally (Ochs & Capps, 2001). Some theorists have
suggested that narratives need to have a beginning, middle, and an end in order to count
as narratives, but Ochs and Capps (2001) posit that narratives are ofien left with no
conclusive ending, especially when the narrators are using the narration to make sense of
an event or inviting others to help them make sense of an event. Others have suggested
that the difference between narrations and recounts lies in whether or not the retelling of
the events contains an evaluation, a term used by Labov, (1972) or moral stance taking, a
term used by Ochs and Capps, (2001) (Juzwik, 2006). For this study, I define a narrative
as an account of an event or events including an element of time (although it may not be
told in chronological order) and consciousness (a social and emotional purpose for telling
the narrative).

When narratives are constructed within a group of people who may be competing
for turns in a conversation they emerge differently, in ways that are more fragmented

than those individuals tell in interviews. In a group, participants may perceive that long

52

stories should be avoided. When telling longer stories, interlocuters in groups seem to
perceive a need to justify their reasons for taking time from the group to tell the stories.
Additionally, the group builds a core body of shared knowledge of these stories, and may
reference prior story-tellings without including all of the details or structures often used
to construct narratives (Georgakopoulou, 2007). This requires an analyst to take a slightly
more flexible approach, attending to retellings that emerge in pieces during ongoing
interactions. Not all stories are told as self-contained units. Therefore, when analyzing the
study-group meetings or the data generated during the CI workshop, I also looked for
retellings of events across the tum-taking interactions and across multiple meetings.

I have categorized the types of stories I collected according to three categories.
Life stories are stories are just that—stories of teachers lives. These were generated when
I asked teachers to talk about their mathematics autobiographies (as in the Phase I
interviews and in the exit interviews for the 2007 cohort of the CI math PD course).
Teachers began by talking about their participation in mathematics as children and talked
about experiences with learning and teaching mathematics up until the time of the
interview. These stories covered the longest time period. The second category of stories I
collected are what I call recounts of episodes. These stories are stories of specific
episodes that occurred during their teaching and learning. Of course, the teachers' life
stories were comprised of several recounts of episodes, but instead of closely analyzing
these episodes, I looked overall at the plot of the entire life story. Most of the recounts of
episodes I analyze in detail in this dissertation are stories teachers told within their
interviews about specific CI mathematics lessons. The third category of narratives are

what Sfard (2007, 2006a) calls direct identifying narratives but which I am calling direct

53

identifizing statements so as not to confuse them with the other types of narratives I am
studying. As mentioned in Chapter 2, these are not narratives according to the definitions
of narrative used by linguists (Juzwik, 2006) but are statements made on the basis of
reified narratives of experience (Sfard, 2006b). The types of stories collected are listed in
Table 2, along with the dates collected (direct identifizing statements are not included in

this table because those were found across all the teacher discourse).

Table 2. Types of Data and Dates Collected

 

 

 

 

 

 

 

Phase Stories Data Date(s)
Two-part life-story interviews with Phase I May-July, 2007
1 participants (Gina, Kelly, Shirley, and
Jonathan)
3 One-part life-story exit interviews with September, 2007
2 '5 participants who were in the 2007 cohort of
a the CI PD course (Glynnis and Joanna)
:13 Short life-story interviews as a part of other March 11, March 21,

interviews for Patricia (because she did not and April 15, 2008
2 participate in Phase I but was in the 2006
cohort and therefore did not have an exit
interview after the 2007 workshop).

 

 

 

 

 

 

2 Individual pre- and post- CI math lesson March 5, 2008-
95 m interviews October 26, 2008
3 g Teacher discussion group sessions April 15, 2008

2 g g May 15, 2008
8 “‘3‘ August 18, 2008

2 9‘ Presentations to teachers/instructors who had October 10, 2008

 

 

 

not yet taken the CI course October 20, 2008

 

Narrative Structures

1 coded the narratives for narrative structures (Gee, 1989; Labov, 1972). Attention
to structure can provide an initial way of cutting into data that can bring ambiguities into
view (Riessman, 1993) as well as to set up interpretive questions and constrain possible
interpretations of them (Gee, 1989). One structural element I looked for is what Gee

refers to as main line vs. non-main line parts of the narrative. Looking for main line

54

 

(plot) and non-main line (asides, descriptions, etc.) parts of the narrative helped me to
interpret the speaker's purpose for telling the story (Gee, 1989). What Labov (1972) calls
evaluation is what Gee's (1989) analysis of main line/off—line of the plot uncovers. It is
the "so what" piece of the story, the part that allows the listener to understand why the
story was told.

I also looked for the unexpected event (Ochs & Caps, 2001) or problem of the story
and the resolution to the problem in the story. This helped me to interpret how the teacher
understood her role and the students' roles in the classroom. I noted issues such as who or

what caused the problem, and to whom does the teacher attribute the resolution.

E thnopoetics and Metaphors

In Chapter 7 I present a micro-discursive analysis of the poetic and linguistic
structures of two stories. I present my method in more detail as I report on the results in
that chapter, but I will give an overview of the methods here. I looked for syntax and
cohesion in the stories (Gee, 1991) such as repetition of words and grammatical
structures. I would compare several similar-sounding sentences side-by-side to see if they
could all fit into one similar template, for example. I also looked at alliteration within
lines of text, repeated words and phrases (Juzwik, 2004).

I looked for shifts in the psychological subjects (Gee, 1991) in two ways; I looked
at the pronouns in the stories to note the pronominal shifts. And I looked at how the
teacher shifted between participant and spectator roles as she told the story (Crespo &
Juzwik, 2006). I also analyzed the stories for clues as to the metaphors of arithmetic the

teachers were indexing (pointing to linguistically) and the way that they shifted between

55

the metaphors. In these two stories, the teachers were both narrating a story about a
lesson in which students were placing numbers (fractions and decimals) on a number line.

Grounding metaphors. Lakoff and Nufiez (2000) theorized four grounding metaphors
that ground arithmetic to physical objects—the metaphor of object collection, the
metaphor of object construction, the measuring stick metaphor, and the metaphor of
motion along a line. Fractions are often understood using the metaphor of object
construction, because they are constructed by dividing a whole into parts. In that case, a
key metaphor is numbers as objects. In the measuring stick metaphor, arithmetic is
conceptualized as the use of a measuring stick. Numbers are physical segments, with the
basic physical segment being one. Longer segments are greater and shorter ones are less.
Arithmetic is putting segments together and taking them apart.

Linking metaphors. Linking metaphors take us beyond the physicality of grounding
metaphors to more sophisticated ideas. The metaphor of numbers as points on a line is an
example. It builds on the measuring stick metaphor and the metaphor of arithmetic as
motion along a path but maps these ideas onto the concept of a line (instead of a physical
object). This requires understanding several other metaphors: a number P1 as point P1 on
a line, zero as a point 0 on the line, one as a point 1 to the right of 0, etc. Sophisticated
mathematical understandings and are built by layering metaphor upon metaphor, a
process called conceptual blending. In contrast, teaching mathematics involves
unpacking the layers of metaphor, in effect, unblending, and deftly moving between

metaphors to facilitate students' conceptual blending.

56

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CHAPTER 4: TO IMPLEMENT OR NOT TO IMPLEMENT: LIFE STORIES

What do you suppose would be his answer if someone told him that what
he had seen before was all a cheat and an illusion, but that now, being
nearer to reality and turned toward more real things, he saw more
truly?...And if he were compelled to look at the light itself, would not that
pain his eyes..And if, said I, someone should drag him thence by force up
the ascent which is rough and steep, and not let him go before he had
drawn him out into the light of the sun, do you not think that he would
find it painful to be so haled along, and would chafe at it?

-Plato, "The Allegory of the Cave"

(Reed & Johnson, 2000, p. 12)

Rita: But don't you realize I want to change.
-Educating Rita (Cartlidge & Gilbert, 1983)

Overview

In the film, Educating Rita, Rita desired to change herself by studying literature.
In Plato's allegory, however, he describes the learner as having to be dragged into the
light. As a teacher and a teacher educator, I have wondered why some teachers seem
eager to take on certain new pedagogies and practices, others are interested in learning
different practices, and some seem to avoid change in their mathematics teaching
practices altogether.

Mathematics teacher educators and professional developers have long been
concerned that teacher PD does not always result in teachers adoption of new pedagogies
as a part of their mathematics teaching practices. As I note in chapter two, the
effectiveness of PD remains a subject of controversy (Wilson & Beme, 1999).

However, I argue that teacher learning and changes in teaching practice involve
more than simply taking in some new information and then doing something new in the
classroom. Learning is impacted by how one identifies. That means that what one

chooses to learn is impacted by one's designated identity—the kind of person one wants

57

 

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to become (Sfard & Prusak, 2005). Just as Rita was compelled to study literature, even
when the implications for her identity were clearly articulated, some teachers choose to
learn about and implement new teaching practices. Why might some teachers be inclined
to innovate and renew their practices by adopting pedagogies taught in PD courses, and
others not? How is it that a teacher comes to see certain invitations to change as being
worth her time and effort?

As I explained earlier, identifying may be defined as participating in a community
of practice (Wenger, 1998). Choosing to implement a given recommendation from a PD
course is then one way of identifying (participating). Adopting this lens as a way to
describe teachers' decisions to implement new practices (or not) means acknowledging
that there is more to changing teaching practice than simply picking up a new teaching
tool and using it. Choosing to implement new mathematics teaching practices necessarily
has implications for how teachers identify themselves.

Another way people identify is through storytelling about their participation. You
may recall from chapter two that two types of narratives that make up identity are life
stories (Drake, Spillane, & Hufferd-Ackles, 2001) and direct identifiing statements
(Sfard & Prusak, 2005).

I examined four teachers' narratives of themselves and their stories of
implementation of CI to teach mathematics to look for parallels and contradictions
between them. I compared how teachers' identified in the present with how they
identified themselves in the past. I compared how they identified as teachers (presently)
with how they identified as students (in the past), and how they identified as teachers

after the CI workshop (presently) with how they identified themselves as teachers before

58

 

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the workshop. I also compared and contrasted their ways of identifying in different
figured worlds.

In this chapter, I report on the life story data, which I collected as Phase I of my
dissertation. During that Phase of the dissertation, I chose four teachers from the first
cohort of the CI course. Two of the teachers had reported implementing CI, and two had
not. I interviewed them about their mathematics life stories because I hypothesized that
their decisions to participate (or not) in implementing CI might parallel the ways they
identified in their life stories. Sometimes this was the case; however, it was not as
straightforward as I originally hypothesized. In this chapter, I demonstrate how the
teachers' narratives function to propose reasons for deciding to implement new practices
(or not).

In the results section of this chapter, I present the cases of each of the four
teachers. For each case, I first summarize the teacher's ways of identifying. I do this in
two ways: a) I summarize the teacher's mathematics life story and describe its plot as
either Failing (Foreclosed), Turning-Point, or Roller Coaster (drawing from Drake, 2006
and Drake, Spillane, and Hufferd—Ackles, 2001); b) I list the teachers direct identifying
statements and the figured worlds that provide the context for each of them. Secondly for
each case, I describe the parallels (and sometimes contradictions) that I discovered
between the teacher's ways of identifying in narrative and stories of implementation (or
not) of CI.

In the discussion section, I theorize that at least sometimes for some teachers, the
decision to implement recommendations from the PD course was made, at least in part,

because there were parallels between the teachers' perceptions of the CI practices and

59

 

principles being implemented and the teachers' ways of identifying in their narratives.
However, this did not always happen, and so I set forth some hypotheses as to why. I also
describe how the results of this Phase of the dissertation led to the problem of study for
the following chapters.

In Chapter 1 I described how the story I uncovered in my dissertation is parallel to
the stories told in Chapter 1. In the present dissertation chapter, I am reporting on the Act
I and Plot Point 1 of this story, in which I describe how teachers' ways of identifying and
their decisions to implement CI to teach mathematics are related. A graphical I
representation of where this chapter fits into the plot structure of this story is shown in
Figure 5. Figure 5 has the same graphic representation of the story plot as Figure 3,

except that the scope of this chapter is highlighted with bold outlines

 
  

 

  
     

Act I: How the Act 11: Stories of Act 111: CI
teachers identify implementation of CI in Teachers
(life stories) math

 

 

 

11

Teachers decide Decision to persist

to implement CI With implementing
(or not) CI for not)

Figure 5. Chapter 4-—Act I and Plot Point 1

Cases of the Four Teachers
Gina
Ways of Identifying

Mathematics life story: failure (foreclosed). Gina told a failure (foreclosed).

mathematics life story (Drake, Spillane, and Hufferd-Ackles, 2001). In failure

60

it]

(foreclosed) narratives the storyteller describes negative mathematics learning
experiences with one or more failures or near-failures. Additionally, these storytellers

accepted their early failures as predictors of their future experiences. Gina began:

The one thing in grade school though that sticks in my mind is having to
memorize my multiplication tables. That year I had rheumatic fever. So I
missed a whole semester of school. And I've always had a hard time and
been nervous. So I think it was more of a nervous thing, but I can
remember standing in front of the teacher's desk after school was out
saying the multiplication tables to her until I had said them all up to twelve
before she'd let me go to the next grade. And I remember I could say them
perfectly at home and then I would get up and stand in front of her desk
and if you made one mistake you had to come back the next day.

In this excerpt, Gina described a failure in school mathematics—a hallmark of a

failure (foreclosed) story.

Another characteristic of failure (foreclosed) stories is the interpretation of
early failures as predicting future failure, or "foreclosing" their mathematics

identities. Gina continued her story:

I can remember after church on Sunday going over to my cousin's
house because she was really good at math, and she tutored me all the way
through 10th grade when we were taking geometry. That's how I made it
through. She helped me all the way.

I just never felt like I could get it. I went to Catholic school and if
you didn't get it that was too bad. They went on and you had to find your
own way. So you know Mary, my cousin helped me, and that worked out
really well.

But I didn't like it at all. When I could quit taking it, I didn't take
any more math either.

Here, Gina described being rescued from near-failure by a more competent cousin,
continuing the plot of her failure (foreclosed) mathematics story. She also described

herself as never understanding mathematics, indicating her acceptance of these early

61

experiences as predictors of lifelong mathematical weakness, another characteristic of

failure (foreclosed) stories.
As a teacher, Gina continued to describe herself as mathematically weak:

When I was teaching sixth grade I was about two chapters ahead of the
kids. And one of the things I was reading about, the negative numbers, and
it all made more sense then. So I always told myself I wasn't
developmentally ready for that when they gave it to me.

Gina explains her early weakness by saying that mathematical ideas were forced on her

before she was ready, which foreclosed her mathematical future.

When Gina told of her participation in teacher PD, she continued to describe

herself as mathematically incompetent. She said of mathematics PD events:

I always have this fear that I'm going to look stupid, you know? So [I]
would sit with the [district] teachers and it would be ok. I've learned how
to kind of hang back and say, "Oh yeah, ok."

As Gina said, "Oh yeah, 0k" she gestured that she would respond as though she
understood what was said. Gina threaded experiences of failure throughout her
mathematics life story from beginning to end. Therefore, I categorized Gina's

mathematics life story as having a failure (foreclosed) plot.

Direct identijying statements and figured worlds. Table one illustrates the direct
identifying statements Gina made during the interview, and also the figured world to

which she was referring when she uttered each (in italics).

62

Table 3. Gina's Direct Identifying Statements and Figured Worlds

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Identifying Statement flured World
As a person or teacher (in weral)
I'm a much better langgge arts teacher than I am math. Her classroom
I am not waiting for it to end like some people Her school
I never pay attention to how long [until retiremerfl. Her school
We're kind of isolated down here. Her school
I always wanted to be a teacher. Schooling (adult)
I knew that 1 could do a little bit better than some of the nuns that I had. Schooling (adult)
I'm a realist. Multiple figured
worlds.
As a mathematics teacher

I could keep that goMrst grade math centers]. Her classroom

I could actually show them and use that eflls sign. Her classroom

Still a little scared and standoffish even thorgh it's just kindergarten math. Her classroom

I was pretty comfortable teaching them how to add and subtract. Her classroom

I would always let them use their fingers. Her classroom

As a math person

1 was never quick. Math class (student)
I never felt like I could get it. Math class (student)
I've always had a hard time and been nervous. Math class (student)

 

I can always remember wanting one of those crowns but never quite being able to Math class (student)
get it.

 

I never cared for it and I always felt that I couldn't do it. And I just don't want [my Math class

 

 

 

 

 

 

 

 

students] to feel that way. (student); Her
classroom

I was just not proficient in that kind of math [upper elementary grades]. Her classroom
I'm never goinMet there. Her classroom
I always have this fear that I'm going to look stupid. Math PD
lcan't do it. Math PD

1 was ok. Math PD

Any time that there's math involved, I feel nervous. Math PD

1 can do more math than what I thought. Cl PD

 

I can understand more math than what I though. And maybe I shouldn't have that Cl PD
big fear, but I do.

 

 

 

I can see that. 1 get it. CI PD
I can't do some of this math. And I'm gonna look, you know. Cl PD
1 had to be one of the explainers. Cl PD

 

 

I can handle this because it's going to be the elementary math. Well then, when it Cl PD
was more, whoa. I don't know about this.

 

 

One notable pattern in Table 1 is that when Gina describes herself as a teacher,
she describes herself as competent. However, when she describes herself as a math
person, she describes herself as incompetent, except when she talked about the CI course.
While she still mentions feeling some inadequacy in that figured world, she also stated

that she learned to see herself as less mathematically incompetent there. However, this

63

 

does not indicate a turning point. A tuming-point experience is one that results in the
teacher realizing that it is possible to succeed at and enjoy mathematics, which Gina did

not indicate.

Mathematics Life Stories, Figured Worlds, and Choosing Not to Implement CI

Gina reported no changes in practice after the workshop. This fits with the
patterns reported by Drake, Spillane, and Hufferd-Ackles (2001), who found that teachers
with failing (foreclosed) mathematics life stories did not implement reform
recommendations.

When asked why she did not implement CI, she stated that she did not believe it
was developmentally appropriate for young children to work together. As you recall, in
her mathematics life story she ascribed her failure to being introduced to concepts before
she was developmentally ready. In that way her mathematics life story and her discussion
of C1 are parallel. These early stories of failure seem to be very powerful influences on
Gina's instructional decision-making.

In her stories about learning mathematics, Gina ascribed her survival (if not
success) to supportive others (her cousin, colleagues). She described herself as less
incompetent in the CI workshop than in other figured worlds. Even though she ascribed
her survival to working with others, she chose not to have students work together. In that
sense, her choice not to implement CI contrasts her own mathematics life story. One
might expect Gina to want to give her students collaborative experiences because they
helped her, but she did not. Again, it appears that Gina's early experiences with failure

and her early foreclosure of her mathematics identity were powerful influences on her

64

instructional decision-making—enough so to make her contradict even her own stories of

survival.

Kelly
Ways of Identifying

Mathematics life story: Two turning points. Kelly, also began her story with
experiences of not understanding mathematics, but later told of a turning point. Again,
teachers with turning point stories describe early failure in mathematics but recent

episodes of positive experiences that changed how they identify in regard to mathematics.
Kelly began:

All of that was just rote. It was paper pencil. You do the problems. You
don't know why you're doing them. . .I took a couple of college prep type
of things. Math wasn't one of them that I was taking because math wasn't
my thing. High school was when it was more frustrating. Doing the
algebra and not really realizing why we were doing all those formulas and
how to apply those formulas. In college, because I didn't take educational
math, I had to go through certain math classes and just take them as an
independent study. I remember a couple of them not passing it right away,
like with fractions and stuff. Because it didn't make sense.

Kelly's descriptions of specific failure in mathematics are characteristic of both
failure (foreclosed) and turning point stories. Kelly continued:

When I really started understanding math was when I was a substitute
teacher for Nancy Glenn. She's in Westfield and she was in the math
department for them for elementary. I went to Westfield so I knew her
quite well. They were doing, what was the math program called? It was
totally hands on. It was first the concrete, and then going from there the
pictorial, and then from pictorial to the abstract. And moving from each
one. And all of that stuff with the multiplication. I'm like, aah, this all
makes sense.

Turning point stories include a recent positive experience that changes the

teachers' views. Tool-based turning point stories are those in which the turning-

65

point came about because of materials. Because of Kelly's emphasis on tools in
this story (manipulatives, pictures), it can be classified as a tool-based turning
point. Kelly's new understandings were based on the tools that she was asking
children to use. As a result of this experience, she believed that she could do and
teach mathematics.
Later, Kelly had another important turning point in her mathematics life
story. She described the CI course as important to her understanding of
mathematics as a discipline:
I was very uncomfortable initially with the Complex Instruction, because I
did not feel like I fit there at all. . .For me to have to do these extensive
math problems it was going to be hard for me to feel positive because
everyone around me understood math concepts, I felt, a lot better than I
did. And how to solve these mathematical problems that were middle
school, high school, you know, higher than what I had been doing in
many, many years. So it was at first very frustrating. And then after a
couple of days it started being, it was just very positive... First, total
frustration. I was probably going to waste my time. To, "You know what, I
can do this. I have the ability to help with finding the answer and I can do
something to benefit my group". . .The way we were taught that there are
many different ways to solve a problem and there's no right way, there are
just many ways. And so having more than one person working on this
makes a lot of sense with math period. You know, it's not a one-person.
Kelly's turning point in the CI workshop was a content-based turning point, because she
started to see mathematics as a discipline as a social activity where diversity of ideas is
important.
Direct identifying statements and figured worlds. Kelly identified in strikingly
different ways in different figured worlds. Table 2 shows her direct identifying

statements and the figured worlds to which she was referring when she uttered them.

66

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Table 4. Kelly's Direct Identifying Statements and Figured Worlds

 

Identifying Statement

Figured World

 

As a person or teacher (in generaD

 

I was a substitute teacher.

The classroom

 

l was second grade.

The classroom

 

I'm very good at change.

Multiple figured worlds

 

I'll tell everybody. I don't, I don't hide anything.

Multiple figured worlds

 

As a mathematics teacher

 

Knowledgeable.

Her classroom

 

I can go to sources.

Her classroom

 

Being able to go with the Investigations and not just fall back to the
ways that I used to teach.

Her classroom

 

Confident and more knowledgeable.

Her classroom

 

I think I'm doing well.

Her classroom

 

More comfortable because 1 know what I'm supposed to be teaching.

Her classroom

 

I know maybe many different avenues to get there.

Her classroom

 

Pretty confident.

Her classroom; mentoring
relationship

 

I will never tell you that I know everything.

Her classroom; mentoring
relationship

 

I'm knowledgeable.

Her classroom; mentoring
relationship

 

I can teach the kids and teach them to understand why they're doing
it.

Reform-based math; her
classroom

 

 

 

I can do this. Reform-based math
I understand the new way of doing math. Reform-based math
I can hflr her. Reform-based math; her family

 

When I'm around adults, I don't see myself as a math teacher.

Her social circle; Her school; PD
events

 

I'm just this little kindergarten teacher who does not have a
background in math.

Her social circle; her school; PD

 

 

 

 

I was Eng ho for it. Commx instruction
As a math person

Math was not my forte. School (as student)

Math wasn't my thing. School (as student)

 

ldon't see myself as confident in math [around adults].

Her social circle; Her school; PD
events

 

 

 

 

 

 

 

 

 

 

 

I did not feel like I fit there. CI PD
1 could do that stuff. CI PD
I can do this. CI PD
1 have the abilityto help with finding the answer. Cl PD
I can do somethinfibenefit my group. Cl PD
1 could answer if I needed to. C1 PD
I can do these kind of things. C] PD
1 could tell [explain how my group solved the problem]. Cl PD
1 don't get better at math. All figured worlds.

 

Most notable is the pattern whereby Kelly identifies as mathematically

incompetent in her social circle and with her colleagues, but competent when in her own

67

 

classroom and mentoring student teachers. It is also notable that this changes when she is
talking about herself in the CI math PD experience, which would be a context in which

she was "around adults," and yet directly identified as capable.

Mathematics Life Stories, Figured Worlds, and Choosing to Increase Students’

Opportunities to Talk to Each Other

Kelly had reported not using CI in her classroom after the PD course. However,
she did report changes in her teaching practice that she attributed to her experience in the
course. She stated:

I always did [group work] with reading and instructional centers. But I did

not do it much with the math. Whereas, now it's pretty much every time

they go work on something, it's with a pair.

Kelly ascribed this change in her practice as being based on a new conception of what it
means to do mathematics, which she gained from the PD course:

Because I realized that two brains are better than one. Again it was just the

way that I was brought up with math, you did it all by yourself. It was

never a group effort. And the grouping makes so much more sense

because of the way math is set up.

According to Drake's (2006) findings, recent, content-based turning point stories,
such as Kelly reported experiencing in the CI course, are often accompanied by content-
based implementations of reform, such as Kelly described.

But Kelly had also described an earlier, tool-based turning point at the beginning
of her teaching career. Interestingly, Kelly reported that one of the constraints to her
mathematics teaching practice was lack of time to prepare tools and lack of resources.
She said:

I enjoy teaching mathematics but it's hard because of the manipulatives

and the time constraints. Just doing the manipulative part takes a long
time.

68

Initially this seemed puzzling, since implementing CI need not require a lot of tools. I
wondered why Kelly had not chosen to implement CI in order to ease this difficulty.
However, it is possible that another challenge to implementation for her is the conflict of
different messages about mathematics reform from the two different figured worlds:
figured worlds that provided the setting for her first tuming—point story (an earlier wave
of mathematics reform) and her latest turning point story (which was set in the figured

world of the CI workshop).
Shirley
Ways of Identifying

Mathematics life story: Turning Point. Shirley's story began:

I don't think I felt I was very good at math. It was probably encouraged
quite a bit by the teaching staff because, in retrospect I think they really
felt that girls, it was just pointless. Why, why were you doing this. It
wasn't as rigorous as it is now either. We didn't have AP, AP was just
starting, so there was no push to keep churning through. When I was in
college I continued thinking I couldn't do math.

Both failure (foreclosed) and turning point stories begin with negative experiences and
one or more stories of failure. Shirley described concluding that she could not do math.

She continued:

For a while I did a lot of things that were applied math because I ran a
radio station, and I did all sorts of things that required a lot of
mathematics. So I actually learned a lot of mathematics from necessity.
After I quit with the radio thing, I really had enjoyed the accounting part
of what I did at the radio station, so I started taking accounting classes and
took quite a number of them before I decided that both [my husband and
me] being in accounting was crazy because our kids just never saw us
from January to May.

69

She stated that she later learned to enjoy mathematics. When asked to elaborate, Shirley

continued:

It was working with decimals and percentages and ratios and graphs and
linear relationships and all that kind of stuff. So the practical application
of it. It made sense.

Shirley did not indicate one short episode as a turning point. However, over time she
began to identify as competent in certain mathematics content. In content-based turning
point stories the turning point involves a new conception of mathematics as a discipline
(Drake, 2006). Because Shirley previously saw mathematics as useless but began to see it

as useful, I call this gradual change in Shirley's perceptions a content-based turning point.

Shirley left accounting school to pursue elementary teacher certification. She said:

I didn't need to take any more math except for the one class that I had with
[the university]. I can't even remember the name of it but it was teaching
mathematics. It was actually a math class... And I enjoyed it.

Shirley now enjoyed mathematics in a different context. One hallmark of a tuming—point

story is that the storyteller has come to see mathematics as something to be enjoyed.

Shirley described her ongoing interest:

If you were talking about my time when I was actually taking mathematics
classes I would say disinterested [in mathematics]. Now that I'm teaching
math I am obsessed. . .Right now I'm trying to teach kids about multiplying
fractions. . ..so I am just obsessed with pictures, of drawing pictures, trying
to show them.

Shirley had developed different conceptions of mathematics and of herself, and
reportedly participates in mathematics differently than before. Therefore, Shirley's story

is a turning point story.

70

Direct identifying statements and figured worlds. Shirley’s direct identifying

statements are in Figure 3, along with the figured worlds she was discussing when they

were uttered (in italics).

Table 5. Shirley's Direct Identifying Statements and Figured Worlds

 

 

 

 

 

Identifying Statement Figured
World
As a person or teacher (in general)
I always thought I wanted to be a teacher. Schooling
I can do science. Her classroom

 

I'm on stage.

Her classroom

 

I‘m willing to put in the time.

Her classroom

 

 

 

 

 

 

 

She said, "You're too controlling," and 1 said, "No, I'm eliminating wasted time." Her classroom
Yes, 1 am controlling. That's true. Her classroom
[My colleagues] probably think I'm silly because I make myself a lot more work. Her school
So I'm sure [my colleague] thinks I'm foolish because his job is much easier. Her SChOOI
I'm resentful that, and it's stupid, it's my fault. I'm resentful that more people don't do Her school
more.

I've always taught in University Heights. District

 

As a mathematics teacher

 

And I think I can hold my own with sixth graders.

Her classroom

 

Now that I'm teaching math I am obsessed.

Her classroom

 

If 1 were describing it now 1 would say excited.

Her classroom

 

Totally obsessed.

Her classroom

 

1 really want to do the best job 1 can, so I'm always looking for other things.

Her classroom

 

And so 1 am just obsessed with pictures, of drawing pictures.

Her classroom

 

I'm always looking for some way to get them excited.

Her classroom

 

I'm always making them get up and stand, you know, stand up, sit down. Do, move
around the room. Demonstrate it, model it, do all this stuff.

Her classroom

 

I‘m demanding. 1 require them to keep working until they understand it.

Her classroom

 

But I think that's why they would probably think I'm demanding. And I'm always saying,
no, you know, you need to do more.

Her classroom

 

I am a work in progress.

Her classroom

 

I know what I expect of them I‘m more comfortable with the process.

Her classroom

 

 

 

 

 

 

 

 

 

 

 

 

 

That was my qualification was that I was here. And I could teach math. Her school
I'm a book junkie. Math PD
I'm self-taught. Math PD
As a math person
If you were talking about my time when I was actually taking mathematics classes 1 Math class
would say disinterested. (student)
I never dreamed that science or math was something I should ever even look at. Math class
(student)
I was one of the ones that was slacking them down back when l was in high school, for Math class
sure. (student)
Just being one of the struggling students myself. Cl PD
I knew I might be responsible for sharing. Cl PD
1 was one who didn't know it all. Cl PD

 

71

 

Shirley directly identified as a competent mathematics teacher. She constructed
herself as a self-motivated, self-taught learner and seeker of mathematical pedagogical
tools, in contrast with her colleagues.

She made fewer direct identifying statements about herself as a math person. In
those, she identified as "disinterested," "slacking them down," and "struggling." She
identified similarly in the figured worlds of the CI event and mathematics class as a
student.

This complexifies the interpretation of Shirley's story as a tuming—point story. She
identified as a competent mathematics teacher, but not a competent student of
mathematics, even within the context of the PD workshop. She identified as a
mathematically competent in the figured worlds of her school and her classroom, but not
in the figured worlds of her childhood school or the CI event. Sometimes she identified in
ways expected of the storyteller of a failure (foreclosed) story. Her direct identifying
statements and mathematics life story plot are not always consistent, which corroborates

theories of identity as dynamic and context-sensitive.

Mathematics Life Stories, Figured Worlds, and Choosing to Implement CI

Shirley reported implementing CI mathematics tasks in her classroom. She
reportedly changed her teaching, randomly assigning students to heterogenous groups for
weekly groupworthy tasks, and implementing interventions taught in the workshop. She

said:

1 allow the groups to struggle more. I allow them to work more carefully
together. I give them that opportunity to talk about math in a structured
way with the tasks much more than I did. I'd never done group tasks for
math. I might have had them doing little tiny things, but nothing like this.
Nothing like they've been doing this year.

72

Shirley reported that several things had come together for her, resulting in her
decision to adopt CI. She had focused on studying grouping practices during another
ongoing PD that year, and had heard Harry Wong (Wong & Wong, 1991) speak about the
importance of teaching procedures and policies to students. She felt that Cl provided
structure that helped students know what was expected of them and saved time:

I learned how to organize groups in a way that makes them efficient. And

I've learned, I learned that by asking each person to be responsible and

holding their feet to the fire that you will get kids to be responsible. You

know, it's not a hundred percent, but it's a lot closer than it had been. This

is another work in progress.

Student responsibility, structure, organization, the opportunity for every student to
contribute to the group and to the class are important in Shirley's discussion of CI. These
are also important ideas in Shirley's life story. These are ways of identifying that are
consistent for Shirley across the figured worlds of her former school (in which she was a
student) and her classroom (in which she is a teacher). For Shirley, implementing CI
allowed her to express these identities.

However, using groupwork may also be viewed as contrasting with her direct
identifying as "self-taught," because students are to work together interdependently.
Group work also contrasts with the way she identified as an outsider among her
colleagues. It is possible that implementing CI in mathematics was one way in which

Shirley tried to redeem her prior experiences and providing a different opportunity for her

students.

73

Jonathan

Ways of Identifiring

Mathematics life story: Roller coaster and turning point. Jonathan's story had a
plot similar to a roller coaster story (he included descriptions of both success and failure
during his K-12 years). Still, most of the episodes were negative. He began:

I wasn't very successful in elementary school or middle school and in high

school they didn't make me take math. They made me take a little bit of

math. . .I can remember being a bit lost in 8th grade. I can remember that

lost feeling where there were kids all around me getting it and the teacher

was trying to explain it to me. I think it was negative numbers. I remember

being left behind too. I didn't get it. I can remember this one day pretty

well. I didn't get it but they just kept going. So after that I tuned out

because they just left without me.
Jonathan described being unsuccessful, alone, and abandoned. He also told of a success
in K-12 mathematics:

In high school I had a pretty motivated teacher who really helped me get a

bit more caught up. I would have to say that was only because of him. He

seemed to care more. I did learn some math. So that was good. He did

give me some of the skills that I needed. He gave me a bit more

confidence that I could learn math.
Although Jonathan gained a bit more confidence, I do not interpret this to be a turning
point in his story. He had a positive experience with this teacher, but he did not feel
competent in mathematics outside of this class. Jonathan still chose not to participate in
mathematics courses for several years.

Jonathan's story does not neatly fit into the category of roller coaster stories
because he includes only one early story of success in mathematics. However, this
success story means that it does not fit all the characteristics of failure (foreclosed) or

turning point stories either, because this one success story indicates that he had not

foreclosed his mathematics identity. He also described something that might be

74

interpreted as a turning point: the mathematics content and methods courses required

during preservice teacher education. He said of these courses:

I don't feel at all negative about [the courses]. Partly, I was a more
motivated learner. You know, when you go to college you, because you
want to college... That's part of it. But those particular classes were in
themselves somewhat motivating. The one was a lot of hands on, a lot of
opportunities to prepare things and present them. Looking at math in a
slightly, less than just textbook oriented. Using manipulatives and using
crayons and colored pencils in math? My goodness. Colored things. So

that was great.

In this excerpt, Jonathan described affective aspects of the class, and talked about

experiences with tools. Therefore, it seemed like this could be a tool-based turning point.

Therefore, it seems that Jonathan's story has elements of both roller-coaster and tuming-

point stories.

Direct identifying statements and figured worlds. Jonathan's direct identifying

statements are in Table 4, along with the figured worlds in which each of these statements

were SCI.

Table 6. Jonathan's Direct Identifying Statements and Figured Worlds

 

IdentifyLng Statement

I Figred World

 

As a person or teacher (in gneral)

 

I always ask the question what can the technology do that you can't do on paper

and pencil.

His classroom

 

I've had kids say they really appreciate that I will take time to explain things
with them.

His classroom

 

I'm the kind of teacher...l like to have things go well and be successful.

His classroom

 

I‘ve always been interested in creating lessons that are for sure engaging. I've
always been interested in engaging kids.

His classroom

 

I think [my students] would say I'm patient. I'm a patient person.

His classroom; all
figured worlds

 

If I were to be highly qualified to be a math teacher, I wouldn't be at a middle
school. At an upper grade. . .So we're really considered self-contained because
we're two people teaching teams.

His school/his district

 

We're pretty lucky.

His school/his district

 

So you are a collaborator whether you like it or not.

His school/his district

 

We have direction.

His school/his district

 

We're a standards based report card.

His school/his district

 

 

I have always taught in Cornerstone

His district

 

 

75

 

 

Table 6 @nt'd)

 

You were just an elementary teacher. You weren't a math teacher or science,
you were everything.

Schooling (adult)

 

I'm 46, so I'm quite a bit older than you.

Multiple figured worlds.

 

I'm an early adopter, it seems like, in just about anything.

Multiple fijured worlds

 

There's just something in me that likes to breakdown those stereotypes.

Multiple figured worlds

 

I'm willing to be reflective and look at what works.

Multiple figured worlds

 

I'm a risk taker so that helps. I'm willing to try things.

Multiple figured worlds

 

As a mathematics teacher

 

(Students) do recognize me as a math teacher. As opposed to being a teacher in
general. I think they think of me as their math teacher.

His classroom

 

I've been willing to tolerate, if that's he right word, I'm not sure, willing to
tolerate kids being at different places, far different places, in the group.

His classroom

 

1 would consider that as being patient.

His classroom

 

I see myself as someone who's willm to take risks with math teaching.

His classroom

 

I see myself as a teacher who wants to bring everyone into the math, and at the
same time, challenge.

His classroom

 

I've been really, I guess strict is the word, really strict about making sure that
everybody in the ggup sort of gets it before they can move on.

His classroom; Cl

 

I have some appreciation having been a struggling student.

His classroom; math
class (student)

 

 

 

 

 

 

[Another teacher] always refers to me as the math guy. His school

It's not an easy job. To be called the math teacher. His school

I'm actually the math and social studies teacher but I sort of identify myself His school

more His school as the math teacher.

I like that idea that I'm the math teacher and I like the idea that we at our school His school

are also in a way the curriculum people.

1 was hired to teach elementary school and part of that was to be a math teacher. Schooling (adult)
1 love that title, to be a teacher of fifth grade mathematics. Schooling (adult)

 

As a math person

 

I wasn't very successful in elementary school or middle school.

Math class (student)

 

1 can remember beinga bit lost in 8th grade.

Math class (student)

 

I remember being left behind.

Math class (student)

 

l was able to tune out.

Math class (student)

 

Adrift. Adrift at times.

Math class (student)

 

Unsupported. I didn't care

Math class (student)

 

My 8th grade teacher, Sister Jane 1 think, she didn't think 1 could learn math
either it seemed.

Math class (student)

 

I didn't think I could learn math.

Math class (student)

 

l was a person who didn't have as much success with math so I should have
though.

Math class (student)

 

I continue to be a learner of math.

Multiple figured worlds

 

I've had to be self-taught in many ways, mathematically.

Multiple figured worlds

 

And the collaboration, the collaborative aspect allowed me to be successful at
the task.

CI PD

 

 

 

 

 

 

 

Be able to do, successfully complete those tasks. C1 PD
Being successful in some of those more difficult tasks. Cl PD
As a student (in general)
I wasn't a stellar student through elementary school but I was a capable, l was a Schooling (student)
capable student.
I was a more motivated learner [in college]. Schoolig (student)

 

76

 

In his direct identifying statements, Jonathan articulated connections between his ways of
identifying in the figured worlds of his former school and his present classroom.
Additionally, he identified as collaborator. Because CI is about teaching children to
collaborate, this is another instance in which the way Jonathan identifies is parallel to his
decision to implement CI.

It is also notable that, when he began to identify as a competent math person, he
identified as "self-taught" (which contrasts with Cl) and as a perpetual learner (which is
consistent with Cl). He also identified as competent in the CI course. However, he seems
to identify as the most competent in the figured worlds of his classroom and present

school.

Mathematics Life Stories, Figured Worlds, and Choosing to Implement CI

Jonathan told stories of implementation of almost all CI principles in his
classroom:

The nature of Complex Instruction, the way I saw it that week, was that
everyone in your group is going to succeed. It's just going to happen. I
don't think I had that notion. I maybe had a notion that it would be nice if
everyone in the group succeeded. And the big argument everyone always
seems to have when I try to have a discussion about Complex Instruction,
about group work, is this idea that one or one person or two people carry
the group. That's legitimate except that I don't look at it that way anymore.
Everyone has a responsibility to succeed and to build this collaborative
sphere. It's not an easy thing to do when kids haven't done it before. But
over time they start to see. That's something, I know I experienced it that
week, and I know that I've seen it in my classroom now... And there was
an increase in student involvement in tasks, in groupwork.

Jonathan told stories of students' increased participation and success in the figured
world of his classroom as a result of CI. These are goals he had originally articulated for
his students, which CI was helping him to meet. He also described himself as successful

in the CI workshop. This makes it unsurprising that Jonathan would use CI to meet these

77

goals. Jonathan identified as a collaborator and a teacher who wants to involve and
challenge all students. Because the principles of C1 are parallel to Jonathan's ways of
identifying, it appears that Cl was a good fit for Jonathan. He did not need to change his
ways of identifying to implement CI.

Jonathan identified as being alone ("adrift," "unsupported") in the figured world
of his childhood school. He reported implementing C1, the goal of which is working
together, in the figured world of his present classroom. He also stated that he should have
been more successful in school when he was a child, and that he wanted his students to
all be successful. In his own story Jonathan correlates loneliness and failure in
mathematics, and togetherness and success in mathematics. Therefore, it is not surprising
that Jonathan tried to bring his experience in the CI workshop into the figured world of

his own classroom.

Discussion

Previous research on the links between teachers ways of identifying and their
implementation of recommendations has attempted to draw a one-way arrow from the
plots of teachers' life stories to their observed implementation of reform (Drake, 2006;
Drake, Spilanne, and Hufferd-Ackles, 2001). The purpose of this chapter is to sketch a
more complex relationship between teachers' ways of identifying and implementation.
The analysis of parallels between teachers' life-stories, direct identifying statements, and
stories of implementing (or not implementing) recommendations from this PD
demonstrates the complexity involved in a teachers' decision to implement these
recommendations. It demonstrates three possibilities for how the teachers' ways of

identifying may translate into implementation of the principles and practices taught in a

78

PD course. These three ways are: 1) implementation allows the performance of an
identity, 2) implementation allows the revision of a life story, and 3) the narrative
becomes a stumbling block to implementation (failure stories). I do not intend to argue
that these are the only three possibilities for how teachers' ways of identifying translate
into their choices to implement certain pedagogies or not. Also, some teachers fall into
more than one of the three categories of these three possibilities, indicating that there are

not hard boundaries between them.

Parallels Between Identijying Stories and Stories of Implementation: When

Implementation Allows the Performance of an Identity

Sometimes teachers' ways of identifying as mathematics teachers and math people
in general paralleled the language they used to describe their implementation of C1 (or the
decision not to implement it).

For example, Shirley identified as independent and self-reliant, and when she
talked about her reasons for implementing CI she evoked language of structure and
responsibility. She seemed to have focused on what CI can bring to her goals for her
students (structure, individual responsibility) and placed less emphasis on other aspects of
CI (interdependence and status), which the professional developers would say were more
important.

Jonathan is another example. Jonathan identified as a teacher who was concerned
about engaging all students in participation in mathematics class while simultaneously
providing challenging content. His perception of CI was that it was a way he could
improve at this. Therefore, he implemented about a dozen carefully planned CI tasks

during the first school year after he took the course.

79

It appears that sometimes teachers choose to implement new practices because
they can construct them as fitting particularly well into their previous ways of identifying.
Another way to say this is that teachers choose to implement pedagogies that help them
to become more of the kind of teacher they want to be. Drake's (2006) research on
teachers with turning point stories alludes to this, in that teachers with tool-based turning
points chose to implement the use of manipulatives (even when doing so resulted in
different implementation of the reform curricula than the curriculum developers
intended), and teachers with content-based turning points made changes related to how
the content was presented (Drake, 2006).

This observation extends the literature on professional development by providing
an alternative to the assumption that teachers who choose not to implement new
pedagogies, or who implement them in ways that are different than what was intended by
PD or curriculum developers, do so because they lack knowledge of the pedagogy or
because they have bad beliefs about students or mathematics. An alternative explanation
is that the teacher did not see how the pedagogy would help her to become the type of
teacher she wants to be. According to Sfard and Prusak's (2005) definition, learning is
closing the gap between current and designated identities, or becoming who one said she
wanted to become (2005). If a particular pedagogy does not resonate with a teacher's
designated identity, perhaps she filters it out, because she perceives that it will not help
her to grow in the future.

A note of caution: I do not intend to imply that only teachers who arrive in our TE
or PD courses with appropriately parallel ways of identifying will be able to implement

the recommendations made in those courses. This would be a dangerous assumption, and

80

could lead to misguidedly pre-judging who can and cannot teach in the ways
recommended by mathematics education reforms. There is evidence in this chapter to
show that the experience of the PD itself can be an experience that changes the ways
teachers identify. Kelly experienced a turning point (Drake, 2006) during the CI course.
As such, the way she spoke of the changes she chose to make in her practice after the CI
course were sometimes parallel with the ways she identified herself in the CI course and
after it. Because the experience of the PD may change how people identify, it is not
possible to predict before a PD who may choose to implement the recommendations and

who will not.

Contrasts Between Identijying Stories and Stories of Implementation: When

Implementation Allows the Revision of a Story

At other times the teachers invoked contrasts in the ways in which they were
identifying as students in school and the ways in which they identified as teachers, or as
as adult learners of mathematics or mathematics teaching, or teachers. For example,
Jonathan—who implemented CI--identified as a student who was alone and left behind,
but he identified as a teacher who values working together and learning from each other
in groups for his students. Such contrasts in language open the possibility that some
teachers adopt the teaching practices they do in order to offer their students ways of
identifying that they never had available for themselves. Another way to say this is that
some teachers want to provide opportunities for their students to have a turning point
story.

One way to theorize this is to use the construct of figured worlds (Holland,

Lachicotte, Skinner, and Cain, 2004). In Jonathan's case, he identified differently when

81

he described himself in the figured world of his childhood school than he did in the
figured world of his current school faculty. Because he identified as a collaborator in the
figured world of the school faculty, he saw a way in which his lonely childhood
schooling experiences could be remedied. In his classroom, he sympathized with students
on the periphery. He saw that he could open new possibilities for his students to
participate in math class, so that they would have a different story to tell about their time
in his classroom. Therefore, he chose to bring CI, a method of collaborative learning, into

the figured world of his own classroom.

The Failing (Foreclosure) Dilemma

At still other times, the teachers' ways of identifying in other figured worlds and
their stories of implementing (or deciding not to implement) CI gave conflicting
messages that are more difficult to explain. For example, Gina identified as incompetent
in mathematics class as a student and attributed her survival to help from others; yet she
chose not to implement CI in her classroom as a teacher, even though it would afford
students opportunities to get help from others.

Perhaps her failure (foreclosed) life story precluded her from attempting new
mathematics teaching practices. This explanation would corroborate the results of
previous work (Drake, Spillane, & Hufferd-Ackles, 2001). How this happens is still
unexplained, however.

If teachers with failure (foreclosed) stories are to implement new pedagogies then,
it seems as though one way this could happen is if they were to experience a turning
point. So, how do professional developers go about trying to provide teachers with

experiences that become turning points in their mathematics life stories? It seems that

82

providing a successful experience is only part of the equation. From a professional
developer's point of view, Gina may have experienced some success in the CI workshop.
After all, she worked with group members to complete some tasks, and was able to
explain the work done in her group. The key seems to be Gina's perception that she
survived (rather than succeeded) in the workshop, a perception that changed very little
about how she identified.

So what might help teachers with failing (foreclosed) mathematics stories
experience turning points that would help them to identify as mathematically competent?
Is such a goal even plausible?

Since the ways teachers identify themselves in their mathematics life stories
seems to be important, one place to start is asking teachers to articulate their stories, their
decisions about implementation, and the connections or contrasts between them. Perhaps,
if teachers could see these connections, they would rethink the decision to not implement
the new practices. Another idea is to help teachers to explicitly author stories of the
successes we see them have in PD courses. Further study would be needed, of course, to
find out whether these would be useful interventions.

It seems that the decision to implement new mathematics teaching practices
involves a negotiation between teachers' ways of identifying themselves at different times
in their lives and in different contexts and their perceptions the goals and principles of the
practices. I suggest that this is a back-and-forth type of negotiation process, whereby at
times, previous ways of identifying seem to be fore-grounded, and at other times the

goals and principles of the practices are fore-grounded. Sometimes these are aligned with

83

each other, and sometimes the alignment and integration of identities and goals is likely

to take some work.

Implications for Further Study

This analysis has focused on teachers' decisions to implement CI after the CI math
PD course. Through this analysis I have argued that teachers' decisions to implement
recommendations from a PD course are likely to be linked to how the teachers identify
themselves as teachers, in the past, in the present, and in different figured worlds.

However, sustaining implementation is another story. How is it that teachers
continue to decide to keep teaching with new practices? How is it that teachers move
from trying something new to becoming a new kind of teacher? And how does continuing
implementation of a new pedagogy impact teachers' ways of identifying? Based on the
literature and the results of Phase I of my dissertation, I hypothesized that teachers would
need to do some identity-alignment work in order to continue implementing these new
practices and eventually add them to their mathematics teaching repertoires. I wondered
what the process of becoming "a CI teacher" might look like. Therefore, I continued this
study with Phase II, in which I studied teachers' stories of teaching mathematics with Cl
over the course of several months. In the chapters that follow, I report on the narratives of
four teachers who were implementing CI during those months.

In this chapter I reported on the results of Phase I of my dissertation. The
remaining results chapters report on data collected for Phase II of my dissertation. The

Phase II participants were a new group of teachers (only Jonathan was in both the Phase I

84

and Phase II groups). In the next chapter, I will begin reporting on the Phase II group of
teachers, all of whom had decided to implement CI to teach mathematics in their

classrooms.

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CHAPTER 5: NARRATIVES OF IMPLEMENTATION: CHALLENGES AND
PERSISTENCE

The people were distinctly bewildered. This silent, cold man, —
was this John? Where was his smile and hearty handgrasp?
"'Peared kind 0' down in the mouf," said the Methodist preacher
thoughtfully. "Seemed monstus stuck up," complained a Baptist
sister. "Of the Coming of John," (DuBois, 1903/ 1994).

I'm beginning to find me. It's great. It is, you know, Frank? It

might sound selfish but all I want for the time being is what I'm
findin' inside me. Educating Rita (Cartlidge & Gilbert, 1983)

Overview

Learning changes how one is perceived by oneself and others. In "Of the Coming
of John," when John returned home, his community perceived the new quietness that
allowed him to succeed in school as being "stuck up." John needed to be a different
person in order to do well in school, and doing so had changed how he identified and was
identified by his community. In Educating Rita, Rita was so enamored with schooling
that her husband accused her of having an affair. What is it that drives an individual to
want to learn, despite the pain that learning can cause?

Based on a review of the literature on identity and learning, as well as on the
results from Phase I of the dissertation, I hypothesized that continuing to implement CI
would change the ways teachers identified, and that the ways they identified would
impact the ways they implemented Cl. As sub-questions for this chapter, I asked, "What
are the difficulties inherent in changing teaching practice that arise from the teachers'
integration of new mathematics teaching identities with their older perceptions of
themselves?" and "In what ways did teachers' new identities allow them to sustain
changes in their mathematics teaching practice despite incredible challenges?" In order to

learn more about the reciprocal nature of identifying and implementation of a new

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mathematics pedagogy, I looked for parallels and contradictions between them. I
compared the teachers' ways of identifying with what they said about implementing CI in
interviews before and after they had implemented individual CI mathematics lessons in
their classrooms.

In this chapter, I report on teachers' stories about CI mathematics lessons
collected in individual interviews, group discussion sessions, and presentations to people
outside of the CI experience that I collected these in Phase 11. During that Phase, I
followed four teachers from both cohorts of the PD course (Glynnis, Joanna, Patricia, and
Jonathan), all of whom had agreed that they would be implementing CI throughout the
upcoming months. (All of these teachers were new participants in Phase 11, except for
Jonathan.) In this chapter, the narratives function to explicate the identity work that had
to happen for these teachers for the new practices to become a part of their teaching
repertoire.

In the results section of this chapter, I describe the difficulties that resulted when
changing pedagogy necessitated an identity shift. Such situations resulted in discomfort
for the teachers, which they recognized as a problem. I report on how the teachers
described this problem as being located in a variety of places. I then describe the ways in
which teachers' identity work motivated their persistence during these challenges.

In this chapter, I tell Act II and plot point 2 of the story of teachers'
implementation of CI for teaching mathematics. A graphic representation of where this

chapter fits into the rest of this dissertation story is included as Figure 6.

87

 

 

 
  
 

  
 
    

Act I: How the
teachers identify
(life stories)

Act 11: Stories of
implementation of CI in
math

Act 111: CI
Teachers

    
    
     
   

 

 

 

 

  

Decision to persist
with implementing
CI (or not)

Teachers decide
to implement C1
(or not)

 

Figure 6 . Chapter 5—Act II and Plot Point 2

The Challenge: Integration and Denial of Identity
Very early in the process of data generation, I was struck by how miserable and
tortured the teachers often sounded. The teachers often found implementing a new
pedagogy to be difficult and painful, sometimes heartbreakingly so. Their narratives
revealed clues as to how, for some teachers, implementing CI necessitated uncomfortable
identity work. Teachers located the source of this discomfort in a variety of places: their

own performance, the students, the CI pedagogy, or their school context.

Joanna: A Giver Steps Back

Shortly after the 2007 C1 course, in which Joanna was a participant, I
conducted a life story interview with her. I asked her how her students would
describe her as a mathematics teacher. She said:

I think they would describe me as being a good math teacher because I
have high expectations, I push them to problem solve and to reason things
through, make sure things make sense, I try to make it fun and engaging
them, not just the paper and pencil or let me show you everything. Let's
investigate. I let them know what my goals are for them as the year
goes on. What I expect them to be able to walk away with. I communicate
to them clearly what's gonna help them the next year, or fruther down the
road. I don't just give them the work to have it be busy work. I explain to
them the reasoning behind it. Of what we're gonna walk away with. What
they'll be able to do if they learn this, how it's gonna help them later. So
those things help them have understanding.

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In this excerpt, Joanna said she had high expectations, but described herself as giving the
students a lot of things. Joanna identified as being a teacher who gives a lot to students.
She said she gives them goals, clear communication, and explanations of reasoning. She
said that those things helped students to understand.

Later, when I conducted my first pre-lesson interview of Joanna, she identified
herself as a teacher that pushes students to think. When asked if she could tell of an event
in which she did this, she told the following story:

We had this story problem: "Mrs. Garcia has taught third grade for

nine years. Mr. Lee taught third grade for seven years and then fourth

grade for six years. How much longer has Mr. Lee taught than Mrs.

Garcia?"

Well a majority of the kids had the right answer. But also a

majority of them didn't show their work. It became clear to me after the

first little girl came up. She wrote down her answer, and when I asked her

if she could show her work she was stuck.

Joanna saw a child who was stuck, and being stuck was troubling for Joanna.
Joanna had set up the problem of this story to be that a student was stuck. This set
the stage for a story in which Joanna might push the student to think or to solve a
problem. She continued:

So when they get stuck I always say, "Ok, you can have a friend come up

and help you." So the friend came up and helped. This is what they got.

"Well, nine take away five equals four. Let's see. What else equals four?

Five take away one equals four, and ten take away six equals four."

As Joanna continued the story, we again see her describing herself as giving to students
to effect student understanding. Joanna saw a child who was stuck, and being stuck was
troubling for Joanna. She portrayed herself as again giving something, this time the

support of a fiiend. She identified herself as a person who "always" does this. She

continued to tell the story:

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I thanked them for sharing that. Then we went through and said, "Yes,
each one of those does equal four. But boys and girls," I asked the class,
"Does this match the story problem?"

Joanna identified as a teacher who thanked children for sharing, even though what they
shared was not what Joanna expected (i.e., not correct). In doing so, she identified as a
person who was appreciative of students' efforts. As she continued the story, she told of
her attempts to correct the students by giving something once again, this time a question.
When Joanna continued the story, she described more students as having trouble
explaining their answer. Then she continued:

And then somebody shared, "No, that doesn't match." And they
explained why.

So then we had another student come up, and she had the right
answer too. She came up and wrote her answer, but she couldn't show her
work either. So I said, "Well, tell me how you solved it," and she told me.

She said, "Well, I started at seven and I counted on with my
fingers. Six more."

I said, "Oh, well. Counting on. What is that?"

She stood there and thought about it for a minute. Then she said,
"Well that's adding."

I said, "So we could make that into an equation, a number
sentence. That's seven plus six equals thirteen. Let's write that down."

So I said, "So really, you were looking for that missing number.
You started at nine and you went up to thirteen."

I said, "So you were filling in this missing number. This is one way
you could've written it. Then I went on to show her, "Or some people
might have thought of writing it as a subtraction problem."

We went on to try to talk about, "How else could we do it if we
wanted to show our work?"

"We could draw a picture."

"Do tallies," somebody came up with. So we showed an example
of that.

Then somebody else said, "Well, I like to put it in words. So we
did an example of how they could put it in words.

Joanna portrayed herself here as giving something once again. This time, she was giving

formal language for students' informal reasoning. Students are portrayed as becoming

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able to understand as a result of her giving. In fact, they were then able to come up with
some interesting ideas about how to represent the problem.

In this story, Joanna had constructed the problem as the students' inagility to
explain their answers. As the tension in the story builds, Joanna portrays herself as giving
a number of things: support, a friend, thanks, pointed questions, formal language, and
another representation. The resolution is that students understand, and are able to then
construct interesting ideas about representing the problem. In telling such a story, Joanna
had identified in a number of ways, but key one way is as a giver whose giving resulted
in student understanding. Being a giver had been a way that Joanna had identified as a
successful teacher.

When I asked Joanna how her students would describe her as a teacher when she
was teaching with Cl, she responded:

How would they describe me? I don't know. I think they feel that they

want more information from me. I think that they feel like I don't give

them enough. Because I want them to do the problem solving and they

don't want that. They want answers right up front. They want me to give

them more. So I think that they get kind of frustrated that they have to do

that work on their own. Because so much in everything else that we do, we

do so much modeling, explaining to them and prepping them. Then, when

it comes to doing something like this, we set them up to do it on their own.

It's opposite of what we do the rest of the time. I think that they get

frustrated with that. I think that they would say I don't prepare them

enough for what they're going to do.

Of course, there is nothing inherently wrong with giving. Teachers' should give students
clear explanations, provide students with formal language for their reasoning, and
facilitate students' understanding. However, when Joanna began to implement CI, this

way of identifying became a hurdle for her. In the CI course, teachers are taught to step

back and observe students for a long time before intervening in groups. Interventions are

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engineered to promote students' autonomy from the teacher and interdependence on each
other. Joanna had identified herself as a giver, and told stories of giving as a successful
part of her teaching practice. Now, stepping back and allowing children to struggle
required her to behave in ways that were different than how she identified. This was
uncomfortable for Joanna. She located the source of this discomfort in her own behavior,
which she perceived as troubling to the students.

Later in the interview, Joanna expressed the magnitude of her discomfort:

Joy: What are you wondering about?

Joanna: Honestly?

Joy: Mm hm.

Joanna: I see value in it but I wonder. Is it worth the effort?

Joy: Mmm. Mm hm.

Joanna: I know, there's value in it but, when I look at the chaos, and all the

work, I really question. Is the effort worth what the benefits are?
It was so uncomfortable that Joanna questioned her commitment to CI. This difficulty
arose because the behaviors CI requires came into conflict with her identity as a math
teacher. Because Joanna had strongly identified as a mathematics teacher who achieved
success by giving, she now wondered whether CI—which did not allow her to "give"—
was successfirl enough to continue. She located the source of her discomfort in the CI
pedagogy.

Joanna decided to continue to use CI. To do so, she employed some strategies to
cope with this discomfort. At first, these seemed strange to me. For example, for one Cl
task, she tried to give her third-grade students a long list of written instructions. Later, she
wondered why she thought that would work. However, considering the work that Joanna

was doing to integrate her old way of identifying as a giver with her new intention to step

back, this move makes sense. It allowed Joanna to give (a long set of directions) and step

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back (because she expected to not need to physically intervene to give directions). If I
had done a traditional pre— post- study of Joanna's learning from the CI course, and I had
returned to Joanna's classroom for a post-course observation on the day that Joanna had
given these directions, I may have concluded that Joanna did not learn from the CI
course. However, in studying the ways in which learning CI required Joanna to integrate
this new way of being a teacher with her existing ways of identifying , a more nuanced,
interpretation is possible.

Several weeks later, she observed an interaction in her classroom that helped
resolve some of these issues for her:

The first table that I went to happened to be this table that I had the

microphone at. And the one little girl was showing the other little girl what

symmetry was. She was helping her. We talked about, in our rules, being a

helper doesn't mean giving the answer. And she took the paper clip that

their stuff came in, and she was just showing her like this. [Holds the

paper clip on the line of symmetry of the figure] I thought that was so

cute. That was so clever, and easy way to show her, without bending up

the cards or anything. As I was walking around, I noticed there were a

couple other tables.

In order for Joanna to identify as the kind of teacher who steps back and lets
students struggle, she needed to do some difficult identity work. Joanna told this story on
subsequent occasions as well. It was an important story for integrating her older, deeply
ingrained identity as a giver with her newly emerging identity as a teacher who allows
students to learn from each other. Joanna had a story of success to tell about a time when
she stepped back and allowed children to work together. She describes the student as
inventing a pedagogy that she herself had not discovered. In stepping back, Joanna had

given something new to her students—the opportunity to learn from each other and draw

on each other's competencies. She had also seen how students could give to each other

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(I!

things she could not necessarily give them. She used the words "showing" and "helping"

to describe the student as a giver.

Jonathan: An Insider Experiences the Periphery

In Jonathan's life story interview, he said:

They trust me a lot and I trust them. We have a lot of meetings every year
so we're pretty lucky. We have these Wednesday early releases, twenty-
eight of them, throughout the year. The students go home at 1:10 and we
end up with an hour-and-a-half of professional development time. Twenty-
eight Wednesdays... And we did a lot of things like creating common unit
assessments and looking at some student work. So I would say that we had
a very collegial, a lot of sharing. So there wasn't anyone that wasn't sort of
equally trusted. And it was a format where you couldn't just sort of show
up and disappear, because it's a small group. There are eight total fifth
grade math teachers. In the building, sometimes we had a building level
meeting so sometimes just 4 people. So you are a collaborator whether
you like it or not. I think it just has to happen. But just given the amount of
sharing that takes place in terms of the workload. I think just that in itself
speaks to how we view each other. Since we all share in sort of products.
We create products together. We have this great book now that has all our
units laid out, and we're working on some anchor projects now. . .It's really
great. It's really great stuff that we're doing.

Jonathan spent significant time talking about himself and his colleagues as a team. In this
excerpt, Jonathan identified as a collaborator. He described his relationships with his
colleagues as being collaborative. He also indicates that this collaboration has been
successful, saying that they are doing "really great stuff".

As with all of the teachers, this is not the only way Jonathan identified. He also
identified as a person who was interested in equity and social justice, and in motivating
students to engage in challenging problems. We would like teachers to identify
themselves and their colleagues as teammates and collaborators, the way Jonathan did.

In his second pre-lesson interview, I asked Jonathan what was lost in

implementing CI. He stated:

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There is one thing that is lost—an understanding from my colleagues. Not

everybody knows what I'm doing. And not everyone values what I'm

doing because they don't fully understand it. And so I lose a little bit. As

much as I wish I could tell them. Deeply, I mean. . .The sort of response I

get is, oh yeah 1, I've done group work. That's group work. That's, oh,

they're calling it Complex Instruction? Oh, yeah, they always change the

name of things, you know, one thing is this, and then it comes around

again, every so often as another, reinvented. It's hard to explain to them

that it's, yes it is group work, and it does have many elements of what they

know as group work, but it's a lot more than that. So, yeah. There is that

loss of understanding.
Jonathan's way of identifying as a collaborator seems at first blush to be a good match
that would make for a smooth adoption of CI to teach mathematics. Unfortunately, in this
case, becoming involved in CI created a small rift between Jonathan and his colleagues.
Because Jonathan identified as a collaborator, this rift created a tension for Jonathan, and
required some difficult identifying work. Jonathan experienced a shift in the ways he
identified as a colleague. Implementing CI, and talking about it with his colleagues,
changed the way he could participate with his colleagues. He identified as a team player
and someone interested in equity and rigor, all important in CI, so implementing it did
not require him to act very differently within the classroom. But his way of identifying in
the figured world of his colleagues was challenged because he felt like his colleagues did
not understand all that he was doing or his reasons for doing it.

This situation was uncomfortable for Jonathan. In the course of data collection, I
was unable to see any real resolution to this problem. However, as I will describe later in
the chapter, Jonathan also identified as a risk-taker and a teacher whose primary goals

were equity, social justice, and rigorous learning. Because those ways of identifying

aligned well with the goals of CI, Jonathan did not need to change how he identified in

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that area. Therefore, he continued to implement CI, even though his identification as a

collaborator continued to be challenged.

Glynnis: A Non-Conformist "Tweaks" CI

Like the others, Glynnis did not identify herself in only one way. However, in
order to describe the challenge Glynnis faced as she began to implement CI, I will tell
about one way Glynnis identified herself—as a non-conformist or dissident. Glynnis used
this way of identifying in her life story interview. As she described how she learned to
teach mathematics, she said:

I began to feel a lot more comfortable with my adequacy in

math. Beginning to take a lot more risk and trying to understand it. I was
very much relieved to learn it's not about the right answer, it's not about,
this whole block of knowledge here that you have to grab hold on. It's an
evolutionary kind of thing, it's growing it's evolving, and it's not done. It's
not ever done... When I first started teaching there was not all the dogma
for tests and standards, so I was able to really grow and learn and do some
things that were comfortable for me, and not feel that I had to be, da da da
da da [gesturing with a vertical hand motion on each "da", to indicate a
row]. . .it became difficult because the stringent, prescribed, curriculum
began to change our [a study group I was in] focus and our shift began

to not be so much on learning, exploring, investigating, as it was on
preparing kids to pass the test. And like it or lump it, that's part of the
mandate. If I want to stay I have to do it.

In this excerpt, Glynnis identified as interested in investigation, learning, and exploration.
She positioned herself against policies that would require her to align herself and her
teaching to a state standardized test. She continued:

I don't think, you can expect kids to, produce. It's a contradictory term. Is
the expectation for you to produce? Or is the expectation for you to grow
and learn and develop? If the expectation is for you do develop, you don't
always get a product. You don't always assess a product. . . .While I may
adhere to a different philosophy the reality that these students face is, they
get told, we get told, you didn't produce, if the scores were not and AYP is
not in a certain percentile.

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Glynnis described herself as having a different philosophy than the state or her school
administration. She identified as having different goals for her students than the district
and the state had for them. She continued:

The best that I can do, given these circumstances and these
constraints, with the development of my kids is to do the da da da da da
[Gesturing with vertical hand motions indicating a row] so that they won't
feel totally inadequate. Because they are being measured by yardsticks,
socially and politically. I tell the kids. I gave them a spiel already. " [The
test is] not important, except people are going to judge you based on it. So
therefore, you have this responsibility saying, 'Yes I can'. And you can.
You can.

You don't have the same opportunities as certain kids have had. As
an urban school, that's the reality. Most of us", and I did this whole, upper
class middle class, lower class, thing. 1 said, "How many of you heard this
'upper class', word?"

A few of them had. "What do you think it means?"

"It means you got money." They can recognize even at 10 that
people that have money usually have something else—prestige and power.

And if you don't have that money, you don't have the same prestige and
power. You already know that there's a difference.

"When I say rich what do think of?"

"You got a big house,"

"You drive a fancy car,"

"You got a lot of money".

"How many of you guys fit in that category? You're not. I'm not.
So therefore we don't have the same power. And prestige. That's human."

So they already got that. And people here have more advantages, because
they have the power to change the rules. They get to make the rules. They
get to do a whole bunch of stuff [laughs] Even if they don't know it. We
talked about that. So that was from whence I come, with the preparation
for the [state test]. Because it is, people with more power, and money, get
a chance to have opportunities... They're speaking a different language
that you're going to see on the [state test].

Glynnis described ways in which she attempted to both prepare students to do well on the
test and simultaneously attend to her other goals. She again identified herself as
somewhat subversive to the state testing mandates, by positioning herself with her

students and against the establishment. This us-against-them positioning is a way in

97

which Glynnis identified on several occasions. She continued to identify in this way as
she described the changing demographics of her school:

My kids are predominantly nonwhite. Predominantly middle to lower
[class]. Historically, it was not like this. So that's caused a paradigm shift
even, as I've been here. It used to be more white. And it used to be all
upper to middle to upper middle class. We've got a lot of white flight.
We've got a population of teachers where I'm the youngest senior teacher
here. Most of them have been here for a long time. So they're not really
accustomed, and have issues of not handling the paradigm, and it's been
very problematic for me. It's been problematic in a different way for them.
I was the second minority teacher here. They've had one she's down in
first grade. She's been the only one, predominantly, for 30 years. Now
there are three African American women here. So it has changed. It's been
slow. It's been painful. It's been very, very filled with angst for me because
I see what a lot of people don't see, and they refuse to admit it. It's a
constant battle—how much push, how much to say, and what's the most
advocacy for kids and their families. Because people don't look at it that
way. They look at it as, "these parents," and "these kids," and the very
negative...

One interesting way in which Glynnis identified herself happened when she said "they"
(the school) had one minority teacher. She also identified her colleagues as "they" in
saying that they view things differently than she does. Glynnis identified as an outsider in
terms of race, in terms of philosophy, and in terms of understanding children and the
changing population in her school. She said:

The collaborative effort, in the building, the climate has changed. When I
started, I wanted to do the open door, and collaborate, and, the door's
closed. I'm honest enough, and shameful enough to say, the door's closed.
And it's simply because, I feel, that my purpose with the students that I
serve here is primary. It has to be first. . .No I'm not doing it. I'm not. It's
not about that. Because, what I do is, I feel, life changing. Everyday. It's
changing my life, and it's changing the destinies of these kids whether it's
academic, social, politically. It's just that important. . .We are life-
changing, life-influencing people. It's scary. I think real hard when I say
that. . .When you are really one—on-one with the overall destiny of
somebody—how they think, how they feel about themselves. It isn't
something I want handled lightly, or irreverently. It's an awesome
responsibility.

98

I see a lot of [teachers for whom] it's a job. Eight to three, it's a

job. Get it my way or hit the highway. You're in and gone and done. I just

don't adhere to that philosophy. They've got a right to think it but I don't

adhere to it.
Glynnis continued to position herself outside of her school faculty in terms of her respect
for the responsibility for the profession. Because she sees her philosophy as so different
from others in her building, she chooses to work alone and not collaborate. Sometimes
her identifying as an outsider is in regards to race, and other times it is because of her
dedication to her students.

In my first individual interview with Glynnis, she said:

Actually and that's what I came away from the class with last year. It isn't

a process that you go through and voila you get an end result. Because it

doesn't work, unless those fundamental principles are in the back of your

mind... in a way, that was one of my biggest frustrations in the class last

year. I think it was getting to me. . .A process of just by doing the task and

the passing out the cards, that it automatically happens. And it doesn't.
She elaborated on this frustration during the first teacher discussion group meeting.

That's what I was having some issues, struggling with Complex

Instruction. Along the devil's advocate, toward that side, is that it is,

sometimes, setting the instruction up, one, two, three, A, B, C, and this is

how you do it. If I have roles, and if I do this, and if I do this, then the

result will be a group that's successful.
Her outsider/dissident identity presented a challenge for Glynnis as she was deciding
whether or not to implement CI. Glynnis had been told about a series of instructional
strategies in the CI course, including assigning roles, teaching norms for participation,
attending to status, and using groupworthy tasks. However, she resisted this as a part of
an organized structure that someone else imposed in the course (even though Glynnis had

volunteered for the course). She described CI as being "one, two, three, A, B, C," similar

to the way she had indicated that the administration at her school wanted her to do "da da

99

da da da". Because Glynnis identified as a non-conformist and a resister, she resisted CI,
even though she had volunteered for the course, questioning whether following it as
prescribed was a good thing. She even identified herself as "the devil's advocate" in the
. first teacher group meeting, positioning herself outside the other teachers in the group.

In the second teacher group meeting, Glynnis continued to talk about her struggle
with Cl as a "package." She said:

I think that the reality of education is that, if you really buy into a process

of learning and teaching, there's never one thing that does the work, for all

the many reasons. You can take this package and just do it. You

mentioned Lucy Calkins and I've mentioned this before, one of the

criticisms now about Lucy Calkins is that teachers are feeling they have to

go and say this word-for-word verbatim. Do this and do like that that.

They don't have any leeway.

Glynnis identified as a person who does not like to have everything lined up for her, who
likes leeway in making instructional decisions. She said paralleled CI as a "package" to a
program that is scripted verbatim. Again, Glynnis pressed the group to consider whether
accepting Cl as a package is helpful. In doing so, she continued to identify as a person
who resisted imposed standards and norms. Glynnis experienced discomfort because she
perceived that Cl did not allow her to be the kind of teacher she wanted to be. She located
the source of this problem in the CI pedagogy itself.

The last data collection session occurred when the CI teachers met to present their

CI teaching at a round table session for new teacher educators who had not taken the CI
course. The teachers who participated in this study came prepared to share something
about their teaching with Cl and to answer questions from the instructors. In that context,

several months after making the previous comments, Glynnis said this about her use of

CI:

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I find a lot of merit in it, so I wanted to use it I've been a very slow upstart

in being able to fully incorporate it into my practice. I keep slowly

working at it though. . .As I grapple with it more and more, I was telling

Joy, I'm starting to get that it can be tweaked for individualism. An

individual style. I looked at Jonathan's tasks and that was like, "Eureka!"

last year. It was like, "Ok, cool. It's not all huggy bears". . .I'm learning the

freedom of tweaking it. That's really big for me, because I was feeling like

I had to do everything. I was really not finding the freedom to change

things because I didn't know if it was still CI if you tweak it. That sounds

really sticky but anyway. So instead of giving them the cubes, I put all the

cubes in the bucket, and part of the task was they had to figure out what

cubes they needed. That was interesting. . .I only had three jobs instead of

four and that made the groups smaller. I think at 4th grade, I think I like

that much better. I told them today there would be sometimes

four, sometimes there'd be three. But since this was our first time I started

out with three.
As she began to see that she could continue to be non-conformist and implement
principles and practices of Cl, she began to be more comfortable with it. When she went
to the teacher discussion groups, she realized that all of the teachers in the group were
doing things a bit differently. Patricia had designed new roles and role cards. Jonathan
had designed tasks that did not require the use of a lot of different cards and materials. So
Glynnis reported that she began understand that she could use the principles of CI to
design tasks that suited her. That freed her up to see CI as something consistent with how

she identified.

Patricia: An Expert Sticks Her Ground

When this study began, both Patricia and Jonathan had been using CI for a year.
However, Patricia had been teaching for over thirty years—about twice as long as
Jonathan. She had also taught one groupworthy task per week out of one of the CI course
resources (citation) during that year, whereas Jonathan had taught about a dozen

groupworthy tasks he had designed or adapted himself. Additionally, Patricia taught first

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grade, which was the youngest grade represented among the teachers participating in this
study. When asked about her goals for students, she said:

It's behavior. It's management of behavior. And the math will be there.
The math will come. But that's because I'm a behaviorist first. You know
my background's special ed. It's emotionally impaired. Behavior plans
were the forefront of everything that I did. Somebody else may be looking
at a different aspect. Jonathan, I understood where he was coming from
because he has only so much time. [His goal has] gotta be the math. You
gotta be learning something with the math, and not so much the behavior.

Patricia saw herself as a behaviorist, and described herself as learning the
importance of teaching behaviors from her special education background. She also
acknowledged that she would have different goals than Jonathan, because he taught an
older grade, only had students for one mathematics class during the day, and had
important mathematics content to teach in that time.

In my next individual interview with Patricia, I asked her to explain what she
meant when she identified as a behaviorist. She replied:

My initial teaching experiences were [that] I was a teacher of
emotionally impaired kids. So you did behavior plans on everybody, to the
nuance of how many times they said something, and then how do you
increase that. So my focus has always been, you get the behavior in
order then learning is more optimal. I certainly understand and know the
opposite of that is you make the learning so engaging the behavior will
take care of itself. But when you're teaching, I think when you're teaching
cooperative learning and teamwork, at the very beginning it has to be
somewhat scripted and it feels a little contrived. But for them to
understand and have a picture in their head, of what expected behavior is,
that it is,

My working knowledge of [behaviorism]? I think about the lesson.
I think ahead of what is going to be a problem in the lesson. How do I
alleviate or run interference with that or preempt it before we even get
started? I am forever the person who goes on the field trip having
everybody in the circle here and we're having a short intense conversation
about my expectations of their behavior. "Are we clear? Can we shut our
eyes and see ourselves sitting on the bus, with our feet on the floor, and
our knees forward? Everybody try that. See it in your head. That's where I

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want you to see." So we do imagery, and I'll do a conversation, and I'll do

vocabulary so that they're clear. "Are we clear? Because I don't want you

to be disappointed when we get on the bus and the natural consequence is

that you cannot join the group." So my feeling always is be very up front

about what is going to happen so that it is not a surprise. It's not a reaction

to the moment. And so they have a heightened awareness of what is

expected of them. For me to take from the emotionally impaired classroom

to this kind of classroom, you don't want it as confining as I had to be in

the other kind of classroom. But on the other hand, I use similar

techniques. I forewam. I give them a model. I want them to visualize it

and see themselves walking through what they're going to do. "Now get up

and everybody do it."

Patricia did not seem to experience the same type of conflicting ways of
identifying or the need to do the types of integrating of identities that the other teachers
did. Throughout the data collection period for this study, Patricia identified herself as a
"behaviorist," which she described as being a person who believes that if you teach
school behaviors explicitly the mathematical learning will follow. This way of identifying
was comfortable for her. As the study continued and she spent more time talking with the
other teachers, she conceded that the mathematical learning is a reasonable focus for
teachers at other grade levels. In doing so, she positioned herself in a place where she
would not need to significantly change her ways of identifying to become a teacher who
was more concerned about mathematical learning during CI tasks. She also claimed that
her students were learning a lot of mathematics during their non-CI mathematics lessons,
but that during the CI time, her focus was behavior. Again, in making these claims,
Patricia created a space for mathematics in her practice without having to integrate any
new ways of identifying with her previous ways of identifying.

One reasonable hypothesis for this is that Patricia had been teaching for much

longer than the other teachers. Therefore, perhaps her ways of identifying were more

congealed, more solid. Another hypothesis is that she could perform her previous way of

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identifying as a "behaviorist" while teaching math using CI to such an extent that she did
not need to change anything. She also identified as an "expert" through her use of How-I-
Teach-Math-With-CI stories (which are the topic of Chapter 6). This led me to wonder
whether "expert" identities might be more reified and less likely to change than other
ways of identifying (which is the focus of Chapter 6).

It is also possible that any conflict between Patricia's previous and newer ways of
identifying were resolved prior to my interviews. Patricia had been teaching using CI for
a year before she began participating in Phase II of the study. This was true for Jonathan
as well; however, Jonathan had taught about 12 CI lessons in the year between the CI
course and Phase II of the study, but Patricia had taught a CI lesson every week of the

school year during that same time.

Persistence: The Expression of Identity
After observing just how challenging implementing these new practices were for
teachers, I became puzzled. Why did they continue to persist in the midst of such
difficulty? Thus, I examined teachers' narratives for evidence as to whether their identity

work was supporting such persistence.

Joanna as Project Person

Joanna had a history of participating in mathematics professional development
events, courses, and research sponsored by our Midwestern University. She came to CI
because she had heard about it from a graduate student with whom she was working on
another university-sponsored project. One way she chose to identify was as a person who
participated in these things and then brought the ideas back to her colleagues. In that way,

participating in CI suited her current ways of identifying.

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I asked Joanna why she participated in projects. She said:

I'm always looking for ways to push the bar higher and trying different

things. I just don't go with the same old thing. I don't use [the textbook] in

the classroom hardly. I pull in all of my own stuff from a variety of

sources that I think engage kids better and get them into that learning

more. [My colleagues] see that, so I think because of that they think that

I'm a good math teacher.
Joanna identified as a teacher who was interested in continuing to change and try new
ideas. This was why she continued to get involved in university-sponsored PD and
research. Joanna identified as a person who had and used a lot of interesting resources.
She perceived that she was viewed positively because of this. It makes sense, then, that
Joanna sought out CI as a place to learn new methods and gain new resources. Later, she
said:

I think [my colleagues] just look at me as being the person that's always

trying new things and willing to say when something doesn't work.

Willing to say when something goes really well. I always share my ideas

but a lot of times they don't want to do it because they think it's too much

to take on... Some of them, I think, are impressed and they wish they had

that time or that energy. We're all in different places to do that or that

comfort zone. Some of them just aren't comfortable with it.
Joanna wanted her colleagues to identify her as dedicated, willing to try new things, and
energetic. Thus, she indicated these as values she has taken on herself, but are also valued
by her colleagues. When I asked her about her motivation for trying new things, she
stated:

For me it's that I'm really trying to be a more effective teacher and this is

my way of doing it. I'm not good with just doing the same thing all the

time. I'm always trying to better my practice
The way Joanna described herself as seeking to better her practice is through getting

involved with projects and PD events. When I asked her to describe herself, she was

quick to respond:

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I'm not satisfied at all. I always feel like ok I could do a lot better than I'm

doing, or cover a concept a lot better than we covered, or maybe I'm the

one that's causing this disconnectedness because I'm not allowing enough

time or, what is a clear connection for me maybe isn't for my kids. No. I'm

just always feeling that I can change or do better. Always. I never feel

fully satisfied. . .I could do better myself or understand more ways. . .As my

daughter would put it I'm a workaholic and I'm just always driving myself

to do more. I'm just never satisfied with the status quo.
In this excerpt, Joanna continued to identify as driven to learn more and new ways of
teaching. At the core seems to be a sense of dissatisfaction with her own competence.

Joanna identified as a teacher who was a joiner when it came to University PD
and projects as the one who tries new things. She identified as dissatisfied with her
competence and always wanting to learn more. As such, it is unsurprising that she joined
the CI course (which was sponsored by the University) or that she volunteered for this
research study.

It also may explain part of the reason she continued to implement CI, even when
it was difficult. In doing so, Joanna could continue to identify as the teacher in her

building who was trying something new, who was striving for something difficult, who

was a "workaholic" working on bettering her practice.

Jonathan as Rigorous and Inclusive

Jonathan stated that he is motivated by a desire to empower students by helping
them to understand the world through mathematics. He said:

I see myself as someone who's willing to take risks with math teaching.
Who wants, I want to find ways that reach more kids and I want to find
ways, I see myself as a teacher who wants to bring everybody into the
math, and yet at the same time challenge. . .I want to be successful. I want
kids to be successful. I think that's probably at the core. There's a great
satisfaction in seeing kids learn. And so I mean just there's a self-
satisfaction in watching kids get it, and becoming. There's power in
knowing things too. I have more of an awareness now when I see things
like people offering bargains that aren't really bargains, you know. Or you

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know looking at data in a way that people say things are the way they are
but you look a little bit deeper in the data you look at the statistics and
they're giving you a false claim. So you know I like that idea of helping
people see that and getting power from that. I think stereotypes have
changed a bit, but there used to be a time when it seemed like guys would
be the mathy students and girls, oh, they'd be the readers and the writers.
But I like that that I can say that the smartest student in the class is a girl,
you know. I don't know why that is. Maybe because I have a daughter. But
there's just something in me that likes to breakdown those stereotypes.

Jonathan identified himself as a risk-taker, especially for the purposes of teaching
more children and facilitating student learning. The central goal of the CI course is
promoting both equity and rigor in mathematics classrooms. Therefore, participating in
the course and implementing CI in his classroom was a way that Jonathan could identify
as an equity-minded, rigorous teacher.

As Jonathan implemented CI, he continued to assess whether or not it was helping
him work toward his goal of being an equitable and rigorous teacher. In a post-lesson
interview, he stated:

I see kids able to show me things they wouldn't be able to show me in

other ways. I think they get deeply engaged in group tasks, much more

than a lot of the work I give them. I can give really meaningful,

challenging work that is supported. I'm twenty-seven [students] this year

in both my sections, which is big. If I had to go twenty—seven-on-one, with

complex stuff like this, I'd have a lot of frustrated kids. I'd have a lot of

kids that didn't get my time, that wouldn't get help, that wouldn't get as

much feedback. Those times that they are in group task there's a lot of

deep thinking, there's a lot of sharing, there's a lot of strategizing.

Jonathan continued to implement CI because doing so allowed him to identify as a

thoughtful teacher who continued to be concerned about rigor, while supporting all

students' learning.

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Glynnis as Subversive

Glynnis identified herself as a dissident who saw injustice in the system and
intended to fight it. She believed that high-stakes testing policies and the ways test scores
were reported and used in her urban school was unjust. Consider the following excerpt
from Glynnis' life story interview:
Glynnis: I believe every human, every teacher, has a desire to impart some
kind of thing. Mine isn't what every other teacher's is. They've
got desires too. I recognize those. They desire for everybody to
conform.
Joy: What would you name your desire?
Glynnis: Hm. Good question. What would I name it?
Joy: You know that it's different than conformity
Glynnis: Yeah.
Joy: And different than what other folks are-

Glynnis: It is, mine is, to use a cliché, to empower. To empower with
identity, to empower with a voice, to empower with
responsibility to change, to empower with the ability, the
responsibility to question. Because even if you don't change you
really have to question, "Is this right?" So that's what I do. The
status quo. Why is it the status quo? Why do kids have to sit in
chairs? Just because. Why is it that girls have to wear skirts and
why is it that boys, you know. You have to question those. And
if you question them, you get to decide, do I want to conform?
Or do I want to be different. And being different sometimes
costs.

As much as her identifying as a resister caused Glynnis to struggle to accept CI,
this also motivated her to persist with CI in mathematics. In the context of a story about
preparing her students for the standardized test, she said:

I think we do a remarkable job at trying to give substantive, I call it a form
of resistance. Closing the door. Doing this project. Doing CI.

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Glynnis made it clearer that it was this way of identifying that motivated her to
persist with using CI in her classroom. Because it afforded her the opportunity to identify

as a resister, she was motivated to persist through many difficulties.

Patricia as Organizer of Materials and Behavior

One way Patricia identified was as organized and orderly. Her method of
implementing CI after the course was to use one of the resource books that contained
tasks (LHS, yr). She began at the beginning of the book, and worked through it, one
activity per week, in order. In order to prepare to do this, she asked parents to copy the
activities, color-code them according to groups, laminate them, and file them according
to her system. When she spoke about her use of CI in the first pre-interview, she said:

I color code because it's so much easier to find the pieces and put them

back. So, it ends up being not difficult to do. . .Once they're made up it's

easy to set up. They're responsible for getting their stuff and, and taking it

back to the table, I have a chart that rotates their jobs and easily rotates

their roles.

Using CI in this particular way, using these particular tasks, allowed her to shine in the
area of organization. Also in the first pre-interview, she said:

At the end of April, I'll reorganize the groups again. I keep the chart like

that and I just move the jobs to the next person so it's all set to go. I keep it

turned around. The orange ones have the actual job names on them. I put

the colorful cards up just before I send them out. Otherwise they're all

going, "It's my turn."

Patricia created sets of color-coordinated, laminated role cards with pictures so that
students could learn and use roles. She set up a chart with color-coordinated cards to help

keep the groups organized from week-to-week. Patricia identified her color-coordination

schemes as being about simplicity and organization.

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Patricia saw another reason for her use of color. She said:

I'm also a first grade person so I like, I make things bright.
She also identified as a teacher of young children, which she claimed was another reason
for organizing materials by color. Color-coordination and organization of materials were
skills that Patricia identified as strengths. Some of the teachers were overwhelmed by the
new and abundant materials; however, Patricia enjoyed that part once the materials were
made.

As you recall from earlier in the chapter, Patricia also identified as a "behaviorist"
(her term). She said:

It's behavior. It's management of behavior. And the math will be there.
The math will come. But that's because I'm a behaviorist first. You know,
my background's special ed. It's emotionally impaired, and behavior plans
were the forefront of everything that I did

Patricia had previously been a special education teacher, and identified that experience as
her reason for being a behaviorist. When I later asked her what it meant to her that she
was a behaviorist, she replied:

My initial teaching experiences, I was a teacher of emotionally impaired
kids. So you did behavior plans on everybody, to the nuance of how many
times they said something, and then how do you increase that. So my
focus has always been, you get the behavior in order, then learning is more
optimal. I certainly understand and know the opposite of that is, you make
the learning so engaging, the behavior will take care of itself. But I think
when you're teaching cooperative learning and teamwork, at the very
beginning it has to be somewhat scripted, and it feels a little contrived, for
them to understand and have a picture in their head, what expected
behavior is. . .I think about the lesson. I think ahead of what is going to be
a problem in the lesson. How do I alleviate, or run interference with that,
or preempt it before we even get started? I am forever the person who goes
on the field trip having everybody in the circle here and we're having a
short intense conversation about my expectations of their behavior. Are
we clear? Can we shut our eyes and see ourselves sitting on the bus, with
our feet on the floor, and our knees forward? Everybody try that. See it in
your head. That's what I want you to see. So we do imagery, and I'll do a

110

conversation, and I'll do vocabulary so that they're clear. Are we clear?
Because I don't want you to be disappointed when we get on the bus and
the natural consequence is that you cannot join the group. So that, my
feeling always is be very up front about what is going to happen that is not
a surprise. It's not a reaction to the moment like this, and they have
heightened awareness of what is expected of them.

To take from the emotionally impaired classroom to this kind of
classroom you don't want it as confining as I had to be in the other kind of
classroom. But on the other hand, I use the same, similar techniques. I
forewam. I give them a model. I want them to visualize it and see
themselves walking through what they're going to do. Now get up and
everybody do it.

The model Patricia described may not align with other conceptions of what behaviorism
is. For her, it is about giving students a clear image of the expected behavior and then
expecting them to behave in the ways she has stated. It is unsurprising, then, that
Patricia's main stated goal for using CI was to help students to learn behaviors for certain
situations.

In the teacher study group, Jonathan had been wondering about the usefulness of
assigning roles to students. In response, Patricia said:

I'm going to play devil's advocate for this. If we think in the real world as
adults, who sit in a team meeting, the roles blend themselves. You tend to
be the person in the meeting that you're most comfortable with. You may
have a different, that you want to be the cheerleader and it seems to
involve itself. But one of the things that we frequently hear business and
real world people say is we're graduating people who do not know how to
work in a group. The struggle of, let it go and leave it the way it is, and the
natural leaders will continue to be natural leaders, the children that are
hitchhikers will continue to be hitchhikers and stand in back letting other
kids go with it. Even though you want to teach the mathematics part, the
role part is integral in that because when else do they have the opportunity
to really get into something different than who they are. Who they
naturally are. So, if I have one child who's the natural cheerleader, but
never volunteers to clean up and never volunteers to pack up, I'm not
giving that opportunity for them to begin to learn that role. . .Now, in an
ideal world, starting at first grade I emphasize the role, but we still do
math, but the math is not the emphasis. By the time you get to fifth grade
you deemphasize the role because you don't need, to do it as much

lll

because you've blended those kinds of things. But if they've never done it,
then they're never going to have it.

Patricia stated that her goal for teaching roles to students is to teach them behaviors that
are valued by the business world, which the students themselves might not gravitate
toward or volunteer for. She justified her continued use of CI:

I'm amazed. Within the first month they are independent. . .They're

responsible for their own clue. They have to get along with each other.

They have to listen to each other. . .The group tends to monitor

themselves. It is pretty amazing to sit here and see them do that. The gym

teacher came in. the gym floor was being finished so that room across the

hall is empty. He's in between classes so he came in, and he says, "I have

to tell you how amazed I am."
Patricia continued to evaluate her use of CI to see whether it was helping her to meet her
goal of teaching behaviors that are privileged in school and business. Patricia was pleased
that, by implementing CI, she had taught her students to behave in ways that impressed
another teacher. It is true that, in the CI course, teachers have a lot of opportunities to
learn about a variety of instructional strategies and interventions. By color-coding and
filing materials, Patricia was able to identify as an organizer of materials. By giving each
student a role and enacting other CI instructional strategies, she was able to identify as a

behaviorist and organizer of students in her classroom. This was a reason for Patricia to

continue to implement CI.

Discussion
The data in this chapter speak to the second research question for this dissertation,
which was, " How do teachers' ways of identifying become obstacles (or aids) to teachers'
attempts to change their teaching practice? How do they make sense of the obstacles and
overcome them?" I have described ways in which changing their practice necessitated

some teachers' integrating new ways of identifying with previous ways of identifying. I

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have described how this identity work was often an obstacle, and how some teachers
worked to overcome the obstacle.

It has been repeatedly argued in the literature that teacher change is difficult, and
that PD events fail to effect change in teaching practice. Often the result is interpreted to
mean that teachers are unable to learn, and/or PD is ineffective. This interpretation of the
data overlooks the complex processes involved in learning reducing change to the simple
decision to implement new practices. One longstanding explanation for the stability of
teaching practice is the apprenticeship of observation theory (Lortie, 1975), which says
that teachers teach in the ways they were taught. However, this explanation does not
account for instances where teachers do learn and sustain new pedagogies. It appears that
the identity work that a teacher must undertake in order to include new teaching
pedagogies into her practice can be significant and difficult for some teachers. As
evidenced in the cases of Joanna, Jonathan, and Glynnis, changes in practice often require
changes in identifying, which can be painful.

As sub-question for this particular chapter, I asked specifically, "What are the
difficulties inherent in changing teaching practice that arise from the teachers' integration
of new mathematics teaching identities with their older perceptions of themselves?" and
"In what ways did teachers' changing identities allow them to persist in making changes
in their mathematics teaching practice despite incredible challenges?" I will address these

two questions in the next two sections.

Difficulties of Identity Integration

Identity integration happened very differently with each. In fact, in the case of

Patricia, I did not see evidence of difficulty. This dissertation, however, has opened up

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possibilities by presenting some varied ways in which this might happen. None of these
ways should be expected to happen in every teacher or in every PD. And it is likely that
there are many more ways in which the identity work necessitated by teacher change may
play out in a teacher's practice.

However, it is clear that adopting teaching practices that do not align with a
teacher's prior self-perceptions can be uncomfortable enough for a teacher to question
whether implementing a new practice is a good idea. In such a case, how do teachers
make sense of their discomfort? It seems that no teacher in this study articulated this
discomfort explicitly as an identity issue, although I argue each case can be described as
such. Interestingly, the teachers located the source of this discomfort in at least four
different locations: 1) their own performance of CI, 2) the students, 3) CI principles, or 4)
external factors, such as administrators, colleagues, or other aspects of the school culture.
For example, Joanna responded that she thought her students saw her as not giving
enough (locating the problem within herself), wondered whether CI was right for her
students, who she described as having "strong personalities" (locating the problem with
her students), and wondered whether CI was worth the time, given the results she was
seeing (locating the problem with the pedagogy). Glynnis, also, responded that Cl was
difficult because she had not had the energy to pay enough attention during the PD course
(locating the problem with her self), she had 34 students in her class (locating the
problem with the students and with the administration and school), and that Cl was too
prescribed and rigid (locating the problem with the pedagogy). Jonathan stated that the
difficulty for him came in changed relationships between himself and his colleagues

(locating the problem with himself, colleagues, and the school culture).

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At first glance, it appears that teachers may be trying to play "the blame game" in
terms of why CI was difficult to implement. That is no surprise. In mathematics
education, researchers frequently play this game. If I had conducted a pre/post study of
teachers' practices before and after the CI workshop, and had come for a post-workshop
observation when Joanna was giving out pages of directions, I may have been tempted to
play that game as well. I may have concluded that something was wrong with the PD, or
that something was wrong with the teacher, or the school, the available resources, etc.
However, reframing this scenario in terms of identity has given me new ways to explain
teaching practice and the outcomes of PD. Instead of placing blame, I have been able to
see the identity-work as part of a longer, ongoing learning process.

If teachers had this same lens available to them, both for looking at students and
themselves, what would change in their teaching? Understanding this identity-integration
process better will allow professional developers to better provide support for teachers as

they anticipate and work through these shifts in identity.

Performing Identities

Additionally, Cl allowed each of these teachers to perform some aspect of their
identities even more fully in the classroom. It allowed each of them to be more of the
type of teacher they wanted to become. It is interesting that this happened for each of
these four teachers, who all had very different ways of identifying to begin with.
Somehow, the scope of CI as a pedagogy is to be broad enough that teachers with very
different ways of identifying were able to perform different ways of identifying by

implementing CI.

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In a textbook used in many elementary mathematics methods courses (Van de
Walle, 2006), teachers read that there are two main ways to adapt instruction to meet the
needs of learners in classrooms heterogeneous in terms of ability: using tasks with
multiple entry points or differentiated tasks. Tasks with multiple entry points are those
rich enough to offer a variety of ways to participate. Differentiated tasks are those for
which the same problem may be done in different versions for students with different
abilities. Similarly, in the CI course, teachers were taught that a "groupworthy" task is a
task that draws on many different student abilities. Because it draws on a range of skills,
no one student in each group would be expected to be able to complete the task without
the skills of others. (Cohen, 1994).

In those terms, the CI course itself may be viewed as a PD course or a pedagogy
with multiple entry points. That is, teachers with a wide variety of ways of identifying
found ways to participate in the course and/or implement the principles and practices of
CI in their classrooms. Four teachers, heterogeneous according to how they identified as
teachers,were all motivated to persist through the challenges of implementation because
implementing CI allowed them to be more of the type of teacher they wished to become.

Understanding the ways certain teaching practices afford teachers' the
opportunities to perform their identities will help professional developers design PD
experiences that make these connections explicit and will resonate with various teachers'
self-perceptions, thereby increasingly the likelihood that the implementation of the

teaching new practices will be sustainable over time.

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CHAPTER 6: GENERALIZED NARRATIVES: HOW-I—T EA CH STORIES AS
INTEGRATED IDENTITIES

Thus he grew in body and soul, and with him his clothes seemed to grow
and arrange themselves; coat sleeves got longer, cuffs appeared, and
collars got less soiled. Now and then his boots shone, and a new dignity
crept into his walk. "Of the Coming of John," (DuBois, 1903/1994).

I believe you're doin' really well at the college now, eh? I hardly
recognized you. You really look the part. Doesn't she, eh? Look the real

student. You'll be on drugs and demonstration next, ah hah. Educating
Rita (Cartlidge & Gilbert, 1983).

Overview

In "Of the Coming of John," Educating Rita, and in my own story, learning
resulted in different ways of speaking, dressing, and relating. As the teachers in Phase II
of my study were implementing CI, they began to tell a new kind of story as well—
stories of themselves as "Complex Instruction teachers." Teachers sometimes told a type
of story that I refer to as how-I-teach stories—general narrations of how they teach and
students respond (as opposed to narrations of a specific event). Crespo & Juzwik (2006)
studied the performative aspects of teachers' expository stories, which are final draft
stories told without invitation for dialogue. How-I-teach stories are a subset of expository
narratives. At various times and in various ways, they began to tell how-I-teach-math
stories specifically about how they use CI to teach mathematics. These narratives
function to claim some new teaching practices as the teachers' own—to identify them as
"CI teachers."

In the results section of this chapter, I present one how-I-teach-math story about
teaching mathematics with Cl from each teacher and report on the timeline for which
they emerged in the teachers' storytelling. Three of the teachers told such stories about

how they go through a CI lesson or several parts of CI lessons. However, in the case of

117

Glynnis, she only told this type of story about one small part of a lesson. In the discussion
section, I theorize these stories as narrative performances of the teachers' emerging CI
teacher identities—that is to say that some teachers began to identify as "CI teachers"
through the reification of experience into integrated (expert) identities.

In this chapter, I am telling Act III of the story of teachers' implementation of CI
to teach mathematics. A graphic representation of the position of this chapter in the plot

of the overall story is included as Figure 7. The scope of this chapter is indicated in bold

 

 

 

 

 

outline on the figure.
Act I: How the Act 11: Stories of
teachers identify implementation of CI in A“ 1113 C1
(life stories) ' math Teachers
Teachers decide to Decision to PCTSiSt
implement C1 (or with implementing CI

not) (or not)

Figure 7. Chapter 6—Act lII

Generalized Narratives

Patricia

As stated in Chapter 3, at the beginning of Phase H Patricia had been using CI for
one school year. When I walked into her classroom for her first individual interview, she
began to talk about CI before I had an opportunity to ask a question. I interrupted her and
asked her if I could turn on the audio recorder. When the recorder was on, I said, "So tell

me what you were going to tell me." She began:

118

I use the Group Solutions book7. I walk through it from the very
beginning. Actually, I don't walk through it from the very beginning. I
start with jobs. I describe the jobs, and we model the four different
jobs. The jobs are not listed as they are in the [the CI course

materials]8. . .The information on the back tells them what to do, and the
model conversation that they can have is still the same on the back. But
on the front I just gave the little picture, because first graders know that, so
that everybody's involved and understands. The jobs rotate. The groups
actually rotate too. About every six weeks I change desks, so they may
end up with different people. Some kids say, "Well, I was material
manager two times in a row," but it all works out in the end. . .We review
[the roles] almost every week. And there'll be an emphasis one too. Like,
I'll say, "The questioner is the person who begins, and then we go
clockwise." That way, we talk more about the questioner and we ask the
other people what the questioner is doing so that they're more aware of, a
little bit of observation, with it. . .At the end, of the activity we all get
together and we talk about how we solved things

This is a how-I-teach story about using CI to teach mathematics. At times, Patricia tells
narratives using present tense terms (I walk... I describe...) indicating how she always or
usually is when doing CI lessons. Remember that direct identifying statements are
reified; that is, that they are spoken of as detemporalized objects (Sfard & Prusak, 2005).
In this story, Patricia is doing something similar with her verbs. By using present tense
verbs, Patricia has detemporalized her use of these practices, thus identifying herself as
the kind of teacher who routinely carries out these practices as a part of teaching. I.e., she

began to identify as a CI teacher.

 

7 . . . . .
This book of group tasks was given to teachers who partrcrpated 1n the Complex
Instruction professional development course (Goodman, 2000).

In using the word "jobs", Patricia is talking about what was called roles in the CI
course. Roles are an important instructional strategy in CI. Students are given roles that
are specifically designed to engage them in the mathematics at stake in a groupworthy
task. The roles introduced in the course are: the resource monitor, recorder/reporter,
facilitator, and team captain. However, Patricia designed a new set of roles for her
students, in order to make them more usable in her first-grade classroom. The roles she
designed are: the questioner, the includer, the material manager, and the cheerleader.

119

Jonathan

Jonathan had also been teaching mathematics using CI for a year when this study
began. Jonathan told he story that follows during the first teacher group meeting. As
stated in Chapter 3, teachers were asked to come prepared to share stories of using CI to
teach mathematics. In the group discussions, teachers took turns sharing these stories
(See Appendix C). In this meeting, Jonathan was the second teacher to share a story.
(Joanna had been first. She told of some students' who violated the groupwork norms

during a recent CI math lesson, and the teachers discussed possible interventions she

could use should it happen again9). When Jonathan took his turn at storytelling in the
group, he said:

I might give the group, inside a laminated or clear sheet, this page with the
other one to the back, so they can see them both. They're on a clear sheet
so they don't really write on them. One of the things I've been working
with is this idea of putting the math skills that that they're going to be
working on the page, and then the groupwork skills that they're going to
be working on below that, and then some of the materials they might
need, and then I put the directions there. . .To back up and tell you howl
do this. I don't do a task every week. I don't always do a task every
month. I tend to do a task toward the end of a unit, because for me it's a
really nice way to bring together a lot of new skills they've learned and
share them out. Sort of showcase. Look at this, we've done, we can do this
now. We can do this. We can do this. And then other people will go, yes.
Yes. No. Yes. You know? And so I find it very powerful to do it like

that. . .They're group one. I don't like names. They're strictly groups one to
seven. Everybody writes their name, whatever seat they're on. And as they
finish the task they are required to call me over. They don't know who in
the group's going to have to explain it. And so I'm, you know. . .I have a
little stamp, and I'll stamp [their task card], and then they can go on to the
next task.

Notice that Jonathan moved back and forth between generalized and specific talk

about his teaching in his first how-I-teach story about using CI to teach mathematics.

 

Groupwork norms and interventions are instructional strategies taught in the CI course.

120

Like Patricia, much of Jonathan's language is in the present tense when he refers both to
how he teaches mathematics with CI (1 don't. . ., I tend to. . ., I find it...) and what his
students do (They don't know. . .,everybody writes. . .). Again, by speaking in this way,
Jonathan identifies as a teacher who does CI, and who does CI in a certain way. He
identifies as a CI teacher.

It is interesting that Jonathan did not tell a how-I-teach story about CI in our first
individual interview, but did tell this type of story in the first teacher group. Within the
teacher group, there were teachers who had only been using CI for a couple of months
and teachers who had been using it for over a year (including Jonathan). One way that
Jonathan and Patricia identified themselves as the old-timers to CI (Sfard & Prusak,
2005) was to tell stories that identified themselves as people who use CI and who use it in

an individualized way.

Joanna

Joanna was one of the teachers who had been teaching mathematics using CI for
only a couple of months when Phase I began. In the first individual interview, she said:

[The students] don't want to follow the rules like, make sure you've talked

this over with everyone in your group before you ask the teacher for

help. There's always, three, somebody from three different tables jumping

up that, you know, that impulsiveness they want to ask me it.
Joanna had used CI to teach mathematics a couple of times, and told this story that
included the type of generalized language that is a marker for a how-I-teach story.
However, it is not really a how-I-teach story but a story about how students respond to
her attempts at teaching mathematics using CI. She only spoke in such a generalized

manner about her students. She did not talk about herself in a generalized way, or identify

herself as a certain kind of CI teacher. She continued the story differently, however:

121

10 . . . .
We have talked about thrs srnce the beginning of the school year, we've
repeated it, since, the beginning of the school year.

When she continued the story to include herself in the action, she switched
back to speaking about the past. "[W]e have talked" sets her story in a
specific time. Her use of past participles indicates it has happened more
than once. However, she is not detemporalizing her behavior, so it is not a
how-I-teach story.

However, at the last data-generation session of Phase II, the round-table

- . . . ll
presentation, Joanna told a how-I-teach story about usrng CI in mathematics . I had

asked her if she would like to tell the group about a task. She said:

I start out modeling with tasks from Group Solutions T 0012, the other
book. It gets my kids accustomed to the norms and roles by using those
tasks. . .But I try to use other tasks too. I try to use tasks from Balanced
Assessment. Something in the classroom that I'm curious about are the
norms, what we just call group rules that I have in the classroom. . .those
are, some of those things are things that we instill throughout the school
day in everything that we do. They're very familiar with them but I keep
them posted when we do any kind of a [CI] task...The kids, in the sharing
process, they'll raise a hand and they'll ask a team, "What do you mean by

 

For a discussion of the use of the pronoun "we" by teachers, see Pimm, 1987.

1 Teachers had been asked to share their work with CI in mathematics with new
instructors. I began the session by asking everyone present to introduce themselves. Some
CI teachers began telling stories during their introductions. Next, I put a slide on the
board that stated the three important components of C1 (the slide had been used in the CI
course), said that I wanted to make sure the guests understood the terms we were using,
and asked the teachers to explain the components. Patricia began by telling a story about
how she deals with status (one of the components) and uses instructional strategies
(another component) in her classroom. Jonathan followed by telling a story about how he
plans tasks (the third component on the slide). Some of the teachers (primarily Patricia
and Jonathan, but sometimes Glynnis) continued telling stories (both how-I-teach stories
and specific stories). When there is was a slight break in the conversation I said, "Joanna,
do you want to tell us about a task?"

12 This book (Goodman & Kopp, 2007) was distributed during the CI math PD course.

122

that. 1 really don't get it. Can you explain that to me?" There's just a lot of
dialoguing that's going on...I rotate [the groups] in my grade book. I go by
whatever their seating arrangement is. I just pass the cards out to each
group and I just put a sticky note on and I just say, "This time you're the
organizer. This time you're the captain. This time you're the
recorder/reporter." I just pass them out. They just hold on to them and it
helps give little reminders about what their responsibility is within their

group.

Joanna told a how—I-teach story about using CI in mathematics. As Patricia and Jonathan
did in their how-I-teach stories, Joanna spoke of herself primarily in the present tense,
indicating that these are things she continues to do. While she had talked this way about
her students at the beginning of the data-generation time period, she had not spoken this
way about herself.

Again, it is interesting that the context and the audience are different here than in
my individual interviews with her or in the CI teacher group meetings. In this context,
with instructors who had not taken the CI course, Joanna identified as a CI teacher who
generally used CI in a certain way. She identified as a CI expert in relation to her

audience for these stories.
Glynnis

At the round-table session with instructors who had not used CI, while Joanna
was sharing materials she used to teach math with Cl, she mentioned that she created
different names for some of the roles than teachers had seen in the CI course. Glynnis
interrupted and suggested that the names for the roles in the CI course were problematic.
She said:

I try to tell the kids that everybody has a role and none of them are more

important than the other. That they each are all very important

roles. Because otherwise they're like, "Oh, she wants me to be the

captain," and the status is already there. Because the 4th grade, especially,
they've had years and years of it.

123

Glynnis only told this one short segment of a story of teaching mathematics with CI in
which she reified her actions. In this story, Glynnis spoke in generalized terms about how
she deals with one possible status problem she sees with the roles. Glynnis did not tell
how-I-teach stories about how she did any other part of the process, including preparing
tasks, organizing her classroom for instruction, or implementing instructional strategies.
Again, it is notable that she only told this type of story about CI teaching in a context
where she was someone who had used CI and there were people in the audience who did
not. In that context, in this small part of a story, she identified herself as a person who
typically deals with this issue in this way. In doing so, she has identified herself as an

expert in dealing with possible conflicts that might come about because of the role cards.

Discussion

I have argued throughout that these teachers' narratives about their mathematics
teaching are related to their decisions to implement C1 (or not) and have explicated the
identity integration work that had to happen in order for them to continue teaching math
with Cl. In this chapter I have shown a particular type of narrative that teachers told
about their use of CI in mathematics—the how-I-teach stories. These narratives function
to claim some new ways of teaching as their own and reify the teachers' experiences into
what I refer to as integrated (expert) identities. In the discussion that follows, I posit how-
I-teach stories as markers of teachers who identify as "CI teachers" with integrated

(expert) identities.

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Integrated (Expert) Identities: What it Means to Identifir as a CI Teacher

The Reification of Experience." Present- Tense Verbs

As stated in the literature review, direct identifying statements are statements
about a person that are reified, endorsed, and significant. A statement is said to be reified
if talk about actions or processes have been replaced with talk about detemporalized
objects. For example, "I gave the students a problem solving task" is not reified because
it talks about an action that a teacher has taken, but "I am a problem-based teacher" is a
reified statement, because the teacher is then describing a characteristic of herself as true
regardless of time (Sfard & Prusak, 2005).

The stories in this chapter are not direct identifying statements because in these
stories the teachers have not made succinct, reified statements. Reification is defined as
turning actions or processes into objects (Sfard, 2008). In these stories, the teachers are
not talking about objects. They are still talking about actions; however, they are not
talking about specific actions. They are portraying these actions to be "the way they act"

regardless of time. For example, instead of saying, "We modeled the first four jobs,"

Patricia stated, "We13 model the first four jobs." That is, she identifies as the kind of

teacher who models the roles with her students on a regular basis, as opposed to having

 

Patricia may be using the word "we" here for a number of reasons. David Pimm (yr)
has argued that this is sometimes done when speaking about relationships with children,
in order to soften the blow of the power differential when an adult is doing something to
a child. However, in this case, I believe that Patricia has used the pronoun "we" because
she brings three students to the front of the class to help her model the different
groupwork roles. Therefore, the "we" refers to herself and three students.

125

 

done that at one specific point in time. Joanna could have identified even more strongly
had she said, "I am a modeler" (or "we are modelers").

If we conceptualize narratives on a continuum of (as opposed to simply narratives
that are reified and those that are not), it is logical that a teacher's narratives would move
along that continuum in the process of her identity work. E. g., she might first say about a
CI lesson "I modeled", then later speak of her CI math teaching practice saying, "1

model," and finally say directly, "I am a modeler."

Integrated Identity

I have argued that persisting in the implementation of CI required the teachers to
integrate new ways of identifying as teachers with their prior ways of identifying. How-I-
teach stories about CI in mathematics are one indicator that some teachers were able to
accomplish this integration of identities.

The case in which identity integration is most evident is Joanna. Joanna struggled
to integrate her identity as a teacher who gives with a designated identity as a teacher
who steps back. However, this designated identity was one that Joanna questioned, even
as she worked toward it. One event that helped her to integrate these two identities was
seeing a child teach another child about symmetry while Joanna was stepping back. That
event was the catalyst for a shift in identifying for Joanna. It enabled her to see that, as a
teacher who stepped back, she was also being a teacher who gave a new resource to
students—the resource of each others' competencies. At the same time, Joanna continued
to modify her use of CI in order to align it with her ways of identifying. As a part of her
struggle to integrate her identity as a teacher who gives withher newer identity as a

teacher who steps back, she incorporated modeling and longer written instructions into

126

her use of Cl. Eventually, she decided to start the year with some tasks from Group
Solutions Too (Goodman & Kopp, 2007), a book she received at the CI course, which she
felt had tasks that were shorter, easier, and better for teaching students how to work as a
group than some other groupworthy tasks she had used. By the end of the Phase 11, she
was telling how-I-teach stories about CI in math class. In one story, she stated, "I start out
modeling with tasks from Group Solutions T 00". In doing so, she is identifying as a
teacher who does the beginning of her year this way. She is also performing her newly
integrated identity: the teacher-who-gives and the teacher-who—steps-back. Because these
types of narratives are indicative of this type of integration of ways of identifying, I call

them integrated identities.

Expert Identities

In telling how-I-teach stories about their use of CI to teach mathematics, the
teachers are identifying as having expertise in one or many areas of teaching with C1.
This is not to say that an outside observer would necessarily label each teacher as an
expert CI teacher. I came to see these particular types of expository narratives as expert
identities because of the contexts in which the stories were told. Patricia and Jonathan,
the two teachers in the group with the most experience using CI to teach math, told how-
I-teach stories about it at the beginning of Phase 11. They used these stories to identify as
the teachers in the group with expertise to share. Joanna and Glynnis told how-I—teach
stories about using CI in math when the audience was a group of instructors who had not
yet taken the CI math PD course. Therefore, in that context, Joanna (and to a lesser
extent, Glynnis) identified as having some expertise to share with people who were more

novice.

127

CHAPTER 7: ZOOMING IN: IDENTIFYING AND MATHEMATICS-FOR-
TEACHING IN NUMBER LINE NARRATIVES

Up in Johnstown, at the Institute, we were long puzzled at the case of John
Jones. For a long time the clay seemed unfit for any sort of moulding. "Of
the Coming of John," (DuBois, 1903/1994).

Overview

Mathematics teacher educators have long been concerned about some elementary
teachers' lack of mathematics knowledge and willingness to identify as "not math
people." As I interviewed teachers about the mathematics lessons they were teaching
using CI, I struggled to get some of them to tell me about the mathematics at stake in the
lessons. When I pushed the teachers to talk about mathematics, Patricia, Joanna, and
Glynnis would often provide a sentence or two at the most, and would repeat it when I
asked them to elaborate. Only Jonathan chose to speak about mathematical content. He
would often talk for several minutes about students' understandings of mathematical
concepts with little or no prompting.

At first, this bothered me terribly. Would my dissertation only serve to reinforce
stereotypes of elementary teachers as not "mathy" enough? Would it reinforce gender
stereotypes, as I described Jonathan's enthusiasm for mathematical content and the
women's reluctance? And even more disconcerting was my fear that maybe the women in
my study had less sophisticated knowledge of mathematics than Jonathan, or that they
were harboring some bad beliefs about who and what mathematics was for.

However, looking at the teachers' discourse through the lens of identity gave
another plausible explanation for the differences in teachers' talk. My theoretical

framework allowed me to look at the teachers' different choices about mathematical

128

 

discourse as different ways of identifying. Some teachers identified as having more
mathematical competence than others. Whether or not a teacher chose to articulate
mathematical concepts explicitly can be seen as a matter of how the teacher positions
herself in relationship to the discipline. In choosing not to speak at length about
mathematical content, Patricia, Joanna, and Glynnis were choosing to position
themselves at a distance from the content (and in proximity to something else). However,
whether the teachers identified as competent mathematics teachers did not entirely
explain their choice to talk about math content or not. Joanna identified as a "math
leader" in her school district, and yet she rarely chose to talk about mathematical content.
At the same time, I was operating under the assumption that all four teachers had
some kind of knowledge of mathematics for teaching, because they all seemed like
reasonably competent teachers. They often talked about mathematics pedagogies in ways
that sounded reasonable, and they had all successfully completed a number of
mathematics tasks during the CI course. But if they were not demonstrating this
knowledge by talking to me about mathematics, how could I access it? And what was this
perplexing relationship between teachers' identifying and their mathematics-for-teaching
competencies? Davis and Simmt (2006) argued that much of teachers' knowledge of
mathematics-for-teaching is tacit. Therefore, I hypothesized that there might be clues in

the form of the teachers' talk, to knowledge that was absent from the content of the

4
teachers' talk (Hymes, 1972)l .

 

l4 . . . .
The form/content dichotomy rs important here because it attends to the oral nature of

the storytelling.

129

In order to examine these questions further, I did a close analysis of the form of
excerpts from two stories. Both Glynnis and Jonathan had taught versions of the same
lesson on fractions and number lines in their fourth and fifth grade classrooms. Students
in both classes were given the task:

Show where the following numbers fit on the number line. Explain how

you know.

Each teacher gave their class a different list of numbers between zero and one, in decimal
form, with the task. Jonathan asked the students to draw their number line on a blank
piece of poster paper. Glynnis gave each group of students a sentence strip (piece of
tagboard about 3 inches by 3 feet) on which was a line segment with the numbers 0 on
the far left and 1 on the far right. She instructed students to write the numbers on that.

Because two teachers used the same lesson, I had an opportunity to look at two
teachers' identifying and knowledge of mathematics-for-teaching in two stories with the
same mathematical content. From each story, I chose an excerpt in which the teachers
were attempting to help students understand the metaphor of numbers as points on a line,
and both teachers were illustrating the number 1 on the number line. I analyzed the
stories for how the teachers were identifying in the stories, as well as how they were
performing knowledge of mathematics-for—teaching. I compared who the teachers said
they were as math people and math teachers with how they said it. In this chapter, the
narratives function to perform knowledge of mathematics-for—teaching that may or may
not be explicitly recognized by the storyteller.

In this chapter, I first report on how Glynnis and Jonathan identified themselves in

the form of these two excerpts from stories of classroom practice. Second, I report on

130

how Glynnis and Jonathan perform knowledge of mathematics-for-teaching
competencies in the form of their storytelling. To conclude, I discuss the interlinked
nature of identifying and mathematics-for—teaching, mathematics-for-teaching as a
culturally-laden term, and implications for moving toward inclusive approaches to

mathematics teacher education.

The Two Stories
In this chapter, I analyze excerpts from two stories, one told by Glynnis and one
told by Jonathan. Glynnis and Jonathan taught the same CI math lesson, in which

students were to place numbers on a number line. Glynnis told me the story about how

 

the lesson had failed because her students did not understand number lines. However, she
went on to tell me the following story of an lesson that took place two days after the
failed lesson:

Actually one of the women that was so upset, recognizing that from here
to here was the one whole. You know, 'cause I think, we were doing it
yesterday and they were ordering one and, one, it was all one and, one
and, and it was still, hundredths and stuff like that. So I says, "Now, let's
write these on the board under each other. Place the decimal here. What do
you notice? What do you really notice? You know. What helps you?" So,
they notice that, you know. "Ok, if I look at the tenths place, it's real easy.
to get it." And so I says, she had went up to plot. She immediately drew a
number line. And I was like, "No you're not gonna draw the number

line." That was really funny. So she drew the number line, which of course
is one of the grade level expectations they're supposed to be able to do. So
I was like, "Whoa." So she drew the number line. And she began to do a
landmark number. Ok, 'cause she was plotting one and three, one and
three tenths lets' just say. I don't remember what it was. So she drew half
here [gestured as to the number line]. And so she was beginning to figure
out. And then she goes, "Oh. It would have to be over here." I says, "So
you're figuring out that between here and here [gestures as to two points
on the number line] represents a whole, and the whole has to be divided

into what?" And so it kind of was a breakthrough and they were kinda
like, "Oh."

131

Jonathan's told a story about his use of the same CI mathematics task (which
Jonathan had originally designed, and Glynnis later adapted). He gave the same number
line task as Glynnis; however, he gave it with a couple of other related tasks. One of the
other tasks was about measurement. Jonathan reported on all of these tasks:

Jonathan: The next [group's work] was very interesting because we had
not done, measurement. And so this was, this had to be based on
what they knew about decimals. And I had meter sticks. And I held
up the meter stick. And I said to them, "This is a meter stick. It's
one meter. So this is the whole."...The number lines, were the most
interesting one. So lets talk about the number lines one...

So the number line, task three, asked them to put the following set
of decimals, from largest, on the number line. Yeah. It had a zero,
and it had a whole, and it had a half marked on it so. Well, it was
very interesting. Because they, what I found was that was very
easy for them, because they knew enough about what a tenth was
and about what a hundredth was, that they could easily put those in
order. No problem. And, you know I, that's good. That's what they
need to be able to do. But take a look at my number lines. And,
you'll see that, oh. ok.

[Looking at one group's work.] Here's actually a number line. It's
actually one of the few that are accurate. And the reason is because
none of them, many of them, are not scale number lines. Ok. Like
here is a great number line. I mean here is a great table that shows
them in order. They're nowhere right where they're, they're. Like,
here's point five which really isn't in the middle. I mean there's no,
and one hundredth is just as far away from zero as two. You know,
in terms of a scale. So they've got no scale. They're not accurately
placed. And I didn't realize they were really doing that until the
end. Yeah, it was very eye opening for me, because I guess I
hadn't done enough with number lines, about what a number line
really means. It's not numbers in order it's places on a line. So it
was very interesting. I think the kids learned a lot that day when
they saw.

[Looking at another group's work, as represented in Figure 8.] This
one, what they did. They took a hundred centimeters. They lined
them up. And they put, lookit, they labeled one hundredth in the
first square.

132

 

101111-11llIlIIIIIlllll-le

Figure 8. The Students' Number Line Work

And they went to the twentieth [square] and made it two

tenths. And forty—ninth. They didn't put the half there. So actually,
what they did is not said where the spot is but they said this square
represents, one [hundredth]. So they thought of it more as a
fraction, right?

Joy: Well, I think,

Jonathan: One out of a hundred. I'm just now thinking about that. 'Cause I
was thinking. Yeah, they put it on the right square, but actually if
you put it on a number line, it would be right, there. [Points to the
line between two squares] So I think what they did is they thought
of it as one out of a hundred and that's one hundredth of the
total. This would equal twenty hundredths or two tenths. 'Cause
they're pointing straight at the square. So in a way there's still a
misconception about a number line when I look at this.

Identifying

Glynnis

In telling her story, Glynnis identified as a certain kind of mathematics teacher. At

first glance, she seemed to hedge a bit, performing some uncertainty about herself by

using words such as "I think" (line 4) and at times looking for affirmation from me by

saying "you know" (lines 4, 18, and 21), and minimizing the effect of this lesson by

saying it was "kind of a breakthrough" (line 52) and students "were kinda like, oh." (line

Participant and Spectator Roles

Because I am interested in how teachers identify, 1 was interested in examining

the different story-telling roles they perform. In the transcript below, I have categorized

Glynnis' story according to when she is performing the role of participant and the role of

133

spectator. When Glynnis is speaking as a participant, I have used normal text. When she

is speaking as a spectator, I use italicized text. I numbered each section (a new section

begins when the teacher shifted from participant to spectator role or back). I also

underlined the pronouns referring to people in the story, because pronominal shifts are

another indicator of changing roles or points-of-view:

1

10

ll

12

l3

14

Actually one of the women that was so upset, recognizing that
from here to here was the one whole.

_XQu know, 'cause I think,
fle were doing it yesterday and Ih_ey were ordering one and, one,
it was all one and, one and, and it was still, hundreds and stuff
like that. So I says, "Now, let'_s write these on the board under each
other. Place the decimal here. What do yo_u notice? What do you
really notice?

_Yo_u know.
What helps you?"

So, t_h_ey notice that, Lo_u know.

"Ok, if I look at the tenths place, it's real easy to get it." And soI
says,

s_h_e had went up to plot.

fl]; immediately drew a number line. And I was like, "No yo_u're
not gonna draw the number line."

That was really funny.
So _s_h_e drew the number line,

which of course is one of the grade level expectations
th_ey're supposed to be able to do.

So I was like, "Whoa." So sh_e drew the number line. And s_he
began to do a landmark number.

Ok, 'cause s_h_e was plotting one and three, one and three

134

tenths let_'s just say. I don 't remember what it was.

15 So she drew half here [gestured as to the number line]. And so fie
was beginning to figure out. And then s13 goes, "Oh. It would
have to be over here." I says, "So m’re figuring out that between
here and here [gestures as to two points on the number line]
represents a whole, and the whole has to be divided into what?"

16 And so it kind of was a breakthrough and flex were kinda
like, "Oh. "
Glynnis moved back and forth between positioning herself as a participant and as a

spectator to the action. However, she constructed herself as a participant more than she

 

constructed herself as a spectator. E

Pronomial Shifts. One interesting pronominal shift occurs in section 3 when
Glynnis was performing the role of participant. She switched from "we" to "they",
saying, "We were doing it yesterday and they were ordering one and..." At the
beginning of this statement, she used the pronoun "we", probably to refer to herself and
the students together. However, in the same statement she stated that "they" (the students)
were doing the mathematics.

Another interesting pronominal shift occurred in section 7. In that section,
Glynnis shifted from talking of the students as a group (they, you [plural]) to speaking of
a student as "I" when she acted out the role of the student ("If I look at the tenths
place. . ."). She did this although she was referring to more than one student, because she
was describing what "they notice" (section 6).

Constructed Dialogue. One way Glynnis identified as a participant is through the
use of constructed dialogue. A look at the constructed dialogue in this same excerpt of

speech further highlights Glynnis' identification in relation to mathematics teaching. In

135

order to highlight her use of dialogue in telling this story, I reorganized the same
transcript. This time, I indented and italicized the dialogue. Sometimes Glynnis is acting
out the role of the students' voice in the dialogue, and sometimes she is acting out the
teacher's voice. I distinguished between these two points of view by also underlining the
dialogue in which Glynnis is performing the role of herself. This second arrangement of
the transcript is included here:

Actually one of the women that was so upset.
Recognizing that from here to here was the one whole.
You know cuz I think we were doing it yesterday
and they were ordering one and, one, it was all one and,
one and, and it was still, hundredths and.stuff like that
so I says
"Now. let's write these on the board. Under each other
place the decimal here.
What do you notice. What do you really notice."
You know.
"What helps you, "
They notice that. You know.
"Ok. If I look at the tenths place, it's real easy. T 0 get it. "
And so, I says.
She had went up to, to plot.
She immediately drew a number line.
And I was like.
"Noyou're not gonna draw the number line. "
That was really funny
so she drew the number line,
which of course is, one of the grade level expectations they're supposed to be able
to do.
so I was like
"Whoa. H
So she drew the number line.
And she began to, do a landmark number.
Ok cuz she was plotting one and three, one and three tenths lets' just say
I don't remember what it was.
So she, drew half here.
And so she was beginning to figure out.
And then she goes,
"Oh. It would have to be, over here. "
I says
"Soyou'reflguring out that between here and here represents. A whole,

136

 

and the whole has to be divided into what?"
and so. It kinda was a breakthrough and

they were kinda like.

"Oh."

Arranging the transcript in this way highlights the way in which Glynnis switches
back and forth between narrating and role-playing. Looking at just the segments of
dialogue, the first thing that is notable is that, Glynnis shifts between dialogue where she
is the speaker and dialogue where one or more students is speaking, alternately acting the
parts of herself and her students. As she does so, she simultaneously constructs herself,
the discipline of mathematics, and her students in particular (though not entirely

consistent) ways. In Table 5, I highlight the ways in which Glynnis' constructed dialogue

constructs students and herself.

137

 

Table 7. Constructed Dialogue in Glynnis' Story

 

 

 

 

 

 

 

 

 

 

Role Dialogue How it constructs How Glynnis
students identifies herself
Glynnis now. let's write these on the Capable of As an arranger of the
board. Under each other understanding environment and an
place the decimal here. mathematical encourager of
What do you notice. What do concepts through students
you really notice. observation
What helps you.
Students Ok. If I look at the tenths Capable of easily As a teacher who
place, it's real easy. To get it. grasping concepts if puts students at case
they are presented (easy). Also a
correctly teacher who
understands her
students and their
thinkin .
Glynnis No you're not gonna draw the As independent, As a teacher who
number line. capable of making allows students'
connections ideas to take center
Glynnis doesn't stage, even when it
intend them to goes against her
make at the time original intentions,
and even when it
makes her
uncomfortable.
Glynnis w_h_Qg Surprisingly Appreciative of
capable. students'
contributions to the
mathematical
understanding of the
class.
A student Oh. It would have to be, over As a problem- As stepping back and
here. solver allowing a student to
come to her own
' realization
Glynnis so you’re [raring out that As people in the As a provider of
between here and here process of figuring more formal
represents. A whole, things out on their language, as a
and the whole has to be own, but still summarizer who
divided into what needing a more articulates students
knowledgeable understandings that
other to provide are not yet formal
formal language
Students Oh. UnderstandirL

 

 

 

 

I38

 

In constructing and performing this back-and-forth dialogue, Glynnis is
identifying as a certain kind of teacher—one that encourages dialogue and includes
students' voices as well of her own. Had Glynnis only acted out the role of herself in her
dialogues, we might conclude that she positions herself in a very different way in
relationship with her students. By performing the role of students' voices in this story,
Glynnis is positioning herself with them in a certain way. She is identifying as a teacher
who empathizes with her students. However, it is also interesting to note that she allots
herself more dialogue than she allots to her students. Glynnis is indicating that, in this
lesson, her voice is the one that is heard the most.

In the first segment of dialogue, Glynnis says, "Now, let's write these on the
board, under each other. Place the decimal here. What do you notice? What do you really
notice? What helps you?" In constructing her voice this way, she does several things. She
constructs students as able to figure out important mathematical concepts through
observation. She constructs mathematical concepts as being available through sensory
observation. She constructs her own role in students' learning as being to arrange the
numbers in order to make the important concepts more easily observable, and then to
challenge and encourage students to do the work of noticing.

In the next line of dialogue, Glynnis plays the part of the student at the board. In
doing so, she shifts the point of vilew of the story to that of the students' experience in the
story. She depicts the student as saying, "Ok. If I look at the tenths place it's real easy to
get it." She constructs the student as being able to easily grasp how to order numbers if
the teacher points out the important place to look. She identifies as a person who can

affect students' mathematical success by pointing out important things to them. She

139

identifies as a teacher who puts students at ease and helps them to achieve confidence in
their ability to do assigned tasks.

The next time Glynnis acted out dialogue, she constructed herself as joking with
the students, "No. You're not gonna draw the number line." This refers to the lesson a few
days previous to this, in which the goal was for students to place numbers on a number
line, but then only put them in order and didn't put them in the right place on the number
line. Glynnis was constructing this student as having made a connection between these
two lessons that not even Glynnis is sure she wanted to have made. She was identifying
as a bit hesitant to revisit a topic that previously was fairly traumatic for her and
unsuccessful for the students. And yet, she is identifying as a teacher who allows the
student to go forward with this line of thinking even though it makes Glynnis a bit
uncomfortable. In fact, she uses the word, "Whoa" to qualify her surprise and discomfort

here.
Jonathan

Participant and Spectator Roles

When examining J onathan's story to look at participant and spectator roles, one of
the reasons his story seemed so different from Glynnis' became obvious to me. Jonathan
spoke from the vantage of a spectator much more than Glynnis did. As I did with the first
transcript of Glynnis' story, I have presented Jonathan's story below with the participant
role in normal text and the spectator role in italics. I have underlined the pronouns that
reference the people in this story, because pronominal shifts may be indicators of the

points-of-view of storytellers (Gee, 1989). I numbered the sections in the excerpt (each

140

section is denoted by a switch from one role to the other). Unlike Glynnis, Jonathan

began this excerpt by performing the role of spectator:

l

The next [group ’s work] was very interesting because me had
not done measurement. And so this had to be based on what
they knew about decimals.

And I had meter sticks. And I held up the meter stick. AndI said to
m, "This is a meter stick. It's one meter. So this is the whole."...The
number lines were the most interesting one. Let'_s talk about the
number lines one...

The number line, task three, asked m to put the following
set of decimals, from largest, on the number line. Yeah. It
had a zero, and it had a whole, and it had a half marked on
it. Well, it was very interesting.

Because what I found was that was very easy for them, because they
knew enough about what a tenth was and about what a hundredth was,
that th_ey could easily put those in order. No problem.

 

And that's good. T hat's what t_hgy need to be able to do. But
take a look at my number lines. And, you'll see that, oh, ok.

[Looking at one group 's work. ] Here’s actually a number
line. It's actually one of the few that are accurate. And the
reason is none of them, many of them, are not scale number
lines. Ok. Here is a great number line. I mean, here is a great
table that shows them in order. They 're nowhere right where
they're, like, here 's point five, which really isn't in the

middle. And one hundredth is just as far away from zero as
two, yo_u know, in terms of a scale. So they ’ve got no scale.

T hey 're not accurately placed.

6 I didn't realize LIE were really doing that until the end.

7

It was very eye opening for £718., because I guess I hadn't done
enough with number lines, about what a number line really
means. It's not numbers in order it's places on a line. So it
was very interesting. I think the kids learned a lot that day
when icy saw.

(Looking at another group's work, as represented in Figure 8)

This one, what Ilfiy did, @331 took a hundred centimeters. They
lined them up. And “146 put, lookit, th_ey labeled one hundredth

141

 

in the first square. And they went to the twentieth (square) and
made it two tenths. And forty-ninth. They didn't put the half
there. So actually, what they did is not said where the spot is
but they said this square represents, one (hundredth).

9 So t_h_e_y thought of it more as a fraction, right? One out of a

hundred. I 'm just now thinking about that. 'Cause I was
thinking. Yeah, Lh_ey put it on the right square, but actually if
M put it on a number line, it would be right, there. (Points
to the line between two squares.) So I think what Ilgy did is
t_h£y thought of it as one out of a hundred and that’s one
hundredth of the total. This would equal twenty hundredths
or two tenths. ’Cause they're pointing straight at the square.
So in a way there's still a misconception about a number line

when I look at this.

Jonathan spent much more time performing the role of spectator than the role of

participant.

Pronomial shifts. The transcript above shows the shifts in pronouns that
accompany the shifts between spectator and participant roles. The scarcity of pronouns
referring to people is noticeable in a couple of sections in which Jonathan is performing
the spectator role—Sections 3 and 5. In those two sections, Jonathan spoke at length
about mathematics with little or no discussion of people. In section 5, there are two
pronouns with human referents, when Jonathan said, "I mean", and "You know".
However, neither of those is referring to a person taking any action. The subjects in the
paragraph are mathematical objects—the number line and numbers. This type of talk was
not a part of Glynnis' story.

Constructed dialogue. Jonathan constructed very little dialogue in this particular
story. The only line of dialogue he constructed was, "This is a meter stick. It's one meter.

So this is the whole." This makes sense, given that Jonathan primarily spoke from the

vantage point of a spectator, as opposed to that of a participant.

142

 

Identijying: Discussion

Glynnis and Jonathan are identifying themselves by the content of their stories.
For example, Glynnis did not talk much about mathematics in any of her stories; in fact,
this was about as "mathy" a story as Glynnis told. In choosing not to talk about
mathematics, Glynnis was identifying as a non-participant in mathematics. (She also did
this directly, at another time, saying that she was not a math person.) However, she
identified herself as a competent mathematics teacher in a number of ways. She identified
herself as a teacher who heard students' voices, when she constructed dialogue for both
herself and the students alternately in her story.

In choosing to speak about mathematics, Jonathan identified in the content of his
story as well. In talking about mathematics, he identified as a participant in mathematics.
Like Glynnis, Jonathan also identified as a teacher who was interested in students. He did
not do this by constructing dialogue of students. However, he did this by talking at length
about students' work and their understandings of mathematical concepts (for example, in
section 9 of the transcript above.

The analysis of the participant and spectator roles in these two stories are telling.
At times, Glynnis and Jonathan are both identifying themselves as participants in the
classroom action, and at other times they identifying as people who can step outside of
the classroom action and interpret it. Glynnis spent more time, in this particular story,
performing the role of participant, an insider to the classroom action. Jonathan spent
more time, in this particular story, performing the role of spectator, or outside narrator of

the classroom action.

143

 

Perhaps this is one reason that I originally intuitively felt that Glynnis' and
Jonathan's stories were different. In stepping outside of the action and providing
relatively lengthy reflection on what was happening and why it was happening, Jonathan
was performing a type of knowing that is valued in teacher education—analytic
reflection. Jonathan was providing an exposition on the classroom events, in addition to a
recount of the events themselves.

It is this type of expository talk that we consider indicative of knowledge in
western cultures. When someone a scholar wants to know more about economic policy,
for example, she is more likely to consult an encyclopedia than an anthology of stories
told by children in poverty. The text in the encyclopedia is privileged as "fact", while the
stories may be brushed aside as just people's perceptions.

However, in the section that follows, I argue that both Glynnis and Jonathan,
along with identifying themselves as certain kinds of teachers in these stories, have also
demonstrated sophisticated understandings of mathematics-for-teaching in their stories.
In the next section, I will describe what my micro-discursive, ethnopoetic analysis

revealed about their performances of mathematics-for-teaching.

Mathematics-For-Teaching Competencies
The knowledge of mathematics needed for teaching is different in content and
form than classical mathematical knowledge. Thus, research on mathematics—for-teaching
seeks to describe the understandings of mathematics that teachers need to teach
mathematics. However, what counts as necessary mathematics-for-teaching is an issue of
debate. Ball, Bass, and Hill (2004) have studied what it is that teachers do in classrooms

as a way to learn about mathematics-for—teaching and include teaching practices such as

144

interpreting students' mathematical work and unpacking the mathematical ideas in a
problem. Davis and Simmt (2006) used a complexity-science framework to expand on
those notions, studying teachers' mathematics-for-teaching by studying what teachers say
in interactions around mathematical content. They said, "We believe that a key (and
perhaps the key) competence of mathematics teachers is the ability to move among
underlying images and metaphors—that is, to translate notions from one symbolic
system to another." (p. 303).

In making these claims, authors have reported on the content of what teachers
have said and done in and about discussions of mathematics and mathematics teaching.
However, the content of discourse is only part of what is said. "How something is said is
a part of what is said" (Hymes, 1972, p. 59). How teachers discursively perform their
understandings of mathematics-for—teaching requires further study. In this paper I report
on the analysis of teachers' narratives about a particular classroom lesson about fractions
on a number line. I focus on teachers' talk to see what can be learned about their
mathematics for teaching, focusing on the form of teachers' discourse mathematics and

mathematics teaching.

E thnopoetics and Mathematical Discourse

Interlocuters construct arguments through both the content and the form of what is
spoken (Hymes, 1972). Because mathematics is a discourse, mathematical knowledge
and arguments are also constructed through both the form and content of the
communication. Consider the following example from the literature. Staats (2008)
highlighted the syntactic parallelism through which a child made a mathematical

argument about passing out cuisinaire rods to a group:

145

1 Yes

2 Cos every fifth one

3 from William

4 is going to be
5 a white,
6 And every fifth one

7 from the next person
8 is going to be
9 a red

10 And every fifth one

11 from the next person
12 is going to be
13 a green
14 eh? (Stats, 2008, p. 26)

She argued that the parallel structure of this speech is the way this child makes his
mathematical argument. The child used a template whereby he placed substitutions of
words into slots (in this case, the only words substituted are "William" with "the next
person" and "white", "red", and "green"). In using this template, he performed performing
understanding of the arithmetic pattern and making a generalization. She points out the
importance of looking across sentences to discern arguments made implicitly by the form
of mathematical discourse. (Staats, 2008).

However, the discourse of mathematics teachers has been primarily studied in terms
of content. Little attention has been paid to the form of teachers' mathematical talk.
Therefore, the field is missing some important knowledge about how teachers
communicate mathematical ideas in their teaching practice.

In the sections that follow, I further analyze excerpts from Jonathan's and Glynnis's
stories. I have chosen to analyze in detail only the parts of those stories in which the

teachers are trying to help students understand the number one and fractions as points on

a number line.

146

Connecting Mathematically Important Ideas and Moving Between Metaphors

Metaphors for fractions and number lines. Davis and Simmt (2006) argued that
the ability to move between metaphors is key to mathematics teaching. Metaphor is a
discursive tool. In fact, mathematics may be conceptualized as a discourse. That is,
mathematics may be theorized as communication (with oneself or others) about
mathematical objects (which are constructed through discourse). Metaphor, or the
mapping of language from one domain (for example, physical objects) onto another (such
as abstract ideas), is central to creating and understanding mathematical ideas (Sfard,
2008). In the sections that follow, I report on teachers' talk about a lesson on fractions
and number lines. Therefore, I will next explain some metaphors that are important to
understanding those two concepts.

Lakoff and Nunez (2000) theorized four grounding metaphors that ground
arithmetic to physical objects—the metaphor of object collection, the metaphor of object
construction, the measuring stick metaphor, and the metaphor of motion along a line.
Fractions are often understood using the metaphor of object construction, because they
are constructed by dividing a whole into parts. In that case, a key metaphor is numbers as
objects. In the measuring stick metaphor, arithmetic is conceptualized as the use of a
measuring stick. Numbers are physical segments, with the basic physical segment being
one. Longer segments are greater and shorter ones are less. Arithmetic is putting
segments together and taking them apart.

Linking metaphors take us beyond the physicality of grounding metaphors to
more sophisticated ideas. The metaphor of numbers as points on a line is an example. It

builds on the measuring stick metaphor and the metaphor of arithmetic as motion along a

147

path but maps these ideas onto the concept of a line (instead of a physical object). This

requires understanding several other metaphors: a number P1 as point P1 on a line, zero as

a point 0 on the line, one as a point 1 to the right of 0, etc. Sophisticated mathematical

understandings and are built by layering metaphor upon metaphor, a process called

conceptual blending. In contrast, teaching mathematics involves unpacking the layers of

metaphor (in effect, unblending) and deftly moving between them to facilitate students'

conceptual blending.

Glynnis. Below, I have re-organized a part of the story I described above in which

Glynnis is describing an event in which she is talking to a student about placing numbers

on a number line. I arranged the following transcript excerpt to highlight parallel ideas. In

this case, the subject and action of the sentences are on the left, and the mathematical

objects are aligned further right as well as highlighted with bold text:

Stanza l

1 actually

2 one of the women that was so upset.
3 Recognizing that from
4 was the

5 You know cuz I think

6 we were doing it yesterday
7 and they were ordering
8 and,

9 it was all

10 and,

11 and it was still,

l2

l3 and stuff like that
Stanza 4

38 So she drew

39 And she began to do

40 Ok cuz she was plotting
41

42 lets' just say

43 I don't remember what

148

here to here
one whole.

 

one
one,
one
one

hundredths

the number line.

a landmark number.
one and three,

one and three tenths

it

 

If.-." ‘I , l

44 So she, drew
45 And so she was beginning to figure out.
46 And then she goes, "Oh.
would have to be over
Stanza 5
47 I says, "So
48 you're figuring out that between
49 represents a
50 and the
51 has to be divided into

was.

half
here.

It
here."

here and here
whole,

whole

what?"

After reading this excerpt arranged in this format, it seemed to me that a pattern

was emerging and that the mathematical objects were generally the objects in each

sentence. Next, I analyzed the same transcript to at the basic structure of the sentences

Glynnis was using. In the following transcript arrangement, 1 divided the story into

sentences, italicized the subject in the sentence, underlined the verbs, and put the object

in bold. I numbered each sentence (which is a different numbering than elsewhere in this

chapter, so I have indicated the difference by numbering the sentences in Roman

numerals):

i Actually one of the women that was so upset, recognizing that from

here to here was the one whole.

ii You know, 'cause I think, we were doing it yesterday and they were
ordering one and, one, it was all one and, one and, and it was

still, hundreds and stuff like that.

iii So I gys, "Now, (you) let's write these on the board under each

 

other.
iv (Yawm the decimal here.
v What do you n_g_tIc_e_?
vi What do you really we?

vii You know. What helps you?"

149

 

viii So, they notice that, you know. "Ok, if I look at the tenths place,
it's real easy to get it."

ix And so I w, she had went up to plot.

x She immediately m a number line.

xi And I wLike, "No you're not gonna draw the number line."
xii That w_a§ really funny.

xiii So she drew the number line, which of course is one of the grade
level expectations they're supposed to be able to do.

xiv So I was like, "Whoa."
xv So she drew the number line.
xvi And she began to do a landmark number.

xvii Ok, 'cause she was plotting one and three, one and three tenths
lets' just say.

xviii I don't remember what it was.

xix So she dr_evy half here (gestured as to the number line).

xx And so she was beginning to figure out.

xxi And then she ggys, "Oh. It would have to be over here."

xxii I s_ay_s_, "So you'ge figuring out that between here and here (gestures
as to two points on the number line) represents a whole, and the

whole has to be divided into what?"

xxiii And so it kind of _wgs a breakthrough and they were kinda like,
"Oh."

If you look at these sentences, Glynnis is using a template, which is <someone>
<does something> <to something>. This pattern indicates a metaphor of mathematics as

people doing things to objects. In order to demonstrate the instances in which Glynnis

150

 

used this template, I put the sentences above that have action verbs into the table below to

demonstrate. I emphasized the mathematical objects in bold:

Table 8. "Someone Doing Something To Something" Template

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sent- Someone Does Something (the To something (object of
ence verb) the verb)

ii We Were doing It (the task)

ii They Were ordering One and hundredths
iii I says

iii You Let us

ii (Us) Write These (numbers)

iv You Place The decimal

v You Notice What

vi You Notice What

vii What Helps you

viii They notice

viii I Look The tenths place

ix I says

ix She had went up

x She Drew A number line

xi I Was like (said)

xi You Are not going to draw The number line

xiii She Drew The number line
Xiv I Was like (said) Whoa

xv She Drew A number line

xvi She Began to do A landmark number
xvii She Was plotting One and three tenths
xviii I Don't remember it

Xix She Drew half

xx She Was beginning To figure out

xxi She Goes (sgysL

Xxii I says

xxii You Are figuring out

xxiii They Were like (said) "Oh"

 

 

 

 

 

 

Below, I have isolated (from the above table) just the phrases that substitute
mathematical objects into template in the position of <to something>. In other words, the
table below isolates the instances in which Glynnis used the template, <someone> <does

something> <to a mathematical object>z

151

Table 9. "Someone Doing Something to a Mathematical Object" Template

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sent- Subject Verb Object

ence

ii They Were ordering One and hundredths

ii (Us) Write These (numbers)

iv You Place The decimal

viii I (the Look The tenths place
students)

x She Drew A number line

xi You Are not going to draw The number line

xiii She Drew The number line

xv She Drew A number line

xvi She Began to do A landmark number

xvii She Was plotting One and three tenths

xviii I Don't remember it

Xix She Drew half

 

 

 

Glynnis is constructing mathematics as people doing things to mathematical objects. It is
interesting to note the verbs she indicates being done to these objects: order, write, place,
look, draw, do, plot, and remember.

Of course, not all of Glynnis's statements fit into the <someone> <doing
something> <to something> template. There are a few sentences above in which
something or someone is being instead of doing. I put the sentences with being verbs into

a similar table, Table 10, to explicate:

Table 10. "Someone/Something Is/W as Something" Template

 

 

 

 

 

 

 

Someone/somethirg Is/was (verb) Something
i One of the women Was upset
viii It (doing the problem) Is easy
xii That (what I said) Was funny
xxi , It (the number) Would have to be here
xxiii It Was A breakthrough

 

 

 

 

 

152

It is notable that Glynnis does put a mathematical object into the subject of a
sentence (it, the number the student was trying to place on the number line). However,
that number does not do anything, it simply "would have to be here".

In an attempt to bring some of the above ideas together into one transcript, I have
rearranged the first transcript from this section (in which I originally aligned the
mathematical objects on the right of the page) so that the verbs acting on the
mathematical objects form a second rough column in the center of the text. It is this

transcript to which I will refer in the analysis of metaphors that follows it:

Stanza 1
actually
one of the women that was so upset.
Recognizing that from here to here
was the one .w_h_ole.

l

2

3

4

5 You know cuz I think

6 we were doing it yesterday
7

8

and they were ordering one

and, one,
9 it was all one
10 and, one
11 and, and it was still,
12 hundredths
13 and stuff like that
Stanza 4
38 So she drew the number line.
39 And she began to, do a landmark number.
40 0k cuz she was plotting one and three,
41 one and three tenths
42 lets' just say
43 I don't remember what it

was.
44 So she, drew half
here.

45 And so she was beginning to figure out.
46 And then she goes, "oh. It

would have to be over here.

Stanza 5
47 I says, "So

153

 

48 you're figuring out that between here and here

49 represents a ghole,
50 and the ghole
51 has to be divided into what.

Glynnis is bringing different metaphors of number into relationship via
parallelism (in the <to something> part of the template) as she tells this story. In Stanzas
l and 5, notice the alignment of the words "here to here", "one", and "whole". Glynnis is
drawing on a metaphor these children understand-the metaphor of number as object in
order to define the space between zero and one as the object on which children should act
(divide) to construct fractions. This metaphor is the metaphor by which children first
understand fractions. Children come to understand a whole (an object) to be acted upon
(divided) to construct fractions (pieces of the whole). Because she is talking about a
"whole" (an object) she is drawing on the metaphor of number as object.

However, in using the words "here to here", she makes a line segment as an object
(the whole), therefore drawing a connection to a second grounding metaphor of
arithmetic—the measuring stick metaphor. In this metaphor, arithmetic is seen as
measuring with a stick. In order to understand this metaphor of arithmetic, one needs to
conceptualize numbers as physical segments and one as the basic physical segment.
Glynnis uses the phrase "here to here" to indicate the parameters of the physical segment.
By calling this physical segment (a one-dimensional line segment) a "whole" (a term
students have likely used for two-dimensional area models of fractions), Glynnis is
facilitating a conceptual blending of the grounding metaphors number as object and
number as physical segment.

In Stanza 4, Glynnis uses parallelism to bring "number line", "landmark number",

"one-and-three—tenths", "half', and "here" into relationship. This is a different metaphor

154

 

of number, numbers as points on a line. This is a more complicated metaphor, and one
that Glynnis indicated that students did not understand. She began by using metaphor in
Stanza 1 with which the students were already familiar—number as object—and moved
to this more difficult metaphor. However, in keeping the words for these mathematical
objects within the same place in the same place in the <someone> <does something> <to
a mathematical object>, Glynnis is bringing the terms into relationship with each other.
She is facilitating the conceptual blending of the metaphors numbers-as-objects,
numbers-as-physical-segments, and numbers-as-points-on-a-line through her use of the
template.

The way Glynnis maps the metaphors of numbers as objects, physical segments,
and points on a line onto each other is in the form of her language—in the properties of
this talk, such as the structural patterns she used to construct the story. She does not
explicitly state the different metaphors on which she is drawing. However, she brings the
ideas into relationship through the form of her speech—by using the language of different
metaphors but using the cohesion strategy of syntactic parallelism (the use of templates
with different words substituted into slots in the template) to help the audience map them
onto each other.

In Stanza 5 in lines 47-50, Glynnis returns to a language of numbers as line
segments, again by placing parameters on the two ends of a one-dimensional segment. In
using the words "here and here" she is indicating that the object to which she is referring
is one-dimensional with two ends—it has a length, but that is all that matters in regards to

its size.

155

However, in line 50-51, when she says " the whole has to be divided into what",
her language is that of the arithmetic as construction of objects metaphor. Children have
previously understood fractions as parts of wholes using this metaphor (a fraction is a
piece of a whole object). Now Glynnis' language proposes that the children take the
metaphor by which they understand fractions, and the new metaphor of numbers as points

on a line, and blend those concepts to understand where to place fractions on a number

 

n
line. By substituting one word for another across lines of speech, using different words

that indicate different metaphors for number in the same place in the <someone> <doing

something> <to something> template, she intends to help students develop sophisticated

metaphorical blends. I highlight Glynnis' shifts between metaphors in the Table 11. l“

Table l I . Glynnis' Metaphorical Shifts

 

Stanza 1 Number as

1 actually object (whole)
2 one of the women that was so upset. Linked to

3 Recognizing that from here to here number as

4 was the one Lhole. physical

segments (from
here to here)

 

 

 

 

38 So she drew the number line. Numbers as
39 And she began to, do a landmark number. points on a line
40 Ok cuz she was plotting one and three,
41 one and three tenths
42 lets' just say
43 I don't remember what it
was.
44 So she, drew half
here.

45 And so she was beginning to figure out.
46 And then she goes, "oh. It

would have to be over here.
47 I says, "So Numbers as line
48 you're figuring out that between here andheLe segments
50 and the lhole Arithmetic as
51 has to be divided into what. object

 

 

construction

 

156

Notice the order in which Glynnis indexes the different metaphors for number: 1)
numbers as objects, 2) numbers as physical segments, 3) numbers as points on a line, 4)
numbers as physical segments, and 5) numbers as objects. I am careful to not read too
much into this order, because I am only looking at an excerpt of the story. However, I
noticed that in this excerpt, Glynnis began with the metaphors in the order in which
Lakoff and Nufiez present them in the book (object, segment, points)——the way they
argue that these metaphors for number come into being. Then she created reversed the
order, creating a symmetrical structure in which she moves from least to most complex
and back.

So far in this section, I have discussed what can be learned about Glynnis'
mathematics-for-teaching competencies from this story. On a related note, I will close
this section with a brief discussion of what can be learned about Glynnis' conceptions of
mathematics as a discipline. In Table 12, I revisit the constructed dialogue in Glynnis'

story in order to point out clues as to Glynnis' conceptions of mathematics.

157

Table 12.

Constructed Dialogue in Glynnis' Story

 

 

 

 

 

 

Role Dialogue How it How Glynnis A Notes
constructs identifies hersel
math
Glynnis Now. let’s write these Can be As an arranger She is rearranging
0n the board. Under known of the numbers with
each other through environment students (let's
place the decimal here. physical write these on the
What do you notice. observation board under each
What do you really other) and
notice. expecting and
What helps you. asking students to
observe.
Students Ok. I] I look at the Observable, As a teacher
tenths place, it’s real easily who puts
easy. To get it. observed students at case
when (easy).
properly
arranged
Glynnis No you're not gonna Scary As a teacher
draw the number line. who allows
students' ideas
to take center
stage, even
when it goes
against her
original
intentions, and
even when it
makes her
uncomfortable.
Glynnis m The Appreciative of
connections students'
are contributions to
complex the
and hard to mathematical
grasp (so understanding
when a of the class.
student
does it, it's
surprising.)

 

 

 

 

 

158

 

 

Table 12 (cont'd)

 

 

 

 

 

 

 

 

 

A Oh. It would have to be, As related As stepping The student comes
student over here. to place, back and to know where the
having a allowing a number should be,
physical student to come even when
position. to her own Glynnis did not
realization arrange this for
her.

Glynnis so ou're z rin out As existing As a provider of There is a case of
that between here in a more formal a mathematical
and here represents. physical language, as a object "here and
A whole location. summarizer here" doing

and the whole hop to be who articulates something
divided into what students "represents".
understandings (Which was
that are not yet overlooked in
formal previous tables
because it is
included as a noun
clause functioning
as a direct
object—a level of
sentence beyond
the
subject/verb/object
analysis above).
Students Oh.

 

 

 

 

 

Glynnis understands mathematics to be an empirical endeavor, similar to other sciences.

Students can learn mathematics and construct mathematical ideas through observation. At

the same time, she does start to talk to her students about a mathematical object as doing

something (it represents). However, almost all of the verbs in this dialogue are attributed

to people (what students should do to mathematical objects).

Jonathan. In the following excerpt, Jonathan is telling the story of the number line

lesson. Although Jonathan and Glynnis used the same number line task (which I included

159

 

. . . . . 15
earlier in this chapter), Jonathan paired that With a measurement task . A more

complete version of this transcript is included near the beginning of this chapter.
However, I will summarize it below, often including re-arrangements of the transcript in
order to highlight patterns I am noticing.

As with Glynnis' story, it is interesting to note the ways in which Jonathan talks
about mathematical objects in the story. In order to highlight this, I have arranged the
transcript below so that all of the words for mathematical objects area aligned on the right

side and highlighted in bold:

1 I had meter sticks.

2 I held up the meter stick.
3 I said to them. "This

4 is a meter stick.
4 It

5 '5 one meter.

6 So this

7 is the whole. "
260 I guess

261 I hadn't done enough with number lines.
262 About what a number line
263 really means.

264 It

265 's not numbers inorder
266 it

267 '5 places on a line.

279 Right so actually,
280 what they did is not said where the spot
28 1 is

 

Jonathan gave the number line task as Glynnis. As a reminder, that task was: "Show
where the following set of numbers fit on the number line. Explain how you know." With
that task Jonathan had listed the following numbers: "0.86, 0.2, 0.99, 0.49, 0.75, 0.01,
0.6." However, on the same page, in order to facilitate students' connections of decimal
numbers across contexts, Jonathan also gave the following task: "A meter stick is made
11p of 100 centimeters or 10 decimeters. If the meter stick is considered 1 whole, how
would you represent the following? 2.17, 0.9, 0.28, 1.11, 0.75."

160

 

282 but they said

283 this square

284 represents, one (hundredth),

285 so they thought of it more as a fraction.

286 Right?

287 one out of a hundred.

Compared with the corresponding arrangement of Glynnis' transcript in the
previous section, it is notable to note the density of the text on the left hand side
of the transcript is different between the teachers. Glynnis has more text in the left
side of the transcript than Jonathan does.

I hypothesized that this was a difference in the ways Glynnis and Jonathan
were treating mathematical objects, so I also analyzed Jonathan's story to look at
the subject, verb, and object in each sentence (the <someone> <does something>
<to something> template, described above). Below, 1 have arranged the transcript
of Jonathan's story according to sentences (numbered with roman numerals). I
indicated the <someone> (subject) in italics, the <does something> (verb) with

underline, and the <to something> (object) in bold text:

i I Mimeter sticks.

ii I h_eld_r_rp the meter stick.

iii 1 gai_d_ to them, "T his jg a meter stick.

iv It's one meter.

v So this jg the whole. "

cclx I guess I hadn't done enough with number lines, about what a

number line really means.

cclxi It's not numbers in order.

161

 

cclxii It's places on a line.

cclxxx Right, so actually what they id is not said where the spot is.
cclxxxi But they fld this square represents one (hundredth).
cclxxxii So they thought of it more as a fraction, right?

cclxxxiii One out of a hundred.

In order to highlight the way Jonathan used this <someone> <doing something>
<to something> template, I display his word substitutions for each of these slots

in table 13. In the table, I have indicated the words for mathematical objects in

 

bold text:

Table I3. Jonathan's "Someone Doing Something to Something" Template

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sentence <Someone> <Doing <To Something>
Something>

i I Had Meter sticks

Ii I Held up The meter stick

iii I Said

cclx I Hadn't done enough

cclxxx They did

cclxxxi They Said

cclxxxi This square Represents One (hundredth)

cclxxxiii They Thought

In Table 14, I have isolated (from the above table) just the phrases that substitute
mathematical objects into template in the position of <to something>. In other words,
Table 14 isolates the instances in which Jonathan used the template, <someone> <does

something> <to a mathematical object>:

162

Table 14. Jonathan's "Someone Doing Something to a Mathematical Object" Template

 

 

 

 

 

 

 

 

 

Sentence <Someone> <Doing <To Something>
Something>

i I Had Meter sticks

Ii 1 Held up The meter stick

cclxxxi This square Represents One (hundredth)

 

By isolating the subjects, verbs, and objects of these sentences, it became apparent that
Jonathan has included a mathematical object in the subject of a <someone> <doing

something> <to something> sentence. Therefore, he has indicated a conception of F

. . . . 16
mathematrcs wherein mathematical objects have agency. They act.

Again, like Glynnis Jonathan also constructed sentences with a different template:

 

<someone/something> <is/was> <something>. I display those sentences in Table 15:

Table I 5. Jonathan's "Someone/ Something Is/W as Something" Template

 

 

 

 

 

 

 

Sentence Someone/somethifl Is/was SomethinL

iii This Is A meter stick

iv It (meter stick) Is One meter

v This (meter) Is The whole

cclxi It Is not Numbers in order
cclxiii It Is Places on a line

 

 

 

 

 

Most of Glynnis' <someone/something> <is/was> <something> sentences were about
people; only one had a mathematical object in the subject position of the sentence.
However, Jonathan put the mathematical object into the subject of the sentence several
times in order to explicate the synonymous nature of two words representing the same

mathematical object.

 

Glynnis did something srmrlar, which I described in the prevrous section in Table 10.
However, because she did it in a noun clause in a compound sentence, it was not as
obvious when looking at verbs in this template.

I63

Like I did with Glynnis' transcript above, I sought to blend some of these ideas
and create a re-organized transcript where several of these observations could be seen at
once. In the transcript below, I aligned the words for mathematical objects in two
columns. When the words appear in the subject of the sentence, that subject is in the
middle column. When the words appear in the object of the sentence, they are placed in

the far right column. Words for mathematical objects are indicated in bold:

1 had meter sticks.

2 I held up the meter stick.

3 I said to them. "This is a meter stick.

4 It's one meter.

5 So this is the whole. "

260 I guess

261 I hadn't done enough with number lines.

262 About what a number line
really means.

263 It's not numbers in order

264 it's places on a line.

279 Right so actually,

280 what they did is not said where the spot
rs
281 but they said
282 this square represents, one (hundredth),
284 so they thought of it more as a fraction.
285 Right?
286 one out of a hundred.

Jonathan also used parallelism to bring ideas together. In lines 1-5 he is bringing

the ideas "meter , one", and "whole" together. Like Glynnis, he draws on two important
metaphors here: numbers as physical segments (lines 1-4) and numbers as
objects/arithmetic as object construction (line 5). Again, these are metaphors that had
become commonplace in the classroom, and he brings them into relationship by

substituting one for another in the template.

I64

In lines 260-264 he adds the concepts, "number line", "places on a line" together.
However, when he starts to explicitly describe a number line as "places on a line", he
makes the number line (it's) into the subject of the sentence (which I indicate in the above
transcript with bold text). He not only brings these ideas into relationship using
parallelism, but also explicitly—he is using both the form and the content of his speech to
make this argument. It is through parallelism that Jonathan is blending the metaphors of
numbers as line segments, arithmetic as object construction, and numbers as points on a
line. Making the metaphor, "number lines as places on a line" explicit is done by
breaking up the pattern and subjectifying the number line.

H H H H

In lines 279-286 he uses parallelism to bring the terms, "spot , square , one
hundredth", "fraction", and "one out of a hundred" together in parallel with previous
ideas. The word "square" (an object) draws on the metaphor of number as object, the
words "one hundredth", "fraction", and "one out of a hundred" draw on the metaphor of
arithmetic as object construction, and the word "spot" (substituted for point) draws on the
metaphor of numbers as points on a line. In lines 279-281 and 283-286 these ideas are
grammatically parallel to one another. However, in line 282 Jonathan subjectifies the
mathematical idea (square) in order to make explicit his understanding of the metaphor
on which his students are drawing. Throughout most of this excerpt, Jonathan uses
parallel syntax as he moves across metaphors in the way he talks. However, at times he

switches this pattern with the result that the metaphors become more explicit. I indicate

Jonathan's shifts between metaphors in Table 16.

165

Table I6. Metaphors in Jonathan's Story

 

 

 

 

 

 

Excerpt Metaphors

l I had meter sticks. Numbers as physical

2 I held up the meter stick. segments

3 I said to them. "This is a meter stick.

4 It's one meter.

5 So this is the whole. " Numbers as objects
(arithmetic as object
construction).

260 I guess Numbers as points on a

261 I hadn't done enough with number lines. line (mathematical

262 About what a number line objects in object

really means. position)

263 It's not numbers in order Numbers as points on a

264 it's places on a line. line (mathematical
objects in both subject
and object positions,
being verbs)

279 Right so actually, Numbers as points on a

280 what they did is not said where the
is
281 but they said

spot

line (mathematical
object in object position)

 

282

this square represents one (hundredth),

Number as object
(mathematical objects in
subject and object
positions, action verb)

 

284 so they thought of it more as
285 Right?
286 one out of a hundred.

 

a fraction.

 

Number as object
(mathematical objects in
object position)

 

Differences between stories. Glynnis and Jonathan are both using similar poetics

to bring concepts into relationship as they move across metaphors. Why, then, was I

initially under the impression that Glynnis's talk was less mathematically sophisticated

than Jonathan's? Was it simply because Jonathan makes metaphors explicit in his story?

There are two main differences I noticed. In the analysis above, there were

differences in the ways Jonathan and Glynnis spoke about mathematical objects. Both

teachers established a rhythm in their story in mathematical objects were inserted in the

<someone/something> <was doing something/was> <to something> template as <to

166

 

 

something> (the object of the sentence). However, Jonathan added complexity by
inserting a counter-rhythm into his story when he put the mathematical objects into the
subjects of sentences (subjectified them).

In Table 17 I place parts of these transcripts side by to highlight other differences
in the form of their talk. Above, I described how Glynnis and Jonathan used parallelism
(templates) in similar ways to bring cohesion to their talk when they move from metaphor
to metaphor. This is indicated in Table 17 as a box with a solid line. However, in Table
13 it is apparent that the rest of their text is structured very differently. Jonathan's appears
to be neat and tidy, only including phrases that line up nicely in parallel (indicated in
boxes with dashed lines). In contrast, Glynnis includes an aside (lines 5-13) in which she
describes the numbers the students were using in the task. Before and after this aside
Glynnis inserts two 3-line stanzas that are grammatically parallel. This brings a sense of
cohesion and connection between lines 1-3 and lines 47-49. It seems that the difference is
in the economy of syntax, as opposed to the level of sophistication of their mathematics-

for-teaching understandings.

167

Table I 7. Comparing Glynnis and Jonathan's Stories

 

 

 

 

 

 

 

 

 

 

 

 

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168

Mathematics-For-Teaching Competencies: Discussion and Conclusions

The difference in the stories' settings could account for some of this variance.
Jonathan was telling me about children's work. Glynnis's story was about what she said in
interaction with children. Perhaps Glynnis, in the form of her story, is performing
knowledge about how her children understand language. Another factor that could
account for some of my initial perception is that Jonathan identified himself as "the math
guy" and Glynnis as "not a math person". This could have impacted my initial
impressions of their mathematics-for-teaching talk. Additionally, a way Jonathan
identifies as mathematically competent is by talking about mathematics-for—teaching
often and in more formal ways, and Glynnis identifies as not mathematically competent
in part by avoiding discussing mathematics—for-teaching.

Does this particular story of Jonathan's perform a more sophisticated
understanding of mathematics for teaching than this particular story of Glynnis's? Maybe.
After all, Jonathan does, at times, make ideas explicit in a way that Glynnis does not.
However, it is also possible that the perception of sophistication upon first read is due to
the comparative elegance of the way Jonathan brings metaphors together, which is an
aesthetic evaluation of the storytelling that parallels one way in which mathematicians
evaluate mathematical proofs. For example, Einstein said of mathematical proofs, "We
are completely satisfied only if we feel of each intermediate concept that it has to do with
the proposition to be proved." (Luchins & Luchins, 1990, p. 38). Perhaps the same
culturally-privileged values of the mathematics community have been used to judge
teachers' mathematics-for-teaching talk and marginalize some mathematics-for-teaching

competencies that facilitate student learning. Is it appropriate to apply the same criteria to

169

 

mathematical pedagogical discourse? After all, an elegant mathematical proof leaves out
any indication of the messy work that led to it. If teachers and students are engaging in
reasoning and proof, it is likely that the majority of their classroom talk will be "ugly" by
mathematical standards. Additionally, mathematical proofs are written with others in the
same discourse community as the intended audience. Teachers' audiences vary widely in
regards to culture, age, children's backgrounds, and the like. Acknowledging that
teaching mathematics requires more than classical mathematical knowledge means
assuming that the discourse of mathematics-for-teaching will be different than the
discourse of classical mathematics. Both Jonathan and Glynnis have tacit knowledge of
the ways of speaking that move their particular set of students toward mathematical
understandings. Looking at teachers' discourse for their mathematics-for-teaching
competencies allows us to see teachers' ways with words as competencies that the field

has formerly overlooked but that contribute to students' mathematical understandings.

Discussion: Identifying and Mathematics-For-Teaching

Throughout this dissertation, I have argued that elementary teachers' narratives
about themselves (their ways of identifying) and their mathematics teaching with Cl
(stories of implementation) are related, based on a review of relevant literature and lore.
In this chapter it is clear that identifying and mathematics-for-teaching are indeed
interlinked. In comparing what teachers said about who they are as math people with
how they said it, I gained insights into identifying and mathematics-for-teaching. In the
section that follows, I discuss ways in which identifying and performing knowledge of

mathematics for teaching are interlinked, knowledge of mathematics-for-teaching as a

170

culturally-laden concept, and the implications for inclusive approaches to mathematics

teacher education practice and research.

How Identifiring and Mathematics-For- Teaching are Interlinked

At the very least, identity and mathematics-for—teaching are performed
simultaneously in these two teachers' stories. In the first part of this chapter, I
demonstrated some of the ways in which the teachers were identifying themselves as
mathematically competent (or not). In the second part, I theorized about what might be
possible to learn about the teachers' mathematics-for-teaching. Therefore, I have
demonstrated the ways in which both of these things were done in very sophisticated
ways simultaneously in the telling of the stories. But do they simply bubble up
simultaneously in these stories? Or are impacting each other? In the sections that follow,
I will make two observations about how identifying and mathematics-for—teaching might

be seen as related—one in Glynnis' case, and one in Jonathan's.

Glynnis: Uncertainty in Her Mathematical Knowledge and the Tentative Nature of

Knowledge

As I described in the results section, a first look at Glynnis' story evidenced verbal
hedges, such as "kind of" and "let's just say", which may be perceived by the audience as
a lack of confidence (whether or not Glynnis would perceive them as such). However,
Glynnis had directly identified in an earlier interview as "not a math person" and had
hesitated to talk about mathematical content or her students' understanding of
mathematics.

In the analysis of the dialogue in her story (Table 12) I demonstrated that one way

she identified her role as a teacher was as an arranger of the environment. She

171

 

constructed students as capable observers, if only the objects to be observed were
carefully arranged.

The analysis of Glynnis' mathematics-for-teaching opened the possibility that

Glynnis views mathematics as an empirical science that is socially constructed”. If this
is the case, then it makes sense for Glynnis to View her own knowledge of mathematics
with the same tentative nature with which a social constructivist views the entire body of
mathematics. As well, a view of mathematics as empirically and socially constructed
could lead Glynnis to believe that students need to be actively involved in discovering
mathematical ideas, and that her role is to let students' voices be heard. The fear that
"untruths" would be discovered would not necessarily be of concern, because

mathematical ideas are open to revision at any time in the process.

Jonathan: Being Verbs and Certainty

In Chapter 2 I presented a definition of direct identifying statements (what Sfard
calls identifying narratives (Sfard 2006a, 2007; Sfard & Prusak, 2005) as statements
about oneself that are reified, endorsed, and significant. "I am" statements are particularly
powerful and decisive direct identifying statements. For example, "I am good at
mathematics" and "I am a mathematics teacher" are more powerful statements of identity
than "I did well in mathematics" and "I teach mathematics" (although no statement is

identity-free, of course). I also presented Sfard & Prusak's (2005) notion of F irst-,

 

l7 . . . . . . . . .
In an earlier rntervrew, Glynnis expressed a View that mathematrcs 1S changing, which

could be further evidence of a social constructivist philosophy of mathematics. Further
analysis into whether there is evidence of this philosophy throughout all of Glynnis'
interviews may be interesting in the future. However, as this chapter is intended to be a
theoretical piece that opens up the possibility of a link between a teacher's ontology and
her identifying (as opposed to an empirical piece proving that this is the case).

172

Second-, and Third- person identifying statements. That is, first-person identifying
statements are statements made about oneself to oneself, second-person identifying
statements are made by someone else to the person being identified, and third-person
identifying statements are statements made by someone to someone else about a third
person.
Remember that earlier in this chapter I noted Jonathan's use of being verbs in

which mathematical objects were the direct object. In Table 15 I presented five such r
sentences, in which Jonathan made statements such as, "This (a meter) is the whole."

Such a statement has something in common with the most powerful forms of direct

 

”78'.

identifying statements, the "I am" statements. As such, these statements of Jonathans ",
could be seen as third-person identifying statements in which Jonathan is talking about
mathematical objects (directly identifying them) to his students. Additionally, Jonathan
had afforded mathematical objects agency, a trait usually assigned to humans.
At the same time, Jonathan is performing his conception of ontology of
mathematical objects, and at the same time, identifying himself as a mathematically
competent person. He is identifying as a person who knows mathematical objects well
enough to identify them to someone else, similarly to one person describing another

person to a third person (third-person identifying).

Identijying and Perceptions of Competence

I had originally identified Jonathan as "mathier" than Glynnis, which is a
perception I made because of how Jonathan and Glynnis identify differently. They both
performed different aspects of their conceptions of mathematics as a discipline—while

both had conceptions of mathematics that could be described as realist, Jonathan

173

 

positioned himself as close to the mathematical objects (identified with them) and
Glynnis positioned herself more distant from them (as objects to be discovered and
examined, not known as one knows another person).

Both of these ontological stances are fairly sophisticated and somewhat different
than one might expect. And both would affect both the ways teachers identify and
perform their knowledge of mathematics-for-teaching. Therefore, they may not be
perceived similarly by an outsider. Glynnis might have been performing a sophisticated
conception of the tentative nature of mathematical knowledge, but uncertainty may be
perceived by others as incompetence. On the other hand, Jonathan identified as closely
knowing mathematical objects in making third-person identifying statements about them.

Such statements are likely to be perceived as confidence, and therefore, competence.

Knowledge of Mathematics-For- Teaching as a C ulturally-Laden Term

Discourses are situated within cultures, which have different discursive routines
and value different forms of communicating. The culture of academic mathematics, for
example, values elegance in mathematical proofs. However, the culture of classrooms is a
very different culture. It is therefore inappropriate to apply the same criteria to
mathematical pedagogical discourse. An elegant mathematical proof leaves out any
indication of the messy work that led to it. If teachers and students are engaging in
reasoning and proof, it is likely that the majority of their classroom talk will be ugly by
mathematical standards.

Additionally, mathematical proofs are written with others in the same discourse
community as the intended audience. However, teachers' audiences vary widely in

regards to culture, age, children's backgrounds, and the like. Acknowledging that

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teaching mathematics requires more than classical mathematical knowledge means
assuming that the discourse of mathematics-for-teaching will be different than the
discourse of classical mathematics, in part because the context is different. Not only that,
teachers' discourse of mathematics-for-teaching is likely to vary between contexts with
different discursive routines. Glynnis' students and Jonathan's students are in different
contexts and likely have different discursive norms. Therefore, the differences in their
performance of mathematics—for-teaching might indicate that each has deep knowledge
about their students, their context, and the discursive routines therein.

Perhaps this is not so different than the way "standard English" is privileged over
"colloquial dialects". It is a prevailing myth that "colloquial dialects" are simplified or
lazy versions of "Standard" English. However, studies have shown that they have

complex rule systems.

Implications For Inclusive Approaches to Mathematics Teacher Education Practice and

Research

A goal in the CI mathematics course was broadening teachers' conceptions of
mathematical competence so that they could see the competencies that students bring to
the mathematics classroom that help them to solve mathematical problems but have not
traditionally been privileged in mathematics classrooms. Teachers are taught to begin to
see students' competencies such as communication through visual rhetoric (color-coding,
use of white space, etc.), verbal skills, geometric imagination, and others as mathematical
competencies. Teachers also think about designing tasks that teach important

mathematical concepts while drawing on these competencies. The purpose is to create

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tasks that engage a larger number of students in mathematics to make participation and
learning more equitable within a particular classroom.

In the same way, mathematics teacher educators should carefully examine the
competencies teachers bring to the classroom in order to understand the competencies
teachers bring to the classroom that may not yet have been privileged in learning to teach
mathematics. It is for this reason that I chose to talk about teachers' mathematics—for-
teaching competencies instead of simply mathematics-for-teaching. In doing this we open
possibilities for designing professional development and teacher education tasks which
validate and extend those competencies.

In designing and facilitating professional development and teacher education
courses, teacher educators have a choice to make about how teachers' are viewed upon
entering the courses. Either teachers are viewed as lacking knowledge and skills—empty
buckets to be filled or ignorant cave dwellers like those in Plato's "Allegory of the Cave,"
or they are viewed as bringing something worthwhile to those courses which can then be
leveraged to help them learn new practices. I argue that this second attitude toward
teachers needs to be explicitly adopted in order to make PD and teacher education

courses more inclusive to a wide variety of teachers.

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CHAPTER 8: DISCUSSION AND CONCLUSIONS
Throughout this dissertation, I have argued that teacher learning may follow a

similar story plot as other stories of learning, in that it results in new ways of identifying
and being able to participate in families and communities (Cartlidge & Gilbert, 1983;
DuBois, 1903/1994; Reed & Johnson, 2000). I have also argued that teacher learning
needs to be viewed as an end unto itself, for the enrichment of teachers' lives, instead of
only as a means to an end, for changes in practice and improved test scores for students
(Featherstone, 2005).

I first made this argument in Chapter 1 by explicating the parallels between the
stories "Of the Coming of John" (DuBois, 1903/1994), Educating Rita (Cartlidge &
Gilbert, 1983), and Plato's "Allegory of the Cave" (Reed & Johnson, 2000), and claiming
that the story I would tell in the dissertation would be parallel to those stories. That is to
say that teacher learning, just as the learning portrayed in those accounts, is challenging
and has unintended consequences.

In Chapter 2, I retraced the history of the problem for study, claiming that theories
on the stability of teaching practice were insufficient to explain why some teachers are
able to make and sustain changes in their teaching practice. I constructed a fi'amework for
this study, drawing on socio-cultural theories of narrative and identity. I stated my
research questions as being about the relationship between teachers' ways of identifying
and their implementation of new practices, how the difficulties inherent in changing
teaching practice and teachers' abilities to sustain implementation of new practices may
be considered identity work, and the dynamic nature of teachers' ways of identifying. I

argued that a narrative study is an effective method for studying identity and teacher

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learning in that it avoids treating teachers only as a means to the end of student
achievement (Featherstone, 2005) by illuminating teachers' voices.

In Chapter 3, I described the design of my dissertation study and explicated
methodological decisions made during the course of the study. In particular, I argued for
the affordances of a study of different kinds of narratives (life-stories, recounts of events,
and direct identifying statements) in different contexts (individually, in groups of CI
teachers, and in groups where teachers had not yet taken the CI course) in order to answer
my research questions regarding teachers' ways of identifying and implementation of new
practices and to speak to my larger problem for study, which is that we do not yet
understand all of what must occur for a teacher to change her practice in order to add
teaching practices and do so in the ways that reformers recommend.

Chapters 4, 5, 6, and 7 are the findings chapters in this study. In Chapter 4, I
described the ways that teachers' life stories functioned to propose reasons for choosing
to implement (or not implement) CI. I described a number of different possible
connections between teachers' life stories and choices of implementation (e. g.
implementing CI to revise a life story).

In Chapter 5, I described the ways that teachers' identifying narratives became
obstacles to be overcome as they implemented CI, and how their ways of identifying
ultimately sustained their persistence. I argued that speaking about the necessary shifts in
ways of identifying provides a way for researchers to avoid speaking of teachers in deficit
terms, and affords an opportunity to learn about teachers in the ways we argue that

teachers should learn about their students. In Chapter 6, I described teachers' generalized

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stories of practice (how-I-teach stories), and theorized about the integration of prior and
new ways of identifying in order for teachers to construct expert identities.

In Chapter 7, I zoomed in to a close-up, ethnopoetic analysis of two teachers'
stories about their implementation of a similar lesson. In this chapter, I theorized that
identifying and mathematics-for-teaching were closely interlinked, and sometimes the
same practice. I also argued that mathematics-for-teaching should be studied using
narrative methodology, in order to attend to teachers' unarticulated knowledge of
mathematics-for-teaching.

In the present chapter, I look across the discussion sections of all four findings
chapters in order to bring them together and answer my research questions. I will first
discuss my research questions in order, drawing from the four findings chapters to answer
them. Then I will explicate the limitations and implications of my study on teacher
professional development practice and research. I will conclude the dissertation with a

discussion of my plans to further this line of research.

Contributions of the Dissertation
As a reminder, the research questions that have guided this dissertation study are

as follows:

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1. How do teachers' ways of identifying and their stories of implementation of a new
mathematics pedagogy interrelate?

2. How do teachers' ways of identifying become obstacles (or not) to teachers' attempts to
change their teaching practice? How do they make sense of the obstacles and
overcome them?

3. How do teachers' ways of identifying change (or remain the same) as teachers
implement CI to teach mathematics?

The next three sections each address one of these three research questions.

The Relationship Between Identijying and Implementation

My first research question asked how teachers' ways of identifying and their
stories of implementation were interrelated. Studies had previously drawn a one-way
connecting arrow between teachers' ways of identifying (in life stories) and their
implementation of a specific reform curriculum (Drake, 2006; Drake, Spillane, &
Hufferd—Ackles, 2001). These studies connected teachers' life stories and their
implementation of the curriculum based on classroom observations. I chose not to
observe in classrooms, but rather to collect teachers' narratives of the classroom lessons
in question. This was a purposeful move, because in doing so, I was able to learn about
more than the activity that I observed in the moment of practice. I was able to learn about
how the teachers understood the relationships between their self-perceptions and
improvisations of practice moment-by-rnoment. How teachers were making decisions
about implementation, how they were thinking about their implementation after the
lessons, and how they were making sense of Complex Instruction were most important to

me. Had I been present in the classroom when the lessons were taught, teachers would

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have made assumptions about what I had seen and thus already knew about the lessons,

and may have had less reason to communicate their impressions to me.

.Identijying Narratives and Implementing or Not Implementing CI

Another advantage to a narrative study was that I was able to look for connections
between identifying and teaching practice that have previously been unexamined and
look for explanations for those connections. As I reported in Chapter 4, the teachers
identifying narratives functioned to propose reasons for choosing to implement (or not
implement) Complex Instruction (CI). This is unsurprising, given the results of Drake's
life story studies (Drake, 2006; Drake, Spillane, & Hufferd-Ackles, 2001). Those studies
spoke to the question of differences in implementation between teachers with different
life stories, but not about the decision to implement the reform or not, or the reasons for
the connections (or apparent disconnections) between life stories and practice.

This study proposes reasons for some of the connections between identifying and
choosing to implement CI. In the discussion section of Chapter 4, I propose that the
choice to implement CI may sometimes be due to a match between the principles of CI
and the ways the teachers' identify. For example, the main purposes of C1 are to promote
equitable and rigorous mathematics instruction in classrooms, and those were salient
aspects of how Jonathan constructed both his designated and current identities; therefore
Jonathan's decision to implement CI was unsurprising. However, it seems that other
teachers choose implementation because it allows them an opportunity to revise their life
stories as they provide different experiences for their students (and for themselves as
teachers) than they experienced as children. And in the case of teachers with failure

(foreclosed) identities, perhaps the way of identifying as a mathematics teacher and

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learner is too scary to revisit, thereby making the consideration of a new pedagogy too
fiightening.

Before going on, however, I will comment on the label "failure (foreclosed)". I
have chosen to use this label in order to build upon and discuss work that has already
made inroads into the relationship between identifying and implementation. As I do so,
however, I am struck by the way that this label implies that failure is a static way of
identifying, and seems to close the door on any hope of a turning point in the future. This
label was likely intentional on the part of previous researchers, who likely noted that
stories of previous failures are often difficult to revisit and overcome. However, the very
existence of tuming-point stories implies that stories of failure do not necessarily
preclude the implementation of new mathematics teaching pedagogies. Additionally,
although Gina did not experience a turning point in the CI course, she did say that it
enabled her to envision herself participating another mathematics PD event in the
future—something that she would not have considered before the course.

It is important to stress again that I do not intend to imply that only teachers who
bring certain ways of identifying with them to PD courses have the possibility of success.
Because identifying is dynamic—both within individual lessons and across longer time
spans—any teacher should be seen as capable of success in the CI math PD course. What
is important is that PD facilitators and designers learn about teachers' ways of identifying
in order to help facilitate the types of connections between identifying and new practices
that will allow teachers' to see the new practices as possibilities. I argue that the CI math
PD course in particular allowed a variety of different teachers to see it as a part of their

designated identities—part of the kinds of teachers they already wanted to become.

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Perhaps the CI math PD course is different than other PD courses in this way. After all,
the use of CI to teach mathematics has been found to allow more students' identities to be
expressed in mathematics classes (Jilk, 2007). Perhaps the same is true of the CI math PD

COUI'SC.

Identifying Narratives and Teachers' Understandings and Implementation of Teaching

Practices

It appears that teachers' ways of identifying and stories of implementation of CI
were connected, if sometimes in unexpected ways. For example, Jonathan saw himself as
rigorous and inclusive, so it was unsurprising that he would choose to implement and
sustain implementation of CI, because rigor and inclusion are its two important
principles. However, Joanna sometimes identified as person who joined projects, Glynnis
often identified as resistant, Patricia identified as a behaviorist—and all three saw CI as a
pedagogy that could help them to become more like their designated identities. Therefore,
the links between a recommended pedagogy and teachers' ways of identifying may not
always be as expected. Therefore, just as teachers are encouraged to learn about their
students in elementary classrooms, it is critical that PD facilitators and designers spend
significant time learning about the kinds of teachers that participants in our courses want
to become.

Even though it seems that the teachers who chose to implement CI had originally
perceived it as helping them to become more of the kinds of teachers they wanted to
become, for three of the four teachers, something troubled this during the time I was
collecting data. Sometimes teachers perceived themselves to be acting in ways contrary

to their identifying narratives. For example, at a point in time, Joanna perceived CI as not

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allowing her to be as giving as she would have liked. Therefore, she changed the way she
was implementing CI in order to ease this discomfort for her. If I had done pre-course
and post-course observations of Joanna to determine what she had learned from the PD
course, and had only come during the time Joanna was giving her lengthy lists of written
directions, I might have concluded that the CI course actually hurt her teaching practice
in some way and assumed that it was because of a lack of knowledge or some bad beliefs.
However, collecting Joanna's stories during that time, as well as over the course of
several months as she went through the cycle of planning, implementing, and reflecting
on three different CI mathematics lessons, I was able to see that each implementation of
CI was changing her perceptions of the pedagogy itself. That is, Joanna was learning

something from implementing CI math lessons, because each implementation was

. . . l8
changing her discourse about her teaching

A final point I would like to make about teachers' ways of identifying and their
stories of implementations of new practices (and the point I think is most important) is
this: Sometimes what has previously been spoken of as teachers' misunderstandings of or
inabilities to implement new practices may sometimes be more aptly described as a
perceived mismatch between teachers' current ways of identifying and the pedagogy.
When teachers' choose not to implement a new practice, or when their implementation

does not look or sound exactly like expected, further investigation is needed before

 

I have argued, with the theorists cited in my literature review in Chapter 2, that
identity and learning are closely intertwined. Because I am studying identity, which is a
part of learning, I feel confident in also claiming that the teachers are learning from their
implementation of CI. In doing so, I am defining learning as a change in discourse (Sfard,
2008). How and what these teachers learned is not the focus of this particular study.
However, it is certainly a topic of interest to me that I hope to pursue further in this data
set, as explained in the Implications section at the end of this chapter.

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assuming that something is lacking. Listening to teachers' narratives is an important start
to understanding all of what goes into teachers' decisions about and implementations of

recommendations made in PD events.

Identifying and Mathematics-For- Teaching Impact Each Other and Are Often the Same
Thing

In Chapter 7, I argued that teachers not only identified and performed knowledge
of mathematics for teaching simultaneously, but that these were often the same practice.
For example, one way to read how Glynnis was identifying was to see her as less
confident in her knowledge of mathematics for teaching than Jonathan. Based on my
analysis, and on other comments made by Glynnis about the nature of mathematics as a
discipline, it is likely that Glynnis viewed the discipline of mathematics as less certain as
well. She had stated that mathematics changes and that it was constructed by people.
Perhaps, then, it is unsurprising that Glynnis regarded her own knowledge of
mathematics with skepticism—not because she was uneasy about her own capabilities (or
perhaps, not only because of that), but because she viewed mathematical knowledge itself
as tentative.

If I had done a survey study of Glynnis' self-efficacy or beliefs about herself as a
mathematics teacher, I would have likely missed this possibility. Perhaps I would have
concluded that Glynnis was lacking self-efficacy or harbored some bad beliefs about
herself as a teacher. A narrative study allowed me to tease out some of the parallels
between how Glynnis was talking about herself and about mathematics as a discipline. It
allowed me wonder if perhaps, in a way, Glynnis was identifying with mathematics, not

against it, in regarding her knowledge as tentative.

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Teaching as Identifying

I have argued repeatedly that participation and narratives of participation are ways
that people identify. I have argued further that narratives are even more telling than an
observation of a teacher's implementation (participation) in regards to identifying,
because it is through narratives that we learn about the ways teachers are making sense of
and understanding their implementation. After all, there are times when all of us act in
ways in which we did not intend (and with which we would not like to identify). In
producing narratives of participation, the teachers were helping me to understand the
ways they were reifying experience into narratives of their practice. One way to say this
is to say that the teachers were teaching me about their learning and identifying. Perhaps
all narratives are a pedagogical practice in some sense.

However, another way to view this is to say that the narratives teachers were
constructing were a part of their teaching practice. Teaching practice may be said to
include more than the improvisation of practice in the presence of children—planning for
lessons and reflecting on them may also be considered practice. In that sense, teachers
identifying narratives were a part of their larger teaching practice. As they told narratives,

they constructed themselves as teachers and constructed their teaching practice.

Obstacles and Persistence

My second research question asked how teachers' ways of identifying created
obstacles to their attempts to implement new practices, and how the teachers overcame
such obstacles. I argued that, although researchers and theorists have noticed that
teaching is difficult to change (e. g. Lortie, 1975), the reasons for the stability of teaching

practice are incomplete, because they have not taken teachers' ways of identifying into

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account. Analyzing teachers' narratives has illuminated some of what makes teaching
with new pedagogies so difficult.

As explained in Chapter 3 (Methodology), all of the teachers in my study had at
least eight years of teaching experience when the study began. Therefore, it is unlikely
that the difficulties associated with implementing new practices were simply about the
difficulties inherent in crafting a new teaching practice for the first time. These teachers
were accomplished teachers, and implementing new practices was still not an easy task.
In this study, I set out to understand why.

In analyzing teachers' life stories, stories of teaching, and stories of teaching
mathematics with Cl, I looked for parallels and contrasts between how teachers were
identifying as mathematics teachers in general and how they were identifying as CI
mathematics teachers in particular. I discovered that when the teachers' ways of
identifying clashed with the ways they saw CI as requiring them to be, discomfort
ensued. However, it seems that the teachers did not always locate the source of this
comfort within their own teaching. Teachers went through times when they were
concerned that the difficulties were because of their own teaching, but also thought they
might be because of their students, because of C1, or because of the context in which they
were teaching. I describe this obstacle as being a clash or inconsistency between the
teacher's previous ways of identifying and their perceptions of the type of teacher they

would need to be as a CI teacher.

Assigning a Location to the Problem

Before continuing to discuss the implication of the gap between these two ways of

identifying, I will comment on the idea that teachers tended to interpret this discomfort as

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a problem located with their own performed knowledge, their students, their context, or
Cl. I argue here that it is unsurprising that teachers would assume that the discomfort they
were feeling was due to a problem located in one of these four locations, because that is
the rhetoric that they hear regularly. Consider the opening excerpt from NCTM's
Principles and Standards for School Mathematics (2000) in which I have emphasized the
factors that are said to make the classroom what it is:

Imagine a classroom, a school, or a school district where all students have
access to high-quality, engaging mathematics instruction. There are
ambitious expectations for all, with accommodation for those who need it.
Knowledgeable teachers have adequate resources to support their work
and are continually growing as professionals. The curriculum is
mathematically rich, offering students opportunities to learn important
mathematical concepts and procedures with understanding. Technology is
an essential component of the environment. Students confidently engage
in complex mathematical tasks chosen carefully by teachers. They draw
on knowledge from a wide variety of mathematical topics, sometimes
approaching the same problem from different mathematical perspectives
or representing the mathematics in different ways until they find methods
that enable them to make progress. Teachers help students make, refine,
and explore conjectures on the basis of evidence and use a variety of
reasoning and proof techniques to confirm or disprove those conjectures.
Students are flexible and resourceful problem solvers. Alone or in groups
and with access to technology, they work productively and reflectively,
with the skilled guidance of their teachers. Orally and in writing, students
communicate their ideas and results effectively. They value mathematics
and engage actively in learning it.

Notice that the teachers, students, curriculum, and context are all constructed as
extremely desirable, and are all constructed as necessary pieces to this puzzle. The
teachers are "knowledgeable" and "skilled", the students are confident, "flexible and
resourceful", "work productively and reflectively", and "value mathematics." The
curriculum is "mathematically rich". The context is ideal. There are "adequate resources"

including technology and opportunities for professional growth.

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The document continues, saying the following about what it will be required for
high-quality mathematics teaching:

The vision for mathematics education described in Principles and

Standards for School Mathematics is highly ambitious. Achieving it

requires solid mathematics curricula, competent and knowledgeable

teachers who can integrate instruction with assessment, education policies

that enhance and support learning, classrooms with ready access to

technology, and a commitment to both equity and excellence.

Based on this statement, one might logically conclude that if classrooms look different
than those described at the beginning of the NCTM document, it might be because the
teachers, the policies, the classroom resources, and/or a lack of commitment.

An explication of the role teachers' integration of ways of identifying renders
more complex this view of what it takes to achieve new visions of mathematics teaching.
There are implications to the statement that "teachers who can. . ." are required. One
implication is that it sounds deceptively static, as if there are simply "teachers who can"
and "teachers who cannot" teach in the way the standards suggest and those are fixed.
The result is a simple categorization of the "cans" and the "can nots", and possibly the
removal of the latter. However, as evidenced in this study, teachers' mathematics teaching
identities and their understandings and implementations of reforms (such as C1) are both
moving targets. Again, if I had observed Joanna only during the time in which she was
providing students with lengthy written directions, for example, I might have too quickly
assumed that she was a "can not". A narrative study allowed me to understand a case in
which this made sense in Joanna's practice, and returning several times allowed me to see
how implementing CI in this way impacted Joanna's implementation of CI after that. If I

had relied only on Kelly's report that she had chosen not to implement CI, 1 might have

too quickly assumed that she was a "can not" who was not willing or able to make

189

changes in her practice. Instead, by analyzing her life story, I was able to understand that
her perceptions of herself and mathematics did have an impact on her teaching. Based on
the results of this study, I anticipate that Kelly will continue to learn about these things as

she continues to teach mathematics in more discussion-oriented ways.

Overcoming the Obstacle and Sustaining Implementation

The teachers all seemed to settle these differences in different ways. For example,
Joanna changed her perception of her students such that she was able to see them as
resources for each other. That led to a change in her perception of her role when she
realized that, in stepping back, she could give students those resources. This allowed her
to perceive CI as a pedagogy that aligned more closely with her ways of identifying.
Jonathan began to see himself and his colleagues as less collaborative. This did not seem
too troubling to Jonathan, who perceived that his colleagues still saw him as competent,
and so he did not alter his perceptions of or his implementation of CI as a result.

While the three teachers whom I observed to be experiencing differences between
their previous and new ways of identifying all found different ways of integrating their
ways of identifying and overcoming this obstacle, they all had one thing in common: they
all sustained their implementation of CI because they were able to find a way in which
being a CI teacher allowed them an opportunity to be themselves. It was the case in all
four teachers that, in order to persist in sustaining implementation of CI, all four had to
come to a place where they perceived being a CI teacher as being consistent with their
self-perceptions. They all had to see being a CI teacher as being a way of identifying that

they could integrate with the other ways they wanted to identify as a teacher.

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Understanding both the obstacles that necessary identity work poses and the ways
that teachers' ways of identifying motivate their persistence is an important step in
treating teachers as more than a means to the end of student achievement (F eatherstone,

2005)

The Dynamic Nature of Identijying and Implementation

In Chapter 2, I noted that the previous link between identifying and
implementation could be viewed as a one-way arrow where teachers' identities impacted
their implementation (Drake, 2006; Drake, Spillane, & Hufferd-Ackles, 2001). Then, I
hypothesized that this arrow would go both ways, so that not only would identifying
impact implementation, but that as teachers continued to implement a new teaching

practice, their ways of identifying would change. I conceptualized this relationship like

A

[ Identifying ] [Implementation ]

V

Figure 9. Original Concept of Identifying and Implementation

Figure 9.

 

Over time, many of the teachers began to tell generalized narratives of their Cl
mathematics teaching. I refer to these generalized narratives as how-I-teach stories,
because the teachers often began them by saying something like, "When I do CI. . .". The
result was that the narratives functioned to claim some certain ways of being CI teachers
as the teachers' own. In a sense, the idea that they were "CI teachers" was assumed, and

they had moved beyond into identifying as a certain kind of CI teacher through these

191

narratives. I have argued that these generalized narratives may be seen as a link between
specific stories of teaching mathematics with Cl and reifications of experience such as
directly identifying as "a successful CI teacher". This is one way I saw implementation
impacting teachers' ways of identifying over several months' time.

However, I after completing this study it is obvious that Figure 10 implies a view
of identifying that is too static. Teachers identified in many different ways, sometimes
within the same story of the same lesson. On the other hand, teachers sometimes
identified in ways that were more stable from story to story. While my study has not yet
untangled all of the relationships involved in identifying and implementation from
moment-to-moment, it is likely that, at the very least, a dimension of time needs to be

added to this conception of identifying and implementation, as shown in Figure 10.

[4‘

Identifying [Implementation

Identifying V‘v { Implementation

Identifying Implementation

 

 

 

 

 
   
 

 

 

Identifying Implementation

 

 

Figure 10. The Relationship Between Identifying and Implementation of CI

The element of time is important because how teachers' identify is, at times, a moving
target. An example is Kelly. Kelly reportedly changed how she identified as a math

person in the context of the CI workshop, and reported that this change influenced the

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changes in her practice (even though she did not report implementing CI in all of the
ways discussed in the course).

However, Figure 10 is still an imperfect representation of the relationship between
implementation and C1, because it appears that identifying and implementation are
changing constantly and at the same rate, which would be beyond the scope of my study
to determine. Additionally, as I stated earlier in this chapter, sometimes identifying and
implementation appear to be almost the same thing, and this diagram portrays them too
separately. A third limitation of the diagram is that it portrays a constant motion forward
through time, when perhaps it is likely that people may go back and forth in reverting to
previous ways of identifying and newer ways of identifying in the process of integrating

older and newer ways of identifying.

Limitations of the Dissertation

In this section, I will describe how the ways I limited the study participants, the
study context, the time period of the study, and the analysis of narratives, and the effects
that these limits have on the conclusions drawn from this study.

I limited the participants in this study to those who could be described as
experienced teachers. I did this purposefully, because I did not want the evidence of
identity shifts inherent in becoming a teacher to be aggregated with the identity shifts
necessary in taking on a new way of being a teacher. For example, if I gleaned evidence
that learning to teach mathematics with C1 was difficult for teachers, I wanted to be
reasonably sure that the difficulty was learning a new pedagogy, not learning to teach for

the first time.

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Among the experienced teachers in this study, there was diversity in gender, race,
grade levels taught, types of locations in which they taught, and years of experience
(from eight to thirty years). However, I did not study new teachers; therefore, what I have
learned about teachers' learning may or may not apply to new teachers.

Another element 1 limited was the PD context. I chose to only study teachers who
were had gone through a course on teaching mathematics with Cl. Focusing on teachers
learning to use CI to teach mathematics had its affordances. Teaching mathematics with
C1 in the ways recommended in the PD requires teachers to take a broad view of
mathematics, to rethink assumptions about mathematical competence, to rethink their
roles as a mathematics teacher, to analyze curriculum, learn new interventions, reflect on
academic and social status among students, and more. Because it is so comprehensive, it
is quite likely that many teachers would need to make many changes in how they identify
in order to succeed. It is also probable that activities such as rethinking what it means to
be mathematically competent would have an effect on teachers' views of their own
mathematical competence (which may account for Kelly's turning point). Therefore, it is
a likely context for seeing changes in ways of identifying.

Of course, although I have reasons to hypothesize that any time teachers are asked
to implement new practices there will be implications for teachers' ways of identifying.
However, how and for whom this may happen in the context of other PD courses,
curriculum guides, or reforms is outside of the scope of this particular study.

Another limitation is due to the nature of participation in the study itself. Teachers
were asked to participate if they indicated that they intended to implement CI to teach

mathematics during the time period of the study. Of course, teachers were informed of

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their rights to discontinue participation. However, knowing that a researcher would be
coming to talk to them about their use of CI to teach mathematics might have been a
reason they continued to do so. At the end of the study, all teachers indicated that they
were glad that they were implementing CI to teach mathematics and that they intended to
do more after the study. However, how much study participation actually motivated them
to teach the lessons they taught is unknown.

Finally, I will describe a limitation in the analysis of Chapter 5 in particular. In
that chapter, I compared teachers' stories of their teaching with C1 to one of the ways I
noticed them identifying as teachers. The limitation of this is that I downplayed the
dynamic nature of identifying in order to look for connections between stories. That is, I
fear that readers may see these teachers as essentialized and two—dimensional. It is the
case that the teachers identified in many different ways, even though I only highlighted

one. The result is a version of the story that is over simplified.

Implications

The results of this study have implications for future research, teacher
professional development practice, and policy. First of all, theories about the stability of
teaching practice are incomplete (at best) and disrespectful (at worst). They are
incomplete because of a lack of attention to the difficult identity work that is required for
teachers to change their practice. In order to be complete, theories of teacher learning
need to take into account issues of identifying, including the loss that may be experienced
as teachers seek to put aside previous ways of identifying and are identified differently by
others and the crises that may arise as teachers seek to integrate identities. They are

disrespectful because, in assuming that teachers are not making the changes professional

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developers might expect, that it may be that they did not understand the proposed
changes or harbored some bad beliefs or incorrect knowledge. Instead, seeing teachers in
the midst of change as engaged in integrating identities helps to explain why changes
may sometimes seem slow, incomplete, or inconsistent. However, those are all a part of
the difficult work of adding new teaching practices to a teacher's repertoire.

Instead, studying teachers' narratives of practice assumes that teachers are making
sense of their teaching and speaking about it in purposeful ways. It is a method that
captures multi-dimensional aspects of the lives of teachers, including cognition, social
interactions, cultural contexts, linguistics, emotions, and more. Without deep and careful
listening to these teachers, I might have assumed that they were not understanding
Complex Instruction, mathematics, or their students. Therefore, more studies of teachers'
narratives are warranted.

This finding has implications for teacher professional development. The
evaluation and assessment of professional development is implicated because traditional
ways of assessing teachers' changes in practice often overlook the identity integration that
is required for teachers adding new teaching practices; therefore, decisions about PD
made on the basis of those assessments may be problematic. Another implication is that
people who support teachers' learning in practice, including administrators, math coaches,
and professional developers should be aware of this identity work and provide support as
teachers experience it. Additionally, more work needs to be done in order to invent ways
to support and mentor teachers that are working toward learning new practices. Perhaps,
with support, these experiences could be leveraged to extend teachers' learning of the

practices they are implementing.

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Additionally, narrativizing experience is one way in which teachers continue to
make sense of their practice (Ochs & Caps, 2001). Therefore, as teachers participate in
narrative studies, they are continuing to refine their understandings. As such, including
teachers' narratives as a part of PD courses or other teacher learning activities seems
desirable. One benefit of doing so would be that it would allow facilitators of PDs to
better learn about teachers in order to draw connections between how they identify and
the practices being recommended. Another reason is that it could provide another
opportunity for teachers to make sense of their practice.

Much has been said in recent times about the need to scale up research in
mathematics education. I argue that, in the midst of such efforts, the use of narrative
methodologies to study teachers and teacher learning be continued. Especially needed are
longitudinal studies of the implications of reforms on teachers' lives and practice. In this
study, I was able to see the ways in which teaching with Cl changed the ways that the
teachers identified, and see some of the interplay between teachers' identifying narratives
and their narratives of teaching practice. However, how the teachers will continue to
make sense of and implement CI as other reforms are recommended and new PD courses
are experienced remains unknown.

New understandings of teachers' ways of identifying as they implement new
pedagogies have implications for policy as well. New calls for reform of mathematics
education seem to be issued regularly. Early on, teachers were asked to be hands-on
teachers, then discussion-based teachers, problem-based teachers, Cognitively Guided
Instruction (F ennema, etal 1995) teachers, and now Complex Instruction teachers.

Teachers are constantly experiencing new waves of PD, curricula, or other

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recommendations in their particular districts as well. Asking teachers to routinely shift
their ways of identifying and become new kinds of teachers has consequences. Based on
this study, it is likely to result in discomfort and conflict between teachers' new and
previous ways of identifying. Teachers may need to integrate those two ways of
identifying. Therefore, it is not surprising that teachers may blend together ideas from
these waves of reform in ways that were not intended by reformers. Attention to what it
takes for teachers to become new kinds of teachers is necessary in order for reforms to be
effective, of course. Perhaps, when recommending changes in practice, reformers and
professional developers can take some of the responsibility for explicating the ways new
practices can be integrated with previous ones. It is possible that doing so might lessen
the burden of integration work on teachers, by demonstrating how new ways of

identifying as a teacher can be integrated with previous ways.

Directions for Future Research

I intend to continue this line of research in the upcoming years. There are a
number of problems and questions that have been raised for me as I carried out this study,
which I will pursue. In this section, I will outline a couple of ways in which I plan to
continue this research.

First of all, I have hypothesized that the CI mathematics PD course itself had
particular characteristics that made it a place where shifts in ways of identifying would be
seen. However, there are theoretical reasons to believe that all learning and changes in
practice would be intertwined with new ways of identifying. Therefore, a similar studying
in and after one or more other PD contexts would be necessary. Carrying out a similar

study with teachers who are newly implementing Cognitively Guided Instruction

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(F ennema etal, 1995), for example, might help me to understand the aspects of the CI
mathematics course and CI as a pedagogy that impacted the relationship between
teachers' ways of identifying and implementation of the practices.

Another extension to this research would be to extend it to different demographic
contexts. I have reported that the teachers in this study came from rural, suburban, and
urban contexts. However, the urban context was in a mid-sized, Midwestern city (not a
large metropolitan area), the suburban schools were suburbs of this town, and the rural
school was within a thirty-minute drive of a research university (not in a remote location,
therefore, teachers there had access to a plethora of university related PD and research
experiences). My experience in rural areas that are more remote leads me to wonder
about a place-based interrelation between identifying and reactions to reforms. A
common theme among my conversations with rural teachers is the perceptions that
reformers (and policy-makers in general) are unaware of, or simply do not care about,
their teaching contexts. Therefore, I hypothesize that it is likely that teachers in more
remote locations may react very differently to suggestions for reform, and that continuing
this line of research in that context would yield a broader picture of teacher learning.

In addition, there are other lines of analysis that I intend to pursue within this
particular set of data. In analyzing the data for this study, I was particularly drawn to look
at instances where teachers' emotions seemed to be most obvious. For example, when
teachers were uncomfortable, or confused, or angry, or excited, I noticed. In particular,
the emotion of fear may be a particularly immobilizing emotion for teachers (fear of not
passing tests, fear of being perceived differently by administrators, etc.) How emotions

play into a study of teacher learning is something about which I plan to theorize.

199

Another line of inquiry I will pursue within this data set is a further analysis of the
ways teachers' discourses of teaching changed over the data collection time period.
Because learning may be defined as a change in discourse (Sfard, 2008), and because
learning and identifying are tightly interwoven and often the same thing, studying
teachers' ways of identifying has been studying a part of teachers' learning. However,
there are other aspects of changes in discourse that could be studied. Whether teachers'
ways of talking about mathematics as a discipline changed, or their ways of speaking
about students, or their descriptions of their teaching context are places I may begin this
process. Additionally, looking at how those ways of speaking are related to teachers'
ways of identifying is an area for further analysis. It makes sense, for example, that if a
teacher develops a new understanding of her role in her classroom, she will also see the

roles of her students differently. This is something I would like to study in the future.

Conclusion

Teachers' learning, like any learning, changes the learner. It changes the way that
the learner can participate in communities of practice, it changes the discourse of the
learner, and it changes the way that the learner constructs his or her identity in the telling
of narratives.

I have argued that this story in teachers' lives parallels other stories of schooling
and shifts in ways of identifying that resonate with many of us. In "Of the Coming of
John" by W.E.B. DuBois (1903/1994) and in the movie Educating Rita (Cartilidge &
Gilbert, 1983), going to school changed the learners and how the learners were perceived

by significant others.

200

In the same way, teachers who participated in the CI mathematics course and
embraced CI as a mathematics pedagogy changed how they identified as teachers and, in
at least some cases, how they were perceived by others such as their students and their
colleagues.

Teachers learning, like all learning, though desirable, can be a dangerous
undertaking. Seeing teachers as a means to an end (instead of being concerned about
teachers themselves) might lead those of us with an interest in mathematics education
reform to simply view teachers as conduits of reform (F eatherstone, 2005). In doing so,
we may rashly push teachers from one reform to another, without attention to the
struggles inherent in becoming a new kind of teacher, then another, then another. And
yet, of course we want teachers to continue learning and improving their practice,
because we care about the lives and learning of students in their classes.

As a partial remedy to this ethical dilemma, I recommend that teacher PD
practitioners, administrators, researchers, and policy makers elicit and deeply listen to
teachers' narratives at every step of the way, from the conceptualizing of new teaching
practices to teachers' consideration of new practices, to their ongoing implementations of
them. The result will be a more ethical and humane approach to teacher learning and

mathematics education reform.

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APPENDIX A: INTERVIEW PROTOCOL FOR PHASE I LIFE STORY INTERVIEWS
(Adapted from Drake, Spillane, & Hufferd-Ackles, 2001)

I. Background Information
a. How many years have you been teaching?
i. In what types of school environments (e.g., suburban, urban, rural)
ii. What grades have you taught?
b. What type of teaching certification do you have?
i. Do you have a mathematics major or minor?
11. Mathematics
a. One of the things I often do with students is to ask them to write their
mathematics autobiographies. I'd like to hear about your math
autobiography. Tell me about your experiences with learning mathematics.
(Possible prompts: What are some adjectives that describe your
experiences with learning math? What are some feelings you had when
learning math?

i. Tell me about a time when you felt (one of the things mentioned in
the last question). (Possible prompts: where did that happen? Who
was involved? What did you do? What were you thinking and
feeling? What effect did that have on you?)

ii. What's another event that stands out to you as important in your
math autobiography? (Possible prompts: where did that happen?
Who was involved? What did you do? What were you thinking and
feeling? What effect did that event have on you?)
111. Teaching Mathematics
a. How would your students describe you as a math teacher and why?

i. Can you tell me about an event that happened that leads you to
believe students would describe you as (one of the ways the
teacher said he/ she'd be described by students)?

b. How might your colleagues describe you as a math teacher and why?

i. Can you tell me about an event that happened that leads you to
believe your colleagues would describe you as (one of the ways the
teacher said)?

c. Describe yourself as a teacher of mathematics.
i. Why do you think that?
ii. Tell me about an experience that leads you to describe yourself as
(one of the ways he/she described him/herself). (Possible prompts:
What were you trying to do? Who was involved? What were you
thinking and feeling? What effect did that have on you?
d. Tell me about your experiences as a teacher of mathematics.
i. What have these experiences been like for you?
ii. What is an example of how they were like (the way he/she just

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describe them)? (Possible prompts: where did that happen? What
did you do? What were you thinking and feeling? What effect did
that have on you?)

e. How have you learned about mathematics teaching?

i. What was an experience that was important to you in learning
about teaching mathematics? (Possible prompts: where did that
happen? Who was involved? Was that a part of your formal
teacher education? What did you learn fiom that? How did that
change your thinking about mathematics teaching and learning?)

IV. The Complex Instruction Workshop

a. Why did you decide to participate in the Complex Instruction workshop
last summer? (Possible prompt: What were you hoping to learn? What
were you hoping would happen there?)

b. I'd like you to reconstruct the experience of the workshop for me so I can
see what the experience was like from your point of view. Tell me the
story of what happened there for you. (Probes: What kinds of activities
did you engage in? What was the nature of the conversation? What did that
feel like for you?)

c. Describe an event from the workshop that stands out in your mind as
important to you.

i. Why was it important? (Possible prompts: What happened? What
were you thinking and feeling? What effect did it have on you?)

(I. What did you learn in the workshop?

i. What event or events helped you to learn that? (The event could be
something structured or could be something like an informal
conversation).

I. What was it about that event or those events that helped
you to learn that?
V. Effect of Complex Instruction on Your Practice

a. I'd like you to think about your mathematics teaching practice before the
Complex Instruction workshop and after the workshop. How do those
compare and contrast with each other? (Possible prompts: Are there
differences in the activities you choose to use with your students? In the
things you think about as you plan? In the things you notice in the
classroom?)

i. Why do you think those things changed (or did not change)?

1. (If the teacher reports making changes)
a. What was it that happened during the workshop
that caused you to make those changes?
b. What factors outside of the workshop have
influenced the changes you've made?
c. Tell me about an example of something that
happened in your classroom when you were trying

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a technique/strategy from the workshop. (Possible
prompts: Why do you think that happened? What
did you do? What did you do differently than you
would have done before the workshop? What were
you thinking about when you did that? How do you
feel about that?)

d. How have your students responded to the changes
in your teaching practice?

e. Tell me about an example of (student response
mentioned above). (Possible prompts: Why do you
think he/she/they responded that way? When did
that happen?)

2. (If the teacher reports not making changes)

a. Why do you think you haven't implemented the
strategies/techniques recommended in the
workshop?

b. What was it that could have happened within the
workshop that might have influenced you to
implement the recommendations?

c. What factors outside of the workshop have
influenced you in this way?

VI. Goals For Mathematics Teaching and Learning in the Future
a. What are your goals for teaching mathematics?

i. Why is that a goal for you?
ii. What things do you already do to reach these goals?

1. Describe an event that illustrates this. (Possible prompts:
where did that happen? Who was involved? What were you
thinking and feeling? What did you do?)

iii. What would you still like to learn about teaching mathematics?

1. How will help you to be a better mathematics teacher?

iv. What types of professional development would be helpful to you?

v. Five years from now how do you see yourself as a teacher of
math? What kind of a teacher would you like to become? Why?
(Possible prompts: Why do you think that is important? What do
you plan to do to learn about that? What do you see being the
outcome of that?)

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APPENDIX B: PROTOCOL FOR PHASE II INDIVIDUAL INTERVIEWS

These interviews will be semi-structured. Other questions may be asked to follow up on the
participants' answers.

Interview 1:
1. Your Classroom Since the Workshop:
a. Tell me about your experiences with teaching mathematics so far this year.
Possible prompts:
i. What have these experiences been like for you?
ii. What is an example of how they have been like (the way he/she
just described them)?
iii. What are you learning from these experiences?
b. Have you done any groupworthy tasks since the CI workshop?
1. If yes, possible questions:

1. What have those experiences been like for you?

2. What is an example of how they have been like (the way
he/she just described them)?

3. How do you think your students would describe you when
you're teaching using CI? How would you describe
yourself?

4. What are you working on? What are you wondering about?

ii. If no, possible questions:

1. Tell me more about why you've chosen not to use CI.

2. How do you think this year's students would describe you
as a teacher? How would you describe yourself as a teacher
this year?

3. Has your teaching changed in other ways since the CI
workshop? If so, in what ways? When I asked you about
doing this interview, you expressed an interest in using a CI
lesson to teach math. What would you need in order to do
this? (A lesson, resources, time for planning?)

2. Planning a groupworthy task
a. When I asked you about doing this interview, you said that you would be
interested in teaching a math lesson using CI. What have you planned (or
thought about) so far in regards to this lesson?
i. Talk me through what you expect will happen during this task. Tell
me the story of how you expect this to go. Possible prompts:

1. What specific math content do you want students to engage
with in this CI lesson?

2. What groupworthy task are you considering using for this
lesson? What makes it groupworthy? What 'smartnesses' do
you expect that this groupworthy task will draw on?

3. How are you thinking about this task in regards to
disrupting status in the classroom? How have you designed

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the task (or can you design the task) to attend to issues of
status?

4. What are the instructional strategies you will use while
teaching using this task? Will you use norms? Roles?
Status interventions?

b. If you had an ideal situation (with all the resources, time, support,
knowledge, etc. that you would need), is this how this lesson would look?
What would be different?

c. What assistance would you like with getting this task prepared, if any?
(Do you need any materials? Do you need copies made? Do you need
classroom resources or ideas for tasks?)

(1. Do you have a lesson plan (notes or jottings are fine) I can keep? Can you
email me a lesson plan before you teach this lesson?

3. Preparing for the next interview

a. When you and I meet again, it will be after you have taught the lesson, to
talk specifically about this particular lesson. In order to do this, I would
like you to collect an artifact during the lesson so that you can use them to
help you think about what happened during the lesson and how it went.
(Examples of artifacts include video or audio of the lesson, student work,
your own joumaling right after the lesson, students' written reflections on
the lesson from right after or a while after the lesson, student test or quiz
items given later which cover the content of the lesson, etc.)

i. What type of a teacher are you trying to become? Is there anything
in particular that you would like to learn about your own practice
by teaching this lesson? If so, is there a certain type of artifact that
you could collect and study to help you learn more about that?

ii. Do you need any assistance and/or resources in order to collect
these artifacts? (For example, video equipment and/or a
videographer, photocopying, etc.?)

iii. When you bring the artifacts to our next conversation, make sure
you deidentify students as much as possible. (Cross their names off
of student work, etc.)

b. What would be a good time and place for our next interview?

Interview 2:
1. The CI lesson
a. What did you notice while you were teaching the lesson? Possible
prompts:
i. What happened? Tell me the story of this lesson. Refer to the
lesson artifacts if they help you to tell the story.

ii. How would your students tell the story of this lesson? How would
they describe you when telling the story of the lesson? How would
you describe yourself?

iii. How are you feeling about the lesson? Why?
b. Tell me about an event that seems important to you in this lesson. Possible
prompts:

206

i. What happened? Who was involved? Why do you think that
happened? What were you thinking/feeling at the time? Are you
thinking differently about that event now?

Tell me about the task and the task card. Do you still think it is
groupworthy? Why or why not? Tell me about an event that happened that
makes you think that.

. What instructional strategies did you use during the lesson?

1. Tell me more about what happened when you used that strategy.

1. Why did you choose to use it? What happened? Did you
get the outcome you expected? How are you thinking about
that now?

What did you notice about status during the lesson?

i. Tell me about something that happened that led you to believe
there was a status issue within a group? (Or alternatively, that led
you to believe there was not a status issue within a group?)

1. What happened? Who was involved?

2. How did you respond? How did the students respond?

3. What were you thinking and feeling about that at the time?
How are you thinking and feeling about that now?

Have you had an opportunity to review the artifacts that you collected
about the lesson?

i. If so. . ..

1. What did you notice when studying the artifacts?

2. Which of your artifacts was most helpful for you in
thinking about your practice using CI?

a. What did that artifact help you to see?
b. What is it that you learned about your practice?
c. What is it that you learned about yourself from this
artifact or from teaching the lesson?
ii. If not. . ..

1. Take some time to review the artifacts during this
interview.

2. What did you notice when studying the artifacts?

Which of your artifacts was most helpful for you in
thinking about your practice using CI?

a. What did that artifact help you to see?

b. What is it that you learned about your practice?

c. What is it that you learned about yourself from this

artifact or from teaching the lesson?

. In the pre-interview, you said that you were trying to become a more
(insert what was said) teacher. How did that go in this lesson? Tell me
about a time when you were (or were not) more (what was said) than
before?
. What would you like to work on or learn about next, in terms of teaching
mathematics using CI?

i. Why is that important for you?

D.)

207

ii.

iii.

What type of teacher are you trying to become? How will that help
you become the kind of teacher you're working to be?

What types of support and/or professional development could help
you to do this?

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APPENDIX C: ROUND ROBIN PROTOCOL FOR TEACHER DISCUSSIONS

Stories of Complex Instruction
Task Card

Each person was asked to think about and be ready to share a story about teaching a
mathematics lesson using Complex Instruction (CI). The story may be about a problem
you encountered, something you are puzzling over, or something you are celebrating.
Each person should tell the group what it is that you would like them to help you think
about when they hear and discuss the story.

The Task:

The task is for each person to share a story of practice and discuss it with the group, for

the group to think together about why things happened during the lesson, and for the

group to summarize their learning and questions about and across stories. To do this:

I First, take a couple of minutes of private think time for group members to organize,
think about, or remember their story and what they would like the group to help them
think about.

I Then, use the following "Round Robin" protocol for telling the stories:

1. Person #1 tells a story about a lesson or something that happened during a
lesson using CI. Others in the group listen, but do not respond yet. (If you
have something that you want to remember to say or ask, jot it down.)

2. The group talks about Person #l's story, focusing on figuring out what was
happening in the classroom and hypothesizing why.

3. Repeat numbers 1 and 2 above with Person #2, Person #3, and Person #4
sharing their stories and the group discussing them.

4. Each person's story should get about 20 minutes of group time, including
both the storytelling and the group discussion around it.

. When every group member has told a story and the group has discussed them, the
group should summarize find questions that occurred across more than one story, and
talk about why those might be important.

209

 

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